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F U N D A M E N T A L PHYSICAL C O N S T A N T S (Reprinted from the Reviews of The numbers in parenthese:s are the standard-deviation uncertainties Units Symbol

Quantity

Value

Error (ppm)

2.9979250(10)

0.33

10 m s e c

1.5 1.5

io-

3

19

Velocity of light

c

Fine-structure constant, l c /4n](e /nc)

a a-

E l e c t r o n charge

e

1.6021917(70) 4.803250(21)

4.4 4.4

io-

Planck's constant

h H -- h/2ir

6.626196(50) 1.0545919(80)

7.6 7.6

1010-

2

N

2

7.297351(11) 137.03602(21)

1

C

3 4

N

6.022169(40)

6.6

10

amu

1.660531(11)

6.6

10

Electron rest mass

m m*

9.109558(54) 5.485930(34)

6.0 6.2

IO" 10

M M*

1.672614(11) 1.00727661(8)

6.6 0.08

1 0 - kg amu

M„

1.674920(11) 1.00866520(10)

6.6 0.10

I O " kg amu

p

N e u t r o n rest mass

K

R a t i o of p r o t o n mass t o electron mass

M /m

Electron charge t o mass ratio

e/m

Magnetic flux quantum, [c]-Hhc/2e) Q u a n t u m of circulation

p

1836.109(11)

e

2 7

1 0

- 2 7 - 2 7

emu esu erg • sec erg • sec . mole-

2 3

24

IO"

4

io-

24

1

g g

-

amu

g

amu 10-24 g

2 7

amu

6.2 10 emu g 1 0 esu g

10

2.0678538(69) 4.135708(14) 1.3795234(46)

3.3 3.3 3.3

IO" 10-

h/2m h/m

3.636947(11) 7.273894(22)

3.1 3.1

1010-

F

9.648670(54) 2.892599(16)

5.5 5.5

10 C k m o l e -

e

2 0

10-28

kg amu

3 1

- 4

1

3

io-

3.1 3.1

h/e

10

1

k g

1.7588028(54) 5.272759(16)

e

e

Faraday constant, Ne

IO ' IO

kmole-

2 6

-27

cm sec-

1 0

1010-

J•sec J•sec

3 4

Avogadro's number

P r o t o n rest mass

10

- 1

io-

A t o m i c mass unit

e

cgs

SI 8

C kg-

1 1

7

1

1 7

1 5 1 5

4 4

T •m J • sec C "

IO" G • c m I O erg • sec e m u IO erg • sec e s u 7

2

J•sec k g J•sec k g -

7

_ 1 _ 1

1

2

- 7

- 1

- 1 7

- 1

erg • sec g erg • sec g

1 1

_ 1 _ 1

10 emu m o l e 1 0 esu m o l e 3

1

1 4

10 c m -

R y d b e r g constant, |> c /47r] (m e /4^ c)

R

1.09737312(11)

0.10

10 m -

B o h r radius,

a

5.2917715(81)

1.5

10-

1 1

m

10-

2.817939(13)

4.6

10-

1 5

m

IO"

1 3

cm

IO"

2 1

erg G -

1

10-

2 1

erg G -

1

[

M o

2

2

0

4

e

x

5

1

1

3

0

9

cm

c /47r]-WM e ) = 2

e

2

Classical electron radius \_li c /A'n']{e /m c ) = a /4nR Q

3

7

- 1 - 1

2

2

e

2

x

Electron magnetic m o m e n t in B o h r magnetons

M e /MB

1.0011596389(31)

0.0031

B o h r magneton, [c]{efi/2m c)

MB

9.274096(65)

7.0

10- 4 J T -

1

Electron magnetic m o m e n t

1

2

e

Me

9.284851(65)

7.0

10-

Gyromagnetic ratio of protons in H 0

yj/2*r

2.6751270(82) 4.257597(13)

3.1 3.1

1 0 rad s e c - • T 10 Hz T -

Yp corrected for diamagnetism o f H 0

y y /^

2.6751965(82) 4.257707(13)

3.1 3.1

10 rad s e c - • T " 10 Hz T -

M /MB

1.52099312(10)

0.066

io-

3

io-

3

P r o t o n magnetic m o m e n t in B o h r magnetons

M /MB

1.52103264(46)

0.30

io-

3

IO"

3

P r o t o n magnetic m o m e n t

M

1 0

10-

2 3

erg G "

1

10-

2 4

erg G -

1

2

2

M a g n e t i c m o m e n t of protons in H 0 in B o h r magnetons

P P

P

2 4

J T"

8

1

7

1

1

7

1

4

1

3

1

8

1 0 rad s e c - • G " 10 Hz G "

1

1 0 rad s e c - • G " 10 Hz G 4

1

3

1

2

Magnetic m o m e n t of protons in H 0 in nuclear magnetons

P

P

Mp/Mn

1.4106203(99)

7.0

2.792709(17)

6.2

2.792782(17)

6.2

_

2 6

j

T

_ i

2

/Ap//i corrected for diamagnetism of H 0 n

2

Nuclear magneton,

[c](en/2M c)

M„

5.050951(50)

A

2.4263096(74) 3.861592(12)

10

10-

2 7

J T-

1

p

C o m p t o n wavelength of the electron, h/m c e

C

\ /2TT c

10-12

3.1 3.1

C o u r t e s y of RCA L a b o r a t o r i e s , Princeton^ N . J

10-

e

1 3

m

m

1010-

1

1

1 0 1 1

cm cm

1

Units Error Quantity

Symbol

SI

Value

(ppm)

C o m p t o n wavelength of the proton, h/M c

1.3214409(90) 2.103139(14)

6.8 6.8

1010-

C o m p t o n wavelength of the neutron, h/M c

1.3196217(90) 2.100243(14)

6.8 6.8

10~ 10-

p

n

1 5 1 6

1 6

m m

1010-

m m

1010-

J kmole - 1 . K - i

Gas constant

R

8.31434(35)

42

10

B o l t z m a n ' s constant, R /N

k

1.380622(59)

43

10-

0

5.66961(96)

170

10-

c

4.992579(38)

0

15

cgs

3

J K-

2 3

1 3 1 4

1 3 1 4

cm cm cm cm

10 erg m o l e " • R 7

1

1 0

1

-i6

E R G

1

K-I

0

Stefan-Boltzman constant, 77 & /607i c 2

4

3

m- K

W

8

2

4

10-

5

10"

1 5

erg s e c

• cm"

- 1

:

2

First radiation constant, 877/ic

x

S e c o n d radiation constant, hc/k

7.6

C

2

1.438833(61)

Gravitational constant

G

6.6732(31)

kx-unit-to-angstrom conversion factor, A = MA)A(kxu); MCuKaJ = 1.537400 kxu

A

1.0020764(53)

5.3

A * - t o - a n g s t r o m conversion factor, A = A ( A ) / A ( A * ) ; XCWKaJ = 0.2090100 A *

A*

1.0000197(56)

5.6

N o t e that the unified atomic mass scale C = 12 has b e e n used througho u t , that a m u = atomic mass unit, C = c o u l o m b , G = gauss, H z = hertz = c y c l e s / s e c , J = j o u l e , K = kelvin (degrees kelvin), T = tesla ( 1 0 G ) , V = volt, and W = watt. In cases where formulas for constants are given (e.g., R ), the relations are written as the p r o d u c t o f two factors. T h e s e c o n d factor, in parentheses, is the expression to b e used w h e n all quantities are expressed in cgs units, with the electron charge in electrostatic a

1 2

4

x

10~

J•m

24

43

10-

2

460

10-

1 1

erg • cm

cm • K

m • K N • m kg" 2

10~ dyn • c m 8

2

2

g-

2

units. T h e first factor, in brackets, is to b e i n c l u d e d o n l y if all quantities are expressed in SI units. W e remind the reader that with the e x c e p t i o n o f the auxiliary constants w h i c h have been taken to be exact, the uncertainties o f these constants are correlated, and therefore, the general l a w o f error p r o p a gation must b e used in calculating additional quantities requiring t w o or m o r e of these constants.

E n e r g y C o n v e r s i o n Factors Value

Quantity

5.609538(24)

1 kg

10

Electron mass

0.5110041(16)

P r o t o n mass

938.2592(52)

N e u t r o n mass

939.5527(52)

1 electron volt

MeV

4.4

MeV

5.5

2 9

931.4812(52)

1 amu

Error (ppm)

Unit

MeV

3.1

MeV

5.5

MeV

5.5 4.4

1.160485(49)

J 1 0 ~ erg 10 Hz 10 m 10 c m 10 K

1.2398541(41j

10-

6

10~

4

1.6021917(70)

10-

1 9

12

2.4179659(81) 8.065465 (27)

1

3

Energy-wavelength conversion R y d b e r g constant,

Bohr m a g n e t o n , p,

R

x

B

1

eV • m eV • c m

Gas constant, R

1 8

10-

1 1

5.788381(18) 1.3996108(43) 46.68598(14)

1 0 - eV T " 10 Hz T "

3.152526(21) 7.622700(42) 2.542659(14)

Q

Standard v o l u m e of ideal gas, V

0

7.6

10-

eV 10 10

3.3 0.35 43

Hz K

1 5

5

5

- i .

T

1

- i

10- c m K T-

1

• T-

1

43

1

1 0 - eV T 10 Hz T 8

6

1

0

-

m

- i .

1010-

8.20562(35)

10~ m

4

4

2

m

3

6.8 5.5 5.5

1

1

2

3.65846(16)

22.4136

3.1 3.1 3.1

1

10

m

3.3

J erg

2.179914(17)

2

n

42

4

13.605826(45) 3.2898423(11) 1.578936(67)

0.671733(29) Nuclear magneton, ji

3.3 3.3

1 4

5

T

_i

cm- • TK T1

1

44

1

3

• atm k m o l e " • R -

kmole-

1

1

1

42

OTHER McGRAW-HXLL HANDBOOKS OF

INTEREST

S O C I E T Y O F M E C H A N I C A L E N G I N E E R S • A S M E Handbooks: Engineering Tables Metals Engineering—Processes Metals Engineering—Design Metals Properties B A U M E I S T E R A N D M A R K S • Standard Handbook for Mechanical Engineers B E R R Y , B O L L A Y , A N D B E E R S • Handbook of Meteorology B L A T Z • Radiation Hygiene Handbook B R A D Y • Materials Handbook B U R I N G T O N • Handbook of Mathematical Tables and Formulas B U R I N G T O N A N D M A Y • Handbook of Probability and Statistics with Tables C A L L E N D E R • Time-Saver Standards C H O W • Handbook of Applied Hydrology C O N D O N A N D O D I S H A W • Handbook of Physics C O N S I D I N E • Process Instruments and Controls Handbook C O N S I D I N E A N D R o s s • Handbook of Applied Instrumentation E T H E R I N G T O N • Nuclear Engineering Handbook F I N K A N D C A R R O L L • Standard Handbook for Electrical Engineers F L U G G E • Handbook of Engineering Mechanics G R A N T • Hackh's Chemical Dictionary H A M S H E R • Communication System Engineering Handbook H A R R I S A N D C R E D E • Shock and Vibration Handbook H E N N E Y • Radio Engineering Handbook H I C K S • Standard Handbook of Engineering Calculations H U N T E R • Handbook of Semiconductor Electronics H U S K E Y A N D K O R N • Computer Handbook I R E S O N • Reliability Handbook J U R A N • Quality Control Handbook K A E L B L E • Handbook of X-rays K A L L E N • Handbook of Instrumentation and Controls K I N G A N D B R A T E R • Handbook of Hydraulics K L E R E R A N D K O R N • Digital Computer User's Handbook K O E L L E • Handbook of Astro nautical Engineering K O R N A N D K O R N • Mathematical Handbook for Scientists and Engineers L A N D E E , D A V I S , A N D A L E R E C H T • Electronic Designers Handbook L A N G E • Handbook of Chemistry M A C H O L • System Engineering Handbook M A N T E L L • Engineering Materials Handbook M A R K U S • Electronics and Nucleonics Dictionary M E I T E S • Handbook of Analytical Chemistry M E R R I T T • Standard Handbook for Civil Engineers P E R R Y • Engineering Manual P E R R Y , C H I L T O N , A N D K I R K P A T R I C K • Chemical Engineers' Handbook R I C H E Y • Agricultural Engineers' Handbook R O T H B A R T • Mechanical Design and Systems Handbook S T R E E T E R • Handbook of Fluid Dynamics T E R M A N • Radio Engineers' Handbook T O U L O U K I A N • Retrieval Guide to Thermophysical Properties Research Literature T R U X A L • Control Engineers' Handbook U R Q U H A R T • Civil Engineering Handbook W O L M A N • Handbook of Clinical Psychology AMERICAN

7

AMERICAN INSTITUTE OF PHYSICS HANDBOOK

American Institute of Physics Handbook Edition

Third

Section Editors BRUCE

H.

BILLINGS,

Ph.D.

Commissioner, Joint Commission on Rural Reconstruction Taipei, Taiwan

D.

F. BLEIL,

K.

COOK,

Ph.D.

Special Assistant for Sound Programs, Office of Deputy Director The National Bureau of Standards

H.

M. CROSSWHITE,

Ph.D.

Adiunct Professor of Spectroscopy, Physics Department The Johns Hopkins University

FREDERIKSE,

Ph.D.

Chief, Solid State Physics Section The National Bureau of Standards

R.

Ph.D.

Associate Technical Director and Head, Research U.S. Naval Ordnance Laboratory RICHARD

H. P. R.

BRUCE LINDSAY,

Ph.D.

Professor of Physics, Emeritus, Brown University

J.

B. MARION,

Ph.D.

Professor of Physics, Department of Physics and Astronomy University of Maryland MARK

W.

ZEMANSKY,

Ph.D.

Professor of Physics The City College of the City University of New York

Coordinating Editor

Dwight E. Gray, Ph.D. American Institute of Physics McGraw-Hill Book Company Dusseldorf New York St. Louis San Francisco Montreal Kuala Lumpur London Mexico Sydney Panama Rio de Janeiro Singapore

Johannesburg New Delhi Toronto

Library of Congress Cataloging in Publication Data American Institute of Physics. American Institute of Physics handbook. Includes bibliographies. 1. Physics~Handbooks, manuals etc. I. Gray, Dwight E., ed. II. Title. QC61.A5 1972 016.5301'5 72-3248 ISBN 07-001485-X

Copyright @ 1972, 1963, 1957 by McGraw-Hill, Inc. All Rights Reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher.

\

1234567890

COCO

765432

The editors for this book were Daniel N. Fischel, Harold B. Crawford, Don A. Douglas, and Winifred C. Eisler, and its production was supervised by George E. Oechsner. It was set in Modern 8A by The Maple Press Company. It was printed and bound by The Colonial Press Inc.

Contents

Contributors

~x

Preface

xiii Section

MATHEMATICS BIBLIOGRAPHY; 81 UNITS Mathematics bibliography.

1

SI units.

MECHANICS . Editor, Dr. R. Bruce Lindsay, Brown University

2

Fundamental concepts of mechanics. Units and conversion factors. Density of solids. Centers of mass and moments of inertia. Coefficients of friction. Elastic constants, hardness, strength, elastic limits, and diffusion coefficients of solids. Viscosity of solids. Astronomical data. Geodetic data. Seismological and related data. Oceanographic data. Meteorological information. Density and compressibility of liquids. Viscosity of liquids. Tensile strength and surface tension of liquid. Cavitation in flowing liquids. Diffusion in liquids. Liquid jets. Viscosity of gases. Molecular diffusion of gases. Compressible flow of gases. Laminar and turbulent flow of gases. Shock waves. ACOUSTICS Editor, Dr. Richard K. Cook, The National Bureau of Standards Acoustical definitions. Standard letter symbols and conversion factors for acoustical quantities. Propagation of sound in fluids. Acoustic properties of gases. Acoustic properties of liquids. Acoustic properties of solids. Properties of transducer materials. v

3

VI

CONTENTS

Frequencies of simple vibrators. Musical scales. Radiation of sound. Architectural acoustics. Speech and hearing. Classical dynamical analogies. Mobility analogy. Nonlinear acoustics (theoretical). Nonlinear acoustics (experimental). Selected references on acoustics. HEAT Editor, Dr. Mark W. Zemansky, The City College of the City University of New York

4

Temperature scales, thermocouples, and resistance thermometers. Thermodynamic symbols, definitions, and equations. Critical constants. Compressibility. Heat capacities. Thermal expansion. Thermal conductivity. Thermodynamic properties of gases. Pressure-volume-temperature relationships of gases. Virial coeffi.cients. Temperatures, pressures, and heats of transition, fusion, and vaporization. Vapor pressure. Heats of formation and heats of combustion. ELECTRICITY AND MAGNETISM. Editor, Dr. D. F. Bleil, U.S. Naval Ordnance Laboratory

5

Definitions, units, nomenclature, symbols, conversion tables. Formulas. Electrical standards.' Properties of dielectrics. Electrical conductions in gases. Magnetic properties of materials. Electrochemical information. Electric and magnetic fields in the earth's environment. ' Lunar, planetary, solar, stellar, and galactic magnetic fields. OPTICS. Editor, Dr. Bruce H. Billings, Joint Commission on Rural Reconstruction, Taipei, Taiwan Fundamental definitions, standards, and photometric units. Refractive index of special crystals and certain glasses. Transmission and absorption of special crystals and certain glasses. Geometrical optics and index of refraction of various optical glasses. Index of refraction for visible light of various solids, liquids, and gases. Optical characterIstics of various uniaxial and biaxial crystals. Optical properties of metals. Reflection. Glass, polarizing, and interference filters. Colorimetry. Radiometry. Wavelengths for spectrographic. calibration. Magneto-, electro-, and )l):otoelastic optical constants. Nonlinear optical coefficients. Speqific rotation. Radiation detection. Radio astronomy. Far infrared. Optical masers.

6

CONTENTS

ATOMIC AND MOLECULAR PHYSICS Editor, Dr. H. M. Crosswhite, The Johns Hopkins University

Vll

7

The periodic system. The electronic structure of atoms. Energylevel diagrams of atoms. Persistent lines of the elements. Important atomic spectra. X-ray wavelengths and atomic energy levels. Constants of diatomic molecules. Constants of polyatomic molecules. Atomic transition probabilities. NUCLEAR PHYSICS. Editor, Dr. J. B. Marion, The University of Maryland

8

Nuclear constants and calibrations. Properties of nuclidcs. Atomic mass formulas. Passage of charged particles through matter. Gamma rays. Neutrons. Nuclear fission. Elementary particles and interactions. Health physics. Particle accelerators. SOLID-STATE PHYSICS Editor, Dr. H. P. R. Frederikse, The National Bureau of Standards Crystallographic properties. Structure, melting point, density, and energy gap of simple inorganic compounds. Electronic properties of solids. Properties of metals. Properties of semiconductors. Properties of ionic crystals. Properties of superconductors. Color centers and dislocations. Luminescence. Work function and secondary emission. Index follows Section 9.

9

Contributors

J. R. Anderson, Ph.D., University of Maryland. Solid State Physics (9d) Gordon Atkinson, Ph.D., University of Maryland. Electricity and Magnetism (5g) . Fred Ayres, Ph.D., Reed College. Mechanics (2n) H. W. Babcock, Ph.D., Hale Observatories. Electricity and Magnetism (5i) Julius Babiskin, Ph.D., U.S. Naval Research Laboratory. Solid State Physics (9d) Stanley Ballard, Ph.D., University of Florida. Optics (6b, 6c) Philip Baumei~ter, Ph.D., University of Rochester. Optics (6i) J. A. Bearden, Ph.D., The Johns H01Jkins University. Atomic and Molecular Physics (7f) E. C. Beaty, Ph.D., The National Bureau of Standards, Boulder. Electricity and Magnetism (5e) L. 1. Beaubien, Ph.D., U.S. Naval Research Laboratory. Mechanics (2e) Leo L. Beranek, Ph.D., Bolt Beranek q,nd Newman Inc. Acoustics (3a, 3b, 3d, 3p) Robert T. Beyer, Ph.D., Brown University. Acoustics (30) Hans Bichsel, Ph.D., University of Washington. Nuclear Physics (8d) Bruce H. Billings, Ph.D., Joint Commission on Rural Reconstruction, Taipei, Taiwan. Optics (6c, 6f, 6h, 61, 60) David T. Blackstock, Ph.D., University of Texas. Acoustics (3n) E. Boldt, Ph.D., NASA-Goddard Space Flight Center. Electricity and Magnetism (5i) R. M. Bozorth, Ph.D., U.S. Naval Ordnance Laboratory. Electricity and Magnetism (5f) Willem Brouwer, Ph.D., Diffraction Ltd. Inc. Optics (6d) James S. Browder, Ph.D., University of Florida. Optics (6b, 6c) R. M. Burley, A.B., Baird-Atomic, Inc. Optics (6p) Constance Carter, M.S., Library of Congress. Mathematics Bibliography; SI Units (la, Ib) Gregg E. Childs, Ph.D., The National Bureau of Standards, Boulder. Heat (4g) R. J. Collins, Ph.D., University of Minnesota. Optics (6s) W. R. Cook, Jr., M.A., Gould, Inc. Optics (6m) H. M. Crosswhite, Ph.D., The Johns Hopkins University. Atomic and Molecular Physics (7a, 7b, 7c, 7d, 7e) Evan A. Davis, Ph.D., Westinghouse Research Laboratory. Mechanics (2f) R. DiPippo, Ph.D., Southeastern Massachusetts University. Mechap.ics (2r) E. S. Domalski, Ph.D., The National Bureau of Standards. Heat (4j) J. D. H. Donnay, Ph.D., The Johns Hopkins University. Solid State Physics (9a) Thomas B. Douglas, Ph.D., The National Bureau of Standards. Heat (4e) J. F. Ebersole, Ph.D., University of Florida. Optics (6b, 6c) Phillip Eisenberg, Ph.D., Hydronautics, Inc. Mechanics (20) Eugene Epstein, Ph.D., Aerospace Corporation. Optics (6q) John Evans, Ph.D., Air Force Cambridge Research Laboratories. Optics (6i) Robley D. Evans, Ph.D., Massachusetts Institute of Technology. Nuclear Physics (8e) ix

x

CONTRIBUTORS

H. P. R. Frederikse, Ph.D., The National Bureau of Standards. Solid State Physics (9b, 9c, ge) Eli Freedman, Ph.D., Ballistic Research Laboratories. Mechanics (2v) R. J. Friaui, Ph.D., University of Kansas. Solid State Physics (9f) Dudley Fuller, Ph.D., Columbia University. Mechanics (2d) George T. Furakawa, Ph.D., The National Bureau of Standards. Heat (4e) John S. Gallagher, A.B., The National Bureau of Standards. Heat (4i) Murrey D. Goldberg, Ph.D., Brookhaven National Laboratory. Nuclear Physics (8f) David T. Goldman, Ph.D., The National Bureau of Standards. Nuclear Physics (8b) Edward F. Greene, Ph.D., Brown University. Mechanics (2v) Martin Greenspan, B.S., The National Bureau of Standards. Acoustics (3e) B. Gutenburg, Ph.D. (Deceased), California Institute of Technology; Mechanics (2i) George A. Haas, Ph.D., U.S. Naval Research Laboratory. Solid State Physics (9j) Lawrence Hadley, Ph.D., Colorado State University. Optics (6g) Thomas A. Hahn, B.S.; The National Bureau of Standards. Heat (4£) William J. Hall, A.B., The National Bureau of Standards, Boulder. Heat (4a) Cyril M. Harris, Ph.D., Columbia University. Acoustics (3j) F. K. Harris, Ph.D., The National Bureau of Standards. Electricity and Magnetism (5c) Miles F. Harris, Ph.D., National Oceanic and Atmospheric Administration. Mechanics (2k) John A. Harvey, Ph.D., Oak Ridge National Labomtory. Nuclear Physics (8f) Georg Hass, Ph.D" U.S. Army Electronics Command. Optics (6g) J. p, Heppner, Ph.D.; NASA-Goddard Space Flight Center. Electricity and Magnetism (5h, 5i) G. Herzberg, Ph.D" National Research Council of Canada. Atomic and Molecular Physics (7h) L. Herzberg, Ph.D. (Deceased), National Research Council of Canada. Atomic and Molecular Physics (7h) D, B. Herrmann, Ph.D., Bell Telephone Laboratories, Inc. Electricity and Magnetism (5d) Joseph Hilsenrath, M.A., The National Bureau of Standards. Heat (4h) David L. Hogenboom, Ph.D., Lafayette College. Mechanics (2m) Robert Howard, Ph.D., Hale Observatories. Electricity and Magnetism (5i) K. P. Huber, Ph.D., National Research Council of Canada. Atomic and Molecular Physics (7g) R. P. Hudson, Ph.D., The' National Bureau of Standards. Electricity and Magnetism (5f) Frederick V. Hunt, Ph.D., Harvard University. Acoustics (3c) Hans Jaffe, Ph.D., Gould, Inc. Optics (6m) T. L. Jobe, B.S., The National Bureau of Standards. Heat (4j) Joseph Kaspar, Ph.D., Aerospace Corporation. Optics (6k) R. Norris Keeler, Ph.D., Lawrence Radiation Laboratory. Heat (4d) George C. Kennedy, Ph.D., University of California. Heat (4d) Joseph Kestin, Ph.D., Brown University. Mechanics (2r) Richard K. Kirby, B.S., The National Bureau of Standards. Heat (4£) Max Klein, Ph.D., The National Bureau of Standards. Heat (4i) C. C. Klick,. Ph.D., U.S. Naval Research Labomtories. Solid State Physics (9h) Karl R. Koch, Ph.D., National Oceanic & Atmospheric Administration. Mechanics (2h) R. Bruce Lindsay, Ph.D., Brown University. Mechanics (2a, 2b, 2c, 2g) Robert Lindsay, Ph.D., Trinity College. Mechanics (21) Go' L. Link, Ph.D.,. Bell Telephone Laboratories, Inc. Electricity and Magnetism (5d)

CONTRIBUTORS

Xl

Ernest Loewenstein, Ph.D., Air Force Cambridge Research Laboratories. Optics (61') Lewis G. Longsworth, Ph.D., The Rockefeller University. Mechanics (2p) Walter Loveland, Ph.D., Oregon State University. Nuclear Physics (Sg) David MacAdam, Ph.D., Eastman Kodak Company. Optics (6a, 6j) Nancy R. McClure, Eastman Kodak Company. Optics (6d) T. R. McGuire, Ph.D., IBM-Watson Resew'ch Center. Electricity and Magnetism (5f) John E. McKinney, Ph.D., The National Bureau of Standards. Mechanics (21) J. B. Marion, Ph.D., University of Maryland. Nuclear Physics (Sa) Robert S. Marvin, Ph.D., The National Bureau of Standards. Mechanics (2m) W. P. Mason, Ph.D., Columbia University. Acoustics (3f, 3g); Solid State Physics (9a) Frank Massa, M.S., Dynamics Corporation of America. Acoustics (3i) W. J. Merz, Ph.D., RCA Laboratories. Solid State Physics (9f) B. M. Miles, Ph.D., Th.e National Bt;reau of Standards. Atomic and Molecular Physics (7i) David Mintzner, Ph.D., Northwestern University. Mechanics (2a) Karl Z. Morgan, Ph.D., Oak Ridge National Laboratory. Nuclear Physics (Si) Edwin B. Newman, Ph.D., Harvard University. Acoustics (3k) Wesley L. Nyborg, Ph.D./University of Vermont. Mechanics (2n, 2q) Harry F. Olson, Ph.D., RCA Laboratories. Acoustics (31, 3m) Norman Pearlman, Ph.D., Purdue University. Heat (4e) Karl B. Persson, Ph.D., The National Bureau of Standards, Boulder. Electricity and Magnetism (5e) Harmon H. Plumb, Ph.D., The National Bureau of Standards. Heat (4a) Robert L. Powell, Ph.D., The National Bureau of Standards, Boulder. Heat (4a, 4g) Martin P. Reiser, Ph.D., University of Maryland. Nuclear Physics (Sj) B. W. Roberts, Ph.D., General Electric Research and Development Center. Solid State Physics (9g) R. C. Roberts, Ph.D., University of Maryland. Mechanics (2s, 2t, 2u) Arthur H. Rosenfeld, Ph.D., University of California. Nuclear Physics (Sh) Bruce D. Rothrock, B.S., The National Bureau of Standards. Heat (4f) Hellmut H. Schmid, Ph.D., National Oceanic & Atmospheric Administration. Mechanics (2h) R. H. Schumm, M.S., The National Bureau of Standards. Heat (4j) Arthur F. Scott, Ph.D., Reed College. Mechanics (2n) Philip A. Seeger, Ph.D., Los Alamos Scientific Laboratory. Nuclear Physics (Sc) J. M. H. Levelt Sengers, Ph.D., The National Bureau of Standards. Heat (4i) J. C. Slater, Ph.D., University of Florida. Solid State Physics (9c) Donald R. Smith, M.S., Air Force Cambl'idge Research Laboratories. Optics (61') W. R. Smythe, Ph.D., California Institute of Technology. Electricity and Magnetism (5a, 5b) George A. Snow, Ph.D., University of Maryland. Nuclear Physics (Sh) Irene A. Stegun, M.S., The National Bureau of Standards. Mathematics Bibliography; SI Units (la, lb) D. E. Stone, A.B., Vertex Corporation. Mechanics (2b, 2e) Daniel R. Stull, Ph.D., Dow Chemical Company. Heat (4k) Masahisa Sugiura, Ph.D., NASA--Goddard Space Flight Center. Electricity and Magnetism (5h, 5i) James F. Swindells, A.B., The National Bureau of Standards. Heat (4a) Paul Tamarkin, Ph.D., RAND Corporation. Mechanics (2a) J. S. Thomsen, Ph.D., The Johns Hopkins University. Atomic and Molecular Physics (7f)

Xll

CONTRIBU'l'ORS

H. M. Trent, Ph.D. (Deceased), U.S. Naval Research Laboratory. Mechanics (2b, 2e) James E. Turner, Ph.D., Oak Ridge National Laboratory. Nuclear Physics (8i) Allyn C. Vine, Ph.D., Woods Hole Oceanographic Institution. Mechanics (2j) D. D. Wagman, Ph.D., The National Bureau of Standards. Heat (4j) David White, Ph.D., University of Pennsylvania. Heat (4c) J. E. White, Ph.D., Globe Universal Sciences, Inc. Mechanics (2i) W. L. Wiese, Ph.D., The National Bureau of Standards. Atomic and Molecular Physics (7i) Randolph C. Wilhoit, Ph.D., Texas A and M University. Heat (41) Ferd E. Williams, Ph.D., University of Delaware. Solid State Physics (9i) E. A. Wood, Ph.D., Bell Telephone Laboratories. Solid State Physics (9a) Cavour Yeh, Ph.D., University of California at Los Angeles. Electricity and Magnetism (5b) Kenneth F. Young, B.S., The National Bureau of Standards. Solid State Physics (9f) Robert W. Young, Ph.D., U.S. Naval Undersea Research and Development Center. Acoustics (3h) Mark W. Zemansky, Ph.D., The City College of the City University of New York. Heat (4b) Fritz Zernike, M.S., Perkin-Elmer Corporation. Optics (6n) Bruno J. Zwolinski, Ph.D., Tm;". A and M University. Heat (41)

Preface

The American Institute of Physics Handbook has won wide acceptance among scientists and engineers. It is just such a degree of acceptance that has stimulated the issuance of this revised and updated third edition. This edition, like the previous two, continues the philosophy of supplying authoritative reference material-including tables of data, graphs, and bibliographies-selected and described with a minimum of narration by leaders in physical methods for research. Among the entirely new sections in this edition are those on nonlinear optics, calibration energies for alpha particles and gamma rays, nonlinear acoustics, atomic mass formulas, particle accelerator principles, atomic transition probabilities, electric and magnetic fields in the earth's environment, and far infrared. Examples of topics in which especially extensive revisions have been made are: optical masers, various optical constants, virial coefficients, heats of combustion and formation, and superconductors. A number of sections were completely rewritten; these include radioastronomy, radiometry, various crystal properties, molecular constants and phase transitions. The mathematics section now consists of a special treatment of 81 units and a bibliography that has been revised to include references to new methods, algorithms, and computer programs. Publication of this Handbook was a mammoth undertaking that required the contributions and cooperation of many individuals and two organizations. Leading the individuals is Dr. Dwight E. Gray, who served as coordinating editor for this 1972 edition, as he also did for the 1957 and 1963 editions. Dr. Gray, who is a master of the pen and is well grounded in physics, was able to coordinate successfully the efforts of the eight section editors and the some 125 contributors. He did this work while also serving as the Washington Representative of the American Institute of Physics. Through his Washington office he was able xiii

XIV

PREFACE

to maintain contact with and coordinate the efforts of the many individuals concerned in the effort, as well as to handle the involvements of the sponsor-the American Institute of Physics-and the publisherthe McGraw-Hill Book Company. Key McGraw-Hill individuals for this project included Mrs. Winifred C. Eisler, who copy-edited the manuscript, and Mr. Don A. Douglas, the Editing Manager. To these individuals, editors and physicists alike, the scientific community is deeply indebted for their painstaking and conscientious contributions and acknowledges their efforts with thanks. As with any user-oriented publication, comments, suggestions, and criticisms are solicited on this edition of the Handbook. Only with such continuing contributions and cooperation can future Handbooks meet their responsibilities. H. WILLIAM KOCH, Director American Institute of Physics

AMERICAN INSTITUTE OF PHYSICS HANDBOOK

Section 1

MATHEMATICS BIBLIOGRAPHY; SI UNITS CONTENTS 1a. Mathematics Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1-2 lb. SI Units .......................................................... " 1-8 The third edition of the AlP Handbook, like the second, presents a bibliography of mathematical references in lieu of an assortment of actual mathematical tables. Selection of such tables necessarily would have been arbitrary; they would have been bound to duplicate many tables already easily available to most physicists; and, most important, including them would have necessitated the omission of significant physics material. The basic pattern of the third-edition bibliography is described at the beginning of Sec. 1a. For reasons outlined in the first paragraph of Sec. 1b, it was believed neither practicable nor desirable to attempt exclusive use of the International System of Units in this edition of the Handbook. Section 1b outlines the background of SI Units, and presents a portion of a National Bureau of Standards bulletin on their interpretation.

1-1

la. Mathematics Bibliography IRENE: A; STEGUN

The National Bureau of Standards CONSTANCE CARTER

liibrary of Congress

In view of the appearance of large compendiums and the increasing use of computers with built-in functions or function subroutines in their compilers, many of the tables of elementary functions have been omitted from this bibliography. An effort has been made to include a dictionary; indexes of mathematical and statistical tables; compendiums of general tables, series, integrals, transforms, and differential equations; and references to numerical methods, new tables, and new disciplines. For algorithms covering a wide variety of subjects such as the evaluation of systems of linear equations, estimations of definite integrals, sorting of data, etc., reference should be made to the "Collected Algorithms from CACM" (Communications of the Association: for Computing Machinery, Inc.). 1. Abramowitz, Milton, and Irene A. Stegun, eds.: "Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables," Dover Publications, Inc.,· New York, 1965, 1046 pages (Republication of National Bureau of Standards, Applied mathematics series, 55. Government Printing Office, Washington, D.C., 1964): A compendium containing most of the tables that have previously appeared in the United States, including the National Bureau of Standards Mathematical Tables, Applied Mathematics, and Columbia University Press Series. Contains mathematical properties, interrelations, and numerical methods, as well as an updated bibliography of textbooks and tables. 2. Arfken, George Brown: "Mathematical Methods for Physicists," Academic Press, Inc., New York, 1968, 704 pages: Includes bibliographies. 3. Bierens de Haan, David: "Nouvelles Tables d'Integrales Definies" (New tables of definite integrals). Corrected 1867 edition, with an English translation of the introduction by J. F. Ritt. Hafner Publishing Company, Inc., New York, 1965,716 pages: A special collection of some 8,400 integrals. 4. British Association for the Advancement of Science: "Mathematical Tables." Prepared under the auspices of the Royal Society, Cambridge University Press, London, 1931-1958: Vol. 1: "Circular and Hyperbolic Functions," 3d ed, 1951. Vol. 2: "Emden Functions," 1932. New edition in preparation. Vol. 3: "Minimum Decompositions into Fifth Powers," 1933. Vol. 4: "Cycles of Reduced Ideals in Quadratic Fields," 1934. Vol. 5: "Factor Table," 1935. Vol. 6: "Bessel Functions," pt. 1, 1958. Vol. 7: "The Probability Integral," 1939. Vol. 8: "Number-divisor Tables," 1940. Vol. 9: ·'Table of Powers Giving Integral Powers of Integers," 1950. Vol. 10: "Bessel Functions," pt. 2, 1952. "Auxiliary Tables," nos. 1-2, 1946. Continued by the Royal Society mathematical tables.

1-2

MATHEMATICS BIBLIOGRAPHY

1-3

5. Burington, Richard S.: "Handbook of Mathematical Tables and Formulas," 4th ed., McGraw-Hill Book Company, New York, 1965, 448 pages: A companion to the "Handbook of Probability and Statistics with Tables," by Richard S. Burington and Donald C. May, the 4th edition includes new sections on sets, relations, and functions; algebraic structures; Boolean algebra; numher systems; matrices; and statistics. A table of derivatives and a comprehensive table of integrals have been included. 6. Burington, Richard S., and Donald C. May: "Handbook of Probahility and Statistics with Tables," 2d ed., McGraw-Hill Book Company, New York, 1970, 450 pages. 7. Byerly, William E.: "An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics," Dover Publications, Inc., New York, 1959, 287 pages. 8. Byrd, Paul F., and Morris D. Friedman: "Handbook of Elliptic Integrals for Engineers and Physicists," (Die Grundlehren der mathematischen Wissenschaften, Band 67), Springer Verlag, Berlin, 1954, 355 pages: A collection of over 3,000 integrals and formulas using Legendre's and Jacobi's notations. 9. Campbell, George A., and Ronald M. Foster: "Fourier Integrals for Practical Applications," D. Van Nostrand Company, Inc., Princeton, N.J., 1948, 177 pages: A large number of the known closed-form evaluations of Fourier integrals are compiled and tabulated in compact form for convenient use. Tables give coefficient pairs, admittances, and transient solutions. 10. "C. R. C. Standard Mathematical Tables," 16th ed, edited by Samuel Selby, Chemical Rubber Co., Cleveland, 1968, 692 pages: An expanded, revised edition of a standard work. The sections involving mensuration, trigonometry, analytic geometry, curves and graphs, and the algebra of sets have been completely rewritten, and sections to cover determinants and matrices have been added. An extension to the octal decimal conversion table to include hexadecimal and decimal conversion increases the usefulness of the volume. 11. David, F. N., M. G. Kendall, and D. E. Barton: "SYmmetric Function and Allied Tables," Cambridge University Press, London, 1966, 278 pages: An elaborate set of 49 major tables accompanied by a detailed introduction of 63 pages, constituting a self-contained treatment of symmetric functions and their applications in statistics. A definitive compilation. 12. Davis, Harold T.: "The Summation of Series," Principia Press of Trinity University, San Antonio, Tex., 1962, 140 pages: Special emphasis placed upon the case of finite limits. 13. Davis, Harold T., comp.: "Tables of the Higher Mathematical Functions," Principia Press, Bloomington, Ind., 1933-1935, 2 vols.: Vol. I: Various tables of the gamma and psi functions as well as sections on classification and history of tables, interpolation and its uses, and interpolation tables. Vol. II: Tables of the polygamma functions (trigamma-hexagamma), the Bernoulli and Euler polynomials and numbers, gram polynomials, and polynomial approximation. 14. Davis, Philip J., and Philip Rabinowitz: "Numerical Integration," Blaisdell Publishing Company, Waltham, Mass., 1967, 230 pages: Includes bibliographies. 15. Doetsch, Gustav: "Handbuch def Laplace-Transformation" (Handbook of Laplace transforms). Verlag Birkhauser, Basel, 1950-1956, 3 vols. ("Lehrbucher und Monographien aus dom Gebiete der exakten Wissenschaften, Mathematische Reihe," vols. 14, 15, and 19): Contents: Vol. 1, "Theory of Laplace Transforms"; vols. 2-3, "Applications of Laplace Transforms," including asymptotic expansions, convergent expansions, ordinary and partial differential equations, integral equations, and whole exponential functions. 16. Dwight, H. B.: "Tables of Integrals and Other Mathematical Data," 4th ed., The Macmillan Company, New York, 1961, 336 pages: Contains derivatives and integrals, classified as algebraic, trigonometric, inverse trigonometric, and exponential functions; probability integrals; logarithmic, hyperbolic, inverse hyperbolic, elliptic, and Bessel functions; surface zonal harmonics; definite integrals; and differential equations. Appendixes: A, Tables of Numerical Values; B, Bibliography. 17. Erdelyi, Arthur, and others: "Higher Transcendental Functions" (Based, in part, on notes left by Harry Bateman and compiled by the staff of the Bateman Manuscript Project, California Institute of Technology), McGraw-Hill Book Company, New York, 1953-1955,3 vols.:

1-4

MATHEMATICS BIBLIOGRAPHY; SI UNITS

An account of the principal properties of such functions as gamma, hypergeometric, Legendre, Bessel, elliptic, automorphic, and generating functions, with extensive lists of references at the end of each chapter. 18. Erdlilyi, Arthur, and others: "Tables of Integral Transforms" (Based, in part, on notes left by Harry Bateman and compiled by the staff of the Bateman Manuscript Project, California Institute of Technology) McGraw-Hill Book Company, New York, 1954, 2 vols: Intended as a companion and sequel to "Higher Transcendental Functions." Contains Fourier, Laplace, and Mellin transforms and their inversions, as well as Hankel transforms. Also included are gamma, Legendre, Bessel, and hypergeometric functions. The entries are arranged in tabular form. 19. Fettis, Henry E., and James C. Caslin: "Elliptic Functions for Complex Arguments," Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio, 1967, 404 pages, ARL 67-0001 (Available from Clearinghouse for Federal Scientific and Technical Information, Springfield, Va. 22151): These unique tables consist of 5D values of the Jacobian elliptic functions sn(w,lc) , cn(w,k) , and dn(w,k), where w = u iv, as functions of Jacobi's nome q, which equals exp (- K' / K), where K and K' are the quarter-periods (the complete elliptic integrals of the first kind of modulus Ie and of complementary modulus Ie', respectively). The range of parameters in the table is: q = 0.005(0.005)0.480, u/K = 0(0.1)1, and v/K' = 0(0.1)1. 20. Fettis, Henry E., and James C. Caslin: "Ten-place Tables of the Jacobian Elliptic Functions," pt. 1, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio, 1965, 562 pages, ARL 65-180 (Available from Clearinghouse for Federal Scientific and Technical Information, Springfield, Va. 22151): This report contains 10D tables of the Jacobi elliptic functions arn(u,k) , sn(u,k) , cn(u,k) , and dn(u,k), as well as the elliptic integral E(am(u),k) , k' = 0(0.01)0.99, u = O(O.Ol)K(k), and for k' = 1, u = 0(0.01)3.69. 21. Fettis, Henry E., and James C. Caslin: "An Extended Table of Zeros of Cross Products of Bessel Functions," Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio, 1966, 126 pages, ARL 66-0023 (Available from Clearinghouse for Federal Scientific and Technical Information, Springfield, Va. 22151) : This report presents 10D tables of the first five roots of the equations: (a) Jo(a) Yo (kOl) - Yo(Ol)Jo(kOl) = 0, (b) Jl(a) Y 1 (kOl) - Y 1 (0l)J 1 (ka) = 0, (c) JO(Ol) Y 1 (kOl) - Y o(a)J 1 (kOl) = O. 22. Fletcher, Alan, and others, eds.: "An Index of Mathematical Tables," 2d ed., AddisonWesley Publishing Company, Inc., Reading, Mass., for Scientific Computing Service, Ltd., 1962, 2 vols.: The second edition is more than double the size of the 1946 edition, and includes, as a new feature, a list of errors found in published tables. Contains an index according to function, giving for each table the range, tabular interval, number of significant figures in the values, 'whether or not tables of proportional parts are given, what orders of differences are shown, etc. Also ineludes an alphabetical list of references by author and publication year. Considered an important index to well-known tables of functions and to other less-known tables appearing in books and periodicals. 23. Forsythe, G. E., and P. C. Rosenbloom: "Numerical Analysis and Partial Differential Equations," John Wiley & Sons, Inc., New York, 1958, 204 pages. 24. Frazer, Robert A., W. J. Duncan, and A. R. Collar: "Elementary Matrices and Some Applications to Dynamics and Differential Equations," The Macmillan Company, New York, 1946, 416 pages. 25. Great Britain, National Physical Laboratory: "Mathematical Tables," H. M. Stationery Office, London, 1956-Vol. 1: "The Use and Construction of Mathematical Tables," by L. Fox, 1956. Vol. 2: "Tables of Everett Interpolation Coefficients," by L. Fox, 1956. Vol. 3: "Tables of Generalized Exponential Integrals," by G. F. Miller, 1960. Vol. 4: "Tables of Weber Parabolic Cylinder Functions and Other Functions for Large Arguments," by L. Fox, 1961. Vol. 5: "Chebyshev Series for Mathematical Functions," 1962. Vol. 6: "Tables for Bessel Functions of Moderate or Large Order," 1962. Vol 7: "Tahles of Jacobian Elliptic Functions Whose Arguments Are Rational Fractions of the Quarter Period," 1964. This series contains tables of mathematical functions which may not come within the range of the more fundamental tables. 'fl. Greenwood, Joseph A., and H, O. Hartley: "Guide to Tables in Mathematical Sta~;istics.J1 Princeton University Press, Princeton; N'.).j 1962, 1014 pages.

+

MATHEMATICS BIBLIOGRAPHY

1-5

27. Grabner, Wolfgang, and N. Hofreiter: "Integraltafel (Integral table), pt. I, "Indefinite Integrals"; pt. II, "Definite Integrals,'~ 2d improved ed., Springer Verlag, Vienna. 1965-1966, 2 vols., 166 and 204 pages: An extensive collection of integrals including a brief survey of methods of evaluation and transformation of integrals. 28. Hart, John F., and others: "Computer Approximations," John Wiley & Sons, Inc., New York, 1968, 343 pages: Extensive in its range in accuracy from a few digits· up to 25D in the approximations; its wide selection of functions includes square root and cube root,exponentialand hyperbolic, logarithm, trigonometric and inverse trigonometric functions, gamma function and its logarithm, error function, Bessel functions, and complete elliptic integrals; and in the range of methods, the book describes and compares them from the general methods of subroutine design t6procedures for the design of maximum efficiency programs for commonly neede'd functions. 29. Hartree, Douglas R.: "Numerical Analysis," Oxford University Press, London, 1952, 287 pages: Includes interpolation and numerical integration formulas, finite differences, harmonic analysis, smoothing. 30. Harvard University Computation Laboratory: "Tables of the Bessel Functions of the First Kind," Harvard University Press, Cambridge, Mass., 1947-1951, 12 vols.: In(x), 0 :::; x :::; 100, 18D for n = 1; 10D for n = 2 through 135. 31. Harvard University Computation Laboratory: "Tables of the Cumulative Binomial Probability Distribution," Harvard University Press, Cambridge, Mass., 1955, 503 pages ("Annals of the Computation Laboratory of Harvard University," vol. 35). 32. Harvard University Computation Laboratory: "Tables of the Function arc sin z," Harvard University Press, Cambridge, Mass., 1956, 586 pages ("Annals· of the Computation Laboratory of Harvard University," vol. 40). 33. Householder, Alston S.: "Principles of Numerical Analysis," McGraw-Hill Book Company, New York, 1953, 274 pages. 34. Jahnke, Eugene, Fritz Emde, and Friedrich Losch: "Tables of Higher Functions," 7th ed, B. G. Teubner, Stuttgart, 1966, 322 pages: Text in German and English. Essentially a corrected version of the sixth edition, containing Bessel functions, circular and hyperbolic functions of a complex variable, cubic equations, miscellaneous conversion tables, Planck's radiation function, powers (2d to 15th), probability integral and related functions, reciprocals and square roots of complex numbers, Riemann zeta function, theta functions, transcendental equations, vector addition, and sine, cosine, and logarithmic integral. 35. James, Glenn, and Robert C. James: "Mathematics Dictionary," 3d ed., D. Van Nostrand Company, Inc., Princeton, N.J., 1968,448 pages: Correlated condensation of mathematical concepts designed for timt) saving reference work. 36. Jolley, Leonard B. W.: "Summation of Series," Dover Publications, Inc., New York, 1961, 251 pages. 37. Kamke, Erich: "Differentialgleichungen, Losungsmethoden und Losungen (Differential equations, methods of solution, and solutions), 6th improved ed., Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1959, 666 pages ("Mathematik und ihre Anwendungen in Physik und Technik," Ser. A, vol. 18): A reference work containing general methods of solution and properties of solution, boundary-, and characteristic-value problems, and a dictionary of some 1,600 equations in lexicographical order with solutions, techniques for solving, and references. Knuth, Donald E.: "The Art of Computer Programming," Addison-Wesley Publishing 38. Company, Inc., Reading, Mass., 1968-1969, 2 vols. 39. Korn, Granino A., and Theresa M. Korn.: "Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review," 2d, enlarged and rev. ed., McGraw-Hill Book Company, New York, 1968, 1129 pages. 40. Lehmer, Derrick H.: "Guides to Tables in the Theory.of Numbers," National Academy of Sciences-National Research Council, Washington, D.C., 1941, 177 pages (National Research Council Bulletin 105). 41. Lehmer, Derrick Norman: "Factor Table for the First Ten Millions Containing the Smallest Factor of Every Number not Divisible by 2, 3, 5, or 7 between the Limits 0 and 1O,017 i OOO."Hafner Publishing Company, Inc., New York, 1956, 476 pages (Carnegie Institution of Washington Publ. 105): Introduction includes a list of errors in former tables by other authors. 42. Lehmer, Derrick Norman: "List of Prime Numbers from 1 to 10,006,721," Hafner Publishing Company, Inc., New York, 1956, 133 pages (Carnegie Institution of Washington Pubt 165):

1-6 43.

44.

45.

46.

47. 48. 49.

50.

51. 52. 53.

54.

55.

MATHEMATICS BIBLIOGRAPHY; SI UNITS

The standard list of primes.· Arranged in such a way that it is easy to find the nth prime for a given n. Lieberman, Gerald J., and D. B. Owen: "Table~ of the Hypergeometric Probability Distribution," Stanford University Press, Stanford, Calif., 1961, 726 pages (Stanford studies in mathematics and statistics no. 3): In addition to the following tables of the hypergeometric probability distribution: N = 2, n = 1 through N = 100, n = 50; N = 1000, n = 500; k = n - 1, n; n = N /2: N = 100, n = 50 through N = 2000, n = 1000, the theory, rationale, and specific applications of the hypergeometric probability are discussed. Luke, Yudell: "Integrals of Bessel Functions," McGraw-Hill Book Company, New York, 1962,424 pages: Designed to provide the research worker with basic information dealing with definite and indefinite integrals involving Bessel functions. Madelung, Erwin: "Die Mathematischen Hilfsmittel des Physikers" (Mathematical tools for the physicist), 6th rev. ed., Springer Verlag, Berlin, 1957, 535 pages ("Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen," vol. IV): Comprehensive collection of formulas used in mathematical physics. Included are numbers, functions and operators, series, algebra, transformations, and statistics. Magnus, Wilhelm, Fritz Oberhettinger, and Raj Pal Soni: "Formulas and Theorems for the Special Functions of Mathematical Physics," 3d enlarged ed., Springer Verlag, Berlin, 1966, 508 pages ("Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen," Band 52) : Survey of the properties of a number of special functions including the following: gamma, hyper geometric, Bessel, Legendre, theta, and elliptic, as well as spherical harmonics, orthogonal polynomials, integral transforms and inversions, and coordinate transforms. Mangulis, V.: "Handbook of Series for Scientists and Engineers," Academic Press, Inc., New York, 1965, 134 pages. Margenau, Henry, and George M. Murphy: "The Mathematics of Physics and Chemistry, 2d ed., D. Van Nostrand Company, Inc., Princeton, N.J., 1956,2 vols. "Mathematics of Computation" (formerly: "Mathematical Tables and other Aids to Computation"), National Academy of Sciences National Research Council, quarterly, Washington, D.C.: Ajournal devoted to advances in numerical analysis, the application of computational methods. mathematical tables, high-speed calculators, and other aids to computation. Meyer zur Capellen, Walther: "Integraltafeln, Sammlung unbestimmter Integrale elElmentarer Funktionen" (Tables of integrals; Collection of indefinite integrals of elemetary functions), Springer Verlag, Berlin, 1950,. 292 pages: Lists some 3,000 integrals of algebraic and transcendental functions, as well as products of algebraic and transcendental functions. Tabulation permits use for differentiation purposes. Morse, Philip M., and Herman Feshbach: "Methods of Theoretical Physics," McGrawHill Book Company, New York, 1953, 2 vols. Oberhettinger, Fritz, "Tabellen zur Fourier Transformation" (Tables of Fourier transforms), Springer Verlag, Berlin, 1957, 213 pages. Parke, Nathan Grier: "Guide to the Literature of Mathematics and Physics including Related Works on Engineering Science," 2d rev. ed., Dover Publications, Inc., New York, 1958, 436 pages: A useful handbook comprising chapters on principles of reading and study, searching the literature, types of materials, library usage, etc.; includes an annotated bibliography of some 5,000 titles arranged by subject with author and subject indexes. Pearson, Karl: "Tables of the Incomplete Beta-function," 2d ed., ·with a new introduction by E. S. Pearson and N. L. Johnson, published for the Biometrika Trustees by the Cambridge University Press, Cambridge, Mass., 1968, 505 pages: Gives I(u,p) with the argument u proceeding by increments of 0.1 from 0 up to that value of u which gives I(u,p) = 1.0000000 to the seventh decimal place. The argument p advances from -1.0 by increments of 0.05, from 1.0 to 5.0 by increments of 0.1, and from 5.0 to 50.00 by intervals of 0.2. Two new tables give some additional values to the integral computed a number of years ago but not hitherto published, and a list of referenoes has been added. Peirce, Benjamin 0.: "A Short Table of Integrals," 4th ed., rev. by Ronald M. Foster, Ginn and Company, Boston, 1956, 189 pages: Fourth revision of Peirce's tables consisting of indefinite integrals, definite integrals, auxiliary formulas, and numerical tables; including common algebraic expressions; functions of angles in radians; differential equations.; exponential functions; hyperbolic-function formulas; elliptic integrals; natural logs;· tables of logs of numbers, logs of sines, cosines, etc.; probability integral and trigonometric formulas.

MATHEMATICS BIBLIOGRAPHY

1-7

56. Riordan, John: "An Introduction to Combinatorial Analysis," John Wiley & Sons, Inc., New York, 1958, 244 pages. 57. Roberts, G. E., and H. Kaufman: "Table of Laplace Transforms," W. B. Saunders Company, Philadelphia, 1966, 367 pages: A comprehensive reference of Laplace transforms and their inverses which should prove useful to pure and applied mathematicians. The volume is in two partsthe first devoted to direct transforms and the §econd to inverse transforms. 58. Royal Society of London: "Royal Society lVlathematical Tables," Cambridge University Press, London, 1950--. Vol. 1: "Farey Series of Order 1025," 1950. Vol. 2: "Rectangular-polar Conversion Tables," 1956. Vol. 3: "Tables of Binomial Coefficients," 1954. Vol. 4: "Tables of Partitions," 1958. Vol. 5: "Representations of Primes by Quadratic Forms," 1960. Vol. 6: "Tables of the Riemann Zeta Function," 1960. Vol. 7: "Bessel Functions," pt. 3, "Zeros and Associated Values," 1960. Vol. 8: "Tables of Natural and Common Logarithms to 110 Decimals," 1964. Vol. 9: "Tables of Indices and Primitive Roots," 1968. Vol. 10: "Bessel Functions," pt. 4, "Kelvin Functions," 1964. Vol. 11: "Coulomb Wave Functions," 1964. 59. Ryzhik, losif M., and 1. S. Gradshteyn: "Table of Integrals, Series, and Products," translated from the 4th Russian ed., Academic Press Inc., New York, 1965, 1086 pages: An inclusive compilation, the work is advertised as the most comprehensive table of integrals ever published. New material on lVlathieu, Struve, Lommel, as well as other special functions, has been added. 60. Slater, L. J.: "Confluent Hypergeometric Functions," Cambridge University Press, New York, 1960, 247 pages. 61. Smithsonian Institution: "Smithsonian Mathematical Formulae and Tables of Elliptic Functions," 3d reprinting, Washington, D.C., 1957, 314 pages. 62. Stroud, A. H., and D. Secrest: "Gaussian Quadrature Formulas," Prentice Hall, Inc., Englewood Cliffs, N.J., 1966, 374 pages: Valuable reference book for use and application of Gaussian quadrature formulas. Text is divided into five parts. Fortran programs to compute the abscissas and weights for quadrature formulas based on classieal Jacobi, Laguerre, and Hermite polynomials are presented. Chapter 5 summarizes the tables of quadrature formulas found in the literature. 63. Todd, John, ed.: "Survey of Numerical Analysis," McGraw-Hill Book Company, New York, 1962, 608 pages. 64. U.S. National Bureau of Standards: "Basic Theorems in Matrix Theory," Marvin Mareus, Government Printing Office, Washington, D.C., 1960, 27 pages (Applied mathematies series, 57). 65. - - : "Experimental Statistics," Mary Gibbons Natrella, Government Printing Office, Washington, D.C., 1963, 1 vol. (various pagings) (Handbook 91): A colleetion of statistical procedures useful in the design, development, and testing of materials; the evaluation of equipment performance; and the conduct and interpretation of scientific experiments. 66. - - : "Guide to Tables of the Normal Probability Integral," Government Printing Office, Washington, D.C., 1952, 16 pages (Applied mathematics series, 21): A ready desk reference to the normal probability integral tabulated in standard statistical textbooks and other important ·sources. Provides a list of available tables as well as the form of the function tabulated. 67. - - : "Matrix Representations of Groups," Morris Newman, Government Printing Office, Washington, D.C., 1968,79 pages (Applied mathematics series, 60). 68. - - : "Probability Tables for the Analysis of Extreme-value Data," Government Printing Office, Washington, D.C., 1953, 32 pages (Applied mathematics series, 22): Introduction outlines the theory and application of extreme values and describes nature, use, accuracy, and method of computation of tables. There are six tables for the asymptotic (cumulative) distribution of the largest value

"IF ij is the total internal force acting on mi (due to all other particles), and F i' is the external force on the ith particle. If we sum over all particles of the system, we obtain, by use of Newton's third law, N

L

Fii

=

(2a-24)

0

i=l MOTION OF THE CENTER OF MASS.

The analog of Newton's second law for the

entire system is therefore

MR

r

(2a-25)

N

where M

=

mi is the total mass of the system,

R is the acceleration of the center

i= 1

of mass of the system, and

~iFi'

is the total external force.

MOMENT OF MOMENTUM AND TORQUE. By forming the cross product of both sides of Eq. (2a-23) with ri and summing over all particles we can show that

~

I

[ri X (m;!i)] =

L

Ti'

= T'

(2a-26)

provided that the internal force Fij acts along the straight line connecting the particles i and j in each case. . In particular, if ri, is the position of the ith particle with respect to the center of mass, so that

it follows from Eq. (2a-26) that N

N

#t l

i=l

fi'

X (m;i:i')

L

ri, X F'

i=l

(2a-27)

FUNDAMENTAL CONCEPTS OF MECHANICS.

UNITS

2-9

That is, the time rate of change of the moment of momentum is equal to the total external torque when both are taken with respect to the center of mass. The above equation is also true if the center of mass is replaced by any point moving with the velocity of the center of mass, which may, of course, also be at rest. Conservation of Momentum. It follows from Eqs. (2a-25) and (2a-26) that: 1. If the total external force is zero, the linear momentum of the center of mass is constant. 2. If the total external torque about a fixed point, or one moving with velocity of the center of mass, is zero, the moment of momentum about that point is constant. Conservation of Energy. WORK-ENERGY THEOREM. The total work done by the external and internal forces acting on the system is equal to the change in the total kinetic energy of the system (the sum of the kinetic energies of all particles) N

\' i 1..

", - " L,\'N lei'r

mi(v i

riO

Vi) =

i=l

(F i'

+ Fii). dr;

(2a-28)

1=1

Z H Q

[Jl

2b. Density of Solids H. M. TRENTl

u.s. Naval Research Laboratory D. E. STONE

Vertex Corporation' R. BRUCE LINDSAY

Brown University

For the definition of density p consult Sec. 2a-3. The cgs unit of density is the gram per cubic centimeter and this is used throughout the tables in this subsection. Densities of the elements in solid form are given in Table 2b-1. All data are takeIt from "Smithsonian Physical Tables" (9th revised edition, 1954) unless otherwi~ stated. The values marked * are calculated densities from X-ray crystallographit data at room temperature and are taken from International Critical Tables (1926}. All others are measured values for poly crystalline condition, save when otherwiS( stated. Standard room temperature is understood, unless otherwise stated. TABLE

2b-1.

DENSITY OF THE ELEMENTS IN SOLID FORM;

Element

Physical state

Density, g/cm 3

Temp., °C

Aluminum ............ Aluminum ............ Antimony ............. Antimony ............. Argon ................ Argon ............. Arsenic ............... Arsenic ............... Barium ............... Beryllium ............. Beryllium ............. Bismuth .............. Bismuth .............. Boron ................ Bromine .............. Cadmium ............. Cadmium ............. Calcium .............. Calcium .............. Carbon ...............

Commercial hard-drawn solid Single crystal Vacuo-distilled solid Single crystal Solid Single crystal Crystallized solid Single crystal Solid Solid Single crystal Vacuo-distilled solid Single crystal Crystallized solid Solid Vacuo-distilled solid Single crystal Solid Single crystal Diamond

2.70 2.692* 6.62 6.73* 1.65 1.645* 5.73 5.75* 3.5 1.85 1.83* 9.78 9.86* 2.535 4.2 8.65 8.56* 1.55 1.54* 3.52

20 20 -233 -253 14 20 20 2(]

-273 20 20 20

Deceased . • H. M. Childers of the Vertex Corporation provided valuable consultant service. 1

2-:\1

---

2-20

MECHANICS TABLE

2b-1.

DENSITY OF THE ELEMENTS IN SOLID FORM '.

Element Carbon .......... ···· . Cerium ............ ·· . Cerium ............. · . Cerium ............... Cesium .......... ···· . Chlorine .............. ChrOnllUDl ............ Chromium ............ Cobalt ............ ·.· . Cobalt ................ Columbium ........... Copper ............... Copper ............... Erbium ............ ·· . Fluorine .............. Gallium .............. Germanium ........... GermaniuDl ........... Gold ................. Gold ................. Gold ............ ··.· . Hafnium .............. Hafnium .............. Helium ............... Hydrogen ............. Indium ............... Indium ............... Iodine ................ Iridium ............... Iridium ............... Iron .................. Iron .................. Krypton .............. Lanthanum ........... Lead ................. Lead ................. Lithium .............. Lithium .............. Magnesium ........... Magnesium ........... Manganese ............ Manganese ............ Mercury .............. Molybdenum .......... Molybdenum .......... Neodymium ...........

Physical state Graphite Solid Cubic crystal Hexagonal crystal Solid Solid Solid Crystal Solid Cubic crystal Solid Vacuo-distilled solid Single crystal Solid Solid Solid Solid Single crystal Vacuo-distilled solid Cast Single crystal Solid Single crystal Solid Solid Solid Single crystal Solid Solid Single crystal Pure solid Single crystal Fe-a Solid Solid Vacuo-distilled Single crystal Solid Single crystal Solid Single crystal Solid Single crystal Mn-a Solid Solid Single crystal Solid

-

(iJO':I/,Unued)

Density, g/cm"

Temp., °C

2.25 6.90 6.90* 6.73* 1.873 2.2 7.14 7.22* 8.71 8.67* 8.4 8.933 8.95* 4.77 1.5 5.93 5.46 5.38* 18.88 19.3 19.4* 13.3 11.3* 0.19 0.0763 7.28 7.43* 4.94 22.42 22.8* 7.86 7.92* 3.4 6.15 11.342 11.48* 0.534 0.534* 1.74 1.71* 7.3 7.21* 14.193 9.01 10.20* 7.00

20 20 20 -273 20 21 20 20 -273 23 20 20 20 -273 -260 20 17

-273 20 20 20

-38.8

2-21

DENSITY OF SOLIDS TABLE

2b-1.

DENSITY OF THE ELEMENTS IN SOLID FORM

Element Neon ................. NickeL ............... Nickel ................ Nitrogen .............. Osmium .............. Osmium ............... Oxygen ............... Palladium ............. ......... Palladium Phosphorus ........... Phosphorus ........... Phosphorus ........... Platinum ............. Platinum ............. Potassium ............ Praseodymium ........ Radium .............. Rhenium ............. Rhodium ............. Rubidium ............. Ruthenium ............ Samarium ............. Scandium ............. Selenium .............. Selenium .............. Silicon ................ Silicon ................ Silver ................ Silver ................ Sodium ............... Sodium ............... Strontium ............. Sulfur ................ Sulfur ................ Sulfur ................ Tantalum ............. Tantalum ............. Tellurium ............. Tellurium ............. Thallium ............. Thallium ............. Thorium .............. Thorium .............. Tin .................. Tin .................. Tin ..................

Physical state Solid Solid Single crystal Solid Solid Single crystal Solid Solid Single crystal Solid, white Solid, red Solid, black Solid Single crystal Solid Solid Solid Solid Solid Solid Solid Solid Solid Solid Single crystal Solid crystal Single crystal Vacuo-distilled Single crystal Solid Single crystal Solid Solid, rhombic Solid, monoclinic Single crystal Solid Single crystal Solid, crystal Single crystal Solid Single crystal Solid Single crystal Solid, white tetragonal Solid, white rhombic Solid, gray

(Continued)

Density, g/cm 3

Temp.,

1.204 8.8 9.04* 1.14 22.5 22.8* 1.568 12.16 12.25* 1.83 2.20 2.69 21.37 21. 5* 0.87 6.48 5(?) 20.53 12.44 1. 53 12.1 7.7-7.8 3.02(?) 4.82 4.86* 2.42 2.32* 10.492 10.49* 0.9712 0.954* 2.60 2.07 1. 96 2.02* 16.6 17.1 * 6.25 6.26* 11.86 11. 7* 11.00 12.0* 7.29 6.55 5.75

-245

DC

-273 -273

20 20

20 19

20 20 20

17 20

20

2-22

MECHANICS TABLE

2b-1.

DENSITY OF THE ELEMENTS IN SOLID FORM

Element Tin .................. Titanium ............. Titanium ............. Tungsten ............. Tungsten ............. Uranium .............. Vanadium ............ Vanadium ............ yttrium .............. Zinc .................. Zinc .................. Zinc .................. Zirconium ............. Zirconium .............

Physical state White single crystal Solid Single crystal Solid Single crystal Solid Solid Single crystal Solid Solid, vacuo-distilled Solid Single crystal Solid Single crystal

Density, g/cm'

7.30* 4.5 4.58* 19.3 19.3* 18.7 5.87 5.98* 3.8 6.92 4.32 7.04* 6.44 6.47*

(Continued)

Temp.,

DC

18

13 15 20 -273

DENSITY OF SOLIDS

2-23

TABLE 2b-2. DENSITY OF COMMON SOLIDS AT 20°C* Substance Agate .................... . Amber .................. . Anthracite ............... . Aragonite ................ . Asbestos ................. . Basalt ................... . Bee!!wax ................. . Beryl .................... . Bone .................... . Brick .................... . Butter ................... . Calcite ................... . Camphor ................. . Caoutchouc .............. . Celluloid ................. . Cement (set) ............. . Chalk ................... . Charcoal, oak ............. . Charcoal, pine ............ . Cinnabar ................. . Clay ..................... . Coal, soft ................ . Coke .................... . Cork .................... . Cork linoleum ............ . Corundum ............... . Dolomite ................. . Ebonite .................. . Emery ................... . Feldspar ................. . Flint .................... . Fluorite .................. . Garnet .......... : .... .' ... . Gelatin .................. . Glass, common ........ ! • • • • Glass, flint ............ 1• • • • Glue ..................... . Granite ................... . Graphite .............. '... . Gum arabic ........... '... .

Density, g/cm"

2.5-2.7 l.06-1.11 1.4-1.8 2.93 2.0-2.8 2.4-3.1 0.96-0.97 2.69-2.7 1. 7-2.0 1.4-2.2 0.86-0.87 2.71

0.99 0.92-0.99 1.4 2.7-3.0 1.9-2.8 0.57 0.28-0.44 8.12 1.8-2.6 1.2-1.5 1.0-1.7 0.22-0.26 0.55 3.9-4.0 2.84 1.15 4.0 2.55-2.75 2.63 3.18 3.15-4.3 1.27 2.4-2.8 2.9-5.9 1.27 2.64-2.76 2.30-2.72 1.3-1.4

Substance

Density, g/cm"

Gypsum ................. . Hematite ................ . Hornblende .............. . Ice ...................... . Ivory .................... . Lava, basaltic ............ . Lava, trachytic ........... . Leather, dry .............. . Leather, greased .......... . Lime, mortar ............. . Lime, slaked .............. . Limestone ................ . Magnetite ................ . Malachite ................ . Marble .................. . Mica .................... . Olivine .................. . OpaL .................... . Paper .................... . Paraffin .................. . Pitch .................... . Porcelain ................. . Pyrite ................... . Quartz ................... . Resin .................... . Rock salt ............... , . Rubber, hard ........ " ... . Rubber, soft .............. . Rutile ................... . Sandstone ................ . Slate .................... . Soapstone ................ . Starch ................... . Sugar .................... . Talc ..................... . Tallow ................... . Tar .. '................... . Topaz ................... . Tourmaline ............... . Wax, sE')aling .............. .

2.31-2.33 4.9-5.3 3.0 0.917 1.83-1.92 2.8-3.0 2.0-2.7 0.86 1.02 1.65-1. 78 1.3-1.4 2.68-2.76 4.9-5.2 3.7-4.1 2.6-2.84 2.6-3.2 3.27-3.37 2.2 0.7-1.15 0.87-0.91 1.07 2.3-2.5 4.95-5.1 2.65 1.07 2.18 1.19 1.1 4.2 2.19-2.36 2.6-3.3 2.6-2.8 1.53 1.61 2.7-2.8 0.91-0.91 1.02 3.5-3.6 3.0-3.2 1.8

'" The density varies with the state and previous treatment of the solids. The figures quoted may be considered reasonable limits (taken largely from" Smithsonian Physical Tables, .. 9th ed.).

2--24

MECHANICS

2b-3. DENSITY OF STEELS· (At room temperature)

TABLE

Composition Type of steel

p,

Condition

g/em' %C

%Si

%Mn

%Cr

-- -- -0.06 0.23 0.435 1.22 0.31 0.315 0.35 1.73 0.80 0.62 0.98 0.20 0.22 0.21 0.30 0.35

0.01 0.11 0.20 0.16 . ... . ... . ... . ... . ... . ... . ... .... . ... .... ....

0.38 . ............... AlInealed at 1700°F 0.635 . ............... Annealed at 1700°F 0.69 . ............... Annealed at 1580°F 0.35 . ............... Annealed at 1470°F Oil-quenched at 1650"F, tempered at 1350°F 0.74 1.00 Annealed at 1580°F 0.69 1.09 0.24 1.56 Annealed at 1580°F 0.30 1.65 Annealed at 1580°F 0.28 1.67 Annealed at 1580°F 0.22 Annealed at 1580°F 1.67 0.28 1.68 Annealed at 1580°F 0.14 Oil-quenched at 1650"F, tempered at 1380°F 1.85 0.10 2.80 Oil-quenched at 1650°F, tempered at 1380°F 0.19 Oij..quenched at 1650°F, tempered at 1380°F 3.88 0.08 Oil-quenched at 1650°F, tempered at 1380°F 5.54 0.59 0.88+0.20Mo Annealed at 1580°F, tempered at 1185°F

Low-alloy Ni-Cr steel. .. 7.85

0.33

....

0.53

0.80

3.38

Low-alloy Ni-Cr steel ... 7.85

0.325

....

0.55

0.71

3 ..41

1.28 1. 28 0.325 0.51 0.34

.... .... .... .... ....

0.24 0.24 0.55 0.22 0.55

1.80 1.80 0.17 1.72 0.78

3.46 3.46 3.47 3.52 3.53 +

%C

%Cr

%Ni %Mo %Zr

Carhon steel. .......... Carhon steel. .......... Carhon steel. .......... Carbon steel. .......... Low-Or steel. .......... Low-Cr steel ........... Low-Or steel ........... Low-Cr steel. .......... Low-Cr steel. .......... Low-Cr steel. .......... Low-Cr steel. .......... Low-Or steel. .......... Low-Cr steel. .......... Low-Cr steel. .. , ....... Low-Cr steel. .......... Low-Cr steel. .. , .......

7.871 7.859 7.844 7.830 7.84 7.84 7.83 7.80 7.82 7.82 7.81 7.84 7.82 7.81 7.79 7.845

. ...

%Ni

Low-alloy Ni-Or steel ... Low-alloy Ni-Cr steel. .. Low-alloy Ni-Or steel. .. Low-alloy Ni-Cr steel. .. Low-alloy Ni-Cr steel ...

7.92 7.82 7.855 7.835 7.86

p,

g/cm'

- - ---

Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainle.. and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels . ..

Annealed at 1580°F, tempered at 1185°F Annealed at 1580°F, tempered at 1185°F Brine quenched at 2190°F Annealed at 1435°F Annealed at 1580°F Annealed at 1435°F 0.39 Mo Annealed at 1580°F, tempered at 1185°F

%Ti %Cu %Mn

-- ---- ---- - -

7.93

0.10

18

9

7.93

....

18

9

7.98

....

23

13

7.98

....

25

20.5

7.98

....

17

12

8.02

....

18

10.5

7.75

....

12.5

7.73

....

13

0.5

2.25

. ..... 0.5

• .. Metals Handhook," 48th ed., American Society for Metals.

Condition

2-25

DENSITY OF SOLIDS TABLE

2b-3.

DENSITY OF STEELS

(Continued)

Composition Type of steel

p,

Condition

g/cm' %C

% Cr

% Ni % Mo % Zr

% Ti % Cn % Mn

------- - - - - ----- - - - - - - -- --·1-------Wronght stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ... Wrought stainless and heat-resisting steels ...

7.70

I

13

7.70

16

7.68

17

7.60

25

7.77

17.88

8.26

7.76

17.55

10.48

7.91

18.40

4.07

7.90

18.50

4.06

6.79

7.78

18.04

2.06

7.90

7.77

17.70

p,

glcm'

......

%W

0.6

0.78

0.68

%Cr

%v

%Mo

%Co

5.33

9.40

%C

Condition

---------1-- - - ---- - - - - ---1-------Toolsteel. ................... Tool steel. ................... Tool steel. ................... Tool steel. ................... Tool steel .................... Tool steel. . ................. Tool steel. ................... Tool steel. ................... Tool steel. ...................

8.67 8.67 7.925 7.93 7.76 8.89 8.68 8.16 7.88

18 18 1.64 5.20 20 18 6 1.5

4 4 3.68 4.60 4.39 4 4

2 1.00 4.00 4.10 2 1 2 1

8.24 4.11 7.75 12 5 5 8

0.80 Quenched at 2200'F 1. 32 Hardened 1. 20 Hardened Annealed Annealed Annealed

---- ---- --%Ni

%Al

20 17 25 28 14 18

12 10 12 12 8 6

%Co

%Cu

---- - --- --Permanent-magnet alloys ....... Permanent-magnet alloys .. _- •.. Permanent-magnet alloys ....... Permanent-magnet alloys ....... Permanent-magnet alloys ....... Permanent-magnet alloys_ ......

6.892 7.086 6.892 7.003 7.307 7.197

Alnico 12.5 5 24 35

- -- - --%Ni

%C

Cast Alnico

...........

8% Ti

%Mn

--- - - --Miscellaneous ferrous alloys ..... Miscellaneous ferrous alloys ..... Miscellaneous ferrous alloys ..... Miscellaneous ferrous alloys ..... Miscellaneous ferrous alloys .....

8.16 8.00 8.3 8.25 7.87

28.37 36 45 50 1.2

13

Quenched at 1740°F lnvar Radio metal ffipernik Austenitic manganese steel. Air-cooled at 1920°F

2-26

MECHANICS TABLE

2b-4.

DENSITY OF ALUMINUM ALLOYS*

(At 20°0) P.

Material

g/cm'

% Al

% Mn

% Cu % Pb

% Bi

% Mg

% Si

% Ni

--------------.---------~

Wrought alloys; Pure aluminum .. (Commercially pure AI) 2S ... 3S ............. l1S ....... ..... R-317 ....... ... 14S ......... ... R-30I (clad) ..... 17S ........ .. 18S ........ '" . 24S ............ 25S ........ .... 32S ........... AS1S ........... 52S ............ 53S ............ 568 ............. 61S ............ 758 ............ R-303 ..........

Material

*

II

% Zn

2.6989 99.996 2.71 2.73 2.82 2.81 2.80 2.78 2.19 2.80 2.77 2.79 2.69 2.69 2.68 2.69 2.64 2.70 2.80 2.82

P.

g/cm'

99.0+ 1.2 98.8 93.5 93.8 0.6 93.6 0.8 93.3 0.8 0.5 95.0 93.5 93.4 0.6 0.8 93.9 84.7 98.15 97.25 97.75 94.6 0.1 97.9 0.20 90.0 89.9

% Al

5.5 4.0 4.4 4.5 4.0 4.0 4.5 4.5 0.9

0.5 0.5

2.66 2.69 2.78 2.79 2.76 2.79 2.91 2.91 2.95 2.68 2.7 2.81 2.81 2.78 2.65 2.65 2.53 2.58 2.77 2.70 2.68 2.73 2.68 2.76 2.89 2.81

88 95 91 93 92 90 89.3 89.5 89.8 83.5 85.1 92.5 95.5 93.0 96.2 94.4 92.0 90.0 90.5 93.2 92.7 89.9 90.0 88.0 91.5 93.2

0.2 1.2 0.7 1.5

4 4 3 4.5 7 1 10 0.8 1.5 4 4.5 4.5

3.8 3.8 8 10

0.3

0.6 0.4 0.4 0.5 0.5 1.5

0.25 1.5 1.2

%Mn % Mg % Cu

0.7

0.5 0.5

0.5 0.3 0.3 0.5

% Zn - -

0.8 12.5 1.0

% Cr

0.9 0.25 0.25 0.25 0.10 0.25 0.30

0.7 0.6

% Si

5.5 6.4

%Ni % Bi % 8n % Ti

-- ---- -- ---12 5 5 3 5.5 2 3.5

1.7

12 12

2.5 2

2.5 1.8

3.5 1.3

6 5 7 8 9.5 8.5

1.5 3.5 1.0

0.6

0.8 1.0

1.0 0.6 2.5 1.3 5.2 1.0 2 5 2.5

-- ---- --

Casting alloys; 13 alloy ..... 43 alloy .... 85 alloy ...... 108 alloy .... Allcast ... A10S alloy .... 113 alloy ... C113 alloy .... 122 alloy. A132 alloy .... Red X-13 ..... 142 alloy ... 195 alloy. B195 alloy .... 214 alloy. A214 alloy. 218 alloy .. ' 220 alloy. 319 alloy ... 355 alloy .. 356 alloy .... Red X-S .... 360 alloy. 380 alloy ... 750 alloy. 40E alloy .....

% Cr

----

1.0 5

0.5

Metals Handbook," 48th ed., American Society for Metals.

6.5

0.2

DENSITY OF SOLIDS TABLE

Material Pure cobalt ........... 61 alloy (cast) ......... Vitallium ............. X-40 alloy ............ 422-19 alloy ........... 8-816 alloy ........... 6059 .................

2b-5.

2-27

DENSITY OF COBALT ALLOYS*

p,

g/cm! % Co %W % Ni % Cr % Mo % Cb % Fe - - - - - - - - - - -- - - - 100 8.9 70.0 5.0 2.0 23.0 8.54 65.0 2.0 27.0 8.30 ... 6.0 8.61 60.0 7.0 10.0 23.0 16.0 23.0 8.31 55.0 ... 6.0 50.0 4.0 20.0 19.0 ... 4.0 8.59 3.0 8.21 39.0 ... 32.0 23.0 6.0

* "Metals Handbook," 48th ed., American Society for Metals.

,

TABJ"E

Material

p,

g/cm 3

:lb-6.

DENSITY OF COPPER ALLOYS*

% Ou %0

%P

% Zn % Pb % 8n % F

---- - --- --- --- --

w rought

alloys: Pure copper .................... 8.96 100 Electrolytic tough-pitch copper ... 8.89-8.94 99.92 Deoxidized copper .............. 8.94 99.94 Gilding metal. ................. 8.86 95.0 Commercial bronze ............. 8.80 90.0 Red brass ...................... 8.75 35.0 Low brass ...................... 8.67 80.0 Cartridge brass ................. 8.53 70.0 YeIlow brass ................... 8.47 65.0 Muntz metal. .................. 8.39 60.0 Leaded commercial bronze ....... 8.83 89.0 Low-leaded brass ............... 8.47 64.5 Low-leaded brass (tube) ......... 8.50 67.0 Medium-leaded brass ............ 8.47 64.5 High-leaded brass ............... 8.47 625 Extra-high-leaded brass ......... 8.50 62.5 Free-cutting brass .............. 8.50 6l.5 Leaded muntz metal. ........... 8.41 60.0 Free-cutting muntz metal ....... 8.41 60.5 Forging brass .................. 8.44 60.0 Architectural bronze ............ 8.47 57.0 Admiralty metal. ............... 8.53 71.0 Naval brass ..... , .............. 8.41 60.0 Leaded naval brass ............. 8.44 60.0 Manganese bronze .............. 8.53 58.5

0.04 .... .. .

.... ....

.... .. .

.... .... ... .... ., .. ....

.... ., .. ... . ....

.... .... ....

... . . ... . ...

... .

0.02 .... . ... . ... . ... .... . ... ..' . . ... . ... . ... . ... . ... . ... .... . ... . ... . ... . ... . ... ... . ....

... .

5.0 10.0 15.0 20.0 30.0 35.0 40.0 9.25 1.75 35.0 0.5 32.5 0.5 34.5 1.0 35.75 1.75 35.0 2.5 35.5 3.0 39.5 0.5 38.4 1.1 38.0 2.0 40.0 3.0 28.0 ...... 39.25 ...... 37.5 1. 75 39.0 •

G

••

~

..

l.00 0.75 0.75 1.00 1.4

8.33 Aluminum brass ................ Aluminum brass ................ 8.33 8.86 Phosphor bronze ................ 8.80 Phosphor bronze 8 % grade C .... 8.78 Phosphor bronze 10 % grade D ... 8.89 Phosphor bronze 1.25% grade E .. 8.94 Cupronickel, 30% .............. Nickel silver, 18% alloy A ....... 8.73 8.70 :Ni-Ag, 18%, alloy B ............ 8.53 Silicon bronze, type A ........... 8.75 Silicon bronze, type B ........... 8.17 5 % aluminum bronze ........... ? 8 % aluminum bronze ........... 10 % aluminum bronze .......... 7.58 7.58 Aluminum bronze .............. 8.9 Constantan .................... Beryllium copper ............... 8.23 ± 0.02 Caating alloys (room temp.): 8.7 Leaded tin bronze .............. 8.80 Leaded tin bearing bronze ....... 8.87 High-leaded tin bronze .......... 8.93 High-leaded tin bronze .......... 8.80 High-leaded tin bronze .......... 9.25 High-leaded tin bronze .......... 9.30 High-leaded tin bronze .......... 8.80 85-5-5-5 ....................... 8.6 Leaded red brass ............... 8.70 Leaded semired brass ............ 8.6 Leaded semired brass ............ 8.50 Leaded yellow brass ............. 8.4 Leaded yellow brass .............

76.0 95.0 92.0 90.0 98.75 70.0 65.0 55.0 97.0 98.5 95.0 92.0 90.0 82.5 55.0 97.65

22.0

88.0 87.0 85.0 83.0 80.0 78.0 70.0 85.0 83.0 81.0 76.0 71.0 66.0

4.5 4.0 1.0 3.0

'" "Metals Handbook," 48th ed., American Society for Metals.

. . . . . . 1 ....

5.0 8.0 10.0 1.25 17.0 27.0

2.50

5.0 7.0 9.0 15.0 25.0 30.0

1.5 1.0 9.0 7.0 10.0 15.0 25.0 5.0 6.0 7.0 6.0 3.0 3.0

6.0 8.0 5.0 7.0 10.0 7.0 5.0 5.0 4.0 3.0 3.0 1.0 1.0

TABLE

Material Leaded yellow brass. . .......... High-strength yellow brass ...... High-strength yellow brass ...... Leaded manganese brass ..... Nickel silver ................... Nickel silver ................... Nickel silver ................... Leaded nickel brass ............. Aluminum bronze ............... Aluminum bronze ............... Aluminum bronze .............. Aluminum bronze ...............

p,

g/cm 3

8.40 7.9 8.2 8.2 8.8-8.9 8.85 8.95 8.95 ?

7.4 7.5 ?

2b-6.

DENSI'l'Y OF COPPER ALLOYS"

% Cu I % 0

I

% P

I%

(Continu.ed)

Zn I % Pb I % Sn

I%

Fe

,---,---,---,---,---,---,--60.0 38.0 I 1.0 1.0 26.0 62.0 3.0 58.0 39.25 1.25 59.0 37.0 0.75 1.25 66.0 2.0 1.5 5.0 64.0 8.0 4.0 4.0 57.0 20.0 9.0 2.0 60.0 16.0 5.0 3.0 89.0 1.0 87.5 3.5 4.0 86.0 79.0 5.0

* "Metals Handbook," 48th ed .• American Society for Metals.

2-31

DENSITY OF SOLIDS

TABLE 2b-7. DENSITY OF LEAD ALLOYS* p,

Material Pure lead .............. Chemically pure lead .... Cable-sheath alloy ...... 1 % antimonial lead ..... Hard lead .............. Hard lead .............. 8 % antimonial lead ..... Grid metaL ............ ASTM-12 bearing metal. ASTM-11 bearing metal. Lead-base babbitt ....... G lead-base babbitt ..... S lead-base babbitt ..... ASTM-lO bearing metal. Lead-base babbitt ...... Lead-base babbitt ...... ASTM-6 bearing metal .. Tin-lead solder ......... Tin-lead solder ......... 50-50 half and half ......

g/cm 3 11.34 11.34 11.34 11.27 11.04 10.88 10.74 10.66 10.67 10.28 iO.24 10.1 10.1 10.07 10.04 9.73 9.33 11.0 10.2 8.89

, %Pb 99.73 99.8 99.0 96.0 94.0 92.0 91.0 90.0 85.0 85.0 83.0 83.0 83.0 80.0 75.0 63.5 95.0 80.0 50.0

% Ca

% Sb

% Sn

% As % Co

---- - - - - - - --- - 0.028

..... ..... .....

1.0 4.0 6.0 8.0 9.0 10.0 15.0 10.0 12.75 15.0 15.0 15.0 15.0 15.0

.....

..... .....

..... .....

..... ..... .....

..... ..... .... . .... . .....

.....

..... . ....

5.0 0.75 1.0 2.0 5.0 10.0 20.0 5.0 20.0 50.0

3.0 1.0

. ..

1.5

* "Metals Handbook," 48th ed., American Society for Metal•. TABLE 2b-8. DENSITY OF MAGNESIUM ALLOYS* Material

p,

% Mg %Al %Mn %Zn % Sn Remarks g/cm 3 --- --- --- -----Magnesium ... 1.74 99.8 AW alloy ..... 1.81 89,9 10.0 0.1 .. . ... Wrought, sand cast, and permanent-mold cast ... Die cast AZ91 alloy .... 1.81 ... . 9.0 0.2 0.7 2.0 AZ92 alloy .. , . 1.82 ... . 9.0 0.1 ... Sand cast and permanent-mold cast 8.0 0.2 A8 alloy ...... 1.80 ... . .. . ... Sand cast . .. Wrought AZ61X alloy .. 1.80 .... 6.0 0.2 1.0 4.0 0.2 ... Sand cast AM244 alloy .. 1.76 ... . .. . 1.25 1 ... Die cast AM11 alloy ... 1.70 ... . .. . 8.5 0.15 0.5 AZ80X alloy .. 1.80 ... . " . Wrought 6.0 0.2 3.0 ... Sand cast AZ63 alloy .... 1.84 ... . 5.0 0.25 1.0 ... Wrought AZ51X alloy .. 1.79 ... . 3.0 0.3 AZ31X alloy .. 1.78 ' " . 1.0 ... Wrought MI .......... 1.76 ... . . .... 1.5 .. . . .. Wrought 3.0 TA54 ........ 1.84 .... 0.5 . .. 5.0 Wrought Mg-Al alloy ... 1.75 98.0 2.0 Mg-Al alloy ... 1. 77 96.0 4.0 Mg-Al alloy ... 1.78 94.0 6.0 Mg-Al alloy ... 1.80 92.0 8.0 Mg-Al alloy ... 1.81 90.0 10.0 I Mg-Al alloy ... 1.82 88.0 I 12.0 I

I

*" Metals Handbook," 48th ed., American Society for Metals.

MECHANICS TABLE

Material

------

2b-9.

DENSITY OF NICKEL ALLOYS'"

p, 3 % Ni % Co % Si % Mn g ( cm -- ---- ----

Nickel. .................. . A nickeL ............. . Cast nickel. .............. . D nickel. ................ . Z nickel.. Monel Cast moneL .. Kmonel. .... S monel.. ............... . HasteUoy A.............. . HasteUoy B .............. . HasteUoy C .............. . HasteUoy D .............. . IlliumG ................. . Inconel .................. . Cast Inconel. ............ . Chromel A ............... . Nichrome ................ . Chromax ............... . Constantan (wrought) ..... . Ni-Fe aUoys .... . Ni-Fe aUoys .............. . Ni-Fe aUoys ............ .. Ni-Fe aUoys ............. . PermaUoy .............. .. Numetal.. .............. .

8.902 8.885 8.34 8.78 8.75 8.84 8.63 8.47 8.36 8.80 9.24 8.94 7.8 8.58 8.51 8.3 8.4 8.25 7.95 8.9 8.8 8.6 8.5 8.35 8.6 8.6

99.95 99.4 97.0 95.2 94 67 63 66 63 60 65 58 85 58 80 77.5 80 60 35 45 90 80 70 60 78 76

1.5

%C

0.5 4.5

0.5

1.0

0.15 0.2

% Al % Cu % Fe

% Mo

% Cr

%W

4.5 1:6

30 32 29 30

1.4

2 20

20 30 17

15

8-11 fl

0.2

24 50

22 14 13.5 20 16 15

55 10 20 30 40 22 16

... Metals Handbook," 48th ed., American Society for Metals. TABLE

2b-10.

DENSITY OF ZINC ALLOYS*

p,

Material

glcm'

% Zn % Al % Cu % Mg % Pb % Cd --- ---

Zinc ........................ Zamak (2) ................ " Zamak (3) .................. Zamak (5) .................. SAE 63, T-ll (cast) .......... Commercial rolled zinc ....... Commercial rolled zinc ....... Commercial rolled zinc ....... Zilloy 40 (rolled) ............. Zilloy 15 (rolled) .............

* UMetals Handbook,"

7.133 6.7 6.6 6.7 6.9 7.14 7.14 7.14 7.18 7.18

100 92 95 94 86 99 99 99 98 98

4 4 4 4

·. ·. ·. ·. ·.

48th ed., American Society for Metals.

--- --- --- --3

·. 1 10

·. ·. ·. 1 1

0.03 0.04 0.04

.... .... . ... . ... 0.01

0.08 0.06 0.3 0.08 0.1

0.06 0.3

2-33

DENSITY· OF SOLIDS TABLE

2b-11.

DENSITY OF WOODS (OVEN-DRY)*

Common name Applewood or wild apple. . . . . . . . .. Aih, black... . . . . . . . . . . . . . . . . . . .. Ash, blue. . . . . . . . . . . . . . . . . . . . . . .. Ash, green. . . . . . . . . . . . . . . . . . . . . .. Ash, white. . . . . . . . . . . . . . . . . . . . . .. Aspen. . . . . . . . . . . . . . . . . . . . . . . . . .. Aspen, large-toothed. . . . . . . . . . . . .. Balsa, tropical American. . . . . . . . .. Basswood. . . . . . . . . . . . . . . . . . .. .. Beech ........................... Beech, blue. . . . . . . . . . . . . . . . . . . . .. Birch, gray. . . . . . . . . . . . . . . . . . . . .. Birch, paper. . . . . . . . . . . . . . . . . . . .. Birch, sweet. . . . . . . . . . . . . . . . . . . .. Birch, yellow. . . . . . . . . . . . . . . . . . .. Buckeye, yellow ................. , Butternut ....................... Cedar, eastern red ................ Cedar, northern white. . . . . . . . . . .. Cedar, southern white. . . . . . . . . . .. Cedar, tropical American .......... Cedar, western red ............... Cherry, black .................... Cherry, wild red. . . . . . . . . . . . . . . .. Chestnut ........................ Corkwood. . . . . . . . . . . . . . . . . . . . . .. Cottonwood, eastern. . . . . . . . . . . . .. Cypress, southern. . . . . . . . . . . . . . .. Dogwood (flowering). . . . . . . . . . . . .. Douglas fir (coast type) ........... Douglas fir (mountain type) ....... Ebony, Andaman marblewood (India) ........................ Ebony, Ebene marbre (Mauritius, East Africa). . . . . . . . . . . . . . . . . .. Elm, American ......... , . . . . . . . .. Elm, rock. . . . . . . . . . . . . . . . . . . . . .. Elm, slippery. . . . . . . . . . . . . . . . . . .. Eucalyptus, Karri (west Australia). Eucalyptm, mahogany (New South Wales). . . . . . . . . . . . . . . . . . . . . . .. Eucalyptus, west Australian mahogany. . . . . . . . . . . . . . . . . . . . . . .. Fir, balsam ...................... Fir, silver ....... '................ Greenheart (British Guiana). . . . . .. See page 2-35 for footnotes.

Botanical name

p,

glcm'

Pyrus malus Fraxinus nigra Fraxinus quadrangulata Fraxinus pennsylvanica lanceolata Fraxinus americana Populus tremuloides Populus grandidentata Ochroma Tilia glabra or Tilia americanus Fagus grandifolia or Fagus americana Carpinus caroliniana Betula populifolia Betula papyrifera Betula lenta Betula lutea Aesculus octandra Juglans cinera Juniperus virginiana ThuJa occidentalis Chamaecyparis thyoides Cedrela odorata ThuJa plicata Prunus serotine Prunus pennsylvanica Castanea dentata Leitneria floridana Populus deltoides Taxodium distichum Comus florida Pseudotsuga taxifolia Pseudotsuga taxifolia

0.717 0.552 0.600 0.714 0.668 0.383 0.404 0.492 0.315 0.352 0.37-0.701' 0.344 0.534 0.425 0.454 0.207 0.433 0.482 0.796 0.512 0.446

Diospyros Kurzii

0.978t

Diospyros melanida Ulmus americana Ulmus racemosa or Ulmus thomasi Ulmus fulva or Ulmus pubescens Eucalyptus diversicolor

0.768t 0.554 0.658 0.568 0.829t

Eucalyptus hemilampra

1. 058t

Eucalyptus marginata Abies balsamea Abies amabilis N ectandra rodioci

0.787t 0.414 0.415 1. 06-1. 2:3t

0.745 0.526 0.603 0.610 0.638 0.401 0.412 0.12-0.20t 0.398 0.655

2-34

MECHANICS TABLE 2b~U. DENSITY OF WOODS (OVEN-DRY)

Common name Gum, black ........................ Gum, blue ....................... Gum, red ........................ Gum, tupelo ..................... Hemlock, eastern ................. Hemlock, mountain ............... Hemlock, western ............. , ... Hickory, bigleaf shagbark ......... Hickory, mockernut .............. Hickory, pignut .................. Hickory, shagbark ................ Hornbeam .................... ·· . Ironwood, black .................. Jacaranda, Brazilian rosewood ..... Larch, western ................... Locust, black or yellow ........... Locust, honey ............ '........ ~agnolia, cucumber .............. ~ahogany (West Africa) .......... ~ahogany (East India) ........... Mahogany (East India) ........... Maple, black .................... Maple, red ...................... Maple, silver .................... Maple, sugar .................... Oak, black ...................... Oak, bur ........................ Oak, canyon live ................. Oak, chestnut .................... Oak, laurel. ..................... Oak, live ........................ Oak, pin ........................ Oak, post ....................... Oak, red ........................ Oak, scarlet ..................... Oak, swamp chestnut ............. Oak, swamp white ................ Oak, white ...................... Persimmon ...................... Pine, eastern white ............... Pine, jack ....................... Pine, Pine, Pine, Pine,

loblolly .................... longleaf .................... pitch ...•.................. red .....•..................

See page 2-35 for footnotes.

*

(Oontinued)

Botanical name Nyssa sylvatica Eucalyptu8 globulu8 Liquidambar styracijlua N ussa aquatica Tsuga canadensis T suga martensiana Tsuga heterophylla Hicoria laciniosa H icoria alba Hicoria glabra H icoria ovata Ostryra virginiana Rhamnidium jerreum Dalbergia nigra Larix occidentalis Robinia pseudacacia Gleditsia triacanthos Magnolia acuminata Khaya ivorensis Swietenia macrophylla Swietenia mahogani Acer nigrum Acer rubrum Acer saccharinum Acer saccharum Quercus velutina Quercus macrocarpa Quercus chrysolepsis Quercus montana Quercus laurijolia Quercus virginiana Quercus palustris Quercus siellata or Quercus minor Quercus borealis Quercus coccinea Quercus prinus Quercus bicolor or Quercus platanoides Quercus alba Diospyro8 virginiana Pinus strobus Pinus banksiana or Pinus divaricata Pinus taeda Pinus palustri8 Pinus rigida Pinus resinosa

p,

g/cm 3

0.552 0.796 0.530 0.524 0.431 0.480 0.43;2 0.809 0.820 0.820 0.836 0.762 1.077 0.85t 0.587 0.708 0.666 0.516 0.668t 0.54t 0.54t 0.620 0.546 0.506 0.676 0.669 0.671 0.838 0.674 0.703 0.977 0.677 0.738 0.657 0.709 0.756 0.792 0.710 0.776 0.373 0.461 0.593 0.638 0.542 0.507

~

2-35

DENSITY OF SOLIDS TABLE

2b-ll.

DENSITY OF WOODS (OVEN-DRY)*

Common name

(Continued)

Botanical name

Pine, shortleaf. . . . . . . . . . . . . . . . . .. Pinus echinata Poplar, balsam ................. " Populus balsamifera or Populus candicans Poplar, yellow ................... Liriodendron tulipifera Redwood. . . . . . . . . . . . . . . . . . . . . . .. Sequoia sempervivens Sassafras. . . . . . . . . . . . . . . . . . . . . . .. Sassafras variafolium Satinwood (Ceylon). . . . . . . . . . . . .. Chloroxylon swietenia Sourwood. . . . . . . . . . . . . . . . . . . . . .. Qxydendrum arboreum Spruce, black. . . . . . . . . . . . . . . . . . .. Picea mariana Spruce, red. . . . . . . . . . . . . . . . . . . . .. Picea T1lbra or Picea rub ens Spruce, white. . . . . . . . . . . . . . . . . . .. Picea glauca Sycamore. . . . . . . . . . . . . . . . . . . . . .. Platanus occidentalis Tamarack. . . . . . . . . . . . . . . . . . . . . .. Larix laricina or Larix americana Teak (India) ..................... Tectona grandis Walnut, black ................... Juglans nigra Willow, black. . . . . . . . . . . . . . . . . . .. Salix nigra

* t

II Handbook of Chemistry and Physics, H 30th ed. Air-dry.

p,

glcm'

0.584 0.331 0.427 0.436 0.473 1. 031 t 0.593 0.428 0.413 0.431 0.539 0.558 0.582t 0.562 0.408

2-36

MECHANICS TABLE 2b-12. DENSITY

0'

OF

PLASTICS* p,

Resin group and subgroup'

Trade names

g/cm l

Lower limit

Upper limit

--- --Acrylate and ~ethacrylate .......... Lucite, Crystalite, Plexiglas Casein ....... ,.... ' ................. Ameroid Cellulose acetate (sheet) ..... " ...... Bakelite, Lumarith, Plastecele, Protectoid "'f Cellulose acetate (molded) .......... Fibestos, Hercules, Nixonite, Tenite '" Cellulose ac~tobutyrate ............. Tenite II Cellulose nitrate ................... Celluloid, Nitron, Nixonoid, Pyralin Ethyl cellulose ..................... Ditzler, Ethocel, Ethofoil, Lumarith, Nixon, Hercules Phenol-formaldehyde compounds: Wood-flour-filled (molded) ........ Bakelite, Durez, Durite, Micarta, Catalin, Haveg, Indur, Makalot, Resinox, Textolite, Formica Mineral-filled (molded) ........... Bakelite, Durez, Durite, Micarta, Catalin, Haveg, Indur, Makalot, Resinox, Textolite, Formica Macerated-fabric-filled (molded) ... Bakelite, Durez, Durite, Micarta, Catalin, Haveg, Indur, Makalot, Resinox, Textolite, Formica Paper-base (laminated) ........... Bakelite, Durez, Durite, Micarta, Catalin, Haveg, Indur, Makalot, Resinox, Textolite, Formica Fabric base (laminated) .......... Bakelite, Durez, Durite, Micarta, Catalin, Haveg, Indur, Makalot, Resinox, Textolite, Formica Cast (unfilled) ................... Bakelite, Catalin, Gemstone, Marblette, Opalon, Prystal Phenolic furfural (filled) ............ Durite Polyvinyl acetals (unfilled) .......... Alvar, Formvar, Saflex, Butacite, Vinylite X, etc. Polyvinyl acetate .................. Gelva, Vinylite A, etc. Copolyvinyl chloride acetate ........ Vinylite V, etc. Polyvinyl chloride (and copolymer) plasticized ...................... Koroseal, Vinylite Polystyrene ....................... Bakelite, Loalin, Lustron, Styron

1.16 1.34 1.27

1.20 1.35 1.60

1.27

1.6()

1.14 1.35

1.23 1.60

1.05

1.25

1.25

1.52

1.59

2.09

1.36

1.47

1.30

1.40

1.30

1.40

1.20

1.10

1.3 1.05

2.0 1.23

1.19 1.34

(?) 1.37

1.2 1.054

1.7 1.070

)

..,

,

* .. Handbook of Chemistry and Physics," 30th ed., p. 1282.

2-37

DENSITY OF SOLIDS

T ABLE 2b-12. DENSITY

OF

PLASTICS (Continued) p,

Resin group and subgroup

Trade names

g/ cm 3

Lower limit

Upper li mit

--- Modified isomerized rubber . ... ... . . Chlorinated rubber .. ... . ...... . .. .. Urea formaldehyde. .. . ..... . . . . . . . . Melamine formaldehyde filled .. .. . .. Vinylidene chloride ..... .... . ... ....

Plioform, P liolite T orneseit, Parlon Bakelite, Beetle, Plascon Catalin, Melmac, P laskon Saran, Velon

TABLE 2b-13. DENSITY ______R_u_b_b_e_r_;_ra_'_v_p_O_I_y_m_e_r_____

OF

I__T_ra_d_e_ N_a_I_n_e_ _

30th ed. , p. 1282.

-

-

( 7)

(7) 1. 55 1. 86 1. 75

RUBBERS*

Natural rubber. . ........... . H evea Butadienestyrene copolyrr.cr . . ..... . .. . . Butadieneacrylonitrile copolymer . . Polychloroprene (neoprene) ... . . . Isobutylenediolefin copolymer (butyl). . . . . . . . . . . . . . . . . . . . . .. . . . .. .. . .. . . . . . . . . . . . . . Alkylene polysulfide . . ..... .

* "Handbook of Chemistry and Physics, "

1.06 1. 64 1.45 1.49 1. 68

___A_t_2_5_ 0C _ I

0 .92 0 . 94 1. 00 1.25 0 .9 1 1.35

__

2c. Centers of Mass and Moments of Inertia R. BRUCE LINDSAY

Brown University

1. 2. 3. 4.

5.

6.

7.

8.

9.

10.

11.

TABLE 2c-1. CENTERS OF MASS * Body Center of Mass Uniform circular wire of radius R, On axis of symmetry distant (R sin e)/e from center sub tending angle 2e at center At intersection of the medians Uniform triangular sheet At intersection of the diagonals Uniform rectangular sheet Uniform quadrilateral sheet From each vertex layoff segments equal to t the length of the corresponding sides meeting at this vertex. Draw extended lines through the ends of the segments associated with each vertex, respectively. These intersect to form a parallelogram. The intersection of the diagonals of this parallelogram is the center of mass of the quadrilateral Uniform circular sector sheet of radius On axis of symmetry distant (2R sin e)/ 38 from center R sub tending angle 2e at center of circular arc Uniform circular segment sheet of On axis of symmetry distant 1'/12A from center, where A = area of segment radius R, subtending angle 2e at R2(2e - sin 2e) center of circular arc and length of 2 chord equal to I = 2R sin e Uniform semielliptical sheet, major On axis of symmetry distant 4a/3rr from center of equivalent ellipse if the semiand minor axes of equivalent ellipse ellipse is bounded by minor axis. The equal to 2a and 2b, respectively distance is 4b /3rr if the semiellipse is bounded by the major axis Uniform quarter-elliptical sheet, major At point 4b/3rr above major axis and and minor axes of equivalent ellipse 4a/3rr above minor axis equal to 2a and 2b, respectively Uniform parabolic sheet segment. On axis of symmetry distant 3h/5 from Chord = 2l perpendicular to axis of vertex symmetry distant h from vertex Right rectangular pyramid (rectan- On axis of symmetry distant h/4 from gular base with sides a and b and with base height h) Pyramid (general) On line joining apex with center of symmetry of base at distance three-quarters of its length from apex

* For definition see Sec. 2a-4.

All bodies cited are homogeneous rigid bodies.

2-38

CENTERS OF MASS AND MOMENTS OF INERTIA

2-39

TABLE 2c-1. CENTERS OF MASS (Continued) Body Center of Mass 12. Frustum of pyramid with area of On line joining apex of corresponding larger base S and smaller base s, and pyramid with center of symmetry of altitude h larger base and distant h(S + 2 VSs + 3s)

13. Right circular cone (height h) 14. Frustum of right circular cone (altitude h, radii of larger and smaller bases Rand r, respectively) 15. Cone (general)

16. Frustum of cone with altitude hand radii of larger and smaller bases R and r, respectively 17. Spherical sector of radius R, with plane vertex angle eqllal to 21i 18. Solid hemisphere of radius R 19. Spherical segment of radius Rand maximum height from base equal to h 20. Octant of ellipsoid with semiaxes a, b, c, respectively, and center of corresponding ellipsoid at origin of system of rectangular coordinates 21. Paraboloid of revolution with altitude h and radius of circular base equal to R 22. Uniform hemispherical shell of radius R (excluding base) 'Z3. Conical shell (excluding base)

4(S + VSs +s) from the larger base On axis of symmetry distant h/4 from base On axis of symmetry distant h[(R 4[(R

+ r)2 + 2r 2] + r)2 - Rr]

from the base On line joining apex with centroid of base at distance three-quarters of its length from apex On line joining apex of corresponding cone with centroid of larger base and distant h[(R 4[(R

+ r)2 + 2r 2] + r)2 - Rr]

from the larger base On axis of symmetry distant 3R ""8 (1 + cos Ii) from the vertex On axis of symmetry distant 3R/8 from center of corresponding sphere . f d' t t h(4R - h) On axIS 0 symmetry IS an 4(3R _ h) above the base of the segment Point with coordinates _ 3c _ 3a _ 3b z =x=g y=S 8 On axis of symmetry distant h/3 from the base On axis of symmetry distant R/2 from center of corresponding sphere On line joining the apex with the center of symmetry of the base at distance twothirds its length from the apex

2-40

MECHANICS TABLE

Body

2c-2.

MOMENTS OF INERTIA*

Axis

Moment of inertia

Uniform rectangular sheet of Through the center parallel to b sides a and b Uniform rectangular sheet of Through the center perpendicular to the sides a and b sheet Uniform circular sheet of Normal to the plate through the center radius T Uniform circular sheet of Along any diameter radius T Uniform circular ring, radii Through center normal rl and r, to plane of ring Uniform circular ring, radii A diameter TI and T, Uniform thin spherical shell, A diameter mean radius T Uniform cylindrical shell, Longitudinal axis radius r, length 1 Right circular cylinder of Longitudinal axis radius r, length 1 Right circular cone, altitude Axis of the figure h, radius of base r Spheroid of revolution, equa- Polar axis torial radius r Ellipsoid, axes 2a, 2b, 2c . ... Axis 2a

a' m 12 a' + b' m~ r'

m

"2

r' m4 rl' + r2' m~-2~m

rl'

+ T2' 4

2r'

m""3 mr'

r'

m'2 m

3 , 10 r 2r'

mS m

(b'

+ c') 5

Uniform thin rod ......... . Normal to the length, l' m3 at one end Uniform thin rod ......... . Normal to the length, l' m 12 at the center Rectangular prism, dimen- Axis 2a (b' + c') m 3 sions 2a, 2b, 2c Sphere, radius r . ......... . A diameter m ~r' 5 Rectangular parallelepiped, Through center pera' + b' edges a, b, and c pendicular to face ab m~ (parallel to edge c) Right circular cylinder of Through center perradius r, length 1 pendicular to the axis of the figure Spherical shell, external ra- A diameter dius rl, internal radius r, Hollow circular cylinder, Longitudinal axis length l, external radius TI, internal radius T, Hollow circular cylinder, Transverse diameter rl' + r,' m ( 4 length l, radii rl and T2

* For definitions see Sec. 2a-5; m

= mass of body.

+ ~)

AI! bodies are homogeneous.

12

2-41

CENTERS OF MASS AND MOMENTS OF INERTIA TABLE

2c-2.

Body Hollow circular cylinder, length l, very thin, mean radius r Right elliptical cylinder, length 2a, transverse axes 2b,2c Right elliptical cylinder, length 2a, transverse axes 2b,2c Frustum of right circular cone with radii of larger and smaller bases, equal to Rand r, respectively Right circular cone, radius of base r, altitude h Solid hemisphere of radius r Spherical sector of radius r, with plane angle at vertex = 28 Spherical segment of radius r and maximum height h Torus or anchor ring mean radius R, radius of circular cross section r Torus mean radius R, radius of circular cross section r

MOMENTS OF INERTIA

Axis

(Continued)

Moment of inertia

Transverse diameter Longitudinal axis 2a (b 2 + c2 ) through center of m 4 mass Transverse axis 2b through center of mass Axis of symmetry 3m(R6 - r 6) lO(RS - r S)

Perpendicular to axis 3m(2+!!!.) 4 of symmetry, through 20 r center of mass 2mr2 Axis of symmetry -5Axis of symmetry through vertex

mr2(1 - cos 8)(2 5

+ cos 9)

Axis of symmetry per( 2 _ 3rh + 3h 2 ) ~ pendicular to base m r 4 20 3r - h Axis of symmetry per- m(4R 2 + 3r2) pendicular to plane of, 4 ring Axis of symmetry in m(4R2 + 5r l ) plane of ring 8

2d. Coefficients of Friction DUDLEY D. FULLER

Columbia University

Symbols

fK fR fs

P r

W

coefficient of kinetic or sliding friction coefficient of rolling friction coefficient of static friction frictional resistance to rolling radius of roller load

2d-1. Static and Sliding Friction. All surfaces encountered in experience are more or less rough in the sense that as bodies move on them they exert forces parallel to the surface and in such direction as to resist motion. Such forces aTe termed "frictional." Frictional force is proportional to the normal thrust between body and surface; however, the coefficient of proportionality, known as the coefficient of friction, can for the same body and surface vary a great deal depending on the nature of the contact and the motion. It is customary to define

f - magnitude of maximum frictional force , -

(2d-l)

magnitude of normal thrust

as the coefficient of static friction if motion is just on the point of starting. On the other hand, fK, called the coefficient of kinetic or sliding friction, is the value of the ratio in Eq. (2d-l), when motion has once been established. In generalfK < fs for the same body and surface or the same two surfaces. The friction between surfaces is dependent upon many variables. These include the nature of the materials themselves, surface finish and surface condition, atmospheric dust, humidity, oxide and other surface films, velocity of sliding, temperature, vibration, and extent of contamination. In many instances the degree of contamination is perhaps the most important single variable. For example, Table 2d-llists values for the static coefficient of friction fs for steel on steel under various test conditions. TABLE

2d-1.

COEFFICIENTS OF STATIC FRICTION FOR STEEL ON STEEL

Test condition

f.

Degassed at elevated temp. in high vacuum ............ . Weld on contact Grease-free in vacuum ............................... . 0.78 Grease-free in air .................................... . 0.39 Clean and coated with oleic acid ...................... . 0.11 Clel1n and coated with solution of stearic acid ........... . 0.Oi3 • Ref crenees follow Tab)e ~(l-4.

2-42

IRef. * 20 1 8 1 21

2-43

COEFFICIENTS OF FRICTION

The' most effective lubricants for nonfluid lubrication are generally those which react chemically with the solid surface and form an adhering film that is attached to the surface with a chemical bond .. This action depends upon the nature of the lubricant and upon the reactivity of the solid surface. Table 2d-2 indicates that a fatty acid such as those found in animal, vegetable, and marine oils reduces the coefficient TABLE

2d-2.

COEFFICIENTS OF STATIC FRICTION AT ROOM TEMFERATURE

Surfaces

NickeL ............. Chromium .......... , Platinum ............ Silver ............... Glass ............... Copper .............. Cadmium ........... Zinc ................ Magnesium .......... Iron ................ Aluminum ...........

Clean

0.7 0.4 1.2 1.4 0.9 1.4 0.5 0.6 0.6 1.0 1.4

Paraffin oil

Paraffin oil

+ 1% lauric

0.3 0.3 0.28 0.8

0.28 0.3 0.25 0.7 0.4 0.08 0.05 0.04 0.08 0.2 0.3

....

0.3 0.45 0.2 0.5 0.3 0.7

acid

Degree of . reactivity of solid Low Low Low Low Low High High High High Mild Mild

of friction markedly only if it can react effectively with the solid surface. Paraffin oil is almost completely nonreactive. The data are taken from ref. 22. It is generally recognized that coefficients of friction reduce on dry surfaces as sliding velocity increases. Dokos (ref. 4) has measured this for steel on steel. It is difficult to screen out the effect of temperature, however, which also increases with sliding velocity so that frequently, under these conditions, both variables are present. Table 2d-3 gives values which are the average of four tests at high contact pressures. TABLE

2d-3.

COEFFICIENTS OF FRICTION, STEEL ON STEEL, UNLUBRICATED

Velocity, in./sec ......... 0.0001 Coefficient of friction/x .. 0.53

I 0.48 0.001 I 0.01 0.39

0.1 0.31

1 110 1100 0.23 0.19 0.18

2-44

MECHANICS

Table 2d-4 presents typical values of the coefficients of static and sliding friction for various materials under a variety of conditions. TABLE 2d-4. COEFFICIENTS OF STATIC AND SLIDING FRICTION* Static friction

Sliding friction

Materials Dry Hard steel on hard steel. .......... 0..78(1)

Mild steel on mild steel. .......... 0..74(19)

Greasy

Dry

Greasy

o..11(l,a) 0..42(2) ....... o..23(1,b) 0. .15(1,c) ....... ....... o..11(l,d) o..o.o.75(18,p) ....... o..o.o.52(18,h) .......

•••••••

0

••••

Hard steel on graphite ............ 0..21(1) o..o.9(1,a) Hard steel on babbitt (ASTM 1) ... 0..70.(11) o..23(1,b) 0. .15(1,c) o..o.8(1,d) o..o.85(1,e) Hard steel on babbitt (ASTM 8) ... 0..42(11) o..17(1,b) o..11(l,c) o..o.9(1,d) o..o.8(1,e) Hard steel on babbitt (ASTM 10.) .. ....... o..25(1,b) o..12(1,c) o..lO(l,d) Mild steel on cadmium silver ...... ....... ........... . Mild steel on phosphor bronze ..... ....... ........... . Mild steel on copper lead .......... ....... ........... . Mild steel on cast iron ............ ....... 0. .183(15,c) Mild steel on lead ................ 1 0..95(11) lo.·5(1,f) Nickel on mild steel. ............. ....... ........... . Aluminum on mild steel. .......... 0..61 (8) ............ Magnesium on mild steel. ......... ....... ........... . Magnesium on magnesium ......... 0..6(22) o..o.8(22,y)

0..57(3)

0..33(6)

....... •

••

00

••

o..o.29(5,h) o..o.81(5,c) o..o.80(5,i) o..o.58(5,j) o..o.84(5,d) o..105(5,k) 0..0.96(5,1) o..108(5,m) o..12(5,a) O.o.9(3,a) o..19(3,u) o..16(1,b) o..o.6(1,c) o..11(l,d)

0. . 14(1,b) o..o.65(1,c) ....... o..07(1,d) ....... o..o.8(11,h) . ...... o..13(1,b) ....... o..o.6(1,c) ....... o..o.55(1,d) . ...... o..o.97(2,f) 0..34(3) 0. . 173 (2,f) . ...... 0. . 145 (2,f) 0..23(6) o..133(2,f) 0..95(11) o..3(11,f) 0..64(3) o..178(3,x) 0..47(3) 0..42(3)

0..35(11) •

0

•••••

* Numbers in parentbeses indicate references to data sources; letters identify lubricant in following

list.

2-45

COEFFICIENTS OF FRICTION

TABLE 2d-4. COEFFICIENTS OF STATIC AND SLIDING FRICTION (Continued) Static friction .Dry Greasy

Materials Cadmium on mild steel. .......... Copper on mild steel. ............ Nickel on nickel. ................ Brass on mild steel. ............. Brass on cast iron ............... Zinc on cast iron ................ Magnesium on cast iron .......... Copper on cast iron .............. Tin on cast iron ................. Lead on cast iron ................ Aluminum on aluminum .......... Glass on glass ...................

.. . . . . .

•••

0

••••••••

0.53(8) ...........'. 1.10(16) 0.28(22,y) 0.51(8) 0.11(22,c)

...... . ............

0.85(16) . ...........

...... . ........... .

.. ........... . ............

1.05(16) ........

....... ...... .

'.,

1. 05(16) 0.30(22,y) 0.94(8) O.35(22,y) 0.1(22,q)

Sliding friction Dry Greasy 0.46(3) 0.36(3) O.18(17,a) 0.53(3) O.12(3,w) 0.44(6) 0.30(6) 0.21(7) 0.25(7) 0.29(7) 0.32(7) 0.43(7) 1.4(3) 0.4(3) O.09(3,a)

........... . ....... ........... . 0.78(8) . ...........

Carbon on glass ................. Garnet on mild steel. ............ Glass on nickel. ................. Copper on glass ................. Cast iron on cast iron ............ Bronze on cast iron .............. Oak on oak (parallel to grain) ....

0.18(3) 0.39(3) 0.56(3) 0.68(8) . . . . . . . . . . . . 0.53(3) 1.10(16) 0.2(22,y) 0.15(9) ....... ........... . 0.22(9) 0.62(9) ............ 0.48(9)

Oak on oak (perpendicular) ....... Leather on oak (parallel) ......... Cast iron on oak ................ Leather on cast iron ............. Teflon on Teflon ................ Teflon on steel. ................. Fluted rubber bearing on steel. ... Laminated plastic on steel ........ Tungsten carbide on tungsten carbide ......................... Tungsten carbide on stee! ........

0.54(9) 0.61(9)

••••

0

••

........... . ............ ....... ........... . ....... ........... .

O.070(9,d) 0.077(9,n) 0.164(9,r) 0.067(9,s) 0.072(9,s)

0.32(9) 0.52(9) 0.49(9) 0.075(9,n) 0.56(9) 0.36(9,t) 0.04(22) ............ 0.04(22,£) 0.04(22) ............ 0.04(22,f) ...... . .......... . ....... 0.05(13,t) ...... . ............ 0.35(12) 0.05(12,t) ~

0.2(22) 0.5(22)

.

0.12(22,a) 0.08(22,a)

Materials

Sliding friction, dry

Nylon 6.6 on mild steel (no fibers) .......................... Nylon 6.6 on mild steel (30% by wt. carbon fibers) ........... Copper-graphite (high copper) on hard steel. ................ Copper-graphite (low copper) on hard steel. ................. ·Carbon-graphite(low graphite) on hard steel. ................ Carbon-graphite (high graphite) on hard steel. ............... Carbon-Teflon on hard steel. .............................. Carbon-copper-Teflon on hard steel. " ......................

0.40(23) 0.35(23) 0.40(23) 0.25(23) 0.50(23) 0.25(23) 0.30(23) 0.29(23)

2-46

MECHANICS

Lubricant References for Table 2d-4 a. - b; c. d. e. f.

g. h. i. j.

k. 1..

Oleic acid Atlan:tic spindle oil (light mineral) Castor oil Lard oil Atlantic spindle oil plus 2 per cent oleic acid Medium mineral oil Medium mineral oil plus -i per cent oleic acid Stearic f!,cid Grease (zinc oxide base) Graphite Turbine oil plus 1 per cent graphite Turbine oil plus 1 per cent stearie acid

Thrbiiie -oil (mediurii- inineralf Olive oil Palmitic acid Ricinoleic acid Dry soap Lard Water Rape oil 3-in-l oil w. Octyl alcohol x. Triolein y. 1 pet cent lauric aCid in paraffin oil

m. n. p. q. r. s. t. u. v.

References for Table 2d-4 1. Campbell, W. E.: Studies in Boundary Lubrication, Trans. ASME 61 (7), 633-641 (1939). 2. Clark, G. L., B. H. Lincoln, and R. R. Sterrett: Fundamental Physical and Chemical Forces in Lubrication, Proc. API 16, 68-80 (1935). 3. Beare, W. G., and F. P. Bowden: Physical Properties of Surfaces. 1, Kinetic Friction, Trans. Roy. Soc. (London), ser. A, 234, 329-354 (June 6, 1935). 4. Dokos, S. J.: Sliding Friction under Extreme Pressures-I, J. Appl. Mech. 13, A-148156 (1946). 5. Boyd, J., and B. P. Robertson: The Friction Properties of Various Lubricants at High Pressures, Trans. ASME 67 (1), 51-56 (January, 1945). 6. Sachs, G.: Versuche nber die Reibung fester Korper .(Experiments about the Friction of Solid Bodies), Z. angew. Math. Mech. 4, 1-32 (February, 1924). 7. Honda, K., and R. Yamada: Some Experiments on the Abrasion of Metals, J.Inst. Metals 33 (1), 49-69 (1925). 8. Tomlinson, G. A.: A Molecular Theory of Friction, Phil. Mag., ser. 7, 7 (46), 905-939 (suppl., June, 1929). 9. Morin, A.: Nouvelles experiences sur Ie frottement (New Experiments on Friction) Acad. roy. 8ci., Paris (a) 57, 128 (1832); (b) 59, 104 (1834); (c) 60, 143 (1835); (d) 63, 99 (1838). 10. Claypoole, W.: Static Friction, Trans. ASME 65, 317-324 (May, 1943). 11. Tabor, D.: The Frictional Properties of Some White-metal Bearing Alloys: The Role of the Matrix and Hard Particles, J. Appl. Phys. 16 (6), 325-337 (June, 1945). 12. Eyssen, G. R.: Properties and Performance of Bearing Materials Bonded with Synthetic Resin, General Discussion on Lubrication and Lubricants, Inst. Mech. Engrs., J. 1, 84-92 (1937). 13: Brazier, S. A., and W. Holland-BowYet:-Ru1515eras a Material for Beatings, General Discussion on Lubrication and Lubricants, Inst. Mech. Engr8., J. 1, 30-37 (1937); India-Rubber J. 94 (22), 636-638 (Nov. 27, 1937). 14. Burwell, J. T.: The Role of Surface Chemistry and Profile in Boundary Lubrication, J. SAE 50 (10), 450-457 (1942). 15. Stanton, T. E.: "Friction," Longmans, Green & Co., Ltd., London, 1923. 16. Ernst, H., and M. E. Merchant: Surface Friction of Clean Metals-A Basic Factor in Metal Cutting Process, Proc. Conf. Friction and Surface Finish (MIT), June, 1940, pp.76-101. 17. Gongwer, C. A.: Proc. Conf. Friction and Surface Finish (MIT), June, 1940,pp. 239244. 18. Hardy, W., and 1. Bircumshaw: Boundary Lubrication-Plane Surfaces and theLimitations of Amontons' Law, Proc. Roy. Soc. (London), ser.A, 108 (A 745), 1-27 (May, 1925). 19. Hardy, W. R., and J. K. Hardy: Note on Static Friction and on the Lubricating Properties of Certain Chemical Substances, Phil. Mag., ser. 6, 38 (233), 32-48 (1919).

COEFFICIENTS OF FRICTION

2-47

20. Bowden, F. P., and J. E. Young: Friction of Clean Metals and Influence of Adsorbed Films, Proc. Roy. Soc. (London), ser. A, 208 (A 1094), 311-325 (September, 1951). 21. Hardy, W. B., and 1. Doubleday: Boundary Lubrication-The Latent Period and' Mixtures of Two Lubricants, Proc. Roy. Soc. (London), ser. A, 104 (A 724), 25-38 (August, 1923). 22. Bowden, F. P., and D. Tabor: "The Friction and Lubrication of Solids," Oxford University Press, New York, 1950. 23. Lancaster, J. K.: Composite Self-lubricating Bearing l\([aterials, Proc. I nst. M echo Engrs. (London) 182,33-54 (1967-1968).

2d-2. Rolling Friction. Rolling is frequently substituted for sliding friction. The resistance to motion is substantially smaller than for sliding under nonfluid film conditions. The frictional resistance to rolling under the action of load W may be designated as P in Fig. 2d-1. The coefficient of rolling friction is then defined as

JE

=

P W

(2d-2)

The frictional resistance P to the rolling of a cylinder under load is applied at the center of the roller and is inversely proportional to the radius r of the roller and proportional to a factor k, a function of the material and its surface condition. Thus

P =~W

..

(2d-3)

If r is in inches, values of k may be taken as follows: hardwood on hardwood, 0.02; iron on iron, steel on steel, 0.002; hard polished steel on hard polished steel, 0.0002 to 0.0004. Noonan and Strange suggest, for steel rollers on steel plates: surfaces well

w

FIG. 2d-1. Rolling friction.

FIG. 2d-2. Load carried on rollers.

finished and clean, 0.005 to 0.001; surfaces well oiled, 0.001 to 0.002; surfaces covered with silt, 0.003 to 0.005; surfaces rusty, 0.00:3 to 0.01. If the load is carried on rollers as in Fig. 2d-2, and k and k' are the respective factors

2-48

MECHANICS

for lower and upper surfaces, the force P is

P

= (k

+ k')W d

(2d-4)

A comprehensive survey of rolling friction may be found in the following references presented at the annual meeting of the American Society of Mechanical Engineers, December 1 to 5, 1968. Hersey, M. D.: Rolling Friction: I, Historical Introduction, Paper 68-LUB-B. Hersey, lVi. D., and M. S. Downes: Rolling Friction: II, Cast Iron Car Wheels, Paper 68-LUB-C. Hersey, M. D.: Rolling Friction: III, Review of Later Investigations. Paper 68-LUB-D.

2e. Elastic Constants, Hardness, Strength, Elastic Limits, and Diffusion Coefficients of Solids H. M. TRENTl

U.S. Naval Research Laboratory D. E. STONE

Vertex Corporation 2 L. A. BEA DBIEN

U.S. Naval Research Laboratory

2e-1. Introduction. For the fundamental ideas connected with elasticity and for the definition of the elastic constants see Sec. 2a-6. For other definitions see Sec. 2e-3. The symbols and abbreviations used in this section are presented below.

E G p

Cj;

So;

T.S. Y.S. Y.P. S.S. El. R.A. Bhn R Vdh, Vhn D v p

Young's modulus modulus of rigidity Poisson's ratio density elastic constant (cf. Sec. 2a-6) elastic coefficient (cf. Sec. 2a-6) tensile strength yield strength yield point shear strength elongation reduction in area Brinell hardness number Rockwell hardness number (often used with sUbscripts) Vickers hardness number diffusion coefficient specific volume pressure

2e-2. Elastic Constants and Coefficients of Crystals. Tables 2e-1 through 2e-6 contain tabulations of the elastic constants Cij and coefficients Si; of cubic, hexagonal, tetragonal, trigonal, orthorhombic, and monoclinic crystals (cf. Sec. 9a for X-ray crystallographic data). All temperatures are room temperatures unless otherwise specified. However, the original sources often contain values for a wide range of temperatures. The two electrical boundary conditions for piezoelectric crystals are as follows: D = 0 denotes ali electric field, generated piezoelectrically, parallel to the direction of wave propagation; E = 0 denotes a field perpendicular to this direction. Boundary 1

2

Deceased. H. M. Childers of the Vertex Corporation provided valuable consultant service.

2-49

2-50

MECHANICS

conditions are given only for those materials for which a change in boundary conditions produces a substantial change in one or more measured values. References for these tables will be found immediately following Table 2e-6. References 1, 2, and 3 are published compilations from which the original sources can be obtained as well as references for values differing slightly from those given in these tables. In those cases in which two references are given, the first is for Gi; and the second for Si;. 2e-3. Elastic Constants, Hardness, Strength, and Elastic Limits of Polycrystalline Solids. Tables 2e-7 through 2e-16 contain data on the Young's modulus, modulus of rigidity, hardness, etc., of various solids, metals, and alloys. The elastic constants, tensile strength, yield strength, shear strength, and all other quantities having the dimensions of stress are expressed in dynes per square centimeter. The definitions of these and other tabulated quantities are given in the following list. 1. Tensile Strength. I "The maximum tensile stress which a material is capable of developing." Note: In practice, it is considered to be the maximum stress developed by a specimen representing the material in a tension test carried to rupture, under definite prescribed conditions. Tensile strength is calculated from the maximum load P carried during a tension test and the original cross-sectional area of the specimen Ao from the formula Tensile strength =

~o

2. Yield Strength. l "The stress at which a material exhibits a specified permanent set." The yield strength is conventionally determined in either of two ways. In the first method, a specimen of the material is repeatedly loaded and unloaded with the load oeing increased at each cycle, the process being continued until a specified permanent set is obtained after one of the unloadings. The stress which produces this specified permanent set is called the yield strength. In the second method, known as the offset method, a load-elongation curve is determined experimentally, the elongation being measured in 'units of extension 'per unit length of the undeformed specimen. A straight line is then drawn having a slope equal to the initial slope of the load-elongation curve and an intercept on the elongation axis equal to the specified offset, which is usually given in units of per cent elongation. The yield strength is taken to be that load defined by the interaction of the added straight line with the load-elongation curve. Further discussion of yield strength can bfl found in ASTM E6-36. 3. Yield Point.' The stress at which a marked increase in deformation takes place without increase in the load. 4. Shear Strength.' "The stress, usually expressed in pounds per square inch, required to produce fracture when impressed perpendicularly upon the cross-section of a material." 5. Elongation.' "In tensile testing the elongation of a specimen is the increase in gage length, after rupture, referred to the original gage length. It is reported as percentage elongation." 6. Reduction in Area.' "In tensile testing the reduction in area of a specimen is the ratio of the difference between the original cross-sectional area of the specimen and the cross-sectional area after rupture, to the original cross-sectional 'area. It is reported as the percentage reduction of area." Standard Definitions of Terms Relating to Methods of Testing, ASTM E6-36. "Metals Handbook," 1948 ed., American Society fOT Metals. 'J. G. Henderson, "Metallurgical Dictionary." 'Nail. Bur. Standards (U.S.) Cire. 0447. 1

2

ELASTICITY, HARDNESS, AND STRENGTH OF SOLIDS

2":"'51

TABLE 2e-1. ELASTIC CONSTANTS AND COEFFICIENTS OF CUBIC CRYSTALS (C.; in units of 1011 dynes/cm2; 8;j in units of 1O-1s cm 2/dyne) Material

Cll C12 ------

8 11

C44

8 12

- - ---

Ag (silver) .............. 12.40 9.34 4.61 22.9 Ag, 25% Au ............. ....... . . . . . . . ...... 20.7 Ag, 50% Au ............. · . . . . . . ....... 19.7 Ag, 75% Au ............. · . . . . . . ....... ....... 20.5 3.3 0.720 31.3 AgBr ................... .. 5.63 AgCl. .................. 6.01 3.62 0.625 30.4 9.25 4.61 23.07 Ag, 1.34% Cd ........... 12.28 Ag, 1.92 % Cd ........... 12.16 9.13 4.59 23.10 8.90 4.50 25.30 Ag, 8.36% In ............ 11.66 Ag, 3.07% Mg ........... 11.98 8.98 .4.60 23.37 Ag, 7.33% Mg ........... 11.59 8.66 4.52 23.94 9.58 4.81 21.93 Ag, 6.22% Pd ........... 12.77 9.22 4.58 24.29 Ag, 3.17% Sn ........... 12.10 9.16 4.58 23.89 Ag, 2.40% Zn ........... 12.09 9.33 4.61 23.54 Ag, 3.53% Zn ........... 12.30 1.07 0.85 52 Alum ................... 2.56 Aluminum .............. 11.2 6.6 2.79 15.7 Al, 5% Cu .............. · . . . . . . ....... ..... . 15 Ammonium alum ........ 2.50 1.06 0.80 53.5 0.59 0.53 36.2 Ammonium bromide ..... 2.96 0.72 0.68 27.2 Ammonium chloride ...... 3.90 Au (gold) ............... 18.6 15.7 4.20 23.3 1.86 1.22 19.4 Barium nitrate .......... 6.04 CaF2 (fluorspar) ......... 16.44 5.02 3.47 7.10 14.4 11.7 4.27 Chromite ............... 32.3 Chromium alum ......... ...... . ...... . ...... 54.2 6.49 Cobalt zinc ferrite ....... 26.6 15.3 7.8 Copper ................. 16.8 12.1 .7.54 15.0 CusAu .................. 19.07 13.83 6.63 13.4 Cu, 4.1% Zn (a-brass) ... 16.33 11.77 7.44 ....... Cu, 9.1 % Zn (a-brass) ... 15.71 11.37 7.23 ...... . Cu, 17.1% Zn (a-brass) .. 14.99 10.97 7.15 ....... Cu, 22.7% Zn (a-brass) .. 14.47 10.71 7.13 ...... . Cu, 47% Zn (a-brass) .... 15.22 11.62 7.19 ....... 10.2 7.44 41.05 Cu, 44.9% Zn «(3-brass) ... 11. 9 Cu, 48.3% Zn «(3-brass) ... 12.91 10.97 8.24 35.3 Cu, 48.9% Zn «(3-brass) ... 12.79 10.91 8.22 36.4 Cu, 4.81 % AI ........... 16.58 12.16 7.49 15.9 Cu, 9.98% AI. .......... 15.95 . 11.76 7.66 16.75 Cu, 1.58% Ga ........... 16.50 .11.92 7.43 . 15.38 Cu, 4.15% Ga ........... 16.5.2 . 12.10 7.41 15.91 Cu, 1.03% Ge .... : ...... 16 ..66 12.10 7.50 15.44 Cu, 1.71% Ge ........... , 16.31 . 11.82 7.50 15.72 Cu, 4.17% Si ....... '" .... 16 .7~r 12.42 7.48 16.10 Cu, 5.16% Si. ........... 16.08 11.88 7.49 16.71 Cu, 7.69%Si............ 16.58 12.64 7.41 17.72 Cu, 4.59% Zn .... " " .. , 16.34 11.92 7.42 15.91 0"

••••

'

8 44

Ref.

--9.83 -8.91 -8.52 -9.09 -11.7 -11.4 -9.91 -9.91 -10.95 -10.01 -10.24 -9.40 -10.51 -10.30 -10.16 -15 -5.8 -6.9 -15.9 -6.0 -4.2 -10.65 -4.6 -1.66 -1.31 -15.3 -2.37 -6.3 -5.65 ........ .

21.7 2 20.5 4 19.7 4 20.6 4 139 2 160 2 21.69 5 21.77 5 22.20 5 21.74 5 22.10 5 20.79 5 21.83 5 21.85 5 21.68 5 118 1 35.9 1 37 6 125 7,3 189 1 147 1 23.8 2 82.0 1 28.8 2 8.56 2 130 3 12.8 1 13.26 1 15.1 2 ..... . 8 ......... ..... . 8 ........ . . ..... 8 ........ . ...... 8 .. ... " ....... 9 -19.0 13.4 2 -16.2 12.2 2 12.2 1 -16.8 -6.73 13 .. 35 10 -7.11 13.05 10 -,6.45 13.46 10 -6.73 13.50 10 -6.50 13.33 10 -6.60 13.33 10 -6.85 .13.37 10 -7.10 13.35 10 -7.66 13.50 10 -6.71 13.48 10 ,

,

2-52

MECHANICS

TA~LE 2e-1. ELASTIC CONSTANTS AND COEFFICIENTS OF CUBIC CRYSTALS (Continued)

C 12 C44 Cl l 8 11 ------- Cu, 28% Zn ............. ....... ...... . ..... . 19.4

Material

Diamond ............... Diamond ............... Fe ............... ··.·· . Garnet 21.8 % FeO ....... Garnet 22.7% FeO ....... Garnet 23.0% FeO ....... Garnet 23.6% FeO ....... Garnet 26.2 % FeO ...... Garnet 28.7% FeO ....... Garnet 33.5 % FeO ....... Fe,04 (magnetite) ....... FeSz (pyrite) ............ GaAs ................... GaSb ................... Germanium ............. Hexamethylene tetramine Lead nitrate ............. Indium antimonide ....... Potassium alum ......... K (potassium) ........... KBr .................... KC!. ................... KF .................... KI.. ................... Li (195°K) .............. LiBr ................... LiC!. ................... LiF .................... LiI ..................... MgO ................... Magnetite .............. Molybdenum ............ Na (sodium) ............ NaBr ................... NaBrO' ................. NaC!. .................. NaClO, ................. NaF ................... NaI. ................... Ammonium alum ........ NH4Br ................. NH4C!. ................ NickeL ................. Palladium ............... Pb (lead) ............... Pb ..................... PbS (galena) ............

107.6 95 23.7 19.7 19.2 22.2 21.0 22.6 27.3 32.7 27.3 36.2 1.192 8.85 12.89 1.5 4.56 6.72 2.54 0.459 3.46 3.98 6.58 2.67 1.320 3.94 4.94 11.12 2.85 28.6 27.5 46 0.945 3.87 5.73 4.87 4.99 9.71 3.035 2.50 2.96 3.90 24.65 22.71 5.03 4.66 10.2

12.50 57.58 0.953 43 1.38 39 7.72 14.1 11.6 9.0 5.7 7.11 8.02 9.9 5.9 10.4 7.0 6.42 10.3 6.7 7.03 12.6 6.2 7.36 15.7 6.8 6.32 12.4 8.9 3.87 9.7 4.7 10.6 -4.4 10.4 2.85 0.599 0.538 126.4 4.04 4.33 15.8 9.78 4.83 6.71 0.3 0.7 70 3.09 1.37 48.5 3.67 3.02 24.2 1.07 0.84 52.5 0.372 0.263 833 0.58 0.505 30.4 0.62 0.625 26.2 1.49 1.28 ....... 0.43 0.421 39.2 1.102 0.960 316.4 1.88 1. 91 ...... . 2.26 2.49 ....... 4.20 6.28 11.35 1.40 1.35 ...... . 8.7 14.8 4.08 10.4 9.55 4.59 17.6 11.0 2.8 0.779 0.618 420 0.97 0.97 28.7 1.76 1.52 20.4 1.24 1.26 22.9 1.41 1.17 22.9 2.43 2.80 ...... . 0.90 0.72 ...... . 1.06 0.8 53.5 0.59 0.53 36.2 0.72 0.68 27.2 14.73 12.47 7.34 17.60 7.173 . . . . . . . 3.93 1.40 63.2 3.92 1.44 92.8 12 3.8 2.5

8 12

Ref. 8 44 ----

-8.4 -0.099 -0.40 -2.85 -2.2 -2.7 -2.1 -2.3 -2.6 -2.3 -1.1 -1.31 0.39 -42.34 -4.96 -2.66 -12 -19.6 -8.55 -15.6 -370 -4.35 -3.5

13.9 11 1. 74 12 2.3 13 9.02 9, 14 17.5 15 15 16.9 14.3 15 14.9 15 16.1 15 14.7 15 11.2 15 2 10.3 9.6 2 2 186 2 23.1 2 14.90 140 1 1 73.0 2 33.1 2 119 380 9,16 2 198 2 160 . ..... 17 2 238 104 18 ...... 17 ..... . 17 2 15.9 . ..... 17 2 6.76 1 10.47 2 9.1 2 162 2 103 2 65.7 2 79.4 2 85.4 ..... . 17 ...... 17 125 '"" 2 189 2 147 2 8.02 ..... . 19 71.4 20 2 69.4 1 40

........ . -5.4 -144

........ . ........ . -3.1

......... -0.95 -1.26 -0.78 -190 -5.8 -4.8 -4.65 -5.05

......... ........ . -15.9 -6.0 -4.2 -2.74

........ . -27.7 -42.4 -3

2-53

ELASTICITY, HARDNESS, AND STRENGTH OF SOLIDS TABLE

2e-1.

ELASTIC CONSTANTS AND COEFFICIENTS OF CUBIC CRYSTALS

Material

C 11

PbS ... ... . ........... . .. . . . . . .. . . .. . RbBr. RbCl. .. ............... . RbF ..... . . . . . . . . . . . . . . RbI. ................... Silicon .................. Strontium ni tra te ........ Thallium bromide ....... Thallium chloride ........ Thallium alum .......... Thallium bromide chloride Thallium bromide iodide .. Thorium ................ W (tungsten) ............ Zinc blende .•........... Zinc sulfide .............

TABLE

2e-2.

12.70 3.185 3.645 5.7 2.585 16.57 4.73 3.78 4.01

...... . 3.85 3.6 7.53 50.1 10.0 10.79

8 11

C 12

C••

---

---

2.98 0.48 0.61 1.25 0.375 6.39 2.18 1.48 1.53

2.48 0.385 0.475 0.91 0.281 7.96 1.46 0.756 0.760

8 12

(Continued)

8 ••

Ref.

--8.7

...... ...... . ....... ...... . .

7.68 29.8 33.9 31.6 . . . . . . . ...... 49.0 1.49 0.737 33.1 1.5 0.555 37 4.89 4.78 27.2 19.8 15.14 2.57 6.5 3.4 20.5 7.22 4.12 20

-1.64

40.3

........ . . . . . . . . ........ ... ... ......... ..... . ........ . ,,",., . '

-2.14 -9.4 -9.5 -8.7 -15.5 -9.2 -11 -10.7 -0.729 -8.1 -8.02

12.56 68.5 132 132

115 136 180 20.9 6.60 29.4 24.3

1 17 17 17 17 2 1 2 2 3 1 1 2 2 1 2

ELASTIC CONSTANTS AND COEFFICIENTS OF HEXAGONAL CRYSTALS

(Cij in units of 10" dynes/cm 2 ; 8 ij in units of 10-13 cm 2 /dyne) Material

C11

C"

C"

C12

C13

811

8"

8"

812

8"

Ref.

-- -- --' - - --- - --- - --- --- -Apatite .............. 16.67 BaTi03 (D = 0) ...... 16.8 BaTi03 (E = 0) ...... 16.6 BaTi03 5 % CaTi03 by wt. (E= 0) ........ 17.41 Beryllium ............ 30.8 Beryl r .............. 27.81 Beryl II .......... '" 29.71 Cadmiulll ............ 11.0 CdS ................. 8.1 Cobalt .............. .30.7 Ice (-16°C) ......... 1.33 rViagnesium .. ........ 5.97 SiG, (600°C) (,,-quartz) 11.66 Yttrium .... " ....... 7.79 Zinc .. - ............. 16.1

13.96 6.63 18.9 5.46 16.2 4.29 16.88 35.7 24.8 26.5 4.69 8.0 35.81 1.42 6.17 11. 04 7.69 6.10

4.74 11.0 6.61 7.54 1.56 1.43 7.53 0.306 1.64 3.606 2.431 3.83

1.31 6.55 7.82 7.10 7.66 7.75 7.93 -5.8 10.01 10.26 4.04 4.9 16.5 0.63 2.62 1.67 2.85 3.42

8.00 8.7 6.77 7.39 3.83 4.8 10.3 0.46 2.17 3.28 2.1 5.01

-4.0 -1.95 -2.85

23 2 2

21.1 -2.45 -2.65 9.09 1.04 -1.17 -1.35 -0.80 15.1 -1.17 -0.78 13.3 -9.3 64.0 -1.5 -8.0 -8.7 70 13.24 -2.31 -0.69 326.5 -41.6 -19.3 -7.85 -5.0 61 27.73 -0.60 -2.62 ....... ... '" ..... 26.1 0.53 -7.31

2 1

7.49 10.9 15.1 8.18 6.76 18.3 8.55 8.93 23.3 8.42 3.37 4.47 4.21 36.9 21. 9 3.19 82.8 19.7 10.62 . . . . . .. , ' . 8.38 28.38

8.05 3.77 4.27 3.97 12.9 22.2 4.72 101.3 22.0 9.41

.

0.97 -2.98 -2.61

"

1 1 2 21 2 22 2 2 24 2

2-54 TABLE

MECHANICS

2e-3.

ELASTIC CONSTANTS AND COEFFICIENTS OF TETRAGONAL CRYSTALS

(C.; in units of

1011

Cl l

Material

dynes/cm 2 )

Caa

C44

C66

C13

C12

Ref.

-- -- - ---- --- ----Ammonium dihydrogen phosphate ... Ammonium dihydrogen phosphate ... Ammonium dihydrogen phosphate (D = 0) ........................ Ammonium dihydrogen phosphate (E = 0) ........................ Ammonium dihydrogen phosphate (deuterated) ..................... Barium titanate (D = 0) ........... Barium titanate (E = 0) ........... Indium ........................... Nickel sulfate ..................... Potassium dihydrogen arsenate ...... Potassium dihydrogen phosphate .... Potassium dihydrogen phosphate (DOC) ........................... Sn (tin) ........................... Sn ............................... Sn ............................... Zircon ............................ TABLE

2e-3A.

6.17 3.28 0.85 7.58 2.96 0.87

0.59 0.72 0.614 -2.43

1.94 1.30

2 2

6.76 3.38 0.867 0.687

0.59

2.0

2

6.76 3.38 0.867 0.608

0.59

2.0

2

6.2 28.3 27.5 4.45 3.21 5.3 7.14

3.0 0.91 17.8 8.05 16.5 5.43 4.44 0.655 2.93 1.16 3.7 1.2 5.62 1.27

8.14 7.85 1.29 8.6 13.3 4.9 7.35 8.7 2.2 8.39 9.67 1. 75 7.35 4.60 1.38

0.61 11.3 11.3 1.22 1.78 0.7 0.628

-0.5 1.4 18.7 14.2 17.9 15.1 3.95 4.05 2.31 0.21 -0.6 -0.2 -0.49 1.29

2 2 2 2 1 1 2

0.63 5.3 2.265 0.741 1.60

3.49 4.07 3.5 3.0 2.34 2.8 4.87 2.81 0.90 -0.54

1 1 2 2 2

ELASTIC CONSTANTS AND COEFFICIENTS

(Continued) (8;; in units of 10-13 cm 2/dyne)

OF TETRAGONAL CRYSTALS

8 11 8 33 8 4 , 8 66 8 12 Ref. 813 ---- -- - --------- - Ammonium dihydrogen arsenate .. 16.9 44.5 152.9 124.0 -17.3 -11.1 3 Ammonium dihydrogen phosphate 20 45.7 117 169 1.7 -12.9 2 Ammonium dihydrogen phosphate 17.5 43.5 114 163 2 7.5 -11 Material

Ammonium dihydrogen phosphate (D = 0) ..................... Ammonium dihydrogen phosphate (E = 0) ..................... Ammonium dihydrogen phosphate (deuterated) .................. Barium titanate (D = 0) ........ Barium titanate (E = 0) ......... Indium ........................ Nickel sulfate .................. Potassium dihydrogen arsenate ... Potassium dihydrogen phosphate. Potassium dihydrogen phosphate (O°C) ........................ Sn (tin) ........................ Sn ............................ Sn ............................ Zircon .........................

18.1

43.5 115.3 145.5

1.9 -11.8

2

18.1

43.5 115.3 164.6

1.9 -11.8

2

44 110 164 19 7.25 10.8 12.4 8.84 8.05 15.7 18.4 .8.84 149.4 187 152.7 82 34.3 86.5 56.2 65 19 27 86.0 152 14.8 19.5 78.7 159.2 17.5 14.6 16.3 18.5 13.9

20 8.5 14.1 11.8 22.1

77.5 159 20.6 19.0 45.4 44.2 57.0 135 72 62

2 -3.15 -2.35 -50.6 -46.8 2 1.7

-11 -3.26 -5.24 -90.2 -1.3 1 -3.79

2 2 2 2 1 1 2

-4 -5.3 -3.6 -9.9 -1.6

-7 -2.07 -4.1 -2.5 -1.4

1 1 2 2 2

2-55

ELASTICITY, HARDNESS, AND STRENGTH OF SOLIDS

TABLE 2e-4. ELASTIC CONSTANTS AND COEFFICIENTS OF TRIGONAL CRYSTALS (Cij in units of 1011 dynes/cm 2 ) Material

C 11

C 33

C 44

C 12

C '3

C 14

11.7 2.61 2.11 4.50 0.79 0.75 0.49 1.6 3.03 1.191 1. 51 4.8 1.60 2.31 3.5 4.9 3.5

10.1 1.05 -0.42 -2.03 0.03 -0.03 -0.03 -1.3 0.5 -1. 791 1.72 3.8 0.82

Ref.

--- -- ------ ------

Alumina (corundum) ... ... . ..... 46.5 56.3 23.3 12.4 Antimony ............. ......... 7.92 4.27 2.85 2.48 Bismuth ....................... 6.28 4.40 1.08 3.50 Calespar (calcite) ............... 13.74 8.01 3.42 4.40 Dextrose sodium bromide ........ 2.06 2.40 0.634 0.53 Dextrose sodium chloride ....... '1 2.20 1. 77 0.771 1.09 Dextrose sodium iodide .......... 2.58 2.06 0.771 1.52 Haematite .......... " ......... 24.2 22.8 8.5 5.5 Mercury ( -190°C) ............. 3.60 5.05 1.29 2.89 a-Quartz ....................... 8.674 10.72 5.79 0.699 a-Quartz ....................... 8.75 10.77 5.73 0.762 Sapphire ........................ 49 . 6 50.2 20.6 10.9 Sodium nitrate ................. 8.67 3.74 2.13 1.63 Tellurium ......... ,. " ......... ..... . 7.00 ...... ..... . Tourmaline .................... 27.2 16.5 6.5 4.0 Tourmaline I ................... 26.3 15.1 5.95 6.1 8.8 Tourmaline II .................. 30.4 17.6 6.5

. ...... -0.68 -0.9 -0.4

-2 2 2 2 1 1 1 2 2 2 2 25 2 2 1 1 1

TABLE 2e-4A. ELASTIC CONSTANTS AND COEFFICIENTS OF TRIGONAL CRYSTALS (Continued) (8ij in units of 10-13 cm 2 /dyne) Material

8 11

8"

--- -2.90 1. 94 16.1 16.1 17.7 33.8 26.9 28.7 11.0 17.3 56.9 52.3 63.8 70.2 60.2 51.6 4.42 4.44 78.7 35.0

Alumina (corundum) ........ Aluminum phosphate ........ Antimony .................. Bismuth ................... Calespar (calcite) ........... Dextrose sodium bromide .... Dextrose sodium chloride .... Dextrose sodium iodide ...... Haematite ................. Lithium trisodium chromate .. Lithium trisodium molybdate ............... 29.5 27.1 Mercury ( -190°C) ......... 154 45 a-Quartz ................... 12.77 9.6 a-Quartz ........... '" ..... 12.69 9.71 Sapphire ................... 2.18 2.02 Sodium nitrate ............. 13.4 30.8 Tellurium ....... " ......... 48.7 23.4 Tourmaline ................ 3.85 6.36 Tourmaline I ............... 4.22 7.34 Tourmaline II ............. '1 3.64 5.89

8 44

8 ,2

8"

8 ,4

5.78 -1.05 -0.38 -0.1 -8.3 53.0 41.0 -3.8 -8.5 104.8 -14.0 -6.2 -3.4 -4.3 39.4 158 -8.6 -16.0 130 -26.1 -16 130 -34.3 -6.2 11.92 -1.02 -0.23

-1.71 8.9 -8.0 16.0 8.6 -3.4 3.6 3.8 0.80

...... . . . . . . . . ...... . ........

...... ........ . . . . . . . . . . . . . . . -119 151 20.04 -1.79 20.05 -1.69 5.04 -0.50 -2.2 51.5 58.1 -6.9 15.4 -0.48 17.1 -0.80 15.4 -1.00

Ref.

--

---

-21 -100 4.50 -1.22 -4.31 -1.54 -0.49 -0.16 -4.8 -6.0 -13.8 ........ 0.45 -0.71 -1.11 0.76 0.29 -0.53 1

2 3 2 2 2 1 1 1 2 3 3 2 2 2 25 2 2 1 1 1

2-56

MECHANICS

'fABLE 2e-5. ELASTIC CONSTANTS AND COEFFICIENTS OF ORTHORHOMBIC CRYSTALS (C'i in units of 1011 dynes/cm 2 ) Material

Cl l

C2 2

CBB

C44

C55

C66

C12

---- ---- -- -- --

Aragonite ........ 16.0 8.7 8.5 Baryte .......... 8.62 9.17 10.84 Celestite ......... 10.44 10.61 12.86 Iodic acid ........ 3.03 5.45 4.36 Lithium ammonium. tartrate .. 3.86 5.39 3.63 Magnesium sulfate ........ 6.98 5.29 8.22 Potassium pentaborate ... 5.82 3.59 2.55 Rochelle salt (D = 0) ....... 2.55 3.81 3.71 Rochelle salt (E = 0) ..... ' . 2.55 3.81 3.71 Rochelle salt (D = 0) ....... 4.25 5.15 6.29 Rochelle salt (E = 0) ....... 4.25 5.15 6.29 Sodium ammonium tartrate .. 3.68 5.09 5.54 Sodium tartrate .. 4.61 5.47 6.65 Staurolite ........ 34.3 18.5 14.7 Strontium formate ....... 4.39 3.48 3.74 Sulfur ........... 2.40 2.05 4.83 Topaz ........... 28.2 34.9 29.5 a-Uranium ....... 21.47 19.86 26.71 Zinc sulfate ...... 4.00 3.22 5.45

4.12 1.20 1.35 1.84

C'3 -~

C2a

Ref.

----

2.56 2.87 2.79 2.19

4.27 2.74 2.66 1.74

3.73 5.23 7.73 1.19

0.17 3.41 6.05 1.17

1. 57 3.56 6.19 0.55

2 1 2 1

1.19 0.67

2.33

1.65

0.87

2.01

1

1.07 2.33

2.22

3.90

2.82

2.83

1

1.64 0.463 0.57

2.29

1. 74

2.31

2

1.34 0.321 0.979 1.41

1.16

1.46

2

..... 0.286 0.960 1.41

1.16

1.46

2

1.25 0.304 0.996 2.96

3.57

3.42

2

0.58 0.278 0.974 2.96

3.57

3.42

2

3.08 3.47 3.20 3.52 6.1 12.8

1 1 26

1.04 -1.49 -0.14 1.33 1.71 1.59 8.8 12.6 8.5 4.65 2.18 10.76 1.32 1.80 1.19

1 2 2 27 1

1.06 0.303 0.87 1.24 0.31 0.98 4.6 7.0 9.2 1.54 1.07 0.43 0.87 10.8 13.3 12.44 7.342 0.50 1. 70

1.72 0.76 13.1 7.433 1. 81

2.72 2.86 6.7

2-57

ELASTICITY, HARDNESS, AND STRENGTH OF SOLIDS TABLE

2e-5A. OF

ELASTIC CONSTANTS AND COEFFICIENTS

(Continued) (8i; in units of 10-13 CIll 2 / dyne)

ORTHORHOMBIC CRYSTALS

8 ..

Material

8"

8"

8"

813

8"

Ref.

- - - - - - - - ---- -- - - - - - - --- --- ----Aragonite ................. Baryte ................... Barium formate .......... . Celestite .................. lodic acid ................. Lithium ammonium tartrate Magnesium sulfate ......... Potassium pentabol'ate ..... Rochelle salt CD = 0) ...... Rochelle salt CE = 0) ...... Rochelle salt CD = 0) ...... Rochelle salt CE = 0) ...... Sodium ammonium tartrate. Sodium tartrate ........... Strontium formate ... ...... Sulfur .................... Topaz .................... a-Uranium. ,. , ............ Zinc sulfate ...............

6.95 18.4 ..... 22.0 39.8 30 24.5 23.2 52.4 52.4 51.8 51.8 57.0 37.1 28.4 71 4.43 4.91 29.5

13.2 17.36 ..... 21. 9 20.1 25.6 34.1 73.6 35.4 35.4 34.9 34.9 38.5 31.6 31 83 3.53 6.73 37.7

12.2 10.96 ..... 11.4 25.6 35 15.0 98.3 33.7 33.7 33.4 33.4 40 26.4 31 30 3.84 4.79 20.4

24.3 83.33 78.5 74.1 54.5 84 93.5 61 74.7 ...... 79.8 174 94.5 80.6 65 232 9.23 8.04 200

39.0 34.84 60.0 35.8 45.6 150 42.9 215 311 350 328 360 330 323 93 115 7.53 13.62 58.8

23.4 36.50 82.5 37.6 57.6 43 45.0 175 102 104 101 103 115 102 58 132 7.63 13.45 55.3

-3.0 -9.45 . ...... -13.9 -7.75 -8.2 -16.6 -10.6 -15.4 -15.4 -15.3 -15.3 -15.5 -12.0

0.4 -2.68 ....... -3.7 -9.7 -2.7 -2.68 -6.1 '-10.3 -10.3

-2.4 -2.73

-4.0 -0.45 -12.2 -6.05 -60 -9.1 -9.1 ~21.1 -10.3 -21.1 -10.3 -22 -15.5 -11.5 -10.9 -2 -8 11 -15 -36 -13 -1.38 -0.86 -0.06 -1.19 0.08 -2.61 -10.8 -3.49 -6.10

2 1 3 2 1

2 2 2 2 2 1

2 2 27 1

TABLE

Material

C ll

2e-6. ELASTIC CONSTANTS AND COEFFICIENTS OF MONOCLINIC (C,; in units of 1011 dynes/em 2 ; 8,j in units of 10-13 em 2 /dyne) C 22

C 33

C 44

C' 5

C 66

C 12

C 13

C 23

C1

---- ------ -- ----------Dipotassium tartrate* ... Ethylene diamine tartrate* ............. Lithium sulfate* ........ Sodium thiosulfate* ..... Tartaric acid* ..........

6.9

3.5

4.4

0.84

1.3

0.96

1.2

3.2

1.4

0

13.4 5.7 3.31 9.30

3.5 7.1 3.02 1.93

6.04 4.9 4..57 4.65

0.53 2.7 0.57 0.81

0.83 2.9 1.11 0.82

0.57 1.4 0.60 1.1

2.7 2.7 1.83 2.0

8.1 1.6 1.84 3.7

2.2 1 1.6 -0 1.68 0 1.4 -1

---- -- ------ ----------811

8 22

8 33

8 44

8"

--------- ----- --- --- --:Dipotassium tartrate* ... "Ethylene diamine tartrate* ............. Lithiumsulfate* ........ Sodium thiosulfate* ..... Tartaric acid* .......... Dipotassiumtartrate** .. Ethylene diamine tartrate .............. Lithium sulfate** ... ~ ...

22.4

33.7

38.8 37 23.9 21.3 50.2 156 21.6 77 47.5 35.3

38.6 119 98 23.1 67.4 38.5 24.0

188 36.9 223 130 113.5

8 66 --~

81.5 104.1 172 41 327 180 102

812

813

-0.8 -16.4 -10.5

174 4.0 -52 74 -9.5 -5 212 -32.3 -6.21 96 -6.1 -15 122.5 -17.4 -8

33.4 36.5 100 192 117 191 22.9 __22.5 !-22.8 _7!.! 64.0 ~6.1

J

81

-19 -3.6 -71.9 -18 -6.2

-6

-7 7 15 28 -7

-3 -30 -18 -17 -5.4 -7.5 -4.6 -2

* The single-starred values of the 8i; correspond to the single-starred values of the O'i; that is, (0*)-1

** The doubJe-starred values are referled to a differently oriented sot of axes.

8 23

---------

= (8*).

ELASTICITY, HARDNESS, AND STRENGTH OF SOLIDS

References for Tables 2e-1 through 2e-6 1. Hearmon: Advances in Phys. 5,323 (1956). 2. Huntington: "Solid State Physics," vol. 7, p. 213, Academic Press, Inc., New York, 1958. 3. Sundara Rao, Vedan, and Krishnan: "Progress in Crystal Physics," vol. 1, p. 73, S. Viswanathan, Madras, India, 1958. 4. Rohl: Ann. Phys. 16,887 (1933). 5. Bacon: Dept. Phys., Case Inst. Technol., Tech. Rept. 15, 1955. 6. Karnop and Sachs: Z. Physik 53, 605 (1929). 7. Sundara Rao: Current Sci. (India) 17,50 (1948). 8. Rayne: Phys. Rev. 112, 1125 (1958). 9. Jones: Physica Hi, 13 (1949). 10. Neighbors and Smith: .Acta Met. 2, 591 (1954). 11. Sundara Rao and Balakrishnan: Proc. Indian Acad. Sci. 28A, 475 (1948). 12. McSkimmin and Bond: Phys. Rev. 105, 116 (1957). 13. Bhagavantam and Bhimasenachar: Proc. Roy. Soc., ser. A, 187,381 (1946). 14. Kimura: Proc. Ph1}s.-Math. Soc. Japan 21, 686, 786 (1939); 22,45, 219 (1940). 15. Ramachandra Rao: Proc. Indian Acad. Sci. 22A, 194 (1945). 16. Seitz: J. Appl. Phys. 12, 100 (1941). 17. Spangenberg and Haussuhl: Z. Krist. ill!), 422 (1957). 18. Nash and Smith:Phys. Chem. Solids 9,113 (1959). 19. Rayne: Phys. Re;. 118, 1545 (1960). 20. Prasad and Wooster: Acta Cryst. 9,38 (1956). 21. Gutsche: Naturwissenschaften 45, 566 (1958). 22. Bass, Rossberg, and Ziegler: Z. Physik 149, 199 (1957). 23. Bhimasenachar: Proc. Indian Acad. Sci. 22 (sec. A), 209 (1945). 24. Smith and Gjevre: J. Appl. Phys. 31,647 (1960). 25. Mayer and Heidemann: J. Acoust. Soc. Am. SO, 756 (1958). 26. Bhimasenachar and Venkata Rao: J. Acoust. Soc. Am. 29, 343 (1957). 27. Fisher and McSkimmin: J. Appl. Phys. 29, 1473 (1958).

AbbTeviations in Tables 2e-7 through 2e-16 Abbreviation H.R ................ C.R.. . . . . . . . . . . . . .. W.Q ................ O.Q ................ A.Q.. . . . . . . . . . . . . .. A.C ................ F.C ................ h-t. . . . . . . . . . . . . . . .. WT .. , . . . . . . . . . . . . . .

ann. . . . . . . . . . . . . . .. art. aged. . . . . . . . . . .. nat. aged ........... spec.. . . . . . . . . . . . .. G.S .................

Definition Hot Tolled Cold rolled Water quenched Oil quenched Air quenched Air cooled Furnace cooled Heat-treated Wrought Annealed Artificially aged Naturally aged Specimen Grain size

TABLE 2e-7. ELASTIC AND STRENGTH CONSTANTS FOR SILVER, GOLD, PLATINUM,

Material

Condition

Ag ..................... 1 Strained 5 %, heated 5 hr ·at 350°C Ag .................... . Ann. Ag + 80 Mo ............ . Ag + 40 Mo ............ . Ag + 20 Mo ............ . Ag +20W ............. . Ag +40W ............. . Ag+80W ............. . Ag + 40 Ni ............ . Ann. Ag + 20 Ni ........... . Ann. Ag + 1 graphite ........ . Ag + 5 graphite ....... . Ag + 10 graphite ....... . Ag+5Cd ............. . Ag + 10 Cd ............ . Ag+20Cd ............ . 33 Ag, 52 Hg, 12.5 Sn, 2 Cu, 0.5 Zn ............ . Au 99.99% ............. . Cast Au 99.99% ............ . Wrought, ann. 58.3 Au, 4.9 Ag, 31.6 Cu, Air cooled 5.2 Ni

E

u

Tensile strength

Yield strength at 0.2% offset

El t

7.1-7.8 X 1011

0.37 55 41 24 34 41 55 26 21

X 10' X 10' X 10' X 10' X 10' X 10' X 10' X 10'

16 X 10' 19 X lOB 20 X 10' 1.0 X 1011 2.8-5.9 X lOB 7.44 X 1011 10.42 12.4 X 10'1 ........ . 8.00 X 1011 0.42 13. 1 X lOB Nil 56.9 X 10' 33. 1 X lOB at 0.1% offset

3 4 4

TAllLE 2e-7. ELASTIC AND STRENGTH CONSTANTS FOR SILVER, GOLD, PLATINUM, PAL

Material

H.6 Au, 4.6 Ag, 43.4 Cu, [ Air cooled 5.0 Ni, 5.4 Zn 69 Au, 25 Ag, 6 Pt ....... Pt 99.99% .............. Pt+5Ir ............... Pt+lOIr .............. Pt + 25 Ir .............. Pt + 3.5 Rh ............. Pt + 5.0 Rh ............. Pt + 10.0 Rh ............ Pt + 20.0 Rho ........... Pt + 5 Ru .. o. Pt + 10 Ru ............. Pt + 1 Ni. .............. Pt + 2 Ni. ..... Pt + 5 Ni. .............. 84 Pt, 10 Pd, 6 Ru ....... 96 Pt, 4 W .............. Pd (pure) ............... 60 Pd, 40 Ag ............ 60 Pd, 40 Cu ............ 95Pd, 4 Ru, 1 Rh ........ 0

•••••

0

0

0

0"

•••

0

•••

* References are on p. 2-76.

Ann. Ann. Ann.

I

E

Condition

[

0

••••••••••••

14.7 X 1011 0.39

............. 0·.·.·.·.·.· .

Ann.

.............

Ann. Ann. Ann.

0·.· ... ·.· ...

Ann.

0·.·.· .......

Ann. Ann.

.............

Ann. Ann. Ann. and rolled

Ann. Ann. Ann.

I

............. [ .... [

Ann.

Ann. Ann. Ann.

(J"

0

••••••••••••

0

••

0

••

•••••••••

•

•

•••••••••

0_.·.·.· . . . · .

............. 0·······.··· . 0·····.····· . 0

••

•••••••••

•

12.1 X 1011

............ . 0

••••••••••••

0

••••••••••••

Tensile strength

Yield strength at 0.2% offset

46.8 X 10 8 [26.7 X 10 8 at 0.1% offset 37.6 X 108 . ........ 12-13 X 10 8 ......... 27 X 10 8 . ......... 38 X 10 8 . ........ 86 X 10 8 . ........ 17 X 10 8 . ........ 21 X 10 8 . ........ 31 X 10 8 . ........ 48 X 10 8 . ........ 41 X 10 8 . ........ 59 X 10 8 . ........ 21 X 10 8 . ........ 28 X 10 8 . ........ 45 X 10 8 ......... 55 X 108 . ........ 48-52 X 10 8 ......... 2::15 X 10 8 ......... 35 X 10 8 ......... . 52 X 10 8 38~1'X 10 8

••

00-

......

TABLE

Alloys Cast alloys: AI, 12 Si. ........................ AI, 5 Si. ......................... AI, 5 Si. ......................... AI, 5 Si, 4 Cu ..................... AI, 4 Cu, 3 Si. .................... AI, 5 Si, 3 Cu ..................... AI, 5 Si, 3 Cu ..................... AI, 5 Si, 3 Cu ........ ............ AI, 5.5 Si, 4.5 Cu .................. AI, 7 Cu, 2 Si, 1.7 Zn .............. AI, 7 Cu, 3.5 Si ..... . .. . . . . . . . AI, 10 Cu, 0.2 Mg ................. AI, 10 Cu, 0.2 Mg ................. AI, 12 Si, 2.5 Ni, 1.2 lVlg, 0.8 Cu .... AI, 12 Si, 1.5 Cu, 0.7 Mn, 0.7 Mg ... AI, AI, AI, AI, AI, AI, AI, AI, AI, AI,

4 Cu, 2 Ni, 1.5 Mg ............. 4.5 Cu ....................... 4.5 Cu, 2.5 Si .................. 3.8 Mg ............ .......... SMg ......................... 10 Mg ........................ 6 Si, 3.5 Cu ..... .......... 6 Si, 3.5 Cu ..... .......... 5 Si, 1.3 Cu, 0.5 Mg ............ 5 Si, 1.3 Cu, 0.5 Mg ............

AI, 7 Si, 0.3 Mg. . . . . . .. _.. _...... AI, 7 Si, 0.3 Mg .................. AI, AI, AI, AI, AI, AI,

S Si, 1.5 Cu, 0.3 Mg, 0.3 Mn .... S Si, 1.5 Cu, 0.3 Mg, 0.3 Mn ... 9.5 Si, 0.5 Mg .................. 8.5 Si, 3.5 Cu .................. 6.5 Sn, 1 Cu, 1 Ni. ............. 5.5 Zn, 0.6 Mg, 0.5 Cr, 0.2 Ti. ..

2e-8.

ELASTIC AND STRENGTH CONSTANTS FOR ALUMINUM A

Condition Die cast Die cast Sand cast Die cast Sand cast Sand cast Sand cast, h-t, aged Perm. mold cast, h-t, aged Perm. mold cast, h-t, aged Sand cast Perm. mold cast Sand cast (ann.) H-t, artificially aged Perm. mold cast, art. aged Perm. mold cast (stress relieved) Ann. (sand cast) H-t, nat. aged H-t, nat. aged Perm. mold cast Die cast Sand cast, h-t, nat. aged H-t, art. aged As cast H-t, art. aged (sand cast) H-t, art. aged (perm. mold cast) H-t, art. aged (sand cast) H-t, art. aged (perm. mold cast) Sand cast (stress relieved) Perm. mold (stress relieved) Die cast Die cast (Perm. mold cast) art. aged Sand cast

E

G

q

Tensile strength

s

7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10

X X X X X X X X X X X X X X X

1011 1011 1011 1011 10" 1011 1011 1011 1011 1011 1011 1011 1011 1011 10"

2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65

X X X X X X X X X X X X X X X

1011 1011 1011 1011 10" 1011 1011 1011 1011 1011 1011 1011 1011 1011 1011

0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33

25.5 20.7 13.1 27.6 14.5 18.6 24.1 2S.9 19.3 16.5 20.7 18.6 27.6 24.8 24.S

X X X X X X X X X X X X X X X

10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10'

12 9. 6. 15 9. 9. 13 15 11 10 16 13 20 19

7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10 7.10

X X X X X X X X X X

1011 1011 1011 10" 1011 1011 1011 1011 1011 10"

2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65

X X X X X X X X X X

1011 1011 1011 1011 1011 1011 1011 1011 1011 1011

0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33

18.6 22.0 27.6 18.6 2S.9 31. 7 24.S lS.6 24.1 29.6

X X X X X X X X X X

10' 10' 10' 10' 10' 10' 10' 10' 10' 10'

12 11 1 11 1 17 16 1 1 lS

7.10 X 1011 7.10 X 10"

2.65 X 1011 0.33 22.7 X 10' 1 2.65 X 10 11 0.33 27.6 X 10' IS

7.10 7.10 7.10 7.10 7.10 7.10

2.65 2.65 2.65 2.65 2.65 2.65

X X X X X X

10" 10" 10" 10 11 10" 1011

X X X X X X

10" 1011 10" 1011 10" 10 11

0.33 0.33 0.33 0.33 0.33 0.33

20.7 24.S 28.9 31.0 15.2 24.1

X X X X X X

10' 14 10' 10' 15 10' 1 10' 6. 10' 17

W:rought alloys:

Aluminum 99.996 AI. .... " ........ Aluminum 99.996 AI. .............. Aluminum 99.0+ AI. ............... Aluminum 99.0+ AI. ............... AI, 1.2 Mn ....................... ........... AI, 1.2 Mn ... AI, 5.5 Cu, 0.5 Pb, 0.5 Bi. AI, 5.5 Cu, 0.5 Pb, 0.5 Bi. ......... AI, 4 Cu, 0.6 Mn, 0.6 Mg, 0.5 Pb, 0.5 Bi ............. AI, 4.4 Cu, 0.8 Si, 0.8 Mn, 0.4 Mg .. AI, 4.4 Cu, 0.8 Si, 0.8 Mn, 0.4 Mg .. AI, 4 Cu, 0.5 Mg, 0.5 Mn .......... AI, 4 Cu, 0.5 Mg, 0.5 Mn .......... AI, 4 Cu, 2 Ni, 0.5 Mg ............ AI, 4 Cu, 2 Ni, 1.5 Mg ......... AI, 4.5 Cu, 1.5 Mg, 0.6 Mn ........ AI, 4.5 Cu, 1.5 Mg, 0.6 Mn ........ AI, 4.5 Cu, 0.8 Mn, 0.8 Si. ......... AI, 12.5 Si, 1.0 Mg, 0.9 Cu, 0.9 Ni .. AI, 1.0 Si, 0.6 Mg, 0.25 Cr. AI, 2.5 Mg, 0.25 Cr ....... AI, 2.5 Mg, 0.25 Cr ............... AI, 1.3 Mg, 0.7 Si, 0.25 Cr. AI, 1.3 Mg, 0.7 Si, 0.25 Cr ......... AI, 5.2 Mg, 0.1 Mn, 0.1 Cr ......... AI, 5.2 Mg, 0.1 Mn, 0.1 Cr ......... AI, 1.0 Mg, 0.6 Si, 0.25 Cu, 0.25 Cr. AI, 1.0 Mg, 0.6 Si, 0.25 Cu, 0.25 Cr. AI, 5.5 Zn, 2.5 Mg, 1.5 Cu, 0.3 Cr, ~2Mn .................... Al, 5.5 Zn, 2.5 Mg, 1.5 Cu, 0.3 Cr, 0.2 Mn ........................ AI, 6.4 Zn, 2.5 Mg, 1.2 Gu ......... References are on p. 2·-76. 7~-in. round specimen. : ;l1-in. round specimen. 10)::

t

1011 10" 1011 10" 1011 1011 10" 1011

2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65

X X X X X X X X

1011 1011 1011 1011 1011 1011 1011 1011

0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33

4.74XlO' 11.2 X 10' 8.96 X 10' 16.6 X 10' 11.0 X 10' 20.0 X 10' 36.5 X 10' 39.3 X 10'

1.2 10. 3.4 14. 4.1 17 32. 30.

H-t, nat. aged Forged, h-t, aged Sand cast Ann. H-t, nat. aged H-t, art. aged H-t, art. aged H-t, art. aged Ann. Strain hardened (H) Ann. H-t, nat. aged Ann. Hard HI! Ann. H-t, nat. aged

7.10 X 1011 7.31 X 10" 7.31X1011 7.17 X 1011 7.17 X 1011 7.10 X 10" 7.10 X 10" 7.31 X 1011 7.31X1011 7.17 X 1011 7.10 X 1011 7.03 X 10" 7.03 X 1011 7.03 X 1011 6.89 X 1011 6.89 X 1011 7.10 X 1011 7.10 X 1011 6.89 X 1011 6.89 X 1011

2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65

X X X X X X X X X X X X X X X X X X X X

1011 1011 1011 1011 1011 1011 1011 1011 1011 1011 10" 1011 1011 1011 10" 1011 1011 10" 1011 1011

0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33

42.1 18.6 48.3 17.9 42.7 43.4 19.3 18.6 46.9 39.3 38.6 32.4 20.0 28.3 11.0 22.8 29.0 40.0 12.4 24.1

24. 9.6 41. 6.8 27. 32. 16. 7.5 31. 24. 31. 27. 9.6 24. 4.8 13. 13. 33. 5.5 14.

Ann.

7.17 X 1011

2.65 X 1011 0.33 22.8 X 10' 10.

H -t, art. aged Ann. (0.064 sheet)

7.17 X 1011 7.17 X 1011

2.65 X 1011 0.33 56.5 X 10' 49. 2.69 X 1011 0.33 20.7 X 10' 10.

Ann. Cold rolled 75% Ann.

Hard HI! Ann. Hard HI! H-t, then cold-worked H-t, then cold-worked, then art. aged Quenched (h-t) Ann.

H-t, art. aged Ann.

6.89 6.89 6.89 6.89 6.89 6.89 7.10 7.10

X X X X X X X X

X X X X X X X X X X X X X X X X X X X X

10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10'

'\[ 10-mm ball, 500-kg load. § 7:i 6-in. sheet specimen. /I H-strain hardened to a prescribed ha

TABLE

Alloy

99.997 Cu, 0.0016 S .................. 99.996 Cu, 0.002 S, 0.002 Fe .......... 99.950 Cu, 0.043 02, 0.002 Fe, 0.002 S 99.92 Cu, 0.04 02 .................... 99.94 Cu, 0.02 P ..................... 95 Cu, 5 Zn ......................... 95 Cu, 5 Zn ......................... 90 Cu, 10 Zn ........................ 90 Cu, 10 Zn ........................ 85 Cu, 15 Zn ...................... 85Cu,15Zn ........................ 80 Cu, 20 Zn ........................ 80 Cu, 20 Zn ........................ 10 Cu, 30 Zn ........................ 70 Cu, 30 Zn ........................ 70 Cu, 30 Zn ............ 65 Cu, 35 Zn ....................... 65 Cu, 35 Zn ....................... 60 Cu, 40 Zn ............... 89 Cu, 9.25 Zn, 1.75 Pb .............. 64.5 Cu, 35 Zn, 0.5 Pb ............... 67 Cu, 32.5 Zn, 0.5 Pb ...... 64.5 Cu, 34.5 Zn, 1.0 Pb .. 62.5 Cu, 35.75 Zn, 1.75 Pb .. 62.5 Cu, 35 Zn, 2.5 Pb ............... 61.5 Cu, 35.5 Zn, 3 Pb ............... 60 Cu, 39.5 Zn, 0.5 Ph ............... 60.5 Cu, 38.4 Zn, 1.1 Pb ............. 60 Cu, 38 Zn, 2 Pb .................. 57 Cu, 40 Zn, 3 Pb .................. 71 Cu, 28 Zn, 1 Sn .............. 60 Cu, 39.25 Zn, 0.75 Sn .............

2e-9.

ELASTIC AND STRENGTH CONSTANTS FOR COPPER ALLO

Condition

}'-in. rod, cold drawn Ann., %-in. rod

Ann., %-in. rod H.R. (0.040-in. flat) 0.040 in. flat spec. (G.S. 0.050 mm) Rolled strip 0.040 in. (G.S. 0.050 rom) Rolled strip 0.040 in. (spring) Flat, 0.040 in. (spring) Flat, 0.040 in. a. H.R. Flat, 0.040 in. (G.S. 0.050 mm) Flat, 0.040 in. (spring temper) Flat, 0.040 in. (G.S. 0.050 mm) Flat, 0.040 in. (spring temper) Flat, 0.040 in. (G.S. 0.070 mm) Flat, 0.040 in. (spring temper) Flat, 0.040 in. (extra epring temper) Flat, 0.040 in., ann.

Flat, 0.040 in. (spring temper) Flat, 0.040 in., ann. Rod, ann. Flat specimen, ann.

Tubular specimen, ann. Rolled, flat spec., ann. Rolled, flat spec., ann. Rolled, flat spec., anD.

Rod, ann. H.R. I-in. plate Light ann. 1.5-in. OD tubing Extruded I-in. rod Extruded 1-in. section All H.R. (I-in. plate) Ae H.R. (I-in. plate)

E

G

12.77 X 1011 11.2 X 1011 10.9 X 1011 11.7X1011 11.7 X 1011 1l.7XlO11 11.7X1011 11.7 X 1011 11.7 X 1011 11. 7 X 1011 11.7 X 1011 11.7X1011 11.0 X 1011 11.0 X 1011 11.0X1011 11.0 X 1011 10.3 X 1011 10.3 X 1011 10.3X1011 11. 7 X 1011 10.3 X 1011 10.3X1011 10.3XlOll 10.3XlOll 9.65XlO11 9.65 X 1011 10.3 X 1011 10.3 X 1011 10.3 X 1011 9.65 X 10" 10.3X10" 10.3 X 10"

4 A .. GS.X.1~11

< 8.96 X 8.96 X 9.65>< 10.7 X

Sand caSt

10.3

XlQ1i

·. .. .. . . .. .

Sand caSt Sand cast Sand caSt Sand casi Sand cast Sand cast, cooled in sand Sand cast Sand .cast, cooled in sand Sand.cast

X wi . ... . . . . . . .

. .......... . .......... . .........,. . . . . . .. .. . . ..........

1011 1011 1011 101i 101i

10.3

........... '~

........... 11. 7 X 1011 12.4 X 1011 1l.7 X 1011

48.2 X 10' 19.3 X

44.8 X 34.4 X 27.6 X 23.4 X ,26.2 X ........... 46.2 X ........... 51.7 X ........... 51. 7 X ........... 65.5 X

·

........ ... ...........

........... 22.0>< 10' 10.3 X ........... 24.1 X 10' 8:27 X · . . . . . . . . . . 2~. 4>< 10' 8.96 >< ............; 2~.6X10· 9.65 X ...........!.. . ;79.2 X 10' 48.2 X

..........

' ~ At 0.01 % offset. § to-nun ball, 3,OOO-kg load.

10' 10' 10' 10' 10' 10' 103 10' 10'

20.7 X 16.5 X 17.,2 X 10.3 X 11.7 X 22.0 X 18.6 X 24.1 X 31'.0 X

TABLE 2e-10., ELASTIC AND STRENGTH CONSTANTS FOR VARIOUS

Material

Condition

Iridium ......... Ann. Osmium ........ Ann. Rhodium ....... Ann. Ruthenium ...... As cast .Antimony ....... ............................... Beryllium ....... Vacuum cast Cadmium ....... Chill cast I-in. section Calcium ........ Cast slab Chromium ...... As cast Cobalt .......... Cast Columbium ..... Sheet, ann. O.Ol-in. section Columbium ..... Sheet, worked O.Ol-in. section Lithium ........................................ Manganese ...... Molybdenum .... Silicon .......... Sodium ......... Tantalum ....... Tantalum ....... Titanium ....... Titanium ....... Tungsten ....... Zirconium .......

Quenched Pressed + sintered (sheet) Chill cast 3.55 X 0.97 X 0.97 in.

~

52 X 10" 56 X 1011 o

•

•

•

•

•

•

•

•

•

•

•

•

41 X 10 11 7.78 X 10 11 29 X 10" 5.5 X 1011t 2-3 X 1011 ............. 21 X 10 11

G

Tensile strength

·....... . . · . . . . . .. . . .......... .......... ·... . . ... .

. ......... . .........

........

'.'

.......... .......... ......... . .......... . . . . . . . . . . . . . ......... . ............ . ......... . ............. ......... . .............

34 X 1011 11.26 X 1011

.........

. ..........

.......... ............................... ............ . ......... . Ann. O.OIO-in. sheet Worked O.OlO-in. sheet Ann. Hard, 60 % reduction

50 X 10 8

... .. ... .. 1.1 X 10 8 12-15 X 10 8 7.1 X 10 8 5.5 X 10 8 . ......... 23.7 X 10 8 34 X 10 8 69 X 10 8

.. . . . . . . . .

50 X 10 8 69 X 10 8 . ......... . .........

34 X 10 8 76 )< 10 8 11.6 X 1011 .......... 54 X 10 8 ............. ......... . 76.82 X 10 8 34 X 1011 13.5 X 1011 . ......... .............................. . 9.99 X 1011 .......... Hard drawn 84 X 10 8

'* References are on p.

t Sand cast. t 3.2-kg load,

E

2-76.

lO-mm ball. Per cent in 4 in.

.............

.............

......... . ......... .

TABLE

%CI

0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.05 0.07 0.05 0.06 0.054 0.025 0.08 0.07 0.03 0.08 0.08 0.10 0.10 0.10 0.10 0.11 0.12 0.15 0.15 0.16 0.15 0.17 0.18 0.18 0.18

0.20

2e-ll.

Alloy

Iron: 2.50 C, 0.79 Si, 0.09 S, 0.04 P 3.52 C, 2.55 Si, 1.01 Mn, 0.215 P, 0.086 S 3.52 C, 2.55 Si, 1.01 Mn, 0.215 P, 0.086 S 1.11>-2.30 C, 0.81>-1.20 Si, 0040 Mn, 0.020P, 0.012 S 2.25-2.70 C, 0.80-1.10 Si Steel: 0.12 Mn, 0.005 Si, 0045 Cu, 0.07 Mo 0.5 Cu 1.0 Cu 1.5 Cu 2.0 Cu 2.5 Cu 3.0 Cu 0.39 Si, 0.25 Mn,'0.014 P, 0.049 S 1.17 Si, 0.32 Mn, 0.013 P, 0.034 S 1.73 Si, 0.35 Mn, 0.014 P, 0.030 S 2.39 Si, 0.16 Mn, 0.010 P, 0.016 S 0.42 Mn, 0.025 Si, 0.031 AI, 0.265 Ti 0.30 Mn, O.OIOY, 0.023 S, 0.09 Ni, 0.09 Cu, 0.26 V 1.01 Cr, 0.41 Cu, 0.80 Si, 27 Mn, 0.145 P, 0.020 S 18.95 Cr, 7.69 Ni 13047 Cr, 0.27 V, 0.04 P, 0.01 S 1.07 Cu, 0.54 Ni, 0043 Mn, 0.16 Si, 0.104 P, 0.022 S 1.46 Si, 0.102 Mn 0.45 Mn, 3.71 Ni, 0.10 S 0.5 Cr, 0.3 Mo, 2.5 Ni 0.6 Cr, 0.3 Mo, 3.3 Ni 0.Q7 Si, 0.69 Mn, 0.092 P, 0.027 S, 0.16 AI, 1.09 Cu, 0.15 Mo, 0.63 Ni 0.6 Mn, 1.4 Cr, 0.17 Mo, 1.0 Ni 0.84 Mn, 0.12 S, 0.099 P, 0.01 Si 0.75 Mn, 0.30 Si, 1.75 Ni, 0.25 Mo 0.75 Mn, 0.30 Si, 3.50 Ni 004 Mn, 1.2 Cr, 0.25 Mo, 4.1 Ni 13.50 Cr, 0.11 Si 0.5 Mn, 0.25 Mo, 1.8 Ni 0.55 Mn, 0.25 Si 2.50 Cr,O.55 Mn, 0.40 Si, 0.40 Mo, 0.20 V 0.92 Mn, 0.115 P, 0.12 S, 0.02 Si 16.17 Cr, 1.06 Mn, 0.30 Si

ELASTIC AND STRENGTH CONSTANTS FOR IRON AND STEEL

Condition

Cast

~s-in. cast, ann. bar 2-in. bar Malleable, cast, ann.

............................ R.R. at 540°C As normalized As normalized As normalized As normalized All normalized As normalized As rolled, All rolled All rolled All rolled R.R., 5 %~Btrained, aged Annealed R.R. %-in. bar C.R. %-in. bar R.R. 3%-ill. bar H.R. %·in. bar W.Q. from 1830°F A.C. from 1550°F O.Q. from 820°C (carburized) O.Q. from 820°C (carburized) R.R. 4 hr at 540°C W.Q. from 900°C (carburized) IJ.jj-in. diam C.R. bar Cast Cast O.Q. from 780 to 180° O.Q. from 1740°F, T at llWOF P.(O.Q.)(carburized)

E

13.8 X 12.1 X 8.27 X 17.2 X

G

Tensile strength

1011 . . . . . . . . . . . . ..... 32.8XI0' 1011 5.IOXI011 ..... 23.5XI0' 1011 43.4 X 1011 ..... 15.5 X 10' 1011 8.61 X 1011 0.17 39.3 X 10'

17.2 X 1011

8.61 X 1011 0.17

· . . . . . . . . . . . ............ ............ ............ ............ ............ ............ ............ ............ ............ · . . . . . . . . . . . ............ ............ ............ ............ ............ .......... ... ......... '"

... .........

... .......... 20.7 X 1011

34.3 34 39 48 .. 55 ..... 56 ..... 56 40.0 46.5 50.0 52.7 48.9 29.2

8.20 X 1011

..... ..... ..... 20.9XI011 ............ ..... ............ ............ . ... ............ ............ .... ............ ............ ..... ............ ........... .......... .... ·20:9'X' iOii ............ ............ ............ .... ............ ............ ....

'2iXx'ioi,

............ ............ . ........... ............ 20.5XlOll ............ 22.6 X 1011

34.4 X 10'

..... ..... ..... ....

17.2 X 10" ............ 18.2 X 1011 8.54 X 10" 20.5 X 1011 7.92 X 1011

............ ·C;.,;t"· ..................•... · . . . . . . . . . . . ............ C.R. O.Q. from 1740°F, T at 840°F

I " I

.... ..... ................

X X X X X X X X X X X X X

I

Yield strength

. ......... . ......... 25."8 x'io"

22.4 X lO't

10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10'

24.0 X lO't 27 X 10' 31 X 10' 43 X lO' 50 X 10' 52 X lO' 52 X lO' 27.9 X lO' 32.7 X lO' 37.6 X lO' 36.9 X lOs 46.9 X 10' 15.8XlO'

54.0 X lOs

41.3 X 10'11

98.5 X 10' 56.8 X 10' 48.8 X lOs

. . . .. . .. .. 38."7 x·io. .

63.7 X 60.4 X 93.1 X 122 X 52.8 X

10' 10' 10' 10' 10'

47.1 X 10' 34.4 X lO't 78.4 X lO't 108 X lO't 39.2XIO't

85.8 57.4 68.9 68.9 135 90.9 83.8 45 96.5 67.6 130

10' 10' 10' 10' 10' 10' 10' 10' 10' 10' 10'

60.8 52.4 44.8 44.8 120 75.8 66.8 25 82.7 45.5 61.2

X X X X X X X X X X X

X X X X X X X X X X X

lO't 10' 10' lO't lO't 10' 10' lO't 10' 10' 10'

o.19 o.20 o.25 o.25 o.15 o. 15 o. 27 o.27 o.19 o.19

o. 10 o. 10

.30 . 34 .38 .91 .04 .37 .60 .45 .33 .33 .34 .37 .31 .43 .40 .32 .32 .27 . 78

1:35 Mn, 0.10 S W.Q. from 1550°F 0.45 Cr, 1.19 Mn, 0.67 Si, 0.033 P, 0.019 S Rolled 0.45 Mn, 0.40 S, 0.03 Si, 0.012 P Rolled, %-in. plate to 0.35 ............................ to 0.25; 0.3-D.6 Mn, 0.045 P, 0.05 S H.R. to 0.25; 0.3-D.6 Mn, 0.045 P, 0.05 S C.R . 0.72 Mn, 0.21 Si, 0.0248, 0.014 P Wr., ann. at 1450°F, F.C. 0.72 Mn, 0.21 Si, 0.024 S, 0.014 P Wr. W.Q. from 1600°F, T at 1l00°F 0.85 Mn, 0.05 (max) S, 0.045 (max) P H.R. (trans. prop.) H.R. (long. prop.) 0.85 Mn, 0.05 (max)S, 0.045 (max) P 0.75 Mn, 0.20 S, 0.10 P H.R. (trans. prop.) 0.75 Mn, 0.20 S, 0.10 P H.R. (long. prop.) 0.70 Mn. 3.5 Ni Ann. 0.88 Mn, 0.35 Si, 0.035 S, 0.019 P lRolled %-in. plate 0.65 Mn, 0.22 Si Wr., ann. at 1450°F; F.O. 0.38 Mn, 0.16 Si, 0.036 P O.Q. from 1575°F, T at 940°F 0.36 Mn, 0.16 Si, 0.018 S, 0.015 P ~i hr at 1550°F, O.Q. from 120°F, T ~, hr at 800°F 0.50 Cr, 1.14 Mn, 0.84 Si, 0.033 S, 0.021 P n.R. %-in. bar 0.56 Cr, 0.62 Mn, 0.26 Si O.Q. from 1470°F, T at 750°F 1.14 Cr, 0.69 Mn, 0.12 Si Nat 1525°F 0.78 Cr, 0.24 Mo, 0.54 Mn, 0.21,Si, 0.025'p, W., F.C. from 1450°F 0.029 S 0.78 Cr, 0.24 Mo, 0.54 Mn, 0.21 Si, 0.025 P, Wr., O.Q. from 1600°F, T at 0.029 S 1l00°F 0.46 Mn, 21.39 Cr, 10.95 Ni, 3.16 W, 1.39 A.C. from 1740°F Si 1.18 Cr, 0.16 V, 0.71 Mn, 0.33 Si, 0.037 S, Wr., F.C. from 1450°F 0.024 P 1.66 Mn, 0.25 Si, 0.024 S, 0.015 P Wr. F.C. from 1450°F 3.47 Ni, 0.64 Mn, 0.20 Si, 0.023 S, 0.015 P Wr. F.C. from 1450°F 1.65 Ni, 0.99 Cr, 0.51 Mn, 0.20 Si, 0.028 S, Wr., F.C. from 1450°F 0.019 P 1.92 Ni, 0.86 Cr, 0.30 Mo, 0.60 Mn, 0.16 Si, Wr., F.C. from 1450°F 0.019 S, 0.014 P 2.42 Ni, 0.49 Cr, 0.38 Mo, 0.88 Mn, 0.23 Si, Cast ann. at 1575°F, 6-in. bar, 0.13 Cu, 0.04 S, 0.03 P Tat 1200°F 12.69 Mn, 0.12 Si W.Q. from 1830°F O.10Mn Ann. at 1472°F

* References are on p. 2-76. t At yield point. tAt 0.2 % offset. 41 % in 70 mm. § %in8in.

............ ................. 89.6)(10 8t 60.8)(10 20.9X10 11 82.0 X 1011 0.276 59.0 X 108 37.4 X 1 20.4 X 1011 ............ 0.306 43.5X10 8 22.3 X 1 20.3 X 1011 78.5 X 1011 0.297 . ......... ........ 20.53 X 1011 78.06 X 1011 0.313 ....... , .. .. . . .. . 20.12 X 1011 78.20 X 1011 0.286 .......... .... .

.

18.9 X 1011 20.4 X 1011

81.3 X 1011 0.316 46.4 X 10 8 82.7 X 1011 0.310 62.8 X 108

.. ... .. . . .. . ............ ..... ............ . . . . . . .. . . . . .....

42.6X10' 44.1 X 108 43.1 X 108 46.1 X 10' 54.7 X 10' 59.5X10 8 8.06 X 1011 0.287 52.2X108 7.44 X 1011 ..... 155 X 10 8 7.44 X 1011 ..... 163 X 108

............ ............ ..... ............ ............ ..... . . '20.'5 X'io,i ............ ............ 0.291 19.8 X 1011 20.8 X 1011 20.5 X 1011 21 X 21.1 X 21 X 19.7'X

1011 ............ ..... 86.1X108 1011 8.27X1011 , , - 0 , . 164 X 108 1011 ............ ... 83.4 X 108 1011 8.27 X 1011 0.288 52.8 X 108

19.8 X 1011

25.8 X 1 37.9 X 1 22.5 25.0 24.7 27.5 39.3

X1 X 1 X 1 X 1 X 1

28.6 24.1 X 110 99.2 X 1 99.2 X 1 55.6 X

10

6i."7 x'ir 29.3 X 1

8.13 X 1011 0.272 86.8 X 108

62.4 X 1

20.1 X 1011 ............ ..... 88.2 X 108

30.9 X 1

20.3 X 1011

8.13 X 1011 0.289 61.1 X 10 8

33.9 X 1

19.2 X 10" 21 X 10" 19.8 X 10"

8.27X1011 0.295 58.5 X 10 8 8.34X10 11 0.308 65.0 X 108 7.78 X 1011 0.299 61. 9 X 108

29.8 X 1 36.5 X 1 30.2 X 1

19.8 X 10"

7.92 X 1011 0.288 66.2 X 108

34.2 X 1

20.2 X 1011

7.92 X 1011 ..... 81.3 X 10'

67.5 X 1

............ . . . . . . . . . . . . . . .. 102 X 108 ............ ............ ..... 68.2 X 108

II At 0.005 % permanent set. •• At 0.05 % permanent set. tt % in 1.5 in. H % in 3.94 in. 'If'lf % in 1.97 in.

~

53.2 X lO 65.4 )( 1

§§ % in 0.7 At 0.001

1111

*** At 0.1 %

ttt m, in 4

,TABLE

Alloy

99.90 Pb ......................... 99.73 Pb ......................... 99.73 Pb ......................... 0.023-{).033 Ca, 0.02-0.1 Cu, 0.0020.02 Ag ....................... 1 Sb ........................... 4Sb ........................... 6 Sb ........................... dSb ........................... 6Sb ...........................

2e-12.

ELASTIC AND STRENGTH CONSTANTS FOR LEAD AND LEAD

E

Condition

Rolled, aged Sand cast Chill cast

~

* References are on p. 2-76.

t M 6-in. ball, 9.85-kg load for 30 seo.

Yield strength at 0.5% offset

Tensile strength

.......... 1. 77 X 10' 0.95 X 10' 1.38 X 1011 1.1-1.3 X 10' 0.55 X 10' .......... 0.40-0.45 1.4 X 10'

.......... Extruded Extruded and aged 1.38 X 10" Rolled, 95 % reduction .......... Chill cast .......... Extruded .......... Cold rolled, 95 % reduc- .......... tion .......... 8 Sb., ........................... Rolled, 95 % reduction .......... 9Sb ........................... Chill cast .......... 4.5-5.5 Sn ...................... .................... 20 Sn .......................... .................... ........... 50Sn .......................... .................... .......... 2.89 X 1011 4.50-5.50 Sn, 9.25-10.75 Sb ...... Chill cast 4.50 5.50 Sn, 14-16 Sb . . . . . . . . . Chill cast 2.89 X 1011 9.3-10.7 Sn, 14-16 Sb . . . . . . . . . . Cast 2.89 X 10" 0.75-1.25 Sn, 0.3-1.4 As, 14.5-17.5 Sb Chill cast 2.89 X 1011 0.6-1.0 Sn, 1.5-3.0 As, 12.0-13.5 Sb Chill cast 2.89 X 10"

+

f1

.......

I ~lo tio in

2 3 4

2.1XlO' 2.1 X 10' 2.77 X 10' 4i.71XlO' 2.27 X 10' 2.82 X 10'

4 5 4 2 6 4

3.20 5.2 2.3 4.0 4.2 6.9 6.9 7.2 7.1 6.8

3 1 5 1 6

X X X X X X X X X X

10' 10' 10' 1.0 X 10' 10' 2.51 X 10' 10' 3.3 X 10' 10' 10' 10' 10' 10'

TABLE

Alloy

99.9+ Mg ............. , .............. 8.3-9.7 AI, 0.10 Mn, 1.7-2.3 Zn, ~0.3 Si, ~0.05 Cu, ~0.01 Ni, 0.3 other 8.3-9.7 AI, 0.10 Mn, 1.7-2.3 Zn, ~0.3 Si, ~0.05 Cu, ~0.01 Ni, 0.3 other 8.3-9.7 AI, 0.10 Mn, 1.7-2.3 Zn, :;:0.3 Si, :;:0.05 Cu, :;:0.01 Ni, 0.3 other 5.3-6.7 AI, ~0.15 Mn, 2.5-3.5 Zn, :;:0.3 Si, ~0.05 Cu, :;:0.01 Ni, 0.3 other 5.3-6.7 AI, ~0.15 Mn, 2.5-3.5 Zn, :;:0.3 Si, :;:0.05 Cu, :;:0.01 Ni, 0.3 other 5.3-6.7 AI, ~0.15 Mn, 2.5·-3.5 Zn, :;:0.3 Si, :;:0.05 Cu, :;:0.01 Ni, 0.3 other 8.3-9.7 AI, ~0.13 Mn, 0.4-1.0 Zn, :;:0.5 Si, ~0.1O Cu, ~0.01 Ni, 0.3 other 8.3-9.7 AI, ~0.13 Mn, 0.4-1.0 Zn, ~0.5 Si, :;:0.10 Cu, :;:0.01 Ni, 0.3 other 8.3-9.7 AI, ~0.13 Mn, 0.4-1.0 Zn, :;:0.5 Si, :;:0.10 Cu, :;:0.01 Ni, 0.3 other 8.3-9.7 AI, ~0.13 Mn, 0.4-1.0 Zn, ~0.5 Si, 0.10 Cu, :;:0.01 Ni, 0.3 other .......... 8.3-9.7 AI, ~0.10 Mn, 0.4-1.0 Zn, ~0.5 Si, :;:0.3 Cu, ~0.01 Ni, 0.3 other ......... 2.5-3.5 AI, ~0.20 Mn, 0.6-1.4 Zn, 0.080.30 Ca, :;:0.3 Si, ~0.05 Cu, :;:0.005 Fe, :;:0.005 Ni, 0.3 other ................. 2.5-3.5 AI, ~0.20 Mn, 0.6-1.4 Zn, 0.080.30 Ca, ~0.3 Si, :;:0.05 Cu, :;:0.005 Fe, :;:0.005 Ni, 0.3 other. . ..............

2e-13.

ELASTIC AND STRENGTH CONSTANTS FOR MAGNESIUM

Condition

E

........................ 4.48 Sand and permanent cast 4.48 molds, as fabricated Sand and permanent cast 4.48 molds, cast and stabilized Sand and permanent cast, 4.48 solution h-t Sand and permanent cast 4.48 molds, as fabricated Sand and permanent cast 4.48 molds, cast and stabilized Sand and permanent cast 4.48 molds, solution h-t Sand and permanent cast 4.48 molds, as fabricated Sand and permanent cast 4.48 molds, solution h-t Sand and permanent cast, 4.48 solution h-t, aged

G

( 10- 7 6.52 X 10- 6 1. 23 >( 10- 5 4.85 X 10- 5 1. 52 X 10- 5 1.68 X 10- 5 1.79 X 10- 5 1.88 X 10- 5 1.98 X 10- 5 2.24 X 10- 5 2.57 X 10- 5 0.72 X 10- 5 6.5 X 10- 5 0.66 0.16 X 10- 5

1 11 11

sec

2 2 2 1 3 4 1 3 3 3 3 5 5 5 5 5 1 7 7 7 6 6 6 1 9 9 9 9 9 9 1 8 8 8 8 8 8 B

1 1

MECHANICS

TABLE 2e-17. DIFFUSION COEFFICIENTS FOR METALS (Continued) Metal

Test temp.

Room Pt into Cu ............................. . Room Pb into Pb ............................. . Room Sb into Ag ............................. . Si into ferrite ........................... . 1435 ± 5°C Si into Cu ............................. . Room Sn into Ag ............................. . Room Room Sn into Cu ............................. . Room Sn into Pb ............................. . 49.27°C Ti into In .............................. . 74: 19°C Ti into In ............................... . Ti.into In .............................. . 101.55°C Ti into In .............................. . 139. 16°C 155. 60°C Ti into In ...•........................... Ti into In .............................. . 155.91°C Ti into In .............................. . 157.80°C Room Ti intd Pb ............................. . N. B. of H.

D

(Cm2) sec

1.02 X 10- 4 6.6 5.31 X 10-6 1.1 X 10- 7 3.7 X 10- 2 7.82 X 10-6 1.13 3.96 1.4 X 10-12 9.2 X 10-12 4.6-4.8 X 10-11 2.8-3.2 X 10-10 2.17 X 10- 9 1.87 X 10-7 2.27 X 10-6 0.025

The values quoted from ref. 1 are for Do in the equation D = Doe-H1BT •

1

1 1 10 1 1 1 1 9 9 9 9 9 9 9 1

Cf. ref. 1 for value.

References for Table 2e-17 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Ref.

No:wick, A. S.: J. Appl. Phys. 22, 1182 (1951). Slifkin, L., D. Lazarus, and T. Tomizuka: J. Appl. Phys. 23, 1032 (1952). Smithells, C. J.: "Metals Reference Book." Martin, A. B., and F. Asaro: Phys. Rev. 80, 123A (1950). Chemla, Marius: Compt. rend. 234, 2601 (1952). Moore, W. J., and Bernard Selikson: J. Chem. Phys. 19, 1539 (1951). Cohen, G., and G. C. Kuczynski: J. Appl. Phys. 21, 1339L (1950). Hoffman, R. E.: J. Chem. Phys. 20,1567 (1951). Eckert, R. E., and H. G. Drickamer: J. Chem. Phys. 20, 13 (1951). Bradshaw, F. J., G. Hoyle, and K. Speight: Nature 171,488 (1953). Kuczynski, G. C.: J. Appl. Phys. 21. 632 (1950).

ELASTICITY, HARDNESS, AND STRENGTH OF SOLIDS

2-79

7. Rockwell Hardness Number.l itA hardness value indicated on a direct-reading dial when a designated load is imposed on a metallic material in the Rockwell hardness testing machine using a steel ball or a diamond penetrator. The value must be qualified by reference to the load and penetrator used. Severllil scales are in common use: Rockwell A hardness is determined with a minor load of 10 kg and a major load of 60 kg using the diamond cone (brale); Rockwell B hardness is determined with a minor load of 10 kg and a major load of 100 kg using a f-1rin. steel ball; Rockwell C hardness is determined with a minor load of 10 kg and a majorload of 150 kg using the diamond cone"; Rockwell D hardness is determined with a minor load of 10 kg and a major load of 100 kg using a diamond cone indenter; Rockwell E hardness is determined with a minor load of 10 kg and a major load of 1QO kg using a -§-in. steel ball indenter; Rockwell F hardness is determined with a minor load of 10 kg and a major load of 60 kg using a ftin. steel ball; Rockwell G hardness is determined with a minor load of 10 kg and a major load of 150 kg, using a -h-in. steel ball indenter. A second set of Rockwell hardness numbers are the Rockwell superficial hardness numbers. One of these is the Rockwell 15T hardness, which is determined with a minor load of 3 kg and a major load of 15 kg, using a y\-in. steel ball. Note: The methods of determining the hardness values can be found in Standard Methods of Test for Rockwell Hardness and Rockwell Superficial Hardness of Metallic Materials, ASTM E18--42.

8. Brinell Hardness Number.2 itA hard spherical indenter of diameter D mm is pressed into the metal surface under a load W kg and the mean chordal diameter of the resultant indentation measured (d mm). The Brinell hardness number (Bhn) is defined as W Bhn = curved area of indentation 2W .~-~---- 7rD(D - VD2 - d2) and is expressed in kg/min 2." 9. Vickers Hardness Number.2 itA pyramidal diamond indenter is pressed into the surface of a metal under a load of W kg and the mean diagonal of the resultant~ind~ta­ tion measured (d mm). The Vickers hardness number (Vhn), or Vickers diamond hardness (Vdh), is defined as Vdh (or Vhn) =

. W. . pyramldal area of mdentatlOn

The indenter has an angle of 136 0 between opposite faces and 146 0 between opposite edges. From simple geometry, this means that the pyramidal area of the indentation is greater than the projected area of the indentation by the ratio 1:0.9272. Hence Vdh =

0.9272W projected area of indentation = 1.8544W /d 2

The value is expressed in kg/mm 2 ." 10. Diffusion Coefficient. If the concentration (mass of solid per unit volume of solution) at one surface of a layer of liquid is d 1, and at the other surface d 2, the thickness of the layer is h, the area under consideration is A, and the mass of a given substance which diffuses through the cross section A in time t is m, then the diffusion coefficient is defined as

1 2

J. G. Henderson, "Metallurgical Dictionary." D. Tabor, "The Hardness of Metals."

2-80

MECHANICS

2e-4. Effect of High Pressure on the Specific Volume of Solids. Tables 2e-18 to 2e-22 present data on the change of specific volume of certain solids as a result of the imposition of very high pressure. The general reference in this field is P. W. Bridgman, "The Physics of High Pressure," G. Bell & Sons, Ltd, London, 1949. Specific references are attached to each table. TABLE 2e-18. VOLUME OF SOLID HELIUM AT OOK* Pressure, kg/em"

Volume, ml/mole

Compressibility (l/v) (avjap)T

52 91 141 207 305 475 718 1,105 1,715 2,240

19.0 18.0 17.0 16.0 15.0 14.0 13.0 12.0 11.0 10.5

184 X 10- 6 135 100 73 52 37 25 16 12 10

"'J. S. Dugdale and F. E. Simon, Proc. Roy. Soc. (London) 218, 291 (1953);

TABLE 2e-19. FRACTIONAL CHANGE OF VOLUME AT 25°0 OF RELATIVELY INCOMPRESSIBLE METALS* ~

c-'~~

Pressure, kg/em' 5,000 10,000 15,000 20,000 25,000 30,000 >I
-' 0' 2 -5 21 9 3 11 24 19 4 -9 -6 -1 -7 -5 -8 -0 -2 -6 -8 -6 -5 -1 -4 -6 -8 -5 7 12 10 -1 -1 -7 -5 -7 -4 -5 -1 -3 -4 1 -0 -9 -6 13 3 -7 2 -7 -10 -2 -6 -7 2 -4 2 1 0 2 o -8 26 18 13 5 -11 11 2-6 6 3 14-2 -3 -13 -3 -3 2 -7 2 1 -0 -3Q -6 -20 ~5 44 21 7 -2 7 4-1 9 13 6 8 -3 -5 -8 -1 -3 0 -4 1 -5 4-2 8 -12 21 -12 -2 48 -8 -7 -18 4 -2 -8 -6 -1 -10 -16 0 6-10 13 -4 21 27 16 10 -6 -2 -4 -1 5 1 8-4 -3 -2 -0 -6 -4 10 9 4 _30' 1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

if'

-"I

4 -2 -3 -6 -3 -3 -1 -.5 -3 7 7 7 -1 -2 -2 -1 -2 -1 10 8 6 3 4 0 -5 -1 3 84M 2 1 1 -1 1 6 -7 -8 -6 -4 -6 -3 3 4 2 5 6 8 1 -1 -1 8 10 11 -12 -10 -10 -Ie -7 -2

-20 -2

-60'

4 4

8 9

11 9

9 9 3 -0

7 0

8 7

6 8

4 -2 9

10

9

U

8 8

5 8 5

5 10

9

14 U

16

7

9

6

9

11

8

9 4

o 3 1 5

5 5 7

o

29 6

2

13

10

Ie

11

7

28 6

5 2 9

7

15 9

11 -0 10

3

6 -4

14 -1 -5 6 -1 1 2 0

1 - - - - - - - - - - - - - - --------,-----------1 -2 -6 -7 -6 -4 -6 -7 -6 -2

-18 -13 -18 -19 -9

-18 -19 -18 -18 20

-17 -24 -22 -15 20

-15 -28 -28 -19 20

_90' • Rapp, op. cit., p, 27.

-14 -31 -12 -14 -11

-15 -32 -15 -12 20

-16 -2D -8 -10 21

I

-16 -24 -22 -7 20

-13 -16 -16 --5 20

o

1

-10 -8 -7 4 -8 -1 5 -5 13 -19 -2 -22 20 20 -6

2 4 8 11 12 9 6 7 8 -2 5 17 -6 -3 3 -1 7 8 10 11 -6 27 20 12 15 40 1 -9 -18 -11 -12 13 16 11 15 22 -2 -23 -1 17 17 13 7 -9 -6 -4 20 20 20 20 20 -23 -9 19 18 18

14 H W H 31 21 30 1 40 32 21 13 -10 8 18 -10 -22 -9 13 6 -1

2-102

MECHANICS

2h-7. Geodetic Reference System: 1967. In 1967 the General Assembly of the International Union of Geodesy and Geophysics recommended replacing the International Ellipsoid and the International Gravity Formula with the Geodetic Reference System 1967 defined byl a = 6378160m GM = 398603 km 3 /sec 2 J 2 = 10827 X 10- 7 with J 2 = - V5 020 • This set of parameters is identical with the parameters adopted by the International Astronomical Union in 1964 as part of a system of new

+90 0

+80· ..... y .. " ...... ...... '.

• ~ ~9I

. .' . . .... ~. '. ::... . . . . ..... "-1-'-'-'.~.1-''-"'-"-1-'---l---1 /1U.-1Q.,t.;;:;:; ~>I\': .···.l40 .....·)~ ."': ·b.~.~.::~:~ :::.~. ~.·20~0.)."""' ..... / ....... ···20,.:--:- ". 'J-.. \.~..p ';. r. JI': ..... .. .••••••••••• "

. ' "',). . . . .

FIG. 2h-1. Geoid obtained by combining satellite and gravimetric data. Units: meters. (W. Kohnlein, Smithsonian Astrophysical Observatory Special Report 264, p. 57, 1967.)

astronomical constants. The values for a, GM, and J 2 , together with the value for the earth's rotational velocity, define an equipotential ellipsoid of revolution completely, so that the shape of the ellipsoid and its external gravity field are determined by the four constants. Only preliminary numerical values for the shape of the ellipsoid and the gravity formula of the Geodetic Reference System 1967 have been published until now. 2 ,3 Bull. Geod. no. 86, p. 367, Paris, 1967. A. H. Cook, The Polar Flattening and Gravity Formula in the Geodetic Reference System 1967, Geophys. J. 15, p. 431, Oxford, 1968. 3 H. Moritz, "The Geodetic Reference System 1967," Allgem. Vermes8., p. 2, Karlsruhe, 1968. 1

2

2i. Seismological and Related Data B. GUTENBERG l

California Institute of Technology J. E. WHITE

Globe Universal Sciences, Inc.

2i-l.List of

~ymbols ,

V v P

,velocity- of longitudinal wave P velocity of transvers.e wave S symbol denoting longitudinal wave S symbol denoting transverse wave k bulk modulus or volume elasticity p. rigidity or shear modulus p density, T Poisson's ratio A 'ratio t temperature in degrees centigrade, time p pressure in bars h depth in the earth T period of seismic disturbance G symbol denoting surface shear waves Ra symbol denoting Rayleigh waves Ll. epicentral distance SH symbol denoting component of S wave in horizontal plane SV symbol denoting component of S wave in vertical plane i actual IJ.ngle of incidence at a discontinuity, 'i apparent angle of iIlicidence at a discontinuity, , u ratio of horizontal ground displacement to incident amplitu~e

VYv

:::

"

2i-2. Fundamental Equations for Elasti~ Consta~ts and Wave Velocities. In purely elastic, isotropic, homogeneous media; the velocity V of longitudinal waves P, v of transverse waves S, the bulk modulus k, the rigidity p., the 'density p, and Poisson's ratio II are connected by the following equations: p

= k

+ -tp.

v2 = ~

(2i-1)

A=Iv

(2i-2)

p

P II

=

iA2 -1 A' - 1

k = p(P 1

-tv') .

Deceased.

2-103

p.

= v2 p

(2i-3)

2-104

MECHANICS

2i-3. Elastic Constants and Wave Velocities in Rocks (Laboratory Experiments). In rocks the elastic constants and the wave velocities usually increase with increasing pressure p (Tables 2i-2 and 2i-3) and decrease with increasing temperature t and with porosity. Phase changes affect all elastic quantities. Many sedimentary rocks show significant anisotropy, with an axis of symmetry. Table 2i-4 gives an example of velocity differences for vertical and horizontal traveP and for shear polarization. TABLE 2i-1. CORRESPONDING V~LUES OF POISSON'S RATIO (]' AND V Iv (]' 1°.0°1°.10 10.20 10.22

0.241°.251°.261°.281°.3°1°.40 1°·50

Vlv 1.414 1.500 1.633 1.670 1.710 1.732 1.756 1.809 1.871 2.449

co

TABLE 2i-2. ELASTIC CONSTANTS AND WAVE VELOCITIES IN ROCKS AT ROOM TEMPERATUREt

I

-

---.---

k, 1011 dynes/cm 2 1011 dynes/cm2 1-',

(]'

1atm

4,000 atm

4,000 atm

1 atm

-----Dunite ............ 4!--6 Gabbro ............ 3-4 Granite ............ 1-}-2j. Obsidian glass ...... 2i-3 1 1 Ice ................ rlI

6-}-7 4-5 3i-3j. ? ?

? 6± 2i-3j.

3i± : 1

V, v, km/sec kID/sec

---

~

12± si±

5i± 3i-4

?

0.25-0.30 0.2-0.3 0.20-0.26 0.l-o.2? 0.3-0.4

7-}-8j. 5-7

5-6i 5±

3i-3!

4-1-4 3-}-4 2-3j. 3j.± 1-}-2

t F. Birch, ed., Handbook of Physical Constants, Geol. Soc. Am., Spec. Paper 36 (1942); L. H. Adams, Elastic Properties of Materials of the Earth's Crust, in "Internal Constitution of the Earth," 2d ed., pp. 50-80, 1951. See also S. P. Clark, Jr., ed., Handbook of Physical Constants, rev. ed., Geol. Soc. Am., Mem. 97 (1966).

TABLE 2i-3. LONGITUDINAL VELOCITIES, KM/SEC, AT PRESSURES P AND TEMPERATURES t CORRESPONDING TO THE DEPTH h IN THE EARTH AF~~R LABORATORY MEASUREMENTt p,

t,

bars

°C

260 1,300 2,600 3,900 6,700

45 135 225 290 400

h, kIn 1

5 10 15 25

Dunite 7.55 7.50 7.22

.... ....

San Marcos gabbro 6.70 6.90 6.96 6.95 6.S0

Texas gray Woodbury granite granite 5.90 6.02 6.02 6.01

5.90 6.15 6.14 6.04

t D. S. Hughes and C. Maurette, Variation of Elastic Wave Velocities in Basic Igneous Rocks with Pressure and Temperature, Geophysics 22, 23-31 (1957).

2i-4. Periods and Amplitudes of Seismic Waves. Seismological instrumentation great advances in fidelity of observation, geographic distribution of stations, and machine data reduction. Strain seismometers have uniform sensitivity from periods of many hours dowIl to a few seconds. Tilt meters and gravimeters also ~as;llfade

1 J. E. White and R. L. Sengbush, Velocity Measurements in Near-surface Formations, GeophY8ic8 18, 54 (1963). -

2-105

SEISMOLOGICAL AND RELATED DATA

~dicate earth motion down to "dc," i.e., periods much greater than the tidal period. A worldwide net of 125 stations has been established, recording three-component motion in 0.1-to-1-sec range and 10-to-100-sec range. A few array stations exist at which signals from dozens of seismometers in an array can be combined. This improved instrumentation gives an improved portrayal of earthquakes and more accurate knowledge of the structure of the earth. Earthquakes create permanent displacements, which may be observed at great distances. 1 Great earthquakes excite the free oscillations of the earth to measurable amplitudes,2 at periods of 3 to 54 min. Love waves and Rayleigh waves in the period range 10 to 100 sec are governed by velocity contrasts in the crust and mantle. Body waves display periods of 0.1 to 10 sec, depending on range, with shear waves tending to longer periods than compressional waves.

TABLE 2i-4. VELOCITIES

Chalk ........... Shale ............

IN

SHALLOW SEDIMENTS, KM/SEC

V vert.

V horiz.

2.6 1.8

3.0 2.4

Vsv

vert.

1.1 0.4

VSH

horiz.

1.2 0.6

Periods of natural microseisms (continuous motion from meteorological sources and ocean waves) range from a fraction of a second to a minute or more. The largest amplitudes of the most frequent types of micrDseisms (periods 4 to 10 sec) are a few microns at inland stations on rock and between 10 and 100 microns at stations near oceans during heavy storms. Mter great earthquakes, waves through the earth's interior may reach the surface at great distances with amplitudes of over 10 microns and periods of the order of 5 sec, while the largest surface waves may have ground amplitudes of 10 mm with periods of 20 sec. Much greater amplitudes occur near the source. In motion from not too close artificial explosions, longitudinal waves usually carry the largest amplitudes; even waves through the earth's core have been identified on such records.' 2i-5. Travel Times of Earthquake Waves. Examples of travel times are given in Table 2i-5. Surface waves traveling a few times around the earth have travel times of several hours. No dispersion has been established for waves through the earth's body except for waves through the transition zone from the liquid outer core to the probably solid inner core. 4 However, the prevailing increase in the velocity of longitudinal and transverse waves with depth results in a prevailing increase in wave velocity of surface waves as their length (depth of energy penetration) increases. Surface waves of first, second, and third modes have been observed. The group velocity of surface waves of first mode has a minimum. for periods of several seconds, depending on the crustal structure. 1 C. J. Wideman and M. W. Major, Strain Steps Associated with Earthquakes, Bull. Seis. Soc. Am. 67, 1429 (1967). 2 L. E. Alsop, Spheroidal Free Periods of the Earth Observed at Eight Stations around the World, Bull. Seis. Soc. Am. 54,755 (1964). 3 B. Gutenberg, Travel Times of Longitudinal Waves from Surface Foci, Proc. Natl. Acad. Sci. U.S. 39, 849 (1953). 4 B. Gutenberg, Wave Velocities in the Earth's Core, Bull. Seis. Soc. Am. 48, 301-314 (1958). • M. Ewing and F. Press, Crustal Structure and Surface-wave Dispersion, Bull. Seis. Soc, Am. 40, 271-280 (1950); 42, 315-325 (1952); 43, 137-144 (1953). Surface Waves and Guided Waves, "Encyclopedia of Physics," vol. 47, pp. 119-139, Springer-Verlag, Berlin,' 1956.

i

2-106

MECHANICS

2i-6. Reflection· arid Refra'ction of Waves. If a longitudinal wave P or a transverse wave S arrives at a discontinuity, one P and one S wave are reflected and one of each type is refracted if the velocity ratioVrlVi of the reflected or refracted (r) and incident (i) wave permits.' " Vr •• (2i-4) . sm ~r = Vi sm ti where ii is the angle of incidence. Examples are given in Table 2i-6. Amplitudes of transverse waves (vibrations .perpendicular to the ray) are frequently resolved into two components, SH in the horizontal plane, and SV (with a vertical component) perpendicular to SH. If an SH wave is incident, the reflected Wave and the refracted wave (if it exists) are always of the SH type. TABLE

2i-5.

TRAVEL TIMES

AND TRANSVERSE WAVES

t

(MIN: SEC) OF DIRECT LONGIT.UDINAL. WAVES

S

THROUGH THE EARTH STARTING AT DEPTH

AND OF SURFACE SHEAR WAVES PERIODS OF ABOUT

G

AND RAYLEIGH WAVES

h

= 25km

P

S

0 2 4 10 20 40 70 100 120 150 180

0:04 0:32 0:59 2:28 4:34 7:36 11:12 13:46 18:54 19:46 20:10

R a,

G, min

min

0:07 0:55 1:56

... .

....

.. , .

. ...

.....

4.1 8.3 16.5 28.9 41.3 49.5 61.9 74.2

4.5 9.0 17.9 31.4 44.8 53.8 67.2 80.6

8:16 13:42 20:20 25:14 28:00 ••

00.

.0 , 0 '

WITH

1 MIN (INDEPENDENT OF FOCAL DEPTH)

(c. = epicentral distance, deg; P waves arriving at C.

tl

Ra

P h,

....

.0.0

h

> 100 deg enter the earth's core)

= 300km

h

= 700 km

P

S

P

S

0:39 0:46 1:07 2:17 4:15 7:11 10:44 13:15 18:19 19:11 19:35

1:08 1:24 1:51 4:03 7:39' 12:52 19:21 24:23 27:09

1:20 1:24 1:32 2:20 3:55 6:44 10:11 12:37 17:38 18:31 18:54

2:24 2:30 2:48 4:12 7:02 12:01' 18:20 23:14 26:01

. .... .0 ...

t B. Gutenberg, Travel Times of"Longitudinal Waves from Surface Foci, Proc. Nat!. Acad. Sci, U.S. ·39, 849 (1953); H. Jeffreys and K.· E .. Bullen, "Seismological Tables," British Associat,ion for the Advancement of Science, 1940; B. Gutenberg, and C. F. Richter, On Seismic Waves, GeTland. Boilr. Geophy •. 4S, 56-133 (1934); 54,94"':136 ( 1 9 3 9 ) . ' . .

If a wave arrives at the earth's surface (actual angle of incidence i) a wave of the same type is reflected (angle i), and one of the other type may be reflected [Eq. (2i4)] (see Table 2i-7). As a consequence of these three waves, the apparent angle of incidence ~ calculated from records of horizontal H and vertical V instruments (tan ~ = H IV) differs from i. In case of incident 'transverse waves the particles move in ellipses,2 if (V sin i) Iv > 1. If an SH wave is incident, the reflected wave has the same amplitude as the incident wave, the' ground displacement is twice the incident amplitude, and ~ = i.' For energy ratios of waves reflected and refracted at the boundary of the earth's core, see Table 2i-8. An SH wave incident upon the core is totally reflected . . 1 M. Ewing,. W. S. Jardetzky, and F. Press, "Elastic Waves in Layered Media," pp.,74-93, : McGraw-Hill Bo.o.k Co.mpany, New Yo.rk, 1957; B. Gutenberg, Energy Ratio. o.f Reflected and Refracted Seismic Waves, Bull. Seis. Soc. Am. 34,85-102 (1944). 2 B. Gutenberg, SV and SR, Trans. Am. Geophys. Union 33,573-584 (1952).

2-107

SEISMOLOGICAL AND RELATED DATA

TABLE 2i-6. SQUARE ROOT OF ENERGY REFLECTED OR TRANSMITTED AT A DISCONTINUITY WITH DENSITY RATIO (UPPER LAYER TO LOWER) 1.103,CORRESPONDING VELOCITY RATIO 1.286 FOR l! AND FOR S, POISSONJS RATIO 0.25 IN BOTH LAYERS (Incident energy taken as unity. Based on Slichter-Gabriel. t 1- indicates values between 0.95 and 1.0. i = angle of incidence. P = longitudinal, SV= component of transverse wave in plane of ray) Refracted waves Pfrom

SVfrom

Pfrom

SVfrom


--------~------_I-------~-------I-------._-------I------_.-------

iO

Above

P

W

0.0 0.1 0.1 0.2 ... 0.3 ... 0.4 .. . 0".0

11-

0.0 0.1 0.1 0.1 0.2 0.3 0.0

1110.5

0 15 30 45 60 75 90

Below

W

P

t

Reflected waves

10.9 0.9 0.8 0.0

Above

Below

W

P

W

P

Above P

0.0 -0.2 0.0 0.1 0.1 0.1 ... 0.2 0.2 ... 0.4 0.3

1- 0.2 1- 0.2 1- - 0.1 1- 0.2 . .. 1- 0.9 . .. 0.8 0.9 . .. 0.0 1.0

.. . .. . .. . .. . .. . ., .

Below

Above

W

P

W

P

0.0 0.1 0.1 0.0 0.1 0.1 0.0

0.2 0.2 0.1 0.1 0.2 0.4 1.0

0.0 0.1 0.1 0.1 0.1 0.1 0.0

1.0 10.9 0.9

W

Below P

W

0.0 0.0 0.2 0.1 0.1 0.1 0.2 0.1 0.0 0.3 . .. 0.2 . .. .. . . , . 0.3 .. . . .. . .. 0.5 .. . . .. . .. 1.0

B. Gutenl:erg, Bull. Seis. Soc. Am. 34,85 (1944).

TABLE 2i-7. SQUARE ROOTS OF RATIO OF REFLECTED TO INCIDENT ENERGY a AT EARTH's SURFACE "AS "FUNCTION OF ANGLE OF INCIDENCE i AND RATIO OF HORIZONTAL" U AND VERTICAL W GROUND DISPLACEMENTS TO INCIDENT AMPLITUDE FOR CONTINUOUS SINUSOIDAL WAVES IF POISSON'S RATIO Is 0.25; i = APPARENT ANGLE OF INCIDENCE CALCULATED FROM OBSERVED HORIZONTAL AND VERTICAL COMPONENTS (Elliptic motion of ground is indicated by *, and corresponding values for 'i are calculated on the assumption that the vertical and horizontal component reach their maximum simultaneously, t = cO:rnponent of transverse wave in plane of ray)

sv

-----

Longitudinal wave P incident

SVincident

i

a of P

a of SV

0° 20 30 35.3 40 45 60 80 90 tB.

1.0 0.8 0.6 0.5 0.4 0.3 0.0 0.1 1.0

U

w

'i, deg a of P

aof SV

U

- - - - - - -----

----- ------

0.0 0.6 0.8 0.9 0.9 0.9 1.0 1.0 0.0

0.0 2.0 0.8 1.9 1.2 1.7 1.3 1.5 1.4 1.4 1.5 1.3 1.7 1.0 1.3 " 0.5 0.0 0.0

0 23 34 39 44 48 60 69

71

0.0 0.9 1.0 0.0

... ... ... ... ...

w

'i, deg

- - - ----1.0 0.4 0.0 1.0 1.0 1.0 1.0 1.0 1.0

2.0 1.8 1.7 4.9 0.7* 0.0 0.5* 0.3* 0.0*

Gutenberg, SV and SH, T,·ans. Am. Geophys. Union' 33, 573-584 (1952).

0.0 0.8 1.0 0.0 1.6* 1.4 1.1* 0.5* 0.0*

0 23 30 ±O -64* ±90 66* 59* 60:"

2-108

MECHANICS

2i-7. Wave Types and Their Symbols. The main discontinuities of the earth (Fig. 2i-1) are its surface, the "Mohorovicic discontinuity" (depth 10 ± km below the surface in the deeper parts of the major oceans, 30 ± km under the lower parts of continents, up to about 70 km under high mountain ranges, e.g. North Pamir 1 ), and the boundary of the earth's core at a depth of 2,900 ± 10 km (radius r = 3,470 km). The transition from the outer to the inner core is probably gradual. At a distance of about 1,500 km from the earth's center, the velocity of longitudinal waves begins to increase more rapidly with depth than in the outer core but becomes approximately constant about 300 km deeper. This transition zone between the outer and the inner core may correspond to a transition from the liquid to the solid state. TABLE 2i-8. SQUARE ROOTS OF ENERGY RATIOS FOR WAVES REFRACTED (REFR.) AND REFLECTED (REFL.) AT THE BOUNDARY OF THE EARTH's COREt [Assumed at the core boundary: densities 5.4 (mantle), 10.1 (core); longitudinal velocities 13.7 and 8.0 km/sec, respectively; transverse velocity in the mantle 7.25 km/sec, 0 in core. i = angle of incidence of the arriving wave]

P incident in mantle i

Refr. P

Refl. P

Refl. S

P incident in core i

Refr. P

Refr. S

Refl. P

SV incident in mantle i

Refr. P

- ---- --- --- - ---- --- --- - ---0 20 40 60 80 83.8 85 89 90

0.999 0.96 0.87 0.79 0.84 0.85 0.85 0.60 0.00

0.04 0.12 0.29 0.42 0.20 0.00 0.10 0.71 1.00

0.00 0.24 0.39 0.44 0.51 0.52 0.52 0.36 0.00

0 20 33-fr 35 35.7 37 50 80 90

0.999 0.90 0.79 0.83 0.00

..... ..... .....

.....

0.00 0.44 0.62 0.55 0.00 0.85 0.92 0.62 0.00

0.04 0.08 0.00 0.10 1.00 0.53 0.40 0.78 1.00

0 20 30 31 32.0 33 40 64 65.0

0.00 0.50 0.61 0.58 0.00 0.84 0.92 0.55

... .

Refl. P

Refl. S --- --0.00 1.00 0.39 0.78 0.47 0.64 0.49 0.65 0.00 1.00 '" . 0.54 . ... 0.40 .., . 0.84 ... 1.00

.

t After S. Dana, The Partition of Energy among Seismic Waves Reflected and Refracted at the Earth's Core, Bull. Bei8. Soc. Am. 34, 189-197 (1944). By international agreement longitudinal waves in the mantle are indicated by P (starting downward at the source) or p (starting upWard), transverse waves by S or s, longitudinal waves through the outer core by K, through the inner core by I, and (hypothetical) transverse waves through the inner core by J (Fig. 2i-2). Some authors use P' == PKP, P" == PKIKP. For a source below the surface, there is one reflection at the surface near the epicenter, another about halfway between source and station. The symbols for these waves are, respectively, pP and PP, sP and SP, pS and PS, sS and SS. Similarly, for twice-reflected waves pPP, PPP, etc., are used. Time differences pP - P, sP - P, 8S - S, etc., give a good indication for the focal depth (Table 2i-9).2 Among observed waves through the core reflected at the 1 I. P. Kominskaya, G. G. Mikhota, and Yu. V. Tulina, Crustal Structure of the PamirAlai Zone from Seismic Depth-sounding Data, Izvest., Geophys. 8er., trans. by Am. Geophys. Un., 1959, p. 673. . . 2 B. Gutenberg and C. F. Richter, Materials for the Study of Deep-focus Earthquakes, Bull. Seis. 80S. Am. 26, 341-390 (1936); see also H. Jeffreys and K. E. Bullen, "Seismological Tables," p. 24, British Association for the Advancement of Science, 1940.

2-109

SEISMOLOGICAL AND RELATED DATA

surface of the earth are pPKP, sPKP, P'P' == PKPPKP, P'P'P', P'P'P'P' (with a travel time of about Ii hr). Waves in the mantle with a reflection at the core surface permit accurate determination of the radius of the core. They are indicated bye, e.g., PcP, PeS, SeS; pPeP, SeSSeS, etc., are in addition, reflected at the surface. All these waves usually have

-E--I---OUTER CORE

- 300, maximum 720 ± km). Most shocks occur in narrow belts (Table 2i-14).' Deep and intermediate shocks are limited to the circumpacific belt and the trans-Asiatic (Alpide) belt. For the magnitude of the largest observed shock and the relative frequency of earthquakes in various depth intervals, see Table 2i-15, which also shows examples of regional differences. 2i-12. Energy E of Earthquakes. Most calculations of E depend on Eq. (2i-13). This empirical formula is based on many observations, but is subject to adjustment. (h ::; 60 km), intermediate (60 < h ::; 300), and deep (h

TABLE 2i-14. NUMBER OF SHALLOW, INTERMEDIATE) AND DEEP-FOCUS EARTHQUAKES, % OF ALL EARTHQUAKES IN THE GIVEN DEPTH RANGE, AND CORRESPONDING ENERGY RELEASE (a) IN THE MAJOR UNITS OF THE EARTH AND (b) IN SELECTED AREAS (Averages 1904-1957) Number, % Region Shallow

IInt~me .

Energy, %

Deep

Shallow

--- ---

--- ---

(a) Circumpacific belt ............ Trans-Asiatic belt ............. Atlantic and Indian Oceans .... All others .................... Total ...................... (b) Pacific region, Alaska to U.S ... North and Central America, West Coast ................ South America, western part ... Kermadec-Tonga Is ........... New Hebrides and Solomon Is .. Marianas Is .................. Japan-Kamchatka ............ Philippine Is ................. Celebes-Sunda Is ............. Hindu Kush ................. Asia Minor to Italy ........... TotaL .....................

82 10 5 3 100 2 12 10 3 12 2 15 5 8 0 2 71

91 9 0 0 --100 --0

100

::;;>_U

~ ~~

w~

300

~ ::E~

~ ~~

~~

I

11

1 1

I

I

LEVELS OF AIRGLOW EMISSION

400

l

W

t>J

>-3

I

300

ffi

iE

~

T ---.--l--

kgn0

k< 1250(7) A Nz+hv-N+N

200

t>J

o ~ o t< o

~

H Q

~

t< k 0, for anticyclonic curvaturerf < O. Zonal Motion. The average motion of the atmosphere is predominantly geostrophic and zonal. The zonal motion between sea level and 50 mb, for s.ummer and winter, Northern and Southern Hemispheres, is shown in Fig. 2k-4, from Mintz [25]. cb

11m

15

°lgi,

w 20 a::

~40 g: sO

10 w

!5

80 101

!;{z

~~ ~~

it", '"

SL

m p

3l0d 'S

310d 'S

FIG. 2k-4_ Zonal circulation of the atmosphere, m sec-I, averaged over all longitudes. represents motion from the west, E is motion from the east. (Frorn Mintz, ref. 25.)

W

For levels above 35 km, see Murgatroyd in ref. [2]. A pronounced 26-month oscillation of the zonal wind in the equatorial stratosphere has been observed (see, e.g., Reed [29]). Eddy Motion. Superimposed on the average zonal motion of the atmosphere are eddy circulations covering a wide spectrum, including cyclones and anticyclones in the lower troposphere and planetary or Rossby waves in the middle troposphere. Under barotropic conditions frequently observed in the middle troposphere, the speed c of planetary waves is given by (2k-9)

2-142

MECHANICS

where U is the west wind speed, {3 the northward change of the Coriolis parameter, and X the wavelength. For an introduction to current numerical techniques of modeling and predicting atmospheric processes, especially atmospheric motions, see . Thompson [31]. Energy Conversions. Figure 2k-5, from Oort [26], shows an est~mate of the generation G dissipation D, and conversion C rates for energy processes m the atmosphere. In the a~erage, the energy cycle proceeds from mean available potential energy PM via eddy available potential energy PE and eddy kinetic energy KE to the mean kinetic energy (KM). 2k-l0. Radiation. Solar Constan:t. The solar constant, the mean value of the total solar radiation, at normal incidence, outside the atmosphere at the mean solar distance = 0.140 w cm- 2 (p.e. = 2%) [18].

FIG. 2k-5. Tentative flow diagram of the atmospheric energy in the space domain. Values are averages over a year for the Northern Hemisphere. Energy units are in 10& joules m- 2 (= 108 ergs cm- 2); energy transformation· units are in watts m- 2 (= 10 3 ergs cm- 2 sec-I). (From Oort, ref. 26.) ! !

Insolation. Figure 2k-6 shows the average daily solar radiation received on a square centimeter of horizontal surface at the ground during January and July on cloudless days [11] (SOlid lines) and on days with average cloudines~ [13] (dotted lines). The units are gram-calories per square centimeter per day. Albedo. Table 2k-6 gives a range of albedo measurements 1 observed for various type of surface. Heat Balance of the Atmosphere. Taking the incident solar radiation as 100 units, Byers [5] has computed the heat budget of the atmosphere as shown in Table 2k-8. 2k-l1. Clouds. 2 The drop-size spectra of typical cloud types are given in Fig. 2k-7.

i

i

2k·12. Climatology. Space limitations preclude the presentation of climatological I data. In addition to standard climatological texts, see [17], [32], [7], [8], [9], and [35];. the reports of World Data Center A, especially the sub centers on Meteorology, Upper Atmosphere Geophysics, and Rockets and Satellites; and various numbers in the key' to Meteorological Records Documentation series, especially No. 4.11 [10].

For a more complete list, including sources, see List, op. cit., Pl). 442-444. Dat,a furnished by Dr. H. J. aufm Kampe, Si~twJ. OQI"P~ ~~!I;i.:q~ering Laboratories, Ft. Monmouth, N.J. I

2

2-143

METEOROLOGICAL INFORMATION

~

...

,

JANUARY

JULY FIG. 2k-6. Average daily solar insolation (g-cal cm- 2 day-I) at the ground on cloudless days (solid lines) and on days of average cloudiness (dotted lines). (After Fritz and MacDonald [11, 13].)

MECHANICS TABLE

2k-6.

ALBEDO MEASUREME])jTS

Forest .............................................. . Fields, grass, etc ..................................... . Bare ground ......................................... . Snow, fresh ......................................... '.. Snow, old .. ,.' ........................................ '. Whole earth, visible spectrum .......................... . Whole earth, total spectrum, .......................... : " Clouds* ............................................ , . Water (reflectivity values are given in the following tablelt ElevatIon of sun ...... 1900 1700 1500 1400 1300 Reflectivity, %... . . . . 2.0 2.1 2.5 3.4 6.0

I

0

20 13.4

I

% 3-10 3-37 3-30 80-90 45-70 39 35 5-85

50 58.4

I

0° 100.0

*

For clouds in the absenee of absorption the albedo is a function of the drop-size distribution, liquid water content, and cloud thickness. See S. Fritz [12]. t The reflectivity of a wat~r surface for solar radiation is a function of the sun's elevation angle. The values given have been computed for a plane surface; however, the observed reflection from diaturbed surfaces shows only small deviation froID these values.

I

3000

" l"\

\

1000 500

:

'.

\ \...

ft>0:

IU

m

100

:::!:

I~

1\.'\ -.:;;:

--

-

STRATUS STRATOCUMULUS

-

ALTOCUMULUS

---FAIRWEATHER CUMULUS_

~I\

I-

, - CUMULUS CONGESTUS -

CUMULONI \1BUS

,:::I

Z

50

'IU

i,2= '

\

.ii

.... 2: H (')

U2

2-195

VISCOSITY OF LIQUIDS TABLE

2m-3.,

VISCOSITIES OF GLYCEROL-WATER SOLUTIONS*

T-m 't: lp eratur e °C ,

Glycerol, wt. %

0

I

40

20

60

I

I

I

80

I

100

Viscosity, centipoise !

10 50 90 98 99 100

* From

I

2.43 14.6 1,310 7,350 9,390 12,000

,

!

I

1. 31 5.98 218 936 1.150 1,410

0.824 3.09 59.8 194 234 283

I I

0.573 1.85 22.43 59.6 68.9, 81.1

I

1. 25 11.0 24.7 27.7 31.8

0.907 5.98 12.2 1:1.2

I

14.8

J, B. Segur and H. E. Oberstar, Ind. Eng. Chem. 43(9), 2117 (September, 1951). Values from original reduced by 0.3 per cent (to adjust basis used to 1.002 for water at 20°0) and rounded to three significant figures. Original tabulation gives values every 10°C fcir 24 compositions.

Dampler,Lakshminarayanan, Lorenz, and Tomkins.! The authors have examined the data up to December, 1966, for 174 single-salt melts, selected the best data in each c~se, and presented these both as tabulations and empirical viscosity-temperature relations. For each compound a concise statement is given, citing the measurements on which the tabulated values are based, and comparing these with other measurements available. A measure of the precision with which each empirical equation represents the data is given, as is an estimated accuracy of the measurements themselves. , This latter estimate is based on an evaluation of the measurement technique, purity of material, and agreement with other values. 2m-7. Viscosity at Elevated Pressure. Absolute measurements of viscosity at elevated pressure depend on factors like variations in dimensions of the instrument which are often not known with certainty. Most measurements have been made with falling-weight or rolling-lt>all viscometers, calibrated at atmospheric pressure. Such instruments can measure a wide range of viscosities with ,a precision of about one per cent, but uncertainties of several per qent may arise in introducing corrections to the calibration constant owing to increased pressure. With the exception of water between 0 and 33°C, the viscosity of liquids increases monoto'nically with increasing pressure. Typically, the log~rithm of viscosity versuS pressure at constant temperature is concave toward the pressure axis at low pressures, becomes almost linear over an appreciable pressure range, and finally, if the sample does not freeze first, reverses curvature and shows an increasing slope at high pressure. The viscosity, especially at elevated pressure, is remarkably sensitive to molecular structure, in contrast to, equilibrium properties such as density or compressibility which tend to follow a rather uniform pattern at high pressure. Water. The lower-temperature isotherms for the viscosity of water versus pressure show minima. At 2°C the minimum value occurs at about 1,000 bars, where the 1 G. J. Janz, F. W. Dampier, G. R. Lakshminarayanan, P., K. Lorenz, and R. P. T. Tomkins, Molten Salts, vol. I, Electrical Conductance, Densit3c and Viscosity Data, Nat!. Bur. Standards Ref. DataSer. 15, October, 1968. See also G. J. Janz, "Molten Salts Handbook," Academic Press, Inc., New York, 1967. (Less detailed, but includes some mixtures.)

If

i-L ~

TABLE

T,K '7, cp

1060 l.149

1070 l.10.

I

1080 l.071

1090 l.03.

Best equation:

I

1100 l.00.

TJ =

2m-4.

I

1110 0.975

Potassium Chloride 1120 1130 1140 0.94. 0.92. 0.901

I

I

55.5632 - 0.127847T

Temperature range, K: 1056.5-1202.0

T,

KI I I

TJ,CP

780 4.41

790 4.17

800 3.95

I

810 3.75

I

820 3.56

I I 830 3.39

840 3.23

I

1070 l.245

1080 l.210

I

1090 l.17.

I

Best equation:

1100 l.149 TJ

=

I

+ 9.99580

I

1110 l.12,

I

860 2.94

I I 870 2.81

I

1150 0.881

1160 0.86,

I

I

1170 118(} 0.847·.0.83.

8,

880 2.69

Uncertainty estimate, %: 1.5

I

890 2.57

I I I 900 2.47

910 920 2.36 ·2.27

I I 930 2.18

I

1120 l.09.

1130 l.07.

64.3240 - 0.152525T 8,

I

1140 l.05.

I

+ 1.23215 X

centipoise: 0.0040

1150 l.03.

I

1160 l.02 2

I

1170 l.00.

I

950 2.02

I

960 l.95

1O- 4 T2 - 3.34241 X lO-.T'

Uncertainty estimate, %: 1.0

~KI~I~I~I~I~I~'I~I~I~ cp 2.53 2.35 2.19 2.04 l.91 l.79 l.68 l.58 l.49 = 8,

4.00 X 10- 2 exp (4531/RT)

centipoise: 0.0367

o

~o >-
1

o

"'l t"'

H

ID

q

H

t:I

UJ.

• I~(X, - X,), ~ n-p

where Xe and Xe are experimental and calculated values, n is the number of experimental values used, and p is the number of coefficients in "Best equation." t Original tabulation gives values every 10 K. :t In 80me cases, as here, the information available was considered insufficient to warrant a quantitative estimate of uncertainty. It is stated that H • • • the data can be considered reliable."

if

'"-"

CO

-..:t

2-198

MECHANICS

viscosity is about 93 or 94 per cent of its value at one atmosphere. This minimum ratio rises and shifts to lower pressures as the temperature is raised. Between 30 and 40°C the minimum disappears; at higher temperatures the isotherms show the normal monotonic increase with pressure. Between about 2 and 20°C, various measurements disagree by as much as 3 per cent, an amount significantly greater than their precision, though within their possible systematic error. References 1. 2. 3. 4.

Horne, R. A. and D. S. Johnson: J. Phys. Chem. 70(7), 2182 (1966). Bett, K. Eo, and J. B. Cappi: Nature :007, 620 (Aug. 7, 1965). Wonham, J.: Nature 215,1053 (Sept. 2, 19(7). Bruges, E. A., B. Latto, and A. K. Ray: Int. J. Heat Mass Transfer 9,465(1966).

Other Liquids. All other liquids for which measurements are available show a monotonic increase in viscosity with pressure. In Table 2m-5 we list values for a few liquids selected from the measurements by Bridgmanl (falling-weight viscometer) and by several investigators at The Pennsylvania State University' (rolling-ball viscometers). These two collections of data are the most extensive that are available on pure compounds. Bridgman obtained his liquids from various sources. Some were the purest available commercial liquids ; some specially purified by various other workers. The Penn State measurements utilized the API-42 compounds mentioned earlier.' There are nO objective grounds for assigning uncertainties to most of these

TABLE 2m-5A. VISCOSITY OF LIQUIDS UNDER ELEVATED PRESSURE a P

{kg/em', .. (atm) ressure bars* ..... (atm) Temperature, °C

500 490

1,000 980

2,000 1,960

6,000 5,880

4,000 3,920

8,000 7,850

10,000 9,810

12,000 11,770

Viscosity in centipoises n-Pentane: b 7J at 30°C, 1 atm = 0.215 cp, from API-44 tables

I

30 75

0.2151°.3261°.4441°.71911.51 0.139 0.222 0.313 0.516 1.02

30 75

0.5191°.724]°.97511.6314.09110.0 125.9178.0 0.324 0.451 0.602 0.959 2.05 4.08 7.96 16.6

30 75

0.991 11. 27 6.68110.4 2 29 4 10 . 1 1.93 . 1 2. 04 4. 27 0.450 0.59 11.57 0.74 1 1.10

8.86 115.1 78 4. 94 1 4.42 2. 1. 74 1 2.83 6.69

Toluene: 7J at 30°C, 1 atm = 0.5187 cp, from API-44 tables 1 35 . 2

Ethyl alcohol: 7J at 30°C, 1 atm = 0.991 cpc

* 1 bar

= 0.9807 kg/em 2 •

1

16 . 1 5 . 94

I

24.3 8.22

Values rounded to closest 10 bars.

1 P. W. Bridgman, in Proc. Am. Acad. Arts Sci. 61, 57 (1926); "The Physics of High Pressure," G. Bell & Sons, Ltd., London, 1952. 2 D. L. Hogenboom, W. Webb, and J. A. Dixon, J. Chem. Phys. 46(7), 258G (1967); D. A. Lowitz, J. W. Spencer, W. Webb, and R. W. Schiessler, J. Chem. Phys. 30(1), 73 (1959); E. M. Griest, W. Webb, and R. W. Schiessler, J. Chem. Phys. 30(1), 73 (1958); results summarized in "Properties of Hydrocarbons of High Molecular Weight Synthesized by Research Project 42 of the American Petroleum Institute," op. cit. 'R. W. Schiessler and F. C. Whitmore, Ind. EnO. Chem. 47(8), 1660 (August, 1955).

TABLE

Pressure, bars ....................

Compound

~-n-Octylheptadecane· ..

C(-C8),

Perhydrochrysene ......

~ S S

l-a-Decalylpentadecane CIS

®tl l-a-N aphthyl-

pentadecane g-C15 , I.

(atm)

2m-5B.

I

200

VISCOSITY OF LIQUIDS UNDER ELEVATED PRESSUREd

I

400

I

600

Temperature, °C

37.78 60.00 79.44 98.89 115.00 37.78 60.00 79.44 98.89 115.00 135.00 60.00 79.44 98.89 115.00 135.00 60.00 79.44 98.89 115.00 4.35.00

I

1,000

I

1,400

I

1,800

I

2,200

I

2,600

I

3,000

I

3,400

Viscosity in centipoises

7.06 3.91 2.60 1.87 1.48 25.6 10.4 5.86 3.80 2.77 2.09 8.56 5.24 3.55 2.64 1.99 8.41 5.05 3.41 2.52 1.90

9.40 5.13 3.35 2.37 1.85 46.8 15.6 8.36 5.15 3.78 2.74 11.8 7.18 4.76 3.57 2.64 10.9 6.49 4.24 3.15 2.36

12.5 6.65 4.26 2.96 2.31 87.5 24.5 12.1 7.05 5.07 3.56 16.1 9.46 6.19 4.63 3.37 14.2 8.18 5.22 3.90 2.89

16.2 8.45 5.34 3.65 2.83 177 40.1 17.9 9.73 6.73 4.60 21.4 12.2 7.89 5.85 4.17 18.2 10.3 6.43 4.75 3.47

26.3 13.1 8.07 5.38 4.17 951 121 41.6 19.1 12.2 7.67 36.6 19.8 12.2 8.75 6.15 29.6 15.8 9.48 6.84 4.83

41.3 63.0 19.5 28.1 11.6 16.2 7.58 10.3 5.80 7.80 9,080 464 2,510 110 345 41.2 95.9 23.0 46.1 13.1 23.5

92.3 39.7 22.6 13.7 10.2

131 55.0 30.7 18.2 13.2

1,450 255 102 43.6

8,650 832 252 87.5

;:1

187 75.5 41.2 23.7 16.7

Ul

o

o

Ul

~

"'1

o

I:z;j

3,670 778 194

2,720 488

E q 8

31.0 18.3 12.8 8.75

47.6 27.0 18.3 12.1

71.8 39.4 25.9 16.7

56.0 36.3 22.6

79.2 49.9 30.3

111 67.9 40.3

23.9 13.7 9.59 6.64

35.3 19.5 13.2 8.87

27.4 18.1 11.7

38.1 24.7 15.3

33.0 20.0

43.8 25.9

Ul

-

If

I-'

c:o c:o

t:Y

~

TABLE

2m-50.

Pressure, bars ..............................

(atm)

400

800

1

Compound

n-Dodecane .. " ................. n-C12

n-Pentadecane. . . . . . .. , ......... n-C15

cis-Decahydronaphthalene ........

~ trans-Decahydronaphthalene ......

~

1,200 1

Temperature, °C

37.78 60.00 79.44 98.89 115.00 135.00 37.78 60.00 79.44 98.89 115.00 135.00 15.56 37.78 60.00 79.44 98.89 15.56 37.78 60.00 79.44 98.89 115.00

o o

VISCOSITY OF LIQUIDS UNDER ELEVATED PRESSURE'

1,600

1

2,400

2,000

1

2,800

I·

3,600

3,200

1

1

1

Viscosity in centipoises

1.102" 0.8026" 0.63 0.5156" 0.41 0.34 1.953" 1.335 1.01 0.7960" 0.67 0.54 3.71 2.310" 1.569" 1.17 0.9162" 2.30 1.546" 1.114" 0.86 0.6960" 0.59

1. 70 1. 23 0.98 0.80 0.69 0.58 3.20 2.10 1.56 1. 24 1.06 0.87 6.59 3.76 2.57 1. 93 1.50 3.76 2.45 1. 76 1. 35 1.08 0.90

2.50 1. 75 1.36 1.11 0.97 0.82 4.85 3.11 2.27 1. 75 1.48 1. 22 10.7 5.81 3.95 2.84 2.19 5.84 3.70 2.58 1. 94 1.53 1. 30

3.52 2.39 1.82 1.46 1. 26 1.07 7.00 4.37 3.14 2.37 1. 98 1. 60 17.3 8.81 5.89 4.05 3.06 8.81 5.38 3.63 2.70 2.12 1. 78

4.78 3.19 2.37 1. 87 1.59 1.34

6.35 4.16 3 05 2.36 1. 99 1. 64

5.29 3.86 2.95 2.45 1. 98

5.93 4.19 3.13 2.58 2.05 27.9 13.3 8.60 5.69 4.17 13.2 7.73 5.06 3.70 2.84 2.36

7.89 5.47 4.02 3.27 2.59 45.3 20.2 12.5 7.93 5.67

10.43 7.02 5.09 4.09 3.22 73.0 30.3 18.2 10.9 7.59

45.5 26.6 15.1 10.2

68.7 38.7 20.9 13.6

11.0 6.89 4.98 3.75 3.09

15.5 9.38 6.58 4.90 3.99

12.7 8.64 6.31 5.07

17.3 11.4 8.11 6.44

6.67 4.47 3.62 2.98 2.39

8.41 5.83 4.38 3.59 2.86

8.90 6.33 5.05 3.93

11.16 7.78 6.20 4.73

7.13 5.23 4.26 3.36

is: t:J

()

::c:

po..

!Z H ()

U2

106 56.5 28.8 18.1

15.0 10.4 8.15

Footnotes to Tables 2m-GA, 2m-5B, 2m-5C a P. W. Bridgman, Proc. Am. Acad. Arts Sci. 61, 57 (1926); "The Physics of High Pressure," G. Bell & Sons, Ltd., London, 1952' G. E. Babb and G. J. Scott [J. Chem. Phys. 40,3666 (1964)] report results which deviate by 4 per cent or less up to 8,000 bars. e T. Titani, as quoted in J. Timmermans, "Physico-chemical Constants of Pure Organic Compounds," vol. 1 American Elsevier Publishing Company, Inc., New York, 1950. d D. A. Lowitz, J. W. Spencer, W. Webb, and R. W. Schiessler, J. Chem. Phys. 30(1),73 (1959). Original also includes data for 7-n ..hexyltridecane, 11-n-decylheneicosane, 13-n-dodecylhexacosane, 1, 1-dipheny lethane, 1, I-diphenylheptane, 1, I-diphenyltetradecane, 9 (2-cYclohexylethyl)heptadecane, 9 (2-phenylethyl) heptadecane, 1,2,3,4,5,6,7,8,13,14,15, 16-dodecahydrochrysene, 1, I-di (a-decaly I) hendecane. e Confirmed measurements of E. M. Griest, W. Webb, and R. W. Schiessler. J. Chem. Phys. 29(4), 711 (1958). Original also includes data for I-phenyl-3 (2-pheny lethyl) hendecane; 1-cyclohexyl-3 (2-cyclohexylethyl) hendecane, 9 (3-cyclopentylpropyl) heptadecane, 1-cyclopentylpropyl)heptadecane, 1·-cyc!opentyl-4(3-cyclopentylpropyl) dodecane, 1,7 -dicyclopentyl-4(3-cyclopentylpropyl) heptane, 9-n-octyl(1 ,2,3 ,4-tetrahydro)naphthacene. Original also includes data for spiro(4,5)decane, spiro(5,5)f D. L. Hogenboom, W. Webb, and J. A. Dixon, J. Chem. Phys. 46(7), 2586 (1967). undecane, cis-octahydroindene, and trans-octahydroindene. g Obtained with Cannon-Fenske capillary viscometers by American Petroleum Institute Research Project 42. "Properties of Hydrocarbons of High Molecular Weight Synthesized by Research Project 42 of the American Petroleum Institute," op. cit., ineludes smoothed data from references d, e, and f for pressures in psi and temperatures in OF. b

;:1 [J2

C':l

o

[J2

>-


2.0

w

Covilolion Incepr,on

1

ll..

G~

:

>

]

Q{:165

"

0.0

0

CD

::

Tempcrolure; 72Q Dc' C:X, ~ 1.2

ID

1&'·

n.

n.

/.0

(J)

1

-'1)

Temperolure. 76°

ex

..

Ie

OI's.Q,96

0 00

(J)

0

W 0::

n. -2.0

p

0

0::

0

n. -3.0 -4.0

~

z

~

w -5.0 0:: => (J)

0.:

(JI

rt

(J)

u

0

~

0 v

(b

(b

0

IV ()

(

t> f'

(J)

w -6.0

I () C:II!~ 9

0::

n.

...J

\

-7.0

w i:':J w

Note. When the last significant digit is shown in boldface type, the conversion factor represents a conventional factor which is accurate by definition and involves no approximation.

~

CI:l CJ.;J

2-234

MECHANICS

2r-1. Definitions. The viscosity of a fluid is defined in relation to a macroscopic system which is assumed to possess the properties of a continuum. To obtain an elementary definition of viscosity (Fig. 2r-l) consider two infinite flat plates, a at rest and b moving at a constant velocity u, the space between them being filled with the fluid under y consideration. In the resulting shear flow the velocity distribution is linear with a constant transverse gradient /I du/dy. It is assumed (Newton's law of fluid friction) that the shearing stress TO at either wall is proportional to the velocity gradient

-

du

TO

a

(2r-l)

= p, dy

FIG. 2r-1. Illustration of Newton's law of fluid friction.

The coefficient of proportionality p, is known as the viscosity, or more precisely, as the dynamic or absolute viscosity of the fluid. The various units of viscosity and their conversion factors are given in Table 2r-1. The ratio v

= !':

(2r-2)

p

is known as the kinematic viscosity; the respective units and conversion factors are given in Table 2r-2.

TABLE 2r-2. KINEMATIC VISCOSITY v; UNITS AND CONVERSION FACTORS Units

m 2/sec

m'/hr

cm 2/sec (stokes)

ft'/hr

ft'/sec

m 2/sec ............ . 1 3,600 1 X 10' 10.7639 299.0 X 10- 5 m'/hr .............. 277.8 X 10-' 2.778 1 cm'/sec (stokes) .... 1 X 10-' 1 10.7639 X 10-' 0.36 ft'/sec ....... ...... 929.03 0.092903 334.45 1 ft2/hr .............. 25.806 X 10- 6 0.092903 0.25806 277.8 X 10- 6

From British Standard Code B.S. 1042: 1943 amended March, 1946.

3.875 X 10' 10.7639 3.875 3,600 1

See Note to Table 2r-1.

In a general field of flow, Ul, U2, u, of a homogeneous Newtonian incompressible fluid, the shearing stresses are proportional to the respective rates of change of strain (Stokes'law). The symmetric stress tensor tii is assumed to be a linear function of the rate of strain tensor eii. Taking into account that in a fluid at rest the stress is an isotropic tensor, we put

where Oii is the Kronecker symbol (0 = 1 for i = j and a = 0 for i trary. Since tii = 0 for ei; = 0, we have tii = -3p and 3" hypothesis). Consequently

~

j) and p is arbi= 0 (Stokes'

+ 2p,

(2r-3)

VISCOSITY OF GASES

2-235

where now p denotes the hydrostatic pressure. The scalar I-' is defined as the absolute viscosity of the fluid. The viscosity is assumed to be a function of the thermodynamic state of the fluid and independent of the velocity field. For a homogeneous fluid I-' is a function of two properties. It is customary to use either of the following two alternative representations: or I-' = l-'(p,T) I-' = l-'(p,T) (2r-4) where T is the absolute temperature, p is the pressure, and p is the density of the fluid. Numerical values of viscosity cannot be calculated with the aid of the equations of thermodynamics. They must be measured directly, the measurement being usually very difficult, particularly at higher pressures and temperatures. In principle, values of viscosity can be calculated by the methods of the kinetic theory of gases and statistical mechanics with quantum corrections where necessary. In relation to a microscopically defined system the viscosity of a gas is assumed to be due to a transfer of momentum effected by molecules, their velocity being composed of the molecular (random) velocity and the macroscopic (ordered) velocity. In shear How (Fig. 2r-2), the shearing stress acting on a small element of area aa is equal to the integral of the change in momentum effected by the particles moving across, both from above and from below it, the integral extending over all --------~~~------~x particles crossing. 2r-2. Variation of Viscosity with Temperature and Pressure. The calculation of the viscosity of gases has so far met with only limited success, extensive experimental determinations still forming the basis for practical applications. The calculation of the viscosity of gases must make FIG. 2r-2. Kinetic interpretation use of a molecular model for the gas, increasing of viscosity. refinements being possible. On the simplest assumption of infinitely small, perfectly elastic molecules with zero fields of force (Maxwell) it is found that the absolute viscosity of a gas is independent of pressure and that it increases in proportion to Tt:

(au) ap

= T

0

(2r-5)

p = const

where Kl and K2 are empirical constants. On the assumption of hard elastic spheres with a weak attraction force (Sutherland), it is found that KT! 1 1-'=-(2r-6) 7=lji C +1' where K and C are empirical constants. Sutherland's equation (2r-6), as well as experimental results, show the increase with temperature to be faster than that in Maxwell's equation (2r-5). The fact that the viscosity of a gas increases with temperature can be understood if it is realized that in gases the effects of molecular motion dominate over those due to intermolecular forces. In liquids cohesion forces are more important, and since the molecular bonds in a liquid are loosened as the temperature is increased, the absolute viscosity of a liquid decreases with temperature; that for a gas increases with

2-236

MECHANICS

temperature. Sutherland's equation (21'-6) is inadequate for the correlation of experimental data over large temperature intervals. In problems of compressible fluid flow it is customary to use the empirical relation /10

;;; =

(T)'" To

(21'-7)

where /100 is the value of /10 at a reference temperature To and w is an empirical constant ranging from 0.6 to 1.5. This correlation is less precise than those given later. All preceding formulas relate to gases at low pressures (say atmospheric). Experimental results (which are still very scarce) show that the viscosity of gases at constant temperature increases with pressure, the increase being of the order of 20 to 40 per cent per 1,000 atm. For moderate pressure ranges it is possible to use a linear interpolation formula (21'-8) where /100 is the viscosity at temperature T, but at zero density, and k is an empirical constant. More precisely;, the viscosity of a gas increases as its density is increased. Since the viscosity of a gas consisting of molecules which exert no forces upon one another (Maxwell) is independent of density, this behavior is talcen as evidence of the existence of intermolecular fields of forces. However, exceptions exist to this rule, notably steam and hydrocarbons, whose viscosity at constant temperature dec/'eases with pressure, and therefore density, in certain ranges of states. In turn this is taken as evidence of the existence of some form of molecular association whose precise nature is not understood. 2r-3. Variation of Viscosity with Temperature and Pressure According to Kinetic Theory. There: exists a rigorous kinetic theory of the equilibrium and transport properties of gases which is based on Boltzmann's equation. Thus, in particular, and in principle, the viscosity, thermal conductivity (see Sec. 4g) and virial coefficients of gases (see Sec. 4i) are calculated in a consistent and unified way. This theory is due to Chapman and Enskog (S. Chapman and T. G. Cowling, "Mathematical Theory of Non-uniform Gases," Cambridge University Press, New York, 1970; J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids," John Wiley & Sons, Inc., New York, 1964.) The calculations are made on the basis of assumed semiempirical force potentials. For nonpolar gases the most widely used potentials have been the Lennard-Jones twelve-six potential and the modified Buckingham exp-six potential; that used for polar gases is the Stockmayer potential. The viscosity at zero density is then calculated from the equation

VmkT fl'(n) (T*) 161ff u 2{}(2,2)*(T*)

5 /100 =

. (21'-9)

or, with the values of the universal constants substituted

!!'..)t fl'(n) (T*)

•

/100.

mlCropOlse

=

M . ( 26.694 gig-mole K

(cr2/.A2) {}(2,2)*(T*)

(21'-10)

Here cr is the molecular distance at which the potential vanishes, M is the molecular weight, k is Boltzmann's constant, and T* = kT IE is a dimensionless temperature with E denoting the depth of the potential well. The collision integral {}(2,2)* and the factor fl'(n), both of which are unique functions of the dimensionless temperature T*, are given in terms of the intermolecular force potential and must be tabulated

2-237

VISCOSITY OF GASES

for each one of them separately. Such tabulations for the more general m - 6 potential can be found in "Tables of Collision Integrals for the (m - 6) Potential for Ten Values of m" by M. Klein and F. J. Smith (Arnold Engineering Development Center Rept.) AEDC-TR-68-!J2, May, 1968, Arnold Air Force Station, Tenn.), with m taking the values m = 9, 12, 15, 18, 21, 24, 30, 40, 50, and 75. Tables for the exp-six potential can be found in "Transport Properties of Gases Obeying a Modified Buckingham (Exp-Six) Potential" by E. A. Mason [J. CAem. Phys. 22, 169 (1954)J. The factor j,,(n) with n = 1, 2, . . . represents successive approximations and it is usual to confine it to the third approximation, j/3), at most. In principle, the form of and the constants in a potential can be determined by quantum mechanics from a knowledge of the structure of the molecule. However, the attendant mathematical difficulties preclude us from doing so, and potentials must be determined by fitting experimental data on a variety of properties to expressions like the one in Eq. (2r-9). The efforts to associate definite potentials and physically meaningful constants with even the simplest molecules have not yet met with complete success. One of the difficulties is connected with the fact that often several alternative potentials give equally good fits to a set of experimental data of a definite property of a gas, but none seems to reproduce all properties to within the experimental error. Thus, there exists no preferred or universal form of the potential, but, as a matter of experience, it can be stated that the viscosity of the simpler gases, except that of helium, is reproduced reasonably well by the potential family

me

E(r) = m - 6

[m]S/(m-S) 6

[(,,)m r - (,,)sJ r

(2r-ll)

in which IT, e, and m are treated as adjustable constants. The viscosity of helium is best reproduced by the exp-six potential with rm = 3.135 A, elk = 9.16 K, and a = 12.4 [E. A. Mason and W. E. Rice, J. Chem. Phys. 22, 522, 843 (1954)J. Average, and to a certain extent preliminary, values of" and € for the Lennard-Jones potential are listed in Table 2r-3. A better representation is obtained with the aid of the 1.5 I I semiempirical formula '.\

Fp.

= 1.0 _

1~0

\

+ o~~

(2r-12)

where

Fp.

0.5

S2p,

=

-

26.694 VMT

a

(2r-12a)

and () =

1.0.

/

'1/ ,

1/

0.,2

gT

A ,

0..4

(2r-12b)

'-

--- -----

0..6 I/T*

0.,8

----

1.0

12

FIG. 2r-3. Dimensionless second virial The optimum values of the constants a, g, coefficient for viscosity b* as a function of and s are listed in Table 2r-4 for several reduced inverse temperature, according gases. to Kim and Ross. [J. Chem. PhY8. 42, Except for the neighborhood of the 263 (1965) J. critical point, the effect of density (i.e., pressure) on the viscosity of gases, even up to pressures of the order of several hundred atmospheres, can be accounted for with the aid of the virial expansion

p,(p,T)

=

p,o(T)

+ b(T)p + C(T)p2 + ...

(2r-13)

containing three or four terms. Kim and Ross [J. Chem. Phys. 42, 263 (1965)J provided a theory for the virial beT). The diagram in Fig. 2r-3 represents the universally valid relation between b*

=

(T*)-lQrei(T*)Bo o(T*)/fI,(2,2)*(T*)

(2r-14)

2-238

MECHANICS

and l/T*.

For the Lennard-Jones model, the expression reduces to

(1) (~)t ( M )t fT E/k gig-mole

b* _ _ 1 (_b ) - 15.20 g/cm'

(2r-15)

In the range where l/T* exceeds 0.2 (T* < 5 approximately), the virial coefficient b is nearly.a constant with b* "" 1. Consequently, Eq. (2r~14) can be simplified.to J,L(p,T) - J,Lo(O,T) =

15.20~ em'

(i) (Eir,kr (g/g~lOler + O(p2)

(2r~16)

This form leads to an approximate equation for the excess viscosity J,L(p,T)- J,Lo(O,T) which has often been used for correlations. This form is (2r-17)

J,L(p,T) - J,Lo(O,T) "" f(p)

in which f(p) is a unique '(empirical) function for each gas. TABLE

2r-3.

MOLECULAR-FORCE CONSTANTS FOR THE

LENNARD-JONES

E(r)

=

4E

(12-6)

[(n

12

-

POTENTIAL

(n

6

]

Symbol

Gas Acetylene .................................... .

Air .......................................... . Argon ....................................... .

Ar

Bromine ..................................... . Carbon dioxide ............................... . Carbon monoxide ............. ' ......•......... Chlorine...... . ....................... . Deuterium ................................... . Ethylene .................................... .

Br. CO. CO

Helium ...................................... .

He

Hydrogen .................................... Iodine .............. , ........................ Krypton ...................................... Methane .....................................

. . . .

H.

Neon ........................................ .

Ne

Nitric oxide .................................. .

NO

C1,

D. C.H.

I. Kr CH.

Nitrogen .................................... .

N,

Oxygen ...................................... . Propane ..................................... . Xenon ....................................... .

0, C,H. Xe

Elk, K

fT,!

Ref.

185 { 84.0} 117.5 { 124.0} 152.8 520 261.1 110.3 257 39.3 205 { 1O.22} 86.20 38.0 550 206.4 144 { 35.7} 60.9 119 { 91. 5} 113.5 113 254 229

4.221 { 3.689} 3.512 { 3.418 } 3.292 4.268 3.705 3.590 4.40 2.948 4.232 { 2. 576} 2.158 2.915 4.982 3.522 3.796 { 2.789} 2.648 3.470 { 3.68l } 3.566 3.433 5.061 4.055

1 1

2 1

2 1

2 3 1 1 1 1 2 1 1 2 1 1

2 1 1

2 1 1 1

Not. 1. Differences in the values in this table and the table in Sec. 4i are a measure of the uncertainties which still exist, as well as of the fact that the best fits to experimental values of virial coefficients and viscosity .are obtained with slightly different values of the constants. Not. 2. In the case of helium the best form of potential function is.that of the modified Buckingham exponential-six with parameters as quoted in the text. Consequently, the values of the parameters shown in the table may not be physically meaningful, especially in tb,e case of those quoted from ref. 2.

References for Table 2r-3 1. Hirschfelder, J. 0., C. F. Curtiss, and R. B. Bird: "Molecular Theory of Gases and Liquids," Table I-A, p. 1110, John Wiley & Sons, ·Inc., New York, corrected edition,

1964. 2. DiPippo, R., and J. Kestin: Viscosity of Seven Gases up to 500°C and Its Statistical Interpretation, Proc. 4th Symp. on Thermophys. Properties, ASME, New York, 1968. 3. Nat!. Bur. Standards Cire. 564, 1955.

2-239

VISCOSITY OF GASES

TilLE 2r-4. PARAMETERS IN VISCOSITY CORRELATION, EQ. (2r-12) Gas

Symbol

Air ...................... Argon ................... Butane .................. Carbon dioxide ........... Ethane .................. Ethylene ................. Helium .................. Krypton ................. Methane ................. Neon .................... Nitrogen .................

Ar C.HIO CO. C,Hs C.H. He Kr CH. Ne N.

....

.

I' g X 10',

a

. (K)-I

1.3034 1.0300 0.91040 0.94147 0.92669 0.71342 1.5779 0.83447 1.0532 1.6602 1. 3127

6.0906 7.5793 5.5145 5.3316 6.2093 3.3598 4.0302 8.4746 5.2434 6.6667 6.2232

8,

A

3.484 2.970 4.730 3.230 3.820 2.235 2.250 2.935 3.208 2.895 3.548

Temp. range, K

298-773 298-573 311-511 298-773 294-511 303-368 298-673 298-473 283-411 298-453 298-773

Unpublisbed correlation prepared by authors of this article.

2r-4. Viscosity in the Neighborhood of the Critical Point. Contrary to earlier views, it has now become accepted that the viscosity of a gas does not increase anomalously in the neighborhood of the critical point, even though the representation in the form of Eq. (2r-13) breaks down there. The viscosity in the neighborhood of the critical point has been measured (rather sketchily) for a very small number of substances only. A qualitative idea of the resulting behavior can be obtained from the diagram for CO 2, given as Fig. 2r-4 [J. Kestin, J. H. Whitelaw and T. F. Zien, Physica SO, 161 (1964)]. 2r-6. Law of Corresponding States. Attempts have also been made to correlate the viscosity of gases with the aid of the law of corresponding states. The most promising correlation [J. M. J. Coremans and J. J. M. Beenakker, Physica 26, 653 (1960)] makes use of molecular constants for the formation of reduced variables. The reference temperature is chosen as T* = kT IE, the reference density being chosen as the fraction of volume occupied by the molecular core p* = ¥ ...n(-~o-)3· where n is the number density. The viscosity p. is referred to iJ.O measured at zero density, so that P.r = p.1P.o and (2r-18) P.r = J(T*,p*) where f is an approximately universal function. series P.r

= 1

+

0.55p*

It can be represented by the power

+ 0.96p*2 + 0.61p*3 T*0.69

(2r-19)

from which it is seen that the relative excess viscosity !ir - 1 is a unique function of relative density p* at constant relative temperature T* according to Eq. (2r-17). Equation (2r-18) reproduces the experimental values for nonpolar or only slightly polar gases, with an error of the order of ±3 per cent over a fairly large range of· temperatures and densities. The error is negligible up to densities of approximately 200 amagat units, and the equation can be used up to about 500 amagat units. 2r-6. Mixtures of Gases. The viscosity of a gaseous mixture cannot be deduced from the knowledge of its composition and of the viscosities of its components by macroscopic methods, and methods of statistical mechanics must be used. In any case it should be noted that the viscosity of a mixture is not equal to the weighted mean of the viscosity of its components, it being possible for the viscosity of a mixture to be higher than that of its components. For example, a mixture of argon (P.Ar = 222 X 10-6 poise) and helium (P.He = 195 X lO- s poise) containing 40 per cent He

2-240

MECHANICS 450~----~------~------~-----r------'------'

400~----~----~~----~----~------~~~~

'"~

.~ 350 e-----~-----+---_+---+--+_I+_--__l

e

.~

:i.

>iii

I-

§ 300 :; o

~

-

TABLE 2r-6. ABSOLUTE VISCOSITY p. OF GASES IN MWROPOISES (10- 6 glom sec = 10- 6 dyne sec/cm2; at 20°C and 1 atm)

I>:)

Gas

Symbol

p., p.poises

Estimated Temp. uncertainty increment ±Llp.,

p.poises Acetylene ............ Air ................. Ammonia ............ Argon ............... Bromine ............. iso-Butane ........... n-Butane ............

C2H 2 ......

NH. Ar Br2 C 4H 1o C 4H 1o

93.5 (at O°C) 181.92 97.4 222.86 149.5 74.8 84.8

Carbon dioxide ....... CO2 Carbon monoxide ..... CO Chlorine ............. Clz

146.63 175.3 133.0

Chloroform .......... Cyanogen ............ Deuterium ........... Deuteromethane ...... Ethane .............. Ethylene ......•.....

CHCla C 2N 2 D2 CD 4 C2H 6 C 2H 4

100.0 100.2 124.68 129.0 91.0 100.0

Helium .............. Hydrogen ............ Hydrogen bromide .... Hydrogen chloride .... Hydrogen deuteride ...

He H2 HBr HCI HD

196.14 88.73 184.3 142.5 111.8

- -

--~~

--~

.... . 0.006 3 0.1 ••• 0.

.... . .... . 0.07 0.1

..... .0 ...

..... 0.07 .0 . . •

0.8 . . • 0.

0.1 0.05 .....

.... . 0.3

Pressure increment (Llp.)p,

(Ll/L)T,

Source

p.poises ;oC p.poises/atm 00

•••

0.536 0.425 0.704 0.500 0.237 0.300 0.450 0.474 0.451 0.340 0.360 0.284 0.580 0.277 0.320 0.464 0.200 0.680 0.500

.....

........ 0.1224

........

0.1753

........ ........ ........

0.0046

........ . .......

. ....... .

....... 0.0082

. ....... ••

0

•••

•

•

........ -0.0093 0.0118

. .......

........ . .......

"International Critical Tables" Bearden, Phys. Rev. 56, 1023 (1939) Wtd. mean of 2 values Ref. 1 Ref. 2 Ishida, Phys. Rev. 21 (1923) Kuenen and Visser, Amsterdam Acad. Sci. 22, 336 (1913) Ref. 1 Wtd. mean of 4 values Rankine, Proc. Roy. Soc. (London), ser. A, 86,162 (1912) Ref. 2 Ref. 2 Ref. 1 Ref. 2 Wtd. mean of 2 values Van Cleave and Maass, Can. J. Research 13B, 140 (1935) Ref. 1 Ref. 1 Ref. 2 Ref. 2 Kestin and Nagashima, Phys. Fluids 7,730 (1964)

~

toJ

Q

~

H

Q

Ul

TABLE 2r-6. ABSOLUTE VISCOSITY p. OF GASES IN MICROPOISES (Continued)

Gas

Symbol

/I,

p.poises

Hydrogen iodide ...... Krypton ............. Mercury ............. Methane .............

HI Rr

Methyl bromide ...... Methyl chloride ...... Neon ............... Nitric oxide .......... Nitrogen ............ Nitrous oxide ........

CHaBr CHaCI Ne NO N2 N,O

132.7 107.0 313.81 189.8 175.69 145.6

Oxygen ............... Propane ............. Sulfur dioxide ........ Xenon ..............

0, CaHs S02 Xe

203.31 80.0 125.0 227.40

Hg CH 4

183.0 249.55 450.0 (200°C) 109.8

Estimated Temp. Pressure uncertainty increment increment (L'.,u}T, ±L'.,u, (L'.,u)Pl ,upoises p.poises;oC ,upoises / atm .....

0.15

0.640 0.735

. ....... 0.2816

.....

.

....

. .......

0.1

0.330

0.016

.....

0.460 0.425 0.697 0.538 0.454 0.475

. ....... . .......

..... 0.15 0.1 0.09

..... 0.1

..... .... .

0.14

0.616 0.22 0.400 0.725

--

0.0354 ........

0.1234

.. . . . . . .

0.1205

... . . ... ........ 0.2624

Source

Ref. 2 Ref. 1 Ref. 2 j Restin and Leidenfrost, "Thermodynamic Prop- w. erties of Gases, Liquids, Solids," p. 321, ASME (') o 1958 W. H Ref. 2 f-3 ~ Breitenbach, Ann. Phys. 5, 166 (1901) o Ref. 1 ":J Wtd. mean of 3 values o Ref. 1 ~ Johnston and McCloskey, J. Phys. Chern. 44, 1038 w. trJ w. (1940) Ref. 1 Ref. 2 Ref. 2 Ref. 1 _._.-

References 1. Kestin, J., and W. Leidenfrost: Physica 25, 1033 (1959). 2. Golubev, I. F.: "Viaz'kost' gazov i gazovykh smesei," Moscow, 1959. difficult to assess.

This reference contains extensive data whose accuracy, however, it is

~

ioI'-

~

~fI:>.

TABLE 2r-7. KINEMATIC VISCOSITY V OF GASES (10- 3 em 2/see; at 20°0 and 1 atm)

fI:>.

Gas

Symbol

v, 10- 3 cm 2jsec

Acetylene ........................... Air ................................. Ammonia ........................... Argon .................. , ........... Bromine ............................ iso-Butane .......................... n-Butane ............................ Oarbon dioxide ...................... Carbon monoxide .................... Chlorine ............................ Chloroform .......................... Cyanogen ........................... Deuterium ........................... Deuteromethane ..................... Ethane ............................. Ethylene ............................ Helium ............................. Hydrogen bromide ................... Hydrogen chloride ................... Hydrogen deuteride .................. Hydrogen ........................... Hydrogen iodide ..................... Krypton ............................ Mercury ............................ Methane ............................

02H2

80.6 (at 0°0) 151.1 138 134.3 22.50 31.0 35.1 80.09 150.6 45.11 20.16 46.35 744.2 154.7 72.9 85.84 1,179 54.79 93.99 889.4 1,059 34.42 72.44 87.12 (at 200°C) 164.8

...... NH, Ax Br2 04H 1O 04H 1O 002 CO Cl, CHCI, C,N, D2 CD 4 02Hs C 2H 4 He HBr HCI HD H2

HI Kr Hg CH4

Estimated uncertainty ±L'>.v, 10- 3 cm 2/sec

0.08 4 0.06

.....

..... ..... 0.04 0.09

..... ..... .....

0.4

.....

0.6

.....

0.6

..... ..... 2.4 0.6

Temp. increment (L'>.v)T, 10-' em 2/ (sec WO)

0.960 1.07 0.882 0.152 0.204 0.244 0.516 0.921 0.307 0.137 0.325 4.24 1.22 0.471 0.997 6.81 0.389 0.651

0.044

6.01 0.237 0.460

0.2

1.06

.....

Pressure increment (L'>.v)p, 10-' cm2j(sec) (atm)

-150.9 -134.1

-80

~

t'j Q

t:ci il> Z H Q

U1

-740

-1,200

-1,060 -72.20 -160

TABLE

Gas

2r-7.

Symbol

KINEMATIC VISCOSITY

v, 10-' cm'/sec

V

OF GASES

(Continued)

Estimated uncertainty ±/lv, 10-' cm'/sec

Temp. increment (/lv h, 10-' cm'/(sec)(OC)

Pressure increment (/lv)p, 10-' cm'/(sec)(atm)

;:S

m C)

Methyl bromide ...................... Methyl chloride ...................... Neon ............................... Nitric oxide ......................... Nitrogen ............................ Nitrous oxide ........................ Oxygen ............................. Propane ............................. Sulfur dioxide ........................ Xenon ..............................

-

CR,Br CH,CI Ne NO N, N,O

0, C,Rs SO, Xe

33.64 50.97 374.1 152.1 150.9 79.57 152.8 43.7 46.94 42.02

..... ..... 0.18 0.08 0.08 ..... 0.08 ..... ..... 0.026

o

m

0.232 0.376 2.11

0.950 0.905 0.531 0.984 0.269 0.310 0.278

~

>/ax ay

similarity parameter stream function 2t.1. Basic Equations in Rectangular Coordinates. The basic equations of motion for a compressible inviscid gas may be written as follows. Momentum Equation. By applying Newton's laws of motion the Euler momentum equation may be derived in the form

+ u au + v au + w au = ax ay az ~ + u av + v av + w~ = at ax ay az aw + u aw + v aw + w aw = au at

at

ax

ay

az

2-253

+X -1 ap + y p ay -1 ap + Z -1 ap p ax

paz

(2t-l)

2-254

MECHANICS

where x, y, z = rectangular coordinates t = time u, v, w = velocity components in direction of x, y, and z axes, respectively p = pressure p = density X, Y, Z = rectangular components of external body force Continuity Equation. The assumption that the gas is a continuous medium expressed by the equation

tf + y;; (pu) + :y (pv) + ;. (pw)

=

0

IS

(2t-2)

Energy Equation. The relationship between the kinetic and internal energy and the work done on the fluid by pressure and external forces is expressed by the equation p DE -I- p D Dt ' Dt

(!2 q2)

= pQ

+ p(uX + vY + wZ)

-

~ax

(pu) -

~ ay

(pv) -

~ az

(pw) (2t-3)

where D == ~ + u ~ + v -.i + w ,i Dt at ax ay az E = internal energy per unit mass = f Cv dT q2 = u2 + v2 + w2 Q = external-heat-production rate per unit mass Cv = specific heat at constant volume Equation of State. For a complete specification of a flow it is necessary to give an equation of state. This commonly takes the form p = f(p,T.)

Many gases obey the equation of state of a perfect gas p

=

pRT

under a great variety of conditions. In this equation R is a constant which depends on the particular gas. If the specific heat can be assumed constant, the gas is said to be calorically perfect and E = cvT where T is the temperature on the absolute scale. A specific case of great importance is that of isentropic flow. If the entropy is constant throughout the flow, the equation of state can be written as

where K is a constant and l' is the ratio of the specific heat at constant pressure Cp to that at constant volume Cv• Now the flow is completely determined by the momentum equations, the continuity equation, and the equation of state. Many practical flow problems are essentially cases of isentropic flow. 2t-2. Dynamic Similarity and Definition of Basic Flow Parameters. In the testing of models it is necessary to maintain a proper scaling of certain dynamic parameters in addition to the geometric scaling. For compressible inviscid flow with no heat sources and in which body forces are neglected, the only dynamic dimensionless parameter is the Mach number_ Definition of Mach Number. Thelocal Mach number.is defined as the ratio of the local flow velocity q to the local sound velocity a; i.e., (2t-4)

COMPRESSIBLE FLOW OF GASES

2-255

Thus in a nonuniform flow the Mach number will vary from point to point. The size of the Mach number indicates whether the flow is subsonic, M < 1; transonic, M ~ 1; or· supersonic, M· > 1. The term hypersonic is often used to describe flows where

M >5. Dynamic Similarity. If the same gas flows around two geometrically similar bodies, it might be expected that under the right conditions the streamline pattern would be similar. This is true if the Mach numbers of the two flows are equal. It then follows that all other dimensionless coefficients such as drag coefficient, lift coefficient, pressure coefficient, etc., are also equal. In determining the Mach number in a flow it is necessary to know not only the flow velocity but the sound velocity as well. For a perfect gas the sound velocity is proportional to the square root of the temperature; i.e.,

a

=

v''YRT

Table 2t-1 is based on this relationship. 2t-3. Basic Idea of One-dimensional Flow. In many cases, as in a pipe of slowly varying cross section, it is possible to make the assumption of constant flow properties across any cross section perpendicular to the pipe axis. Although strictly speaking there are no one-dimensional flows, because of viscous effects on the boundaries, it is still possible to get much valuable information of a practical nature from the assumptions. TABLE

2t-1.

Basic Equations.

VARIATION OF VELOCITY OF SOUND WITH TEMPERATURE

T, oK

a, fps

a, m/sec

150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350

805 832 857 882 907 930 953 975 997 1,019 1,040 1,060 1,081 1,100 1,120 1,139 1,158 1,176 1,195 1,213 1,230

246 254 261 269 276 283 290 297 304 311 317 323 329 335 341 347 353 359 364 370 375

On the assumption of isentropic flow the equations of motion are (momentum)

(2t-5)

(continuity)

(2t-6)

2-256

MECHANICS

where A is the cross-sectional area. For unsteady one-dimensional flow in general and in particular for an excellent treatment of flow in pipes of constant area see ref. 3. The above equations also cover the case of cylindrical and spherically symmetric flow; i.e., 1 iiA 1 (for cylindrical flow)

x Aax = x

Aiix = 1iiA 2

(for spherically symmetric flow)

In the important case of steady flow the equations can be integrated to give _'Y_l!. ~ u' = const 'Y-1p 2 puA = m = const

+

(2t-7)

(2t-8)

where m is the mass flow. By taking logarithmic derivatives and remembering the definition of the Mach number M, the continuity equation may be written du (1 _ M') u

+ dA A

= 0

(2t-9)

Thus, if du T" 0 and M = 1, we see that dA = O. In other words, the Mach number becomes equal to unity only in a section of the pipe where the area is a minimum. This fact is of prime importance in the design of supersonic wind tunnels. The dependence of the various flow variables on the Mach number for steady onedimensional isentropic flow is given in Table 2t-2. 2t-4. Two-dimensional and Axially Symmetric Flow. Many important types of flow belong to the class of two-dimensional or axially symmetric flows. These include flows past wedges, cones, bodies of revolution, etc. The important distinctions to be made are those between subsonic and supersonic flow. Purely subsonic flow is qualitatively quite similar to incompressible flow, while supersonic flow exhibits many startlingly different properties. Among these are the appearance of shock waves (see Sec. 2v) and the existence of wavefronts. A general discussion of the above topics can be found in refs. 2, 3, and 6. The greater bulk of the literature on two-dimensional and axially symmetric flow is concerned with steady flow. The unsteady cases are usually extremely difficult to solve. Velocity Potential and Stream Function. In cases ofirrotational or steady flow it is convenient to introduce the velocity potential or the stream function. This reduces the number of equations to one. The velocity potential exists whenever there is a state of steady or unsteady irrotational flow; i.e., the velocity components satisfy the equations

Then the velocity components u, v, w can be expressed as the components of the gradient of the velocity potential cf>. Thus iicf>

U=-

iix

v

iicf>

=-

iiy

w

=

iicf>

iiz

(2t-1O)

For steady isentropic flow the equations of motion reduce to the single equation for cf>,

ar

cf>zz ( 1- cf>z·)

+cf>UY (cf>u') 1-(i2 +cf>•• (cf>l) 1- ar (2t-ll)

where

2-257

COMPRESSIBLE FLOW OF GASES

TABLE2t-2. DEPENDENCE OF FLOW VARIABLES ON MACH NUMBER FOR ONl!:-DIMl!:NflWNAL ISEN'l'lWPIC FUlIY*

pu'/2po

pu/poao

pip~

T/To

a/ao

5.822 2.9635 2.0351 1. 5901

0.00000 0.00695 0.02723 0.05919 0.10031

0.00000 0.09940 0.19528 0.28437 0.36393

1.00000 0.99502 0.98028 0.95638 0.92427

1.00000 0.99800 0.99206 0.98232 0.96899

1.00000 0.99900 0.99602 0.99112 0.98437

0.48795 0.57950 0.66803 0.75324 0.83491

1. 3398 1.1882 1.0944 1.0382 1.0089

0.14753 0.19757 0.24728 0.29390 0.33524

0.43192 0.48704 0.52880 0.55739 0.57362

0.88517 0.84045 0.79161 0.73999 0.68704

0.95238 0.1)3284 0.91075 0.88652 0.86059

0.97590 0.96583 0.95433 0.94155 0.92768

0.52828 0.46835 0.41238 0.36091 0.31424

0.91287 0.98703 1.0574 1.1239 1.1866

1.00000 1.0079 1.0304 1.0663 1.1149

0.39670 0.41568 0.42696 0.43114

0.57870 0.57415 0.56161 0.54272 0.51905

0.63394 0.58170 0.53114 0.48290 0.43742

0.83333 0.80515 0.77640 0.74738 0.71839

0.91287 0.89730 0.88113 0.86451 0.84758

1.5 1.6 1.7 1.8 1.9

0.27240 0.23527 0.20259 0.17404 0.14924

1.2457 1. 3012 1.3533 1.4023 1.4479

1.1762 1.2502 1.3376 1.4390 1.5553

0.42903 0.42161 0.40985 0.39476 0.37713

0.49203 0.46288 0.43264 0.40216 0.37210

0.39484 0.35573 0.31969 0.28684 0.25699

0.68966 0.66138 0.63371 0.60680 0.58072

0.83045 0.81325 0.79606 0.77904 0.76205

2.0 2.1 2.2 2.3 2.4

0.12780 0.10935 0.09352 0.07997 0.06840

1.4907 1. 5308 1.5682 1.6033 1.6360

1.6875 1. 8369 2.0050 2.1931 2.4031

0.35785 0.33757 0.31685 0.29614 0.27579

0.34294 0.31504 0.28863 0.26387 0.24082

0.23005 0.20580 0.18405 0.16458 0.14719

0.55556 0.53135 0.50813 0.48591 0.46468

0.74535 0.72894 0.71283 0.69707 0.68168

2.5 2.6 2.7 2.8 2.9

0.05853 0.05012 0.04295 0.03685 0.03165

1.6667 1. 6953 1.7222 1. 7473 1.7708

2.6367 2.8960 3.1830 3.5001 3.8498

0.25606 0.23715 0.21917 0.20222 0.18633

0.21948 0.19983 0.18181 0.16534 0.15032

0.13169 0.11788 0.10557 0.09463 0.08489

0.44444 0.42517 0.40683 0.38941 0.37286

0.66667 0.65205 0.63784 0.62403 0.61062

3.0 3.1 3.2 3.3 3.4

0.02722 0.02345 0.02023 0.01748 0.01512

1.7925 1.8135 1.8329 1. 8511 1.8682

4.2346 4.6573 5.1210 5.6287 6.184

0.17151 0.15774 0.14499 0.13322 0.12239

0.13666 0.12426 0.11301 0.10281 0.09359

0.07623 0.06852 0.06165 0.05554 0.05009

0.35714 0.34223 0.32808 0.31466 0.30193

0.59761 0.58501 0.57279 0.56095 0.54948

3.5 3.6 3.7 3.8 3.9

0.01311 0.01138 0.00990 0.00863 0.00753

1.8843 1.8995 1. 9137 1.9272 1.9398

6.790 7.450 8.169 8.951 9.799

0.11243 0.10328 0.09490 0.08722 6.08019

0.08523 0.07768 0.07084 0.06466 0.05906

0.04523 0.04089 0.03702 0.03355 0.03044

0.28986 0.27840 0.26752 0.25720 0.24740

0.53838 0.52763 0.51723 0.50715 0.49740

4.0 4.1 4.2 4.3 4.4

0.00659 0.00577 0.00506 0.00445 0.00392

1. 9518 1. 9631 1.9738 1. 9839 1.9934

10.72 11.71 12.79 13.95 15.21

0.07379 0.06788 0.06250 0.05759 0.05309

0.05399 0.04940 0.04524 0.04147 0.03805

0.02766 0.02516 0.02292 0.02090 0.01909

0.23810 0.22925 0.22084 0.21286 0.20525

0.48795 0.47880 0.46994 0.46136 0.45305

4.5 4.6 4.7 4.8 4.9

I

0.00346 0.00305 0.00270 0.00239 0.00213

2.0025 2.0111 2.0192 2.0269 2.0343

16.56 18.02 19.58 21.26 23.07

0.04898 0.04521 0.04177 0.03862

0.01745 0.01597 0.01464 0.01343 0.01233

0.19802 0.19113 0.18457 0.17832

0.44499 0.43719 0.42962 0.42228

0.03572

0.03494 0.03212 0.02955 0.02722 0.02509

5.0

I 0.00189

2.0412

25.00

0.03308

0.02315

0.01134

M

p/po

0.0 0.1 0.2 0.3 0.4

1.00000 0.99303 0.97250 0.93947 0.89561

0.00000 0.09990 0.19920 0.29734 0.39375

0.5 0.6 0.7 0.8 0.9

0.84302 0.78400 0.72093 0.65602 0.59126

1.0 1.1 1.2 1.3 1.4

u/ao

I

A/A*

00

I 0.36980

I

* A more complete table may be found in refs.

4, 5, and 7.

I

0.17235 0.16667

I

0.41516 0.40825

2-258

MECHANICS

and qmax is the velocity with which the gas flows into a vacuum. Other forms of this equation in different numbers of dimensions and for unsteady flow can be found in ref. 2. - - TRANSONIC THEORY 0.6

0.4

10.0'

Cp 0.2

1'

7.5 0

o~

4.5 0

1.2

1.0

0.8

0.6

1.6

1.4

M", fa) M 2-1

~:1;!1lf

4

o

\

\

2 WEDGE SEMI-ANGLE

c

X= [(r+l)r;!(t!c)]%

\

08:P~~

3

4.5 0 7.5" 10.00

\

Cp

LJNEA~',

THEORY _

CPo

- .......

[(r+])M~]llJ (tlc)%

CP

----

(I

o X (b) FIG. 2t-1. Comparison of the extended transonic similarity law with experiment. (a) Plotted in conventional coordinates. (b) Plotted in transonic similarity coordinates. (After J. R. Spreiter, NACA; taken from ref. 6.)

In compressible flow a stream function if; exists only for steady two-dimensional or axially symmetric flow. The introduction of the function if; causes the continuity equation to be satisfied identically. In two dimensions u If cylindrical coordinates (x, function if; may be defined by

T,

=

1

pif;y

v

=

-

1

- if;x p

(2t-12)

e) are used and the flow is independent of e, then the v

-

1

-if;x

pf

(2t-13)

COMPRESSIBLE FLOW OF GASES

2-259

Note that u and v are now the velocity components in the x and r directions and y2 + Z2. Further details are given in ref. 2. Eq1wtions of Small-pertw'bation Them·y. For many slender or flat two- and threedimensional bodies it may be assumed that the flow is disturbed very little from uniform flow. Thus if the free-stream velocity U is parallel to the x coordinate and M ro is the free-stream Mach number, the velocity components can be written in the form r =

-vi

u

=

U

+ '1' + 2'1 = '1B + : '1 = '1 + '::) = '1'0 '0 "" 3~ + '1B = 2 + :L "" viscosity number '1 ",

(3c-1O)

Putting (3c-7), (3c-8), (3c-9) into (3c-6) yields the vector force equation in the following equivalent forms: au,

aUi

Pat: +pu; ax; =pFi =

pF, (3c c lla)

PROPAGATION OF SOUND IN FLUIDS p

~~

p

~~

=

=

+ (-'I' + 1]) grad (div u) + 1]V2(U) + (div u) grad 1]' + 2 (grad 1] - grad) u + grad 1] X curl u p(u . V)u - vp + (1]' + 21])V(V . u) - 1]V X (v X u) + (V' u)V1]' + 2(V1] . V)u + V1] X (v X u)

3-:-41

pF - grad P

pF -

(3c-lIb)

(3c-He)

The vorticity, defined by R = j curl u = j(v X u), and the dilatation rate, ,:; == V . u, can be introduced as useful abbreviations. A somewhat more symmetrical expression in terms of the mass transport velocity pU is obtained if the last form of the continuity equation (3c-5) is multiplied by u and added to (3c-lIe), giving

a~~) + u(V' pU) + (pu' v)u

=

pF - vP

+ 1]'OV':; - 2~v X R + ,:;V1]' + 2(V1] . v)u + 2V1] X R

(3c-lId)

These equations reduce to the so-called Navier-Stokes equations when it is assumed that 1] and 1]' are constant (V1] = V1]' = 0) and that the Stokes relation holds (1]B = 0, '0 = ~); and still further simplification follows if the motion is assumed irrotational so that R = O. If the viscosity coefficients are to be regarded as functions of one or more of the state variables, however, the gradients of the ~'s must be retiLinEid so that the implicit functional dependence can be introduced by writing, for example, V1] = (a1]/aT)vT + .... Energy Relations and Equations ofllState. The conservation of energy requires that the following power equation be sati"Sfied:

+

D(EkDt Er) =

r pFiUi dV + lv

f

it tiiui dai -

1 A

qi da;

(3c-12)

where Ek is the kinetic energy associated with the material velocity, Er is the total internal energy, V is a volume bounded by the surface A, da; is the projection of a surface element of A on the plane normal to the +Xi axis, F, is the extraneous body force (per unit mass), and qi is the total heat flux vector (mechanical units). After the surface integrals are converted to volume integrals by using the divergence theorem, and with the help of (3c-6), this equation reduces to the Fourier-Kirchhoff-C. Neumann' energy equation, D. 'aqi p Dt = tiidii - ax,

(3c-13)

where. is the local value of the specific internal energy (per unit mass) defined through Er =

Iv

P€

dV.

It is now postulated that the state of the fluid is completely specified

by • and two other local state variables, which can be taken as the specific entropy s (per unit mass) and the specific volume v = in terms of which the thermodynamic pressure and temperature, and the specific heats can be defined by

p-"

• = .(s/v)

p (:;)p

(3'c-14)

C == T

The second law of thermodynamics can be introduced in the form of an equality, which replaces the classical Clausius-Duhem inequality, through the expedient of accounting explicitly for the creation of entropy Sirr (per unit volume) by irreversible 1

See footnote, p. 3-39.

ACOUSTICS

dissipative processesjl thus D Dt

f

V

psdV = -

f

~da·

AT'

+ Jv ( DBirrdV Dt

(3c-15a)

This relation states that the increase of entropy in a material element is accounted for by the influx of heat and by the irreversible production of entropy within the element. The left-hand side of (3c-15a) can also be written, with the help of the continuity relation, as

Iv

p(Ds/ Dt) dV.

Then, after converting the surface integral to a volume

integral, the second law can be given in differential form as , Ds _ ...!!- ~ + DBirr p- = ox, T Dt De _ 1. oq, + q, oT + DBirr T ox,

T2 ox,

Dt

(3c-15b)

A thermal-dissipation function q,k can be defined by (3c-16) whereupon multiplying (3c-15b) by T yields the second-law equality in the form (3c-15c)

Taking the material derivative of the basic equation of .state (3c-141) (where the subscript added to an equation number indicates the serial number of the equality sign to which reference is made when sev:eral relations are grouped under one marginal identification number), introducing the definitions for P th and T, multiplying by p, and using (3c-4), gives Ds D. (3c-17) pT Dt = p Dt +Pthtl The energy equation (3c-13) can be recast, using (3c-7) an(3cr9), in the form (3c-18) in which V,jd'j, the dissipative component of the stress power t'jd,j, is defined as the viscous dissipation function q,~. The usefulness of specifying the arbitrary scalar in (3c-7) as the thermodynamic pressure, so that P; = P th, becomes apparent when p~l)./Dt-is eliminated between (3c-18) and (3c-17), giving pT Ds = (Pth _ P)tl Dt = ~ _

+ q,~

oq. ox;

_ Oqi

ox;

(3c-19)

The viscous dissipation function (dissipated energy per unit volume) is thus seen to account for either an effiux of heat or an increase of entropy. Subtracting (3c-19) from (3c-15c) then allows the rate of irreversible production of entropy to be evaluated directly in terms of the two dissipation functions,

TD~t

=

~ + q,.

(3c-20)

The total heat:iflux vector q., whose divergence is the energy transferred away from the volume element, must account for energy transport by either conduction or radii Tolman and Fine, Revs. Modern Phys. 20, 51-77 (1948).

37""43

PROPAGATION OF SOUND IN FLUIDS

ation. The part due to conduction is given by the Fourier relation, which serves also to 'define the heat conductivity K,

(3,c-21a) The last term, containing the gradient of K, must be retained if implicit dependence of Kon the state variables is to be represented. On the other hand, if Kis assumed to be constant, (3c-21a) reduces to the more familiar form

The component of heat flux due to radiation can be approximated, for small temperature differences, by Newton's law of cooling,

a(q')r.d = P.q C (T -,,-uXi

- T 0)

= V'qrad .

(:;k-21b)

where (T - To) is the local temperature excess and q is a radiation coefficient introduced by Stokes'! The foregoing thermal relations can be combined with the equations of continuity and momentum more readily if the term T(D8/ Dt) appearing in (3c-19) is expressed in terms of the variables u, v, and T. The defining equations (3c-14) establish that P = P(v,8)'and T = T(V,8) , from which it follows that one may also write 8 = 8(T,v) or B = 8(T,P). Using both of the latter leads; after some, manipulation,' to the identity !:J. pT DB Dt = pC. [ ('Y - 1) ~

+ DT] Dt

(3c-22)

in which # is the coefficient of thermal expansion, # == p(av/aT)p. Mter (3c-22) and (3c-21) are'combined with (3c-19), the energy equation can be written in the alternate, forms, pC.DT + pC. 'Y - 1 au; + aqi _ q", = 0 Dt # ax, aXi

pC~ (~~ + U' VT)

+ p(C p ; ; C.) !:J. - V • (KVT) + pC.q(T - To) -

aT +u'VT + ('Y -1)!:J. __K_V2T _ VT'VK +q(T _ To) at # pC. pC. The viscous dissipation function (3c-9) in the explicit form ~ =

=

_.:hi.... = pC.

0 (3c-23) 0

can be evaluated, with the aid of (3c-8) and

= '1/'d d;; + 2'1/diidi; + i'1/ [(aul)2 + (aU2)2 + (aus)'

Viidii

= '1/B!:J.'

~

~

kk

3

_ aUl aU2 _ aU2 aus _ aus aU1] aXl aX2 axs aXl aX2 aX2 axs axs aXl U + '1/ [(aUl + aU2)2 + (a 2 + aus)2 + (aus + aUl)2] (3c-24a) aX2 aXl axs aX2 aXl axs

The thermal dissipation function . due to heat conduction can be evaluated, with the aid of (3c-16) and (3c-21a), in the form .=

qi aT= -Tax,

(aT)2 +-Tax, K

K • =_(VT)2

T

(3c-24b)

It does not appear explicitly in (3c-23), but it is there implicitly as a consequence of the heat-transfer processes described by (3c-23). lPhil. Mag. (4) 1,305-317 (1851). • See, for example, Zemansky, "Heat and Thermodynamics," 3d ed., pp. 246-255, McGraw-Hill Book Company, New York, H151.

3-"44

ACOUSTICS

Summary of Assumptions. The fluid considered is assumed to be continuous except at boundaries or interfaces, locally homogeneous and isotropic when at rest, viscous, thermally conducting, and chemically inert, and its local thermodynamic condition is assumed to be completely determined by specifying three "state" variables, any two of which determine the third uniquely through an equation of state. No structural or thermal "relaxation" mechanism has been presumed up to this point in the analysis, except to the extent that ordinary heat conduction and viscous losses may be described in such terms. Local thermodynamic reversibility has been assumed in using conventional thermodynamic identities based on the second law, but the irreversible production of entropy by dissipative processes has been accounted for explicitly. It is also assumed that the stress tensor is a linear function of the rate of deformation, and that the tractions due to viscosity can be represented by the linear terms of an expansion in powers of the viscosity coefficients. The viscosity and heat-exchange parameters of the fluid '1], '1]', K, and q. may depend in'any continuous way on the state variables and hence may be implicit functions of time' and the spatial coordinates. Within the scope thus defined the equations given are exact. The functional dependence on time and the spatial coordinates of the condition and motion variables P, T, p, and u can be evaluated, in a formal sense at least, by solving the set of four simultaneous equations connecting these variables [Eqs. (3c-5), (3c-11), (3c-23), and (3c-15) or one of its alternates]. No general solution of these complete equations has been given, however, and one or another of the least important terms is usually omitted in order to render the equations tractable for dealing with specific problems. 3c-3. The Small-signal Acoustic Equations. The physical theory of sound waves deals with systematic motions of a material medium relative to an equilibrium state and thus comprises the variational aspects of elasticity and fluid dynamics. Such perturbations of state can be described by incremental, or acoustic, variables and approximate equations governing them can be obtained by arbitrarily "linearizing" the general equations of motion. These results, as well as higher-order approximations, can be derived in an orderly way by invoking a modified perturbation analysis. 1 This consists of replacing the dependent variables appearing in (3c-5), (3c-11), and (3c-23) by the sum of their equilibrium or zero-order values and their .first- and secondorder variational components, and then forming the separate equations that mus~. be satisfied by the variables of each order. Two of the composite state variables, for example P and T, can be defined arbitrarily, whereupon the third, P, is determined by the functional equation of state. These definitions, some self-evident manipulations, and the subscript notation identifying the orders can be exhibited as follows:

+ + +

+ + + + + + [G~)Jo (T - T~) +

P "" Po Pl P. T "" To 01 02 Vp = VPl VP. VT = VOl VO. P(p,T) "" Po(po,To) PI p.

PI

K=

KT "" P

+ P2

=

(iJP) iJp

[(~~)Tl (p

- po)

Co' ""

T

[(iJP) ] "" (K.)o iJp • PO 0

Co' Co 2 PI = - (PI ~OPOOl) p. = - (P' ~OP002) 7 7 U "" 0 Ul U. V • U "" LI. "" Ll.l LI.. = V • Ul V • U2 pU = [poudl [PlUl poU.]. V' (pU) = [poV • ulh (PIV • Ul UI • VPl poV • U2].

+

+

+

+

1

+

+

Eckart, PhY8. Rev. 73, 68-76 (1948).

+ +

+

+ ... +

+ +

(3c-25)

PROPAGATION OF SOUND IN FLUIDS

3-45

Terms containing Vpo have been omitted in writing out V • (pu), on the assumption that po, To, and Po are constant and Uo = O. The reference state need not be so restricted to one of static equilibrium provided its time and space rates of change are presumed small in comparison with the corresponding change rates of the acoustic variables. The extraneous body force F will also be omitted hereafter; it would become important in cases involving electromagnetic interaction, but it usually derives from a gravitation potential and affects primarily the equilibrium configuration. l Little generality is sacrificed by omitting F and assuming a static reference, moreover, since the basic equations characterize directly the equilibrium condition and since the "cross-modulation" effects brought in by nonlinearity are dealt with adequately through second- or higher-order approximations. Notice that the foregoing represents a mathematical-approximation procedure that is concerned only with the precision achieved in interpreting the content of the basic equations. The accuracy with whioh the basic equations themselves delineate the behavior of a real fluid is an entirely different quesLion that must be considered independently on its own merits. It follows that,while good judgment may restrain the effort, there is no impropriety involved in pursuing higher-order solutions of the acoustic equations, even though the equations themselves may embody first-order approximations to reality such as that represented by assuming linear dependence on the viscosity coefficients and the deformation rate. When the appropriate relations from (3c-25) are substituted in (3c-5), (3c-ll), and (3c-23), the first-order acoustic equations can be separated out in the form apl at + po(V • Ul) co 2 (1V OVPl l ) VPl at + -:y + {3opo

po aUl

poC.

alit at + poCv('Y{30 -

-

=

+ 1) V X (V X Ul) = KOV 20 l + POCvqOl = 0

(1)o'D)V(V' Ul)

1)

(V . Ul) -

(3c-26a)

0 O

0 (3c-26b) (3c-26c)

Inasmuch as the first-order effects of both shear and dilatational viscosity and of heat cOllduction and radiation have been included, these equations comprehend a viscothermal theory of small-signal sound waves. The sound absorption and velocity dispersion predicted by this theory are discussed below. Note especially that taking heat exchange into account explicitly by including (3c-26c) has precluded the conventional adiabatic assumption and denied the simplifying assumption that P = pep). Adiabatic behavior would be assured, on the other hand, if it were assumed at the outset that K = q = 0, but the behavior would not at the same time be strictly isentropic so long as irreversible viscous losses are still present and accounted for. The difference between adiabatic and isentropic behavior in this case is of second order, however, as indicated by the fact that the second-order dissipation functions '" do not appear in the first-order energy equation (3c-26c), which is thereby reduced to yielding just the isentropic relation between dilatation and excess temperature. It is allowable, therefore, in this first-order approximation, to replace the quotient (vedVpl) appearing in (3c-26b) with the isentropic derivative (aT lap). = (I' - 1) I p{3, whereupon the first-order equation of motion for an adiabatic viscous fluid can be written as (3c-27) If the effects of viscosity, as well as of heat exchange, are to be neglected, the divergence of what is left of (3c-27) can be subtracted from the time derivative of (3c-26a) 1 But, for a case in which F and Vpo cannot be neglected, see Haskell, J. Appl. Phys. 22,157-168 (February, 1951).

3-46

ACOUSTICS

to yield the typical small-signal scalar wave equation of classical acoustics,

e:),

a;;'1 =

(3c-28a)

V'P1

and, with the help of the first-order isentropic relation P1 = CO'(P1)" this wave equation becomes, in terms of the sound pressure, (3c-28b) 3c-4. The Second-order Acoustic Equations. The same substitution of composite variables that delivered (3c-26a), (3c-26b), and (3c-26c) will also yield directly the second-order equations of acoustics, which can now be marshaled as follows:

ap. at Po

+ po(V . u.) + V' (P1U1)

=

(3c-29a)

0

au. + --ata(P1ul) at + POU1 (V • u,) + Po (U1 . V) 111 + Co''Y (1 + (3oPo vp, VII,) vp, - '7o'Ov(v . u,) + 2'70(V

X R.)

- (V'7~)(V . U1) - 2(V'71 . V)U1 - 2(V'71) X R1

ao. + 111 • (V01) + 'Y at

-;;

1 (V . 112)

,.,0

The subscripts appended to generic form '7(T,

p, . • • ) =

K

-

=

0

(3c-29b)

KC V'1I 2 O

Po

v

and the '7's imply that each may be expressed in the

'7o(T o, po, . . . )

+ '71

a'7 aT 111

'71 = -

a'7 + -ap P1 + . ..

(3c-30)

No general solution of these complete second-order equations has been given, but they provide a useful point of departure for making approximations and for investigating some second-order phenomena that cannot be predicted by the first-order equations alone. 3c-6. Spatial and Material Coordinates. Equations (3c-26) and (3c-29) are couched in terms of'the local values assumed by the dependent variables p, P, T, and u at places identified by their coordinates Xi in a fixed spatial reference frame, commonly called Eulerian coordinates (in spite of their first use by d' Alembert). As an alternate method of representation, the behavior of the medium can be described-in terms of the sequence of values assumed by the dependent condition and state variables pertaining to identified material particles of the medium no matter how these particles may move with respect to the spatial coordinate system. The independent variables in this case are the identification coordinates ai, rather than the position coordinates; the latter then become dependent variables that describe, as time progresses, the travel history of each particle of the medium. Such a representation in terms of material coordinates is commonly called Lagrangian (in spite of its first introduction and use by Euler). The Wave Equation in Material Coordinates. The use of material coordinates can be demonstrated by deriving the exact equations governing one-dimensional (planewave) propagation in a nonviscous adiabatic fluid. Consider a cylindrical segment of the medium of unit cross section with its axis along +x, the direction of propagation, and let X and x + ilx define the boundaries of a thin laminar "particle" whose undisturbed equilibrium position is given by a and a + aa. The difference x - a = ~ defines the displacement of the a particle from its equilibrium position and provides a convenient incremental, or acoustic, dependent variable in terms of which to describe

PROPAGATION OF SOUND IN FLUIDS

the position, velocity, and acceleration of the particle; thus

x(a,t)

=

a

ax _ L _ a~ at - u (a,t) - at

+ ~(a,t)

(3c-31)

Continuity requires that the mass of the particle remain constant during any dis. placement, which means that (3c-32a) or, for three-dimensional disturbances and in general,

Po a(Xl,X2,XS) pL = a(al,a2,aS)

(3c-32b)

in which the symbolic derivative stands for the Jacobian functional determinant. The superscript L is used here and below as a reminder that the dependent variable so tagged adheres to, or "follows" in the Lagrangian sense, a specific particle, and that it is a function of the independent identification coordinates. When not so tagged, or with superscript E added for emphasis, the state variables p, P, T and the condition variable u are each assumed to be functions of time and the spatial coordinate x. The net force per unit mass acting on the particle at time't is - (pL)-laPL / ax, where pL and pL are the density and pressure at x, the" now" position of the moving particle. However, inasmuch as x is not an independent variable in this case, the pressure gradient must be rewritten as (apL/aa) (aa/ax), from which the second factor can be eliminated by recourse to (3c-32a). The momentum equation then becomes just poa2~

at2

-aPL

=aa

(3c-33)

The adiabatic assumption makes available the simplified equation of state, P = P(p), and this relation, in turn, allows the material gradient, apL / aa, to be written as (3c-34)

aa from which the last factor can be eliminated by using (3c-32a) again; once to the exact wave equation 1 a2~ = (CpL) 2 a2~ = c2

at 2

aa 2

Po

(1 + aaa~)-2 aaa2~2

This leads at

(3c-35)

The pressure-density relation for a perfect adiabatic gas is P = Po(p/ po)'Y, from which it can be deduced that

c2 = (ap) ap.

=

,,(Po Po

(.!!-)' Y-l = Po

co2

(.!!-)' Y-l Po

(3c-36)

No generalization of comparable simplicity is available for liquids. 2 )Vb.en (3c-36) is introduced in (3c-35), the exact "Lagrangian" wave equation fo>; an adiabatic perfect gas becomes a2~ (pL)'Y+1 -a2~ = co 2 ( 1 J.. -a~)-('Y+1) -a2~ (3c-37) = co 2 at 2 Po aa 2 aa aa 2 In the Lagrangian formulation illustrated above, the choice of a, the initial-position coordinate, as the independent variable is useful but any other coordinate that Rayleigh, "Theory of Sound," vol. II, §249; Lamb, "Hydrodynamics," §§13-15, 279-284. But see Courant and Friedrichs, "Supersonic Flow and Shock Waves," p. 8, Interscience Publishers, Inc., New York, 1948. 1

2

3-48

ACOUSTICS

identifies the particles would serve the same purpose. For example, the particle located momentarily at x can be uniquely identified by the material coordinate h ""

fox

p

dx, whcre h represents the mass of fluid contained between the origin and

the particle. Inasmuch as this included mass will not change as the particle moves, the use of h as an independent" mass" variable automatically satisfies the requirements of continuity, with some attendant simplification in the analysis of transient disturbances. In the undisturbed condition, P = PO and x = a, whence the relation a = hiPO allows the independent variables to be interchanged by direct substitution in (3c-37). Material and Spatial Coordinate Transforms. It is useful to have available a systematic procedure for converting a functional expression for one of the state variables from the form involving material coordinates to the corresponding form in spatial coordinates, or the inverse. One should avoid, however, the trap of referring to the state variables themselves as Lagrangian or Eulerian quantities; density and pressure, for example, are scalar point functions that can have only one value at a given place and time. On the other hand, it is of prime importance to distinguish carefully (and to specify!) the independent variables when computing the derivatives of these quantities. The E and L functions are tied together by the displacement variable ~, which provides a single-valued connection between the a particle and its instantaneous position coordinate x and which may therefore be regarded as a function of either of its terminal coordinates a or x. This can be indicated [cf. (3c-31)] by writing x(a,t) = a + Ha,t), or the inverse relation a(x,t) = x - Hx,t) , from which follow the alternate expressions a = x -

Ha,t)

x = a

+ Hx,t)

(3c-38)

The desired coordinate transforms can then be established by means of Taylor series expansions, the two forms following according to whether the expansion is centered on the instantaneous partiCle position or spatial coordinate x, or on the particle's equilibrium position or material coordinate a. Thus, if q is used to represent anyone of the variables p, P, T, or u, one of the expansions can be based on the obvious identity qL(a,t) =

qE(X,t)"~a+~(".t)

= qE (X,t)"~a

+ [Hx,t)

aqE (x,t)] ax "~a

+ 1: [~2(x,t) a2qE (~,t)] 2

ax

"~a

+...

(3c-39)

Note that all terms on the right of (3c-39) are functions of the spatial coordinates and that each is to be evaluated at the equilibrium position coordinate a. This transform yields, therefore, the instantaneous value in material coordinates of the variable represented by q, in terms of the local value of q modified by correction terms (comprising the succeeding terms of the series) based on the spatial rate of change of q and the instantaneous displacement. The inverse transform is derived in a similar way from the identity qE(X,t) =

[qL(a,t)]a~"_i;(a.t)

qE(X,t) =

[qL(a,t)]a~" - [Ha,t) aqL(a,t)] aa

a~"

+1: [~2(a,t) a2qL(~,t)J 2

aa

a~"

-

. . . (3c-40)

In symmetrical contrast with (3c-39), all terms on the right in (3c-40) are functions of the material coordinates and are to be evaluated for a = x. This transform, therefore, yields the instantaneous local value of the variable q at the place x, in terms of the instantaneous value of q for the now-displaced particle whose equilibrium position or material coordinate is a = x, modified by the succeeding terms of the series in accordance with the material-coordinate rate of change of q and the instantaneous displacement.

3-49

PROPAGATION OF SOUND IN FLUIDS

The transforms (3c-39) and (3c-40) indicate that the differences between qL and qE are of second order, which explains why the troublesome distinction between spatial and material coordinates does not intrude when only first-order effects are being considered. It also follows that the first two terms of these transforms are sufficient to deliver all terms of qL or qE through the second order. The use of these transforms can be illustrated by writing them out explicitly for u and p, including all second-order terms, 'ilL ;: ~, u E = 'ilL - ~UaL = ~, - ~~ta (3c-4Ia) pL = Po (I + ~a)-' = po(l - ~a + ~a2 - . • . ) pE = po(l - ~a + ~a2 + Haa) = poll - ~a + (Ha)aJ (3c-4Ib) in which the subscripts indicate partial differentiation with respect to a or t. The product of (3c-4la) and (3c-4Ib) gives at once the relation between the material and spatial coordinate expressions for the mass transport pu; thus, through second order, pEU E = pLu

-

HpLuL)a

+ e(paLU a L )

= po[~t -

(H,)a]

=

po[~ -

Halt

(3c-42)

It is then straightforward to show that, if the particle velocity w is simple harmonic, the time average of the local mass transport pEU E will vanish through the second order, even though the average value of u E is not zero. Note, however, that the displacement velocity ~, is measured from an equilibrium position that is here assumed to be static; the average mass transport may indeed take on nonvanishing values if the wave motion as a whole leads to gross streaming (see Sec. 3c-7). :"c-6. Waves of Finite Amplitude.' A distinguished tradition adheres to the study of .the propagation of unrestricted compressional waves. That the particle velocity is forwarded more rapidly in the condensed portion of the wave was known early (Poisson, 1808; Earnshaw, 1858; Riemann, 1859); and that this should lead eventually to the formation of a discontinuity or shock wave was recognized by Stokes (1848), interpreted by Rayleigh,2 discussed more recently by Fubini,3 and has been reviewed still more recently with heightened interest by modern students of blast-wave transmission. 4 By virtue of the adiabatic assumption underlying P = P(p), the speed of sound is also a function of density alone and may be approximated by the leading terms of its expansion about the equilibrium density: c2 =. co 2 [ 1 -

2~a Po Co

(DC) Dp

0

+ ... ]

(3c-43)

When (3c-43) is introduced in the exact wave equation in material coordinates, (3c-35), the latter can be recast in the following form, using the subscript convention for partial differentiation and retaining only, but all, terms through second order:

~tt

-

C02~aa

= -co 2 [

1

+ ~ (~~)

J (~a2)a

(3c-44)

If it is then assumed that an arbitrary plane displacement HO,t) = f(t) is impressed at the origin, it can be verified by direct substitution that a solution of (3c-44) is

Ha,t) =

f

(t -!::) + ~ [1 + i'.!l (DC) ] [f' (t - !::)J2 Dp CO

2c0 2

Co

0

(3c-45)

Co

The density variations associated with these disphwements are to be found by entering (3c-45) in (3c-32), and the variational pressure can then be evaluated in terms of the adiabatic compressibility of the medium. Relatively more attention has been devoted to the analysis of solutions of (3c-37) for the case of an adiabatic perfect gas. For an arbitrary initial displacement, as 1 For more recent developments see Sec. 3n, Nonlinear Acoustics (Theoretical), pp. 3-183 to 3-205. 2 "Theory of Sound," vol. II, §§249-253. Proc. Roy. Soc. (London) 84, 247-284 (1910). 3 Alta Frequenza 4, 530-581 (1935). 'See also Sec. 2y of this book, Shock Waves, pp. 2-273 to 2-278.

.

3-50

ACOUSTICS

above, the solution of the corresponding wave equation (3c-37), again including all terms through second order, is Ha,t)

=

I

(t -!!.) + ~ 2co' Co

'Y

+ 1 [I' (t 2

- !!.) J'

(3c-46)

Co

Technological interest in this problem centers on the generation of spurious harmonics, which can be studied by assuming the initial displacement to be simple harmonic, viz., I(t) = ~o(l - cos wt) at the origin. The solution then takes the explicit form Ha,t) =

~o[l

- cos (wt - koa)]

+ -1 ko'~o2a[1 + -'Y 8

- cos 2(wt - koa)]

(3c-47)

in which ko is written for the phase constant, ko = w/co = 27r/Ao, The most striking feature of the solutions (3c-45) and (3c-47) is the appearance of the material coordinate a in the coefficient of the second-harmonic term. As a consequence, the condensation wave front becomes progressively steeper as the wave propagates, the energy supplied at fundamental frequency being gradually diverted toward the higher harmonic components. The compensating diminution of the fundamental-frequency component would be exhibited explicitly if third-order terms had been retained in (3c-46) and (3c-47) inasmuch as all odd-order terms include a "contribution" to the fundamental. When such higher terms are retained it is predicted that propagation will always culminate in the formation of a shock wave at a distance from the source given approximately by a == 2~o/ (-y + l)M', where M is the peak value of the particle-velocity Mach number.l On the other hand, when dissipative mechanisms are taken into account, the fact that attenuation increases with frequency for either liquids or gases leads to the result that, except for very large in tial disturbances, the wavefront will achieve a maximum steepness when the propagation distance is such that the rate of energy conversion to higher frequencies by nonlinearity is just compensated by the increase of absorption at higher frequencies. Note, however, that this steepest wave front does not qualify as a "disturbance propagated without change of form." When attention is centered on the fundamental component, the diversion of energy to higher frequencies appears as an attenuation and accounts for the relatively more rapid absorption sometimes observed near a sound source.' The variational or acoustic pressure, in material coordinates, can be expressed generally as a function of the displacement gradients by using the adiabatic pressuredensity relation pL = Po(pL /po)'Y in conjuction with the continuity relation (3c-32); thus, (3c-48) in which the last member identifies the steady-state alteration of the average pressure and the fundamental and second-harmonic components of sound pressure. When the harmonic solution (3c-47) is introduced in (3c-48), the two alternating components of pressure for a' » (A/47r)' can be shown, after some algebraic manipulation, to be

+ V2 P, sin (wt - koa) + 1) sin 2(wt - koa) = V2 P sin 2(wt -

PI L

= +yPoM sin (wt - koa) =

P2 L

=

'YPoM'koa-Hy

2

koa)

(3c-49a) (3c-49b)

in which P, and P 2 are the rms values of the fundamental and second-harmonic sound pressures, and M = ko~o = w~o/co is again the peak value of the particle-velocity Mach number at the origin. The relative magnitude of P 2 :ncreases linearly with distance from the origin and is directly proportional to the peak Mach number, as may be deduced from (3c-49a) and (3c-49b); thus

P2 P,

=

1

4;

('Y

+ l)Mkoa

'Fubini, Alta Frequenza 4, 530-581 (1935). Fox and Wallace, J. Acoust. Soc. Am. 26, 994-1006 (1954). Soc. Am.:36, 534-542 (1964). 2

(3c-50)

Blackstock, J. Acoust.

3-51

PROPAGA'I'IONOF SOUND IN FLUIDS

Various experimental studies of second-harmonic generation have given results in reasonably good agreement with the predictions of (3c-50).1 The sound-induced alteration of mean total pressure, or "average" acoustic pressure, is given by the time-independent terms yielded by the substitution of (3c-47) in (3c-48), viz., (3c-51) Note that this pressure increment is given as a function of the material coordinates, which means that it pertains to a moving element of the fluid. The local value of the pressure change can be found by means of the transform (3c-40), which gives, through second-order terms, the following replacement for (3c-48), (3c-52) When (3c-47) is introduced in (3c-52), the time-independent terms give the local change in mean pressure as (3c-53) and since 'Y is usually less than 2, it follows that the local value of mean pressure will be reduced by the presence of the sound wave, in striking contrast to the increase of mean pressure that would be observed when following the motion of a particle of the medium. Negative pressure increments as large as 10 newtons m- 2 (100 dynes cm- 2) have been reported experimentally, in reasonably good agreement with (3c-53). The mean value of the material particle velocity, u L == ~t, vanishes, as may be seen by differentiating (3c-47). The local particle velocity that would be observed at a fixed spatial position does not similarly vanish, however, and may be shown, by using the transform (3c-40) again, to be UN

= 1;, -

H'a

( E u)

= - -1 C Oll!2 = - pocow2~02 --= 2 2

2poco

(pOC0 2)-1

( )

J

(3c-54)

where (J) is the average sound energy flux, or sound intensity.' Sc-7. Vorticity and Streaming. As suggested above, and with scant respect for the traditional symmetry of simple-harmonic motion, sound waves are found experimentally to exert net time-independent forces on the surfaces on which they impinge, and there is often aroused in the medium a pattern of steady-state flow that includes the formation of streams and eddies. The exact wave equation considered in the preceding section has been solved only for one-parameter waves (i.e., plane or spherical), and these solutions do not embrace some of the gross rotational flow patterns that are observed to occur. It is necessary, therefore, to revert for the study of these phenomena to the perturbation procedures introduced by the first- and second-order equations (3c-26) and (3c-29). It is plausible that vortices and eddies should arise, if there is any net transport at all, inasmuch as material continuity would require that any net flow in the direction of sound propagation must be made good in the steady state by recirculation toward the source. Streaming effects can be studied most usefully, therefore, in terms of the generation and diffusion of circulation, or vorticity. More specifically, the time average of the second-order velocity U2 will be a first-order measure of the streaming 1 Thuras, Jenkins, and O'Neil, J. Acoust. Soc. Am. 6, 173-180 (1935); Fay, J. Acoust. Soc. Am. 3, 222-241 (October, 1931); O. N. Geertsen, unpublished (ONR) Tech. Report no. III, May, 1951, D.C.L.A.; D. T. Blackstock, Report of the Fourth International Congress on Acoustics, Part I, 1962. 2 Westervelt, J. Acoust. Soc. Am. 22, 319-327 (1950).

3-52

ACOUSTICS

velocity. The vector function describing u. can always be resolved into solenoidal and lamellar components defined by V'A. = - (v Xu.)

(3c-55)

The irrotational component that represents the compressible, or acoustic, part of the fluid motion is derived from the scalar potential '1',. The vector potential A. is associated with the rotational component comprising the incompressible circulatory flow that is of primary interest in streaming phenomena. The failure of the first-order equations to predict streaming can be demonstrated by writing directly the curl of the first-order force equation (3c-26b). The gradient terms are eliminated by this operation, since V X v( ) == 0, leaving just (3c-56) Thus the first-order vorticity, RI == i(V X UI), if it has any value other than zero, obeys a typical homogeneous diffusion equation. On the other hand, it would appear to follow that, if RI were ever zero everywhere, its time derivative would also vanish everywhere and RI would be constrained always thereafter to remain zero. This is not a valid proof of the famous Lagrange-Cauchy proposition on the permanence of the irrotational state, but the absence of any source terms on the right-hand side of (3c-56) does indicate correctly I that first-order vorticity cannot be generated in the interior of a fluid even when viscosity and heat conduction are taken into account. Instead, first-order vorticity, if it exists at all, must diffuse inward from the boundaries under control of (3c-56).j A notably different result is obtained when the second-order equations are dealt with in the same way. It is useful, before taking the curl of (3c-29b), to eliminate the second and third terms of this equation by subtracting from it the product of (Pr/ po) and (3c-26b), and the product of UI and (3c-26a). In effect this raises the first-order equations to second order and then combines the information in both sets. The augmented second-order force equation can then be arranged in the form PO a~.

+ 21/o(V

X R.)

-2[(V1/I· V)UI

+ po'llpIV(V . UI)

+ V1/1

X (V X UI)] - Biv

- 2POPI(V X R I) - 2pO(UI X R I)

+ 2(V1/1

(~PI')

X R I)

+ PoV (~UI. UI) + B.vp.

- 1/o'llV(V . U2) -

V1/~ (V· UI)

= 0

(3c-57)

The following abbreviations have been used for the coefficients of VPI in (3c-26b) and of VP2 in (3c-29b): BI

co' [ == -;y 1

+ {jopo (D(h) DPI

0

]

(3c-58)

in which the quotients (VBr/VPI) and (vB./vp,) have been replaced by the corresponding material derivatives DB I Dp, which must be evaluated, of course, for the particular conditions of heat exchange satisfying the energy equations (3c-26c) and (3c-29c). This evaluation can be evaded temporarily (at the cost of neglecting vB I and VB,) by observing that each of the last five terms of (3c-57) contains a gradient. These disappear on taking the curl of (3c-57), whereupon the vorticity equation emerges as

aR. at -

poV'R.

=

aS I )+_ "21 Po'll ( VSI X V at Po IV X (UI . V)V'71

+ po81V'RI

-- POVSI X (V X R I) - V X (UI X RI)"+ PO-IV X (V1/1 X R I) I

St. Venant, Compt. rend. 68, 221-237 (1869).

(3c-59)

PROPAGATION OF SOUND IN FLUIDS

in which

·3-53

has been introduced as an abbreviation for the first-order condensation, This inhomogeneous diffusion equation puts in evidence various secondorder sources of vorticity: four vanish if the first-order motion is irrotational (R l = 0), and two drop out when the shear viscosity is constant (V7)l = 0). It is notable that the dilatational viscosity 7)' does not appear in any of these source terms except through the ratio 7)'/7) that forms part of the dimensionless viscosity number '0 "" 2 + (7)'/7). Except for the third source term, which (3c-56) shows to be one order smaller than the change rate of R l , all the vorticity sources would vanish -and the streaming would "stall"-if the wave front were strictly plane with Ul, 81, and 7) functions of only one space coordinate. Wave fronts cannot remain strictly plane at grazing incidence, however,' and rapid changes in the direction and magnitude of Ul will occur near reflecting surfaces, in the neighborhood of sound-scattering obstacles, and in thill viscous boundary layers. As a consequence, the "surfa,ce" source terms containing Rl become relatively more important in these cases. 2 In other circumstances, when the sound field is spatially restricted by source directionality, the first source term in (3c-59) dominates and leads to a steady-state streaming velocity proportional to the ratio of the dilatational and shear viscosity coefficients-and hence to a unique independent method of measuring this moot ratio. s Both the force that drives the fluid circulation and the viscous drag that opposes it are proportional to the kinematic viscosity, which does not therefore control the final value of streaming velocity but only the time constant of the motion, i.e., the time required to establish the steady state. 4 Evaluating the second-order vorticity source terms in any specific case requires that the first-order velocity field be known, and this calls in the usual way for solutions that satisfy the experimental boundary conditions and the wave equation. Unusual requirements of exactness are imposed on such solutions, moreover, by the fact that even the second-order acoustic equations yield only a first approximation to the mean particle velocity. The analysis of vorticity can be recast, by skillful abbreviation and judicious regrouping of the elements of (3c-57), in such a way as to yield a general law of rotational motion, according to which the average rate of increase of the moment of momentum of a fluid element responds to the difference between the sound-induced torque and a viscous torque arising from the induced flow. 5 A close relation has also been shown to exist in some cases between the streaming potential and the attenuation of sound by the medium without regard for whether the attenuation is caused by viscosity, heat conduction, or by some relaxation process; in effect the average momentum of the stream "conserves" the momentum diverted from the sound wave by absorption. 6 This principle has so far been established rigorously only for the adiabatic assumption under which P = P(p), and under restrictive assumptions on the variability of 7) and '0, but its prospective importance would appear to justify efforts to extend the generalization. 3c-8. Acoustical Energetics and Radiation Pressure. If the kinetic energy density that appeared briefly in (3c-12) is restored to (3c-18), the change rate of the specific 81

81

= PI! PO.

1 Morse, "Vibration and Sound," 2d ed., pp. 368-371, McGraw-Hill Book CompanY4 New York, 1948. 2 Medwin and Rudnick, J. Acoust. Soc. Am. 25, 538-540 (1953). 3 Liebermann, Phys. Rev. 15,1415-1422 (1949); Medwin, J. Aco",,;8t. Soc. Am. 25, 332-341 (1954). 4 Eckart, PhY8. Rev. '13, 68-76 (1948). 5 Nyborg, J. Acoust. Soc. Am. 25, 938-944 (1953); Vvestervelt, J. Acoust. Soc. Am. 25, 60-67 and errata, 799 (1953). 6 Nyborg, J. Acoust. Soc. A.m. 25, 68--75 (1953); Doak, Proc. Roy. Soc. (London), ser. A, 226, 7-16 (1954); Piercy and Lamb, Proc. Roy. Soc. (London), ser. A, 226, 43-50 (1954).

3-54:

AconSTICS

total energy density (per unit mass), E/p, can be formulated in terms of D(E/p) _ D(ju u) p -----nt - p Dt 0

+

D(iu u)

D. p Dt

Dv

0

=p

Dt

(3c-60)

-pPDt-Voq+cfJ>J

Material derivatives are used here so that the energy balance reckoned for a particular volume element will continue to hold as the derivatives "follow" the motion of the material particles. The mechanical work term on the right in (3c-60) can be resolved into two components by writing P = Po + p, where the excess, or sound, pressure p' now represents the sum of the variational components of all orders (p = Pi

+ P2 + ...)

Thus D(E/p) D(ju u) Dv Dt - pp Dt p -----nt = p 0

Dv

+ PP o Dt

- V

0

q

+

zw

I-

!;t ~

///

0.2

w

f3 n =3.0

~

,//

(!)

/

/

()

0 H, kilogauss

FIG. 3f-9. Magnetic-field variation of the attenuation of shear waves in annealed Nb-25 atom. % Zr at 4.2 K. Dashed curves are calculated values. (After Y. Shapira.)

+6r-------------------~----~

+4

+2

-14.'-;--__:~-__:~-L_:!_lLL1JlJ..,.....iLlJ

1.5

2

3

4 5 678 10 20 (X)

~H,

gauss - em FIG. 3f-1O. Relative attenuation in pure single-crystal copper as a f]Inction of the, product of the wavelength times the magnetic field for several orientations of magnetic field and wave direction. (After Morse.)

superconducting field H c2• Above Hc2 the material is in the normal state, and tha attenuation rises rapidly with the field. ' M agnetoacoustic8 and Fermi Surface Determinations. In the presence of a magnetic field, the attenuation in metals in the normal state shows variations which are cyclic when plotted as a function of }.H, where H is the magnetic field. Figure 3f-l0 shows

ACOUSTIC PROPERTIES OF SOLIDS

3-117

measurements in a very pure copper single crystal at 4.2 K. These cyclic variations can be related to the shape of the Fermi surface, which is a constant-energy surface that bounds the occupied states of electrons in momentum space. The electrical effects in a metal are primarily determined by the electrons whose energy is near the Fermi surface, since these are the only ones free to move. For free electrons, such surfac~s are spherical with a radius determined by the Fermi energy. The effect of the periodic crystal potential in the band-theory approximation is to distort the Fermi surface from a spherical surface. Electrons of the same energy (which all lie on the Fermi surface) will then have different momenta. Figure 3f-11 shows the probable Fermi surfaces for monovalent copper, gold, and silver, and their relation to the Brillouin zone. If an electron's orbit in momentum space carries it

FIG. 3f-l1. Fermi surfaces for copper, gold, and silver, and their relation to the Brillouin zone. (After Pippard.)

to the Brillouin zone face, the electron will be refracted to the opposite Brillouin zone face. In momentum space, this has the effect of repeating the zone over and over in an extended zone scheme. The effect of a magnetic field is to localize the electrons that can move onto a plane perpendicular to the magnetic field in momentum space. It can be shown that the periodicity of the attenuation-l\H curves can be related to the linear dimension of the Fermi surface perpendicular to the magnetic field and perpendicular in momentum space to the direction of wave propagation in real space. The various measurements of Fig. 3f-10 give details of the Fermi surface for different directions in momentum space. Several other types of oscillations in the attenuation occur.' These are the de Haas-van Alphen oscillations of the attenuation, the giant quantum oscillations, acoustic cyclotron resonance, and open orbit resonances. ,'See B. W. Roberts, "Physical Acoustics," vol. lVB, chap. 10, Academic Press, Inc. New York, 1968.

3g. Properties of Transducer Nlatel'ials W. P. MASON

Columbia University

To determine the acoustic properties of gases, liquids, and solids and to utilize them in acoustic systems, it is necessary to generate the appropriate waves by means of transducer materials which convert electrical energy into mechanical energy and vice versa. For liquids and solids, the most common types of materials are piezoelectric crystals, ferroelectric materials of the barium titanate type, and magnetostrictive materials. 3g-1. Piezoelectric Crystals. The static relations for a piezoelectric quartz crystal producing a single longitudinal mode are for rationalized mks units (3g-1)

where S2 and T2 are' the longitudinal strain and stress, respectively, the elastic compliance along the length measured at constant electric field, d21 the piezoelectric constant relating the strain with the applied field Ex, Dx the electric displacement, and f1T the dielectric constant measured at constant stress. Equations of this type suffice to determine the static and low-frequency behavior of piezoelectric crystals. Using the first equation, one finds that the increase in length for no external stress and the external force for no increase in length are, respectively, S22 E

(3g-2)

where V is the applied potential, l, w, and t are the length, width, and thickness of the crystal, and F is the force which is considered positive for an extensional stress. From the second equation one finds that the open-circuit voltage and the short-circuited charge for a given applied force are, respectively, V

=

_

(! (w Q = )0 )0 Dx dl dw

(d 21 ) IF fT

tw

=

d 21

Fl

t

(3g-3)

Another Important criterion for transducer use is the electromechanical-coupling factor k whose square is defined as the ratio of the energy stored in mechanical form' to .the total input electrical energy. Using Eqs, (3g-1), this can be shown to be (3g-4)

,It is readily shown that the clamped dielectric constant f S , obtained by setting S2 =' 0, and the constant-displacement elastic cOD2pliancesD , obtained by setting Jj~ == 0, are related to the constant-stress dielectric constant fT and the constant-field elastic compliance S22 E by the equations " S

filter section can efficiently transform mechanical into electrical energy and vice versa with a loss determined only by the dissipation in the elements of the crystal.

PROPERTIES OF TRANSDUCER MATERIALS

'rhe simplest method for mechanically resonating the crystal is to use it near its natural mechanical resonance. An exact equivalent circuit for a vibrating crystal is shown by Fig. 3g-1B. Near the first resonant frequency, the equivalent circuit for a clamped quarter-wave crystal is shown by Fig. 3g-1C while the equivalent circuit for a half-wave crystal is shown by Fig. 3g-1D. When the half-wave crystal resonated by a shunt coil is applied to converting electrical into mechanical energy, the same formulas given in Eqs. (3g-14) and (3g-15) are applicable except that- k2/(I- k 2) is replaced by (8/1I'2)[k 2/(I - k 2)]. By using the complete representation of Fig. 3g-IB the effect can be calculated by using various backing plates on the radiation from the front surface. The general form of Eq. (3g-1) holds for any single mode whether it is longitudinal or transverse as long as the appropriate constants are used. For longitudinal thickness modes when the radiating surface is a number of wavelengths in diameter, 822 E is replaced: by I/cuE and d 21 by e21/cuE, the appropriate thickness piezoelectric constant. For a thickness shear mode, the appropriate shear_Jltiffness (C4~" C66, or C66)

f. f2

f.

f2

FREQUENCY FREQUENCY FIG. - 3g-2. Use of equivalent circuit in determining _the optimum conditions for- energy transmission. replaces 1/822 and the appropriate shear piezoelectric constant replaces d 2l• Table 3g-1 lists the constants in mks units for a number of standard crystal cuts. 3g-2. Electrostrictive and Magnetostrictive Materials. Other types -oLmaterials that have been used in transducers are ferroelectric crystals and ceramics of the barium titanate type and ferromagnetic crystals, polycrystals, and sintered materials of the ferrite type. All these materials have changes in lengths proportional to squares and even powers of the polarization and to obtain a linear response they have to be polarized. These polarized materials have relations between stresses, strains,_electric and magnetic fields, and electric displacement and magnetic fiux similar to those for a piezoelectric crystal shown by Eq. (3g-I) and hence these materials can be said to have "equivalent" constants which depend not only on the material but also on the degree of poling and in some cases on aging effects. The dielectric and permeability constants are those associated' with the polarized medium as are also the elastic constants. To obtain these equivalent piezoelectric and piezomagnetic constants; one can start with the more fundamental potential equations which have the same form for. either eleCtrostrictive or magnetostrictive materials. For polycrystalline or sinteredmaterials, these potential equations can be written in the form'

Cf 3g-1.

'TABLE

Crystal and cut

Quartz X cut, length Y

Mode

L.L.

X cut

T.L.

Y cut

T.S.

Elastic constant, 10-11 m2/ newton 822 E

1 cuE

1

~

C66

f-"

PROPERTIES OF PIEZOELECTRIC CRYSTALS IN MKS UNITS

Piezoelectric constant d, 10-12 coulomb/ newton

Open-circuit Dielectric Electrovoltage capacitivity mechanical g = die, 11 coupling E, 10volt- meters / k farad/m newton

I:Y

Force factor

t-:)

I

dis,

Density,

newtons/ voltmeter

kg/m s

lOs

----

= 1.27

d 21 = 2.25

4.06

0.099

0.055

0.177

2.65

= 1.16

cuE

~ = -2.04

4.06

0.093

0.050

0.175

2.65

~ = +4.4

4.06

0.137

0.108

0.171

2.65

0.78

0.098

6.5

1.77

Q

0.288

0.29

0.287

1. 77

Ul

= 2.57

C66 E

Rochelle salt, 45-deg X cut L.L.

s~f = 6.7

d 14 = 435

45-deg Y cut

L.L.

s~f = 9.89

d225 = -28 . 4

ADP, 45-deg Z cut

L.L.

sf;

= 5.. 3

d236 = 24 . 6

13.8

0.29

0.178

0.465

1.804

KDP, 45-deg Z cut

L.L.

sf;

= 4.85

d236 = 10 . 7

19.6

0.12

0.058

0.22

2.31

EDT, Y cut, length X DKT, 45-deg Z cut L.H., Y cut

L.L. L.L. T.L.

3.88 sf; = 4.25 _1_ = 2

7.4 5.8

0.215 0.245

0.152 0.21

0.29 0.287

1.538 1.988

~E = 15

9.15

0.35

0.165

0.75

2.06

d21 + d 22 + d 2s = 13

9.15

~ = -1.84

6.65

0.3

3.1

d3l + d 33 = -2.16

6.65

S11 E =

C22 E

L.B., hydrostatic

H.

Tourmaline, Z cut

T.L.

Tourmaline, hydrostatic

H.

1 C33 E

2

d21 = 1l.Z d;l = -12.2

444.0 9.85

C22

= 0.61

CaaE

0.143 0.092

0.0275 0.0325

I

Abbreviations: L.L. ~ length longitudinal; T.r.. ~ thickness longitudinal; T.S. = thickness shear; ADP dihydrogen phosphate; EDT ~ ethylene diamine tartrate; L.R. = lithium sulfate monohydrate.

~

ammonium dihydrogen phosphate; KDP

~

potassium

i>

o

c::i 1-3 ....,

Q

Ul

3-12,3

PROPER'fIES OF TRANSDUCER MA'fERIALS

.G

-j[SllD(TI2 + T,2 + T3 2 ) + 2S12 D(T 1T, + TIT3 + T,Ta) + 2(SllD - SI,D)(T 4 2 + T 5' + T 6'») - {Qll(D1'TI +'D,'T, + D32T3) + QdT1(D2' + D 3') + T,(D 1' + D a') + T 3(DI' + D,'») + 2(Qll - Q12)(T.D,D 3 + TsDID3 + T 6D 1D,)} + ~,611T(DI' + D,2 + D,2) + K l1T(D 1' + D,' + D,4) + K12T(DI2D22 + D1'D,' + D 22D3') + Kl1IT(D 1S + D 2s + D,S) + K 112 T[D I4(D,' + D,2) + D,4(D I' + D,2) + D a4(D I' + D22») + K123 T DI2 D22 D,2 (3g-16),t

where T I, T" T, are the three extensional stresses, T., T s, T6 the three shearing stresses, D I, D" D, the three components of the electrical displacement for ferroelectric materials or the three components of the magnetic flux B for ferromagnetic materials, the s constants are the compliance constants for an isotropic material measured at constant electric or magnetic displacement, the Q's are the electrostrictive or magnetostrictive constants, ,611T the inverse of the initial dielectric constant or permeability measured at constant stress, and the KT'S are constants determining the total energy stored for higher polarizations. The static equations can ,be ohtain\ld by differentiation of G according to the relations E

=~~

(3g-17)

maD",

Since linear equations are obtained only if a permanent polarization Po is introduced, we assume that (3g-18) Da = Po + D,* where Da* is a small variable component superposed on P~. Also, DI and D, are small so that their squares and higher powers can be neglected compared with Po. Introducing these into (3g-16) and differentiating, we have Sl1 DT I + s12 D(T, + T I) + QI'(P O' + 2PoD,*) SllDT 2 + sl,D(TI + T,) + Q12(P O' + 2PoD,*) SllDT, + S12D(TI + T,) + Qll(P 02 + 2PoD,*) 2(SllD - S12D)T4 + 2(Ql1 - QI2)P oD 2 2(SllD - SI2D)Ts + 2(Ql1 - QdP oD 1 2(Sl1D - 812D)T6 El = -2(Ql1 - QdPoTs + D I(,611T + 2K12TP 0 2 + 2K 112 TP O') E2 = -2(Ql1 - Q12)PoT 4 + D,(,611 T + 2K1,TP O' + 2K 112 TP 04) E3 = -2Ql1PoT, - 2Q I2P O(T I + T,) + D3*(,611T + 12KllTP o'

81 = 8, = 8, = 84 = 85 = 86 =

(3g-19)

+ 30K

l11

TP o')

It is obvious that the variable components of Eq. (3g-19) follow the same rule as for a piezoelectric crystal. There are three longitudinal modes and a shearing mode. The length longitudinal mode has the following constants: 2QI,Fo d n = ,6"T(P o)

1 .3,T(P s) - - - ,6,,(P o)

(3g-20)

where ,6"T(P o) = (,611 T + 12K l1 TP o' + 30K I11 TP o4) is the dielectric impermeability of the ceramic when it has a permanent polarization Po 2Ql1P O d" = ,6"T(P o) -

I:: 2:

V

30

lt)

W

Q:

0

//

40

50

V

/ I

(0

100

200

TOTAL BEAM ANGLE IN DEGREES

FIG. 3i-3. Directivity index of a piston or ring as a function of total beam angle where beam angle is defined as the included angle of the main beam between the lO-decibel-down points in the directional response. (Computed from Massa, "Acoustic Design Charts," The Blakiston Division, McGraw-Hill Book Company, Inc., New York, 1942.)

Si-4. Directivity Index. It has already been mentioned that a directional transducer has an advantage over a nondirectional structure whenever it is desired to send or receive signals from a particular localized direction only . The fact that the directional transducer is less sensitive to sounds coming from random undesired directions makes it possible for it to detect weaker signals than would be possible with a nondirectional unit. The measure of this improvement in decibels corresponds to the directivity index of the transducer, which is 10 times the logarithm (to the base 10) of the ratio of intensity of the response along the axis of maximum sensitivity to the average intensity of the response over the entire spherical region surrounding the transducer. See Sec. 3a for a more detailed definition. The directivity index of a transducer is expressed in decibels, and a plot of the directivity index as a function of beam width for a piston or ring is shown in Fig. 3i-3.

3j. Architectural Acoustics CYRIL M. HARRIS

Columbia University

3j-1. Sound-absorptive Materials. When sound waves strike a surface, the energy may be divided into three portions: the incident, reflected, and absorbed energy. Suppose piane waves are incident on a surface of infinite extent. For this case, the absorption coefficient a of the surface may be defined as

a

Is Ip· ds

1.

IA

•

(3j-l)

ds

where Ip is the time average of the intensity vector of the sound field at the absorptive surface, ds is the vector surface element-the positive direction being into the material from the incident side, and IA is the time average of the intensity vector which would exist at the surface element if the surface were removed. The absorption coefficient defined above is a function of angle of incidence and frequency. For acoustical designing in architecture, it is convenient to employ an absorption coeffieient a (at a given sound frequency) which represents an average over all angles of incidence. But a depends also on the area of the absorbent surface; the larger the area of a sound absorber on a wall, floor, or ceiling of a room, the smaller is its sound absorption coefficient. The data for a presented in this section are for measurements made on areas of about 72 sq ft, but we assume these are valid for all areas. A surface of S ft2 is said to have an absorption of as sabins. Thus the sabin (sometimes called a square-foot unit of absorption) is the absorption equivalent of 1 ft 2 of material having an absorption coefficient of unity. A quantity which describes the acoustical properties of a material that is mme fundamental than absorption coefficient is its acoustic impedance, defined as the complex ratio of sound pressure to the corresponding particle velocity at the surface of the material. Because of the complexities involved in the solutions to problems of roem acoustics by boundary-value theory in terms of boundary impedances, the simpler concept of absorption coefficient is usually employed in calculating the acoustical properties of rooms, as indicated in the following section. Most manufactured acoustical materials depend largely on their porosity for their acoustic absorption, the sound waves being converted into heat as they are propagated into the interstices of the material and also by vibration of the small fibers of the material. Another important mechanism of absorption is panel vibration; when sound waves force a panel into motion, the resulting flexural vibration converts a fraction of the incident sound energy into heat. The average value of absorption eoefIicient of a material varies with frequency. Tables usually list the values of a at 125, 250, 500, 1,000, 2,000, and 4,000 Hz, or at 3-144

3-145

ARCHITECTURAL ACOUSTICS

128, 256, 512, 1,024, 2,048, and 4,096 Hz, which for practical purposes are identical. In comparing materials which are used for noise-reduction purposes in offices, banks, corridors, etc., it is sometimes useful to employ a single figure called the noisereduction coefficient (abbreviated NRC) of 1-1.00,-------------. the material which is the average of the NRC iii .90 ---.10 absorption coefficients at 250, 500, 1,000, .80 ---.67 and 2,000 Hz, to the nearest mUltiple of B:w .70 --.64 0.05. 8.60 z .50 Figures 3j-1 through 3j-3 give the ab~ .40sorption coefficient vs. frequency for sev~ .30 eral types of acoustical material.1 The ab- - CEMENTED ~ .20 ---- ON FURRING sorption-frequency characteristics of regu- - - ON SUSPENSION « .10 larly perforated cellulose fiber tile ! in. __SYSTEM _ __J 125 250 500 1,0002,0004,000 thick is shown in Fig. 3j-1. These curves FREQUENCY, Hz represent average coefficients for materials of the same type, thickness, and method of FIG. 3j-1. The absorption vs. frequency characteristic for regularly perforated celmounting but of different manufacture. Similar data are shown in Fig. 3j-2 for lulose fiber acoustical tile. These data represent average values for ~-in.-thick fissured mineral tile i ~ in. thick. Values tile, mounted in the same way but of difof noise-reduction coefficient are shown to ferent manufacture. (After H. J. Sabine, the right of the graph. Values of absorp- chap. 18 in "Handbook of Noise Control," M. Harris, ed., McGraw-Hill Book tion coefficient for various types of building C. Company, New York, 1957.) 1 materials are given in Table 3j-1. The equivalent absorption of individuals and seats, expressed in sabins, is given in Table 3j-2. More complete data and data for other types of material are given in the literature. 1,2 Sound-absorptive materials and structures may be classified in the following way: (1) prefabricated units, including acoustical tile, tile boards, and certain mechanically O~·~--L--~-~

I-

tJ

1.00 .90

~ .80

~~

~ .60 ~ .50 2 .40 ll: .30 a:: o 20

[:2 '10 « .0

~~:... --

I-

1.00

Q .90 80

.69

ii:'

.66

~~

8 - - CEMENTED

- - - ON SUSPENSION

SYSTEM 125 250 500 1,000

4,000

FREQUENCY, Hz FIG. 3j-2. The absorption vs. frequency characteristic for fissured mineral tile. These data represent average values for H-in.-thick tile, mounted in the same way but of different manufacture. (After H. J. Sabine, chap. 18 in "Handbook of Noise Control," C. M. Harris, ed., McGraw-Hill Book Company, New York, 1957.)

~RC .73--1" .68-3/4"

I.

.58----112"

l~

,....-__~. . . . .

l/

.60

z .50 ~ .40 c.. .30 !5 tI) .20

L_~

~

',,I IJ

A"

,&1

b,l

~ .10 ,/ OL--:-!:-=-."-!.,,.....-:~-:-c~--l---,,...}.

125 250 500 1,000

4,000

FREQUENCY, Hz FIG. 3j-3. The absorption vs. frequency characteristic for regularly perforated cellulose fiber acoustical tile which has been spot-cemented to a rigid surface. These data represent the average value for tiles of different manufacture, mounted in the same way and having different thicknesses. (After H. J. Sabine, chap. 18 in "Handbook of Noise Control," C. M. Harris, ed., McGraw-Hill Book Company, New York, 1957.)

perforated units backed with absorptive material; (2) acoustical plasters; (3) acoustical blankets, consisting of mineral wool, glass fibers, hair felt, or wood fibers held together in blanket form by a suitable binder; (4) panel absorbers, including panels of plywood, paperboard, and pressed-wood fiber; (5) membrane absorbers consisting of a membrane of negligible stiffness backed by an enclosed air space; (6) resonator absorbers of the Helmholtz type; and (7) special types. 1 Acoust. jJlatetials Assoc., Bull. XXIX, New York, 1969. 2 For example, see V. O. Knudsen and C. M. Harris. "Acoustical Designing in Architecture," John Wiley & Sons, Inc., New York, 1950.

3-'-146

ACOUSTICS

Frequency, Hz Materials 125

250

500

1,000 2,000 4,000

-------------
-~-I·-~I--------

Brick, unglazed ...... , .............. . Brick, unglazed, painted.......... , .. . Carpet, heavy, on con~rete ...... , .... . Same, on 40-oz hairfelt orfoam rubber. Same, with impermeable latex backing on 40-oz hairfelt or foam rubber .. . Concrete block, coarse ..... .' .... , .... . Concrete block, painted .............. . Fabrics: Light velour, 10 oz/yd 2 hung straight, in contact with wall ............. . Medium velour, 14 oz/yd 2, draped to half area .......................... Heavy velour, 18 'oz/yd 2, draped to half area .................... : ... Floors: Concrete or terrazzo ................ Linoleum, asphalt,rubber, or cork tile on concrete ......... .' ... : ......... Wood ... :'......................... Wood parquet in asphalt on concrete. GIas.s: Large panes of heavy plate glass·.... . Ordinary window glass ............. . Gypsum board, i in. itailed to 2 X4's 16 in. o.c ......... : ........ ; ...... . Marble or glazed tile.......... " ...... . Openings: Stage, depending on furnishings ..... . Deep balcony, upholstered seats...... . Grills, 'ventilating .. > ..... .- ........ . Plaster, gypsum or lime, smooth finish . on tile or brick .................... , .. l'laster, gypsum, or lime, r01.lghfinish on, lath .................. , ......... ", Same, with smooth unish ........... , Plywood paneling, t in. thick, , , .. : , .. . Water surface, as in a swimming pool. : '.

* From Acoust. Materia.!s Assoc. Bull.

0~04

0.03 0:03 0·010.01 0.02 '0.06 0.98 . 0.2.4

0.03 0 .. 02 0.14 0.57

0.02 0.37 0.69

0.05 0.02 0.60 0.71

0.07 0.03 0.65 0.73

(r08 0:36 0:10

0.27 0.44 0.05

0.39 0.31 0.06

0.34 0.29 0.07

0.48 0.39 0.09

0.63 0.25 0.08

0:03' .0.04

0.11

0.17

0.24

0.35

0.07

0.31

0.49

0~75.

0.70

0.60

o.h

0.:35

0 .. 55' 0.72

0.70

0.65,

0.01

0.01.0.015 0.02

0.02

0.02

0.02 0.15 0.04

0 . 03 0.11 0.04

0,03 0.03 0.10 0.07 0.07; 0.06

0.03 0.06 0.06

0.02 0.07. 0.07

0.18 0.35

0.06 '0.25

0.29 0.01

0.10 0.01

0.03 0.02 0.12 '0.07 0.05 0.01

0 .. 04 0.01

0.02 0.04

0.07- 0.09 0.02. 0.02

0,25-0.75 0.50--1.00 0.15-0.50 0.013 0.015 0.02. .0.03

0.04

0.02 0.03 0.02 0:02 0.28 0.22 0.008 0.008

0.04 0.04

0.04 0.03 0.17 0.013

.0.05

0.03 ' 0.03 0·99 0.10 0.11 0.015 0.020 0.025 0.05 0.04

XXIX, New York, 1969.

'Some tables list the ','ceiling attep.uation factor" of acoustical materials designed for use hi suspended ceilings. This factor is a measure of the reduction of sound level between two contiguous rooms when the transmission path of the sound is througb the two su~pended ceilings and the plenum common to both. 3j-2. Reverberation-time Calculations. Mter sound has. been produced in or enters an enclosed space, it will be refiectedhy the Qoundaries .of the enclosure.

3-147

ARCHITECTURAL ACOUSTICS

Although some energy is lost at each reflection, several seconds may elapse before the sound decays to inaudibility~ This prolongation of sound after the original source has stopped is called reveTberation, a certain amount of which is found to add a pleasing characteristic to the acoustical qualities of a room. On the other hand, excessive reverberation can ruin the acoustical properties of an otherwise well-designed room. TABLE

3j-2. ABSORPTION OF SEATS AND AUDIENCE* (In sabins per person or unit of seating) 125 Hz

250 Hz

500 Hz

1,000 2,000 4,000 Hz Hz Hz

3.3

4.1

4.8

5.3

5.1

4.7

2.7

3.6

4.4

4.8

4.5

3.9

2.4 3.1 0.15

3.0 3.4 0.19

3.3 4.1 0.22

3.4 4.7 0.39

3.2 5.0 0.38

2.6 4.7 0.30

- - - - - - - - - - - - --- --Audience, seated in upholstered seats ... Unoccupied seats, cloth-covered, upholstered .......................... Unoccupied seats, leather-covered, upholstered .......................... Wooden pews, occupied ............... Chairs, metal or wood seats ...........

I

* Based on values given in AcoUBt. Mate:rials Assoc. Bull. XXIX, New York, 1969, modified by author.

Materials and methods of fabrication can greatly influence the above values.

Because of the importance of the proper control of reverberation in rooms, a standard of measure called reverberation time (abbreviated tao) has been established. It is one of the important parameters in architectural acoustics. This is the time required for a specified sound to die away to one-thousandth of its initial pressure, a drop in sound pressure level of 60 dB. It is given by the following equation:

t

_ 60 -

0.049V Sa + 4mV

sec

(3j-2)

where V = volume of the room, fts S = total surface area, ft' a = average absorption coefficient given by

+ Ot,S, + OtsSs + ... = !!. (3j-3) 8 1 + S. + 8 s + . . . S Otl = absorption coefficient of area Sl, etc. a = total absorption in the room, sabins The quantity m is the attenuation coefficient for air given by Fig. 3d-6. For relatively small auditoriums and frequencies below 2,000 Hz, the mV term can usually be neglected so that Eq. (3j-2) reduces to a = OtlSl

t60

=

o.O:!V

sec

(3j-4)

3j-S. Optimum Reverberation Time. A certain amount of reverberation in a room adds a pleasing quality to music. Since the reverberation time one would consider to be optimum is a matter of personal preference, it is not a quantity that can be calculated from a formula. On the other hand, useful engineering-design data can be obtained from a critical evaluation of empirical data based upon the preference evaluations of large groups of individuals. The results of such information from all available sources considered reliable, in this country and abroad, have been carefully evaluated by Knudsen and Harris,l who have published the curves for optimum 1

Ibid.

3-148

ACOUSTICS

reverberation time shown in Figs. 3j-4 and 3j-5. The data in Fig. 3j-4 give the optimum reverberation times at 500 Hz as a function of volume for rooms and auditoriums that are used for different purposes. Since the optimum reverberation time for music depends on the type of music, it is represented by a broad'band. The optimum reverberation time for a" room used primarily for speech is considerably shorter; a reverberation time longer than those shown results in a decrease in speech intelligibility. I

2.v

'\l?-~-~~

1.6

c

'"

='= 1.0 w

::;:

i= O.B

0.6 0.4

'lI~i(~ilFC""W/' ~ ~f@~~E8~

.-. ~ ~ 0; ~::/, ::/,

~ ~ r0; ~ f%:' f,%

1.4

~ ~ ~ ~ ~ 1«1% z ~ ~ 0;'0 ~ ~ l?Z ~~ o u 1.2 ~~ ~ ~ I72Z rz;; ~ w

60

20

10

PER50SQUARE METER

i-------+---i----+.--1,---t-i n

45 m

-0

:i

'--_ _ _ _ _ 40 •

" ~--1/2

I

I

35 r------f--..---l--~1/2" frIll"

Ci5 (J)

"

I ll"

IrJ' ~~'" . 1/2"

3/8 GYPSUM .LATH

'2"

13"1'

~'"

"

-oil

I--

"i

iiI';:

. ...~

'lifii.,jii.:·r· HARDWOOD _1t1 FLOOR

I

:l.:t-4'-'CTIN-DE-R-Bt-L-OC-K+-+-+'-+-'

f'fTER '. .-1\PU~STER-I'METAL LATH

11 m

1

~ DOOR,13i4" SOLID OAK

20

l.l

D. Berendt, G. E. Winzer, and C. B. Burroughs, "A Guide to Airborne, Impact, and Structure Borne Noise-Control in Multifamily Dwellings," U.S. Department of Housing and Urban DeVelopment, Washington, D.C., September, 1967. For average values for other types of construction including doors and windowpane materials, see Fig. 3i-6. For the definition of STC used to obtain the ratings (above), see ASTM Rept. E90-66T, Tentathe Reoommended Practice for Laboratory Measurement of Airborne Sound Transmission Loss of Building Walia and Floors.

1-:3 H

l.l

U2

ARCHITECTURAL ACOUSTICS

8-153

if a value of pC = 40.8 rayls is assumed for air; W = power of the sound source in watts, and a = total absorption of the room in sabins. A consideration of the above formula shows that, if the acoustic-power output of the noise source remains constant, and if the total absorption in the room is increased from al to a2, the reduction in noise level is given by Noise reduction = 10 log ~

5

within the external meatus. The values in Table 3k-2 are representative but arp. subject to wide variations among individuals. 1 Sk-S. Minimum Audible Sound. Table 3k-3lists the minimum audible (threshoiu, sound pressures of pure tones measured at the entrance to the external meatus. The pressure measurements were made when the subject heard the tone one-half the time it was presented via an earphone applied to his ear with a standard static force. Observations were made on young persons, eighteen to twenty-five years of age, with no record of hearing impairment. Sound pressures were determined with a probe-tube microphone a:i:J.~d are given in decibels relative to 2 X 10-4 dyne/cm 2 • The results of such measurements made in various laboratories show a considerable amount of variation. The pressures in Table 3k-3 are based on measurements made in two indep~endent laboratories; see the first footnote for details. The variance in the threshold sound pressures measured at the entrance to the meatus has :been so great that such pressures cannot serve usefully as standards for audiometry. Experience has shown that the most accurate method for storing audiometric standard threshold information is as follows. Measurements of threshold voltages on an earphone applied to a number of young persons at the various audiometric frequencies are the primary data. The sound pressures which are produced by these voltages when the earphone is applied to an artificial ear (coupler) then serve as the standard thresholds for that particular earphone-coupler combination. This method of measuring and storing standard threshold sound pressures is now in use in several countries. A comparison of the standard thresholds was completed under the auspices of Technical Committee 43 on Acoustics of the International Organization for Standardization (ISO). An internationally agreed-upon standard threshold has been issued by ISO in its Recommendation R389, Standard Reference Zero for the Calibration of Pure Tone Audiometers. The standard data in it are sound pressures corresponding to the threshold of hearing for five earphone-coupler combinations now in use in several countries. 2 Sk-4. Threshold of Feeling or Discomfort. The upper limit for a tolerable intensity of sound rises substantially with increasing habituation. Moreover, a variety of subjective effects are reported, such as discomfort, tickle, pressure, and pain, each at a -alightly different level. As a simple engineering estimate it can be said that naIve listeners reach a limit at about 125 dB SPL and experienced listeners at 135 to 140 dB. These are overall measures of sound falling within the audible range and are roughly independent of frequency. 3k-5. Differential Thresholds for Pure Tones and Noise. A differential threshold represents a cl),reful determination by laboratory methods of the ability of a subject to just detect, and report, a difference in any specific property of a sound, all other factors presumably being held constant. The method for determining the differential threshold for intensity of pure tones employed one tone beating with a second tone at 3 beats per second. 3 Much evidence is available to support what should be kept always in mind, that thresholds determined by other methods are a function of numerous psychological parameters and will differ systematically from the values in Table 3k-4. A more conventional method was used to determine the thresholds for white noise, with the results given in the last column. 4 1 E. Waetzmann and L. Keibs, Horschwellenbestimmungen mit dem Thermophon und Messungen am Trommelfell, ~ Ann. Physik 26, 141-144 (1936); O. Metz, The Acoustic Impedance Measured on Normal and Pathological Ears, Acta Oto-Laryngol., Suppl. 63, 1-254 (1946); A. H. Inglis, C. H. G. Gray, and R. T. Jenkins, A Voice and Ear for Telephone Measurements, Bell System Tech. J. 11,293--317 (1932). 2 P. G. Weissler, International Standard Reference Zero for Audiometers, J. Aco"st. Soc. Am. 44, 264-275 (1968). 'R. R. Reisz, Differential Intensity Sensitivity of the Ear for Pure Tones, Phys. ReI'. 31,867-875 (1928). 4 G. A. Miller, Sensitivity to Changes in the Intensity of White Noise and Its Relation to MasKing and Loudness, J. Acoust. Soc. Am. 19, 609-619 (1947).

3-156

ACOUSTICS

The ability to distinguish pitch is subject to a greater range of individual variability than other functions reported here. The data given are for three trained listeners and have been smoothed in both directions. Untrained listeners usually require a greater TABLE 3k-2. ACOUSTIC IMPEDANCE OF THE EAR IN ACOUSTIC OHMS, MEASURED JUST WITHIN THE 11EATUS Frequency

Total impedance

Resistive component

250 350 500 700 1,000

200 150 125 70 55

50 40 35 25 25

Reactive component -190 -145 -1l5 -65 -50

Above 1,000 Hz measurements depend increasingly on the method of measurement.

TABLE 3k-3. MINIMUM AUDIBLE (THRESHOLD) PRESSURE AT ENTRANCE TO EXTERNAL EAR OANAL (MAO) * (In dB re 2 X 10- 4 dyne/cm 2 ) Frequency, Hz 125

250

500

1,000

1,500 2,000

- - ---- ----MAO

22

35

14

8

3,000

4,000

6,000 8,000

10,000

------ --- --- --- ---

9

9

10

9

14

17

16

The following quantities are to be added in order to obtain threshold pressures for other conditions: a. MAO to Threshold Pressure at Tympanic Membranet

Frequency, Hz 125

250

500

1,000

2,000

4,000

6,000

8,000

--- --- --------- ------ --Add ...........

0.0

0.0

-0.5

-1.0

-4.5

-10.5

-4.0

-2.5

b. MAO to Free Field (MAF) (plane wave, 0 0 azimuth in absence of head)t Frequency, Hz 125

250

500

1,000

2,000

--- ------ ------ ---

Add ....

*

+1.0

+0.5

-2.0

-4.0

4,000

6,000

8,000

10,000

--- --- --- ---

-11.0 -12.5

-7.0

-3.0

-3.0

J. P. Albnte, R. E. Shutts, M. B. Whitlock, R. K. Cook, E. L. R. Corliss, and M. D. Burkhard, Research III Normal Threshold of Hearing, AMA Arch. Otolaryngol. 68, 194-198 (1958). t F. M. Wiener and D. A. Ross, The Pressure Distribution in the Auditory Canal in a Progressive Sound Field, J. Acoust. Soc. Am. 18, 401--408 (1946). :t: L. J. Sivian and S, D. White, On Minimum Audible Sound Fields, J. Acoust. Soc. Am. 4, 288-321 (1933).

SPEECH AND HEARING TABLE

3k-3.

3-157

MINIMUM AUDIBLE (THRESHOLD) PRESSURE AT ENTRANCE TO EXTERNAL EAR CANAL (MAC)

(Continued)

c. Mean Monaural to Mean Binaural Listening § Frequency, Hz 125-2,000

4,000

6,000

8,000

10,000

-2.0

-3.0

-4.0

-5.0

-6.0

Add ...................

d. Reference Age Group (18-25) to Older Age Groups 'if

Frequency, Hz

Add for: Men 3D-39 ......... Men 40-49 ......... Men 50-59 ......... Women 30-39 ...... Women 40-49 ...... Women 50-59 ......

125-1,000

2,000

4,000

6,000

8,000

10,000

+1.0 +2.0 +5.0 +1.0 +3.0 +5.0

+2.0 +5.0 +13.0 +2.0 +5.0 +9.0

+5.0 +13.0 +27.0 +3.0 +6.0 +13.0

+6.0 +13.0 +32.0 +4.0 +8.0 +18.0

+6.0 +11.0 +35.0 +4.0 +9.0 +20.0

+7.0 +13.0 +35.0 +4.0 +9.0 +22.0

§ H. Fletcher, "Speech and Hearing in Communication," p. 131, D. Van Nostrand Company, Inc., Princeton, N.J., 1953. ~ J. C. Steinberg, H. C. Montgomery, and M. B. Gardner, Results of the World's Fair Hearing Tests, J. Acoust. Soc. Am. 12,291-301 (1940); J. C. Webster, H. W. Himes, and M. Lichtenstein, San Diego County Fair Hearing Survey, J. Acoust. Soc. Am. 22,473-483 (1950).

TABLE

3k-4.

DIFFERENTIAL THRESHOLD FOR INTENSITY, IN DECIBE.LS

Sensation level, dB above absolute threshold 5 10 20 30 40 50 60 70 80 90 100 110

Pure tones, frequency in 35

.... 7.24 4.31 2.72 1. 76

.... .... .... .... ... . ... . ... .

70

200

Hz

1,000 4,000 7,000 10,000

------------ --. ... 4.75 3.03 2.48 4.05 4.72 4.22 2.38 1.52 1.04 0.75 0.61 0.57

. ...

.... ... . ... .

3.44 1.93 1.24 0.86 0.68 0.53 0.45 0.41 0.41

.... ....

2.35 1.46 1.00 0.72 0.53 0.41 0.33 0.29 0.29 0.25 0.25

1.70 0.97 0.68 0.49 0.41 0.29 0.25 0.25 0.21 0.21

2.83 1.49 0.90 0.68 0.61 0.53 0.49 0.45 0.41

3.34 1.70 1.10 0.86 0.75 0.68 0.61 0.57

White noise 1.80 1.20 0.47 0.44 0.42 0.41 0.41

ACOUSTICS

3-158

frequency difference than that reported here. Note also that individual listeners commonly show idiosyncrasies at particular frequencies. TABLE 3k-5.

DIF~'ERENTIAL

Pure tones, frequency in Hz

Sensation level, dB above absolute threshold

125

50 0.0252 0.0140 0.0092 0.0073

5 10 15 20 30 I.

* J.

THRESHOLD FOR FREQUENCY, IN f1F IF"

0.0110 0.0060 0.0040 0.0032 II 0.0032

250

500

0.0097 0.0053 0.0035 0.0028 0.0028

0.0065 0.0035 0.0024 0.0019 0.0019

1,000

2,000

0.0049 0.0027 0.0018 0.0014 0.0014 I

0.0040 0.0022 0.0014 0.0012 0.0011

I

4,000 0.0077 0 .0042 0.0028 0.0022 0.0022

I

I

D. Harris, Pitch Discrimination, J. Aco".t. Soc. Am. 24, 750-755 (1952).

Sir-5. Masking. M:asking reff)m to OUI' inability to hear a weak sound ill the presence of a louder sound. It is usually measured by t.he amount of change in the threshold of the weaker sound, i.e., how much more intense the weak sound must be made in order to be heard over the masking sound than it needed to be when the masking sound was not present. The masking of one pure tone by another is a complex function of the particular frequencies and of the absolute level of the respective tones. See any standard text on hearing for the curves describing this relationship. The masking of a pure tone by a noise with a reasonably flat and continuous spectl'UlIl is a linear function (except at levels below 10 dB) of the total intensity within a "critical band" centered on the masked tone. The width of the critical band of frequencies whose total energy is just equal to the energy of the masked tone is given by Table 3Ic-5.

TABLE 3k-5. WIDTH OF "CRI'l'ICAL BAND" f1F AS A FUNC1'ION OF CENTER FREQUENCY F (10 log f."F)* Frequency, Hz 100

M, dB

~II~

1,000

2,000

4,000

8,000 10,000

19.417.117.118.019.923.127.729.2

* N. R. French and J. C. Steinberg, Factors Governing the Intelligibility of Speech Sounds, J. Acoust. Soc. Am. 19, 90-119 (1947). The III asking of a narrow-band noise by two tones, one higher and one lower than the noise, shows a similar relationship. The masking produced by the two tones overlaps unless the tones are separated by more than a "critical band," at which point the masking begins to fall off sharply. The critical band measured in this way is 3 to 4 dB wider than the values given in Table 3k-6. 1 The masking of one continuous noise by another can be thought of as a case of differential sensitivity to change in the intensity of a noise (see last column of Table 3k-4). Thus, above 40 dB SPL, if a weak noise is more than 10 dB less intense than a very similar masking noise, the weak noise will not be heard; its presence or absence 1 E. Zwicker, G. Flottorp, and S. S. Stevens, Critical Band Width in Loudness Summation, J. Acoust. Soc. Am. 29, 548-557 (1957). See especially the summary of the concept of critical bands, pp. 554-557.

3-159

SPEECH AND HEARING

does not produce a discriminable difference in intensity. If the spectral compositions of the two noises, masking and masked, are quite different, then the critica;I-band concept must be employed. Sk-7. Sounds of Short Duration. AcolLstic disturbances of very short duration, i.e., less than 0.0001 sec, are heard only to the extent that they transmit energy to the ear. Short pulses at ultrasonic frequencies are generally not heard unless they are rectified. Impulse or step functions excite the ear, but not efficiently. At the opposite extreme, tones, or continuous noise, of duration greater than from 0.2 to 0.5 sec are generally heard independently of duration. Between these limits relatively complex relations are found.! As a first approximation for both tones and noise, the effective intensity of short sounds is a function of total energy integrated over the duration of the sound. More accurately, the threshold is defined by2 It = kItO.8

(3k-1)

For some short tones and for many types of impulse noise, account must be taken of the frequency distribution of energy. Inasmuch as the ear varies in sensitivity as a function of frequency, any change in the shape or duration of a short acoustic pulse will also change its effectiveness because of the altered spectral composition. 3k-S. Loudness. Loudness and pitch are ways in which a listener reacts to sounds. Furthermore, within limits, a listener can use numbers to describe how much of a response he makes to the sound. These numbers usefully describe how loud or how high in pitch a sound seems to be. It is then necessary to relate how loud it is (subjective response) to how intense it is in physical terms. The loudness of a pure tone of 1,.100 Hz is described by the following relationship: log L = 0.0301N - 1.204

1

(3k-2)

in which L is the loudness measured in sones and N is the loudness level in phons (equal to the sound pressure level of the tone in decibels above 0.0002 dyne/cm 2).' Another way of putting this is to say that loudness doubles for each 10-dB change in sound pressure level. There is some evidence that the loudness of a noise grows more rapidly than that of a tone with an increase in sound pressure level, especially at low levels. The exact relations are less well known than those for a tone. The loudness of tones at other frequencies than 1,000 Hz is given by determining the loudness level in the manner described below and converting to sones by Eq. (3k-2). The loudness of noises can be measured by direct subjective comparison with a standard, such as a tone of 1,000 Hz, but such comparisons are difficult and need to be repeated by a number of judges. An approximation to the loudness of a noise can be calculated from measurements of the sound pressure level in a series of bands, usually a third-octave, a half-octave, or an octave in width, covering the audible spectrum. The total loudness of the noise is given by the formula i

Lt = 8m

+ F (18i

- 8m)

(3k-3)

! S. S. Stevens, ed., "Handbook of Experimental Psychology," pp. 1020-1021, John Wiley & Sons, Inc., New York, 1951. 2 D. B. Yntema, "The Probability of Hearing a Short Tone Near Threshold," Ph.D. Dissertation, Harvard University, 1954, 43 pp. 3 S. S. Stevens. The Measurement of Loudness, J. AcouBt. Soc. Am. 27, 815-82\1 (1955).

3-160

ACOUSTICS

The calculated loudness It should be qualified by the width of the bands used for its calculation. The terms 8 are empirical values of a loudness index shown as the parameter of the curves in Fig. 3k-1. The figure is entered with the geometric mean frequency of each band and the band pressure level as arguments. The loudness index 8 i is estimated for each of the i bands. The band having the greatesi , index 8 m is determined by inspection.

0.5 0.3 0.2 0.1 1OL.L":'5...Ll.L.Lf-j-""2!:-L.J.-!-1...J..lJt+--~.L...J+L.I..l..!..!---l j 1 5 2 5

100

1,000

10,000

FREQUENCY, Hz FIG. 3k-1. Loudness index S, as a function of geometric mean frequency of band measured and band pressure level (sound pressure level in third-octave, half-octave, or octave band under measurement). (Taken from S. S. Stevens, "Procedure for Calculating Loudness," Mark VI, Psycho-Acoustic Laboratory Report PNR-253, Mar. 1, 1961.)

As the formula indicates, total loudness is linearly additive except for a constant factor F that represents the reduction due to mutual masking of all bands except the loudest. The value of F depends on the width of the bands used. It has the value of 0.15 for third-octave, 0.2 for half-octave, and 0.3 for octave ba,nds. The loudness L t can be converted to loudness level by Eq. (3k-2).

SPEECH AND

3-161

HEARING

3k~9. Loudness L e v e l The loudness level of a tone of 1,000 Hz, expressed in phons, is denned as the sound pressure level in decibels above the reference level of 0.0002 d y n e / c m . T h e loudness level of tones of other frequencies is given b y the empirical relations in Table 3k-7. 2

TABLE

3k-7.

LOUDNESS L E V E L

AS A F U N C T I O N

AND

Sound pressure level

10 20 30 40 50 60 70 80 90 100 110

OP S O U N D

PRESSURE

LEVEL

FREQUENCY*

Frequency, Hz 125

250

500

1,000

2,000

4,000

8,000

10,000

4.0 17.0 34.0 52.0 70.0 86.0 98.0 108.0 118.0

6.3 18.0 31.0 45.5 59.5 72.5 84,5 95.5 105.5 115.5

16.0 26.5 38.5 52.0 64.5 76.0 86.0 96.0 105.0 113.0

10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0

18.0 28.0 37.0 45.5 55.0 64.0 73.5 84.5 95.0 106.0 117.0

18.0 28.0 36.5 45.0 54.0 63.5 72.5 83,0 94.5 106.0 117.5

11.0 20.5 29.5 38.0 47.0 56.0 66.0 77.0 88.0 101.5

17.0 26.0 35.0 43.5 53.5 63.5 73.5 85.5 98.0

* A m e r i c a n S t a n d a r d f o r N o i s e M e a s u r e m e n t , A S A Z24.2—1942.

N o t e that this table is based on the A S A standard and presumes the ' 'free-field" measurement of sound pressure. This requires a measurement of a plane progressive w a v e at the listener's position before the listener is placed in the field. M o r e meaningful measurements w o u l d doubtless be obtained from pressure measurements at the ear. For this purpose, apply the corrections contained in Table 3k-36 to the ear canal pressures before entering Table 3k-7. T o enter the table with sound pressure levels measured under other conditions, first add the corrections in Table 3k-36, then subtract rather than add corrections in Tables 3k-3a and 3k-3c. N o t e , however, that corrections given for presbycusis in Table 3k-3d m a y give quite misleading results because of recruitment at high frequencies in some elderly people. 3k-10. Pitch. The relation between frequency and the subjective magnitude of perceived pitch is shown b y Table 3k-8. B y definition, the pitch of a tone of 1,000 Hz at 40 d B SPL is 1,000 m e l s . 3 k - l l . Localization of Sound. T h e localization of complex sounds is primarily a function of time differences of arrival at the t w o ears, and t o a first approximation, such differences can be calculated b y assuming the ears on either end of the diameter of a sphere of 7.5 c m radius. T h e localization of tones of low frequency (below 1,500 H z ) is possible on the basis of phase differences, which m a y b e interpreted in terms of time differences. 1

T h e localization of tones of high frequency is possible on the basis of intensity differences resulting from the sound shadow of the head. Exact measurements here are difficult at best. S. S. Stevens and J. Volkmann, The Relation of Pitch to Frequency: A Revised Scale, Am. J. Psychol. 53, 329-353 (1940). 1

3-162

ACO U STICS

TAB LE 3k -8. PITCH OF A P URE T ONE, IN ME LS, Frequency

Mels

Frequency

AS A

F UNC'l'ION OF F l1 EQUE NCY

Frequency

Mels

Mels

"

20 30 40 60 80 100 150 200 250 300

0 24 46 87 126 161 237 301 358 409

350 400

460 508 602 690 775 854 929 1 , 000 1 , 154 1 , 296

.~OO

600 700 800 900 1 , 000 1 , 250 1 , 500

1, 750 2 , 000 2 , 500 3 , 000 3 , 500 4 , 000 5 , 000 6 , 000 7 , 000 10 , 000

1, 428 1 , 545 1 , 771 1 , 962 2 , 116 2 , 250 2 , 478 2 , 657 2 , 800 3,075

Sound loca lization is great ly aided when the head or body can be rotated or moved abou t in the sou nd field while t he observer hears t he appropriate sequence of so und s. ! Sound localiza t ion in reverberan t rooms or with so-ca lled " stereop honic-sound sources" depends critically upon a "precedence effect," by which t he localization determ ined by t he primary sound or sound from t he nearer of two sound sources is overriding in its effect .2 In experimen ts where tim e differences are used t o balance out intensity differences in the opposite direction, 1.0 X 10- 5 sec priority offsets a 6-dB difference in intensity ; 2.3 X 10 - 5 sec offsets a 14-dB difference in intensit y between t he two ears.3 Sk-12. Speech Power. The total radiated speech po wer , averaged over a 15-sec interval for a sample including bo th m en and women at con versational levels used fo r telephone ta lking, has been estima ted as 32 m icrowatts. When m easured a t t he face of a telephone t ransmitter, t his power produces t he sound pressure levels given in T able 3k-9 for different distances from t he mouth of the speaker .4 T ABLE 3k-9. AVERAGE SOUND PRESSURE LEVEL PRODUCED BY CONYER ATIO NAL SPEECH AS A FUNCTION OF D ISTA CE FROM LrPS TO MICROPHO E Dist ance, cm T ouching

0.5

1.0

2.5

5.0

-- -- -- Sound pressure level , . ' . ,

104

102

99

95

90

10.0 -

85

25.0

- 78

50.0 100.0 -

-

--

72

66

A second source of variability lies in the essen tially st at istical distribution of speech power in time. If speech power is m easured in successive -§--sec in tervals (a tim e slightly shor ter than a syllable and slightly longer t han a phoneme ), a distribu t ion is obtained wit h t he mean values given in T ab le 3k-9 and va riability t hat can be ! H . Wallach , U eber die Wahrnehmu ng d er Sch allricht ung, P sychol. Fo!'sch. 22, 238-266 (1938) . 2 H. Wallach, E. B. Newman, an d M. R . R osenzweig, T he Precedence Effect in Sound Lo ~ali zat i on , Am. J . P sychol. 62, 313-336 (1949). • J . H . Shaxby and F. H. Gage , St udies in the Localization of Sound. A. T he L ocalization of Soun ds in t h e M edian P lan e: An Experiment al I nvestigat ion of t h e Physical Processes Concerned, M ed. R esearch Council (Brit.) S pec. R ept. S er . n o. 166 (1932), 32 pp. , , 4 M . H . A b rams, S. J. Goffard, J. M iller, F. H. Sanford, a nd S. S. Stev ens , The Effect of Microphone P osition on the Intelligi bility of Speech in Noise, OS RD Rept. 4023 (1944) . 16 pp.

3-163

SPEECH AND HEARING

attributed to time sampling equalto a standard deviation of 7.0 dB.1 The distribution is badly skewed, so that the value 7.0 dB indicates only a rough order of magnitude. The variability is also greater when particular frequency bands are measured. A third source of variability is the variation in effort expended by the person who is talking. As a rough approximation, a raised voice level is 6 dB above conversational level, the loudest level that can be maintained is 12 dB above conversational level, and the loudest shout .is 18 dB above conversational level. In the other direction, a whisper may be 20 dB below conversational level. 3k-13. Speech Sounds TABLE

Symbol

3k-l0.

CHARACTERISTICS OF SOUNDS IN GENERAL AMERICAN SPEECH

Example

Power, * dB re long time averaget

Relative frequency of sound, %~

Formant frequencies for men and women 'If First

Second

--M W M

-

-

~-

W

Third M

W

- -- - --

cool cook cone talk

+0.6 +2.3 +2.5 +4.1

1.60 0.69 0.33 1.26

300 370 870 950 2,240 2,670 440 470 1,020 1,160 2,240 2,680 500 ... 820 570 590 840 920 2,410 2,710

D

cI~th}

Q

calm

+3.7

{2.81} 0.49

730 850 1,090 1,220 2,440 2,810

a

~Sk}

+2.5

3.95

660 860 1,720 2,050 2,410 2,850

+1.6 +1.4 0.0 0.0 -0.5

3.44 1.84 8.53 2.12 0.53 4.63 2.33

530 610 1,840 2,330 2,480 2,990

u U

0 0

III

e e I i (J'

a A

eI aI ju ou au 01

bat bet tape bit beet bird sofa bun laid bite you soap about boil -

..... +2.9 +1.4 +2.5 +0.6 +2.5 +2.3 +3.0

390 430 1,990 2,480 2,550 3,070 270 310 2,290 2,790 3,010 3,310 490 500 1,350 1,640 1,690 1,960 640 760 1,190 1,400 2,390 2,780

see e

1.59 0.31 1.30 0.59 0.09

* The power measurements do not represent the peak instantaneous power but the average over the ""stained portion of the phoneme where such a period can be defined. In this ca.se, as with the formant frequencies, the absolute values are highly variable, but intercomparisons among the vanous soun9.B are generally more reliable. I t H. Fletcher, "Speech and Hearing in Communication," p. 86, D. Van Nostrand Company. nc .. Princeton, N.J., 1953. ., C b 'd t G. Dewey, "Relative Frequency of English Speech Sounds." Harvard Umverslty Press, am n ge, Mass., 1923. . In N Y k '\[ E. G. Richardson, ed., "Technical Aspects of Sound," pp. 215-217, ElseVler Press, c., ew or. 1953. 1

H. K. Dunn and S. D. White, Statistical Measurements on ConversatiQn!l1' ~:PElElQ~ •.

J. Acoust. Soc. Am. 11,278-288 (1940).

3-164

ACODSTICS TABLE

3k-1O.

CHARACTERISTICS OF SOUNDS IN GENERAL

AMERICAN SPEECH

Symbol

Power, * dB Example re long time averaget

w

lip me nip sing we

r

~p

1 m n 1]

j

-

:res

p

~ie

t

tie 'key by die [uy

k b d g

v f

e ti s z

S :5

h tS dS

VIe

-

foe thin then ~ip

isshy measure hit

-

~op

Joe

-

-3.0 -5.8 -7.4 -4.4 0.0 -l.0 0.0 -15.2 -11.2 -11.9 -14.6 -14.6 -11.2 -12.2 -16.0 -23.0 -12.6 -11.0 -11.0 -4.0 -10.0 -13.0

-6.8 -9.4

(Continued)

Relative frequency of sound, %t

3.74 2.78 7.24 0.96 2.08 6.35 0.60 2.04 7.13 2.71 l.81 4.31 0.74 2.28 l.84 0.37 3.43 4.55 2.97 0.82 0.05 l. 81 0.52 0.44

"B'ormant frequencies for men and women ~ First

Second

~~-.

-----

450 140 140 140

1,000 1,250 1,450 2,350

2,550 2,950 2,250 2,750 2,300 2,750 2,750

500 270

1,350 2,040 800 1,700 Variable 800 1,700 Vl1riable 1,150 1,150 1,450 1,450 2,000 2,000 2,150 2,150

1,850 3,500

...

... ...

140 140 140 140

... ... 140 ...

140

... 140

Third Fourth ~--

~~-

1,350 2,450 1,350 2,450 2,500 3,650 2,500 3,650 2,550 2,550 2,700 2,700 2,650 2,650

* The power measurements do not represent the peak instantaneous power but the average over the sustained portion of the phoneme where such a period can be defined. In this case, as with the formant frequencies, the absolute values are highly variable, but intercomparisons among the various sounds are generally more reliable. t H. Fletcher, "Speech and Hearing in Communication," p. 86, D., Van Nostrand Company, Inc" Princeton, N.J., 1953. :t G. Dewey, ,"Relative Frequency of English Spee-ch Sounds," Harvard University Press, Cambridge, Mass., 1923. 'If E. O. Richardson, ed., "Technical Aspects of Sound," pp. 215~217, Elsevier Press, Inc., New York, 1953. ' , ' Sk-14. Articulation Index. The articulation index is a set of numbers that makes possible the prediction of the efficiency of some types of voice-communication systems by the addition of suitably chosen values. The operations involve (1) dividing the speech spectrum into a series of bands having an equal possible contribution L'>.A to the total efficiency, and (2) determining what proportion of the L'>.A each band will contribute under the particular noise and speech conditions being tested. Under (1) it is customary to use no more than 20 such bands. The frequeney limits of 20 such bands are given in Table 3k-ll.

3-165

SPEECH AND HEARING TABLE

3k-l1.

TWENTY FREQUENCY BANDS CONTRIBUTING EQUALLY TO EFFICIENCY OF SPEECH COMMUNICATION*

Band No.

Frequency range

1 2 3 4 5 6 7

395 395-540 540-675 675-810 810-950 950-1,095 1,095-1,250

,~

Band No.

Frequency range

Band No.

Frequency range

8 9 10

1,250-1,425 1,425-1,620 1,620-1,735 1,735-2,075 2,075-2,335 2,335-2,620 2,620-2,930

15 16 17 18 19 20

2,930-3,285 3,285-3,700 3,700-4,200 4,200-4,845 4,845-5,790 5,790

11

12 13 14

* H. Fletcher, "Speech and Hearing in Communication," D. Van Nostrand Company, Inc., Princeton, N.J., 1953. For conditions where substantial wide-band noise is present, the second requirement may be approximated by the formula

(3k-4) in which Wi is a weight having a maximum value of 1.0, Si is the signal level in band i in decibels, Ni is the noise level in the same band i in decibels referred to the same base as Si. 1 TABLE

3k-12.

ARTICULATION SCORES AS A FUNCTION OF ARTICULATION INDEX*

Articulation index

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Monosyllabic CVC syllables, % words (PB lists), %

7 22

38 55 68 79 87 93 96 98

7 22 40 61 77 87 93 96 98 99

* E. G. Richardson, ed., "Technical Aspects of Sound/' Elsevier Press, Inc., New York, 1953. "ove syllables are estimated from sets of words that vary the initial consonants, the vowel, and the final consonant separately. PB words are lists of monosyllables phonetically balanced so that the proportion of phonemes roughly equals that in general speech. ll

The articulation index A is then described by the summation

(3k-5) Articulation scores are related to the articulation index according to the Table 3k-12. 1 N. R. French and J. C. Steinberg, Factors Governing the Intelligibility of Speech Sounds, J. Acoust. Soc. Am. 19, 90-119 (1947).

31, Classical Dynamical Analogies HARRY F. OLSON

RCA Laboratories

Analogies are useful when it is desired to compare an unfamiliar system with one that is better known. The relations and actions are more easily visualized, the mathematics more readily applied, and the analytical solutions more readily obtained in the familiar system. Analogies make it possible to extend the line of reasoning into unexplored fields. In view of the tremendous amount of study which has been directed toward the solution of circuits, particularly electric circuits, and the engineer's familiarity with electric circuits, it is logical to apply this knowledge to the solutions of vibration problems in.other fields by the same theory as that used in the solution of electric circuits. The objective in this section is the establishment of analogies between electrical, mechanical, and acoustical systems. 31-1. Resistance. Electric Resistance. Electric energy is changed into heat by the passage of an electric current through an electric resistance. Electric resistance RE, in abohms, is defined as e (31-1) =-;: t

where e = voltage across the electric resistance, abvolts i = current through the electric resistance, abamp Mechanical Rectilineal Resistance. Mechanical rectilineal energy is changed into heat by a rectilinear motion which is opposed by mechanical rectilineal resistance (friction), Mechanical rectilineal resistance (termed mechanical resistance when there is no ambiguity) RM, in mechanical ohms, is defined as fM RM = -

(31-2)

U

where fM = applied mechanical force, dynes u = velocity at the point of application of the force, em/sec Mechanical Rotational Resistance. Mechanical rotational energy is changed into heat by a rotational motion which is opposed by a rotational resistance (rotational friction). Mechanical rotational resistance (termed rotational resistance when there is no ambiguity) R a, in rotational ohms, is defined as (31-3)

where fa = applied torque, dyne-em n = angular velocity about the axis at the point of the torque, radians/sec Acoustic Resistance. . Acoustic energy is changed into heat either by a motion in a fluid which is opposed by acoustic resistance due to a fluid resistance incurred b'y viscosity or by the radiation of sound. Acoustic resistance RA, in acoustical ohms, 1S defined as

3-166

CLASSICAL DYNAMICAL ANALOGIES

R.a

=

E.. U

3-167 (31-4)

where p = pressure, dynes/cm' U = volume velocity, cm 3/sec 31-2. Inductance, Mass, Moment of Inertia, Inertance. Inductance. Electromagnetic energy is associated with inductance. Inductance is the electrio-cirouit element that opposes a change in current. Inductance L, in abhenrys, is defined as di

e=L Iit

(31-5)

where e = voltage, emf, or driving force, abvolts di di = rate of change of current, abamp/sec Mass. Mechanical rectilineal inertial energy is associated with mass in the mechanical rectilineal system. Mass is the mechanical element which opposes a change in velocity. Mass m, in grams, is defined as

1M

=

du dt

m-

t(31-6)

' where du dt = accel eratlOn, cm / sec 2

1M = driving force, dynes Moment of Inertia. Mechanical rotational energy is associated with moment of inertia in the mechanical rotational system. Moment of inertia is the rotational element which opposes a change in angular velocity. Moment of inertia I, in gram (centimeter)2, is defined as (31-7)

dn

where dt

=

angular acceleration, radians /sec'

fR = torque, dyne-cm

Inertance. Acoustic inertial energy is associated with inertance in the acoustic system. Inertance is the acoustic element which opposes a change in volume velocity. Inertance M, in grams per (centimeter)', is defined as p

where

~~

dU

=

Mdt

(31-8)

= rate of change of volume velocity, cm 3/sec 2

p = driving pressure, dynes/em' 31-3. Electric Capacitance, Rectilineal

Compliance, Rotational COpJ.pli@,nce, Acoustic Capacitance. Electric Capacitance. Electric capacitance is associated with capacitance. Electric capacitance is the electric-circuit element which opposes a change in voltage. Electric capacitance CE , in abfarads, is defined as (31-9) (31-10)

where Q = charge on the electrical capacitance, abcoulombs e = emf, abvolts Rectilineal Compliance. Mechanical rectilineal potential energy is associated with the compression of a spring or compliant element. Rectilineal compliance is the

3-168

ACOUSTICS

mechanical element which opposes a change in the applied force. Rectilineal compliance (termed compliance when there is no ambiguity) CM , in centimeters per dyne, is defined as (31-11)

where x = displacement, om f M = applied)orce, dynes Rotational Compliance. Mechanical rotational potential energy is associated with the twisting of a spring or compliant element. Rotational compliance is the mechanical element that opposes a change in the applied torque. Rotational compliance CR , in radians per centimeter per dyne, is defined as fR =

where

-'GtR-

(31-12)

angular displacement, radians applied torque, dyne~cm Acoustic Capacitance. Acoustic potential energy is associated with the compression of a· fluid or a gas. Acoustic capacitance is the acoustic element which opposes a change in the applied pressure. The acoustic capacitance CA , in (centimeters)5 per dyne, is defined as cf> =

JR

=

x

(31-13)

p = CA

where X = volume displacement, cm 3 p = pressure, dynes/cm 2 31~4. Representation of Electrical, Mechanical Rectilineal, Mechanical Rotational, and Acoustical Elements. Electrical, mechanical rectilineal, mechanical rotational, RR

RE

-'\Nv--

RA ~

ezZZZzz! l

---'000'-CE

-If-

= =

~

m

M

D CA

CM

:J fA,

RM

/

:W I

W CR

~

~

RECflLlNEAL

ROTATIONAL

MECHANICAL ELECTRICAL ACOUSTICAL FIG. 31-1. Graphical representation of the three basic elements in electrical, mechanical rectilineal, mechanical rotational, and acoustical systems.

and acoustical elements have been defined in the preceding sections. Figure 31-1 illustrates schematically the three elements in each of the four systems. The electrical elements, electric resistance, inductance, and electric capacitance, are represented by the conventional symbols . . Mechanical rectilineal resistance is represented by sliding friction which causes dissipation. Mechanical rotational resistance is represented by a wheel with a sliding-

CLASSICAL DYNAMICAL ANALOGIES

3-169

friction brake which C8,uses dissipation. Acoustic resistance is represented by narrow slits which cause dissipation due to viscosity when fluid is forced through the slits. These elements are analogous to electric resistance in the electrical system. Inertia in the mechanical rectilineal system is represented by a mass. Moment of inertia in the mechanical rotational system is represented by a flywheel. Inertance in the acoustical system is represented as the fluid contained in a tube in which all the particles move with the same phase when actuated by a force due to pressure. These elements are analogous to inductance in the electrical system. Compliance in the mechan1cal rectilineal system is represented as a spring. Rotational compliance in the mechanical rotational system is represented as a spring. Acoustic capacitance in the acoustical system is represented as a volume which acts as a stiffness or spring element. These elements are analogous to electric capacitance in the electrical system. Table 31-1 shows the quantities, units, and symbols in the four systems. :n-5. Description of Systems of One Degree of Freedom. Electrical, mechanical rectilineal, mechanical rotational, and acoustical systems of one degree of freedom are shown in Fig. 31-2. In one degree of freedom the activity in every element of the

'M:[~I ACOUSTICAL

f~~Rd ~

:;dJ

RR

'-"

FREQUENCY

ROTATIONAL MeCHANICAL

FIG. 31-2. Electrical, mechanical rectilineal, mechanical rotational, and acoustical systems

of one degree of freedom and the current, velocity, angular velocity and volume velocity response characteristics. system can be expressed in terms of one variabie.In the electrical system an electromotive force e acts upon an inductance L, an electric resistance R E , and an electric capacitance CE connected in series. In the mechanical rectilineal system a driving force iM acts upon a particle of mass m fastened to a spring of compliance eM and sliding upon a plate with a frictional force which is proportional to the velocity and designated as the mechanical rectilineal resistance RM. In the mechanical rotational system a driving torque iR acts upon a flywheel of moment of inertia I connected to a spring or rotational compliance CR and the periphery of the wheel sliding against a brake with a frictional force which is proportional to the velocity and designated as the mechanical rotational resistance RR. In the acoustical system, an impinging sound wave of pressure p acts upon an inertance M and an acoustic resistance RA comprising the air in the tubular opening which is connected to the volume or acoustical capacitance CA. The acoustic resistance RA is due to viscosity. The differential equations describing the four systems of Fig. 31-2 are as follows: Electrical (31-14)

3-170

ACOUSTICS

Mechanical rectilineal (31-15) Mechanical rotational (31-16) Acoustical (31-17) E, F M, FR, and P are the amplitudes of the driving forces in the four systems. EE;"'t = e, FME;"'t = JM, FRE;"'t = JR and PE;"" = p. The steady-state solutions of Eqs. (31-14) to (31-17) are: Electrical e (31-18) ZE

Mechanical rectilineal i;

= RM

FE;"" JM - (i/v)C M) = ZM

(31-19)

FE;"" JR - (j/wC R) = ZR

(31-20)

+ jwm

Mechanical rotational c/>

= RR

+ jwI

Acoustical (31-21) The vector electric impedance is

~ WCE

(31-22)

RM +jwm - - j wCM

(31-23)

ZE = RE

+ jwL

-

The vector mechanical rectilineal impedance is

ZM

=

The vector mechanical rotational impedance is

ZR = RR

+ jwI

-

~

(31-24)

~c·

(31-25)

WCR

The vector acoustic impedance is

ZA = RA,+ jwM -

w A

CLASSICAL DYNAMICAL ANALOGIES

3-1'71

:U-6. Applications of Classical Electrodynamical Analogies. The fundamental principles relating to electrical, mechanical rectilineal, mechanical rotational, and acoustical analogies have been established in the preceding sections. Employing these fundamental principles, the vibrations produced in mechanical and acoustical systems owing to impressed forces can be solved as follows: Draw the electrical network which is analogous to the problem to be solved; solve the electrical network by conventional electrical circuit theory; convert the electrical answer into the original system. In this procedure any problem involving vibrating systems is reduced to the solution of an electrical network. In the illustrations in the preceding sections, the elements in the electrical network have been labeled rE.L and CEo However, when analogies are used in actual practice, the conventional procedure is to label the elements in the analogous electrical network with rM, IV, and CM for a

TO ENGINE Ml

CROSS-SECTIONAL VIEW

ACOUSTICAL NETWORK

31-3. Cross-sectional view and acoustical network of an automobile muffler. In the acoustical network: 111" M2, M s, and M 4, the inertances of the series elements; TAl, ,'A2, and r Aa, the acoustical resistances of the shunt elements; M 5, M 6, and M 7, the inertances of the shunt elements; CAl, CA2, and CAa, the acoustical capacitances of the shunt elements. FIG.

(After Olson, "Dynamical Analogies," D. Van Nostrand Company, Inc., Princeton, N.J., 1959.)

mechanical rectilineal system; with rR, I, and CR for a mechanical rotational system; and with r A, M, and CA for an acoustical system. This procedure will be followed in this section in labeling the elements in the analogous electrical network. The customary procedure is to label the network with the caption mechanical network or rotational network or acoustical network as the case ma.y be. When there is only one path, the term circuit will be used instead of network. A complete treatment of the examples of the use of analogies in the solution of problems in mechanical and acoust.ical systems is beyond the scope of this section. However, a few typical examples will serve to illustrate the principles and method. Acoustical-Automobile ]}!'uf!lm·. The sound output from the exhaust of an automobile engine contains all audible frequencies in addition to frequencies below and above the audible range. The purpose of a muffler is to reduce the sound output in the audible frequency range without increasing the exhaust back pressure. By the application of acoustical principles employing analogies improved mufflers have been developed in which the following advantages have been obtained: smaller size, higher attenuation in the audible frequency range, and reduction of bad, pressure at the engine. A cross-sectional view of the improved muffler is shown in Fig. 31-3. The acoustical network shows that the system is essentially a low-pass acoustical filter. The main channel is of the same diameter as the exhaust pipe. Therefore, there is no increase in the direct flow of exhaust gases as compared with a plain pipe. In order not to impair the efficiency of the engine, the muffler should not increase the acoustical impedance to subaudible frequencies. The system of Fig. 31-3 can be designed so that the subaudible frequencies are not attenuated and at the same time high attenuation is introduced in the audible frequency range.

3-172

ACOUSTICS

T h e terminations at the two ends of the network are not ideal. Therefore, it is necessary to use shunt arms tuned to different frequencies in the low-frequency range. Acoustical resistance is obtained b y employing slit-type openings into the side chambers. In a development of this kind, the frequency spectrum of the sound which issues from the exhaust is usually determined. F r o m these data the amount of suppression required for each part of the audible frequency range can b e ascertained. The acoustical network can b e determined from these data and the terminating acoustical

*M—|

ZMF FLOOR

Z

M

MECHANICAL RECTILINEAL SYSTEM

MACHINE

ZMFH

F

MECHANICAL CIRCUIT

F I G . 31-4. Schematic view, mechanical rectilineal system, and mechanical circuit of a machine mounted directly upon the floor. In the mechanical circuit: /M, the vibrating force developed by the machine; m, the mass of the machine; ZMF, the mechanical impedance of the floor. (After Olson, "Dynamical Analogies," D. Van Nostrand Company, Inc., Princeton, N.J.,

1959.)

— f|| —

•KM CM

DAMPED SPRING

^Hflff FLOOR MACHINE

Z

M

F

-MF MECHANICAL RECTILINEAL SYSTEM

MECHANICAL NETWORK

F I G . 31-5. Schematic view, mechanical rectilineal system, and mechanical network of a machine mounted upon a vibration isolating system. In the mechanical network: fu, the vibrating force developed by the machine; m, the mass of the machine; CM, the compliance of one of the four spring mounts; RM, the mechanical rectilineal resistance of one of the spring mounts. (After Olson, "Dynamical Analogies," D. Van Nostrand Company, Inc., Princeton,

N.J.,

1959.)

networks. In general, changes are required to compensate for the approximations. In this empirical w o r k the acoustical network serves as the guide in directing the appropriate changes. Mechanical Rectilineal—Machine Vibration Isolator. T h e vibration of a machine is transmitted from its supports to all parts of the surrounding building structure. I n m a n y cases, the vibrations are so intense as to b e intolerable. T h e reduction of the transmission of machinery vibrations is one of the most c o m m o n problems in noise control. For these conditions, the solution of the problem is to provide suitable vibrational isolation between the machine and the floor u p o n which it is placed. A machine m o u n t e d directly on the floor is shown in Fig. 31-4. T h e mechanical rectilineal system and the mechanical circuit for vertical vibrations are shown in Fig. 31-4. T h e driving force / M is due to the vibrations of the machine. The

CLASSICAL DYNAMICAL ANA LO GIES

3-173

m echanical circuit shows that t he only isolation in the system of Fig. 31-4 is due to the mass of t he machine. In the simple isolating system of Fig. 31-5 t he machine is mounted on springs with mechanical resistance added to serve as damping. The com pliance and mechanical resistance of each support are C M a nd rM . Since there are foUl' supports, these values become Cl>d 4 and 4rM in t he mechanical rec tilineal system and the mechanical network for vertical vibrations. The mechanical network depicts t he act ion of the shunt circuit CMrM in reducing the force of the vibra tion transmitted to t he floor ZM F . llfechanical Rotational-Vibration Damper. In recipro cating engines and ot her rotat ing machinery, ro tational vibrations of large amplitudes occur at certain speeds. These rotational vibrations are sometimes of such 'high amplitude that t.he shaHs MOMENT OF INERTIA 12

FLYWHEEL II

SHAFT

END VIEW

SIDE VIEW

ROTATIONAL NET WORK

FIG. 31-6. End and side views and the rotational network of a vibration damper. I n t he rot ational n etwork: h , t he moment of inertia of t he flywheel; 1 2, the moment of inertia of the damper ; CR, the rotation al complia n ce of t he damper ; rR, the mech anical rotation al resist ance between the damper and the sh aft. (A.fter Olson, " D ynamical A.nalogies," D. Van Nostrand Company, Inc., Princeton, N .J., 1959. )

will fail after a few hours of opera ting. A number of various rotational dampers have been developed for reducing these ro tational vibrat ions. A typical example of a vibration damper used to control the vibra tions of t he flywheel is hown in Fig. 31-6. The damper consists of a rotational element having a moment of inertia I , rotating on a shaft with a mechanical ro tational resistance rR b etween t he inertia element and shaft. The inert ial element is coupled to the flywh eel by means of a spring of compliance CM. The ro tational compliance is C R = CM/ a', where a is the radius at t he point of attachment of the spring with respect to the center line of t he shaft. Referring to the rotational network it will be seen that the rotational damp er forms a shunt mechanical ro tational system. The shun t ro tational circuit CRTR I , is tuned to t he frequ ency of t he vibration. Since t he mechanical ro tat.ional im pedance of the shunt resonant rotational circuit is very high at the resonant frequency, the angular velocity (or amplitud e) of vibration of t he flywheel will be reduced. A considera t ion of the rotational network illustrates t he principle of t he device. Electrical jy[echanical-Direct Radiator Dynamic Loudspeaker. The direct radiator dynamic loudspeaker shown in Fig. 31-7 is almost universally used for radio, phonograph, television , and other small-scale sound reprodu ctions. The mechanical circuit of t he loudspeaker is shown in Fig. 31-7. T he mechanical reetilineal impedance at the voice coil, where a forc e f M is applied, can be determined from the constants of the elements of the mechanical circuit . The mass m, and t he mechanical resistance rM2 of t he air load can be obtained from Sec. 3i-2 on the acoustic impedan ce of vibrating pistons. The elect rical circuit of the loudspeaker is also shown in Fig. 31-7. The motional

3-174

ACOUSTICS

ELECTRICAL CIRCUIT

MECHANICAL CIRCUIT

CROSS-SECTIONAL VIEW

FIG. 31-7. Cross-sectional view, electrical circuit. and mechanical circuit of a direct radiator loudspeaker. In the electrical circuit: e, the open-circuit voltage of the generator or vacuum tube; rEG, the electrical resistance of the voice coil; L, the inductance of the voice coil; ZEM, the motional electrical impedance of the driving system. In the mechanical circuit: m" the mass of the cone; rMI, the mechanical resistance of the suspension system; eM, the compliance of the suspension system; m2, the mass of the air load; r M2, the mechanical rectilineal resistance of the air load. (After Olson, "Dynamical Analogies," D. Van Nostrand Company, Inc., Princeton, N.J., 1959.)

electrical impedance in the electrical circuit is given by (Bl) ,

(31-26)

ZEM = - ZJ.ldT

where

ZEM =

motional electrical impedance, abohms

B = flux density in air, gauss l = length of conductor in voice coil ZMT = mechanical impedance at location 1M in mechanical circuit, mechanical

ohms The mechanical driving force is given by 1M =

Bli

(31-27)

where 1M = driving force, dynes i = current in voice coil, abamp The velocity can be determined from the mechanical circuit of Fig. 31-7 and th6' following equation:

x= where

1M ZMT

x is the velocity in centimeters per second. P = rMx 2

where P

(31-28)

The sound output is given by (31-293

sound power output, ergs/sec mechanical ohms x = velocity of cone from Eq. (31-28) The object is to select the constants so that the power output as given by Eq. (31-29) is practically independent of the frequency over the desired frequency range. =

rM =

3-175

CLASSICAL DYNAMICAL ANALOGIES TABLE

31-1.

QUANTITIES, UNITS, AND SYMBOLS FOR ELECTRICAL, MECHANICAL

RECTILINEAL, MECHANICAL ROTATIONAL, AND ACOUSTICAL ELEMENTS

Electrical Quantiy

Unit

Mechanical rectilineal Symbol

Electromotive force ........ Volts X 10 8 e Coulombs X 10- 1 Q Charge or quantity Current ........ Amperes X 10- 1 i Electric imped-' ance Electric resistance Electric reactance Inductance .... Electric capacitance Power .........

Ohms X 10 9

ZE

Ohms X 10 9

RE

Ohms X 10 9

XE

Henrys X 10 9 Farads X 10- 9

L

Ergs per second

PE

CE

Quantity

Force Dynes Linear disCentimeters placement Linear velocity Centimeters per second Mechanical Mechanical impedance ohms Mechanical Mechanical resistance ohms Mechanical Mechanical reactance ohms Mass Grams Centimeters Compliance per dyne Ergs per second Power

Unit

x i;

or u

I ZM RM XM m

CM PM

Acoustical Symbol

Torque ........ Dyne-centimeters JR

Quantity Pressure

q,

Angular Radians displacement Radians per Angular velocity second

Volume displacement ¢ or r.l Volume velocity

Rotational impedance Rotational resistance Rotational reactance Moment of inertia Rotational compliance Power .........

Rotational ohms

ZR

Rotational ohms

RR

Rotational ohms

XR

---

JM

I

Mechanical rotational Quantity

Symbol

Unit

(Gram) (centiI meter) 2 Radians per dyne CR per centimeter Ergs per second PR

Acoustic impedance Acoustic resistance Acoustic reactance Inertance Acoustic capacitance Power

Symbol

Unit Dynes per square centimeter Cubic centimeters Cubic centi'[ meters per second Acoustic ohms

p

X kor U

,

ZA

Acoustic ohms RA Acoustic ohms XA Grams per M (centimeter) 4 (Centimeter) 5 CA per dyne Ergs per second PA

,

3m. Mobility Analogy HARRY F. OLSON

RCA Laboratories

The analogies that have been presented and considered in Sec. 31 have been formal ones owing to the similarity of the differential equations of electrical, mechanical and acoustical vibrating systems. For this reason these analogies have been termed the classical impedance analogies; they are, however, not the only ones possible of development for useful applications. For example, mechanical impedance has been defined by some authors-in addition to the ratio of force to velocity as developed in Sec. 31as the ratio of pressure to velocity, the ratio of force to displacement, and the ratio of pressure to displacement. During the past three decades the developments in the field of analogies have been reported in publications" by many investigators. In this connection a useful analogy, developed by Firestone and designated by him as the "mobility analogy," has been employed on a wide scale to solve problems in mechanical vibrating systems. In the mobility analogy mechanical mobility is defined as the complex ratio of velocity to force. Although the mobility analogy can be applied and used with all types of vibrating systems, its most direct and useful application is in the field of mechanical vibrating systems. Therefore, in order to make the subiect of analogies complete in this handbook, it seems logical to include the mobility analogy. Accordingly, it is the purpose of this chapter to develop the mobility analogy, particularly as applied to mechanical rectilineal systems. 2 3m-1. Mechanical Rectilineal Mobility. Mechanical rectilineal mobility is the inverse of mechanical rectilineal impedance. Nlechanical rectilineal mobility ZI, in mechanical mhos, is defined as the complex ratio of linear velocity to linear force as follows: Zj

V

=-

JM

(3m-I)

where v = velocity, em/sec JM = force, dynes It will be evident that a mechanical element in the mechanical mobility sense is analogous to the electric element if velocity difierence across the mechanical element is analogous to the voltage difference across the electric element and if the force through the mechanical element is analogous to the electric current through the electric element. See the end of Section 3 for a list of references. The considerations in this section will be confined to mechanical rec'cilineal systems. The mobility analogy is equally applicable to mechanical rotational systems. In this connection mechanical rectilineal and mech,mical rotational systems are not sufficiently different to warrant a separate treatment for the mechanical rotational system, particularly in view of the faet that fundamental aspects of the two systems have been considered from the classical impedance analogy viewpoint in this book. 1

2

8-176

MOBILITY ANALOGY

Mechanical rectilineal mobility may be written as follows:

ZI,

3-177

in mechanical mhos, is a complex quantity and (3m-2)

where rI = responsivity, mechanical mhos XI = excitability, mechanical mhos 3m-2. Responsivity (Mobility Resistance). In the mechanical rectilineal mobility system mechanical rectilineal responsivity (mobility resistance) rI, in mechanical mhos, is defined as v 1 rr = - = (3m-3) 1M

rM

w here v = velocity, cm/sec 1M = force, dynes rM = mechanical impedance, mechanical ohms Sm-S. Mass (Mobility Capacitance). In the mechanical rectilineal mobility system the mass (mobility capacitance) mI, in grams, is analogous to electric capacitance CEo The mechanical rectilineal excitability Xl of mass (mobility capacitance), in mechanical mhos, is defined as

a

(3m-4) where w =

27r1

I = frequency, hertz Equation (3m-4) shows that the mass (mobility capacitance) mI in the mechanical rectilineal mobility system is analogous to electric capacitance CE in the electric system. Mass (mobility capacitance) mI in the mechanical rectilineal mobility system may also be defined as follows: (3m-5) (3m-6) In the electric system electric capacitance CE may be defined as follows:

.

z=

C de

Edt

(3m-7)

where i = electric current, abamp CE = electric capacitance, abfarads e = electromotive force, abvolts t = time, sec (3m-8) where i = current in abamperes. It will be seen that Eqs. (3m-5) and (3m-6) in the mechanical rectilineal mobility system are analogous to Eqs. (3m-7) and (3m-8) in the electric system. 3m-4. Compliance (Mobility Inertia). In the mechanical rectilineal mobility system the compliance (mobility inertia) CI , in centimeters per dyne, is analogous to electric inductance L. The mechanical rectilineal excitability XI of a compliance (mobility inertia), in mechanical mhos, is defined as (3m-9) where w = 27rJ I = frequency, Hz Equation (3m-9) shows that compliance (mobility inertia) CI, in centimeters per dyne, is analogous to inductance.

3-178

ACOUSTICS

Compliance (mobility inertia) C[ in the mechanical rectilineal mobility system may also be defined as v = C J dilk[

(3m-IO)

dt

In the electric system inductance may be defined as

e=Lr!i

(3m-H)

dt

where L = inductance in abhenrys. It will be seen that Eq. (3m-IO) in the mechanical rectilineal mobility system is analogous to Eq. (3m-H) in the electric system. 3m-Ii. Representation of Electrical and Mechanical Rectilineal Mobility Elements. Electric elements have been defined in Sec. 31. Elements in the mechanical rectilineal mobility system have been described in this sep,t.ion.

m

D MECHANICAL ELEMENTS

mr

CE

II

--II

MOBILITY ELEMENTS

ELECTRICAL ELEMENTS

FIG. 3m-I. Graphical representation of the three basic elements in mechanical rectilineal, mobility, and electric systems. rlk[ = mechanical rectilineal T[ = responsivity rE = electrical resistance resistance Clk[ = compliance C[ = mobility inertia L = inductance m = mass m[ = mobility capacitance CE = electric capacitance (After Olson, "Solutions of Engineering Problems by Dynamical Analogies," D. Van Nostrand Co., Princeton, N.J., 1966.)

Figure 3m-1 illustrates schematically the mechanical elements and the analogous elements in the electric and mechanical rectilineal mobility systems. Mechanical rectilineal resistance rlk[ in the mechanical rectilineal system is represented as sliding or viscous friction. Mechanical rectilineal responsivity (mobility resistance) r[ in the mechanical rectilineal mobility system is the reciprocal of mechanical rectilineal resistance Tlk[ and is analogous to electrical resistance rEo Compliance Clk[ in the mechanical rectilineal system is represented as a spring. Compliance (mobility inertia) C[ in the mechanical rectilineal mobility system is analogous to inductance L in the electric system. Mass m in the mechanical rectilineal system is represented as a mass or weight. Mass (mobility capacitance) m[ in the mechanical rectilineal mobility system is analogous to electric capacitance CE in the electric system.

3-179

MOBILITY ANALOGY

The electrical and the mechanical rectilineal quantities in the mobility system are shown in Table 3m-I. The units and the analogous elements and symbols also are shown in Table 3m-I. 3m-5. Mechanical Vibrating System Consisting of a Mass, Compliance, and Mechanical Resistance. The vibrating system! of one degree of freedom consisting of a mass, compliance, and mechanical resistance has been considered from the standpoint of the classical mechanical impedance analogy in Sec. 31. It is the purpose of this section to consider the same mechanical vibrating system from the standpoint of the mechanical mobility analogy.2 TABLE 3m-I. CORRESPONDENCE BETWEEN ELECTRICAL AND MECHANICAL QUANTITIES IN THE MOBILITY SYSTEM Electrical

Quantity

Electromotive force

Unit

Symbol

Volts X 10-8

e

Velocity

q

Impulse or mo- Gram-centimeter mentum per second

i

Force

Dynes

Charge or quanCoulombs X 10- 1 tity Current

Mechanical rectilineal mobility

Amperes X 10- 1

---

Quantity

Unit

Symbol

Centimeters per second

x or v Q

1M

Electrical impedance

Ohms X 10'

ZE

Mechanical mobility

Mechanical mhos

itl

Electrical resistance

Ohms X 10'

TE

Responsivity

Mechanical mhos

T[

Electrical reactance

Ohms X 10'

XE

Excitability

Mechanical mhos

Inductance Electrical capacitance Power

XI

---. Henrys X 10' Farads X 10' Joules per second

L CE PE

Compliance or mobility inertia

Centimeters per dyne

Mass or mobility Grams capacitance Power

Ergs per second

CI

--m[

---PI

The mechanical system consisting of a mass, compliance, and mechanical resistance is shown in Fig. 3m-2A. The mechanical vibrating system may be rearranged to form the equivalent as shown in Fig. 3m-2B. From the mechanical vibrating system of Fig. 3m-2B it is a relatively simple matter to develop the mobility analogy of Fig.3m-20. 1 The preceding paragraphs have been concerned with fundamental considerations. Therefore, the modifier rectilineal has been employed for the sake of accuracy. Since the remainder of this section will be concerned with applications of the mechanical rectilineal mobility, the modifier rectilineal will be dropped. 2 In view of the fact that this section is concerned with mechanical systems, the modifier mechanical in relation to the mechanical mobility analogy is also superfluous and need not be used.

3-180

ACOUSTICS

~mr MECHANICAL SYSTEM

B

C

~ i~

rr

fMl

mr

v

i

fM3 Cr

i

MOBILITY ELECTRIC NETWORK NETWORK FIG. 3m-2. A mechanical vibrating system consisting of a mass, compliance, and mechanical resistance. A. Mechanical sys_tem. B. Mechanical system equivalent to the mechanical system of A. C. Mobility network of the mechanical system. D. Electric network analog of the mobility system. (Ajter Olson, "Solution oj Engineering Problems by Dynamical Analogies,"' D. Van Noctrand Company, Princeton, N.J., 1966.) MECHANICAL SYSTEM

The sum of the forces through the three branches of the mobility network' of Fig. 3m-2C is (3rn-12) 1M = 1M' + 1M2 + 1M3 V

where

(3m-13)

rI

dv

1M2

=

m[

Iii

(3m-14)

1M3

=

~I

J v dt

(3m-IS)

From the sum of Eqs. (3m-13) to (3m-IS) the differential equation of the mobility network of Fig. 3m-2C is

1M

= mr -dv

dt

+ -v + -C[I r[

J

v

dt

(3m··Hi)

The sum of the electric currents of the electric network of Fig. 3m-2D is

where

i = i, + i2 . e 2,

+ i3

= -

rE

(3m-17) (3m-IS)

.

C de

Iii

(3m-19)

is =

LJedt

(3m-20)

22 =

E

'In establishing analogies between electric and mechanical systems the elements in the electric. network have been labeled rE, L, and CEo However, in using analogies in actual practice, the conventional procedure is to label the elements in the analogous electric network as rM, m, and CM for the classical mechanical rectilineal system and as fl, CI, and mI for the mobility mechanical rectilinear system. This procedure will he followed in this section in labeling the elements of the analogous electric network. It is literally accurate to label the network with the caption "Analogous electric network of the mechanical rectilineal system" (or, of the mobility mechanical rectilineal system). For the sake of brevity, these networks will be labeled "mechanical network" and "mohility network." vVhere there is only one path, "circuit" will be used instead of "network."

3-181

MOBILITY ANALOGY

From the sum of Eqs. (3m-IS) to (3m-20) the differential equation of the electric network of Fig. 3m-2D is . de e I f edt ~=CE-+-+(3m-2I) dt TE L Oomparing the variables and coefficients of the mobility and electric networks in the differential equations (3m-I6) and (3m-2I) establishes the analogous variables and quantities in the two systems as given in Table 3m-I. The classical mechanical impedance analogy of the mechanical system of Fig. 3m-2 has been considered in Sec. 31 and will not be repeated here.

A

e MECHANICAL ,SYSTEM SCHEMATIC VIE'W OF THE ELECTRIC AND MECHANICAL SYSTEMS

C

L:j"I

rII

_ _ _ _ _...1

L

ELECTRIC AND MOBILITY NETWORKS

D

:..'!!L B2t 2

L e2t 2

e

8pa2

~ 3B2 t 2

ELECTRIC NETWORK FIG. 3m-3. Cross-sectional view, the mechanical system, the electric and mobility networks, and the electric network of a direct radiator dynamic loudspeaker. In the electric and mechanical networks: e, the electromotive force of the electric generator. rEG, the electrical resistance of the electric generator. L, the inductance of the voice coil. REl, the electrical resistance of the voice coil. ml, the mass of the cone. CM, and rMl, the compliance and mechanical resistance of the suspension. m. and rM', the mass and mechanical resistance of the air load. m{, the mobility capacitance of the cone. C[ and r /1, the mobility inertia and responsivity of the suspension. m12 and r12, the mobility capacitance and responsivity of the air load. B, the flux density in the air gap. l, the length of the voice coil conductor. a, the radius of the cone. p, the qensity of !'Iir. (After Olson, "Solutions of Engineering Problems by Dynamical Analogies," D. Van Nostrand Company, Princeton, N,J" 1966.)

3-182

ACOUSTICS

Sm-7. Direct Radiator Loudspeaker. The direct radiator dynamic loudspeaker shown in Fig. 3m-3 is almost universally used for radio, phonograph, television, and other small-scale sound reproduction. The electric and mechanical systems of the complete loudspeaker are shown in Fig. 3m-3A. The mechanical vibrating system consisting of the voice coil, cone, suspension, and air load is presented in Fig. 3m-3B. The mass ml of the cone and voice coil, and the compliance CM and mechanical resistance of the suspension system, can be obtained from measurements of the vibrating system. The mechanical system of the air load-namely, the mechanical resistance TM2 and masS m2 of the air load upon the front of the cone-is depicted in Fig. 3m-4A and A r==~-

rill .~

0

Iml '-----'

MECHANICAL SYSTEM

MECHANICAL SYSTEM

B

E

I"

fill

~~ '---I fr

mI

~~

,m

MECHANICAL NETWORK

I

MOBILITY CIRCUIT

C

_1_

tpa

Spa2

~J

2

MECHANICAL

MOBILITY CIRCUIT FIG. 3m-4. Air load upon a loudspeaker cone. A. Mechanical system: m, the mass of the air load. rM, the mechanical resistance of the air load. B. Mechanical network of the air load upon a loudspeaker cone. C. Mechanical network of the air load upon a loudspeaker cone: a, the radius of the cone. p, the density of air. c, the velocity of sound. D. Mechanical system same as A. E. Mobility circuit of the air load upon a loudspeaker cone: mI, the mobility capacitance of the air load. rl, the responsivity of the air load. F. Mobility circuit of the "ir load upon a loudspeaker cone: a, the radius of the cone. p, the density of air. c, the velocity of sound. (After Olson, "Solution of Engineering Problems by Dynamical Analogies." D. Van Nostrand Company, Princeton, N.J., 1966.) r~ETWOFII
nce and mass of the air load upon the front of the cone are shown in the mechanical network of Fig. 3m-4C. The mobility circuit of the air load upon the front of the cone appears in Fig. 3m-4E. The constants of the responsivity and compliance are given in the mobility circuit of Fig. 3m-4F. The electric and mobility networks with the ideal transformer connecting the electric and mobility sections are shown in Fig. 3m-3. In Fig. 3m-3D the ideal transformer has been eliminated, and the entire vibrating system reduced to an electric network. The electrical impedance due to the mechanical system is given by Eq. (31-26) as follows: ZEM

(El)2 =-ZM

(3m·22)

NONLINEAR ACOUSTICS (THEORETICAL)

where

ZEM

3-183

= electrical impedance due to the mechanical system, abohms

mechanical impedance of the mechanical system, mechanical ohms flux density in the air gap, gauss l = length of the voice coil conductor, em Since IjzM = ZI, Eq. (3m-22) may be written as ZM =

B

=

ZEM =

(3m-23)

(BI)2Z1

where ZI = mobility in mechanical mhos. By means of Eq. (3m-23) it is possible to convert the combined electric and mobility networks to the electric network, as shown in Fig. 3m-3. The process employing the mobility analysis of this section may be compared with the classical impedance analysis of Sec. 31. References 1. Olson, H. F.: "Dynamical Analogies" 2d ed., D. Van Nostrand Company, Inc., Prince-

ton, N.J., 1958. 2. Olson, H. F.: "Solution of Engineering Problems by Dynamical Analogies," Van Nostrand Reinhold Co" New York, N.Y., 1968.

3u. N onliuear Acoustics (Theoretical) DAVID T. BLACKSTOCK

University of Texas

Until the early 1950s most of what was known about sound waves of finiteamplitucle was confined to propagation, and to a lesser extent reflection, of plane waves in lossless gases. Since that time a great deal has been learned about propagation in other media, about nonplanar propagation (still chiefly in one dimension), about the effect of losses, and about standing waves. Inroads have been made on problems of refraction. Diffraction is still relatively untouched. In this section the exact equations of motion· for thermoviscous fluids will first be stated. Various retreats from the full generality of these equations will then be discussed. No attempt will be made to cover streaming and radiation pressure. See Sees. 3c-7 and 3c-8 for a discussion of those topics. GENERAL EQUATIONS FOR FLUIDS The basic conservation equations will be stated briefly for viscous fluids with heat flow. Other compressible media, such as solids and relaxing fluids, are discussed later in the section. 3n-1. Conservation of Mass, Momentum, and Energy. In Eulerian (spatial) coordinates the continuity and momentum equatiJns are respectively Dp Dt p -Du,

Dt

+ ap -

ax,

+ a?t, =

= 0

Pax, - a (' 'Y/ dkkO" aXi

.,

(3n-l)

+ 2'Y/ d0'•. )

3-184

ACOUSTICS

An entropy equation is stated here in place of the usual energy equation: (3n-3) Here P is the density, Ui is the ith (cartesian) component of particle velocity, p is pressure, Oi; is the Kronecker delta, eli; = t(auilaxj + aU;/aXi) is the rate-of-deformation tensor, YJ and r/ are the shear and dilatational coefficients of viscosity, Cvand Cp are the specific heats at constant volume and pressure, 3 is absolute temperature, S is entropy per unit mass, 'Y = CpIC" is the ratio of specific heats, (3, = -p- 1 (apla3)p is the coefficient of thermal expansion, 1/;(;1) = 2YJel i ;el;i + YJ'elk/,elii is the viscous energy dissipation function, and Qi is the ith component of the total heat flux. The material derivative D ( ) I Dt stands for a ( ) I at Uia ( ) I aXi. If the flow of heat is due to conduction, a3 (3n-4) Qi = aXi

+

-/C~

where K is the coefficient of thermal conduction. For heat radiation the relation between q and 3 is generally quite complicated; see, for example, Vincenti and Baldwin (ref. 1). The model used by Stokes (ref. 2) amounts to Newton's law of cooling and may be expressed by aQi

pC vq(3 - J o)

:;- = UXi

(3n-5)

where J o is the ambient temperature, and q is the radiation coefficient. Although too simple to describe radiant heat transfer in a fluid adequately, this equation is worth considering because of (1) its analytical simplicity and (2) its application as a convenient model for relaxation processes. 3n-2. Equation of State. To the conservation equations must be added an equation of state. Perfect Gas. The gas law for a perfect gas is (3n-6)

p = RpJ

where R is the gas constant. An approximate form of this equation will now be derived. For a perfect gas the small-signal sound speed Co is given by co 2 = 'YRJo = 'YPol Po, where po and PO are the ambient values of p and p. Let J = (3,0(1 + e), p = po + poc0 2P, and p = po(l + 8), where (3,0 is the ambient value of (3, (for perfect gases (3'°30 = 1). Assume that 0, P, and 8 are small quantities of first order. Expansion of Eq. (3n-6) to second order yields

e=

'YP -

8

+

82 -

'YP8

(3n-7)

First-order relations are now defined to be those that hold in linear, lossless acoustic theory; examples are Pt = -poll' U and p - po = co 2 (p - po). At this point we assert that any factor in a second-order term in Eq. (3n-7) may be replaced by its first-order equivalent. The justification is that any more precise substitution would result in the appearance of third- or higher-order terms, and such terms have already been excluded from Eq. (3n-7). Thus in the l&st second-order term in Eq. (3n-7) P may be replaced by 8 to give () =

COlTect to second order.

'YP - 8 -

(1' - 1)8 2

(3n-8)

This is a useful approximate form of the perfect gas law.

NONLINEAR ACOUSTICS (THEORETICAL)

3-185

One of the most fruitful special cases to consider is the isentropic perfect gas. When a perfect gas is inviscid and there is no heat flow, Eq. (3n-3) can be used to reduce the gas law, Eq. (3n-6), to P (3n-9) po = Po

(p)7

The square of the sound speed, which by definition is,

OJ "" becomes cfi = 'YP = p

(aapp ) s co" (E-) (7-1)1'11 po

(3n-1O) (3n-ll)

An expanded form of Eq. (3n-9) is aR follows:

= 8 + t('Y

P

- l)s'

+

(3n-12)

Other Fluids. For liquids and for gases that are not perfect, one can start with a general equation of state J = J(p,p). Recognizing that (B'J/ap)p = 'Y(pc 2i3.)-t, one obtains the exact expression Ot = ; :

+ 8)-1 [I' (~r P t -

(1

(3n-13)

8tJ

In order to obtain an approximation analogous to Eq. (3n-8), it is first necessary to set down a general isentropic equation of state, p -

po = poCo 2

(8 + 2A ~ 82+ 3A !Z. S3 + . . .)

(3n-14)

where the coefficients B/A, CIA, etc., are to be determined experimentally (see Sec. 30). With the help of this expression and some elementary thermodynamic relations, one invokes the approximation procedure described following Eq. (3n-7) and reduces Eq. (3n-13) to (ref. 3)

o=

'YP - s - (h - 1)8 2

(3n-15)

correct to second order, where

h = 1

+

;! +

2~)

!(-y - 1) (1 -

- (I' -

1)2(4i3,oJ)-1

(3n-16)

If Egs. (3n-14) and (3n-12) are compared, it will be seen that B/A replaces the quantity I' - 1 in describing second-order nonlinearity of the p - p relation. For a perfect gas, therefore, replace B/A by I' - 1 and i3 is given by x - X(cf»

cf> =

t - u ± co(l

+ c,U +

C

2U2 . . . )

(3n-51)

where U is to be interpreted as co-Ix,(cf». Solids. The mathematical formalism for plane, longitudinal elastic waves in solids, either crystalline or isotropic, is very similar to that for liquids and gases (refs. 11-13). The wave equation is given in Lagrangian coordinates as ~tt = co2G(~aHaa

G(~a)

where

=

1

+ C:~:) ~a + C:~:) ~aa

(3n-52) ...

(3n-53)

Here a represents the rest position of a particle; ~ is partical displacement; and M 2 , M 3, M 4, etc., are quantities involving the second-, third-, fourth-, and higher-order elastic coefficients (ref. 12). The quantity co 2G plays the same role that (pc/ PO)2 does for fluids (ref. 14). By the Lagrangian equation of continuity, po/p = 1 + ~a; thus replace Eq. (3n-1S) by A= =

- Co

!o

-Co[~a

1;.

-

[G(~a')Jt d~a' tm3~a2

+ (t -

(3n-54) tm4)m32~aS . . . J

(3n-55)

where ms = -JYI 3/M 2, m4 = 1 - M4/M2m32, etc. Riemann invariants are defined as before by Eq. (3n-24). Note that u = ~, in Lagrangian coordinates. Simple-wave fields are again specified by Eq. (3n-21), which when combined with Eq. (3n-5) leads to (3n-56) ~a = + U + tm3U2 + tm4m32Us The propagation speed for simple waves is

_ ±coG! (da) cit u~con't-

(3n-57)

The factor u, which appears in Eq. (3n-23), is absent here because the coordinate system is Lagrangian. Equation (3n-57) expanded in series form is (3n-58)

3-192

ACOUSTICS

Therefore, the solution of the piston problem, given u(O,t) =

t

+ 1

=

a/co

±

~maU

+ ima'(l

- 2m.) U'

X,(t), is

(3n-59)

where U is to be interpreted, as in Eq. (3n-S1), as co-1X,( Ao) is equivalent to a plane wave in a medium in which the dissipation increases with distance. Conversely, for a converging wave (A < Ao) the dissipation seems to decrease with distance (refs. 17, 18). gn-lS. Equations for Other Forms of Dissipation. If dissipation is due to an agency other than the thermoviscous effects discussed in the last section, it may still be possible to derive an approximate unidirectional-wave equation similar to Burgers'. Relaxing Fluids. An elementary example of a relaxing fluid is one that radiates heat in accordance with Eq. (3n-.5)(ref. 38). For simplicity take the fluid to be a perfect gas, and let it be inviscid and thermally nonconducting. At very low frequencies infinitesimal waves travel at the isothermal speed of sound, given by bo2 = 'Pol po. At very high frequencies the speed is the adiabatic value, given by boo' =

3-201

NONLINEAR ACOUSTICS (THEORETICAL)

'YPo/ Po (the notation boo is used here in place of Co to emphasize the role played by frequency). The dispersion m, defined by

(3n-99) is equal to 'Y - 1 for the radiating gas. If the dispersion is very smail, i.e., m « 1 (which in this case implies 'Y == 1), the following approximate equation for plane waves can be derived:

(q

+ a~/) u x -

bo- 2 (!3iq +!3a

a~/) uu; = ± 2~o u""

(3n-l00)

where t' = t =+= x/boo It is seen that the radiation coefficient q [see Eq. (3n-5)] is the reciprocal of a relaxi1tion time. Subscripts a and 1: used with (3 indicate adiabatic and isothermal values, respectively; that is, !3a = ('Y 1)/2 and (3i = (1 1)/2 = 1. The two values are essentially the same, since it has been assumed that 'Y == 1. At either very low frequencies (wq-l « 1) or very high frequencies (wq-l » 1) the lefthand side of the equation takes on the same form as Eq. (3n-47). If the equation is linearized, a dispersion relation can be found that gives the expected behavior for a relaxation process (the actual formulas for the attenuation and phase velocity agree with the exact ones for a radiating gas only for m « 1). Polyakova, Soluyan, and Khokhlov considered a relaxation process directly and obtained a pair of equations that can be merged to form a single equation exactly like Eq. (3n-lOO) except that {Ji and!3a are equal (ref. 39). Some solutions (refs. 39,40) have been found. One represents a steady shock wave. The shock profile is singlevalued for very weak shocks. But when the shock is strong enough that its propagation speed [see Eq. (3n-72)] exceeds boo, the solution breaks down (a triple-valued waveform is predicted). This illustrates an important fact about the role of relaxation in nonlinear propagation: Relaxation absorption can stand off weak nonlinear effects, but not strong ones. In frequency terms, relaxation offers high attenuation to a broad mid-range of frequencies. If the wave is quite weak, the distortion components are easily absorbed because their frequencies fall in the range of high attenuation. But if the wave is strong, many more very high frequency components are produced, and these are not attenuated efficiently by the relaxation process. To keep the waveform from becoming triple valued, it is necessary to include a viscosity term in the approximate wave equation. In ref. 40 the problem of an originally sinusoidal wave is treated. Quantitative approximate solutions are obtained for cases in which the source frequency is either very low or very high, and a qualitative discussion is given for source frequencies in between. Marsh, Mellen, and Konrad (ref. 30) postulated a "Burgers-like" equation for spherical waves. It is similar to Eq. (3n-100) but is corrected to take account of spherical divergence. A viscosity term is added, and {Ji and !3a are the same. At either very low or very high frequencies the equation takes on the form of Eq. (3n-98) [for spherical waves (A/Ao)! = r/ro = ez1ro ], and some initial attempts at solving this equation were described. Boundary-layer Effects. Consider the propagation of a plane wave in a thermoviscous fluid contained in a tube. The wave can never be truly plane because the phase fronts curve a great deal as they pass through the viscous and thermal boundary layers at the wall of the tube. If the boundary-layer thicknesses are small compared with the tube radius, however, the curvature of the phase fronts is restricted to very narrow regions, and the wave may be considered quasi-plane. The boundary layers still affect the wave, causing an attenuation that is proportional to V;;; and a comparable dispersion. If the frequency is low, the attenuation from this source is much

+

+

3-202

ACOUSTICS

more important than that due to thermoviscous effects in the mainstream (central core of the fluid), and so it makes sense to find a Burgers-like equation for this case. A one-dimensional model of time-harmonic wave propagation in ducts with boundary-layer effects treated as a body force has been given by Lamb (ref. 41). Chester (ref. 42) has generalized this model and applied it to compound flow in a closed tube. His method can be used to obtain the following equation for simple-wave flow:

u. -l!...uu,. co'

=

=+=

1

+ (I'

-

1)/VPr

coD /2

(!:..)t 10('" u,.(x,t' 7r

,,).!£..v;.

(3n-101)

where D is the hydraulic diameter-ef the duct (four times the cross-sectional area divided by the circumference). No solutions are presently available. But the equation does have proper limiting forms. If the effect of the boundary layers (right-hand side) is neglected, the result is Eq. (3n-47). If the nonlinear term is dropped, the time-harmonic solution can be found, and this solution yields the correct attenuation and dispersion. Because of the relative weakness of boundary-layer attenuation (the dimensionless attenuation all. varies as 1lVw), the higher spectral components generated as a manifestation of steepening of the waveform are not efficiently absorbed. Thus discontinuous solutions, modified somewhat by the attenuation and dispersion, are to be expected.

REFLECTION, STANDING WAVES, AND REFRACTION 3n-14. Reflection and Standing Waves. For plane interacting waves in lossless fluids we return to Eqs. (3n-24) to (3n-26). For perfect gases the Riemann invariants are given by t=_c_+~

(3n-102a)

ll=_c_+~

(3n-102b)

1'-1

1'-1

2

2

Equations (3n-26) tell us that the quantity t is forwarded unchanged with speed u + c = t('Y + 1)r - i(3 - 1')6. Similarly, the speed for the invariant 6 is u c = t(3 - 'Y)r - t('Y + 1)6. The roles of independent and dependent variables can be reversed to give the following differential equation for the flow: (3n-103) where N = t('Y + 1)/(1' - 1). For monatomic and diatomic gases N = 2 and N = 3, respectively. An exact solution of this equation in terms of arbitrary functions f(r) and g(13) is known, but it is usually difficult to determine f and g from the initial conditions (ref. 4). Reflection. Certain valuable information about reflection can be obtained without solving for the entire flow field. Consider the problem of reflection from a rigid wall. For the moment we need not be specific about the equation of state. Let the incident wave be an outgoing simple wave. The Riemann invariant t for a particular signal in this wave is, by Eqs. (3n-21) and (3n-24), 2r =

"i +

Ui = 2"i

But rcan also be evaluated at the wall during the interaction of the incident and reflected waves: i.e., 2t = "wall + Uwall = "wall Elimination of r between these two expressions gives

"wall

= 2"i

NONLINEAR ACOUSTICS (THEORETICAL)

3-203

This is an exact statement of the law of reflection for continuous finite-amplitude waves at a rigid wall: The quantity A doubles, not the acoustic pressure. To see what happens to the pressure, we must specify an equation of state. Take the case of a perfect gas, for which A = 2(c - co)/(1' - l)(thus c - Co doubles at a rigid wall). Using Eq. (3n-ll), we obtain p) ( po wall

where

p. =

21'/(1' - 1).

=

[(Pi)l/~ 2. Po

- 1

JIL

(3n-105)

Now define a, wall amplification factor a by :l

NH.Ol at 25°0 [9]

NH.Ol at -78.5°0 [10]

NH.IO. [6J

NH4N03 [9]

1.000 0.973 0.052

1.000 0.978 0.960

1.000 0.980 0.963

1.000 0.972 0.948

0.805 0.794

0.933 0.917

0.045 0.931

0.946 0.931

0.928 0.912

0.885 0.885 0.873 0.862 0.851 0.842

0.773 0.773 0.764 0.754

0.906 0.906 0.895 0.885 0.875 0.867

. .... . .... . ....

. .... . ....

0.888 0.888 0.875 0.864 0.853 0.843

0.897 0.882 0.870 0.857 0.846 0.835

. .... . .... . .... . .... ..... . .... . .... ..... ..... . ....

. .... . .... . .... . .... . . .. . . .... . .... . .... . .... . ....

0.835 0.826 0.S18 0.810 0.803 0.796 0.790 0.783 0.776 0.769

. .... . ....

. .... . ....

. ....

. .... . ....

NH.Br NH.OH0 2 at -78.5°0 [5] [10]

kilobars

NaIO. [6]

NaNH.0.H.06 [21]

NaN03 [9]

0 5 10

1.000 0.981 0.966

1.000 0.974 0.952

1.000 0.982 0.965

1.000 0.964 0.938

1.000 0.973 0.950

1.000 0.973 0.951

1.000 0.965 0.932

15 20

0.953 0.942

0.933 0.917

0.950 0.937

0.917 0.900

0.929 0.910

0.032 0.915

25 30 35 40 45 50

. . .. ..... .....

.... . . .... . .... . .... . ....

0.912 0.912 0.001 0.890 0.881 0.871

0.845 0.845 0.834 0.824

0.878 0.878 0.863 0.850 0.838 0.827

55 60 65 70 75 80 85 90 95 100

.... . .... . .... . .... . .... . .... . ..... .... . .... .

..... . .... . .... ..... .... . . .... . .... .... . . .... . ....

0.852 0.943 0.836 0.830 0.823 0.817 0.812 0.807 0.802 0.797

0.817 0.808 0.800 0.892 0.885 0.878 0.872 0.865 0.859 0.852

P,

!:

(Continued)

aa

.

.....

..... .....

....

.

0.889

z

. .... . ....

. . .... . .... ""

y

* For references, see p. 4-96. "Transition at 53.9; volumes 0.864 and 0.853. , Transition at 22.8; volumes 0.892 and 0.868. aa Transition at 11.2; volumes 0.926 and 0.815.

. .... .....

.....

. .... . .... ..... . .... . .... .... . . ....

. .... ..... . .... . .... . ... . .... .....

. .... . .... . .... ..... . .... . ....

0.826 0.817 0.810 0.804 0.797 0.792 0.787 0.784 0.780 0.777

iIi t'=J

i>-

>-'3

TABLE

P,

kilobars

0

4d-2. V IVa

OF INORGANIC COMPOUNDS*

NH4P0 4 [5J

NiSO. [21]

PbI z [5]

NH2S0aH [21]

NH4010 4 [6]

1.000

1.000

1.000

1.000

1.000

0.979 0.963 0.948 0.935

0.971 0.948 0.927 0.910

NH.I at 25°0 [9]

NH.I at -78.5°0 [10]

NH4IOa [6]

PbS at 25°0 [2]

PbS at -78.5°0 [10]

1.000

1.000

1.000

1.000

1.000

1.000

0.982 0.966 0.952 0.940

0.832 0.807 0.781 0.767

0.967 0.941 0.920 0.901

0.986 0.973 0.961 0.950

0.983 0.969 0.956 0.945

0.989 0.980 0.971 0.962

NH4H,PO. [21]

bb

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0.981 0.965 0.951 0.939 0.877 0.867 0.857 0.848

....

. .... . .... . .... .

.... . .... . .... . ..

".,

.... . .... . .... . . ... .

0.983 0.967 0.953 0.940 0.886 0.861

.... . .... . . .... . .... . .... .. -.,

. .... . .... .... . .... . . .... ..

'"

.

.... .

.... .

0.897 0.878 0.863 0.850 0.838 0.827 0.818

. .... . .... . .... . .... ..... . .... . .... . .... .... . . .... .... . . .... .... .

'*' For references, see p. 4-96. bb Transition at 5.0; volumes 0.963 and 0.924. "Transition at 0.5; volumes 0.997 ± and 0.856. ad Volume at 24.2 = 0.958 and at 22.3 = 0.937.

(Continued)

cc

0.923 0.913

..... ..... ..... ..... ..... ..... .... . ..... ..... ..... .... . ..... ..... .0"

•

dd

0.895

..... . .... . .... . .... . .... . .... . .... . .... .... . . .... . .... . .... . .... . ....

.... .

0.929 0.919

. .... . .... . .... . .... . .... . .... ... " . . .... . .... . .... ..... . .... . .... . ....

0.754 0.740 0.728 0.716 0.705 0.695 0.686 0.678 0.670 0.662 0.655 0.648 0.642 0.635 0.628 0.622

0.885 0.870 0.858 0.846 0.837 0.828

. .... . .... . .... . .... . .... . .... . .... . .... . .... . ....

0.940

. .... ..... . .... . .... .....

. .... . .... . .... . .... . .... ..... . .... . .... . .... . ....

0.935 0.928 0.921 0.915 0.909 0.903 0.899 0.896 0.892 0.890 0.887 0.885 0.882 0.880 0.878 0.876

0.933 0.925 0.918 0.913 0.909 0.905

()

o

~

;g trJ m

m

td

>-
>-3

Q

P>

'"d

P>

Q

>-
-3 trI

>-< lf1

t

f-'

0 ""l

TABLE

4e-2.

l\10LAR HEAT CAPACI1'Y AT CONSTANT PR~JSSURE OF THE CHEMICAL ELEMENTS AT HIGHER THAN ROOM TEMPERATURE, CAL/MOLE'

t

R

i-'

o

Ae revised December, 1967 Element

298.15

400

500

600

700

800

00

1000

1200

7.59(1) 7.50(1) 4.968

7.59(1) 7.50(1) 4.968

1500

2000

2500

3000

---Aluminum AI 26.9815 ................ Antimony Sb 121.75 ................. Argon Ar 39.948 .................... Arsenic As 74.9216 .................. Beryllium Be 9.0122 ................. Bismuth Bi 208.980 ............... , . Boron (crystalline) B 10.811 .......... Boron (amorphous) B 10.811. ....... Bromine Br, 159.818 ................ Cadmium Cd 112.40 ................. Calcium Ca 40.08 ................... Carbon (graphite) C 12.01115 ........ Carbon (diamond) C 12.0115 ........ Cerium Ce 140.12 ................... Chlorine CJ, 70.906 .................. Chromium Cr 51.996 ............... Cobalt Co 58.9332 ................. Copper Cu 63.54 ..•................. Erbium Er 167.26 ................... Europium Eu 151.96 ................ Fluorine F, 37.9968 ................. Germanium Ge 72.59 ................ Gold Au 196.967 .................... Hafnium Hf 178.49 .................. Helium He 4.0026 ................... Holmium Ho 164.930 ................ n-Hydrogen H, 2.01594 .............. n-Deuterium D, 4.02820 ............. Indium In 114.82 ................... Iodine I, 253.8088 ... '" ............. Iridium Ir 192.2 .................... Iron Fe 55.847 ...................... Krypton Kr 83.80 ................... Lanthanum La 138.91. .............. Lead Pb 207.19 .................... Lithium Li 6.939 .................... Lutetium Lu 174.97 ................. Magnesium Mg 24.312 ...............

5.81(e) 6.03(e) 4.968(g) 5.89(a)

3.93(e) 6.20(e) 2.65(e) 2.86(e) 18.09(1) 6.20(e) 6.26(a)

2.04(e) 1. 46(e) 6.44(1/) 8. 11 (g) 5.58(e) 5.93(a)

5.84(e) 6.71(e) 6.48(e) 7.49(g) 5.58(e) 6.06(e) 6.15(a)

4.968(g) 6.49(a)

6.892(g) 6.978(g) 6.39(e) 13.01(e) 6.00(e) 5.97(a)

4.968(g) 6. 65(/l) 6.32(e) 5.78(e) 6.40(e) 5095(c)

6.16 6.22 4.968 6.15 4.73 6.45 3.72 3.78 8.78(g) 6.49 6.62 2.85 2.45 6.76 8.44 6.02 6.34 6.08 6.79 6.68 7.90 5.8.5 6.17 6.34 4.968 6.65 6.975 6.989 6.93(e) 19.28(1) 6.14 6.54 4.968 6.81 6.56 6.62(c) 6.42 6.24

6.45 6.38 4.968 6.32 5.20 6.69(e) 4.49 4.40 8.86 6.78(e) 7.03 3.50 3.24 7.10 8.62 6.41 6.74 6.25 6.87

*

8.21 5.95 6.29 6.52 4.968 6.74 6.993 7.018 7.03(1) 8.95(g) 6.27 7.10 4.968 6.97 6.79 7.20(1) 6.46 6.52

6.72 6.56 4.968 6.50 5.54 7.6(1) 4.99 4.88 8.91 7.10(1) 7.45 4.04 3.85 7.46 8.74 6.73 7.09 6.39 6.97 7.24 8.43 6.03 6.40 6.70 4.968 6.76 7.008 7.078 6.99 8.98 6.41 7.66 4.968 7.13 7.02(c) 7.06 6.50 6.80

7.00 6.88 4.968 6.74 5.82 7.6 5.32 5.26 8.94 7.10 7.87(a)

4.44 4.31 7.84 8.82 7.00 7.42(a)

6.52 7.11 7.52 8.59 6.10 6.51 6.88 4.968 6.80 7.035 7.171 6.96 9.00 6.55 8.27 4.968 7.29 7.25(1) 6.93 6.61 7.08

7.37(e) 7.15(e) 4.968 7.02(a)

6.06 7.6(1) 5.56 5.57 8.97 7.10 7.86(1/) 4.74 4.66 8.25 8.88 7.22 7.7.5(1/) 6.62 7.27 7.88 8.71(g) 6.19 6.65 7.06 4.968 6.95 7.078 7.288 6.93(1) 9.02(g) 6.69 9.07 4.968 7.45(/l) 7.17 6.92 6.79 7.36(e)

6.54(e) 5.95 6.05 9.01(g) 7.10(1) 9.32(1/) 5.15 5.16 9. 14(/l) 8.95 7.66 8.84 6.82 7.67 9.09(e) 6.50 6.89 7.43 4.968 7.61 7.215 7.557 ......... 6.96

4.968

......... ......... 4.971(g: 4.968

4.968

4.968(g:

......... . . . . . . . . . . . . . . . . . . ......... 5.020(g: 6.26 6.39

6.67 6.75

7.12(c) 7.07(e)

4.968(g) 4.968 4.968 4.969(g: 4.968 ........ 7.4(1) 5.008(g) 5.219 5.796(g: 6.06(e) 5.43 5.67 5.97 5.86 5.60(e) 9.35(1) 9.01(g) 9.68(e) 8.48 ......... . . . . . . . . . 7.358(g 9.50(1/) 10.33 7.00(e) 7.5(1) . . . . . . . . . ......... 6.010(g: 8.18 9. 14(e) ......... 5.1O(g) 9.11(1) 5.61 6.74(g) 6.86(e) 7.13(e)

6.60(1) 7.0(1)

7.79(a)

4.968 8.58 7.401 7.824 .........

4.968

7.24 8. 13Cy) 4.968

7.03 6.89 7.24 7.8(1)

6.88(1) 6.87(1) 7.85 7.8(1)

13.01(a)

.........

4.968 1O.69(a) 7.706 8.164 . ........

4.968 8.162(g) 8.568(g) . ........

4.968

4.968(g:

5.709(g) 5.509(g:

7.65(e) 8.73(1')

4.968

4.968

......... ......... 9.09(e) 4.968(g)

4.969

4.968

4.968(g:

6.951(g) 7.925(g; 4.977

5.022(g;

~ ~ ~

>-:3

Manganese Mn 54.9380 .............. Mercury Hg 200.59 .................. Molybdenum Mo 95.94 .............. Neodymium Nd 144.24 ............. Neon Ne 20.183 .................... Nickel Ni 58.71 ..................... Niobium (columbium) Nb(Cb) 92.906 .. Nitrogen N, 28.0134 ................. Osmium Os 190.2 ............... ' ... Oxygen 0, 31.9988 .................. Ozone 0,47.9982 ................... P •. lladium Pd 106.4 ................. Phosphorus (red, trielinie) P 30.9738 .. Phosphorus (white) P 30.9738 ........ Platinum Pt 195.09 .................. Plutonium Pu[239] .................. Potassium K 39.102 ................. Radon Rn [222] ..................... Rhenium Re 186.2 .................. Rhodium Rh 102.905 ................ Ruthenium Ru 101.07 ............... Samarium Sm 150.35 ................ Scandium Se 44,956 ................. Selenium (metallic) Se 78.96 •......... Silicon Si 28.086 .................... SHver Ag 107.870 ................... Sodium Na 22.9898 .................. Sulfur S 32.064 ..................... Tantalum Ta 180.948 ................ Tellurium Te 127.60 ................. Thallium TI 204.37 .................. Thorium Th 232.038 ................. Thulium Tm 168.934 ................ Tin (white) Sn 118.69 ............... Titanium Ti 47.90 ................... Tungsten (wolfram) W 183.85 ........ Uranium U 238.03 .................. Vanadium V 50.942 ................. Xenon Xe 130.30. . ................ Ytterbium Yb 173.04 ................ Yttrium Y 8.905 ...... , ............. Zinc Zn 65.37 ....................... Zirconium Zr 91.22 ..................

6.28(e) 6.76 7.21 6.69(1) 6.54 6.48 5.73(e) 6.05 6.25 6.55(",) 7.24 6.88 4.968(g) 4.968 4.968 6.23(e) 7.37 6.80 5.88(e) 6.18 6.09 6.96(g) 7.07 6.99 5.90(e) 5.99 6.09 7.02(g) 7.20 7.43 9.38(g) 10.46 11.30 6.21(e) 6.35 6.49 5.07(e) 5.54 5.85 6.29(1) 6.29(1) 5.70(1l) 6.18(e) 6.31 6.44 7.64(",) 8.03(tJ) 8.53(')') 0.70(e) 7.53(1) 7.34 4.968(g) 4.968 4.968 6.16(e) 6.22 6.32 5.97(e) 6.21 6.45 5.75(e) 5.82 5.91 7.06(",) 7.93 8.94 6.10(",) 6.29 6.41 6.06(e) 6.65(e) 8.40(1) 4.78(e) 5.30 5.61 6.07(e) 6.18 6.30 7.53(1) 6.72(e) 7.30 5.40(rh) 7.73(1) 9.08 6.06(e) 6.22 6.30 6.14(e) 6.68 7.21 6.29(a) 6.57 7.03(a) 6.53(",) 7.15 6.85 6.51 6.46(e) 6.49 7.32(e) 6.45(e) 6.89 6.31 6.53 5.98(a) 5.81(e) 5.96 6.06 7.65 6.61(a) 7.10 5.95(e) 6.27 6.44 4.968(g) 4.968 4.968 7.41 6.39(a) 6.60 6.34(a) 6.49 6.65 6.31 6.55 6.07(e) 6.06(",) 6.54 6.78

8.35(",) 7.63 8.01 6.48(1) 4.968(g) 4.968 6.38 6.48 6.55 7.66 8.14 8.71 4.968 4.968 4.968 8.31 7.37 7.44 6.28 6.38 6.48 7.20 7.35 7.51 6.18 6.27 6.63 7.67 7.88 8.06 11.92 12.70 12.37 6.62 6.90 6.76 6.16 6.50(e) 6.82 6.70 9.00(0) 8.40(.) 7.13 7.11 7.20 4.968 4.968 4.968 6.69 6.43 6.56 6.69 6.93 7.17 6.04 6.23 6.42 9.75 10.19 10.52 6.57 6.75 6.96 8.40(1) 6.13 5.82 5.99 6.42 6.56 6.72 7.12 7.00 6.92 8.20 7.80(1) 6.45 6.33 6.39 8.26(e) 9.00(1) 7.73 7.2(1) ...... , .. .. .... .. 7.76 8.06 7.45 6.59 6.76 7.08 6.87(1) 6.85 6.85(1) 7.01 6.77 7.25 6.16 6.27 6.37 9.08 9.99(",) 8.31 6.70 6.57 6.85 4.968 4.968 4.968 7.25 7.13 7.37 7.00 7.18 6.82 6.79(e) 7.5(1) 7.5 7.01 7.23 7.45

9.01(1l) 4.968 6.70 10.03(",) 4.968 7.88 6.68 7.82 6.54

8.34 13.15 7.17

6.57

7.08

9.21(f'l) 4.968 6.93 10.65(')') 4.968 8.34 6.88(e) 8.06 6.72 8.53 13.43 7.44

10.99(0)

4.968 7.47 4.968 8.65(e)

4.968 8.63 4.968 10.30(1)

8.33 7.00(e) 8.74 13.68(g) 7.86(e)

8.60

5.039(g) 5.252(g) 4.968 4.968(gJ 1O.46(e) 4.968

4.968(g:

8.76

8.86(g)

4.968

4.968(g·

9.03(g)

7.34

7.72

8.37(e)

4.968 7.22(e)

4.968

4.968

9.00(0)

7.26(1) 4.968 6.95 7.65(e) 6.75 10.82(",) 7.46 6.35 7.15 6.92(1)

'

6.57 9.00(1) . ........ 8.67 7.52 ........ .

7.73(",) 6.59 10. 26(iJ) 7.27 4.968 7.64(a)

7.53 7.5(1) 7.90(,,)

il1 trJ

~

>-:l 7.11 11.22(1l) 8.06 6.49 7.62(e)

7.24(e) 6.06(g)

..........

6.08(g)

9.14(a)

('1

~

~

('1

6.66(e) , .......

4.968(g) 4. 969 (g'

..

H

>-:l

H

trJ

6.69 . ........

6.87 .........

U2

7.17

7.46(e)

5.420(g)

5.830

6. 173 (g'

5.20(g)

5.32(g) 6.266(g:

8.30

9.80(e)

4.968 5.01

4.968(g: 5.16(g)

4.968

4.969(g)

9.28(a)

7.89 ....... - . 7 .10CtJ) 6.80 9.15(')') 7.85 4.968 8.79(1) 7.90 4.968(g) 7. 50 ell)

8.43(e) . ......... 7. 85(tJ) 7.14 7.71 11.45(1) 8.69(e) 4.968 4.968 4.97(g) 4.97 8.43(a)

4.968 7.50(1l)

4.968

t

f-'

o

CO

·4-110

REA'!'

gases). With the exception of B (amorphous), C. (diamond), Se, Te, and the gases R 2, D 2, Eu, Sm, Tm, and Yb, the tabulated values are based on (1) RHultgren, R.L. Orr, P. D. Anderson, and K. K. Kelley, "Selected Values of Thermodynamic Properties of Metals and Alloys," John Wiley & Sons, Inc., New York, 1963 (and later looseleaf supplements); (2) JANAF Thermochemical Tables, Clearinghouse, U.S. Dep~rtment of Commerce, Springfield, Va. (PB Rept. 168370, 1965; PB Rept. H,8370-1, 1966) (and later looseleaf supplements); and (3) J.Hilsenrath, C. G. Messina, lJ.nd W. H. Evans, Ideal Gas Thermodynamic Functions for 73 Atoms and Their First and Second Ions to 10,000K, Air Force Weapons Lab. Rept. TDR-64-44, Kirtland Air Force Base; N.Mex., 1964. TABLE 4e-3. HEAT CAPACITY OF WATER (Osborne, Stimson, and Ginnings, National BUreau of Standards) Temp,oC

--

J g·K

Temp.,oC

--

0 5 10 15 20 25 30 35 40 45 50

4.2177 4.2022 4.1922 4.1858 4.1819 4.1796 4.1785 4.1782 4.1786 4.1795 4.1807

50 55 60 65 70 75 80 85 90 95 100

4.1807 4.1824 4.1844 4.1868 4.1896 4.1928 4.1964 4.2005 4.2051 4.2103 4.2160

J g·K

As a first approximation in explaining the temperature dependence of the heat capacity of solids, Einstein made the assumption that all oscillators in the lattice ~ibratlld with the same frequency /In. If h is Planck's constant and k is Boltzmann's constant; let

and denote the zero-point energy per mole by U o• Then the Einstein theory· of specific heat yields for the molar energy U at the temperature T fE' .) U - U o x ~. mstem 3RT = e" - 1

(4e-l)

where R is the universal gas constant. The Einstein molar heat capacity at constant volume is given .by Cv = dU/dT, or x 2e" .. . . . Cv (4e-2) (Emstem) 3R = (e" - 1)2 The inolarentropy SIs equal to f(Cv/T) dT, whence (Einstein)..Ii 3R

=

_x_. - In (1 - e-") e" - 1

(4e-3)

Numerical values of the quantities in Eqs. (4e-l), (4e-2), and (4e-3) are given in Tables 4e-4, 4e-5 and 4e-6, taken from '~Contributions to the Thermodynamic Functions by a Planck-Einstein Oscillator in One Degree of Freedom," prepared by Herrick L. Johnston, Lydia Savedoff, and Jack Belzer, of the Cryogenic Laboratory of the

4-111

HEAT CAPACITIES TABLE

eE T

0.0

0.1

0.2

4e-4.

0.3

U - Uo 3RT 0.4

(EINSTEIN)

0.5

0.6

0.7

0.8

0.9

- - - - - - - - - - - - - - -- - - - - - - - 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1.00000 0.95083 0.90333 0.85749 0.81330 0.77075 0.72982 0.69050 0.65277 0.61661 0.58198 0.54886 0.51722 0.48702 0.45824 0.43083 0.40475 0.37998 0.35646 0.33416 0.31304 0.29304 0.27414 0.25629 0.23945 0.22356 0.20861 0.19453 0.18129 0.16886 0.15719 0.14624 0.13598 0.12638 0.11739 0.10898 0.10113 0.09380 0.08695 0.08057 0.07463 0.06909 0.06394 0.05915 0.05469 0.05055 0.04671 0.04314 0.03983 0.03676 0.03392 0.03128 0.02885 0.02658 0.02450 0.02257 0.02079 0.01914 0.01761 0.01621 0.01491 0.01371 0.01261 0.01159 0.01065 0.00979 0.00899 0.00826 0.00758 0.00696 0.00639 0.00586 0.00538 0.00494 0.00453 0.00415 0.00381 0.00349 0.00320 0.00293 0.00269 0.00246 0.00225 0.00206 0.00189 0.00173 0.00158 0.00145 0.00133 0.00121 0.00111 0.00102 0.00093 0.00085 0.00078 0.00071 0.00065 0.00059 0.00054 0.00050 0.00045 0.00042 0.00038 0.00035 0.00032 0.00029 0.00026 0.00024 0.00022 0.00020 0.00018 0.00017 0.00015 0.00014 0.00013 0.00012 0.00011 0.00010 0.00009 0.00008 0.00007 0.00007 0.00006 0.00006 0.00005 0.00005 0.00004 0.00004 0.00004 0.00003 0.00003 0.00003 0.00002 0.00002 0.00002 0.00002 0.00002 0.00002 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001

TABLE

eE T

0.0

0.1

0.2

0.3

Cv 4e-5. 3R 0.4

(EINSTEIN)

0.5

0.6

0.7

0.8

0.9

----- --- - -- -- -- -- ---- - -- 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1.00000 0.99917 0.99667 0.99253 0.98677 0.97942 0.97053 0.96015 0.94833 0.93515 0.92067 0.90499 0.88817 0.87031 0.85151 0.83185 0.81143 0.79035 0.76869 0.74657 0.72406 0.70127 0.67827 0.65515 0.63200 0.60889 0.58589 0.56307 0.54049 0.51820 0.49627 0.47473 0.45363 0.43301 0.41289 0.39331 0.37429 0.35584 0.33799 0.32073 0.30409 0.28806 0.27264 0.25783 0.24363 0.23004 0.21704 0.20462 0.19277 0.18149 0.17074 0.16053 0.15083 0.14162 0.13290 0.12464 0.11683 0.10944 0.10247 0.09588 0.08968 0.08383 0.07833 0.07315 0.06828 0.06371 0.05942 0.05539 0.05162 0.04808 0.04476 0.04166 0.03876 0.03605 0.03351 0.03115 0.02894 0.02687 0.02495 0.02316 0.02148 0.01993 0.01848 0.01713 0.01587 0.01471 0.01362 0.01261 0.01168 0.01081 0.01000 0.00925 000855 0.00791 0.00731 0.00676 0.00624 0.00577 0.00533 0.00492 0.00454 0.00419 000387 0.00357 0.00329 0.00304 0.00280 0.00258 0.00238 0.00219 0.00202 0.00186 0.00172 0.00158 0.00145 0.00134 0.00123 0.00114 0.00104 0.00096 0.00088 0.00081 0.00075 0.00069 0.00063 0.00058 0.00054 0.00049 0.00045 0.00042 0.00038 0.00035 0.00032 0.00030 0.00027 0.00025 0.00023 0.00021 0.00019 0.00018 0.00016 0.00015 0.00014 0.00013 0.00012 0.00011 0.00010 0.00009 0.00008 0.00008

-

Ohio State University, under contract between the Office of Naval Research and the Ohio State University Research Foundation, 194.9. Debye assumed that the oscillators occupying the lattice points in a crystalline solid vibrated with a continuous spectrum of frequencies from zero to a maximum value Pm. Defining the "Debye temperature" eD and y by the equations

4-112

HEAT TABLE

eE T

S 4e-6. 3R

(EINSTEIN)

I

0.0

0.1

0.3

0.2

0.4

0.5

0.6

0.7

0.8

0.9

-- - - - - - - - - - - - - -- - - - - - - 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

00 3.30300 2.61110 2.20772 1.92293 1.70350 1.5256~ 1.37684 1.24939 1.13845 1.04066 0.95363 0.87560 0.80521 0.74139 0.68331 0.63027 0.58171 0.53714 0.49617 0.45845 0.42367 0.39158 0.36194 0.33455 0.30921 0.28579 0.26410 0.24403 0.22546 0.20826 0.19234 0.17760 0.16396 0.15133 0.13964 0.12884 0.11883 0.10958 0.10102 0.09312 0.08580 0.07905 0.07281 0.06704 0.06172 0.05681 0.05228 0.04809 0.04423 0.04068 0.03740 0.03438 0.03159 0.02903 0.02666 0.02450 0.02249 0.02064 0.01896 0.01739 0.01596 0.01464 0.01343 0.01232 0.01130 0.01035 0.00949 0.00869 0.00797 0.00730 0.00669 0.00613 0.00562 0.00514 0.00470 0.00431 0.00394 0.00361 0.00330 0.00303 0.00276 0.00252 0.00231 0.00211 0.00193 0.00176 0.00162 0.00148 0.00135 0.00123 0.00113 0.00103 0.00094 0.00086 0.00078 0.00072 0.00065 0.00060 0.00055 0.00050 0.00046 0.00042 0.00038 0.00035 0.00032 0.00028 0.00026 0.00024 0.00022 0.00020 0.00019 0.00016 0.00015 0.00014 0.00013 0.00012 0.00011 0.00010 0.00009 0.00008 0.00008 0.00007 0.00006 0.00005 0.00005 0.00004 0.00004 0.00004 0.00003 0.00003 0.00003 0.00002 0.00002 0.00002 0.00002 0.00002 0.00002 0.00001 0.00001 0.00001 O.qOOOl 0.00001 0.00001 0.00001 0.00001 0.00001 O.OOOCI 0.00001 0.00001

TABLE

eD T

0.0

0.1

-

0.2 --

4e-7.

0.3

U - Uo 3RT 0.4

(DEBYE)

0.5

I

0.6

0.7

0.8

0.9

- - - - - - - - - - - -- -

0 1.000000 .963000 .926999 .891995 .857985 .824963 .792923 .761858 .781759 .702615 1 .674416 .647148 .620798 .595351 .570793.. 547107 .524275 .502280 .481103 .460726 2 .441128 .422291 .404194 .386816 .370137 .354136 .338793 .324086 .309995 .296500 3 .283580 .271215 .259385 .248070 .237252 .226911 .217029 .207589 .198571 .189959 4 .181737 .173888 .166396 .159246 .152424 .145914 .139704 .133780 .128129 .122739 5 .117597 .112694 .108016 .103555 .099300 .095241 .091369 .087675 .084152 .080789 6 .077581 .074520 .071598 .068809 .066146 .063604 .061177 .058858 .056644 .054528 7 .052506 .050573 .048726 .046960 .045271 .043655 .042109 .040630 .039214 .037858 8 .036560 .035317 .034126 .032984 .031890 .030840 .029834 .028869 .027942 .027053 9 .026200 .025380 .024593 .023837 .023110 .022411 .021739 .021092 .020470 .019872 10 .019296 .018741 .018207 .017692 .017196 .D16718 .016257 .015812 .015384 .014970 11 .014570 .014185 .013813 .013453 .013106 .012770 .012445 .012131 .011828 .011534 12 .011250 .010975 .010709 .010452 .010202 .009960 .009726 .009499 .009279 .009066 13 .008859 .008658 .008463 .008275 .008091 .007913 .007740 .007572 .007409 .007251 14 .007097 .006947 .006801 .006660 .006522 .006388 .006258 .006132 .006008 .005888

- - - - - - - - - - - - - - - - -- -

eD T

-. 10 20 30 40

0

1

2

3

4

5

6

7

8

9

- -- - - - - - - - - - - -- .019296 .014570 .011250 .008859 .002435 .002104 .001830 .001601 .000722 .000654 .000595 000542 .000304 .000283 .000263 .000245

.007097 .005771 .004756 .003965 .003340 .002840 .001409 .001247 .001108 .000990 :000887 .000799 000496 .000454 .000418 .000385 .000355 .000328 .000229 .000214 .000200 .000188 .000176 .000166

4-113

HEAT CAPACITIES

eD

-

T

0.0

0.1

0.2

TABLE

Cv 4e-S. 3R

0.3

0.4

(DEBYE)

0.5

--- --- --- ---

0.6

0.7

0.8

0.9

--- ------ ---

0 1.000000 .999500 .998003 .915514 .992046 .987611 .982229 .975922 .968717 .960643 1 .951732 .942020 .931545 .920346 .908467 .895950 .882842 .869186 .855031 .840422 2 .825408 .810034 .794347 .778392 .762213 .745853 .729355 .712759 .696103 .679424 3 .662758 .646137 .629593 .613154 .596848 .580700 .564732 .548966 .533421 .518113 4 .503059 .488272 .473763 .459543 .445620 .432002 .418693 .405700 .393024 .380669 5 .368635 .356922 .345529 .334456 .323698 .313255 .303121 .293293 .283767 .274536 6 .265597 .256943 .248568 .240466 .232631 .225056 .217735 .210662 .203828 .197229 7 .190856 .184704 .178766 .173035 .167505 .162169 .157021 .152055 .147264 .142644 8 .138187 .133889 .129744 .125746 .121890 .118172 .114585 .111126 .107790 .104572 9 .101467 .098472 .095583 .092795 .090105 .087509 .085004 082585 .080251 .077997 10 .075821 .073719 .071690 .069729 .067835 .066005 .064236 .062526 .060874 .059276 11 .057731 .056237 .054791 .053393 .052039 .050730 .049462 .048235 .047046 .045895 12 .044780 .043700 .042653 .041639 .040655 .039702 .038777 .037880 .037010 .036166 13 .035347 .034552 .033781 .033031 .032304 .031597 .030910 .030243 .029595 .028964 14 .028352 .027756 .027177 .026613 .026065 .025532 .025013 .024508 .024016 .023537

-

--- ---

eD

-

T

0

1

2

--- --- --- --- --- --4

3

5

6

7

8

9

------ --- ------ --- -----10 20 30 40

.075821 .057731 .044780 .035347 .028352 .023071 .019018 .015859 .009741 .008414 .007318 .006405 .005637 .004987 .004434 .003959 .002886 .002616 .002378 .002168 .001983 .001818 .001670 .001538 .001218 .001131 .001052 .000980 .000915 .000855 .000801 .000751

.013361 .003550 .001420 .000705

.011361 .003195 .001314 .000662

the values of molar energy, molar heat capacity at constant volume and molar entropy were found to be U - Uo 3 (Y Z3 dz (4e-4) (Debye) 3RT = D(y) = Y')o e' - 1 Cv (Debye) 3R

S (Debye) -

3R

=

3y 4D(y) - eY - 1

4

= -

3

D(y) - In (1 - c

(4e-5) Y)

(4e-6)

Values of the quantities in Eqs. (4e-4), (4e-5) and (4e-6) are given in Tables 4e-7, 4e-8 and 4e-9, prepared by John E. Kilpatrick and Robert H. Sherman, of the Los Alamos Scientific Laboratory, under contract with the U.S. Atomic Energy Commission, 1964. To calculate U - U o in joules/mole, and C v and S in joules/mole kelvin, take as the value of R R = 8.3143 joules/mole kelvin To convert to calories, it must be kept in mind that there are three different calories: The 15-degree calorie = 4.1858 joules The International steam table calorie = 4.1868 joules The thermochemical calorie = 4.1840 joules

4-114

HEAT TAllLE

ev

-T

-

0.0

0.1

0.3

0.2

S

4e-9. 3R

0.4

(DEBYE)

0.5

0.6

0.7

0.8

0.9

--- --- --- --- --- --- --- --- --- --0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

'"

1.357896 0.733585 0.429176 0.260801 0.163557 0.105924 0.070920 0.049083 0.035057 0.025773 0.019444 0.015007 0.011814 0.009463

3.636168 1.267635 0.693684 0.407715 0.248562 0.156373 0.101605 0.068257 0.047393 0.033952 0.025029 0.018928 0.014639 0.011546 0.009263

2.943771 1.186113 0.656362 0.387463 0.236970 0.149554 0.097495 0.065715 0.045775 0.032892 0.024313 0.018431 0.014284 0.011286 0.009069

2.539553 1.111987 0.621403 0.368341 0.225990 0.143077 0.093583 0.063289 0.044227 0.031873 0.023623 0.017950 0.013940 0.011034 0.008881

2.253613 1.044212 0.588616 0.350279 0.215585 0.136926 0.089858 0.060972 0.042744 0.030895 0.022959 0.017485 0.013607 0.010790 0.008697

2.032703 0.981958 0.557832 0.333211 0.205724 0.131083 0.086310 0.058760 0.041324 0.029956 0.022318 0.017037 0.013284 0.010552 0.008519

1.853102 0.924550 0.528900 0.317077 0.196375 0.125530 0.082930 0.056646 0.039963 0.029053 0.021701 0.016603 0.012972 0.010321 0.008345

1.702152 0.871435 0.501685 0.301819 0.187510 0.120252 0.079709 0.054626 0.038658 0.028184 0.021106 0.016184 0.012669 0.010097 0.008176

1.572296 0.822152 0.476064 0.287386 0.179103 0.115234 0.076639 0.052695 0.037407 0.027349 0.020532 0.015778 0.012375 0.009880 0.008011

1.458656 0.776313 0.451928 0.273728 0.171126 0.110462 0.073712 0.050849 0.036208 0.026546 0.019978 0.015386 0.012090 0.009669 0.007851

-- --- --- --- --- --- --- --- --- --- --€Iv T

0

1

2

3

4

5

6

7

8

9

--- --- --- --- --- --- --- --- --- ---

10 20 30 40

0.025773 0.003247 0.000962 0.000406

0.019444 0.002805 0.000872 0.000377

0.015007 0.002439 0.000793 0.000351

0.011814 0.002135 0.000723 0.000327

0.009463 0.001879 0.000661 0.000305

0.007695 0.001662 0.000606 0.000285

0.006341 0.001478 0.000557 0.000267

0.005287 0.001320 0.000513 0.000250

0.004454 0.001183 0.000473 0.000235

0.003787 0.001065 0.000438 0.000221

At values of the temperature less than eD/1 00, the Debye theory can be relied upon to give correctly the COllLl'iLutioll Lo the heat capacity attributable to lattice vibrations. In some cases it holds well at temperatures up to e D /50. At these low temperatures the Debye expression for Cv reduces to (4e-7) or

Cv

=

mJI 124.8 moe· K

(T)3 "15>

tYD

(4e-8)

For metals there is a contribution to the heat capacity due to the free electrons equal to "T, where " is known as the electronic constant. The total heat capacity of a metal is therefore

(4e-9) The most reliable values of eD and" are obtained from heat-capacity measurements in the liquid helium region. It is customary in such work to plot Cv /T against Ta. On such a plot a nonmetal gives a straight line through the origin, whereas a metal gives a straight line with a positive intercept. Values of eD and" are given in Table 4e-IO. They were obtained in almost all cases by calorimetric measurements in the liquid helium range or lower. Values of , ElD are given in kelvins, and those of "in millijoules per mole kelvin squared.

4-115'

HEAT CAPACITIES TABLE

4e-10.

VALUES OF

8D

AND 'Y, IN THE HEAT-CAPACITY EQUATION

Substance

Symbol

Aluminum .... · . · . · . .. . .. . Antimony. .. . 0.' • . ..... .. . Argon ....... .. . ., . ·. Arsenic .. " .. ..... ..... . · . Barium. .... .. . · . ... . · . Beryllium. . .. · .', ... ·. Bismuth ...... ... .. . . .. . .. Bismuth telluride ..... · . .. . . .. ... Cadmium. . . ... ',' . · . . .. Calcium .... . .. . .. · . ·. Calcium fluoride .. '" . ' " . Carbon (graphite) . ·.· . ·. Carbon (diamond) .... · . .. . · . .. Cesium ...... · . . .... ... . Chlorine ...... ... .. . .. . Chromium ..... ... " ',' · . Cobalt ... . . ... · . · . Copper ..... .. . . .. . ·.·., . .. Dysprosium. · . ·. . . . Gadolinium ...... · . .. . · . Gallium ..... .. . ... Germanium ..... · .. .... . .. . Germanium telluride .... Gold ....... ... .. .. . · . Hafnium. .. .. · . ·. ·. Helium 4 (hcp) ...... · . ·. 3 Helium (bcc) .. '.. ... . · . · . Hydrogen. .. . ... .. · . · . Hydrogen2 ..... ... . . . ..... ·. Ice. .... . .. Indium ...... ... ·. " Indium antimonide. .. . · . · . · . Iodine. . . .. . .. ' .. ·. ·. Iridium. ... . ..... '," . · . · . Iron. .... · . · . . . . .. . .. Iron oxide .... ... . .. ·. ·. Iron selenide ... · . ... ·.·. Iron sulfide .. . .. .. '.',' · . Krypton. .. ·. .. ·. · Lanthanum ... · .... '.' . .. . · . Lead ....... ... . ... · . · . . .. Lead selenide . ., . · . · . ·. Lead sulfide. · . · . . .. .. . Lead telluride ... . .. ·. Lithium .. . . " . .... · . .. . · . Lithium chloride. ... .. . · . ... . Lithium fluoride .... ·. ·. Magnesium ... ... ·. ·.· . Magnesium cadmide .. " ·.·. Magnesium oxide ..... ·. Manganese. · . · . ·. ·. Mercury . .. . ... ·. · . Molybdenum. ... . ... · . .. . · . Neon. ... . ...... . ..... .. . · . Nickel. . . ...... . .. .. . . .. · . Nickel selenide .... ....... .. . .. . Niobium. ' " . ... . · . .... . '" . · .

Al Sb A As Ba Be Bi Bi,I'e, Cd Ca CaF, C C Cs Cl Cr Co Cu Dy Gd Ga Ge GeTe Au Hf He 4 (hcp) He 3 (bcc) H H2 H 2O In InSb I Ir Fe Fe203 FeSe2 FeS2 Kr La Pb PbSe PbS PbTe Li LiC!; LiF Mg Mg,Cd MgO Mn Hg Mo Ne Ni NiSe2 Nb

•

0,0

'

,

0

•••••

,

mJ

428 211 93 282 110 1440 119 155 209 230 510 420 2230 38 115 630 445 343 210 195 320 370 166 165 252 26.4 16 . 105 97 192 108 200 106 420, 467 660 366 637 72 142 105 135-160 194 124-135, 344 422 732 400 290 946 410 71.9 450 75 450 297 275

(4e-9).

'Y, mole. K2

Refs.

1. 35 0.112

1, 2, 89 3

0.19 2 ..7 0.17 0.021

3, 4 5 6, 7 8, 9 31

0.69 2.. 9

3.2 1.40 4.7 0.688 0.60 1..32 0.69 2.16

10

11, 5 32 12 13, 14, 90 15 16 17, 18, 19 20, 21, 22, 54 23 93 10, 24 25, 26 85 20,27, 91 28 84

1.,6

29, 30 44

.3.J 5.0

59 60, 61, 92 33 66 66

10 3.0

62, 95 56

1..63 1.3 0 .. 8 14 1. 79 2.0 7.1 7.79

15 86 34, 35, 36 63, 64 37 38 48, 49, 50 65, 75 59, 67, 68, 94 45 16, 69 66 52, 58, 94

.

4-116 TABLE

HEAT

4e-lO.

VALUES OF

E>

AND 'Y, IN THE HEAT-C.HACITY EQUATION

(4e-9)

(Continued)

Substance

Niobium-tin. . . . . . . . . .. . . . . . . . . .. Nitrogen ........................ Osmium ......................... Oxygen ........................ Palladium. . . . . . . . . . . . . . . . . . . . . .. Platinum. . . . . . . . . . . . . . . . . . . . . . .. Potassium ............ " .... " ... Potassium bromide. . . . . . . . . . . . . .. Potassium chloride. . . . . . . . . . . . . .. Potassium fluoride .............. " Potassium iodide ............... " Rhenium ........................ Rhodium ........................ Rubidium. . . . . . . . . . . . . . . . . . . . . .. Rubidium bromide. ............. Rubidium chloride.... . . . . . . . . . . .. Rubidium iodide. . . . . . . . . . . . . . . .. Ruthenium ...................... Scandium. ..................... Selenium. . . . . . . . . . . . . . . . . . . . . . .. Silicon .......................... Silicon dioxide. . . . . . . . . . . . . . . . . .. Silver ...................... '.' ... Silver bromide. . . . . . . . . . . . . . . . . .. Silver chloride. ................. Sodium ......................... Sodium bromide.. . . . . . . . . . . . . . . .. Sodium chloride ............... '" Sodium fluoride. . . . . . . . . . . . . . . . .. Sodium iodide .................... Strontium. . . . . . . . ..... .... . . . . . . .. Tantalum. . . . . . . . . . . . . . . . . . . . . .. Tellurium. . . . . . . . . . . . . . . . . . . . . .. Thallium. . . . . . . . . . . . . . . . . . . . . . .. Thorium. . . . . . . . . ... . . . . . . . . . . .. Tin (white) .................... " Tin (gray) ....................... Titanium. . . . . . . . . . . . . . . . . . . . . . .. Titanium dioxide ................. Tungsten. . . . . . . . . . . . .. . . . . . . . . .. Uranium ........................ Uranium dioxide ................. Vanadium ....................... Xenon .......................... yttrium ......................... Yttrium iron garnet. . . . . . . . . . . . .. Zinc. . . . . . . . . . . . . . . . . . . . . . . . . . .. Zinc sulfide. . . . . . . . . . . . . . . . . . . . .. Zirconium. . . . . . . . . . . . . . . . . . . . . ..

mJ

Symbol

Nb,Sn N Os 0 Pd Pt K KBr KCI KF KI Re Rh Rb RbBr RbCI RbI Ru Sc Se Si SiO, Ag AgBr AgCI Na N aBr NaCI N aF NaI Sr Ta Te TI Th Sn Sn Ti Ti0 2 W U U02 V Xe Y YIG Zn ZnS Zr

'Y, mole. K'

228 68 500 91 274 240 91 174 235 336 132 430 480 56 131 165 103 600 360 90 640 470 225 144 183 158 225 321 492 164 147 240 153 78.5 163 199 210 420 760 400 207 160 380 64 280 510 327 315 291

Refs.

13.1

87

2.4

59

9.42 6.8 2.1

53 27 15 39 39,40,41 86 39 59,70, 94 59 15 86 86 86 59 46, 96 71 25,26 42 54

2.3 4.9 2.4

3.3 10.7

0.650 1.4

3.6 5.9 1.47 4.3 1. 78 3.5 1.3 10.0 9.8 10.2 0.65 2.80

5, 72, 73 86 39 86 39 5

59, 68, 74, 94 71 75 76, 97 67,77,78 41 28,79, 80 43 59, 68, 81 57,97 47 45 51,94 24, 82, 83 35 28

HEAT CAPACITIES

4-117

References for Table 4e-l0 1. Phillips, N. E.: Phys. Rev. 114, 676 (1959). 2. Zavaritskii, N. V.: Zhur. Eksp. i. Teoret. Fiz. 84, 1116 (1958) [transl. Soviet Phys. JETP 7, 773 (1958)]. 3. Culbert, H. V.: Phys. Rev. 157, 560 (1967). 4. Taylor, W. A., D. C. McCollum, B. C. Passenheim, and H. W. White: Phys. Rev. 161, 652 (1967). 5. Roberts, L. M.: Proc. Phys. Soc. (London), ser. B, 70, 738 (1957). 6. Ahlers, G.: Phys. Rev. 145,419 (1966). 7. Gmelin, M. E.: Compt. rend. 259,3459 (1964). 8. Kalinkina, 1. N., and P. G. Strelkov: Zhur. Ek8p. i. Teoret. Fiz. 84, 616 (1958) [trans!. Soviet PhY8. JETP 6, 426 (1958)]. 9. Phillips, N. E., PhY8. Rev. 118, 644 (1960). 10. Phillips, N. E.: Phys. Rev. 184, A385 (1964). 11. Griffel, M., R. W. Vest, and J. F. Smith: J. Chem. Phys. 27, 1267 (1957). 12. Flubacher, P., A. J. Leadbetter, and J. A. Morrison: Phys. Chem. Solids 18, 160 (1960). 13. Burk, D. L., and S. A. Friedberg: Phys. Rev. 111, 1275 (1958). 14. Desnoyers, J. E., and J. A. Morrison: Phil. Mag. 8(8), 42 (1958). 15. Martin, D. L.: Phys. Rev. 189, A150 (1965). 16. Rayne, J. A., and W. R. G. Kemp: Phil. Mag. 1(8), 918 (1956). 17. Arp, V., N. Kurti, and R. Petersen: Bull. Am. PhY8. Soc. 2(II), 388 (1957). 18. Duyckaerts, G.: PhY8ica 6, 817 (1939). 19. Heer, C. V., and R. A. Erickson: Phys. Rev. 108,896 (1957). 20. Corak, W. S., M. P. Garfunkel, C. B. Satterthwaite, and A. Wexler: Phys. Rev. 98, 1699 (1955). 21. Phillips, N. E.: Proc. 5th Intern. Con!. Low Temp. Phys. Chem., pp. 414-416, University of Wisconsin Press, Madison, Wis., 1958. 22. Rayne, J. A.: AU8tralian J. Phys. 9, 189 (1956). 23. Dreyfus, B., B. B. Goodman, G. Troillet, and L. Weil: Compt. rend. 253, 1085 (1961). 24. Seidel, G., and P. H. Keesom: Phys. Rev. 112, 1083 (1958). 25. Flubacher, P., A. J. Leadbetter, and J. A. Morrison: Phil. Mag. 4(8), 273 (1959): 26. Keesom, P. H., and G. Seidel: Phys. Rev. 113, 33 (1959). 27. Ramanathan, K. G., and T. M. Srinivasan: Proc. Indian Acad. Sci. 49, 55 (1959). 28. Kneip, G. D., Jr., J. O. Betterton, Jr., and J. O. Scarbrough, PhY8. Rev. 130, 1687 (1963) . 29. Bryant, C. A., and P. H. Keesom: Phys. Rev. Letters 4, 460 (1960). 30. Clement, J. R., and E. H. Quinnell: Phys. Rev. 92, 258 (1953). 31. Itskevich, E. S.: Zhur. Eksp. i. Teoret. Fiz. 38, 351 (1960). 32. Huffman, D. R., and M. H. Norwood: Phys. Rev. 117,709 (1960). 33. Kouvel, J. S.: Phys. Rev. 102, 1489 (1956). 34. Jones, G. 0., and D. L. Martin: Phil. Mag. 45(7), 649 (1954). 35. Martin, D. L.: Phil. Mag. 46(7), 751 (1955). 36. Scales, W. W.: Phys. Rev. 112, 59 (1958). 37. Bergenlid, U. M., R. S. Craig, and W. E. Wallace: J. Am. Chem. Soc. 79,2019 (1957). 38. Barron, T. H. K., W. T. Berg, and J. A. Morrison: Proc. Roy. Soc. (London), ser. A, 250, 70 (1959). 39. Berg, W. T., and J. A. Morrison:Proc. Roy. Soc. (London), ser. A, 242, 467, 478 (1957). 40. Keesom, P. H., and N. Pearlman: Phys. Rev. 91, 1354 (1953). 41. Webb, F. J., and J. Wilks: Proc. Roy. Soc. (London), ser. A, 230, 549 (1955). 42. Jones, G. H. S., and A. C. Hollis-Hallett: Can. J. Phys. 38,696 (1960). 43. Keesom, P. H., and N. Pearlman: Phys. Rev. 112, 800 (1958). 44. Keesom, P. H., and N. Pearlman: Unpublished data. 45. Fenichel, H., and B. Serin: Phys. Rev. 142,490 (1966). 46. Wohlleben, D. (quoted by M. A. Jensen and J. P. Maita: PhY8. Rev. 149,410 (1966). 47. Radebaugh, R., and P. H. Keesom: Phys. Rev. 149, 209 (1966). 48. Guthrie, G. L., S. A. Friedberg, and J. E. Goldman: PhY8. Rev. 139, A1200 (1965). 49. Stetsenko, P. N., and Y. 1. Avsebt'ev: Soviet Phys. JETP 20,539 (1965). 50. Scurlock, R. G., and W. N. R. Stevens: Proc. PhY8. Soc. (London) 86,331 (1965). 51. Heiniger, F., E. Bucher, and J. Muller: Phys. Kond. Jl.laterie, 5,243 (1966). 52. van der Hoeven, B. J. C., and P. H. Keesom: Phys. Rev. 134, A1320 (1964). 53. Veal, B. W., and J. A. Rayne: Phys. Rev. 135, A442 (1964). 54. Martin, D. L.: Phys. Rev. 141,576 (1966). 55. Zimmerman, J. E., A. Arrott, and S. Shinozaki: in Proc. Low Temp. Calorimetry Con!. (Hel8inki), O. V. Lounasmaa, ed., p. 147, 1966. 56. van der Hoeven, B. J. C., and P. H. Keesom, Phys. Rev. 137, Al03 (1965).

~118

HEAT

Ho, J. C., and N. E. Phillips: Phys. Rev. Letters 17, 694 (1966). Leupold, H. A., and H. A. Boorse: Phys. Rev. 134, A1322 (1964). Wolcott, N. M.: Bull. inst. intern. du froid, Annexe 1955-3, PP. 286-289. Duyckaerts, G;: Physica 6, 401 (1939). Keesom, W. H., and B. Kurrelmeyer: Physica 6, 633 (1939). Berman, A., M. W. Zemansky, and H. A. Boorse: Phys. Rev. 109,70 (1958). Logan, J .. K., J. R. Clement, and H. R. Jeffers: Phys. Rev. 105, 1435 (1957). Smith, P. L.: Phil. Mag. 46(7), 744 (1955). Douglass, R. L.,W. H .. Lien, R. G. Peterson, and N. E. Phillips: Proc; 7th Intern. Conf. Low Temp. Phys. Chem., pp. 242-243, University of Toronto Press, 1960. 66. Gr¢nvold, F. C., and E. F. 'Westrum, Department of Chemistry, University of Michigan. 67. Bryant, C. A.: Thesis, Purdue University, 1960 (unpublished). 68. P. H. Keesom and N. Pearlman: in "Encyclopedia.of Physics," S. Flugge, ed., vol. XIV, "Low Temperature Heat Capacity of Solids," Tables 1-13, Springer-Verlag OHG, Berlin, 1956. 69. Keesom, W: H., and C. W. Clark:Physica 6, 513 (1939). 70. Keesom, P. H., and C. A. Bryant: Phys. Rev. Letters 2, 260 (1959). 71. Smith, P. L.: Bull. inst.intern. du froid, Annexe 1955-3, pp. 281-283. 72. Gaumer, R. E., and C. V. Heer: Phys. Rev. 118, 955 (1960). 73. Lien, W. H., and N.E. Phillips: Phys. Rev. 118,958 (1960). 74. Chou, C., D. White, and H. L. Johnston: Phys. Rev. 109,788 (1958). 75. van derHoeven, B. J. C., and P. H. Keesom: Phys. Rev. 135, A631 (1964). 76. Smith, P. L., and N. M. Wolcott: Bull .. inst. intern. du froid, Annexe 1955-3, pp. 283286. 77. Corak, W. S., and C. B. Satterthwaite: Phys. Rev. 102, 662 (1956). 78. Zavaritskii, N. V.: Zhur. Eksp. i. Teoret. Fiz. 33, 1085 (1957) [transl. Soviet Phys. JETP 6, 837 (1958)]. 79. Aven, M. H., R. S. Craig, T. R. 'Waite, and W. E. Wallace: Phys. Rev. 102, 1263 (1956). 80. Wolcott, N. M.: Phil. Mag. 2(8),1246 (1957). 81. Waite, T. R., R. S. Craig, and W. E. Wallace: Phys. Rev. 104, 1240 (1956). 82. Garland, C. W., and J. Silverman: J. Chem. Phys. 34,781 (1961). 83. Martin, D. L.: Phys. Rev. 167, 640 (1968). 84. Edwards, D.O., and R. C. Pandorf: Phys. Rev. 140, A816 (1965). 85. Finegold, L.: Phys. Rev. Letters 13, 233 (1964). 86. Lewis, J. T., A. Lehoczky, and C. V. Briscoe, Phys. Rev. 161,877 (1967). 87. Vieland, L. J., and A. W. Wicklund: Phys. Rev. 166,424 (1968). 88. Rayne, J. A.: Phys. Rev. 95, 1428 (1954). 89. Berg, W. T.: Phys. Rev. 167, 58il (Hllol::I). 90. van der Hoeven, B. J. C., and P. H. Keesom: Phys. Rev. 130, 1318 (1963). 91. Martin, D. L.:Phys. Rev. 170, 650 (1968). 92. Shinozaki, S. S., and A. Arrott: Phys. Rev. 152,611 (1966). 93. Donald, D. K., L. T. Crane, and J. E. Zimmerman, unpublished [quoted by O. V. Lounasmaa and L. J. Sundstrom, Phys. Rev. 150,399 (1966)]. 94. Morin, F. J., and J. P. Maita: Phys.Rev. 129, 1115 (1963). 95. Ohtsuka, T., and T. Satoh: in Proc. Low Temp. Calorimetry Can/. (Helsinki), O. V. Lounasmaa, ed., p. 92, 1966. 96. Flotow, H. E., and D. W. Osborne: Phys. Rev. 160,467 (1967). 97. Gordon, J. E., H. Montgomery, R. J. Noer, G. R. Pickett, and R. Tobon: Phys. Rev. 152, 432 (1966).

57. 58. 59. 60. 61. 62. 63. 64. 65.

4f. Thermal Expansion RICHARD K. KIRBY, THOMAS A. HAHN, AND BRUCE D. ROTHROCK

The National Bureau of Standards

In Table 4f-1, the coefficients of linear thermal expansion, a = (l/L m )dL/dT, are given in units of 10- 6 K-l; and the expansion, E = (LT - L 293 ) /L293, is given in units of 10- 6• When data are given for two or more crystalline forms of the element, the forms are designated (a), (fJ), ('Y), etc. The coefficient of cubical expansion may be computed from the following equations: fJ = 3a

+

fJ = 2aa ac fJ=aa+ab+ac

where aa, ab, and ac are the coefficients of linear expansion in the a, b, and c directions. In Table 4f-2, the coefficients of linear thermal expansion, a = (l/L o)dL/dT, are given in units of 10-8 K-l. The data designated by the symbols II or .L are in directions parallel or perpendicular to the c axis of the crystal. An (S) denotes data for the material in the superconducting state. The data for the material in the normal state in the snperconducting region were measured in a magnetic field high enough to destroy superconductivity. An asterisk (*) denotes a region where large errors may result because of the coefficient changing rapidly. In Table 4£-3, the coefficient of linear thermal expansion, a = (1/L 20 )dL/dt, is given in units of 1O- 6 /K-l; and the expansion, E = (L, - L 20 ) /L 20 is given in units of 10-5 where t stands for the Celsius temperature and L 20 is the length at 20°C. References used in the compilation of these tables and data on or references to publications on the thermal expansion of other materials can be obtained from the National Bureau of Standards. An approximate relation between the coefficient of volume expansion fJ

=

1.V

(aV) aT

p

and the temperature is given by Gruneisen's equation fJ =

Cv Qo[l - k(U /Qo)l'

where Cv is the molar heat capacity at constant volume, U is the energy of the lattice vibrations, and Qo and k are constants. If the Debye temperature ElD is known, both Cv and U may be calculated at any temperature T from the equations* Cv

=3R[12(:!..-)3 {eDIT y 3dy

U =

ElD

loT

10

Cv dT

* This material is concluded on page 4-142. 4-119

e" - 1

-3

ElD/T

eeD1T

-

1

]

4-120 TABLE

Temperature, K

25 50 75 100 150 200 250 293 350 400 500 600 700 800 1000

TABLE

HEAT

4£-1.

COEFFICIENTS OF LINEAR THERMAL EXPANSION,

Antimony

Aluminum

a (lO-a) (K-l),

Antimony a axis

Antimony c axis

a

€

a

€

a

€

a

0.5 3.5 8.1 12.0 17.1 20.2 22.0 23.0 24.1 24.9 26.5 28.2 30.4 33.5

-4160 -4120 -3970 -3720 -2980 -2040 -980 0 1340 2560 5150 7890 10890 14110

..... . ......

..... . ..... .

...... ..... .

..... . ..... .

..... . . .....

8.2 9.3 10.1 10.5 10.8 11.0 11.2 11. 3 11.6 11.8 11.9 12.0

-2220 -2000 -1520 -1000 -470 0 630 1200 2350 3520 4700 5890

4.7 6.0 7.1 7.7 8.1 8.4

-1580 -1440 -1120 -750 -360 0

15.1 15.8 16.1 16.1 16.2 16.2

-3500 -3110 -2310 -1500 -700 0

......

...... ...... . .....

4£-1.

..... . ..... . ..... .

..... .

..... . ..... . ..... .

..... .

. ..... ...... ......

a (lO-a) (K-l),

Bismuth c axis

Bismuth a axis

Bismuth

. ..... ...

..,

. ... . . ..... . ..... . . ..... . .....

COEFFICIENTS OF LINEAR THERMAL EXPANSION,

Beryllium c axis

Tempera-

...... ..... .

..... . ..... .

E

tUTS ,

K a

25 50 75 100 150 200 250 293 350 400 500 600 700 800 1000 1200 1400

I

a

E

'"

.... . .... .

0.1 0.7 2.9 5.2 7.4 8.9 10.2 11.1 12.3 13.4 14.4 15.7

-970 -960 -870 -670 -350 0 550 1080 2260 3540 4940 6440

...

....

.... .

""

'"

. . .

.

.... . .... .

I

4.8 9.2 11.1 11.9 12.6 12.9 13.1 13.2 13.4 13.4 13.5

E

E

a

... . ... .

. ..... . .....

. ... ... .

. ..... ......

9.0 9.9 10.9 11.2 11.5 11.7 11.9 11.9 12.1

-2390 -2150 -1630 -1070 -500 0 680 1270 2470

15.3 15.7 16.1 16.2 16.2 16.2 16.3 16.3 16.4

-3500 -3110 -2320 -1510 -700 0 930 1740 3380

I -3200 -3010 -2760 -2470 -1860 -1220 -570 0 760 1430 2780

... ... ... ...

. . . .

. ..... ..... . ..... . ..... .

...

. .

..... . ..... .

...

a

... . . ... . . ... . ... . "

... .

. ..... ..... . . ..... . ..... ..... . . .....

. ... . ... ... . ... . . ... . ...

I

E

...... ...... ...... ...... . .....

. .....

4-121

THERMAL EXPANSION , AND THE EXPANSION, • (10- 6), OF ELEMENTS

Argon

ex

Arsenic

ex

e

Barium

ex

e

e

--- --- --- ------

220 460 590

.. . .. . ... .. .

.. . .. . .. . .. . ... .. . .. . .. .

1950 10870 24140

.... . . .... .... . ..... ..... ..... ..... .... . ..... .... . . .... .....

.. . . .. .. . .. . .. .

5.6

. ..

.. . .. . .. . . .. .. . . ..

AND THE EXPANSION, •

30.7

31. 3 32.0 33,0 38.4

..

-7370 -7010 -6420 -5760 -4340 -2860 -1330

o

1800 3420 6940

-3.8 +2.7 7.8 10.7 14.4 16.5 18.2 19.8 22.1 24,9 36.7

...

. ... . ... . ... . ... . ...

13 18 21 24

0 880 1850 4110

. .. . ..

. ...

,

,

0 320

. .. ... .. ., . ... .. .

.. . . ..

,

(10- 6),

. ... . ... . ...

OF ELEMENTS

-3340

-3110 -2460 -1690 -820

o

1190 2360 5350

43.5 58.8 60.4 59.9 58.7 57.5 55.8 54.3 51. 7 49.1 41. 9

•

ex

e

0,5 1.3 4,1 7.1 9.6 11.2 12.7 13.7 15.2 16.4 17.7 19,0 21. 6

-1300 -1280 -1150 -870 -450 0 690 1350

0,7 1.6 4,7 8.0 10.7 12.3 13.9 15.0 16.6 18.0 19.3 20.6

-1470 -1440 -1290 -970 -500 0 750 1480 3060 4790 6650 8650

2790 4380

6080 7920 11970

(Continued)

Cadmium c axis

-3460 -3480

ex

---

.. . . .. . .. "

Cadmium a axis

Cadmium

12.0 21.4 25.3 27.1 29.2 30.2

... . .. ... ...

Beryllium a axis

Beryllium

Carbon (diamond)

Calciurn

-15450

-14090 -12580 -11080 -8110 -5200 -2370

o

3020 5550 10120

14.0 16.7 18.9 20.4

21.4 22.1 22.7 23.0 23,5 23.8 24.0

-4260 -3870

-2970 -1990 -940

o

1280 2420 4750 7120 9510

0.05 0.20 0.41 0,70 1.00 1.5 1.8 2.5 3,0 3.4 3.7 4.3 4.7 5,1

-84 -78 -63 -36

o

71 153 369 640 960 1320 2120 3030 4020

4-122 TABLE

Temperature

HEAT

4f-1.

COEFFICIENTS OF LINEAR THERMAL EXPANSION, a

Carbon (graphite) a axis

Carben (graphite)

Carbon (graphite) c axis

.(lO-:-6,)(K-l),

Cerium

K

25 50 75 100 150 200 250 293 350 400 .500 600 700 800 1000 1200 1400 1600 1800 2000

TABLE

Temperature, K

25 50 75 100 150 200 250 293 350 400 500 600 700 800 1000

a

a

E

.... ... .

. .....

....

......

o.

-1300 -1030 -700 -330. 0 460 870 1730 2640 3600 4580 6630 8730 10890 13090 15330 17610

-0.4 -0.6 -0.8 -1.0 -1.2 -1.2 -1.1 -0.7 -0.2 +0.2 0.5 0.8 0.9 1.0 1.1 1.2 1.2

... .

4.9 6.1 7.0 7.6 7.8 8.1 8.4 8.9 9.4 9.8 10.0 10.4 10.6 10.9 11.1 11.3 11.5

4f-1.

0

•••

0.

E

"0 ••

.

,.,

..... ..... ..... 152 127 92 47 0 -68 -128 -226 -276 -278 -246 -118 +52 250 460 690 930

Dysprosium .c axis

a

a

5.5 5.6 5.7 5.9 6.1 6.3 6.7

...

0"

E

•

'"

.

0"

•

15.4 19.5 22.6 24.8 25.9 26.8 27.4 28.2 28.6 28.9 29.1 29.6 30.1 30.6 31.1 31.6 32.1

..0.· . .. 0.· .

. .....

-4220 -3340 -2280 -1090 0 1500 2860 5640 8480 11360 14260 20140 26100 32180 38340 44620 50980

a

E

. .. . .. . .. ... ... .. . ...

E

•••

0'

... .

... . ... . 0 320 600 1170 1770 2390 3040

... .

0

E

••• •

0

•••

.... ....

. ... . ...

. ...

0.' •

14.8 15.8 16.5 18.2 19.6 21. 6 23.5

0 870 1680 3410 5310 7370 9620

'0'

.

... .

... . 8.4 9.0 9.3 9.5 9.7 9.8 10.1 10.5 10.9 11.6 14.8

. ... . ... ,0 ••

·

...

.

'"

. ... 0 310 590 1210 1870 2600 3390 5140

...

.0 .•

...

. ...

(lO-6) (K-l), Erbium .a axis

Erbium

a

..0.

5.2 5.6 5.8 6.4 7.0 7.6 8.2 9.4

COEFFICIENTS OF LINEAR THERMAL EXPANSION, a

Dysprosium a axis

... . .. .. . .. .

a

E

a

. ...

. .. . .. . .. . ..

-1300 -860 -410 0 550 1040 2040 3070 4140 5260 7830

6.0 6.1 6.2 6.4 6.7 7.1

... . ..

E

"0 •

· .,. ·

'0'

..0.

0 340 650 1280 1930 2620

. ... · ...

4--,.123

THERMAL EXPANSION AND THE EXPANSION,

(10- 6),

E

OF ELEMENTS

Chromium

Cesium

(Continued)

Cobalt

Copper

Dysprosium

a

a

E

a

•

a

E

.. . .. . .. . .. .

0.1 0.6 1.5 2.5 4.0 5.1 5.6 5.0 7.1 8.0 9.0 9.7 10.4 10.9 12.0

-980 -980 -950 -900 -740 -510 -240 0 290 670 1530 2470 3470 4530 6830

. .....

..... .....

0.6 3.8 7.6 10.5 13.6 15.2 16.1 16.7 17.3 17.6 18.3 18.9 19.6 20.4 22.4 24.8

-3252 -3214 -3067 -2836 -2218 -1492 -707

.. . · .. .. .

100

... · .. · .. · .. ...

... ... .. .

....

..... . ...... ..... . ..... . 13.7 13.8 13.9 14.2 14.9 18(a)

14.3((3) 14.5 14.7

. ....

AND THE EXPANSION, E

Erbium c axis

. ..... . .....

(10- 6),

Europium

.....

..... . .... ..... .....

0 780 1480 2880 4330 5900 8380 11260 14170

OF ELEMENTS

•

a

.. .

0 310 590 1170 1790 2450 3180

....

E

... .

...

.

'"

... . .. . ..

5.2 6.2

...

.

. ... '" .

-460 -160 +150 350 0 220 530 1270 2080 2940 3850 5890

16.6 16.9 17.1 17.7 18.2 18.7

... . ... .

'"

. .

0 960 1800 3540 5340 7180 '"

'"

. .

... 25

... ... ... ... .. , . .. '"

6.0(a)

-1 -2 +5.5((3) 6.7 7.8 8.4 8.9 9.4 11.7

-1120 -940 -760 -360 0 500 960 1920 2950 4050 5230 7870

Gadolinium c axis

a

a

E

(Continued) GadolinlUm a axis

a

... .

0

970 1840 3640 5500 7420 9420 13700 18410

4.9(a)

1.9 7.7((3) 8.3 8.ti 9.0 9.3 10.0 10.6 11.4 12.3 14.2

Gadolinium

E

a

a

5.5 5.6 6.0 6.4 6.9 7.7

5.4 9.0 11. 5 12.4 12.8 13.0

E

0 40 410 1490 2690 3950 5240

4-124 TABLE

HEAT

4f-1.

COEFFICIEN1.1S OF LINEAR THERMAL EXPANSION, a

Gallium

Temperature,

Gal- Gal- Gallium lium lium a axis b axis c axis

(10- 6) (K-l) ,

Germanium

Gold

K a

a

a

a

--25 50 75 100 150 200 250 293 350 400 500 600 700 800 1000 1200

TABLE

19.2 19.7

4f-1.

-840 0

a

11

34 47 55 65 71 79 87

11.5

a

€

31.0

-0.1 +0.2 1.1 2.4 4.1 4.9 5.6 5.7 6.0 6.2 6.5 6.7 6.9 7.2

-950 -950 -930 -890 -720 -490 -240 0 330 640 1280 1940 . 2620 3320

3.2 7.8 10.5 11.9 13.1 13.6 14.0 14.2 14.5 14.7 15.2 15.8 16.4 17.1 18.8 21.1

-3250 -3120 -2880 -2600 -1970 -1300 -610 0 820 1550 3040 4590 6200 7870 11440 15400

COEFFICIENTS OF LINEAR THERMAL EXPANSION, a

Iodine a axis

Iodine

Temperature, K

25 50 75 100 150 200 250 293 350 400 500 600 700 800 1000 1200 1400 1600 1800

16.6

a

---

a

-16600 -16100 -15000 -13800 -10700 -7300 -3600 0 5360

133

0 7580

(10- 6) (K-l), Iodine c axis

Iodine b axis

a

€

a

94

0 5870

34

0 2560

4-125

THERMAL EXPANSION AND THE EXPANSION, e

(Continued)

OF ELE!y[ENTS

Holmium

:aafnium

a

(10-'),

Indium a axis

Indium

e

a

a

.. . .. . ...

. ..... ...... ..... .

... ...

4.8 5.4 5.7 5.9 6.0 6 ..1 6.2

-1090 -830 -550 -260 0 350 660

10.1 19 .. 3 23 ..0 24.9 26.7 28.2 30.0 32.1 35.9 39,9

. .. ... ... ... ... 10 ...

...

Indium c axis

e

a

e

a

e

-6970 -6580 -6050 -5450 -4150 -2780 -1330 0 1930 3830

4.4 17.7 24.2 28.4 34.0 39.5 46.0 52.9 64.6 77.3

-9130 -8840 -8310 -7650 -6080 -4250 -2120 0 3320 6860

21.4 22.6 20.5 18.0 12.2 5.6 -2.0 -9.6 -21.5 -34.8

-2660 -2090 -1550 -1060 -300 +150 +240 0 -870 -2240

I

I AND THE EXPANSION, e

Iridium

(10-'),

Krypton

Iron

I

-~------~--~--

a

e

a

(Continued)

OF ELEMENTS

E

Lead

Lanthanum

,

I

I

eo

a

e

a

2170 8230 16040 25780

. ..... ..... . ..... . ..... .

.....

14.2 21. 7 24.4 25.4 26.6 27.5 28.2 28.7 29.3 29.8 32.1

ao

e

---

... . ... .

..... . ..... . ..... .

4.4 5.3 5.9 6.2 6.5 6.7 6.8 7.2 7.4 7.7 7.9 8.4 8.8 9.2 9.6 10.1

-1110 -860 -580 -270 0 380 710 1410 2140 2900 3680 5320 7040 8840 10730 12700

... .

0.2 1.3 3.5 5.7 8.4 10.1 11.1 11.8 12.6 13.2 14.3 15.2 16.1 16.5 15.5(= f-

I~

Region of anomalous effecls I kg/cm2 • 9.80665x 104 N/m2

100

:;:

80

:;

6::J 60 0

Z 0

U

-' 40

-3

TABLE

4h-11.

ENTHALPY OF ARGON,

(H - EoO)/RTo

70 atm i 100 atm 40atm 7 atm 10 atm T, K 1 atm 4atm - - - - - - - - - - - -- - - - - - - - - - - 100 200 300 400 500

0.8935 1. 8236 2.7422 3.6590 4.5750

1. 8029 2.7319 3.6532 4.5718

1. 7819 2.7217 3.6476 4.5686

1. 7606 2.7114 3.6418 4.5654

1.53 2.610 3.586 4.535

1.3 2.512 3.533 4.506

600 700 800 900 1000

5.4907 6.4063 7.3218 8.2372 9.1525

5.4891 6.4057 7.3220 8.2380 9.1538

5.4874 6.4052 7.3222 8.2388 9.1551

5.4859 6.4047 7.3226 8.2396 9.1564

5.471 6.400 7.326 8.249 9.170

5.457 6.397 7.330 8.258 9.184

4h-12.

TABLE

T, K

4atm

1 atm

ENTROPY OF ARGON,

7 atm

10 atm

40 atm

SIR 70 atm

100 atm

---- - - - - -- - -- - -- - - - - -- - -

2.42 3.48 4.48

100 200 300 400 500

15.8425 17.6069 18.6245 19.3449 19.9032

16.2012 17.2308 17.9548 18.5146

15.6218 16.6637 17.3913 17.9527

15.245 16.2995 17.0308 17.5937

13.64 14.8389 15.6067 16.1850

12.83 14.2067 15.0118 15.6037

12.2 13.781 14.618 15.2261

5.445 6.395 7.335 8.268 9.198

600 700 800 900 1000

20,3593 20.7449 21.0787 21.3733 21.6368

18.9715 19.3575 19.6917 19.9864 20.2500

18.4104 18.7969 19.1313 19.4263 19.6900

18.0522 18.4391 18.7739 19.0690 19.3328

16.6513 17 .0426 17.3802 17.6772 17.9423

16.0776 16.4732 16.8134 17.1122 17.3785

15.7072 16.1070 16.4498 16.7503 17.0179

1100 1200 1300 1400 1500

10.0679 10.9832 11.8985 12.8138 13.7291

10.0696 10.9852 11.9007 12.8162 13.7316

10.0712 10.9871 11.9029 12.8186 13.7342

10.0729 10.9891 11. 9051 12.8210 13.7367

10.090 11.009 11.927 12.845 13.763

10.107 11.029 11.950 12.869 13.788

10.125 11.049 11.972 12.894 13.815

1100 1200 1300 1400 1500

21. 8751 22.0926 22.2927 22.4780 22.6505

20.4884 20.7060 20.9062 21.0916 21.2640

19.9285 20.1462 20.3464 20.5318 20.7043

19.5715 19.7892 19.9895 20.1749 20.3474

18.1819 18.4003 18.6010 18.7869 18.9597

17.6190 17.8381 18.0394 18.2256 18.3988

17.2592 17.4789 17.6807 17.8673 18.0408

1600 1700 1800 1900 2000

14.6443 15.5595 16.4749 17.3901 18.3053

14.6470 15.5624 16.4778 17.3931 18.3085

14.6497 15.5652 16.4808 17.3962 18.3116

14.6524 15.5680 16.4837 17.3992 18.3147

14.680 15.597 16.513 17.430 18.346

14.707 15.625 16.543 17.460 18.377

14.735 15.654 16.572 17.491 18.409

1600 1700 1800 1900 2000

22.8119 22.9635 23.1064 23.2415 23.3698

21.4254 21. 5771 21. 7200 21. 8551 21. 9834

20.8657 21.0174 21.1603 21.2955 21.4238

20.5089 20.6606 20.8035 20.9387 21.0670

19.1214 19.2733 19.4165 19.5518 19.6802

18.5607 18.7128 18.8561 18.9915 19.1201

18.2029 18.3552 18.4987 18.6343 18.7630

2100 :2200 2300 2400 2500

19.2206 20.1358 21.0510 21. 9662 22.8815

19.2238 20.1390 21.0543 21. 9696 22.8849

19.2269 20.1423 21.0576 21. 9729 22.8884

19.2301 20.1456 21.0609 21. 9763 22.8918

19.262 20.178 21.094 22.010 22.926

19.294 20.211 21.127 22.044 22.960

19.326 20.243 21.160 22.077 22.994

2100 2200 2300 2400 2500

23.4917 23.6080 23.7192 23.8256 23.9276

22.1053 22.2217 22.3329 22.4393 22.5413

21.M57 21. 6620 21.7732 21. 8797 21. 9817

21.1890 21.3053 21.4165 21.5229 21. 6249

19.8022 19.9187 20.0299 20.1364 20.2385

19.2422 19.3587 19.4701 19.5766 19.6787

18.8851 19.0017 19.1131 19.2197 19.3218

2600 2700 2800 2900 3000

23.7967 24.7120 25.6272 26.5424 27.4576

23.8002 24.7154 25.6307 26.5459 27.4612

23.8036 24.7189 25.6342 26.5495 27.4647

23.8071 24.7224 25.6377 26.5530 27.4683

23.842 24.757 25.673 26.5S9 27.504

23.876 24.792 25.708 26.624 27.540

23.911 24.827 25.743 26.659 27.575

2600 2700 2800 2900 3000

24.0257 24.1200 24.2109 24.2987 24.3834

22.6394 22.7337 22.8246 22.9124 22.9971

22.0798 22.1741 22.2650 22.3528 22.4375

21.7231 21. 8174 21. 9083 2l. 9961 22.0808

20.3366 20.4310 20.5219 20.6098 20.6945

19.7769 19.8713 19.9623 20.0501 20.1349

19.4201 19.5145 19.6055 19.6934 19.7782

>-3

::q ~

~ o

tJ >-1

>z: P>

:s:,.... o

>tJ

!;U

o

>tJ ~

!;U

>-3

H

~

[J2

o

""J

---_ ...

_

... _ - - -

.

-----

-

-

-

-

~

[J2

~

[J2

t

f-L (J)

CO

TABLE

4h-13.

Z T, X

l'atoo

'f'

COMPRI!lSSIBILITY FACTOR FOR HYDROGEN,

4atoo

=

PVjRT

7atoo

10 atm

TABLE

4Oatoo

70e.tm

100 atoo

--- - - - - - - - - -- - - - - - - - - - - -

4h-14.

RELATIVE DENSITY OF HYDROGEN,

T, K

1 atm

4atm

7atm

10atm

40atm

70 atm

pjpo

I-'

-..:t

o

100 atm

--- - - -- - -- - - - - - - - - - - - - - -

40 60 80 100 120

0.9845 0.9955 0.9986 0.9998 1.0003

0.9362 0.9822 0.9946 0.9992 1.0012

0.8853 0.9691 0.9908 0.9987 1.0021

0.8317 0.9564 0.9872 0.9983 1.0030

0.8757 0.9682 1.0029 1.0176

0.8700 0.97.82 1.0222 1.0405

0.9395 1.0174 1.0560 1.0726

40 60 80 100 120

6.9408 4.5761 3.4214 2.7338 2.2771

29.195 18.552 13.740 10.942 9.0999

54.029 32.905 24.138 19.158 15.910

82.160 47.632 34.609 27.379 22.709

208.08 141.15 109.01 89.532

366.53 244.49 187.17 153.23

484.88 335.82 258.83 212.36

140 160 180 200 220

1.0005 1.0006 1.0007 1.0007 1.0007

1.0020 1.0024 1.0028 1.0028 1.0028

1.0036 1.0043 1.0048 1.0048 1.0048

1.0052 1.0062 1.0067 1.0068 1.0067

1.0243 1.0271 1.0283 1.0283 1.0276

1.0488 1.0516 1.0523 1.0513 1.0497

1.0786 1.0798 1.0785 1.0760 1.0730

140 160 180 200 220

1.9514 1.7073 1.5174 1.3657 1.2415

7.7937 6.8167 6.0569 5.4512 4.9557

13.617 11.907 10.578 9.5206 8.6551

19.422 16.978 15.084 13.574 12.341

76.240 66.528 59.067 53.160 48.361

130.30 113.71 101.01 90.995 82.849

181 :01 158.21 140.80 127.01 115.79

240 260 280 300 320

1.0007 1.0006 1.0006 1.0006 1.0006

1.0027 1.0024 1.0024 1.0024 1.0024

1.0047 1.0044 1.0042 1.0042 1.0041

1.0066 1.0064 1.0061 1.0059 1.0057

1.0269 1.0259 1.0247 1.0238 1.0229

1.0480 1.0459 1.0439 1.0420 1.0402

1.0698 1.0667 1.0636 1.0607 1.0579

240 260 280 300 320

1.1381 1.0506 0.97559 0.91055 0.85364

4.5431 4.1949 3.8953 3.6356 3.4084

7.9347 7.3265 6.8045 6.3509 5.9546

11.314 10.446 9.7026 9.0575 8.4931

44.361 40.988 38.105 35.596 33.401

76.068 70.358 65.457 61.204 57.479

106.46 98.553 91.780 85.896 80:740

340 360 380 400 420

1.0005 1.0005 1.0005 1.0005 1.0005

1.0021 1.0020 1.0020 1.0020 1.0019

1.0037 1.0036 1.0035 1.0034 1.0033

1.0054 1.0052 1.0050 1.0048 1.0046

1.0217 1.0209 1.0201 1.0193 1.0185

1.0384 1.0367 1.0353 1.0339 1.0325

1.0553 1.0529 1.0507 1.0486 1.0466

340 360 380 400 420

0.80351 0.75887 0.71893 0.68298 0.65046

3.2088 3.0309 2.8714 2.7278 2.5982

5.6065 5.2956 5.0174 4.7670 4.5404

7.9959 7.5532 7.1571 6.8006 6.4780

31.473 29.748 28.204 26.815 25.558

54.192 51.265 48.632 46.286 44.120

76.178 72.110 68.458 65.165 62.181

440 460 480 500 520

1.0004 1.0004 1.0004 1.0004 1.0004

1.0017 1.0016 1.0016 1.0016 1.0016

1.0030 1.0029 1.0028 1.0028 1.0028

1.0045 1.0043 1.0041 1.0040 1.0039

1.0180 1.0172 1.0165 1.0160 1.0155

1.0314 1.0301 1.0289 1.0280 1.0271

1.0448 1.0431 1.0415 1.0400 1.0385

440 460 480 500 520

0.62095 0.59396 0.56921 0.54644 0.52542

2.4806 2.3729 2.2741 2.1831 2.0991

4.3353 4.1473 3.9749 3.8159 3.6691

6.1842 5.9165 5.6711 5.4448 5.2359

24.408 23.365 22.407 21.522 20.704

42.160 40.377 38.740 37.223 35.823

59.457 56.964 54.675 52.563 50.615

540 560 580 600

1.0004 1.0004 1.0003 1.0003

1.0016 1.0015 1.0013 1.0012

1.0026 1.0026 1.0024 1.0023

1.0037 1.0036 1.0035 1.0034

1.0148 1.0144 1.0140 1.0136

1.0260 1.0252 1.0244 1.0237·

1.0372 1.0360 1.0348 1.0337

540 560 580 600

0.50596 0.48789 0.47112 0.45541

2.0214 1.9494 1.8826 1.8200

3.5339 3.4077 3.2908 3.1815

5.0430 4.8634 4.6961 4.5400

19.951 19.246 18.590 17.977

34.533 33.326 32.202 31.150

48.801 47.113 45.541 44.070

------

~ ~

4-171

THERMODYNAMIC PROPERTIES OF GASES

300

.4

.5

.6

.7

.8

.9

I

I

I

I

I

I

I

I"'M!:::'::'UI'fC.

I II

P, atm

~f-'TIRTo

TABLE

4h-22.

ENTROPY OF NITROGEN,

t ,....

SIR

-1 T, K

--100 200 300 400 500

4 atm 1 atm --- --1.2589 2.5535 2.5358 3.8302 3.8385 5.1244 5.1203 6.4194 6.4178

7 atm

10 atm

--- ---

40 atm 70 atm 100 atm - - - ---- - - -

2.5179 3.8221 5.1164 6.4162

2.4999 3.8140 5.1125 6.4147

2.3140 3.7351 5.0756 6.4005

2.125 3.662 5.0420 6.3891

1.94 3.596 5.013 6.3802

7 atm -----------16.55 19.1705 17.607 100 21.6249 20.2208 19.6431 200 23.0482 21.6549 21.0884 300 24.0586 22.6687 22.1055 400 24.8479 23.4595 22.8977 500 T, K

1 atm

4atm

10 atm

40 atm

- - -- - -

100 atm - - ---70 atm

19.2682 20.7248 21. 7454 22.5390

17.6905 19.2706 20.3246 21.1322

16.932 18.6461 19.7322 20.5532

16.382 18.230 19.3448 20.1781

600 700 800 900 1000

7.7334 9.0735 10.4428 11.8416 13.2683

7.7333 9.0744 10.4444 11.8438 13.2708

7.7332 9.0752 10.4460 11.8459 13.2734

7.7331 9.0762 10.4477 11.8482 13.2760

7.7332 9.0861 10.4647 11.8705 13.3025

7.73549.0977 10.482£ 11. 8937 13.3296

7.7393 9.1103 10.5020 11.9177 13.3573

600 700 800 900 1000

25.5020 26.0662 26.5656 27.0154 27.4260

24.1144 24.6790 25.1786 25.6286 26.0393

23.5534 24.1184 24.6183 25.0685 25.4793

23.1953 23.7607 24.2609 24.7113 25.1223

21.7958 22.3654 22.8682 23.3203 23.7323

21.2236 21. 7970 22.3022 22.7561 23.1693

20.8548 21.4319 21.9396 22.3949 22.8094

1100 1200 1300 1400 1500

14.7203 16.1950 17.6894 19.2014 20.7288

14.7232 16.1982 17.6929 19.2050 20.7325

14.7261 16.2014 17.6963 19.2086 20.7363

14.7290 16.2046 17.6997 19.2122 20.7400

14.7588 16.2369 17.7343 19.2486 20.7779

14.7891 16.2697 17.7691 19.2851 20.8159

14.8197 16.3029 17.8043 19.3221 20.8542

1100 1200 1300 1400 1500

27.8039 28.1543 28.4811 28.7872 29.0751

26.4173 26.7678 27.0947 27.4007 27.6887

25.8574 26.2080 26.5349 26.8410 27.1290

25.5004 25.8511 26.1780 26.4842 26.7721

24.1114 24.4627 24.7901 25.0965 25.3848

23.5491 23.9010 24.2289 24.5357 24.8242

23.1899 23.5424 23.8707 24.1779 24.4666

1600 1700 1800 1900 2000

22.2695 23.8219 25.3848 26.9568 28.5370

22.2734 23.8259 25.3889 26.9610 28.5413

22.2773 23.8299 25.3930 26.9652 28.5455

22.2812 23.8340 25.3971 26.9693 28.5498

22.3203 23.8742 25.4382 27.0113 28.5924

22.3597 23.9146 25.4795 27.0533 28.6352

22.3992 23.9550 25.5209 27.0954 28.6779

1600 1700 1800 1900 2000

29,3467 29.6037 29.8477 30.0799 30.3013

27.9603 28.2173 28.4613 28.6936 28.9150

27.4006 27.6577 27.9017 28.1339 28.3553

27.0438 27.3009 27.5449 27.7772 27.9986

25.6567 25.9140 26.1582 26.3905 26.6120

25.0964 25.3537 25.5981 25.8306 26.0522

24.7390 24.9965 25.2410 25.4736 25.6953

2100 2200 2300 2400 2500

30.1246 31. 7187 33.3187 34.9240 36.5342

30.1290 31. 7230 33.3231 34.9284 36.5387

30.1333 31. 7274 33.3275 34.9329 36.5432

30.1376 31. 7318 33.3319 34.9374 36.5477

30.1808 31.7755 33.3761 34.9819 36.5926

30.2241 31.8193 33.4203 35.0266 36.6377

30.2674 31.8632 33.4647 35.0712 36.6827

2100 2200 2300 2400 2500

30.5129 30.7154 30.9097 31.0963 31.2759

29.1266 29.3291 29.5234 29.7100 29.8896

28.5670 28.7695 28.9638 29.1504 29.3300

28.2102 28.4128 28.6071 28.7937 28.9733

26.8238 27.0264 27.2207 27.4074 27.5870

26.2640 26.4667 26.6611 26.8478 27.0275

25.9072 26.1100 26.3043 26.4911 26.6708

2600 2700 2800 2900 3000

38.1488 39.7676 41.3901 43.0160 44.6452

38.1533 39.7722 41.3947 43.0206 44.6499

38.1579 39.7767 41.3993 43.0252 44.6545

38.1624 39.7813 41.4039 43.0298 44.6591

38.2076 39.8268 41.4496 43.0758 44.7053

38.2530 39.8723 41.4954 43.1218 44.7514

38.2983 39.9179 41.5413 43.1678 44.7976

2600 2700 2800 2900 3000

31.4488 31.6157 31.7769 31. 9327 32.0836

30.0625 30.2294 30.3906 30.5464 30.6973

29.5029 29.6698 29.8310 29.9868 30.1377

29.1462 29.3131 29.4743 29.6301 29.7810

27.7600 27.9269 28.0882 28.2440 28.3949

27.2004 27.3674 27.5287 27.6846 27.8355

26.8438 27.0108 27.1721 27.3280 27.4790

--

Ol

iII t>=J

p... >--3

TABLE

4h-23.

COMPRESSIBILITY FACTOR FOR OXYGEN,

Z T, K

1 atm

4atm

=

PV/RT

7 atm

10 atm

TABLE

4h-24.

RELATIVE DENSITY OF OXYGEN,

p/po

40 atm

70atm

OOatm

.--- - - -- - - - - - - - -- - - ----

T, K

1 atm

4stm

7 stm

10 stm

40 atm

70 stm

100 atm

--- - - -- - ----- - - -- - -- - ----

100 200 300 400 500

0.97724 0.99701 0.99939 1.00001 1.00022

0.98796 0.99759 1.00006 1.00088

0.97880 0.99580 1.00012 1.00154

0.96956 0.99402 1.00019 1.00222

0.8734 0.97731 1.00161 1.00942

0.7764 0.9636 1.0042 1.0173

0.6871 0.9541 1.0079 1.0256

100 200 300 400 500

2.79257 1.36860 0.91023 0.68225 0.54568

5.5245 3.6474 2.72885 2.18129

9.7584 6.39455 4.77519 3.81474

14.073 9.151 6.8212 5.4459

62.4 37.231 27.246 21.628

123 66.082 47.557 37.556

198.5 95.34 67.69 53.217

600 700 800 900 1000

1.00029 1.00031 1.00031 1.00030 1.00029

1.00116 1.00124 1.00124 1.00121 1.00115

1.00204 1.00218 1.00218 1.00211 1.00202

1.00292 1.00312 1.00311 1.00302 1.00288

1.01205 1.01275 1.01265 1.01223 1.01167

1.0216 1.0227 1.0224 1.0216 1.0206

1.0314 1.0328 1.0323 1.0312 1.0296

600 700 800 900 1000

0.45470 0.38974 0.34102 0.30313 .0.27282

1.81723 1.55750 1.36282 1.21143 1.09035

3.17736 2.72307 2.38269 2.11809 1.90646

4.5351 3.8864 3.4006 3.0231 2.7211

17.9767 15.3980 13.4746 11. 9823 10.7901

31.165 26.684 23.355 20.776 18.717

44.098 37.747 33.045 29.404 26.505

1100 1200 1300 1400 1500

1.00027 1.00026 1.00025 1.00023 1.00022

1.00109 1.00104 1.00098 1.00093 1.00088

1.00192 1. 00182 1.00172 1.00163 1.00155

1.00274 1.00260 1.00246 1.00233 1.00221

1.01107 1.01047 1.00991 1.00938 1.00890

1.0195 1.0184 1.0174 1.0165 1.0156

1.0281 1.0265 1.0250 1.0237 1.0224

1100 1200 1300 1400 1500

0.24802 0.22736 0.20987 0.19488 0.18189

0.99129 0.90872 0.83887 0:77899 0.72710

1. 73331 1.58903 1.46694 1.36228 1.27157

2.4741 2.26828 2.09409 1.94476 1.81533

9.8150 9.0024 8.3145 7.7247 7.2131

17.034 15.631 14.443 13.423 12.539

24.131 22.154 20.480 19.041 17.794

1600 1700 1800 1900 2000

1.00021 1.00020 1.00019 1.00018 1.00017

1.00084 1.00080 1.00076 1.00072 1.00069

1.00147 1.00140 1.00133 1.00127 1.00121

1.00210 1.00200 1.00190 1.00181 1.00173

1.00845 1.00803 1.00765 1.00728 1.00696

1.0149 1. 0141 1.0134 1. 0128 1.0122

1.0213 1.0202 1.0193 1.0183 1.0175

1600 1700 1800 1900 2000

0.17053 0.16050 0.15158 0.14361 0.13643

0.68168 0.64161 0.60599 0.57412 0.54543

1.19219 1.12214 1.05987 1.00415 0.95400

1.70206 1. 60210 1.51324 1.43373 1.36215

6.7653 6.3700 6.0184 5.7037 5.4202

11.764 11.080 10.472 9.927 9.436

16.700 15.735 14.874 14.105 13.410

2100 2200 2300 2400 2500

1.00017 1.00016 1.00015 1.00015 1. 00014

1.00066 1.00063 1.00061 1.00058 1.00056

1.00116 1. 00111 1.00107 1.00102 1.00098

1.00166 1.00159 1.00152 1.00146 1.00141

1.00666 1.00638 1.00610 1.00586 1.00564

1.0117 1.0112 1.0107 1.0103 1.0099

1.0167 1.0161 1.0153 1.0147 1.0142

2100 2200 2300 2400 2500

0.12993 0.12403 0.11863 0.11369 0.10915

0.51947 0.49587 0.47432 0.45457 0.43640

0.90862 0.86736 0.82968 0.79515 0.76337

1.29737 1.23849 1.18473 1.13543 1.09007

5.1637 4.9303 4.7173 4.5218 4.3419

8.991 8.587 8.217 7.878 7.566

12.781 12.208 11.686 11.206 10.763

2600 2700 2800 2900 3000

1.00014 1.00013 1.00013 1.00012 1. 00012

1.00054 1.00052 1.00050 1.00049 1.00047

1.00095 1.00091 1.00088 1.00085 1.00082

1.00135 1.00130 1.00126 1. 00122 1.00117

1.00543 1.00523 1.00505 1.00488 1.00471

1.0095 1.0092 1.0089 1.0086 1.0083

1.0136 1. 0131 l.0127 l.0122 l.0118

2600 2700 2800 2900 3000

0.10495 0.10106 0.09745 0.09409 0.09096

0.41962 0.40409 0.38966 0.37623 0.36370

0.73404 0.70688 0.68165 0.65817 0.63625

1.04820 1.00943 0.97342 0.93989 0.90861

4.1758 4.0219 3.8790 3.7458 3.6216

7.278 7.010 6.762 6.531 6.315

10.355 9.976 9.624 9.296 8.990

>-3

p::

IC"J

~

o

t::I

~

~

H

..

-

o

"d :;:d

o

"d IC"J

:;:d

-_.-

>-3

H

IC"J

w

o

"'J Q.

po. w IC"J w

t

i-'

'"'l '"'l

TABLE

4h-25.

SPECIFIC HEAT OF OXYGEN,

4atm

7atm

TABLE

CpjR

4h-26.

ENTHALPY OF OXYGEN,

10 atm

(H - EoO)jRTo

40atm

70 atm

100 "tm

1atm

4atm

7 atm

100 200 300 400 500

1.254 2.5523 3.8424 5.1523 6.5000

2.5308 3.8319 5.1464 6.4968

2.5091 3.8213 5.1406 6.4936

2.4871 3.8108 5.1349 6.4905

2.248 3.705 5.078 6.460

1.972 3.602 5.023 6.431

1.659 3.505 4.971 6.403

4.052 4.120 4.180 4.232 4.277

600 700 800 90a 1000

7.8919 9.3254 10.7951 12.2949 13.8198

7.8903 9.3250 10.7956 12.2960 13.8213

7.8888 9.3245 10.7960 12.2970 13.8228

7.8873 9.3242 10.7965 12.2981 13.8243

7.873 9.321 10.802 12.309 13.840

7.860 9.319 10.807 12.321 13.857

7.848 9.318 10.814 12.333 13.874

4.3085 4.3442 4.3771 4.4076 4.4369

4.316 4.350 4.382 4.412 4.440

1100 1200 1300 1400 1500

15.3653 16.9285 18.5067 20.0985 21.7025

15.3672 16.9307 18.5092 20.1012 21.7054

15.3691 16.9329 18.5116 20.1038 21.7082

15.3710 16.9351 18.5141 20.1065 21.7111

15.391 16.958 18.539 20.134 21.740

15.411 16.981 18.565 20.161 21.769

15.431 17.004 18.591 20.189 21.799

4.4621 4.4905 4.5185 4.5464 4.5739

4.4652 4.4933 4.5209 4.5485 4.5758

4.468 4.496 4.523 4.551 4.578

1600 1700 1800 1900 2000

23.3181 24.9447 26.5820 28.2299 29.8880

23.3211 24.9479 26.5852 28.2333 29.8915

23.3241 24.9510 26.5885 28.2366 29.8949

23.3271 24.9541 26.5917 28.2399 29.8983

23.358 24.986 26.625 28.274 29.933

23.388 25.018 26.658 28.308 29.968

23.419 25.050 26.691 28.342 30.003

4.5999 4.6272 4.6544 4.6812 4.7074

4.6016 4.6287 4.6558 4.6824 4.7085

4.6032 4.6301 4.6570 4.6835 4.7095

4.605 4.631 4.658 4.685 4.710

2100 2200 2300 2400 2500

31.5566 33.2353 34.9239 36.6229 38.3314

31.5601 33.2389 34.9275. 36.6266 38.3352

31.5636 33.2424 34.9312 36.6303 38.3389

31.5671 33.2460 34.9348 36.6340 38.3426

31.6!}2 31.638 33.282 33.318 34.971 . 35.008 36.671 ' 36.708 38.418 38.380

31.674 33.355 35.045 36.745 38.455

4.7331 4.7582 4.7826 4.8064

4.7341 4.7590 4.7834 4.8072

4.7349 4.7598 4.7841 4.8077

4.736 4.761 4.785 4.808

2600 2700 2800 2900 3000

' 40.0500 41.7778 43.5151 45.2614 47.0165

40.0537 41.7816 43.5189 45.2653 47.0204

40.0575 41.7854 43.5227 45.2691 47.0243

40.0613 41.7892 43.5266 45.2730 47.0282

40.099 41.827 43.565 45.312 47.067

10atm

40atm

70atm

100 "tm

T, K

1 atm

200 300 400 500 600

3.519 3.5403 3.6243 3.7415 3.8611

3.5681 3.5584 3.6335 3.7470 3.8648

3.6196 3.5766 3.6427 3.7526 3.8685

3'.6739 3.5951 3.6520 3.7582 3.8722

4.415 3.7862 3.7453 3.8134 3.9087

5.66 3.981 3.836 3.8677 3.9445

7.6 4.165 3.921 3.920 3.980

700 800 900 1000 1100

3.9681 4.0583 4.1332 4.1952 4.2472

3.9707 4.0603 4.1347 4.1964 4.2481

3.9733 4.0622 4.1361 4.1975 4.2491

3.9759 4.0641 4.1376 4.1987 4.2500

4.0016 4.0830 4.1521 4.2101 4.2591

4.0266 4.1017 4.1664 4.2213 4.2681

1200 1300 1400 1500 1600

4.2915 4.3302 4.3653 4.3976 4.4283

4.2922 4.3308 4.3658 4.3981 4.4287

4.2930 4.3315 4.3663 4.3985 4.4291

4.2937 4.3321 4.3669 4.3990 4.4295

4.3012 4.3382 4.3721 4.4034 4.4332

1700 1800 1900 2000 2100

4.4579 4.4869 4.5154 4.5437 4.5716

4.4582 4.4872 4.5156 4.5439 4.5717

4.4586 4.4875 4.5159 4.5441 4.5719

4.4589 4.4878 4.5161 4.5443 4.5721

2200 2300 2400 2500 2600

4.5993 4.6268 4.6540 4.6808 4.7071

4.5995 4.6269 4.6542 4.6810 4 ..7072

4.5997 4.6271 4.6543 4.6811 4.7073

2700 2800 2900 . ;;000

4.7328 4.7579 4.7824 4.8062

4.7329 4.7580 4.7825 4.8063

4.7330 4.7581 4.7826 4.8064

-.--- - - - - - - - - - - - - - - - - - - - - -

T, K

- - - - - -- - -- - -- - - - - - - - - - - -

40.137 ' 40.175 41.904 41.866 43.643 43.604 45.351 45.390 47.107 47.146

t

I-'

"" 00

il:i t 5.328(3) ~ H

o

T = 158,490 K

":I

-5 -4 -3 -2 -1 0 +1

~

2.000( -5) 2.000( -4) 2.003( -3) 2.025( -2) 0.2194 2.8280

5.9999 5.9994 5.9937 5.9392 5.5589 4.5360

28.5422 28.5379 28.4949 28.0852 25.2956 18.8699

6. 9627( -2) 0.6962 6.9641 6.9784(1) 7.0764(2) 7.4431(3)

1.667(-5) 1. 669( -4) 1. 692( -3) 1. 835( -2) 0.2175 2.8~02

log T -5 -4 -3 -2 -1 0 +1

.. ......... 1. 998( -5) 2.000( -4) 2.000( -3) 2.002( -2) 0.2022 2.1844

=

6.9990 6.9902 6.9085 6.4490 5.5974 4.4719

5.4

T

=

42.1981 42.1008 41.2041 36.2326 27.9500 19.7684

6.769( -2) 0.6770 6.7823 6.8664(1) 7.0637(2) 7.4734(3)

.. ........

..... . . ......

...........

1.1031(-1) 1.1036 1.1034(1) 1.1035(2) 1.1057(3) 1.12054(4)

1. 667( -5) 1. 6(17( -4) 1. 607( -3) 1. 6~'2( -2) 0.1715 1.9859

7.0000 6.9998 6.9981 6.9814 6.8305 6.0354

0.1073 1.0731 1. 0728(1) 1.0734(2) 1. 0772(3) 1.10221(4)

6.0038 6.0003 5.9995 5.9945 5.9465 5.5780

21.4332 21.3407 21.3298 21. 3127 21.1549 19.9145

9.595( -6) 9. 599( -5) 9.6395(-4) 9.9069( -3) 0.1095 1.4194

12.422 12.418 12.374 12.094 11.134 9.045

62.862 62.814 62.367 59.617 51. 714 38.120

6.92( -2) 0.692 6.92 6.95(1) 7.074(2) 7.449(3)

9.5( -7) 9. 59( -6) 9. 594( -5) 9. 596( -4) 9. (l09( -3) 9:140( -2) 1.070

12.481 12.429 12.423 12.421 12.407 12.267 11. 350

48.109 46.705 46.559 46.537 46.467 45.827 41.941

1.090( -2) t':I w. 0.110 1.097 1. 097(1) 1. 097(2) 1. 099(3) t-.:) 1. 1170( 4)

251,190 K

..... . . . . . . . . . . . . . . . . . . .

30.5070 30.5059 30.4954 30.3921 29.4678 24.9055

o

":I

t':I

..,

~ H

t':I

w.

o

";I

§2 [f2

t

o '"""'

TABLE

4h-31.

THERMODYNAMIC PROPERTIES OF HIGHLY IONIZED NITROGEN, OXYGEN, AND

log C. p/po

Z*

I E*/RT I

P,atm

-1

0 +1

1. 553( -5) 7.4409 1. 673( -4) 6.9782 1. 822( -3) 6.4876 (967( -2) 6.0831 0.20046 5 9883 2.0749 5.8194

43.5716 34.2161 25.3735 18.2776 16.9174 16.5657

0.1684 1. 7016 1. 7227(1) 1. 7437(2) 1. 7496(3) 1. 7599( 4)

=

-1

0 +1

-1

0 +1

7.2854 7.0382 7.0037 6.9973 6.9700 6.7442

= 5.8

T

..... . ...... .

.. ......... .. ........ .....

1.429( -5) 1. 430( -4) 1. 438( -3) 1. 499( -2) 0.1657 1.8873

7.9995 7.9953 7.9547 7:6706 7.0362 6.2987

0.2640 2.6410 2.6423(1) 2.6560(2) 2.6932(3) 2.7459(4)

39.3190 39.2610 38.7050 34.8243 26.4910 18.8607

1. 253( -5) 1. 277( -4) 1.371( -3) 1. 485( -2) 0.1621 1. 7010

.

.......... ..... . ...... . .. .........

1.429( -5) 1. 429( -4) 1.429( -3) 1. 4286( -2) 0.1436 1.498

8.0000 8.0000 7.9996 7.9961 7.9618 7.6748

29.2413 29.2410 29.2388 29.2170 29.0039 27.2651

0.4184 4.184 4.1839(1) 4.1820(3) 4.1856(3) 4.2090(4)

=

T

29.2764 23.9473 23.2082 23.1256 23.0593 22.5449

1. 689( -1) 1.6987 1. 7006(1) 1.7001(2) 1.70157(3) 1. 7112( 4)

= 630,960 K . ...... . ...........

8.9802 8.8308 8.2929 7.7337 7.1689 6.8787

6.0

P,atm

p/po

Z*

1~*/RT I

P,atm

T =c 398,110 K

. ..........

log T -5 -4 -3 -2

5.6

1. 591( -5) 1. 656( -4) 1. 666( -3) 1. 667( -2) 0.1675 1.7409

log T -5 -4 -3 -2

I E*/RT I

Z*

p/po

log T -5 -4 -3 -2

Air

Oxygen

Nitrogen

!g

AIR (Continued)

=

50.7353 48.1188 38.7624 29.6449 21.2538 18.6828

0.2599 2.6049 2.6264 2.6530(2) 2.6844(3) 2.7028(4)

14.816 81.096 0.1685 7.80(-6) 8.346( -5) 13.982 64.088 1.701 8.934( -4) 13.194 49.831 1.718(1) 9.476( -3) 12.553 38.606 . 1.734(2) 9.623( -2) 12.392 36.433 1.739(3) 12.300 35.661 1. 7500(4) 0.9970 16.422 83.609 2.62(-2) 6.9(-7) 6.94(-6) 16.414 83.468 0.2631 6. 972( -5) 16.344 82.269 2.633 7.116( -4) 16.052 77.434 2.639(1) 7 .481( -3) 15.368 67.457 2.655(2) 0.08245 14.128 50.766 2.691(3) 0.9225 12.843 37.646 2.7367(4)

1,000,000 K

. .........

. ..... ...... .

1. 250( -5) 1. 250( -4) 1. 250( -3) 1. 254( -2) 0.1288 1. 4231

9.0000 8.9997 8.9971 8.9719 8.7657 8.0268

37.2123 37.2095 37.1809 36.9010 34.6534 27.3929

........... 6.93( -7)

0.41186 4.1186 4.1186(1) 4.1186(2) 4.133(3) 4.1819(4)

6.933( -6) 6.933(-5) 6.934( -4) 6. 942( -3) 0.07011 0.74082

16.423 16.423 16.423 16.421 16.405 16.264 15.498

61.855 61.855 61.853 61.838 61.685 60.398 54.584

4.17(-2) 0.417 4.17 4.17(1) 4.17(2) 4.175(3) 4.2033(4)

~

L:9

>-

.."

TABLE

4h-31.

THERMODYNAMIC PROPERTIJilS OF HIGHLY IONIZED NITROGEN, OXYGEN, AND

AIR (Continued)

Oxygen

Nitrogen

Air

log C, . p/po

Z*

!E*/RT!

P,atm

Z*

p/po

log T = 6.2

!E*/RT!

P,atm

p/po

Z*

! E*/RT!

P,atm

>-'3 ~

t9

~

T = 1,584,900 K

o

I:;:!

-5 -4 -3 -2 -1 0 +1

>H

Process Initial

NpF, ........

P

(Continued)'

liq g

0.044 760 ............ 46.6 760

823 1193 1236 253.86 315.4

kcal/mol

52.5

3.306 7.0

219.7

13.832 29.3

0.045 760 ............

850 1243 1303

18.47

g

2.5(E - 3)

1080

77.3

323.4

g

0.43 ............

750 1070

36.5

152.7

liq

c, III c, II c, I

c, II c, I liq

............ ............ 87

525 565 2263

0.0 0.0

0.0 0.0

tr tr fus

c, III c, II c, I

c, II c, I liq

. ........... ............

533 850 910

2

4.4

·.... .... . . .

fus yap yap

c liq liq

liq

327.92 327.92 328.33

4.189 7.133

tr tr fus Yap yap

c, III c, II c, I liq liq

c, II c, I liq

£1.18

e liq

li'! -

yap fus sub

c c

liq

fus yap yap

c liq liq

liq

tr fus sub Yap

e e e liq

e liq

tr fus yap

c, II c, I liq

c, I liq

tr fus Yap yap

c, IV c, III liq liq

c, III liq

fus yap

e liq

liq

fus Yap

c liq

liq

g g

g g

g

g

g g

g g

g

g g

g

g

748.6 748.6 760 ............ ............ 1.14 1.14 760 O.Rfi

760 ............ 6.2(E - 6)

23.85 43.77 54.363 54.363 90.180 80.65 161.3 3323 2550

53.0

kJ/mol

0.022 0.178 0.106 1.828 1.630 0.5 3.58 187.4

221.7 77 .28

17.527 29.844 0.0920 0.745 0.4435 7.648 6.820 2.1 14.98 784.1

0.566 15.1 760

343.1 400 499.0

15.69

65.647

81.3 463.6 463.6 760

272.7 306.5 306.5 320.6

2.0 1.6 8.40 6.70

8.37 6.69 35.15 28.03

305.6 332.3 354.0

1.62 8.74

6.778 36.57

· . . . . . . . .. . . ·. . . . . . . . . . . ·. . . . . . . . . . .

195.35 317.30 317.30 530

0.500 0.628 13.32 12.48

2.092 2.628 55.731 52.216

........ ....

232.7 446.4

9.33

39.04

183 348.3

7.17

30.00

·. . . . . . . . . . . 175.6 394.6

760 760

............ 760

TRANSITION, FUSION AND VAPORIZATION TABLE

4j-1.

State SubBtance

P

T

mmHg

K

Final

PClo .........

fus Bub

c c

liq g, equil.

............ 760

437.7 432

PF..........

tr tr fUB yap

c, III ,c, II c, I Jiq

II c, I liq g

c,

............ ............ 9.80 760

83.7 110.6 121.8 171.8

PF •••.••••...

fUB yap Yap

c Iiq Iiq

liq' g g

427 427 760

PH •.•••.....

tr tr tr fUB Yap

c, IV 'c, III c, II c, I liq

c, III c, II c, I liq' g

P.O •..••......

fUB Yap yap

c liq liq

liq g g

P'OlO •.......

fUB Bub fUB sub

0, 0, 0,

hexag. hexag. rhomb. c, rhomb.

liq g liq g

Pb ......... ·.

fus yap

c liq

liq g

............

tr fUB yap

c, II c, I liq

c, I liq g

............

Pb(CH.) •....

fUB vap

c liq

Iiq g

............ 760

PbC!, ........

tr fUB yap

c, a c, (J liq

c, (J liq g

............ ............

tr fUB yap

c, rhomb. 0, cubic liq

0, cubic liq g

............ 723 ............ 1099

tr fUB Bub

c, II c, I c, I

c, I liq g

tr fua yap

0, red c liq

PbS .........

fUB

PbSO •.•••••.

tr fUB

Pd .......... PdCl........

PbI ••••••....

PbO .........

............ ............ ............ 27.2 760 1.7 1.7 760 3690 3690 570 570 760 0.011 760

kcal/mol 6.1 18.1

kJ/mol 25.5 75.73

0.060 0.55 0.224 3.48

0.251 2.30 0.9372 14.56

179.4 179.4 188.7

2.7 4.2 4.1

11.3 17.6 17.2

30.31 49.46 88.15 139.40 185.43

0.0196 0.186 0.115 0.270 3.486

0.08200 0.7782 0.4812 1'.130 14.585

i

PbF.........

I!.H

Process

Initial

PbBr•..•....

4-239

(Continued)

TEMPERATURES, PRESSURES, AND HEATS

i

297.1 297.1 448.5

3.36 11,14 10.38

693 693 844 844

5.0 13.9 16.1 36.4

20.9 58.16 67.36 152.3

600.45 2023

1.147 42.5

4.7990 177'.8

617 643.1 1166

5.0 30.2

20.9 126.4

2'.58 7.87

10.79 32.93

5.25 30.4

21.97 127.2

1566

3.0 38.4

12.6' 160.7

............. 0.23 0.23

645 685 685

3.9 36.8

16.3 154.0

0, yellow liq g, equil.

............

762 1158 1813

c

liq

c, II c, I

c, I liq

fUB vap

c liq

liq g

0.031 760

1825 3237

fUB

c

liq

............

953

760

760

242.92 383.2

14.06 46.610 43.430

695 773 1227

0.394 6.57

1.648 27.49

............ 1382

4.2

17.6

............ 1139

4.06 9.6

16.99 40.2

4.20 85.4

17.56 357.3

0.35 760

............ 1360

5

21

HEAT

4-240 TABLE

4hl.

TEMPERATURES, PRESSURES, AND HEATS

State Substance

P

T

mmHg

K

(Continued) aH

Process Final

Initial

kcal/mol

kJ/mol

Po ..........

tr ius yap

c, II c, I liq

c, I liq g

. ........... ........... ,. 760

327 527 1235

3.0

12.5

Pr ...........

tr fus yap

c, a c, (3 liq

c, (3 liq g

............ ............

1068 1204 3785

0.76 1.65 70.9

3.18 6.904 296.6

PrBro ..... ·· .

fus sub

c c

liq g

............ ............

966 966

11.3 68.1

47.28· 284.9

PrCIa ........

ius sub yap

c c liq

liq g g

3.5(E - 3) 3.5(E - 3) 23

1059 1059 1523

12.1 70.3 54.7

50.62 294.1. 228.9

PrF •.........

fus sub

c c

liq g

............ 1.3(E - 3)

1668 1400

82.3

344.3

PrI •....•....

ius sub

c c

liq g

1011 1011

12.7 66.5

.53.14 278.2

Pt ...........

Ius yap

c liq

liq g

............ 2043 4097 760

4.7 121.8

19.7 509.6

PtF •.........

tr fus yap

c,orthorh. c, cubic liq

c, cubic liq g

. ........... ............ 760

Pu ..........

tr tr tr tr tr fus yap

c, VI c, V c, IV c, III c, II c, I liq

c, V c, IV c, III c, II c, I liq g

. ........... ............ ............ ............ ............

760

. . . . . . . . .. . . ............

............ 760

PuB .........

fus yap

c liq

liq g

2.1(E-3) 2.1(E-3)

PuCl .........

Ius y ..p

c liq

liq g

1.9(E - 3) 1.9(E - 3)

PuF.........

fus sub

c c

llq g

PuF.........

sub fus

c c

g liq

PuF .........

fus y ..p

c liq

liq g

533.0 760

2.14 1.08 7.06

8.954 4.519 29.54

395 480 588 730 753 913 3503

0.80 0.14 0.13 0.02 0.44 0.68 82.1

3.35 0.586 0 ..544 0.084 1.84 2.85 343-.7

954 954

11.6 57.3

48.53 239.7

1033 1033

13.3 58.6

55.65 245.2

276.15 334.45 342.29

1698 0.72 2.33(E - 3) 1400 4.3(E - 4) 8.2(E - 3)

1123 1310 324.74 335.31

93.0

389.1

45.9

192,0

4.456 7.03

18.644 .. 29 .. 41

R ............

fus

c

liq

............

973

Rb ..........

fus v ..p

c liq

liq g, equil.

............ 760

312 967

0.54

.' .. 2 ..26

RbBr ........

fus v ..p

c liq

liq g

760

965 1625

5.57 37.1

23.30 155.2

RbC!. .......

fus yap

c liq

liq g

0.27 760

995 1654

5.67 36.9

23.72 154.4

RbF .........

ius sub

c c

iiq

0.6 0.6

1068 1068

5.5 52.3

23.0 218.8

g

. . . . . . . . . . ..

4-241

TRANSITION, FUSION AND VAPORIZATION TABLE

4j-1.

TEMPERATURES, PRESSURES, AND HEATS

State Substance

Final

libI ...... ...

Ius sub yap

c c liq

liq g g

RbNO, ......

tr tr tr Ius

c, c, c, c,

c, III c, II c, I liq

RbOH ......

tr Ius

c, II c, I

Re .........

Ius Yap

(ReB,,), .....

sub

(ReC!,)' ......

sub

ReF, ........

fus yap yap

ReF, ........

tr Ius yap

ReF7 ........

Rb .......... Rn .......... Ru ....... '" RuF, ........

RuF, ........

RuO •........ S ............

SF •..........

T

mmHg

K

t,H

Process

Initial

Re'07 ........

P

(Continued)

kJ/mol

920 920 1578

5.27 46.7 35.9

22.05 195.4 150.2

............ ............ ............ . . . ..... . ..

437 501 564 589

0.90

3.77

0.88 1.10

3.68 4.602

c, I liq

............ ............

518 656

1. 70

7.113

c liq

liq g

0.024 760

c

g

e

g

e liq liq

liq g g

0.37 5.61 760

e, II e, I liq

e, I liq g

153.1 426.5 760

271.2 291.8 306.9

2.09 1.10 6.8

8.745 4.602 28.5

tr Ius yap

e, II e, I liq

e, I liq g

163 321.4 321.4

1.80 7.35

7.531 30.75

Ius sub yap

e e liq

liq

Ius yap

e liq

liq

Ius yap

e liq

liq

Ius yap

c liq

liq

Ius yap

e liq liq

liq

yap

tr sub Ius

e, II e, I e, I

Ius Yap

c liq

liq

tr tr Ius yap

c, rhomb. e, rbomb. c, moneel. liq

c, monoel. c, monoel. liq g, equil.

Ius yap

e liq

liq

IV III II I

g g

g

g

g

g g

e, I g

liq g

g

0.4 0.4 760

kcal/mol

.

3453 5960

7.9 171

33.1 715.5

............

550

47.6

199.2

............

550

49

205

321.1 367 494

13.9

............ 311.6 311.6 72 72 760

573.5 573.5 634

............ 2233 4000 760 502 760

202 211

............ 2700 760 4390 5.71 5.71 760

379 379 500

14.7 32 16.8

61.50 134 70.29

5.15 118

21.55 493.7

0.69 4.0

2.89 16.7

6.2 141 10.4 15.6

............ 40 ............

275.6 281 327

9.1

10.6 10.6

298.5 298.5

2.6 10.6

3.8(E - 3)

. ........... 760

368.46 374.15 388.33 717.75

0.096 0.0 0.411 2.2

0.54 41. 7

152.1 192

6.3

............

58.16

25.9 589.9 43.51 65.27

38.1 10.9 44.35 0.402 0.0 1. 711 9.20

26.4

4-2·42

HEAT TABLE

4j-1.

TEMPICRATURES, PRES.SURES, AND HEATS

State

I

t;H

P

T

mmHg

K

kcal/mol

94.26 209.5 222.5

0.384 5.70 1.20

1.607 23.85 5.021

197.69 263.13

1.769 5.955

7.4015 24.916

4.75

19.87

Process

Substance

Final

Initial

SF, ..........

SO, ........ .. Sb ...... ..... SbBr •........ SbCIa ........ SbC], ........ SbF, .........

SbH •........ Sbl •......... Sb.O, ........

Se ...........

SeBr •........ SeCt, ........ SeF •.........

ScI .......... Se ...........

SeF4. ........ SeF6 .........

SeO' ......... Si.

(Continued)

.......

I

. ........ 760 1,700

c, I

tr sub lils

c, II c, I c, I

Ius yap

c liq

liq

Ius yap

c liq

liq g, equil.

I

fus yap

c liq

liq

I

fus

c liq

liq

yap

Ius yap

c liq

liq

Ius yap yap

c liq liq

liC[

IUB yap

C

liq

liq

g

Ius yap

c liq

liq

tr Ius Yap

0, cubic c, orthorh. liq

0, orthorh. liq

tr IUB Yap

e, II e, I liq

e, I Iiq

g

liq

12 56 760

g

g

...

.....

...

.... .

562

3 5 12 6

14.6 52 72

...

346.4 494

3.0 10.80

12 5 45 187

...

276.2 358

2.4 11.7

10.0 48.95

281.4 281.4 416

11.1

46.44

5.1

21.3

14.8

61. 92

2.8 27 17 .8

11.7 113 74.48

760 ......... 30

1.47 1.47 760

g g

............ 760

843 928 1729

0.52 2.5 760

g

179 255 443 3 675

1.6 760

g

904 1860 369.8

1.65

760

..........

1608

0.96

4.02

O.OM

181~

3.37

14.10

g

760

3104

75.1

314.2

g

162 1,530

1134 1233

63.0

263.6

1,260 1,260

1240 1240

63

264

1290 1620 1803

89

372

112 1100 . . . . . . . . . . . 1218

61

255

sub Ius

e e

Ius sub

e e

sub tr Ius

e, II c, II c, I

e, I liq

sub ius

c c

liq

tr Ius Yap

c, II c, I liq

c., I liq g, equil.

Ius yap

c liq

liq

sub Ius yap

c c liq

g

liq g

760 1,500 1,500

Bub

c

g

760

Ius

c liq

liq

yap

.. . .

760

g

g

'-"

kJ/mol

liq liq g g

g

1.8(E - 3) ............ .......... ..

.

1 65 760

263.6 380

............

g

g, eqlliL

760

398 494 958

............

.

........... 760

226.6 238.6 238.6

0.18 1.25

10.0 6.27 1. 78 4.30

0.753 5.230

41.84 26.23 7.448 17.99

629

21.1

88.28

1685 3540

21.1

50.62

4-243

TRANSITION, FUSION AND VAPORIZATION TABLE

4j-1.

.TEMPERATURES, PRESSURES, AND HEATS

State Substance

(Continued)

P

T

mmHg

K

kcal/mol

I1H

Process Initial

Final

kJ/mol

Si6r.........

fus yap

c liq

liq g

1.83 760

278.0 426

9.1

38.1

Si(OH.) •.....

fus yap

c liq

liq

0.2 760

174.12 299.8

1.648 5.79

6.8952 24.23

fus yap

c liq

liq

............ 760

205 330.4

1.84 6.81

7.699 28.49

fus sub

c c

liq

1340 1340

183.0 183.0

2.27 6.33

9.498 26.48

tr fus yap

c, II c, I liq

c, I liq

760

63.5 88.5 161.8

0.147 0.159 2.9

0.6150 0.6653 12.1

SiH.F .......

yap

liq

g

760

185.1

4.3

18.0

SiO •.........

tr tr tr fus tr tr tr tr fus tr fus

quartz, III quartz, II quartz, I quartz, I tridym., IV tridym., III tridym., II tridym., I tridym., I cristob., II cristob., I

quartz, II quartz, I tridym., I liq tridym., III tridym., II tridym., I cristob., 1 liq cristob., I liq

Sm ..........

tr fus vap

c, II c, I liq

c, I liq

Sm,O •.......

tr fus

0, 0,

tr fus Yap

c, white liq

g

fus vap

c liq

liq

tr fUB yap vap

c, II c, I liq liq

c, I liq

fus vap

SiOI •........ SiF •......... SiH •.........

Sn ...........

SnBr. ........

C,

g g

g

monool. cubic grey

g

g

~

............ ............

............ ............ ............ ............ ............ ............ ............

·. . . .. . . . . . . ............

91 846 1140 1883 390 436 498 1743 1953 522 2001

............ 1190 3.18 1345 2064 760

cubic

............

1148 2535

grey

............ ............

286.2 505.06 2896

0,

liq 0,

·........... · .... ......

liq

............

760

0.15 0.12 2.04 0.07 0.04 0.05 0.05

0.628 0.502 8.535 0.29 0.18 0.21 0.21

0.20 1.84

0.837 7.699

0.74 2.06 39.8

3.10 8.619 166.. 5

0.500 1.67 70.8

.. 2.092 6.987 296.2.

1.7 2.2

7.11 92.1 1.272 11.71 51.04 .44.77

............ ............

505 911

g g

............ 0.66 0.66 760

288.5 302.5 302.5 477

0.304 2.80 12.2 10.7

c liq

liq

............

521 888

3.0 21.0

ius vap

c liq

liq g

SnF..........

vap

liq

g, equil.

SnH .........

fus yap

c liq

liq g

............

SnI•.........

ius vap

c liq

liq g

............ 760

593 1000

SnI •.........

fUB vap

c liq

liq

............

417 621

SnB..........

SnOh ........ SnOI.. .......

g

g

g

760

·........ ... 760 760 760

760

239.9 386.8

12.5 87.86

2.19 8.5

9.163 35.5

4.4

18.4

22.4

93.72

4.53 12.4

18.95 51.88

1126 123.3 220.8

4-244

HEAT TABLE

4j-1.

TEMPERATURES, PRESSURES, AND HEATS

State Substance

Final

SnS., .......

tr Ius yap

c, II c, I liq

c, I liq g, equil.

Sr ...........

tr tr IUB yap

c, a c, f3 C, "y

c, f3 c, "Y liq

liq

g

, tr Ius yap

c, II c, I liq

c, I liq

tr tr IUB

c, 'III c, II c, I

tr Bub fUB yap

c, II c, I c liq

Bub fUB

c

fUB yap

SrCO •......

SrCI, .......

SrF, ......... SrI. .... , ....

T

mm Hg

K

875 1153 1500

760

g

918 930 1200

2.90 2.50 58.2

12,1310.46 243.5

c, II c, I Iiq

· . . . . . . . . . . 1689 · . . . . . . . . . . 1770

. ...........

1203

4.7 0.8

19.7 3.3

6) 6) 4) 3)

1003 1003 1146 1245

0.65 71.5 3.80 66.0

2.72 299.2 15.90 276.1

liq

3.02(E - 6) ............

1270 1736

98.4 7.13p

411.7 29.853

C

liq

·......... .

Iiq

g

811 1200

4.70 56.8

19.66 237.7

12.7

5.3

~

"

1.07(E 1.07(E ll.O(E 4.0(E

c, I g

liq g g

-

0.040

liq

............

891

liq

............

2688

SrSO •.......

tr fUB

c, II c, I

c, I Iiq

............

............

1425 1878

c

liq

..........

2313

liq

............

1843

fUB

Ta ..........

Jus yap

TaBr' .......

Ius

TaCh ........

fus

Tb .. , ......

TbCIa ........

;

c liq

Iiq

c

Iiq

c Yap : Iiq

,tr fUB .tr· ius Yap fus Yap

TbF, ........

fUB

Tb20' .......

fus

I, c,c tetrag. I !

0.669 31.59

. ......... ............ 5.9(E - 3)

IUB

Ius

0.160 7.55

138.9

fUB

SrTiO •....

k,J/mol

33.2

Sr(NO.), .....

SrWO •.... '...

kcal/mol

............ 505 · . . . . . . . . . . . 893 1043 1.8 760 1648

SrO., .... '...

Ta'O' ........

~H

Process

Initial

SrBr, ........

P

(Continued)

g

Iiq g

,

760 ............ • • • • • • • • • • I' •

760

c, tetrag. Iiq

............

c, II. C, I Iiq

c, I Iiq

............

c Iiq

Iiq

c

............

3250 5638

7.5 182.1

31.4 761. 91

7.1 12.8

29.7 53.56

528 489.0 506.0 1633 2160

8.1(E-4) 760

1560 1630 3496

1.20 2.58 79.1

5.021 10.79 331.0

g

............ 0.275

855 1223

42.0

175.7

Iiq

............

1446

Iiq

............

2565

............

2610

2.0(E - 6) 2.0(E-4)

2150 2443

g

Tb,,07 .... :...

fus

Iiq

Tc ...........

sub ius

g

Iiq

164

686.2

4-245

TRANSITION, FUSION AND VAPORIZATION TABLE

4j-1.

TEMPERATURES, PRESSURES, AND HEATS

State Substance

(Continued)

P

T

mmHg

K

kcal/mol

............

267.8 310.5 310.5 328.4

1.8 1.1 7.44 7.22

I!.H

Process Initial

Final

kJ/mol

tr fus Yap Yap

c, I c, II liq liq

c, II liq g g

Tc.O, ........

sub yap

c liq

g g

Te .....•....

fus yap

c liq.

liq g, equil.

TeF •........

fus

c

liq

TeO' ..........

fus yap

c liq

liq g

Th ..........

tr' fus yap

c, fJ liq

c, fJ liq g

tr fus

c, a c, fJ

c, fJ liq

ThF •........

fus yap

c liq

liq g

ThI ..........

sub

c

g

ThO •........

. sub fus

c c

g liq

1.8(E - 4) 2400 ............ 3490

162

677.8

Ti. ..•.....••

tr fus vap

c, a. c, fJ liq

c, fJ liq g

............ 4.4(E - 3) 760

0.99 3.7 100.6

4.15 15.5 420.91

TiB" ........

fus yap yap

c liq liq

liq g g

0.411 0.411 760

311.4 311.4 506.6

3.08 13.10 10.60

12.89 54.810 44.350

fus Yap

c liq liq

liq g g

............

yap

249.9 249.9 410.6

2.23 10.34 8.15

9.330' 43.263 34.10

TcFI ........

ThC!,. ......

TiCI.. .......

C,

a

400 400 760 0.7 0.7

392.6 392.6

0.176 760

722.95 1261

.............

126.4 78.66

4.18

17.49

5.5

23.0

1006 1006

6.95 53.1

29.08 222.2

............ ............ 760

1636 2028 5061

0.65 3.85 123.0

2.72 16.11 514.63

............

679

1.20 14.69

0.11 0.11

402.7

30.2 18.8

7.53' 4.60 31.13 30.21

............ 1042

5.021 61.463

0.52 0.52

1375 1375

4 71.3

17 298.3

7.2(E - 4)

623

36.1

151.0

............ 760

1167 1943 3562

456.3

21.6

90.37

379 428 650

2.37 4.68 13.98

9.916 19.58 58.492

............

1264

0.82

3.43

liq

............

2113

11

c, fJ liq g

............ ............ 760

507 577 1760

0.09 0.98 39.4

0.38 4.10 164.8

g liq

1.9 1.9 760

733 733 1092

30.5 3.92 23.9

127.6 16.40 100.0

............ 760

704 1093

3.72 24

15.56 100

TiF..........

sub

c

g

Til •.........

tr fus yap

c, a c, fJ liq

c, fJ liq g

............ ............

TiO .........

tr

c, II

c, I

TiO, .........

fus

c

Tl. ..........

tr fus yap

C, a.

c, fJ liq

TIBr .........

sub fus yap

c c liq

TICl. ........

fus yap

c liq

I:~'"""

g, equil.

760

760

46.0

4-246

HEAT TABLE

4j-1.

TEMPERATURES, PRESSURES, AND HEATS

State Substance

P

T

mmHg

K

(Continued) f>.H

Process Initial

Final

kcaI/mol

kJ/mol

fus yap

c liq

liq g, equil.

· . . . . . .. . . . .

TII.. ........

tr fus yap yap

c, II c, I liq liq

c, I liq g g, equil.

. ........... 1.0 1.0 760

TINO, .......

tr fus

c, II c, I

c, I liq

............ ............

416 479.8

0.91 2.264

ThO .........

fus

c

liq

............

852

7.24

30.29

ThO •........

fus

c

liq

............

998

3

12.5

Tm ..........

fus yap

c liq

liq g

............ 1818 2220 760

4.02 45.6

16.82 190.8

TmOla .......

fus yap Yap

c liq liq

liq g g

............. 0.554 760

1103 1173 1763

77.5

324.3

tr fus sub

c, a c, (3 c, (3

liq g

· . . .. . . . . . . . 2.7(E-3) 2.7(E - 3)

1316 1431 1431

88.9

372.0

Tm20 •.....•.

fus

c

liq

............

2665

U ...........

tr tr fus Yap

c, a c, (3 c, 'Y liq

C,

(3 c, 'Y

............

liq g

TIF .........

TmF •........

C,

(3

760

595.4 1099 451 715 715 1099

941 ........... , 1048 ........... , 1405 760 4407

3.315 0.22 3.52 27.3

0.667 1.137 2.036 110.9

13.870 0.92 14.73 114·.2 3.81 9.473

2.791 4.7572 8.5186 464.01.

UBra ........

tr

c

liq

1003

15

UB« ........

fus yap yap

c liq liq

liq g g

5.7 5.7 760

792 792 1039

16 33.9 30.5

UOl •.........

tr fus yap

c, II. c, I Iiq

c, I liq g

............ 32.6 760

820 863 1075

11 20.4

46.0 85.35

UOl,. ........

sub

c

g

1.8

370

17.3

72.38

UF ..........

tL fus yap

c c liq

c liq g

·..... ......

1110 1330 1330

3.4 10.24 57.1

14.2 42.844 238.9

tr fus yap Yap

c, II c, I liq liq

c, I liq g g

............ 13.4 13.4 760

408 621 621 776

11.1 25.1 23.2

46.44 105.0 97.07

UF6 .........

sub sub yap

c c liq

g g g

329.7 337.2 337.2

11.5 11.4 6.9

48.12 47.70 28.9

UI.. .........

fus

c

liq

4.5

779

19.3

80.75

UO, .........

fus

c

liq

·. ..........

3115

18.2

76.15

V ...........

fus yap

c liq

liq g

2.0(E - 3) 760

2175 3682

5.00 108.0

20.92 451. 87

UF ...........

0.013

7.03 7.03

760 1,138 1,138

62.8 66.9 141.8 127.6

4-247

TRANSITION, FUSION AND VAPORIZATION TABLE

4j-1.

TEMPERATURES, PRESSURES, AND I-ImATS

I

State Substance

p

T

mmHg

K

(Continued) b.H

Process

Initial

Final

kcal/mol

kJ/mol

VOl ..........

fUB yap

c liq

liq g

............ 760

252.6 426

2.3 9.5

9.62 39.7

VOOl, .......

fus Yap

c liq

liq g

............ 760

196 400

2.29 8.45

9.581 35.35

V,O, .........

fus

c

liq

.. ..........

947

15.6

65.27

W ...........

fus yap

c liq

liq

0.039 760

3653 5828

8.46 197.0

35.40 824.25

fUB yap

c liq

liq

............ 760 I

5GS 665

4 13.9

17 58.16

fus

c liq liq

liq g

595.5 595.5 604.5

14 13.4 13.2

58.6 56.07 55.23

213 213

458 503.1 554.6 554.6

3.39 1.6 15.0

14.18 6.69 62.76

240 240 413 413 760

264.9 264.9 275.1 275.1 290.2

1.0 8.8 1. 3' 7.70 6.25

4.18 36.8 5.44 32.22 26.15

25.1 25.1 760

377.8 377.8 459.0

WBr' ........ WOB" ......

Yap vap

WOI, ........

WF, .........

WOF" .......

WO, ......... Xe .......... XeF, ........ XeF4. ....... XeF, ........ y ...........

yO], ......... yF, .........

yI, .......... y,O •........

tr tr fus

g

g

g

yap

c, III c, II c, I liq

tr sub fus yap yap

c, II c, II c, I liq liq

c, I

fus yap yap

c liq liq

liq

tr fus

c, f3

fus

C,

a

c, II c, I

liq g

g

liq g g

g

I:, ~ IIq

Iiq

640 640' 760

· ......... , ....

.. .....

"

............ ............

yap

c liq

g

fus sub

e c

g

fus sub

c c

liq

fUB sub

c c

liq

tr fus yap

C,

a

c,

~

liq

g

fus yap

c liq

liq

tr sub fus

c, II c, I c, I

liq

sub fus

c c

liq

............ · , ..........

890 1237

fus

c, cubic

Iiq

............

2556

Iiq

g

g

611 760

1050 1745 161. 36 165.03

1412 1412

1.4 14.8 13.8

5.86 . 61. 92 57.74

0.410 17.03

1.715 73.43

0.548 3.021

2.293 12.640

402.2 402.2

13.0

54.39

811.3 811.3

390.25 390.25

14.40

60.250

159 159

319 319

15.5

64.85

......... . 2.2(E - 3) 760

1752 1799 3611

1.193 2.724 86.8

4.9915 11.397 363.2

g

............ · ...........

973 1100

30.9

129.3

c, I

.,'"

c, ~ liq

g

g

.......

1325 · . . . . . . . . . . . 1325 · . . . . . . . . . . 1420

.

100

53.6

418.4 224.3

4-248

HEAT

(Continued)

TABLE4j-1. TEMPERATURES, PRESSURES, AND HEATS

State Substance

Initial

Final

mmHg

K

I

b.H

kcal/mol

kJ/mol

1033 1097 1467

.......... 1.41

981 1573

59.8

250.2

c

g

· . . . . . . .. . . .

1362

85.5

357.7

e, cubic

liq

· . . . .. . . . . . .

2645

C

liq g

0.15 760

692.65 1184

1.765 27.62

7.3848 115.56

675.2 928.6

3.74

15.65

760

0.021 760

590 989.4

2.45

10.25

tr fus yap

c, a c, f3 liq

c, f3 liq g

YbCb .......

fus yap

C

liq

liq g

YbF •........

sub

Yb,O •.......

fus

Zn ..........

fus yap

liq

fus yap

liq

fus Yap

liq

liq g,.equil.

ZnC[, ........

T

............ 19.8 760

yb ..........

ZnBf2 .......

P

Process

C

C

liq g, equi!.

"

· . . . . . . . . .. .

0.418 l.83 30.8

1.749 7.657 128.9

ZnO .........

fus

C

iiq

'

...........

2248

ZnSO ........

tr

c, a

c, f3

............

1007

4.8

20.1

Zr ...........

tr fus yap sub

c, a c, f3 liq c, a

c, f3 liq g g

l.8(E - 18) l.2(E - 5) 760 l.8(E - 18)

1136 2125 4682 1136

0.94 4.0 139 144.7

3.93 16.9 58l.6 605.42

ZrBr .........

sub

c

g

550

27.2

113.8

ZrC ...... '"

fus

c

liq

· ........... 3765

ZrC[, ........

fus

c

liq

· ...........

995

ZrCl;. .......

sub fus

C

g

C

yap

liq

liq g

760 15,800 15,800

605 710 710

2·1.4 6.9 16:8

102.1 28.9 70.29

tr sub sub fus

c, a

c, f3 c, f3 c, fJ

g liq

......... , .. 2.1(E - 3) 760 819

678 800 1181 1185

53.0 50.4

221.8 210.9

ZrI4 .........

sub

c

g

l.3(E - 3)

425

26.2

109.6

ZrN .........

fus

c

iiq

·

ZrO' .........

tr sub fus

c, II c, I c, I

c, I

· . . .. ....

g

3(E - 4) ......... , ..

ZrF4 .........

c, g

40

fJ

iiq

............

.. .

3225 1473 2400 2979

1.42 165 20.8

5.941 690.4 87.0

TRANSITION, FUSION AND VAPORIZATION TABLE

Substance

4j-2.

4-249

SELECTED REFERENCES*

Reference

Ac

164

Ag AgBr AgCN AgCl AgF AgI AgNO, Ag,S Ag,SO. Ag,Se

164 31, 33, 411 294 31, 33, 205, 209 417 15, 24, 178, 223, 239, 271, 289 5,84,90,169,170,204,308,318 321,383 152 10, 275, 321, 383, 412

Al Al,Br. AI,Cl. AIF. All, AI,O, AlPO.

164 97, 175, 387 112,267 46, 93, 104, 215 387 48, 57, 125, 183, 270, 324, 336 329

Am

164

Ar

129

As AsCla AsF, AsF5 AsF.O AsH, AsI, A~.05

164 212, 266 384 320 249 352, 385 75, 120 374

Au

164

B BBra B(CH,)a BCla BF. B,H 6 B,Hg B20,

164, 196 14, 160 117 4, 133 214 287,407 143, 176,408 30, 262, 313, 333, 368

Ba BaBr, BaCO, BaCh BaF, BaI, Ba(NO.), BaO BaTiO,

164 100, 165, 171 11,220,307 100, 217, 273, 171 19, 150, 292, 293, 301, 317 100, 165 203 166, 260 105, 354, 372, 389

Be BeCr,

164 111, 116, 132, 154, 208, 231

Substance

Reference

BeF, BeO BeSO.

130, 157, 343, 384 8, 103, 131, 182, 363 20

Bi BiBr, BiCla BiF. Bi,O, BioS.

164 76, 305, 391, 410 81, 83, 173, 246, 305, 390, 399 81 76, 119, 222, 370 71, 127

Br, BrF, BrF5

129 276 225,315

C CBr, CCI, CF. CH. CH.Br CH,CI CH,F CH,I CH,OH CH,CI, CH,F, CH,I, CH,O CHBr. CHCI, CHF, CO CO2 COBr, COCI. COF, CS, COS

164 320 320 320 320· 320 320 108 253 110, 230, 359, 398 320 240 320 320 320 219, 312 162 320 320 320 124 284 320 320

Ca CaB,O. Ca,B,O, CaBr2 CaC, CaCO, CaCI, CaF, CaO CaSO. CaSiO, Ca2SiO. CaTiO,

164 200 200 100, 165, 171 320 306 58, 100, 156 37, 86, 301, 340 9,270, 335 135, 144, 394 126 41,70 66, 186

Cd CdBr, CdCl 2 CdF, CdI,

164 33, 391 35, 36, 391 28 33, 391

* Numbers in Reference coluilln refer to items in the list that follows this table.

4-250

HEAT TABLE

4j-2.

SELECTED REFERENCES

Reference

Substance

(Continued)

Substance

Reference

Feo.9.70 Fe20. Fe.O. FeS

388 67 67 68

129 320 137; 232 139

Ga (GaCl,), Gal.

164 133 311,362

Co Cocr. CoF, CoO

164 325 29. 181 320

Gd GdBr. GdCl, GdF, Gd,O.

164 101 101 301, 369 254, 401

Cr CrBn Cr(CO). CrF. Cr,O,

164 357 65, 309 149, 416 335

Cs CsBr CsCI CsF CsI CsNO, CsOH C8'SO.

164 99, 332, 365 99, 339, ?65 99, 332 99, 332, 335 257 319 22, 291

Ge GeBr. GeCl. GeF. GeH. Gel. GeO,

164 320 12 320 320 177 236, 238, 261

Cu (CuBr), (CuCI), CuF, (CuI). Cu,O Cu,S

164 151, 234, 145, 250, 237 163,

Dy

164

Er ErCla ErF.

164 101, 255, 274 369

Eu EuCl. Eu,O.

164 255,298 254, 334; 401

F, F,O

129 337

Fe FeBr, Fe (CO)& FeCr. (FeCla), FeF, FeF. FeI,

164 233 227 328 243 56, 192 416 326

Ce CeO, Ce,O.

164 218,254 254

Cr. CIF CIF. CIO,

351 351 190 351 310

H, HBr HCN HCI HF HI ,HNO. ·H,O H,S H,SO. H,Se H,Te H.PO. 'H2H 'H2HO 'H,O

129 129 62 129 129 129 98 320 320 129 129 320 129 129 129 129

He

129

Hf HfBr. HfCI. HfF. HfI. HfO,

164 331 268,285 111 376 270

Hg HgBr, HgCr. HgF, HgI, HgS

164 168 77, 174, 391 320 120,234 320

Ho HoCla HoF. Ho,Oa

164 101, 255 27 254

TltANSITION, FUSION AND VAPORIZATION TABLE

Substance

4-251

4j-2. SELECTED REFERENCES (Continued)

Reference

Substance

Reference

---I, IF, IF7 In InBr, InC] InCIa Inl, In,O,

129 52 316 47 164 362 106, 362 362 109, 362 334

Ir IrF,

164 50

K KBr KCN KC] KF KI KNO, KOH K 2S04

164 34, 99 320 18, 35, 339, 392 292, 304 35, 99 204, 367 320 320

Kr KrF, KrF.

129 141, 142 138

La LaBra LaCIa

164 101, 353 100, 353

Ie]

I.JaF?

244, 301

Lala La20a

353 254

Li LiBr LiC] LiF Lil LiNOa LiOH Li 2S04

164 99 99, 314 91, 301 99 114, 204 355 290, 397

Lu LuCIa LuFa LU20a

164 255 369,415 254

Mg MgBr2 MgCr, MgF2 MgI, MgaN2 MgO lVIgSO,

164 26 155, 339 155, 317 26 320 335 320

.

----

Mn MnBr2 MnC]' MnF, Mnl, MnO Mn,O.

164 320 235, 325 136, 191 320 360 146

Mo Mo(CO), MoF, MoF 6 MoO,

164 320 51 282 140, 201

N2 NH, N 2H. NH4Br NH.C] NH.F NH,I NH.NO, NO N20

320 320 129 320 129 129 320 129 320 320

Na NaBr NaCN NaC] NaF Nal N,,"iVT,,0, NaNO, NaOH Na,SO, Na2TiO,

164 99, 118 320 85, 99 113, 301, 304 99, 118 320 114, 204, 258 92 69, 300 320

Nb NbCl, NbF, Nb02 Nb20,

164 3, 189, 267 40, 107 199, 347 121, 281

Nd NdBr, NdCl, NdFa Ndl a Nd20,

164 353, 100, 369, 353, 254,

Ne

129

Ni NiBr2 Ni(CO), NiCl, NiF2 NiO

164 229, 327 371 49, 229, 325 55, 102 197

101 272, 353 418 100 286

4-252

HEAT TABLE

4j-2.

SELECTED REFERENCES

Reference

Substance Np NpF,

164 283,405

0, 0,

129 129

Os OsF, OsF, OsOF,

.54, 164, 395 51 50 17

P. PBr, PCI, PCI, PF, PF, PH, P40, P,O'O

129 288 288 288 288 129 129 129 129

Pb PbBr2 Pb(CH')4 PbC!' PbF2 FbI 2 FbO PbS PbSO,

164 32,33 373 13, 18, 35, 251 13 32,96,252 207 358 320

Fd FdC!'

164 21, 280

Po

129

Pr PrBr, FrCls FrFs FrI,

164 101, 100, 369, 100,

Pt PtF,

164 403

Pu FuBra PuCls FuF, PUF4 FuF,

164 296 296 296 296 296

Ra

320

Rb RbBr RbCI RbF

164 99 99, 392 99, 304, 344, 365

353 299, 353 379 353

(Continued)

Substance

Reference

RbI RbNO, RbOH

42,99 6, 115,204 38

Re (ReBr,), (ReC!'), ReF, ReF, ReF, Re,O,

164 45 45 51 50,241 241 128, 366

Rh

164

Rn

129

Ru RuF, RuF, RuO,

164 159 61 264

S SF, SF, SO,

129 43 259 320

Sb SbBr, SbC!' SbCh SbF, SbH, Sbl, Sb,O,

164 78, 356 266 279 158

Sc ScBrs ScCls ScF, ScI,

164 320 64 193, 213 320

Se SeF, SeF, SeO,

288 79, 129 129 245

Si SiBn Si(CH,). SiCI, SiF, SiH, SiF,H SiO,

164 44, 322 382 288 288 320 378 320

Sm Sm20,

164 254,270

25

120, 356 288

TRANSITION, FUSION AND VAPORIZATION TABLE

4]-2,

SELECTED REFERENCES

Reference

Substance Sn SnBr2 SnBr. SnCh SnCI, SnF 2 SnR, Snl2 Snl. SnS

164 320 185 122 265, 277 111,414 320 184 180 63, 206

Sr SrBr2 SrCO, SrCI, SrF 2 Srl2 Sr(NO')2 SrO SrSO, SrTiO, SrWO,

164 100, 165 11 100, 171, 224, 273 19, 293, 301 100, 165 203 320 320 95 361

Ta TaBr, TaC]' Ta2O,

164 23 3, 263, 345 400

Tb TbCI. TbF, Tb20, Tb.O,

164 255 369 254 254

Tc TcF, Tc 2O,

216 341 364

Te TeF. Te02

164 179 247, 302, 303, 413

Th ThCI, ThF. Thl. ThO,

164 58 80,301 123 1, 82

Ti TiEr, TiCI, TiF, Til, TiO Ti02

164, 396 147, 185, 322 226, 256, 277, 402 148 202 320 323

TI TlBr TICI TIF TlI TINO, Tl 20 ThO,

164 16, 205, 419 16, 72, 205, 420 188, 419 16, 73, 419 7, 204 76 348

(Continued)

Substance

Ilefer811ce

Tm TmCI, TmF, Tm20,

164 255 369, 415 254

U UBr, UBn

164 134 134

UCI 4 UCI, UF. UF, UFo UI, U0 2

134, 194, 350 172 194, 198, 221 2,409 195, 393 134 153

V VCI. VOC], V,O,

164 277 277, 278 161, 210

W WBr, WOB" WCI, WF6 WF,O WO,

164, 380 346 211 349, 375, 406 50 51 201

Xe XeF, XeF, XeF,

248 338 338 242,404

Y YC]' YF, YI, YeO,

164 94, 255 193, 301, 386 89 270

Yb YbCI, YbF, Yb,O,

39, 164 297 415 254

Zn ZnBr, ZnCh ZnO ZnSO,

164 74, 187 74, 122, 187 320 167

Zr ZrBn ZrC ZrCl, ZrCl, ZrF,

164 330 320 381 87, 88, 269, 285 53, 59, 111, 342 123 320 60, 228, 270

ZrL:I

ZrN Zr02

4-253

4-254

HEAT

References Ackerman, R. J., R. J. Thorn, and P. W. Gilles: J. Am.Chem. Soc. 78, 1767 (1956). Agron, P. A.: U.S. AEC Rept. TID 5290, 1958. Ainscough, J. B., R. J. W. Holt, and F. W. Trowse: J. Chem. Soc. 1967, 1034. Apple, E. F., and T. Wartik: J. Am. Chem. Soc. 80, 6158 (1958). Arell, A.: Ann. Acad. Sci. Fennicae, Ser. A, VI, 100 (1962). Arell, A., and M. Varteva: Ann. Acad. Sci. Fennicae, Ser. A, VI, 88 (1961). Arell, A., and M. Varteva: Ann. Acad. Sci. Fennicae, Ser. A, VI, 98 (1962). ll. Austerman, S. B.: U.S. AEC Rept. NAA-SR-7654, 1963. 9. Babeliowsky, T. P. J. H.: J. Chem. Phys. 38,2035 (1963). 10. Baer, Y., G. Busch, C. Frolich, and ,E. Steigmeier: Z. Naturforsch. 17a, 886 (1962). 11. Baker, E. H.: J. Chem Soc. 1962, 2525. 12. Balk, P., and D. Dong: J. Phys. Chem. 68, 960 (1964). 13. Banashek, E. 1., N. N. Patsakova, and 1. S. Rassonskaya: Izvest. Sektora Fiz.-Khim. Anal. Akad. Nauk S.S.S.R. 27, 223 (1956). 14. Barber, W. F., C. F. Boynton, and P. E. Gallagher: PB Rept. 148374, 1959. 15. Barrall, E. M., and L. B. Rogers: Anal.Chem. 36, 1405 (1964). 16. Barrow, R. F., E. A. N. S. Jeffries, and M. Swinstead: Trans. Faraday Soc. 61, 1650 (1955). 17. Bartlett, N., and N. K. Jha: J. Chem. Soc. A1968, 536. 18. Barton, J. L., and H. Bloom: J. Phys. Chem. 60, 1413 (1956). 19. Bautista, R. G., and J. L. Margrave: J. Phys. Chem. 69, 1770 (1965). 20. Bear, 1. J., and A. G. Turnbull: Australian J. Chem. 19, 751 (1966). 21. Bell, W. E., V. Merten, and M. Tagami: J. Phys. Chem. 65, 510 (1961). 22. Belyaev, 1. U., and N. N. Chikova: Zhur. Neor(j. Khim. 8, 1442 (1963). 23. Berdonosov, S. S., A. V. Lapitskii, and E. K. Bakov: Zhur. Neor(j. Khim. 10, 322 (1965). 24. Berger, C., M. Richard, and L. Eyrand: Bull. soc. chim. France 5, 1491 (1965). 25. Berka, L., T. Briggs, M. Millard, and W. L. Jolly: J. Inor(j. Nuclear Chem. 14, 190 (1960). 26. Berkowitz, J., and J. R. Marquart: J. Chem. Phys. 37, 1853 (1962). 27. Besenbruch, G., T. V. Charles, K. F. 2mbov, and J. L. Margrave: J. Less-Common Metals 12, 335 (1967). 28. Besenbruch, G., A. S. Kana'an, and J. L. Margrave: J. Phys. Chem. 69, 3174 (1965). 29. Binford, J. S., J. M. Strohmeyer, and T. H. Herbert: J. Phys. Chem. 71, 2404 (1967). 30. Blackburn, P. E., and A. Buchler: J. Phys. Chem. 69,4250 (1965). 31. Blanc, M.: Compt. rend. 2'7, 273 (1958). 32. Blanc, M., and G. Petet: Compt. rend. 248, 1305 (1959). 33. Bloom, H., J. O'M. Bockris, N. E. Richards, and R. G. Taylor: J. Am. Chem. Soc. 80, 2044 (1958). 34. Bloom, H., and J. W. Hastie: Australian J. Chem. 21, 583 (1968). 35. Bloom, H., and S. B. Tricklebank: Australian J. Chem.19, 187 (1966). 36. Bloom, H., and B. J. Welsh: J. Phys. Chem. 62, 1594 (1958). 37. Blue, G. D., J. W. Green, R. G. Bautista, and J. L. Margrave: J. Phys. Chem. 67,877 (1963). 38. Bogart, D.: J. Phys. Chem. 58, 1168 (1954). 39. Bohdansky, J., and H. E. J. Schins: J. Less-Common Metals 12, 248 (1967). 40. Brady, A. P., O. E. Myers, and J. K. Clauss: J. Phys. Chem. 64, 588 (1960). 41. Bredig, M. A.: J. Am. Ceram. Soc. 33, 188 (1950). 42. Bridgers, H. E.: Thesis, Ohio State University, 1953. 43. Brown, F., and P. L. Robinson: J. Chem. Soc. 1966, 3147. 44. Brown, H. C., and W. J. Wallace: J. Am. Chem. Soc. 75, 6279 (1953). 45. Buchler, A., P. E. Blackburn, and J. L. Stauffer: J. Phys. Chem. 70, 685 (1966). 46. Buchler, A., E. P. Marram, and J. L. Stauffer; J. Phys. Chem. 71,4139 (1967). 47. Burbank, R. D., and F. R. Bensey: J. Chem. Phys. 27, 981 (1957). 48. Burns, R. P.: J. Chem. Phys. 44,3307 (1966). 49. Busey, R. H., and W. F. Giauque: J. Am. Chem. Soc. 74,4443 (1952). 50. Cady, G. H., and G. B. Hargreaves: J. Chem. Soc. 1961, 1563. 51. Cady, G. H., and G. B. Hargreaves: J. Chem. Soc. 1961, 1568. 52. Calder, G. V., and W. F. Giauque: J. Phys. Chem. 69, 2443 (1965). 53. Cantor, S., R. F. Newton, W. R. Grimes, and F. F. Blankenship: J. Phys. Chem. 62, 96 (1958). 54. Carrera, N. J., R. F. Walker, and E. R. Plante: J. Research NBS 68A, 325 (1964). 55. Catalano, E., and J. W. Stout: J. Chem. Phys. 23, 1284 (1955). 56. Catalano, E., and J. W. Stout: J. Chem. Phys. 23, 1803 (1955). 1. 2. 3. 4. 5. 6. 7.

TRANSITION, FUSION AND VAPORIZATION

4-2;} 5

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TRANSITION, FUSION AND VAPORIZATION 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225.

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317. 318. 319. 320. 321. 322. 323. 324. 325. 326. 327. 328. 329. 330. 331. 332. 333. 334. 335. 336. 337. 338.

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4-260

HEAT

Schrier, E. E., and H. M. Clark: J. Phys. Chem. 67, 1259 (1963). Schultz, D. A., and A. W. Searcy: J. Phys. Chem. 67, 103 (1963). Selig, H., and J. G. MaIm: J. Inorg. & Nuclear Chem. 24, 641 (1962). Sense, K. A., M. J. Snyder, and R. B. Filbert, Jr.: J. Phys. Chem. 58,995 (1954). Sense, K. A., and R. W. Stone: J. Phys. Chem. 62,453 (1958). Sense, K. A., and R. W. Stone: J. Phys. Chem. 62, 1411 (1958). Shchukarev, S. A., and A. R. Kurbanov: Vestnik Leningrad Univ. 17, Fiz. Khim. 144 (1962). 346. Shchukarev, S. A., G. I. Novikov, and G. A. Kokovin: Zhur. Neorg. Khim. 4, 2185 (1959) . 347. Shchukarev, S. A., G. A. Semenov, and K. E. Frantseva: Zhur. Neorg. Khim. 11,233 (1966). 348. Shchukarev, S. A., G. A. Semenov, and I. A. Ratkovskii: Zhur. Neorg. Khim. 6,2817 (1961). 349. Shchukarev, S. A., and A. V. Suvorov: Vesinik Leningrad Univ. 16,87 (1961). 350. Shchukarev, S. A., I. V. Vasil'kova, A. I. Efimov, and V. P. Kerdyashev: Zhur. Neorg. Khim. 1,2272 (1956). 351. Shelton, P. A. J.: Trans. Faraday Soc. 57, 2113 (1961). 352. Sherman, R. H., and W. F. Giauque: J. Am. Chem. Soc. 77,2154 (1955). 353. Shimazaki, E., and K. Niwa: Z. anorg. allgem. Chem. 314, 21 (1962). 354. Shirane, G., and A. Takeda: J. Phys. Soc. Japan 7, 1 (1952). 355. Shomate, C. H., and A. J. Cohen: J. Am. Chem. Soc. 77,285 (1955). 356. Sime, R. J.: J. Phys. Chem. 67, 501 (1963). 357. Sime,_R. J., and N. W. Gregory: J. Am. Chem. Soc. 82, 93 (1960). 358. Simpson, D. R.: Econ. Geol. 59, 150 (1964). 359 .. Singh, J., and G. C. Benson: Can. J. Chem. 46, 1249 (1968). 360. Singleton, E. J., L. Carpenter, and R. V. Lundquist: U.S. Bur. Mine8 Rept. Inve8t. 5938,1962. 361. Smirnova, I. N., and I. P. Kislyakov: Izvest. Akad. Nauk S.S.S.R. Neorg. Mater, 1, 1162 (1965). 362. Smith, F. J.,and R. F. Barrow: Trans. Faraday Soc. 1i4, 826 (1958). 363. Smith, D. K., C. F. Cline, and V. D. Frechetti: J. Nuclear Mater. 6,265 (1962). 364. Smith, W. T., Jr., J. W. Cobble, and G. E. Boyd: J. Am. Chem. Soc. 75,5773 (1953). 365. Smith,D. F., C. E. Kaylor, G. E. Walden, A. R. Taylor, and J. B. Gayle: U.S. Bur. Mines Rept. Invest. 5832, 1961. 366. Smith, W. T., L. E. Line, and W. A. Bell: J. Am. Chem. Soc. 74,4964 (1952). 367. Sokolov, V. A., and N. E. Schmidt: Izvest. Sektora. Fiz.-Khim. Anal. Inst. Obshch. i. Neorg. Khim. Akad. Nauk S.S.S.R. 27, 217 (1956). 368. Sommer, A.: Thesis, Ohio State University, 1962. 369. Spedding, F. H., and A. H. Daane: Iowa State Coli. Rept. IS-902, 1957. 370. Speranskaya, E. I., and A. A. Arshakuni: Zhur. Neorg. Khim. 9,414 (1964). 371. Spice, J. E., L. A. K. Staveley, and G. A. Harrow: J. Chem. Soc. 1955,100. 372. Statton, W. 0.: J. Chem. Phys. 19,33 (1951). 373. Staveley, L. A. K., J. B. Warren, H. P. Paget, and D. J. Dowrick: J. Chem. Soc. 1954, 1992. 374. Stevenson, F. D., and C. E. Wicks: U.S. Bur. Mines Rept. Invest. 6212, 1963. 375. Stevenson, F. D., C. E. Wicks, and F. E. Block: U.S. Bur. Mines Rept. Inve8t. 6367, 1964. 376. Stevenson, F. D., C. E. Wicks, and F. E. Block: J. Chem. Eng. Data 10, 33 (1965). 377. Stull, D. R., ed: JANAF Thermochemical Tables, PB Rept. 168370, 1965. 378. Sujishi, S., and S. Witz: J. Am. Chem. Soc. 79,2447 (1957). 379. Stivorov, A. V., E. V. Krzhizhanovskaya, and G. I. Novikov: Zhur. Neorg. Khim. 11, 2685 (1966). 380. Swarc, R., E. R. Plante, and J. J. Diamond: J. Research NBS 69A, 417 (1965). 381. Swaroop, B., and S. N. Flengas: Can. J. Chem. 44, 199 (1966). 382. Tannenbaum, S., S. Kaye, and G. F. Lewenz: J. Am. Chem. Soc. 75,3753 (1953). 383. Tavernier, B. H., J. Vervechev, P. Messieu, and lYI. Baiwir: Z. anorg. allgem. Chem. 356, 77 (1967). 384. Taylor, A. R., and T. E. Gardner: U.S. Bur. Mines Rept. Invest. 6664, 1965. 385. Thiloaud, E., and P. Flogel: Z. anorg. allgem. Chem. 329, 244 (1964). 386. Thomas, R. E., C. F. Weaver, H. A. Friedman, H. Imsley, L. A. Harris, and H. A. Yokel, Jr.: J. Phys. Chem. 65, 1096 (1961). 387. Thonstad, J.: Can. J. Chem. 42, 2739 (1964). 388. Todd, S. S., and K. R. Bonnickson: J. Am. Chem. Soc. 73, 3894 (1951). 389. To"dd, S. S., and R. E. Lorenson: J. Am. Chem. Soc. 74, 2043 (1952). 390. Topol, L. E., S. W. Mayer, and L. D. Ransom: J. Phys. Chem. 64, 862 (1960). 391. Topol, L. E., and L. D. Ransom: J. Phys. Chem. 64, 1339 (1960). 339. 340. 341. 342. 343. 344. 345.

TRANSITION, FUSION AND VAPORIZATION

4-261

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4k. Vapor Pressure DANIEL R. STULL

The Dow Chemical Company W¥!'"

Tables 4k-l to 4k-4, Vapor Pressures of Inorganic and Organic Compounds, were compiled by the author at The Dow Chemical Company and were published in Ind. Eng. Chem. 39(4), 517 (April, 1947), and 30(12), 1684 (December, 1947). A much more extensive list and references can be found in this journal. The numbers represent temperatures in degrees Celsius at which the vapor pressure is the value appearing at the top of the column. Symbols d decomposes d dextrorotatory dl inactive (50 % d and 50 % I) e explodes 1 levorotatory

M.P. P, p s T,

melting point critical pressure polymerizes solid critical temperature

TABLE

4k-1.

VAPOR PRESSURE OF INO:"GANIC COMPOUNDS-PRESSURES LESS THAN

1

~

ATMOSPHERE

0;,

Temp., °0 Formula

M.P. 1 mm

A B,H12 ....... A Bra. ........ A Ck ......... Al Fa. ......... A I, ........... A ;lOa .......... N ff, .......... N D, .......... N ff,N, ........ N E!,Br ........ N ff,CO,NH2 ... N E!,Cl. ....... N 'i,HS ........ N E!4l ......... N 'i,CN ...... Sb Bra ........ . Sb C!' ......... Sb Clo ......... Sb I, ......... " Sb 20 3 . . . . . . . . . A Bra ......... A Cr, ......... A. F3 .......... A. F, .......... A H, .......... A 20 3 . . . . . . . . . B B,H, ....... B Bn ......... B Cb ......... B I, .......... B Br~ ......... B Cl, .......... B l,CO ........ B ~r8 .......... B JI 3 • . . • • • • • • . B ~~ ...........

t-:J

Name

Aluminum borohydride Aluminum bromide Aluminum chloride Aluminum fluoride Aluminum iodide Aluminum oxide Ammonia Deutero ammonia Ammonium azide Ammonium bromide Ammonium carbamate Ammonium chloride Ammonium hydrogen sulfide Ammonium iodide Ammonium cyanide Antimony tribromide Antimony trichloride Antimony pentachloride Antimony triiodide Antimony trioxide Arsenic tribromide Arsenic trichloride Arsenic trifluoride Arsenic pentafluoride Arsenic hydride (arsine) Arsenic trioxide Beryllium borohydride Beryllium bromide Beryllium chloride Beryllium iodide Bismuth tribromide Bismuth trichloride Borine carbonyl Boron tribromide Boron trichloride Boron trifluoride

,

5mm

20mm

40mm

60mm

-

-

81 3, 100.0, 1238 178.0, 2148 -109.1,

52.2 103.8 116.4, 1298 207.7 2306 - 97.5,

42.9 118.0 123.8, 1324 225.8 2385 - 91.9,

32.5 134.0 131. 8, 1350 244.2 2465 - 85.8,

20.9 150.6 139.9, 1378 265.0 2549 - 79.2,

29.2, 198.3, 26.1, 160.4, 51.1 210.9, 50.5, 93.9 49.2, 22.7 163.5, 574, 41.8 11.4

49.4, 234.5, - 10.4, 193.8, - 36.0 247.0, - 35.7, 126.0 71.4, 48.6 203.8 626, 70.5 11.4

59.2, 252.0, 2.9, 209.8, - 28.7 253.5, - 28.6, 142.7 85.2 61.8 223.5 666 85.2 23.5

+

69.4, 270.6, 5.3, 226.1, - 20.8 282.8, - 20.9, 158.3 100.6 75.8 244.8 729 101.3 36.0

-108.0, -130.8, 242.6, 19.8, 325, 328, 322, 251 242 -127.3 - 20.4 - 75.2 -145.4,

-103.1, -124.7, 259.7, 28.1, 342, 346, 341, 282 264 -121.1 - 10.1 - 65.9 -141.3,

80.1, 290.0, 14.0, 245.0, - 12.3 302.8, - 12.6, 177.4 117.8 91.0 267.8 812 118.7 50.0 2.5 - 92.4, -110.2 299.2, 46.2, 379, 384, 382, 327 311 -106.6 14.0 - 47.8 -131.0,

,

-

-

,

-117.9, -142.6, 212.5, 1.0, 289, 291, 283,

+

,

-139.2, - 41.4 - 91.5 -154.6,

-

10 mm

,

+

,

-

,

,

,

+

,

- 98.0, -117.7, 279.2, 35.8, 361, 365, 361, 305 287 -114.1 1.5 - 57.9 -136.4,

+

,

100mm

200 mm

400 mm

760 mm

-

-

+

+

320.0, 26.7, 271.5, 0.0 331.8, - 0.5, 203.5 148.3 114.1 303.5 957 145.2 70.9 13.2 - 84.3, - 98.0 332.5 58.5, 405, 411 411, 360 343 - 95.3 33.5 - 32.4 -123.0

28.1 227.0 171. 6, 1496 354.0 2874 - 45.4 - 45.4 120.4, 370.9, 48.0, 315 5, 21 8 381.0, 20.5, 250.2 192.2 ........ 368.5 1242 193.5 109.7 41.4 - 54.0 - 75.2 412.2 79.7, 451, 461 461, 425 405 - 74.8 70.0 3.6 -108.3

45.9 - 64. 5 256.3 97. 5 180.2, 192. 4 1537 1040 385.5 2977 2050 - 33.6 - 77. 7 - 33.4 - 74. o 133.8, 39B.0, 58.3, 337.8, 520 33.3 404.9, 36 31. 7, 275.0 96. 6 73. 4 219.0 2. 8 ........ 167 401.0 1425 555 220.0 130.4 - 18 56.3 5. 9 - 52.8 - 79. 8 - 52.1 -115. 3 457.2 312. 8 90.0, 123 474, 490 405 487 487, 488 461 218 441 230 - 64.0 -137. o 91. 7 - 45 12.7 -107 -110.7 -125. 8

13.4 161. 7 145.4, 1398 277.8 2599 - 74.3 - 74.0 86.7, 303.8, 19.B, 256.2, 7.0 316.0, - 7.4, 188.1 128.3 101.0 282.5 873 130.0 58.7 4.2 - 88.5, -104.8 310.3, 51. 7, 390, 395, 394, 340 324 -101. 9 22.1 - 41.2 -127.6,

3.9 176.1 152.0, 1422 29'1. 5 2665 - 58.4 - 67.4 95.25

11.2 199.8 151. 8, 1457 322.0 2765 - 57.0 - 57.0 107.7, 345.3, 37.2, 293.2, 10.5 355.8, 9.6, 225.7 165.9

+ +

d

333.8 1085 157.7 89.2 25.7 - 75.5 - 87.2 370.0 69.0, 427, 435 435, 392 372 - 85.5 50.3 - 18.9 -115.9

+

iI1

toJ i» >-3

B,H5 ... : ....... B,BrH5.... : ... BaH5N, .... :: . .' B,Hlo.: ....... B5H, ....... B5Hll ... :.: ... BIOHl4. . . . . . . .

BrF5. CdC], ......... CdF, .......... Cdr, .......... CdO .......... CBr4- ......... CCk ......... CF4 ........... C,O, .......... CS, ........... C,S,. ......... CSSe .......... CO ........... COCl, ......... COSe ......... COS. CCJ,NO, ...... CCIF, ......... C,N, .......... CBrN ......... CCIN ......... CFN .......... CIN. _ CDN .......... CC12F2 ........ CHChF ....... CHCIF, ...... _ CCIsF ......... CsBr .... CsCI. .... CsF ........... Csl. _... ,. -. ... CIF ..... CIF, ......... C),O .... CIO, .......... CI,06 .......... CIzO, ..........

Dihydrodiborane Diborane hydro bromide Triborine triamine Tetrahydrotetraborane Dihydropentaborane Tetrahydropentaborane Dihydrodecaborane BrOlnine pentafluoride Cadmium chloride Cadmium fluoride Cadmium iodide Cadmium oxide Carbon tetra bromide Carbon tetrachloride Carbon tetrafluoride Carbon sub oxide Carbon disulfide Carbon subsulfide Carbon selenosulfide Carbon monoxide Carbonyl chloride Carbonyl selenide Carbonyl sulfide Chloropicrin Chlorotrifluoromethane Cyanogen

Cyanogen bromide Cyanogen chloride Cy&nogen fluoride Cyanogen iodide Deuterocyanic acid Dichlorodifluoromethane Dichlorofluoromethane Chlorodifluorornethane Trichlorofluoromethane Cesium bromide Cesium chloride Cesium fluoride Cesium iodide Chlorine fluoride Chlorine trifluoride Chlorine monoxide Chlorine dioxide Dichlorine hexoxide Chlorine heptoxide

-159.7 - 93.3 63.0, - 90.9

,

-

50.2 60.0, 69.3,

-

,

1385 416 1000,

,

- 50.0, -184.6, - 94.8 - 73.8 14.0 - 47.3 -222.0, - 92.9 -117.1 -132.4 - 25.5 -149.5 - 95.8, - 35.7, - 76.7, -134.4, 25.2, - 68.9, -118.5 - 91.3 -122.8 - 84.3 748 744 712 738

-149.5 - 75.3 - 45.0 - 73.1 - 40.4 - 29.9 SO.8, - 51.0 618 1504 481 1100.

-144.3 - 66.3 - 35.3 - 64.3 - 30.7 - 19.9 90.2, - 41.9 656 1559 512 1149,

-138.5 - 56.4 - 25.0 - 54.8 - 20.0 9.2 100.0 - 32.0 695 1617 546 1200,

- 19.6 -169.3 - 71.0 - 44.7 54.9 - 16.0 -215.0, - 69.3 - 95.0 -113.3 7.8 -134.1 - 76.8, - 10.0, - 53.8, -118.5, 57.7, - 46.7, - 97.8 - 67.5 -103.7 - 59.0 887 884 844 873 -139.0 - 71.8 - 73.1 - 59.0 42.0 - 13,2

8.2 -164.3 - 62.2 - 34.3 69.3 - 4.4 -212.8, - 60.3 - 86.3 -106.0 20.0 -128.5 - 70.1, 1.0, - 46.1, -112.8, 68.6, - 38.8 5 - 90.1 - 586 - 96.5 - 49.7 938 934 893 923 -134.3 - 62.3 - 64.3 - 51.2 54.3 - 2.1

,

-

98.5

- 30.0, -174.1 - 79.0 - 54.3 41.2 - 26.5 -217.2, - 77.0 -102.3 -119.8 3.3 -139.2 - 83.2, - 18.3, - 61.4, -123.8, 47.2, - 54.0, -104.6 - 75.5 -110.2 '--- 67.6 838 837 798 828 -143.4 - 80.4 - 81.6

+ -

7.5 45.3

-

, ,

,

,

30.5 23.8

,

+

,

-131.6 - 45.4 - 13.2 - 44.3 - 8.0 2.7 117.4 - 21.0 736 1673 584 1257, 96.3 4.3 -158.8 - 52.0 - 22.5 85.6 8.6 -210.0, - 50.3 - 70.4 - 98.3 33.8 -121. 9 - 62.7, 8.6, - 37.5, -106.4, 80.3, - 30.1, - 81.6 - 48.8 - 88.6 - 39.0 993 989 947 976 -128.8 - 51.3 - 54.3 - 42.8 68.0 10.3

+

+ +

+

+

-127.2 - 38.2 - 5.8 - 37.4 - 0.4 10.2 127.8 - 14.0 762 1709 608 1295, 106.3 12.3 -155.4 - 45.5 - 15.3 96.0 17.0 -208.1, - 44.0 - 70.2 - 93.0 42.3 -117.3 - 57.9, 14.7, - 32.1, -102.3, 88.0, -

-

24.78

76.1 42.6 83.4 32.3 1026 1023 980 1009 -125.3 - 44.1 - 48.0 - 37.2 76.3 18.2

-120.9 - 29.0 4.0 - 2S.1 9.6 20.1 142.3 - 4.5 797 1759 640 1341, 119.7 23.0 -150.7 - 36.9 - 5.1 109.9 28.3 -205.78 - 35.6 - 6l.7 - 85.9 53.8 -111.7 - 51. 8, 22.6, - 24.9, - 97.0, 97.6, - 17.5, - 68.6 - 33.9 - 76.4 - 23.0 1072 1069 102,; 1055 -120.8 - 34.7 - 39.4 - 29.4

+ +

-111.2 ' - 15.4 18.5 - 14.0 24.6 34.8 163.8 9.9 847 1834 688 1409, 139.7 38.3 -143.6 - 23.3 10.4 130.8p 45.7 -102.3 - 22.3 - 49.8 - 75.0 71.8 -102.5 - 42.6, 33.8, - 14.1, - 89.2, 111.5, - 5.4, - 57.0 - 20.9 - 65.8 - 9.1 1140 1139 1092 1124 -114.4 - 20.7 - 26.5 - 17.8

+

+

87.7

104.7

29.1

44.6

-

99.6 0.0 34.3 0.8 40.8 51.2

+

-

+

86.5 16.3 50.6 16.1 58.1 67.0

d

........

25.7 908 1924 742 1484, 163.5 57.8 -135.5 - 8.9 28.0

40.0 967 2024 796 1559, 189.5 76.7 -127.7 6.3 46.5

p

65.2 -196.3 - 7.6 - 35.6 - 62.7 91.8 - 92.7 - 33.0 46.0, 2.3 - 80.5, 126.1, 10.0 - 43.9 6.2 - 53.6 6.8 1221 1217 1170 1200 -107.0 4.9 - 12.5 4.0 123.8 62.2

+ +

+

........

85.6 -191.3 8.3 - 21.9 - 49.9 111.9 - 81.2 - 21.0 61.5 13.1 - 72.6, 141.1, 26.2 - 29.8 8.9 - 40.8 23.7 1300 1300 1251 1280 -100.5 11.5 2.2 11.1 142.0 78.8

+

+ +

+ + +

-169 -104. 2 - 58. 2 -119. 9 - 47. 0 -

99. 6 61. 4 568 520 385

90. 1 - 22. 6 -183. 7 -107 -110. 8 O. 4 - 75. 2 -205. 0 -104

+

-138. 8 - 64

'd

0

~

'd

~

I:tJ

-

-

34. 4 58 6. 5

-

12

U2 U2

q

!:d I:tJ

-135 -160 636 646 683 621 -145 - 83 -116 - 59 3.5 -91

t

tv

0;,

IX!

TABLE

4k-1.

VAPOR PRESSURE OF INORGANIC COMPOUNDS-PRESSURES LESS THAN

1

ATMOSPHERE

(Continued)

~

0)

Temp.,oC

Formula

~

M.P.

Name

1 mm

10 mm

20 mm

53.5 58.0 3.2

64.0 68.3 13.8

75.3 79.5 25.7

666 645 610 221. 8, ........ -186.6 67.8, -151.0 43.3 - 24.9 - 22.3 - 54.6 - 69.8 - 12.8

86.3 718 702 656 235.5, 7'00 -182.3 76.5, -145.3 56.8 - 15.0 - 13.0 - 45.2 - 60.1 - 0.9

5 mm

40 mm

60 mm

100 mm

200 mm

400 mm

760 mm

87.6 91.2 38.5 770 11.0 121.5 844 838 786 256 8, 779 -173.0 107.5 -131. 6 88.1 8.0 8.8 - 23.4 - 38.2 26.3 -258.2 -108 3, -123 8, - 30.9, - 45.0 - 85.6, 77.0 - 84.7, -102.3, 6.0 - 59.1, 64.6 32.2 - 45.3, 30.3 686 725

95.2 98.3 46.7 801 18.5 133.2 887 886 836 263.7, 805 -170.0 118.0 -126.7 98.8 16.2 16.2 - 16.2 - 30.7 35.5 -257.6 -103.8, -119.6, - 25.1, - 37.9 - 79.8, 85.8 - 80.2, - 97.9, 12.8 - 53.7, 70.0 40.0 - 39.4, 39.1 711 750

105.3 108.0 58.0 843 29.0 148.5 951 960 907 272.5, 842 -165.8 132.0 -120.3 113.2 27.5 26.5 - 6.3 - 20.3 47.9 -156.6 - 97.7, -114.0 - 17.8, - 28.2 - 72.1, 97.9 - 74.2, - 91.6, 22.0 - 45.7, 77.5 50.0 - 31.9, 50.3 745 784

120.0 121.8 75.2 904 44.4 172.2 1052 1077 1018 285.0, 897 -159.0 152.8 -111.2 135.4 44.4 41.6 8.8 4.7 67.0 -255.0 - 88.1, -105.2 - 5.3 - 13.2 - 60.3, 116.5 - 65.2, - 82.3 35.3 - 32.4 87.9 65.4 - 20.7, 68.0 796 833

136.1 137.2 95.2 974 62.0 198.0 1189 1249 1158 298.0, 961 -151.9 176.3 -100.2 161.6 63.8 58.3d 26.0 13.3 88.6 -253.0 - 78.0 - 95.3 10.2 2.5 - 48.3 137.4d - 53.6 - 71.8 49.6 - 17.2 99.2 81.2 - 8.3, 86.1 856 893

151.0d 151.0 117.1 1050 80.0 225.0 1355 1490 1336 319.0 1026 -144.6 200.0 - 88.9 189.0 84.0 75.0d 44.0 31.5 110.8 -251.0 - 66.5 - 84.8 25.9 19.7 - 35.1 158.0d - 41.1 - 60.4 64.0 2.0 110.0 97.0 4.0, 105.0 914 954

---

HSO,Cl. ...... Cr(CO)' ....... CrO,CIz ....... COCIz ......... Co(OO),NO .... CbF........... Cu2Br ........ . Cu,CIz ........ Cu,1, ......... FeCb ......... FeCI, ......... F,O .. '" ...... GaCh ......... GeR •......... GeB" ......... GeCI .......... GeHOb ....... Ge(CH,).. ....' Ge,H6 ......... Ge,Hs ....... :. HD ........... HBr .......... HCI. .......... HON .......... liF ........ : .. HI. .......... '. H,O, ........ :. H,Se .......... H,S ........ '::. H,S, .......... H,Te ........... NH,OH ....... IF, ........... IF, ........... Fe(CO)' ....... PbBn ......... PbC!,. ........

Chlorosulfonic acid Chromiulll carbonyl Chromyl chloride Cobaltous chloride Cobalt nitrosyl tricarbonyl Columbium pentafLuoride Cuprous bromide Cuprous chloride Cuprous iodide Ferric chloride Ferrous chloride Fluorine monoxide Gallium trichloride Germanium hydride Gerrnanium bromide Germanium chloride Trichlorogermane Tetramethylgermanium Digermane Trigermane Hydrogen deuteride Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen fluoride Hydrogen iodide Hydrogen peroxide Hydrogen selenide Hydrogen sulfide Hydrogen disulfide Hydrogen teluride Hydroxylamine Iodine pentaftuoride Iodine heptafluoride Iron pentacarbonyl Lead bromide Lead chloride

-

32.0 36.0 18.4

, ,

+

,

572 546

,

194.0, ........

-196.1 48.0, -163.0

,

-

45.0 41.3 73.2 88.7 36.9

, , ,

,

,

........

........

-'~59.8

-138.8, -150.8, - 71.0,

-127.4, -140.7, - 55.3, - 74.7 -109.6, 38.8 -103.4, -122.4, - 24.4 - 82.4, 39.0 1.5, - 70.7, - 6.5 578 615

-121.8, -135.6, - 47.7, - 65.8 -102.3, 50.4 - 97.9, -116.3, - 15.2 - 75.4, 47.2 8.5 - 63.0, 4.6 610 648

,

-123.3, 15.3 -115.3, -134.3, - 43.2 - 96.4,

,

- 15.2, - 87.0, ........ 513 547

+

+

,

-

1.3 103.0 777 766 716 246.0, 737 -177.8 91.3 -139.2 71. 8 - 4.1 - 3.0 - 35.0 - 49.9 11.8 -259.1 -115.4, -130.0, - 39.7, - 56.0 - 94.5, 63.3 - 91.8, -109.7, 5.1 - 67.8, 55.8 20.0 - 54.5, 16.7 646 684

+

+

+ +

+

+

+ + +

+

-

80

735 -11 75.5 504 422 605 304 -223.9 77.0 -165 26.1 - 49.5 - 71.1 - 88 -109 -105.6 - 87.0 -114.3 - 13.2 - 83.7 - 50.9 0.9

-64 -

85.5 89.7 49.0 34.0 8.0 5.5 - 21 373 501

~

i:'j ~

>-3

PbF, .......... PbI,. ......... PbO .......... PbS ........... LiBr .......... LiCI. .......... LiF ........... LiI. ........... MgC\, ......... MllC\, ........ HgBn ......... HgC\, ......... HgI, .......... MoF6 ...... MoO, ......... NiCb ......... Ni(CO), ....... NF, ........... NO ........... N,O ........... N,O, .......... N,O ........... NOCI. ........ NOF .......... NO,F ......... OsO, .......... Os04. .........

0, ............ PBr' .......... PCI, .......... PCl. .......... PH, ........... PH,Br ........ PH,CI. ........ PH,I. ......... P,O, .......... POC!,. ... P,O, .......... P,O, .......... PSB" ........ PSCI, ........ KBr ........

Lead fluoride Lead iodide Lead oxide Lead sulfide Lithium bromide Lithium chloride Lithium fluoride Lithium iodide Magnesium chloride Manganous chloride Mercuric bromide Mercuric chloride Mercuric iodide Molybdenum hexafluoride Molybdenum trioxide Nickel chloride Nickel carbonyl Nitrogen trifluoride Nitric oxide Nitrous oxide Nitrogen tetroxide Nitrogen pentoxide Nitrosyl chloride Nitrosyl fluoride Nitroxyl fluoride Osmium tetroxide (white) Osmium tetroxide (yellow) Ozone Phosphorous tribromide Phosphorous trichloride Phosphorous pentachloride Phosphorous hydride (phosphene) Phosphonium bromide Phosphonium chloride Phosphonium iodide Phosphorous trioxide Phosphor'ous oxychloride Phosphorous pentoxide (stable form) Phosphorous pentoxide (metastable form) Phosphorous thiobromide Phosphorous thiochloride Potassium bromide

136.5, 136.2, 157.5, - 65.5, 734, 671,

861 540 1039 928, 840 880 1156 802 877 736 165 3, 166.0, 189.2, - 49.0, 785, 731,

904 571 1085 975, 888 932 1211 841 930 778 179.8, 180.2, 204.5, - 40.8, S14 759,

950 605 1134 1005, 939 987 1270 883 988 825 194.3, 195.8, 220.0, - 32.0, 851 789,

-184.5, -143.4, - 55.6, - 36.8,

-175.5 -180.6, -133.4, - 42.7, - 23.0,

-170.7 -178.2, -128.7, - 36.7, - 16.7,

-165.7 -175.3, -124.0, - 30.4, - 10.0,

-120.3 -132.1 15.6, 22.0, -168.6 34.4 31.5 74.0,

-114.3 -126.2 26.0, 31.3, -163.2 47.S 21.3 83 25

-107.8 -119.8 37.4, 41.0, -157.2 62.4 10.2 92 5,

28.5, 79.6, 9.0, 39.7

21.2, 74.0, 1.1, 53.0 2.0

13 3, 68 0, 7.3, 67.8 13.6

479 943 852. 748 783 1047 723 778

-132.0 -143.7, 5 6, 3.2, -180.4 7.8 51.6 55.5, 43.7, 91.0, 25.2,

+

+

384,

424,

442,

462,

189 50.0 - 18.3 795

220

236 -83:6 16.1 940

253 95.5 29.0 994

72.4

+ 8924.6

1003 644 1189 1048, 994 1045 1333 927 1050 879 211. 5, 212.5, 238.2, - 22.1, 892 821, - 23.0 -160.2 -171.7, -118.3, - 23.9, 2.9, - 60.2 -100.3 -112.8 50.5 51. 7s -150.7 79.0 2.3 102.5,

+

-129.4 5.0, 61.5, 16.1, 84.0 27.3 481, 270 108.0 42.7 1050

1144 750 1330 1160 1147 1203 1503 1049 1223 1028 262.7 256.5, 291.0 4.1, 1014 904, 8.8 -145.2 -162.3, -103.6, 5.0 15.6, 34.0 79.2 93 ..5 89.5 89.5 -132.6 125.2 37.6 131. 3,

1036 668 1222 1074, 1028 1081 1372 955 1088 913 221.0, 222.2, 249.0, - 16.2, 917 840, - 15.9 -156.5 -168.9, -114.9, 19.9, 1.8, 54.3 95.7 -108.4 59.4 59.4 -,146.7 89.8 10.2 10S.3,

lOS0 701 1265 1108, 1076 1129 1425 993 1142 !l60 237.8 237.0, 261.8 8.0, 955 866. 6.0 -152.3 -166.0, -110.3, 14.7, 7.4, 46.3 88.8 -102.3 71.5 71.5 -141. 0 103.6 21.0 117.0,

-125.0 0.3, 57.3, 21.9, 94.2 35.8

-118.8 7.4, - 52.0, 29.3, 108.3 47.4

-109.4 17.6, - 44.0, 39.9, 129.0 65.0

510 s

294 126.3 63.8 1137

+

+

493, 280 116.0

51.8 1087

+ +

1219 807 1402 1221 1226 1290 1591 1110 1316 1108 290.0 275.5, 324.2 17.2 1082 945, 25.8 -137.4 -156.8, 96.2, 8.0 24.4, 20.3 68.2 83.2 109.3 109.3 -122.5 149.7 56.9 147.2,

+

1293 872 1472 1281 1310 1382 1681 1171 1418 1190 319.0 304.0 354.0 36.0 1151 987, 42.5 -129.0 -151. 7 88.5 21.0 32.4 6.4 56.0 72.0 130.0 130.0 -111.1 175.3 74.2 162.0,

855 402 890 1114 547 614 870 446 712 650 237 277 259 17 795 1001 - 25 -183. 7 -161 90. 9 9. 3 30 t)4. 5 -134 -139 42 56 -251 - 40 -111. 8

98.3 28.0, 35.4, 51.6, 150.3 84.3

87.5 -132. 5 38.3d 27.0, 28. 5 62.3, 173.1 22. 105.1 2

532,

556,

591

314 141.8 82.0 1212

339 157.8 102.3 1297

358 175.0d 124.0 1383

~"t:I

o

l=d

~

tfo3

CC :a F .......... Cl HB r ............ HC 1. ........... HC N ........... HI ...... 0

••

••

H, ,........... )

•

.

H" ~e ........... Kr ..... . ...... NO N2 ) ............ N,' ) •........... SiF 4. • • • • • • • • • • • • SiC IF a .......... SiC lzF, ......... SiClaF .......... Sn :a 4 . . . . . . . . . . .

SO : ........... . SO I··········· .

Trichlorofluoromethane Chlorine Hydrogen bromide Hydrogen chloride Hydrogen cyanide Hydrogen iodide Hydrogen sulfide Hydrogen selenide Krypton Nitric oxide Nitrous oxide Nitrogen tetroxide Silicon tetrafluoride Chlorotrifluorosilane Dichlorodifluorosilane Fluorotrichlorosilane Stannic chloride Sulfur dioxide Sulfur trioxide

23.7 - 33.8 - 66 ..5 - 84.8 25.9 - 35.1 - 60.4 - 41.1 -1.52.0 -151. 7 - 88.5 21.0 - 94.8, - 70.0 - 31.8 12.2 113.0 - 10.0 44.8

44,.1 77.3 - 16.9 + 10.3 - 51.5 - 29.1 - 71.4 - 50.5 75.8 41i.8 - 18.9 + 7.3 - 4;"5.9 - 22.3 - 2:5.2 0.0 -14:3.5 -130.0 -145.1 -135.7 - 76.8 - 58.0 37.3 59.8 - 84.4 - 67.9 - 57.3 - 37.2 - 15.1 + 11.6 32.4 64.6 141. 3 184.3 + 6.3 32.1 60.0 82.5

108.2 146.7 172.0 194.0 84.8 101.6 35.6 65.0 - 8.4 + 16.8 48.1 33.9 17.8 - 31. 7 - 8.8 + 5.9 102.7 135.0 153.8 169.9 62.2 32.0 83.2 100.7 - 0.4 + 25.5 41. 9 55.8 + 23.4 .50.8 69.7 84.6 -118.0 -101. 7 - 88.8 - 78.4 -127.3 -116.8 -109.0 -103.2 - 40.7 - 18.8 - 4.3 + 8.0 79.4 100.3 112.3 121.4 - 52.6 - 33.4 - 21.2 ." .... - 18.6 + 4.1 19.4 ....... 36.6 66.2 86.0 . " ' " . 94.2 131. 8 156.0 ....... 223.0 270.0 299.8 ....... 83.8 102.6 118.0 55.5 104.0 138.0 167.8 175.0

·..... . 115.2 60.0 27.9 183.5 116.2 66.7 97.2 - 66.5 - 99.0 18.0 127.0 ...... . ...... . ·...... ...... . ·. . . . . . 130.2 187.8

· . . . . . . 198.0 43. 2 127.1 144.0 76. 1 90.0 84. 4 70.6 51.4 81. 6 36.2 ....... 183.5 .50. o 127.5 1.51. 0 82. o 76.3 100.3 88. 9 91. 108.7 137 ....... - 63 .54 - 94.8 - 92.9 64. 27.4 36 . .5 71. 132.2 158 99 ...... . - 14.2 36. ...... . 34.8 34. . ...... 95.8 34. · . . . . . . 165.3 35. · . . . . . . 318.7 37. 141.7 157.2 77. 198.0 218.3 83.

.,.

>-3

,H4Cl, ........ 1,2-Dichloroethane ,H,O, ........ Acetic acid 2H402.· ... -.... Methyl formate ,H,Br •....... Ethyl bromide ,H,Cl..· ...... Ethyl chloride ,H,F ......... Ethyl fluoride zH 6 .·.·.'.·.- . . . . . Ethane .c ,H,O.·....... Ethanol .C ,H,O .. ·...... Dimethyl ether ,C ,H,S .... ·...... Dimethyl sulfide C ,H,S.,'.· ...... Ethanethiol Ethylamine C ,H7N .. ,'.· .. ,C ,H7N.·.·.·.· .. ·.. ·Dimethylamine C ,N, .. ,' ......... Cyanogen ,0 aH4 .. '.. '.'.'.'.'.'. Propadiene (0 3,H.L ... ·.'.'.·.·.· .. Propyne C ,H,N,Og ..... Nitroglycerine ,C ~H6 ... ','.', ...... Propylene .C ,H,O ......... Acetone .C ,H,O, ......... Propionic acid .C ,H,O" ........ Methyl acetate C ,H,O, ........ Ethyl formate C 3H8 ............... Propane ,H,O ......... I-Propanol ,H,O ......... 2-Propanol ,HsO ........ 'Eth}Cl methyl ether ,HsO, ........ Glycerol· , ,H,N ... : .... Propylamine ,H,N ......... Trimethylamine ~H2 ....... ; .. 1,3-Butadiyne ~H6 ....... :.'. 1,:2.,.Butadiene ,C 4H6 . . . . . . . :. 1,3-Butadiene C ~H6 ... : .. :; .. Cyclobutene ,C ~H6 ... :: .. .' ... 1-Butyne ,H, .......... 2-Butyne c .H,O., ........ Acetic anhydride ,H,O ......... Dimethyl oxalate 4HsO, ........ Butyric acid ,H,02 ........ Isobutyric acid .j.HS02 ........ Ethyl acetate ,H,O, ........ Methyl propionate

.44.5. 17..2, 74.2 74.3 89.8 -117.0 -159.5 - 31.3 -115.1 75.6 76.7 82.3. 87.7 95.8, ~ 120 ..6 -111.0, 127 ~ 131.9 59.4 4.6 57.2 60.5 -128.9 - 15.0 - 26.1 - 91.0 125.5 64.4 - 97.1 -

82.5 8

- 89.0 -102.8 99.1 92.5 73.0, 1.7 20.0 25.5 14.7 43.4 42.0

24.0 6.3. 57.0 56.4 73. 9 -103.8 -148.5 - 12.0 -101.1 58.0 59.1 66.4 72.2 83.2, -108.0 - 97.5 157 -120.7 40.5 2S.0 38.6 42.2 -115.4 5.0 7.0 75.6 153.8 46.3 81.7 68.0. 72.7' 87.6 83.4 76.7 57.9. 24.8 44.0 49.8 39.3 23.5 21.5

+

+

t

i

13.6 17.5 48.6 47.5 65.8 97.7 -142.9 2.3 93.3 49.2 50.2 58.3 64.6 76.8, -101.0 - 90.5 188 ~112.1

31.1 39.7 29.3 33.0 -108.5 14.7 2.4 - 61.8 167.2 37.2 73.8 61.2. 64.2 79.7 75.4 68.7 50.5. 36.0 56.0 61.5 51.2 13.5 1l.8

+

- - - - - - -- - _...

_ ... _-

2.4 29.9 39.2 37..8 56.8 90.0 -136.7 8.0 85.2 39.4 40.7 48.6 56.0 70.1, 93.4 82.9 210 -104.7 20.8 52.0 19.1 22.7 -100.9 25.3 12.7 59.1 182.2 27.,1 65.0 53.8, 54.9 71.0 66.,6 59.9 42.5, 48.3 69.4 74.0 64_0 3.0 1.0

+

+

10.0 43.0 28.7 26.7 47.0 81.8 -129.8 19.0 76.2 28.4 29.8 39.8 46.7 62.7. 85.2 74.3 235 96.5 9.4 65.8 7.9 11.5 92.4 36.4 23.'8 49.4 198 .. 0 16.0 55.2 45.9. 44.3 61.3 56.4 50.0 33.9, 62.1 83.6 88.0 'l7.8 9.1 1l:0

+ +

18.1 51. 7 21.9 19.5 40.6 76.4 -125.4 26.0 70.4 21.4 22.4 33.4 40.7 57.9. 78.8 68.8 251d 91.3 2.0 74.1 0.5 4.3 87.0 43.5 30.5 43.3 20S.0 9.0 48.8 4.1.0. 37.5 55.1 50.0 43.4 27.8 70.8 92.8 96.5 86.3 16.6 18.7

29..4 53:0 12.9 10.0 32.0 69.3 -119.3 34.9 62.7 12.0 13.0 25.1 32 6 51 8, 72.5 61.3 84.1 7.7 85.S 9.4 5.4 79.6 52.8 39.5 34.8 220.1 0.5 40.3 34.0 28.3 46.8 41.2 34.9 18.8 82.2 104.8 108.0 98.0 27.0 29.0

+ + +

+

45.7 80.0 0.8 4.5 18.6 58.0 -110.2 48.4 50.9 2.6 1.5 12.3 20.4 42.6,' 61.3 49.8

+ +

+ +

-,- 73.3 22.7 102.5 24.0 20.0 68.4 66.8 53.0 22.0 240.0 15.0 27.0 20.9 14.2 33.9 27.S 21.6 5.0 100.0 123.3 125.5 115.8 42.0 44.2

+

-

64.0 99.0 16.0 21.0 3.9 45.5 99.7 63.5 37.8 18.7 17.7 2.0 7.1 33.0 48.5 37.2

60.9 39.5 122.0 40.0 37.1 55.6 82.0 67.8 7.8 263.0 31. 5 12.5 6.1 1.8 19.3 12.2 6.9 10.6 119.8 143.3 144.5 134.5 59.3 61.8

+ +

+

82.4 118.1 32.0 38.4 12.3 32.0 88.6 78.4 23.7 36.0 35.0 16.6 7.4 21.0 35.0 23.3

+ -

47.7 56.5 141.1 57.8 54.3 42.1 97.8 82.5 7.5 290.0 48.5 2.9 9.7 18.5 4.5 2.4 8.7 27.2 139.6 163.3 163.5 154.5 77.1 79.8

+ + .+ + +

35.3 16.7' 99.8 -117.8 -139 -183.2 -112 -138.5 - 83.2 -121 - 80.6 - 96 - 34.4 -136 -102.7 11 -185 94.6 22 98.7 79 -187.1 -127 85.8 17.9 83 -117.1 - 34.9

..q ". "d

o ~

"d ~

l'J

l/2 l/2

c:: ~

l'J

-108.9 -130 32.'5 73 4.7 47 82.4 87.5

t

--.:r

i-'

TABL~J

-

4k-3.

VAPOR PRESSURE OF ORGANIC COMPOUNDS-PRESSURES LESS THAN

1

t

(Continued)

ATMOSPHERE

tV "'1 tV

Temp.,oC Formula

Name

M.P. 1 mm

C'HS02 ........ C'HlO ......... C'HlO ......... C,H1OO ........ C'HlOO ........ C,HlOO ....... C'H1OO ........ C'HlOO ........ C'H1OS ........ C4.HuN ........ C4Hl:!Si ........ C'H1OO2 ....... C'HIOO2 ....... C'HlOO2 ....... C'HlOO2 ....... C5HJo02 ....... eliHu ........ . C'HI" ........ C,H" ......... C'HI'O ........ C,H,Br ........ C,H,C!.. ...... C,H,F ........ C,H,I. ........ C,H, .......... C,H,O ........ C,H,N ........ C,H12 ......... C6H14 ......... C,H14 ......... C,H, ......... C'HI O • • • • • • • • • CSHIO......... C,Hls ......... C12Hz 6 . . . . . . . . .

-

Propyl formate Butane 2-Methylpropane Butyl alcohol Me-Butyl alcohol Isobutyl alcohol tert-Butyl alcohol Diethyl ether Diethyl sulfide Diethylamine Tetramethylsilane Ethyl propionate Propyl acetate Methyl butyrate Methyl isobutyrate Isobutyl formate Pentane 2-Methylbutane 2,2-Dimethylpropane Ethyl propyl ether Bromobenzene Chlorobenzene Fluorobenzene Iodobenzene Benzene Phenol Aniline Cyclohexane Hexane 2,3-Dimethylbutane Toluene Heptane Ethylbenzene Octane Dodecane

- 43.0 -101. 5 -109.2 - 1.2 - 12.2 - 9.0 - 20.4, - 74.3 - 39.6

,

- 83.8 - 28.0 - 26.7 - 26.8 - 34.1 - 32.7 - 76.6 - 82.9 -102.0, - 64.3 2.9 - 13.0 - 43.4, 24.1 - 36.7, 40.1, 34.8 - 45.3, - 53.9 - 63.6 - 26.7 - 34.0 - 9.8 - 14.0 47.8

+

5 mm

-

+ + + -

-

-

-

+ +

22.7 85.7 94.1 20.0 7.2 11.0 3.0, 56.9 18.6

,

66.7 7.2 5.4 5.5 13.0 11.4 62.5 65.8 85.4, 45.0 27.8 10.6 22.8 50.6 19.6, 62.5 57.9 25.4, 34.5 44.5 4.4 12.7 13.9 8.3 75.8

10 mm

-

+ -

-

+ + + -

-

+ -

12.6 77.8 86.4 30.2 16.9 21. 7 5.5, 48.1 8.0 33:0 58.0 3.4 5.0 5.0 2.9 0.8 50.1 57.0 76.7, 35.0 40.0 22.2 12.4 64.0 11. 5, 73.8 69.4 15.9, 25.0 34.9 6.4 2.1 25.9 19.2 90.0

20 mm

-

-

1.7 68.9 77.9 41. 5 27.3 32.4 14.3, 38.5 3.5 22.6 48.3 14.3 16.0 16.7 8.4 11.0 40.2 47.3 67.2, 21.0 53.8 35.3 1.2 78.3 2.6, 86.0 82.0 5.0, 14.1 24.1 18.4 9.5 38.6 31. 5 104.6

+ -

+ + -

+

40mm

+ -

-

-

+ + + -

10.8 59.1 68.4 53.4 38.1 44.1 24.5, 27.7 16.1 11.3 37.4 27.2 28.8 29.6 21.0 24.1 29,2 36.5 56.1, 12.0 68.6 49.7 11.5 94.4 7.6 100.1 96.7 6.7 2.3 12.4 31.8 22.3 52.8 45.1 121. 7

60mm

-

-

-

-

-

18.8 52.8 62.4 60.3 45.2 51. 7 31.0 21.8 24.2 4.0 30.3 3,5.1 37.0 37.4 28.9 32.4 22.2 29.6 49.0, 4.0 78.1 58.3 19.6 105.0 15.4 108.4 106.0 14.7 5.4 4.9 40.3 30.6 61. 8 53.8 132.1

+ -

100 mm

-

-

29.5 44.2 54.1 70.1 54.1 61.5 39.8 11.5 35.0 6.0 20.9 45.2 47.8 48.0 39.6 43.4 12.6 20.2 39.1, 6.8 90.8 70.7 30.4 118.3 26.1 121.4 119.9 25.5 15 8 5.4 51.9 41.8 74.1 65.7 146.2

+ -

-

+

+

200 mm

45.3 31.2 41.5 84.3 67.9 75.9 52.7 2.2 51.3 21.0 - 6.5 61.7 64.0 64.3 55.7 60.0 1.9 5.9 - 23.7, 23.3 110.1 89.4 47.2 139.8 42.2 139.0 140.1 42.0 31. 6 21.1 69.5 58.7 92.7 83.6 167.2

-

+

+

400 mm

62.6 16.3 27.1 100.8 83.9 91.4 68.0 17.9 69.7 38.0 10.0 79.8 82.0 83.1 73.6 79.0 - 18.5 10.5 - 7.1 41.6 132.3 110.0 65.7 163.9 60.6 160.0 161.9 60.8 49.6 39.0 89.5 78.0 113.8 104.0 191.0

-

+

+

760 mm

-

81.3 0.5 11.7 117.5 99.5 108.0 82.9 34.6 88.0 55.5 27.0 99.1 101.8 102.3 92.6 9S.2 36.1 27.8 9.5 61. 7 156.2 132.2 84.7 188.6 SO.l 181.9 184.4 80.7 68.7 58.0 110.6 98.4 136.2 125.6 216.2

+

- 92.9 -135 -145 - 79.9 -114.7 -108 25.3 -116.3 - 99.5 - 38.9 -102.1 - 72.6 - 92.5 - 84.7 - 95.3 -129.7 -159.7 - 16.6 -

30.7 45.2 42.1 28.5 5.5 40.6 - 6.2 6.6 - 95.3 -128.2 - 95.0 - 90.6 - 94.9 - 56.8 9.6

+

+

;:q trJ il>

>-3

VAPOR PRESSURE

4-273

Tables 4k-5 to 4k-14, Vapor Pressures of Special Gases, listing values of th~ vapor pressures of He., He" normal and equilibrium H 2,Ne, N 2, and O 2, were taken from Thermometry at Low Temperature, a master's essay at the University of Pittsburgh, 1965, by Edward R. Simco. This booklet is also entitled Research Report 4 and was supported in part by the National Science Foundation. , Table 4k-15, Vapor Pressures of the Chemical Elements, lists values of the vapor pressure, :temperature, and heat associated with the phase transitions for the chemical elements. The numbers represent temperature in degrees Celsius at which the vapor pressure is the value appearing at the top of the column. A circled dot between columns indicates a change of phase. The six columns on the right side list ith~ following iriformation: .

Tm I1B m T, I1B, _

Trans

heat of vaporization at 25°C, or atmospheric boiling temperatUlie if the vahie contains an asterisk (*), cal/mol ' melting temperature heat of melting, cal/mol transition temperature heat of transition, cal/mol designates solid-state transition

Equilibrium vapor pressures are listed for substances with polymorphic' v~por or condensed forms (As, Sb, Bi, P, Po, S, Se, Te). The basic sources should be: consulted for vapor pressures of the various polymorphic forms. The sources for this table are: (1) Ralph Hultgren, RaymondL. Orr, Philip D. Anderson, and Renneth K. Kelley, "Selected Values of Thermodynamic Properties of Metals and Anoys," John Wiley & Sons, Inc., New York, 1963 (updated by privately distributed supplements); (2) Daniel R. Stull and Gerard C. Sinke, "Thermodynamic Properties of the Elements," Advances in Chem: Ser .. No. 18: (3) Richard E.Honig, "Vapor Pressure Data for the Solid and Liquid Elements," RCA Rev. 23 (4), 567-586 .(1962); (4) Richard E. Honig and H. '0. Hook, "Vapor Pressure Data for Some Common Gases," RCA Rev. 21(3), 360-368 (1960). ' Table 4k-16, Vapor Pressure of Ice, has been taken from the NBS Circ. 564, Tables of Thermal 'Properties of Gases, by J. Ril.senrath, C. W. Beckett,W. S. Benedict, L. Fano, H. J. Hoge,' J. F. Masi, R. L. Nuttall, Y. S. Touloukian, and :H. W. Woolley, U.S. Government Printing Office, Washington, D.C., 1955. The: values were smoothed, and adjusted to agree with the ice-point value adopted in Table ~~

,

Table 4k-17, Vapor Pressure of Liquid Water below 100°C, and Table 4k-18; Vapor Pressure of Liquid Water above 100°C, have been taken from the recent ",ork of M. R. Gibson and E. A. Bruges, J. Mech. Eng. Sci., 9(1), 24'-35 (February,: 1967). Table 4k-19, Vapor Pressure of Mercury, is taken from the compilatioljl of J. Johnston, F. Fenwick, and H. G. Leopold, "International Critical Tables," vol. III, McGraw-Hill Book Company, New York, 1928. Table 4k-20, Vapor Pressure of Carbon Dioxide, is from C. H. Meyers and M. S. Van Dusen, J. Research NBS, 10, 409 (1933). Table 4k-21, Vapor Pressure of Ethyl Alcohol, and Table 4k-22, Vapor Pressure of Methyl Alcohol, are reprinted by permission from the "Smithsonian Physical Tables," 9th ed. Smithsonian: Institution, Washington, D.C., 1954. , Table 4k-23, Constants in the Equation for the Rate 6f Evaporation of Metals, is taken by permi~sion from pages 752-754 of "Scientific Foundations of V/Lcuum Technique," by S. Dushman, John Wiley & Sons, Inc., New York, 1949.

!

TABLE 4k-4. VAPOR PRESSURE 'OF ORGANiC COMPOUNDS-PRESSURES GREATE:R THAN 1 ATMOSPHERE

'"'-l ~

Formula OCIFa.... CChF 2 • • • CC1,O .... CClaF .... 001 4 • • • • • OHOIF2.. OH01,F .. OHCla.... OHN ..... CHaBr ... OHaCI. ... OHaF .... CHal .... ; CR •...... CH.O .... CH.S ..... CH.N. '" CO ...... OS2 ...... 02CIFa... 02Cl2F 4 •••

Temp.,oC

Name

Chlorotrifluoromethane Dichlorodifluoromethane Carbonyl chloride Trichlorofluoromethane Carbon tetrachloride Chlorodifluoromethane Dichlorofluoromethane Trichloromethane Hydrocyanic acid Methyl bromide Methyl chloride Methyl fluoride Methyl iodide Methane Methanol Methanethiol Methylamine Carbon monoxide Carbon disulfide l-Chloro-l ,2,2-trifluoroethylene 1,2-Dichloro-l,I,2,2tetrafluoroethane C 2ClaFa... 1,1,2-Trichloro-1,2,2trifluoroethane

1 atm 81.2 29.8 8.3 23.7 76.7 - 40.8 8.9 61.3 25.8 3.6 - 24.0 - 78.2 42.4 -161.5 64.7 6.8 - 6.3 -191.3 46.5 - 27.9 3.5 -

47.6

2.atm - 66.7 - 12.2 27.3 44.1 102.0 - 24.7 28.4 83.9 45.5 23.3 - 6.4 - 64.5 65.5 -152.3 84.0 26.1 10.1 -183.5 69.1 - 11.1 22.8

+

70.0

5 atm 42.7 16.1 57.2 77.3 141.7 0.3 59.0 120.0 75.5 54.8 22.0 - 42.0 101.8 -138.3 112.5 55.9 36.0 -170.7 104.8 15.5 54.0 -

+ +

+

+

105.5

Tc

10 atm 20 atm 30 atm 40 atm - 18.5 34.8 52.8 12.0 42.4 74.0 95.6 . . . . . . . 85.0 119.0 141.8 159.8 108.2 146.7 172.0 194.0 178.0 222.0 251.2 276.0 24.0 52.0 70.:3 85.3 87.0 121.2 144.0 162.6 152.3 191.8 216 ..5 237.5 103.5 134.2 154.0 170.2 84.0 121.7 147.5 170.2 47.3 77.3 97.5 113.8 - 21.0 2.6 15.5 26.5 138.0 176.5 206.0 228.5 -124.8 -108.5 - 96.3 - 86.3 138.0 167.8 186.5 203.5 83.4 117.5 140.0 157.7 59.5 87.8 106.3 i21.8 -161.0 -149.7 -141.9 ....... 136.3 175.5 201.5 222.8 40.0 71.1 91.9 82.3 117.5 140.9 .......

50 atm 60 atm

+

· . . . . . . ....... 174.0 · . . . . . . · . . . . . . ....... ....... ...... . · . . . . . . ....... 177.5 · . . . . . . 254.0 ....... 183.5 · . . . . . . 190.0 ....... 126.0 137.5 43.5 36.0 248.0 ...... .

+

....

.

138.0

177.7

205.0

0

•••

•• "G ••

· . . . .. . · . . . . . . 214.0 172.0 133.7

224.0 185.0 144.6

· . . . . . . ....... 240.0

256.0

...... . ...... . ...... . ••••

•

0

0-

•••

••

0

•

......

.

Pc

----

53 40. 111.5 39. 181.7 56. 198.0 43. 283.1 45. 96 48. 178.5 51. 260 54. 183.5 50. 194 51. II: t 44.9 62. 1-'3 255 54 . - 82.1 45. 8 240.0 78. 7 1fJ6.8 71. 4 156.9 73. 6 -138.7 34. 6 273.0 72. 9 107.0 39 . o 145.7 32. 3 214.1 33 . 7

02H2..... 02H20b " D2H.Ob .. O.H•..... O.H.Br2 .. 02H.0l2 .. 02H.012 .. 02H.02... 02H.02... 02H sBr ... OoH.Ol. .. 02H•F .... OoH •..... OoH.O .... O.H.O .... 02H .S .... 02H.S .... 02H 7N ... OoH 7N ... OoNo..... O.H •..... O.H •..... O.H •..... O.H.O .... O.H.02...

Acetylene cis-l,2-Dichloroethylene trans-l,2,Dichloroethylene Ethylene 1,2-Dibromoethane 1,I-Dichloroethane 1,2-Dichloroethane Acetic acid Methyl formate Ethyl bromide Ethyl chloride Ethyl fluoride Ethane Ethanol Dimethyl ether Ethanethiol Dimethyl sulfide Ethylamine Dimethylamine Oyanogen Propadiene Propyne Propylene Acetone Propionic acid

- 84.0, 59.0 47.8 -103.7 131.5 57.3 83.7 118.1 32.0 38.4 12.3 - 32.0 - 88.6 78.4 - 23.7 35.0 36.0 16.6 7.4 - 21.0 - 35.0 - 23.3 - 47.7 56.5 141.1

- 71.6 - 50.2 - 32.7 - 10.0 + 4.8 82.1 119.3 152.3 194.0 221.5 69.8 104.0 135.7 174.0 199.8 - 90.8 - 71.1 - 52.8 - 29.1 - 14.2 157.7 200.0 237.0 269.0 286.0 80.2 117.3 150.3 192.7 220.0 108.1 147.8 183.5 226.5 254.0 143.5 180.3 214.0 252.0 276.5 51.9 83.5 112.0 147.2 169.7 60.2 95.0 126.8 164.3 188.0 32.5 64.0 92.6 127.3 149.5 - 16.7 + 7.7 30.2 57.5 75.7 - 75.0 - 52.8 - 32.0 - 6.4 + 10.0 126.0 151.8 183.0 203.0 97.5 - 6.4 + 20.8 45.5 75.7 96.0 56.6 90.7 121.9 159.5 184.3 57.8 92.3 124.5 163.8 188.5 35.7 65.3 91.8 124.0 146.0 25.0 53.9 80.0 111.7 132.2 - 4.4 + 21.4 44.6 72.6 91.6 33.2 - 18.4 + 8.0 64.5 85.5 74.0 - 7.1 94.0 4;3.8 19.5 - 31.4 - 4.8 + 19.8 49.5 70.0 78.6 113.0 144.5 181.0 205.0 160.0 186.0 203.5 220.0 228.0

+

-

16.8 26.8 244.5 260.0 220.0 236.5 1.5 + 8.9 295.0 300.0 243.0 261.5 272.0 285.0 297.0 312.5 188.5 213.0 206.5 220.0 167.0 180.5 90.0 ....... 23.6 ....... 218.0 230.0 112.1 125.2 204.7 220.0 209.0 224.5 163.0 176.0 149.8 162.6 106.5 118.2 108.5 118.0 111.5 125.0 85.0 ....... 214.5 ....... 233.0 238.0

34.8

36.0 62. o

...... . .......

243.3 54 . 5 9.6 50. 7 309.8 70. 6 261.5 50 . o 288.4 53 . o 321.6 57 . 2 214.0 59. 1 230.8 61. 5 187.2 52 . o 102.2 49 . 6 32.3 48. 2 243.5 63. 1 126.9 52. o 225.5. 54. 2 229.9 54. 6 183.2 55 . 5 164.5 52 . 4 126.6 58. 2 120.7 51. 8 128 52. 8 91.4 45. 4 235.0 47 . o 239.5 53 . o

...... . 271.0 57 . 9 304.5 ...... .

...... . ...... . • • • • • • 'I

229.5

...... . ...... . .......

242.0 .......

....... ....... ...... . ...... "

••••••

o.

.......

....... ....... ...... .

...... .

liY H

• = In PV _ 2BN _ ~ CN' NRT V 2 V'

and

La is heat of vaporization of liquid He 4 at 0 K, Sl and Vz are the molar entropy and volume of liquid He', m is the mass of a He' atom, and Band Care virial coefficients of He'. The scale has been approved by the International Committee on Weights and Measures and is used for temperature measurements between the boiling point of helium (4.2150 K) and about 1.0 K and can be used up to the critical point of helium (T = 5.1994 K, P = 1,718 mm Hg). It is in agreement with the thermodynamic scale to within ±2 millikelvins. The vapor-pressure-temperature relation for He' is based on the equation In P,

=

2.4~174

+ 4.80386

+ 0.198608T2 - 0.0502237T3 + 0.00505486T4 + 2.24846 In T 0.2 :::; T

- 0.286001T

:::; 3.324 K

which defines theT" He' temperature scale. The T 6, He' temperature scale is the result of the work done by Sydoriak, Roberts, and Sherman 2 at the Los Alamos Scientific Laboratory. Table 4k-7, which gives the temperature as a function of vapor pressure for He s, is taken from the work of R. H. Sherman, S. G. Sydoriak, and T. R. Roberts." Temperature measurements using the vapor pressure of liquid hydrogen are complicated by the phenomenon of ortho-para conversion. For ortho-hydrogen the proton spins are parallel while for para-hydrogRD the) spins are antiparallcl. Due to the different energies of the two states, the equilibrium composition varies from 75 % ortho-H, and 25% para-H, at room temperature to 99.79% para-H, and 0.21% ortho-H, at 20.4 K.' However in the absence of a catalyst the rate of conversion from the ortho to the para form is very slow; thus it is possible to liquefy hydrogen and preserve for many hours the equilibrium composition at room temperature. 5 Hydrogen having the composition 75 % ortho-H, and 25 % para-H, is generally called normal hydrogen and hydrogen having the composition 99.79% para-H, and 0.21 % ortho-H, equilibrium hydrogen. The vapor-pressure-temperature relation for equilibrium hydrogen (99.79 % para-H, and 0.21 % ortho-H,) is based on an equation proposed by Durieux,· log P (mm) = 4.635384 - 44,;'674

+ 0.021669T -

0.000021T2

1 F. G. Brickwedde, H. Van Dijk, M. Durieux, J. R. Clement, and J. K. Logan, J. Research NBS G4A, 1 (1960). , S. G. Sydoriak, T. R. Roberts, and R. H. Sherman, J. Research NBS G8A, 559 (1964). 'R. H. Sherman, S. G. Sydoriak, and T. R. Roberts, Los Alamos Rept. LAMS 2701, pp. 17-21,1962; J. Research NBS G8A, 579 (1964). 'G. K. White, "Experimental Techniques in Low Temperature Physics," p. 41, Oxford University Press, London, 1959. 5 R. P. Hudson, in "Experimental Cryophysics," p. 224, F. E. Hoare, L. C. Jackson, and N. Kurti, eds., Butterworth & Company (Publishers), Ltd., London, 1961. 5 M. Durieux, Thesis, p. 95, Leiden, 1960.

4-279

VAPOR PRESSURE TABLE

.4k-5.

VAPOR PRESSURE QF HELWM

4 (1958

SCALE)

Vapor pressure of 'He. Unit 1O~3 mm Hg at O°C, y = 980.665 em/sec' T

0.00

0.01

' 0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .022745 .031287 .35649 .44877 2.7272 3.2494 13.187 15.147 46.656 52.234

.042561 .057292 .56118 .69729 3.8549 4.5543 17.348 19.811 58.355 65.059

.076356 .86116 5.3591 22.561 72.386

.10081 1.0574 6.2820 25.624 80.382

.13190 1.2911 7.3365 29.027 89.093

.17112 1. 5682 8.5376 32.800 98.567

.22021 1.8949 9.9013 36.974 108.853

1.0 120.000 292.169 '625.025 , 1208.51 2155.35

132.070 316.923 670.411 1284.81 2274.99

145.116 343.341 718.386 1364 .83 2399.73

159.198 371.512 769.057 1448.73 2529.72

174.375 401.514 822.527 1536.61 2665.09

190.711 433.437 878.916 1628.62 2805.99

208.274 467.365 938.330 172jl. 91 2952.60

227.132 503.396 1000.87 1825.58 3105.04

247.350 541.617 1066,67 1930.79 3263.48

269.006 582.129 1135.85 2040.67 3428.07

1.5 3598.97 5689.88 8590.22 12466.1 17478.2

3776.32 5940.76 8931. 18 12913.7 18047.7

3960.32 6199.90 9282.06 13372.8 18630.1

4151.07 6467.42 9643.02 13843.6 19225.5

4348.79 6743.57 10014.3 14326,.1 19834.1

4553.58 7028.47 10395.9 14820.7 20455.9

4765,68 1322.31 10788.2 15327.3 21091.1

4985.18 7625.21 11191.2 15846.3 21739.7

5212.26 5447.11 7937.40 8259.02 11605.1 12030.1 16377.7 16921.7 22402.0 23077.9

2.0 23767.4 31428.1 40465.6 51012.3 63304.3

24470.9 32271.1 41446.6 52160.2 64635.2

25188.1 33128.0 42443.5 53325.8 65985.4

25919.2 33998.6 43456.5 54509.2 67354.8

26664.2 34882.8 44485.7 55710.5 68743.,5

27423.3 35780.3 45531:3 56930.0 70152.0

28196.3 36690.9 46593.5 58167.8 71580.2

28983.2 37614.3 47672.5 59423.8 73028.1

29784.,2 38550.2 48768.6 60698.8 74496.0

0.5

.016342 .28121 2.2787 11. 445 41. 581

30599.1 39500.3 49881.8 61992.0 75984.2

2.5 77493.1 79022.2 80572.2 82142.9 83734.6 85347.2 86981.2 88636.7 90313.8 92012.6 110228 93733.4 95476.0 97240.8 99028.2 100838 102669 104525 106403 108304 130765 112175 114145 116139 118156 120198 122263 124353 126465 128603 151349 1'53763 139663 141949 144260 146597 148961 132952 135164 137401 179364 156204 158671 161164 163684 166230 168802' 171402 174028 176682 3.0

266166 303008 343141

204755 235714 269706 306871 347341

207719 238074 273278 310768 351575

373269 377714 382194 419324 424128 ' 428968 469038 474218 479435 522564 '"528132 533739 580059 586034 592051

386710 433846 484691 539387 598110

391262 438760 489985 545075 604210

395849 443713 495317 550805 610352

635345 700851 770772 845255 924573

641700 707643 778010 852966 932778

648099 714479 785294 860725 941033

654541 721360 792623 868533 949338

661926 728285 799999 876390 957693

667554 735255 807422 884296 966099

674125 742269 814893 892252 974556

1008905 1098449 1193407 1293991 1400429

1017621 1107699 1203209 1304367 1411404

1026390 1117002 1213066 1314802 1422438

1035213 1126359 1222981 1325297 1433533

1044087 1135772 1232955 1335850 1444690

1053014 1145239 1242983 1346462 1455911

1061995 1154761 1253069 1357136 1467191

198jl14 229285 262658 299178 338976

201820

368860 414556 463897 517036 574126

629033 694103 763579 837592 916418 1000239 1089254 1183662 1283673 1389516

182073 210711 242266 276880 3i4697

184810 213732 245587 280516 318659

187574 216783 248939 284183 322654

190366 252322 287883 326684

3.5

355844 400471 448702 500688 5,56574

360147 405130 453729 506098 562383

364485 409825 458794 511547 568234

4.0

616537 680740 7;19328 822411 900258

622764 687399 756431 829978 908313

4.5

983066 1071029 1164339 1263212 1367870

991628 1080114 1173972 1273414 1378662

2111864

193187 222975 255736 ,291615 330747

196037 226115 259182 295380 334845

232'484

5.0 1478535 1489940 1501409 1512940 1524535 1536192 1547912 1559698 1571546 1583458 1595437 1607481 1619589 1631761 '1644000 1656305 1668673 1681108 1693612 1706180 1718817 1731521 1744290

4-280

HEAT

TABLE 4k~6.

1958

p

0

1

K

HE4 VAPOR-PRESSURE-TEMPERAT1JRE SCALE, TIN

FUNCTION OF

P

IN MILLIMETERS MERCURY AT GRAVITY, 980.665 CM/SEC 2

4

3

2

O°C

7

6

5

AS A

AND STANDARD

8

9

--- ------ --- --- --- --- --39 28 22

18 9106 179279 149427

19 9125 . 18 9143 159294 169310 14 9441 149455

18 15 13

13 9559 12 9677 109784

129571 11 9688 10 9794

12 9583 11 9699 10 9804

'12 11 10

109873 . 10 9883

99892

99901

10

99983 80067 70146

89991 80075 , 80154

9 8 8

65 7972 388445 278754

61 8033 358480 26.8780.

578Q90 538143 358515 338548 258805 .258830

50 8193 328580 248854

478240 328612 .248878

458285 308642 .238901

0.04 0.8967 0.05 0.9161 0.06 0.9325

21 8988 189179 159340

219009 17 !H96' 159355

209029 .209049 179213' 179230 159370 159385

199iJ68 169246 14 ~399

209088 179263 149413

0.07 0.94:68 0.08 0.9595 · 0.09 0.9710

14 9482 129607 119721

139495 129619 11 9732

139508 12 9631 109742

139521 . 139534 129'643 11 9654 11 9753 10 9763

129546 11 9665 11 9774

0.10 0:9814

109824

.109834

109844

109854 .

99863

99965 90051 80131

99974 80059 80139

438328 29 8671 22.8923

0.11 0.9911 0,12 1.0000 0.13 l.O083

99920 90009 80091

99929 80017 80099

99938 90026 80107

99947 ·80034 80115

99956 80042 80123

· 0.14' 1. 0162 0.15 1. 0236 0.16 I. 0'305

70169 70243 70312

80177 70250 70319

70184 70257 7032.6

80192 70264 60332

70199 70271 70339

70206 70278 70346

80214 70285 60352

'70221 70292 70359

70228 70299 60365

8 6 7

0.17 1. 0372 0.18 1.0435 0.19 ~.0496

'60378 60441 60502

70385 70448 6 (J508

60391 60454 50513

70398 60460 '.60519

60404 60466 60525

60410 60472 60531

70417 60478 60537

60423 60484 50542

60429 6. 0490 60548

6 6 6

520715 1150

490764 381188

490813 361224

460859 36 1260

450904 35 1295

44 0948 341329

430991 33 1362

41 33

311490 261770 222011

301520 261796

· 0.2 ' .1.0554 0.3 1.1032

:

408368 28869.9 228945

0.01 0.7907 0.02 0.8407 0.03 0.8727

550609 54 0663 411073 ·39 1112

3~

0:4 0.5 ,0.6

1: 1395 1.1691 1.1942

321427 271718 241966

232Q34

301550 25 1821 222056

291579 251846 222078

291608 251871 212099

281636 24 1895 212120

271663 24 1919 212141

28 23 21

0.7 0.8 0.9

i .2162 1. 2359 1. 2536

21 2183 202203 202223 ' '20 2243 182377 182395· 192414 182432 172553 ' 172570 162586 :172603

202263 172449 162619

202283 182467 162635

192302 182485 16 2651

192321 172502 162667

192340 172519 162683

19 17 16

321459 261744 23 1989

LO 1.2699

152714'

162730

152745

152760

152775

152,790

152805

152820

142834

15

1.1 1.2 1.3

1. 2849 1.2989 1. 3119

142863 133002

la3132

152878 133015 133145

142892 143029 12 3151

142906 .133042 123169

142920 133055. 13 3182

142934 133068 123194

142948 13 3081 12 3206

132961 133094 12 3218

142975 133107 123230

14 12 12

104 1: 3242 1,.5 L3359 1.6 , 1. 3469

123254 11 3370 11 3480

123266 123278 11 3381; 12 3393 11 3491 103501

123290 11 3404 11 3512

11 3301 11 3415 11 3523

12 3313 10 3533

12 3325 11 3437 11 3544

11 3336 11 3448 10 3554

123348 11 3459 10 3564

11 10 11

'1.7 ,I. 3575 1,8 • 1. 3675 1.9 1. 3771

103585 103685 93780

103595 10 3695 10 3790

103605 93704 93799

,103615 10 3714 10 3809

103625 10 3724 93818

10 3635 93733 93827

103645 103743 93836

103655 93752 93845

10 3665 103762 93854

10 9 9

113~6

3

2.

1.3863 1.4632

893952 654697

864038 63 4760

824120 634823

804200 604883

77 4877 604943

754352 585001

734425 565057

71 4496 565113

694565 55/i108

67 53

4 5 6

1. 5221 535274 1:5707 445751 1..612.3, 39 6162

515325 445795 386200

51 5376 435838 386238.

495425 42 5880 376275

49 $474 425922 376312

48 5~22 41 5963 366348

475569 416004 366384

475616 406044 366420

465662 40 6084 356455

45 39 35

7 8 9

1:6490 1;:6820 1. 7120

356525 31 6851 287148

346559 31 6882 297177

346593 31 6913 .28 7205

336626 306943 287233

336659 306973 287261

336692 307003 277288

326724 297033 287316

326756 297062 277343

326788 297091 277370

32 29 26

10

1. 7396

277423

267449

267475

267501

267527

257552

267578

257603

257628

25

11 12 13

1.7653 2576711 L7893 . 23 7916 1.8119 228141

247702 237939 2.28163

257727 23 7962 21 8184

247751 237985 228206

247775. 238008 218227

247799 228030 228249

247823 238053 218270

237846 223075 21 8291

24 7870 228097 21 8312

23 22 21

14 15 16

1. 8333 1. 8536 1. 8729

208353 198555 198748

218374 208575 198767

218395 208595 188785

208415 19 8614 198304

208345 208634 198823

21 8456 19 8653 188841

208476 198672 19 8860

208496 198691 188878

208516 198710 188896

20 19 18

17 18 19

1. 8914 1. 9092 ' 1. 9262

188932 179109 17 9279

188950 179126 179296

188968 189144 16 9312

188986 179161 17 9329

189004 179178 169345

189022 179195 17936.2

179Q39 179212 169378

18 9057 179229 169394

179074 179246 179411

18 16 16

20

1. 9427

16 9443

16 9459

169475

16 9491

169507

16 9523

16 9539

159554

169570

16

4-281

VAPOR PRESSURE TABLE

4k-6: 1958

HE'

FUNCTION OF

VAPOR-PRESSURE-TEMPERATURE

SCALE, TIN

K

AS A

P

IN MILLIMETERS MEHCURY AT O°C AND STANDARD GRAVITY, 980.665 CM/SEC 2 (Continued)

p

0

21 22 23

1. 9586 1. 9740 1. 9889

1 2 3 4 5 6 7 ' 8 9 --- - - - - . - - - - - - - - - - --~ - 169602 159617 15 9632 169648 15 9663 16 9679 15 9694 15 9709 159724 16 15 9755 159770 15 9785 15 9800 15 9815 15 9830 149844 15 9859 15 9874 15 149903 15 9918 149932 15 9947 14 9961 15 9976 149990 150005 140019 14

24 25 26

2.0033 2.0174 2,0311

150048 140188 130324

140062 140202 140338

14 0076 13 0215 13 0351

140090 14 0229 14 0365

140104 140118 140243 140257 13 0378 ' 13 0391

140132 13 0270 140405

.14 0146

140284 130418

14 0160 13 0297 130431

14 14 13

27 28 29

2.0444 2.0575 2,0702

140458 130588 130715

130471 120600 120727

130484 130613 130740

130497 13 0626 12 0752

13 0510 13 0639 130765

130536 120664 13 0790

130549 130677 120802

130562 130690 120814

13 12 13

30

2.0827

120839

120851

12 0863

130876

12 0888

120900

120912

120924

120936

13

31 32 33

2.0949 2.1068 2.1185

120961 121080 12 1197

120973 12 1092 111208

120985 111103 12 1220

120997 12 1115 111231

12 1009 12 1127 12 1243

12 1021 12 1139 111254

111032 111150 12 1266

12 1044 12 1162 111277

12 1056 121174 12 1289

12 11 11

34 35 36

2,1300 2,1413 2,1524

12 1312 111424 111535

111323 12 1436 11 1546

111334 111447 111557

12 1346 111458 11 1568

111357 111469 111579

111368 111480 111590

111379 11 1491 111601

12 1391 111502 111612

111402 111513 111623

11 11 11

37 38 39

2.1634 2.1741 2.1848

10 1644 111752 ,10 1858

111655 111763 111869

111666 10 1773 111880

111677 111784 10 1890

111688 111795 10 1900

10 1698 10 1805 10 1910

111709 111816 111921

111720 10 1826 10 1931

111731 111837 111942

10 11 10

---

13 0523 13 0652 12 0777

40

~.1952

10 1962

111973

10 1983

10 1993

112004

102014.

10 2024

10 2034

102044

11

41 42 43

2.2055 2.2156 2,2255

10 2065 10 2166 102265

102075 102176 10 2275

10 2085 102186 102285

10 2095 10 2196 92294

102105 10 2206 102304

10 2115 10 2216 10 2314

11 2126 92225 10 2324

10 2136 10 2235 10 233!

10 2146 10 2245 92343

10 10 10

44 45 46

2.2353 2.2450 2.2544

10 2363 92459 102554

92372 92468 92563

10 2382 102478 10 2573

10 2392 102488 92582

92401 10 2498 92591

10 2411 102421 92507 92516 10 2601 , 92610

92430 92525 92619

10 2440 10 2535 10 2629

10 9 9

47 48 49

2,2638 2,2730 2,2821

92647 92739 92830

92656 92748 92839

102666 92757 92848

92675 102767 92857

92684 92776 92866

92693 92785 92875

102703 92794 92884

92712 92803 92893

92721 92812 92902

9 9 9

50 60

. 2.2911 2.3745

882999 783823

873086 783901

863172 763977

853257 754052

843341 754127

833424 744201

82 3506 73 4274

81 3587 734347

793666 71 4418

79 71

70 80 90

2.4489 2,5163 2,5781

71 4560 645227 605841

694629 64 5291 58 5899

694698 635354 595958

684766 635417 576015

684834 625479 586073

674901 61 5540 57 6130

664967 61 5601 56 6186

66 5033 61 5662 576243

66 5099 605722 556298

64 59 56

100

2.6354

556409

556464

54 6518

54 6572

536625

54 6679

536732

526784

526836

52

110 120 130

2.6888 2.7390 2.7865

526940 497439 467911

51 6991 487487 467957

51 7042 487535 458002

51 7093 487583 468048

507143 48 7631 458093

50 7193 477678 45 8138

507243 477725 448182

497292 477772 458227

497341 467818 44 8271

49 47 44

140 150 160

2.8315 2.8744 2.9153

448459 418785 409193

438402 428827 409233

44 8446 428869 40 9273

438489 41 8910 39 9312

438532 41 8951 409352

438575 41 8992 39 9391

428617 409032 39 9430

42 8659. 41 9073 399469

43 8702 409113 399508

42 40 38

170 180 190

2.9546 2,9924 3,0287

399585 37 9961 360323

38 9623 369997 350358

38 9661 370034 350393

389699 370071 360429

389737 360107 350164

37 9774 360143 350499

389812 360179 35 0534

38 9850 360215 340568

36 9886 36 0251 350603

38 36 34

200

' 3,0637

350672

340706

340740

340774

340808

340842

340876

33' 0909.

340943

33

210 220 230

3.0976 3.1304 3.1622

341010 33 1337 32 1654

331043 32 1369 311685

331076 321401 311716

331109 32 1433 311747

33 1142 321465 311778

321174 31 1496 31 1809

33 1207 32 1528 31 1840

331240 32 1560 301870

321272 311591 311901

32 31 30

240 250 260

3.1931 3,2231 ,3.2524

311962 30 2261 282552

30 1992 302291 29 2581

302022 '292320 292610

302052 292349. 282638

30 2082 302379 292667

302112 292408 282695

30 2142 292437 292724

302172 292466 282752

302202 292495 282780

29 29 28

270 280 290

3.2808 3.3086 3.3357

282836 273113 273384

282864 283141 263410

282892 273168 273437

282920 273195 263463

282n48 273222 273490

282976 273249 26 3516

273003 273276 273543

283031 273303 26 3569

273058 273330 26 3595

28 27 27

300

3.3622

263648

263674

263700

263726

263752

263778

253803

26

382~

263855

25

310 320 330

3.3880 3.4134 3.4382

263906 254159 244406

25 3931 254184 254431

26 3957 254209 244455

25 3982 244233 244479

264008 254258 254504

254033 254283 244528

254058 254308 244552

254083 244332 244576

264109 254357 25 4601

25 25 24

"HEAT TAlJLE

4k-6. 1.958

1! I;N,I( AS A

H]jJ4 VAPPR-ERESSURE-TEMPERATURE SCALE,

FUNCTION OF P IN 'MILLIMETERS MERCURY AT O°C AND STANDARD," GRAVITY, 980.665 CM/SEC 2 (Continued) , '

._.- .. -p

.-

,_

'

..

..

0

~

340 . 350 ,360

....

,370 380 390 G

,,'

. -

: .

4

5

---

--~

3

2

1

7 '

6

8

9

- - - - - - ------.--:----,244792 235027 ~3 5258

244816 235050 -225280

244649 234886 235120

244673 244910 23 5143

2446g7 23 4933 23 5166

24 4721 24 4957 235189

23 4744 23 4980 23 5212

244768 245004 235235

3.5326 ' 235349 3.5551 225573 3,5772, 225794

2253n 235596 225816

23 5394 22 5618 2~ 5838

235417 225640 225860

22 5439 22 5662 21 5881

235462 ,225664 225903

225484 225706' 22 5925

225506 23 5529 225728 '225750 225941. 21 5968

2 2 2

2,1 6140

21 6161

21 6182'

2

'8,4621; 3,4863 3.5097

,400:, " 3'.5990

,gc~~!

21 6011

23 4839 235073 23 5303

24 24 23

226033,

'2i 6054

~2 607,6

21 6097

226119

216267 21 6477 21 6683

21 6288, 206497 206703

21 6309 21 6518 21 6724,

21 6330 .. 21 6351 21 6539 : 206559 206744 206764

216372" 2i 6393' 216580 ' 21 6601 ' 216785, 20 6805

2 2 2

206946 . 20 6966 207165 207145 19 7341 197360

206986 197184 207380

207006 '207204 19 7399

2o 2 2

410 420 .430

· 3.6621

21 6225-. 21 6246 216435 . 216456 ' 216642 206662

440 4.50 460

· 3,6825 3.7026 3.7224

206845 2() 7046 19 7243

21 6866 207066 207263

2(} 6886 207086 197282

206906 , 20 6926 197105 ' 20 7125 20 7322 207302

470 480' 490,

3.7419 3.7611 3.7800

197438 197630 19 7819

19 7457 19 7649 19 7838

207477 19 7668 19 7857

19 7496 197687 187875

19 7515 19 7706 19 7894

197534 19 7725 19 7913

19 7553 "20 7573 197744 19 7763 18 7931 197950

19 7592 18 7781 19 7969

19 19 18

, 500

3.7987

198006

188024

198043

188061

19 8080

188098

198117,

18 8135

188153

1

510 520 530

3.8172 · 3.8354 3.8533

188190 188372 188551

188208 188390 188569

198227 188408 188587

188245 188426 188605

18 8263 18 8444 17 8622

188281 188462 188640

188299 188480 188658

188317 188498 188676

19 8336 188516' 178693

18 17 18

,540 , 550 560

3.8711 3.8886 · 3.9059

178728 178903 179076

188746 18 8921 17 9093

188764 178938 189111

178781 178955 179128

18 8799 18 8973 17 9145

178816 178990 179162

188834 ' 178851 179007 18 9025 179179 , 17 9196

18 8869 179042 179213

17 17 17

570 580 590

3.9230 3: 9399 3.9566

179247 17 9416 179583

179264 179433 16 9599

179281 169449 179616

179298 179466 16 9632

17 9315 17 9483 17 9649

179332 169499 16 9665

179349 17 9516 179682

169365 179533 16 9698

179382 169549 17 9715

17 17 16


22 2740 2857 2972

12 12 11 11 12'

650 660 670 680

8997 9152 9305 9456 9606

12 12 12 12 12

12 12 11 12 12

12 12 12 12 11

15 15 15 15 15

85013.2984 11 12995 12 3087 11 13018 1213030 11 13041 1213053 11 13064 12 3076 11 3087 12 860 3.3099 11 3110 11 3121 12 3133 11 3144 12 3156 11 3167 11 317.8 12 3190 11 3201 11 870 3.321212 3224 11 3235 11 3246

and for normal hydrogen (75% ortho-H 2 and 25% para-H 2 ), log P (mm)

=

4.658334 -

44.~93

+ 0.021276T

- 0.0000021T2

The equation for equilibrium hydrogen is within El'lCperimental accuracy in agreement with the vapor-pressure data of Hoge and Arnold' over the whole temperature region (14 to 33 K). The equation for normal hydrogen is in agreement with the vapor-pressure data of Woolley, Scott, and Brickwedde. 2 For solid equilibrium hydrogen the vapor-pressure-temperature relation is based on the equation, log P (mm)

I 2

=

4.62438 -

47.~172

+ 0.03635T

R. J. Rage and R. D. Arnold, J. Research NBS, 47, 63 (1951). H. W. Woo!ley,R. B.Scott, and F. G. Brickwedde, J. ReseaTch NBS 41,379 (1948).

VAPOR PRESSURE and for solid normal hydrogen, log P (mm) = 4.56488 -

47.~059

+ 0.039'391'

These equations were obtained by Woolley, ~cott, and: Brickwedde ' from their measurements on the vapor pressure of hydrogen. Tables 4k-8 to 4k-ll give the temperature in kelvins for integral values of vapor pressure in millimeters of mercury at-O°C and standard gravity, 980.665 cm/sec 2, as calculated, from the above equations. TABLE 4k-8. SOLID EQUILIBRIUM HYDROGEN TEMPEJ:tATURES IN K FOR ,INTEGRAL VALUES OF VAPOR PRESSURE, P, IN MILLIMETERS' 'OF MERCURY AT O°C -i\-ND , STANDARD GRAVITY; 980.665 CM/SEC 2 '5 0 2 3 4 1 --P

6

7

8

9

--- ---------

,-

0 10 20 30 40 50

9.463 10:029 10.391 10.662 10.881 11.067 11.228 11.371 11.501 11.619 11.727 11.. 828 11.922 12.010 12.093 12.172 12.247 12.318 12.386 12.452 12.514 12.575 12.633 12.689 12.743 12.795 12.846 12.896 12.944 12.990 13.035 13.080 13.123 13.165 13.206 13.246 13.285 13.323 13.361 13.398 13.434 13.469 13.504 13.,538 ~3 ..571 13.604 13.636 13.668 13.699 13.730 13.760 13.789 13.819 ......

TABLE 4k-9. SOLID NORMAL HYDROGEN TEMPERATURES IN K FOR INTEGRAL VelLUES OF VAPOR PRESSURE P, ,IN MILLIMETERS' OF MERCURY AT O°C AND STANDARD GRAVITY, 980.665 CM/SEC 2 ' P

0,

·1

'

,

2

3

4

- - - - - ---, - - - - - - -,-0

10 20 30 40 50

5

6

7

- - --- ---

8

9

---

...... ' 9.554 10.124 10.488 10.761 18.982 1i'.169 11. '331 11.475 11.605 11. 723 11.832 11.933 12.028 12 ~ 116 12.200 12,279 12.354 12.426 12.494 12.559 12.622 12.683 12.741 12.797 12.852 12.904 12.95,5 13.004 13.052 13.099 13.145' 13.189 13.232 13.274 13.315 1'3.355 13.39i 13.433 13.470 13.507 13.543 13.579 13.613 13.647 13.681 18.713 13.746 13.777 13,808 13:839 13.869 13.899 18.928 13.957 13.985

The vapor-pressure-temperature relation for,neon is based on the equation logP

(m~)

:0

8.746376 _ 126;80 - 0.04368341",

This ~qiIatibn ~~s obtained by Henning and Otto 2 from their experimental measurements on the vapor pressure of neon. Table 4k~12 gives the temperature in K f:I

>

>-3 1287 234 1363

1200 90 654

a-fJ a-fJ cc-fJ

882

1017

a-fJ

668 775

667 1137

fJ-c'Y

760 1479

418 1193

a-fJ

cc-fJ

crfJ

~301

VAPOR PRESSURE

4k-16. VAPOR PRESSURE OF ICE (Pressure of aqueous vapor over ice from -120 to O°C) TABLE

Temp.,oC

Temp., DC

I

Bars

mm of Hg

0.000 0000001

0.00000009 0.00000011 0.00000015 0.00000019 0.00000025

-70 -69 -68 -67 -66

o.000 002 577 0.000 002 992 o.000 003 469

-65 -64 -63 -62 -61

o.000 005 360

0.000 000 0011

0.00000032 0.00000041 0.00000052 0.00000066 0.00000084

0.000006179 0.000 007 113 0.000008178 O. 000 009 389

0.004021 0.004635 0.005336 0.006135 0.007043

-110 -109 -108 -107 -106

0.0000000014 0.000 000 0018 0.000000 0023 o .000 000 0028 o.000 000 0035

0.00000107 0.00000135 0.00000169 0.00000213 0.00000266

-60 -59 -58 -57 -56

0.000010765 0.000 012 328 0.000 014 098 O. 000016103 0.000 018 369

0.008076 0.009248 0.010576 0.012080 0.013780

-105 -104 -103 -102 -101

o.000 000 0044 o. 000 000 0055 o.000 000 0068 o.000 000 0085 0.000 000 0105

0.00000332 0.00000413 0.00000513 0.00000636 0.00000785

-55 -54 -53 -52 -51

0.00002093 0.00002382 0.00002707 0.000 030 73 0.00003485

0.01570 0.01787 0.02031 0.02305 0.02614

-100 -99 -98 -97 -96

0.000 0.000 0.000 a .000 O. 000

0129 0159 0194 0238 0290

0.00000968 0.000011 90 0.00001459 0.00001785 0.000 02178

-50 -49 -48 -47 -46

0.00003947 0.00004466 0.000 050 47 0.00005697 0.000 064 24

0.02961 0.03350 0.03786 0.04274 0.04819

-95 -94 -93 -92 -91

o .000 000 0354 a .000 000 0430 0.0000000521 0.000 000 0630 0.0000000761

0.00002653 0.00003224 0.00003909 0.00004729 0.00005710

-45 -44 -43 -42 -41

0.000 072 36 0.00008142 0.00009152 0.00010277 0.00011528

0.054 28 0.06108 0.06866 0.07709 0.08648

-90 -89 -88 -87 -86

0.000000 0.000 000 0.000 000 0.000 000 o .000 000

0917 1103 1323 1584 1894

0.00006879 0.00008271 0.00009924 0.000 118 85 0.00014205

-40 -39 -38 -37 -36

0.00012918 0.00014462 0.00016174 0.00018072 0.00020172

0.09691 0.10849 0.12133 0.13557 0.15133

-85 -84 -83 -82 -81

O. 000 000 O. 000 000 O. 000000 O. 000 000 O. 000 000

2259 2689 3196 3792 4490

0.000 1694 0.000 2018 0.000 2398 0.000 2844 0.0003368

-35 -34 -33 -32 -31

O. 0002250 O. 000 2506 0.000 2790 O. 000 3103 O. 0003447

0.1688 0.1880 0.2093 0.2328 0.2586

-80 -79 -78 -77 -76

o.000 000 5307 6262 7376 8673 0182

0.0003981 0.0004697 0.0005533 0.000 6506 0.0007638

-30 -29 -28 -27 -26

0.0003827 0.0004245 0.0004704 0.0005209 0.0005762

0.2871 0.3184 0.3529 0.3907 0.4323

-75 -74 -73 -72 -71

0.000001 1934 0.000 001 3964 0.000 001 6314 0.000 001 9030 0.0000022162

0.0008952 0.0010476 0.0012238 0.0014275 0.001 6625

-25 -24 -23 -22 -21

0.0006370 0.0007035 0.0007764 0.0008561 0.0009433

0.4778 0.5277 0.5824 0.6422 0.7076

-120 -119 -118 -117 -116 -115 -114 -113 -112 -111

o.000 000 0002 o.000 000 0002 o.000 000 0003 o.000 000 0003 o.000 000 0004 o.000 000 0005 O. 000 000 0007 o.000 000 0009

000 000 000 000 000

0.000 000 0.000 000 0.000 000 0.000001

Bars

0.000004017 O. 000 004 643

mm of Hg

0.001933 0.002245 0.002603 0.003013 0.003483

4-302

HEAT TABLE

Temp., °0

Bars

4k-16.

VAPOR PRESSURE OF

mm of Hg

-20 -19 -18 -17 -16

0.0010385 0.0011424 0.0012558 0.0013794 0.0015140

0.7790 0.8570 0.9421 1.0348 1.1358

-15 -14 -13 -12 -11

0.001661 0.001820 0.001993 0.002181 0.002386

1.246 1.365 1.495 1.636 1.790

-10 -9

0.002607 0.002847 0.003107 0.003389 0.003694

1.956 2.136 2.331 2.542 2.771

-8 -7 -6

Temp., °0

ICE (Continued) Bars

I mm of Hg

-5 -4 -3 -2 -1

0.004023 0.004379 0.004763 0.005178 0.005625

3.018 3.285 3.573 3.884 4.220

0

0.006107

4.581

4-303

VAPOR PRESSURE 4k-17. VAPOR PRESSURE OF WATER BELOW 100°0 (Pressure of aqueous vapor over water from -15.0 to 100.0°0) TABLE

Bars

Temp.,-"O.

mm of Hg

Temp., °0

Bars

mm of Hg

-14.8 -14.6 -1404 -14.2

0.001914 0.001946 0.001978 0.002011 ,0.002044

1.436 1.459 1:484 1.508 1.533

-5.0 -4.8 -4.6 -4.4 -4.2

0.004216 0.004280 0.004345 0.004411 0.004478

3.162 3.210 3.259 3.308 3.359

-14.0. -13.8 -13.6 -13.4 -13.2

0.002078 0.002112 0.002147 0.002182 0.002218

1.558 1.584 1.610 1.637 1.663

-4.0 -3.8 -3.6 -3.4 -3.2

0.004545 0.004614 0.004684 0.004754 0.004826

3.409 3.461 3.513 3.566 3.620

-,13.0 -12.8 -12.6 -12.4 -12.2

0.002254 0.002291 0.002328 0.002366 0.002404

1.691 1.718 1.746 1. 775 1.803

-3.0 -2.8 -2.6 -2.4 -2.2

0.004898 0.004972 0.005046 0.005121 0.005198

3.674 3.729 3.785 3.841 3.899

-12.0 -11.8 -11.6 -11.4 -11.2

0.002443 0.002483 0.002523 0.002564 0.002605

1.833 1.862 1.893 1.923 1.954

-2.0 -1.8 -1.6 -1.4 -1.2

0.005275 0.005353 0.005433 0.005513 0.005595

3.957 4.015 4.075 4.135 4.196

-11.0 -c 10.8 -10.6 -10.4 -,-10.2

0.002647 '0.002689 0.002732 .0.002776 . 0.002820

1:985 2.017 2.049 2.082 2.115

-1.0 -0.8 -0.6 -0.4 -0.2

0.005677 0.005761 0.005846 0.005932 0.006019

4.258 4.321 4.385 4.449 4.515

-10.0 -9.8 -9.6 -9.4 -9.2

0.002865 0.002 III 1 0.002957 0.003003 0.003051

2.149 :t.1!>3 2.218 2.253 2.288

0.0 0.4 0:6 0.8

0.006107 10.006190 0.006287 0.006379 :0.006472

4.581 4.648 4.716 4.785 4.854

-9.0 -'-8.8 -8.6 -8.4 -8.2

0.003099 0.003148 0.003197 0.003248 0.003298

2.324 2.361 2.398 2.436 2.474

1.0 1.2 1.4 1.6 1.8

'0.006566 0.006661 0.006758 0.006856 ; 0.006955

4.925 4.996 5.069 5.142 5.217

-8.0 -7.8 -7.6 -7.4 -7.2

' 0.003350 0.003402 0.003455 ' 0.003509 : 0.003564

2.513 2.552 2.592 2.632 2.673

2.0 2.2 2.4 2.6 2.8

0.007055 0.007 157 0.007260 0.007364 • 0.007469

5.292 5.368 5.445 5.523 5.602

-7.0 -6.8 -6.6 '-6.4 -,6.2

0.003619 0.003675 0.003732 0.003790 0.003848

2.715 2.757 2.799 2.842 2.886

3.0 3.2 3.4 3.6 3.8

0.007576 0.007684 0.007794 0.007905 0.008017

5.683 5.7M 5.846 5.929 6.013

-6.0 -5.8 -5.6 -5.4 -5.2

0.003907 0.003967 : 0.004028 0.004090 .0.004152

2.931 2.976 3.021 3.067 3.114

4.0 4.2 4.4 . 4.6 4.8

0.008131 0.008246 0.008363 0.008481 0.008600

6.099 6.185 6.273 6.361 -6.451

~15.0

0.2

I

1-

I

-

4-304 TABLE

Temp.,oC

HEAT

4k-17, -VAPORPRESSURE , ,

Bars

5.0 5.2 , 5:4 5:6 5:8

:0.008721 '0.008844 :0.008968 '

6:0 6.2 6:4 6:6 6:8

iO.009349 ' '0.009479 :0.009611 ' ,0.009745 0.009880 '

7.0 7.2 7:4 7::6 7.8

'0.010 016 :0.010 155 10.010295 ' '0.010437 0.010 580

8.0 8.2 8:4 8:6 8.8 9:0 9.2 9.4 9.6 9:8

~m

OF WATER BELOW

100°0 (Continued)

Temp.,oC

of Hg

Bars

mm of Hg

i

,

6.542 6.633 6.726 6:821 6:9i6

15.0 ,15.2 15:4 15:6 15:8

'0.017049 0.017270 ,0.017493 0.017719 ,0.017947 '

, 13.121

7:012 7.110 7.209 7.309 7.410

i6:0 16.2 16.4 16:6 16:i,j

iO.018178 ' :0.018412 !0.01864;8 '0.018887 ' ;0.019128

13:635 13.810 13:987 :, 14:166, 14:347

7:513 7.617 7.722 7:828 7.936

17:0 17.2 17A 17:6 17.8

0.019373 ' 0.019620 0.019869 ;0.020122 ;0.020377

0.010 725 0.010872 0.011 021 0.011172 0.011324

8.045 8.155 8.267 8:379 8.494

18:0 18.2 18.4 18.6 18.8

iO.020 635 ' : 0.020896 0.021160 0.021427 0.021696

15:478 15.673 15:87i' 16.071 16.274

0.011 478 ' ,0.011 634 0.011 792 0.011952 0.012113

8.609 8.726 8:845 8:965 9.086

19.0 19.2 19.4 19.6 19:8

0.021969 0.022245 0.022523 0.022805 0.023090

16.478 16.685 16:894 17:105 17:319

10.0 10.2 10.4 10:6 10:8

0.012277 ,0.012442 0.012610 0.012779 0.012951

9.209 9.333 9.458 9.585 9:714

20.0 20.2 20.4 20:6 20.8

0.023378 ' : 0.023669 ,0.023963 ' 0.024261 ' '0.024562

17'.535 17.753, 17'.974 18.,197 18.42;1

11.0 11.2 11.4 11'.6 11.8

0.013124 0.013300 ; 0.013 477 0.013657 0.013838

9:844 9.976 10.109 10.243 10.380

21.0 21.2 21.4 21.6 21'.8

0.024866 ' 0.025173 0.025483 0.025797 ' 0.026115

18.651 18.881 19.114 19.350 19.588

12'.0 12.2 12.4 12.6 12'.8

0.014022 0.014208 ,0.014396 ' : 0.014587 0.014779 '

10.518 10.657 10.798 10.941 11'.085

22.0 22.2 22.4 22.6 22.8

0.026435 ' ,0.026759 0.027087 ,0.027418 : 0.027753 '

19.828 20.071 20.317 20.565 20.816

13.0 13.2 13.4 13.6 13.8

'0.014974 0.015171 0.015370 ' 0.015572 ' 0.015776

11.231 11.379 11'.529 11.680 11.833

23.0 23.2 23.4 23.6 23.8

' 0.028091 ' 0.028433 0.028778 ' ; 0.029 127' ,0.029480

21.070 21.326 21.58,5 21.847 22.1],'2

14.0 14.2 14.4 14'.6 14.8

0.015982 0.016191 0.016402 ' 0.016615 0.016831

11.988 12.144 12.302 12.462 12'.624

24.0 24.2 24.4 24.6 24.8

' 0.029836 0.030197 : 0.030561 0.9309,28 0.031300

22.379 22.6:19 22'.922 23.1~8 23':477

iO.009093

I

!

10.009220

, ,

"

,I

!

..

12.788 12.954,

13.290 13:462 I i

, ;

14.531 14.716 14:903 15:093 15.284..

"

-

4-305

VAPOR PRESSURE TABLE

Temp.,oC

4k-17.

VAPOR PRESSURE OF WATER BELOW

Bars

mm of Hg

25.0 25.2 25.4 25.6 25.8

0.031676 0.032055 0.032439 0.032826 0.033217

23.759 24.043 24.331 24.621 24.915

26.0 26.2 26.4 Z6.6 26.8

0.033613 0.034013 0.034416 0.034824 0.035236

27.0 27.2 27.4 27.6 27.8

Temp.,oC

100°0 (Continued) Bars

mm of Hg

35.0 35.2 35.4 35.6 35.8

0.056237 0.056862 0.057493 0.058130 0.058773

42.181 42.650 43.123 43 .. 601 44.083

25.212 25.512 25.814 26.120 26.429

36.0 36.2 36.4 36.6 36.8

0.059422 0.060077 0.060739 0.061407 0.062081

44.570 45.062 45.558 46.059 46.565

0.035653 0.036073 0.036498 0.036928 0.037361

26.742 27.057 27.376 27.698 28.023

37.0 37.2 37.4 37.6 37.8

0.062762 0.063449 0.064143 0.064843 0.065549

47.075 47.591 48.111 48..636 49.166

28.0 28.2 28.4 28.6 28.8

0.037800 0.038242 0.038689 0.039141 0.039597

28.352 28.684 29.019 29.358 29.700

38.0 38.2 38.4 38.6 38.8

0.066263 0.066983 0.067710 0.068443 0.069184

49 .. 701 50.241 50.786 51. 337 51.892

29.0 29.2 29.4 29.6 29.8

0.040058 0.040524 0.040994 0.041469 0.041948

30.046 30.395 30.748 31.104 31.464

39.0 39.2 39.4 39.6 39.8

0.069931 0.070686 0.071447 0.072216 0.072991

52.453 53 .. 019 53.590 54.166 54.748

30.0 30.2 30.4 30.6 30.8

0.042433 0.042922 0.043417 0.043916 0.044421

31.827 32.195 32.565 32.940 33.318

40.0 40.2 40.4 40.6 40.8

0.073 774 0.074564 O. U75 36~ 0.076166 0.076979

55.335 55.928 56.5:l6 57.130 57.739

31. 0 31.2 31.4 31.6 31.8

0.044930 0.045444 0.045964 0.046488 0.047018

33.700 34.086 34.476 34.869 35.267

41. 0 41.2 41.4 41.6 41.8

0.077798 0.078626 0.079460 0.080303 0.081153

58.354 58.974 59.600 60.232 60.870

32.0 32.2 32.4 32.6 32.8

0.047553 0.048094 0.048639 0.049190 0.049747

35.668 36.073 36.483 36.896 37.313

42.0 42.2 42.4 42.6 42.8

0.082011 0.082876 0.083750 0.084631 0.085521

61.513 62.162 62.818 63.479 64.146

33.0 33.2 33.4 33.6 33.8

0.050309 0.050876 0.051449 0.052028 0.052612

37.735 38.160 38.590 39.024 39.462

43.0 43.2 43.4 43.6 43.8

0.086418 0.087324 0.088237 0.089159 0.090090

64.819 65.498 66.184 66.875 67.573

34.0 34.2 34.4 34.6 34.8

0.053201 0.053797 0.054398 0.055005 0.055618

39.904 40.351 40.802 41.257 41. 717

44.0 44.2 44.4 44.6 44.8

0.091028 0.091975 0.092931 0.093894 0.094867

68.277 68.987 69.704 70.427 71.156

I

4-306 TABLE

HEA'I'

4k-17.

VAPOR PRESSURE OF WATER BELOW

Temp.,oC

100°C (Continued)

Bars

mm of Hg

45.0 45.2 45.4 45.6 45.8

0.095848 0.096838 0.097837 0.098844 0.099861

71.892 72.635 73.384 74.139 74.902

55.0 55.2 55.4 55.6 55.8

0.15745 0.15896 0.16049 0.16203 0.16358

118.09 119.23 120.38 121.53 122.70

46.0 46.2 46.4 46.6 46.8

0.100886 0.101921 0.102964 0.104017 0.105079

75.671 76.447 77.230 78.019 78.816

56.0 56.2 56.4 56.6 56.8

0.16515 0.16672 0.16831 0.16992 0.17153

123.87 125.09 126.25 127.45 128.66

47.0 47.2 47.4 47.6 47.8

0.106150 0.107231 0.108321 0.109421 0.110530

79.619 80..430 81..248 82.072 82 .. 904

57.0 57.2 57.4 57.6 57.8

0.17il16 0.17481 0.17646 0.17813 0.17981

129.88 131.12 132.36 133.61 134.87

48.0 48.2 48.4 48.6 48.8

0.111649 0.112777 0.113916 0.115064 0.116222

83.744 84.590 85.444 86.305 87.174

58.0 58.2 58.4 58 .. 6 58.8

0.18151 0.18322 0.18494 0.18668 0.18843

136.14 137.43 138.72 140.02 141. 34

49.0 49.2 49.4 49.6 49.8

0.117390 0.118568 0.119757 0.120955 0.122164

88.050 88.934 89.825 90.724 91. 630

59.0 59.2 59.4 59 .. 6 59.8.

0.19020 0.19198 o .19B 77 0.19558 0.19740

142.60 144.00 145.34 140.70 148.. 06

50.0 50.2 50.4 50.6 50.8

0.12338 0.12461 0.12585 0.12710 0.12837

92.545 93.467 95.336 96.282

60.0 60.2 60.4 60.6 60.8

0.19924 0.20109 O.;W~ 96 0.20484 0.20673

149.44 150.83 152.2il 153.64 155.06

51.0 51.2 51.4 51.6 51.8

0.12964 0.13092 0.13221 0.13352 0.13483

97.236 98.198 99.169 100.147 101.134

61.0 61.2 61.4 61. 6 61.3

0.20864 0.21057 0.21251 0.21447 0.21644

156.50 157.94 159.40 160.86 162.34

.52.0 52.2 52.4 52.6 52.8

0.13616 0.13750 0.13885 0.14021 0.14158

102.129 103.133 104.145 105.166 106.195

62.. 0 62.2 62.4 62.6 62.8

0.21842 0.22043 0.22244 . 0.22448 0.22653

163.83 165.33 166.85 168.37 169.91

53.0 53:2 53.4 53.6 53.8

0.14296 0 . .14436 0.14577 0.14718 0.14861

107.232 108.278 109.333 110.397 111.4'70

63.0 63.2 63.4 63.6 63.8

0.22859 : 0.23067 0.23277 0.23488 0.23701

171.46 173.02 174.59 176.18 177.77

54.0 54.2 54.4 54.6 54.8

0.15006 0.15151 0.15298 0.154 45 0.15594

112.551 113.642 114.741 115.850 116.967

64.0 64.2 64.4 64.6 64.8

0.23916 0.24132 0.243.50 0.245.69 0.24791

179.38 181. 00 182.64 184.29 185.94

Temp.,oC

-

-,.

-

-~-

94.il9~

I

Bars

mm of Hg

4-307

VAPOR PRESSURE TABLE

4k-17.

VAPOR PRESSURE OF WATER BELOW

100°0 (Continued)

Bars

mm of Hg

65.0 65.2 65.4 65.6 65.8

0.25013 0.25238 0.254 64 0.25692 0.25922

187.62 189.30 191. 00 192.71 194.43

75.0 75.2 75.4 75.6 75.8

0.38553 0.38877 0.39203 0.39532 0.39862

289.17 291.60 294.05 296.51 298.99

66.0 66.2 66.4 66.6 66.8

0.26154 0.26387 0.26622 0.26859 0.27097

196.17 197.92 199.68 201.46 203.25

76.0 76.2 76.4 76.6 76.8

0.40195 0.40531 0.40868 0.41208 0.41551

301.49 304.00 306.54 309.09 311.66

67.0 67.2 67.4 67.6 67.8

0.273 38 0.27580 0.27824 0.28070 0.28317

205.05 206.87 208.70 210.54 212.40

77.0 77.2 77.4 77.6 77.S

0.41896 0.42243 0.42592 0.42945 0.43299

314.24 316.85 319.47 322.11 324.77

68.0 68.2 68.4 68.6 68.8

0.28567 0.28818 0.29071 0.29327 0.29584

214.27 216.15 218.05 219.27 221.90

78.0 78.2 78.4 78.6 78.8

0.43656 0.44015 0.44377 0.44742 0.45109

327.45 330.14 332.86 335.59 338.34

69.0 69.2 69.4 69.6 69.8

0.29843 0.30103 0.30366 0.30331 0.30897

223.84 225.79 227.76 229.75 231.75

79.0 79.2 79.4 79.6 79.8

0.45478 0.45850 0.46225 0.46602 0.46982

341.12 343.91 346.71 349.54 352.39

70.0 70.2 70.4 70.6 70.8

0.31166 0.31437 0.31709 0.31984 0.32260

233.76 235.79 237.84 239.90 241.97

80.0 80.2 80.4 80.6 80.8

0.47364 0.47749 0.481 37 0.48527 0.48920

355.26 358.15 361. 05 363.98 366.93

71.0 . 71.2 71.4 71.6 ". 71.8

0.32539 0.32820 0.33102 0.33387 0.33674

244.06 246.17 248.29 250.42 252.57

81.0 81.2 81.4 81.6 81.8

0.49315 0.49713 0.50114 0.50518 0.50924:

369.89 372.88 375.89 378.92 381.96

72.0 72.2 72.4 72.6 72.8

0.33963 0.34254 0.34547 0.34842 0.35139

254.74 256.92 259.12 261.34 263.57

82.0 82.2 82.4 82.6 82.8

0.51333 0.51745 0.52160 0.52577 0.52997

385.03 388.12 391. 23 394.36 397.51

73.0 73.2 73.4 73.6 73.8

0.35439 0.35740 0.36044 0.36350 0.36658

265.81 268.07 270.35 272.65 274.96

83.0 83.2 83.4 83.6 83.8

0.53420 0.53846 0.54275 0.54706 0.55140

400.68 403.88 407.09 410.33 413.59

74.0 74.2 74.4 74.6 74.8

0.36968 0.37281 0.37596 0.37913 0.38232

277.29 279.63 281. 99 284.37 286.76

84.0 84.2 84.4 84.6 84.8

0.55578 0.56018 0.56461 0.56907 0.57356

416.87 420.17 423.49 426.84 430.20

Temp., °0

I

I

Temp., °0

I

Bars

mm of Hg

4-308 TABLE

Temp.,oC

HEAT

4k-17.

VAPOR PRESSURE OF WATER BELOW

Bars

mm of Hg

Temp.,oC

100°C (Continued) Bars

mm of Hg

85.0 85.2 85.4 85.6 85.8

0.57808 0.58262 0.58720 0.59181 0.59645

433.59 437.00 440.44 443.89 447.37,

93.0 93.2 93.4 93.6 93.8

0.78491 0.79078 0.79669 0.80263 0.80861

588.73 593.14 597.57 602.02 606 .. 51

86.0 86.2 86.4 86.6 86.8

0.60112 0.60582 0.61055 0.61531 0.62010

450.88 454.40 457.95 461.52 465.11

94.0 94.2 94.4 94.6 94.8

0.81463 0.82068 0.82678 0.83290 0.83907

611.02 615.56 620.13 624.73 629.36

87.0 87.2 87.4 87.6 87.8

0.62492 0.62978 0.63467 0.63958 0.644 53

468.73 472.37 476.04 479.73 483.44

95.0 95.2 95.4 95.6 95.8

0.84528 0.85152 0.85780 0.86412 0.87048

634.01 638.69 643.40 648.14 652.91

88.0 88.2 88.4 88.6 88.8

0.64951 0.65453 0.65957 0.66465 0.66976

487.18 490.94 494.72 498.53 502.36

96.0 96.2 96.4 96.6 96.8

0.87687 0.88331 0.88979 0.89630 0.90285

657.71 662.54 667.39 672.28 677.20

89.0 89.2 89.4 89.6 89.8

0.67491 0.68008 0.68529 0.69053 0.69581.

506.22 510.10 514.01 517.94 521.90

97.0 97.2 97.4 97.6 97.8

0.90945 0.91608 0.922.76 0.92947 0.93622

682.14 687.12 692.12 697.16 702.23

90.0 90.2 90.4 90.6 90.8

0.70112 0.70646 0.71184 0.71725 0.72270.

525.88 529.89 533.93 537.98 542.07

98.0 98.2 98.4 98.6 98.8

0.94302 0.94986 0.95673 0.96365 0.97061

707,32 712.45 717.61 722.80 728.02

91.0 91.2 91.4 91.6 91.8

0.728.18 0.73369 0.73924 0.744 83 0.75045

546.18 550.32 554.48 558.67 562.88

99.0 99.2 99.4 99.6 99.8

0.97761 0.98466 0.99174 0.99887 1.00604

733.27 738.55 743.87 749.21 754.59

92.0 92.2 92,4 92.6 92.8

0.75610 0.76179 0.767.52 0.77328 0.77908

567.12 571.39 575.69 580.01 584.36

100.0

1.01325

760.00

4-309

VAPOR PRESSURE

TABLE 4kc 18. VAPOR PRESSURE OF WATER ABOVE 100°C (Pressure of aqueous vapor over water from 100° to the critical temperature, 374.15°C)

Temp.,oC

Bars

Temp.,oC

mm of Hg

Bars

mm of Hg

100 101 102 103 104

1. 0133 1.0500 1. 0878 1.1267 1.1667

760.0 787.5 815.9 845.1 875.1

150 151 152 153 154

4.7597 4.8887 5.0205 5.1551 5.2926

3,570.1 3,666.8 3,765.7 3,866.7 3,969.8

105 106 107 108 109

1.2080 1.2504 1.2941 1.3390 1. 3851

906.1 937.9 970.6 1,004.3 1,038.9

1.55 156 157 158 159

5.4331 5.5765 5.7228 5.8723 6.0248

4,075.1 4,182.7 4,292.5 4.404.6 4,519.0

110 111 112 113 114

1.4326 1.4814 1. 5316 1.5831 1. 6361

1,074.5 1,111.1 1,148.8 1,187.4 1,227.2

160 161 162 163 164

6.1805 6.3393 6.5014 6.6668 6.8355

4,635.8 4,754.9 4,876.5 5,000.5 5,127.1

115 116 117 118 119

1.6905 1.7064 1.8038 1.8627 1.9232

1,268.0 1,309.9 1,353.0 1,397.2 1,442.5

165 166 167 168 169

7.0076 7.1831 7.3621 7.5446 7.7306

5,256.1 5,387.8 5,522.0 5,658.9 5,798.4

120 121 122 123 124

1.9853 2.0490 2.1144 2.1815 2.2503

1,489.1 1,536.9 1,585.9 1,636.2 1,687.8

170 171 172 173 174

7.9203 8.1136 8.3107 8.5115 8.7161

5,940.7 6,085.7 6,233.5 6,384.2 6,537.6

125

1,740.7 1,795.0 1,850.6 1,907.7 1,966.1

175 176 177 178 179

8.9247

127 128 129

2.3208 2.3931 2.4673 2.5433 2.6213

9.3535 9.5739 9.7985

6,694.0 6,853.4 7,015.7 7,181.1 7,349.5

130 131 132 133 134

2.7011 2.7829 2.8667 2.9525 3.0405

2,026.0 2,087.4 2,150.2 2,214.6 2,280.5

180 181 182 183 184

10.0271 10.2599 10.4969 10.7383 10.9839

7,520.9 7,695.6 7,873 .4 8,054.4 8,238.6

135 136 137 138 139

3.1305 3.2226 3.3170 3.4136 3.5124

2,348.1 2,417.2 2,487.9 2,560.4 2,634.5

185 186 187 188 189

11.234 11.489 11. 748 12.0n 12.279

8,426 8,617 8,811 9,009 9,210

140 141 142 143 144

3.6135 3.7170 3.8228 3.9310 4.0417

2,710.3 2,787.9 2,867.3 2,948.5 3,031. 5

190 191 192 193 194

12.552 12.830 13.113 13.399 13.693

9,415 9,623 9,835 10.050 10,270

145 146 147 148 149

4.1549 4.2706 4.3889 4.5098 4.6334

3,116.4 3,203.2 3,292. 3,382.7 3,475.4

195 196 197 198 199

13.989 14.291 14.598 14.910 15.228

10,492 10,719 10,949 11,184 11,422

126

°

I

iJ .1371

4-310 TABLE

HEAT

4k-18.

VAPOR PRESSURE OF WATER ABOVE

100°C (Continued)

Temp., °0

Bars

mm of HIl:

Temp., °0

Bars

mm of Hg

200 201 202 203 204

15.550 15.879 16.212 16.551 16.895

11,664 11,910 12,160 12,414 12,672

250 251 252 253 254

39.776 40.452 41.137 41.830 42.533

29,834 30,341 30,855 31,375 31,902

205 206 207 208 209

17.245 17.601 17.962 18.329 18.701

12,935 13,202 13,472 13,748 14,027

255 256 257 258 259

43.244 43.965 44.695 45.434 46.182

32,436 32,976 33,524 34,078 34,640

210 211 212 213 214

19.080 19.464 19.855 20.251 20.654

14,311 14,599 14,892 15,190 15,492

260 261 262 263 264

46.940 47.707 48.484 49.270 50.066

35,208 35,783 36,366 36,955 37,553

215 216 217 218 219

21.062 21.477 21.899 22.326 22.760

15,798 16,109 16,425 16,746 17,072

265 266 267 268 269

50.872 51.687 52.513 53.349 54.195

38,157 38,769 39,388 40,015 40,650

220 221 222 223 224

23.201 23.648 24.102 24.562 25.030

17,402 17,738 18,078 18,423 18,774

270 271 272 273 274

55.051 55.917 56.794 57.681 58.579

41,292 41,941 42,599 43,264 43,938

225 226 227 228 229

25.504 25.985 26.473 26.968 27.470

19,129 19,490 19,856 20,227 20,604

275 276 277 278 279

59.487 60.406 61.336 62.277 63.229

44,619 45,308 46,006 46,712 47,426

230 231 232 233 234

27.979 28.495 29.019 29.550 30.088

20,986 21.373 21,766 22,164 22,568

280 281 282 283 284

64.192 65.166 66.151 67.147 68.155

48,148 48,878 49,617 50,365 51,121

235 236 237 238 239

30.634 31.188 31.749 32.318 32.895

22,978 23,393 23,814 24,241 24,674

285 286 287 288 289

69.175 70.206 71.249 72.304 73.370

51,885 52,659 53,441 54,232 55,032

240 241 242 243 244

33.480 34.073 34.673 35.282 35.899

25,112 25,557 26,007 26,464 26,926

290 291 292 293 294

74.449 75.539 76.642 77.757 78.884

55,841 56,659 57,486 58,322 59,168

245 246 247 248 249

36.524 37.157 37.799 38.450 39.109

27,395 27,870 28,352 28,840 29,334

295 2.96 297 298 299

80.024 81.177 82.342 83.521 84.712

60,023 60,888 61,762 62,646 63,539

4-311

VAPOR PRESSURE TABLE 4k-18.VAPOR PRESSURE OF WATER ABOVE 100°C

(Continued)

Tem'p.,·C

'Bars

mm of Hg

Temp."oO

300 301 302 303 304

85.916 87.133 88.363 89.606 90.863

64,442 65,355 66,278 . 67,210 08,153

340 341 342 343 344

146.08 147.92 149.78 151.66 153.56

109,569 110,949 112,344 113,753 115,177

305 306 . 307 308 ,. ',309

,92.134 93.419 94.717 96.029 97-.356

69,106 70,070 71,044 72,028 73,023

345 346 347 348 349

155.48 157.41 159.37 161. 35 163.35

115,616. 118,070 119,539 121,023", 122,523

310 311 312 ,313 • 314, .

98.696 100.050 101.418 102.801 104.199

74,028 75,044 76,070 77,107 78,156

3050 351 352 353 354

165.37 167.40 169.46 171.54 173.64

124,038 125,563 127,106 128,665 130,242

3105 316 317 318 319,

105;611, 107.039 108 :481 109.939, 111.412

79,215 80,286 81,368 82,461 83,566

355 356 357 358 359

175.77 177.91 180.08 182.28 184.50

131,835 133,446 135,075 136,721 138,385

320 321 322 323 324

112.900 114.403 115.921 117.4;56, 119.006,

84,682 85,809 86,948 88,099 89,262

360 361 362 363 364

186.74 189.00 191.28 193.60 195.,93

140,067 141,761 143,41,5 145,209 146,963

325 326 327 ' 328 329

120.57 122.15 123.75 125.37 127.00

90,437 91,624 92,823 94,035 95,259 ,

365 366 367 3,68 369

198.30 200.69 203.11 205.55 208.03

148,736 150,530 152,344 154,179 156,034

330 331 332 333 334

128.65 130.31 131.99 133.69 135.41

96,495 97,743 99,003 100,277 " 101,564

370 371 372 373 374

210.53 213.06 215.62 218.21 220.84

157,911 159,808 161,728 163,67-1 165,644

335' 336 337" 338 ' 339

137.14 138.89 140,66 142.45 144.26

102,864 104,178 105,505 ' 106,846 108,201

374.15

221.23

165,936

,

;

,

Bars

mm of Hg

1

,

4-312

HEAT

TABLE4kc19. VAPOR .PRESSURE OF MERCURY* (Vapor'pressureof mercury in mm of Hg for temperatures from -38 to 400°0. Note that the values for the first four lines only are to be multiplied by 10- 6 ) I

Temp., °0

-30 -20 -10 - 0 + 0 +10 20 30 40 50 60 70 80 90 . 100 110 120 130 140

0 10- 6 4.78 .18.1 60.6 185

2 10- 6 3.59 14.0 48.1 149

4 10- 6 2.66 10.8 38.0 119

6

8

10- 6 1.97 8.28 29.8 95.4

1O~6

1.45 6.30 23.2 76.2

0.000185 0.000490 0.001201 0.002777 0.006079

0.000228 0.000588 0.001426 0.003261 0.007067

0.000276 0.000706 0.001691 0.003823 0.008200

0.000335 0.000846 0.002000 0.004471 0.009497

0.000406 0.001009 0.002359 0.005219 0.01098

0.01267 0.02524 0.04825 0.08880 0.1582

0.01459 0.02883 0.05469 0.1000 0.1769

0.01677 0.03287 0.06189 0.1124 0.1976

0.01925 0.03740 0.06993 0.1261 0.2202

0.02206 0.04251 0.07889 0.1413 0.2453

0.2729 0.4572 0.7457 1.186 1.845

0.3032 0.5052 0.8198 1.298 2.010

0.3366 0.5576 0.9004 1.419 2.188

0.3731 0.6150 0.9882 1.551 2.379

0.4132 0.6776 1.084 1.692 2.585

150 160 170 180 190

2.807 4.189 6.128 8.796 12.423

3.046 4.528 6.596 9.436 13.287

3.303 4.890 7.095 10.116 14.203

3.578. 5.277 7.626 10.839 15.173

3.873 5.689 8.193 11. 607 16.200

200 210 220 230 240

. 17.287 23.723 32.133 42.989 56.855

18.437 25.233 34.092 45.503 60.044

19.652 26.826 36.153 48.141 63.384

20.936 28.504 38.318 50.909 66.882

22.292 30.271 40.595 53.812 70.543

250 260 270 280 290

74.375 96.296 123.47 156.87 197.57

78.381 101.28 129.62 164.39 206.70

82.568 106.48 136.02 172.21 216.17

86.944 111. 91 142.69 180.34 226.00

91.518 117.57 149.64 188.79 236.21

300 310 320 3.30 340

246.80 305.89 376.33 459.74 557.90

257.78 319.02 391.92 478.13 579.45

269.17 332.62 408.04 497.12 601.69

280.98 346.70 424.71 ' 516.74 624.64

293.21 361. 26 441.94 537.00 648.30

350 360 370 380 390

672.69 806.23 960.66 1138.4 1341.9

697.83 835.38 994.34 1177.0 1386.1

723.73 865.36 1028.9 1216.6 1431.3

400

1574.1

750.43 896.23 1064.4 1257.3 1477.7

* From the compilation of J. Johnston, F. Fenwick, and H. G. Leopold. Tables," Vol. III, p. 206, McGraw-Hill Book Company, New York, 1928.

777.92 928.02 1100.9 1299.1 1525.2

"International Crit'cal

4-313

VAPOR PRESSURE TABLE

4k-20.

VAPOR PRESSURE OF CARBON DIOXtDE* SOLID

Pressure, Microns of Mercury ~

Temp.,oC

-180 -170 -160 -150 -140

-

0

1

2

4

3

5

6

7

8

9

--- --- --- .--- --- --- --- ---

--- ---

0.013 0.008 0.006 0.004 0.003 0.37 0.27 0.14 0.20 0.10 5.9 4.6 2.7 3.6 2.1 60.5 48.8 39.2 25.1 31.4 431 359 298 247 204

0.0005 0.026 0.67 9.8 92

0.0017 0.074 1. 58 19.9 168

0.0011 0.052 1.19 15.8 138

0.0007 0.037 0.90 12.4 113

0.0003 0.018 0.50 7.6 75

Pressure, Mm of Mercury

-130 -120 -110 -100 - 90 - 80 - 70 - 60 - 50

2.31 9.81 34.63 104.81 279.5 672.2 1486.1 3073 .1 . ,-,."_...

1. 97 8.57 30.76 94.40 254.7 618.3 1377.3 2865.1 ........

1

1

1

1.2) 1.0) 1.68 1.43 0.87 1 0.73 0.51 0. 61 1 7.46 6.49 5.63 4.88 4.22 3.64 3.13 2.69 27.27 24.14 21.34 18.83 16.58 14.58 12.80 11.22 84.91 76.27 68.43 61.30 54.84 48.99 43. 71 1 38.94 231.8 210.8 191.4 173.6 157.3 142.4 128.7 116.2 568.2 521.7 478.5 438.6 401.6 367.4 335.7 306.5 1275.6 1180.5 1091.7 1008.9 931.7 859.7 792.7 730.3 2669.7 2486.3 2314.2 2152.8 2001. 5 1859.7 1726.9 1602.5 . ...... ',._'--'-' . -'- - . ...... . " .... ....... 3780.9 3530.2 3294.6 ~

LIQUID

Temp.,oC

-50 -40 -30 -20 -10 - 0 0 10 20 30

0

1

2

5127.8 4922.7 4723.9 7545 7271 7005 10718 10363 10017 B331 1138GI 14781 19312 19872 18764 1 26142 25457 24786 26142 26840 27552 33763 34607 35467 42959 43977 45014 54086 55327

I

3

4

5

4531.1 4344.3 4163.2 6250 6746 6494 9679 9350 9029 13461 j13C!O 12530 18228 17703 117189 24127 23482 22849 28277 29017 29771 38146 36343 37236 46072 47150 48250

7 _61_

8

9

3987.9 3818.2t 3653.9t 3495.0t 5557 5339 6012 5781 7826 8412 8115 8716 11082 11455 111838 16194 115712 115241 20443 21026 21622 22229 30539 31323 32121 32934 39073 40017 40980 41960 49370 50514 51680 52871

I~!!!~

* From C. H. Meyers and M. S. Van Dusen, Nat!. Bur. Standards J. Research 10, 409 (1933: Mercury column density = 13.5951 g/cm'; u = 980.665 em/sec'. t Undercooled liquid. Critical temperature = 31.0°C. Triple point, - 56.602 ± 0.005°C; 3885.2 ± 0.4 mm.

4-314

,HEAT

4k-21.

TABLE

1

I

6

Vapor pressure, mm Hg at

ooe

4

·3

\I

1

0 Temp.,

VAPOR PRESSURE OF ETHYL. ALCOHOL*

1

1

5

1

I

7

s

.1

I

I I

II

°0

0 10 20 30

12.24 '23.78 44.00 78.06

13.18 25.31 46.66 82.50

14.15 27.94 49.47 87.17

40

133.70 220.00 350.30 541.20

140.75 230.80 366.40 564.35

148.10 242.50 383.10 588.35

50 60 70

I

I

TABLE

0

Temp., °C 0

30

40 50

60

16.21 30.50 55.56 97.21

155.80 -163.80 253.80 265.90 400.40 418.35 613.20 638.95

I

* Ramsay and Young,

10 20

15.16 28.67 52.44 92.07

I

18.46 17.31 32.44 34.49 58.86 62.33 102.60 108.24

19.68 36.67 65.97 114.15

20.98 38.97 69.80 120.35

22.34 41.40 73.83 126.86

172:20 278.60 437.00 665.55

190.10 305.65 476.45 721.55

199.65 319.95 497.25 751. 00

209.60 334.85 518,85 781.45

181.00 291.85 456.45 693.10

I

I

I

I

Trans. Roy. Soc. (London) 17'1', part I, 123 (1886).

4k-22. 1

VAPOR PRESSURE OF METHYLALCOHOL*

2 1

1

3

4 1

5 1

1

6

1

8

7 1

I

9

Vapor pressure, mm Hg at O°C

29.97 53.8 94.0

31.6 33.6 35.6 37.8 40.2 42.6 45.2 '47.9 50.8 57.0 60.3 63.8 67.5 71.4 75.5 79.8 84.3 89.0 99.2 104.7 110.4 116.5 122.7 129.3 136.2 143.4 151.0

167.1 175.7 184.7 194.1 203.9 214.1 271.9 285.0 298.5 312.6 327.3 342.5 409 . 4 1427.71446. 6 1466.31486. 6 1507.71529.5 1 624.3 650.0 676.5 703.8 732.0. 761.1 791.1 158.9 259.4

224.7 235.8 247.4 358.3 374.7 391.7 552.01575.31599.4 822.0

* RamBay and Young, Trans. Roy.',Soc. (London),. 178, 313 (1887); Bee also Young, Sci. Proc. Roy. Dublin Soc., 12, 374-443 (1910).

4-315

VAPOR PRESSURE Table 4k-23 concerns the evaporation of metals. a metal is given by the equation log W = A -

B 1 T - 2 log T

The rate of evaporation W of

+c

where W is expressed in g/sec cm 2• The values of A, B, and c given in Table 4k-23 are chosen to yield the value of W in these units. TABLE

4k-23.

CONSTANTS IN THE EQUATION FOR THE RATE OF EvAPORATION OF ~ETALS*

10- 3 X B

10- 3 X B

c+4

Metal

A

10.50(1) 10.71(1) 10.36(1) 10.42(1) [10.53(1) 9.86(1) Cs ...... [10.02(1)

7.480 5.480 4.503 4.132 4.291] 3.774 3.883]

0.1867 0.4468 0.5621 0.7319

Si .......

Th ...... Ge ......

13.20(s) 12.55(1) 11.25(s) 11.98(1) 12.38(s) 13.04(1) 12.52(1) 10.94(1)

19.72 18.55 18.64 20.11 25.87 27.43 28.44 15.15

18.06 16.58 14.85 14.09 18.52 18.22 16.59 7.741

0.6678 Sn ...... Pb ...... V ....... Nb ...... Ta ......

9.97(1) 10.69(1) 13.32 14.37(s) 13.00(s)

13.11 9.60 26.62 40.40 40.21

0.8032 0.9242 0.6195 0.7500 0.8947

Mg ......

12.81(s) 11.72(1) 12.28(s) 11.66(1) 11.65(1) 12.99(s) 11.95(1) 11.82(s)

Ca ...... Sr ....... Ba ...... Zn ...... Cd ......

11.30(s) 11. 13(s) 10.88 11. 94(s) 11.78(s)

9.055 8.324 8.908 6.744 5.798

Sb 2 . . . . . . Bi. ...... Cr ...... Mo ...... W .......

11.42 11.14(1) 12.88(s) 11. 80(s) 12.24(s)

9.913 9.824 17.56 30.31 40.26

0.9592 0.9260 0.6240 0.7570 0.8983

B ....... AI ....... Sc ....... y ....... La ......

14.13(s) 11.99(1) 11.94 12.43 11.88(1)

21.37 15.63 18.57 21.97 18.00

12.88(1) 12.25(s) 12.63(s) 13.41(1) 12.43 13.28(s) 12.55(1)

25.80 14.10 20.00 21.40 21.96 21.84 20.60

0.9544 0.6359 0.6395

Ce ...... Ga ...... In ....... Tl. ...... C .......

13.74(1) 10.79(1) 10.93(1) 11.15(1) 14.06(s)

20.10 13.36 12.15 8.92 38.57

13.50 13.55 11.46 13.59 13.06 12.633

33.80 30.40 19.23 37.00 34.11 27.50

0.7696 0.7722 0.7801 0.9056 0.9089 0.9112

Metal

A

Li. ...... Na ...... K ....... Rb ......

Cu ...... Ag ...... Au ...... Be ......

Ti ....... Zr .......

0.8278

0.7825 0.9135 0.2436 0.4590

0.5675 0.7373 0.8349 0.6737 0.7914 U ....... Mn ...... 0.2831 Fe ....... 0.4814 0.5931 Co ...... 0.7405 Ni ...... 0.8374 0.8392 Ru ...... 0.6877 Rh ...... 0.7959 Pd ...... 0.9212 Os ...... 0.3056 Ir ....... Pt .......

c+4 0.4900 0.4900 0.6061 0.7460 0.9488 0.6965

0.6512 0.6503

* From Saul Dushman, "Scientific Foundations of Vacuum Technique," pp. 752-754, John Wile) & Sons, Inc., New York, 1949.

·41. Heats of Formation and Heats of Combustion BRUNO J. ZWOLINSKI AND RANDOLPH C. WILHOIT

Thermodynamics Research Center, Texas A&M University

Tables 41-],41-2, and 41-3 list values of the enthalpy of formation, !J.Hr, and enthalpy of combustion, 4Hco, of pure elements and compounds in their standard states at one atmosphere pressure and 25°C in units of kilocalories per mole. Data on "key" substances, which play important roles in evaluating the data on other compounds, are collected in Table 41-1. Enthalpies of formation of elements and inorganic compounds are given in Table 41-2. They are arranged in a standard order, based on the order of elements in the periodic table. The organic compounds in Table 41-3 are arranged first by standard order of the elements of which they are composed and then by classes which have certain common molecular structural features or functional groups. 41-1. Sources of Data. All reported values were derived from published experimental measurements, and most of the data were selected from the following compilations: (1) Selected Values of Chemical Thermodynamic Properties: part 1, NBS Tech. Note 270-1, 1965; part 2, NBS Tech. Note 270-2, 1966; (2) Selected Values of Properties of Hydrocarbons and Related Compounds, Am. Petroleum Inst. Research Proj. 44, Thermodynamics Research Center, Texas A&lVI University, College Station. Texas (looseleaf nat.a sheets, extant 1967); (3) Selected Values vf Propertiel:! of Chemical Compounds, Thermodyn. Research Center Data Proj., Texas A&lVI University, College Station, Texas (looseleaf data sheets, extant 1967). These sources were supplemented by information in the files of the Thermodynamics Research Center at Texas A&M University. Data in all three tables are internally consistent, and, wherever necessary, original data have been converted to the units and conventions listed below. 41-2. Symbols and Units calorie the thermochemical calorie defined as equal to 4.184 joules (exactly) mole a unit of mass equal to the formula (molecular) weight in grams, calculated from the 1961 table of unified atomic weights based on carbon-12 standard state for condensed phases, the specified crystal or liquid form at one atmosphere pressure; for gases, the hypothetical ideal gas at one atmosphere pressure g gas l liquid c crystal aq aqueous (water) solution enthalpy, H = U + PV, for a change from an initial to a final state, H 4H = H(final) - H(initial), which is equal to the heat absorbed by the system at constant pressure 4-316

HEATS OF FORMATION AND HEATS OF COMBUSTION

4-317

the heat of formation of one mole of compound or element in its standard state from the elements in their reference states. [For an organic oxygen compound this corresponds to the chemical reaction, aC(graphite) -!bH2(gas) -!cOc(gas) -> CaHbOc(standard state). Reference states for elements are identified by a zero enthalpy of formation in the tables.] AHco, gross the heat of combustion of a compound with excess oxygen gas to produce pure, thermodynamically stable products at 25°C and one atmosphere, with all components in their standard states. [The pronucts of combustion are: CO 2{gas), H 20 (lIquid); HF(gas);Ch(gas), Br2(liquid), 12 (crystal), H 2S0 4 (liquid), and N 2(gas), as appropriate for the stoichiometry of the combustion reaction.] AHco, net the heat of combustion of a compound with excess oxygen to produce the following products: CO. (gas), H.O(gas), HF(gas), Ch(gas), Br2(gas), 12(gas), S02(gas), and N 2 (gas). (These are the principal products formed when a compound is burned in an open flame in the air.) 41-3. Uncertainties. The number of significant figures used in reporting a value of AHr or AHco is related to the estimated uncertainty according to the following scheme. AHr

+

Estimated" uncertainty in AHf" or AHco, kcal mole- 1 0.005--0.05 0.05-0.5 0.5--2 2~1O

+

Value written to 0.001 0.01 0.1 1.

4-318 TABLE

HEAT

41-1. HEATS OF FORMATION AND HEATS OF COMBUSTION OF KEY COMPOUNDS kcal mole- 1 at 25°C

Substance name

Formula and state

Mol. weight

-!J.Hco !J.HjD Gross

H20,g H2O.! Hydrogen fluoride ......... HF,g HF,l in 00 H20 .............. HF,aq Hydrogen chloride ......... HCl,g in 00 H2O .............. HCl,aq Hydrogen bromide ........ HBr,g in 00 H 2O .............. HBr,aq Hydrogen Iodide .......... HI,g Sulfur dioxide ............. S02,g S02.1 Sulfuric acid .............. H,SOd in 00 H 20 .............. H 2S04,aq in 115 H2O ............. H 2S04,aq Orthophosphoric acid ...... H,P04,C H,POd in 00 H2O .............. H,P04,aq Carbon dioxide ........... C02,g Butanedioic acid (succinic C4H,04,g C4H,04,C acid) Benzoic acid .............. C 7H,02,g C7H ,02,C Carbon tetrafluoride (tetrafluoromethane) ........ CF4,g p-Fluorubell~o;c acid ...... C BH,02F,c a,a ,a -Trifluoro-m-toluic acid ................. C BH,02Fa,c Carbon disulfide .......... CS2,g CS2.1 Thianthrene (diphenylene disulfide) ............. C'2H 8S 2 ,C N-Benzoylaminoethanoic acid (hippuric acid) ... C,H,O,N,c Boric oxide ............... B20a,c amorphous ............. B20a,c Boron trifluoride BFa,g Silicon dioxide quartz ................. Si02,C cristo balite ............. Si02,c tridymite ............... Si02,C amorphous ............. Si02,C Silicon tetrafluoride ........ SiF 4,g

Water ................ · ..

· . .. . . .

-57.796 - 68.315 -64.8 71.65 -79.54 -22.062 -39.952 -8.70 -29.05 6.33 -70.944 -76.6 -194.548 -217.32 -212.192 -305.7 -302.8 -307.92 -94.051 -196.8 -224.86 -70.19 -92.04

88.005 140.115

18.015

....... 20.006 ....... · .. . . . . 36.461

.......

12.096

Net

6.836

25.46

16.50

40.49

27.77

384.35 356.29 793.11 771.26

352.79 324.73 761. 55 739.70

-221 -139.56

720.22

6\J!i .1\)

190.123 76.139

761.44

·.... ..

-253.68 21.44 28.05

........ ........

750.92 263.99 257.38

216.326

43.12

1,697.46

1,544.80

179.177 69.620

-145.49 -304.20 -299.84 -271.03

1,008.39

961. 05

80.917

.......

127.912 64.063

.......

98.078

....... .......

97.995

....... .......

44.010 118.090

·......

122.125

·... ... 67.806 60.085

....... ....... .......

104.080

-217.72 -217.37 -217.27 -215.95 -385.98

HEATS OF FORMATION AND HEATS OF COMBUSTION TABLE

41-2.

4-319

HEATS OF. FORMATION OF ELEMENTS AND INORGANIC COMPOUNDS

Substance

Formula

nan~e

Mol. weight Gas

Liquid

Solid

Oxygen and Hydrogen Oxygen ........... " .... Ozone .................. Hydrogen ............... Hydrogen peroxide .......

31.999 47.998 2.016 34.015

. . . .

0.0 34.1 0.0 -44.88

Halogens Fluorine ................ . Chlorine ................ . Chlorine monoxide, ...... . Chlorine dioxide ......... . Dichlorine monoxide ..... . Perchloric acid .......... . Chlorine monofluoride .... . Chlorine trifluoride ....... . Bromine ................ . Bromine monoxide ....... . Bromine dioxide ......... . Bromine trifluoride ...... . Bromine pentafluoride .... . Bromine chloride ........ . Iodine .................. . Todic aCid .............. . Iodine monofluoride ...... . Iodine pentafluoride ...... . Iodine heptafluoride ...... .

F2 C]'

CIO CIO, Cl,O

HCIO.

CIF CIF, Br, BrO BrO, BrF, BrF, BrCI 12 HIO, IF IF5 IF7 ~odillB IllonOChlor. ~u....le . . '. .. . '1 Iel Iodine trichloride ........ . ICla Io_dine mono bromide ..... . IBr

37.997 70.906 51.452 67.452 86.905 100.459 54.451 92.448 159.818 95.908 111.908 136.904 174.901 115.362 253.809 175.911 145.903 221. 896 259.893

0.0 0.0 24.36 24.5 19.2 -9.70 -11.92 -38.0 7.39 30.06

-44.3 0.0 11.6

-102.5 3.50 14.92 -2.10 -196.58 -225.6

152.357

4.2-5

233.263 206.813

9.76

-71.9 -109.6 0.0 -55.0 -206.7 -5.71

I

-8,4 -21.4 -2.5

Sulfur Sulfur ................... rhombic ................ monoclinic ............. Sulfur ................... Sulfur ................... Sulfur trioxide ............ Hydrogen sulfide .......... Sulfur tetrafluoride ........ Sulfur hexafluoride ........ Disulfur dichloride ........ Thionyl'chloride .......... Sulfuryl chloride .......... Thionyl bromide ..........

S

32.064

66.64

.......... . ...... . .......... . ...... .

....... . . ....... ....... . . .......

S, SO, H,S SF, SF, S,C]' SOC]' SO,C]' SOBr,

30.68 31. 7 -94.58 -4.93 -185.2 -289. -4.4 -50.8 -87.0 -17.7

8,

64.128 96.192 80.062 34.080 108.058 146.054 135.034 118.969 134.969 207.881

0.0 0.08 -108.63

-105.41

-14.2 -58.7 -94.2

Nitrogen Nitrogen ................. N, Nitric oxide ..... . . . . . . . . . NO Nitrogen dioxide ......... ; . NO,

28.013 30.006 46.006

0.0 21.57 7.93

I I I

4-320

HEAT TABLE

41-2.

HEATS OF FORMATION OF ELEMENTS AND

INORGANIC COMPOUNDS

Substance name

Formula

I

(Continued)

Mol. weight

!:.HjD, kcal mole- 1 at 25°C Liquid

Gas

I

Solid

Nitrogen (Cont.) Nitrous oxide ............. Nitrogen trioxide ......... Nitrogen tetroxide ........ Nitrogen pentoxide ........ Ammonia ................ Hydrazine ............... Hydrogen azide ........... Nitrous acid .............. Nitric acid ............... Hydroxylamine ........... Ammonium hydroxide ..... Ammonium nitrate ........ Nitrogen trifluoride ....... Nitrosyl fluoride .......... Ammonium fluoride ....... Nitrogen trichloride ....... Nitrosyl chloride .......... Ammonium chloride ....... Hydrazine hydrochloride ... Ammonium perchlorate .... Nitrosyl bromide .......... Ammonium bromide ...... Ammonium iodide ........ Ammonium hydrogen sulfide ............... Sulfamic acid ............. Sulfamide ................ Ammonium hydrogen sulfate ............... Ammonium sulfate ....... '1

N,O N 2O, N 2 O. N,O, NH, N 2H. HN, HNO, HNO, NH20H NH.OH NH.NO, NH, NOF NH.F NCI, NOCl NH.CI N 2H,Cl NH.CIO. NOBr NH.Br NH.!

44.013 76.012 29.011 108.010 17.031 32.045 43.028· 47.014 63.013 33.030 35.046 80.044 71.002 49.005 37.037 120.366 65.459 53.492 68.506 117.489 109.915 94.924 144.943

19.61 20.01 2.19 2.7 -11. 02 22.80 70.3 -19.0 -32.28

I

12.02 -4.66 -10.3 12.10 63.1 -41.61 -27.3 -86.33 -87.37

-29.8 -15.9 -110.89 55 12.36 -75.15 -47.0 -70.58 19.64 -64.73 -48.14

NH.HS H,NSO,H S02(NH')2

51.111 97.093 96.108

-37.5 -161.3 -129.3

NH.HSO. (l'l"H.)2S0.

115.108 132.139

-245.45 -282.23

Phosphorus phosphorus a, white ............... triclinic, red ............ black .................. amorphous, red ......... Phosphorus ............... Phosphorus .............. Phosphorus trioxide ....... Phosphorus pentoxide ..... Phosphine ................ Metaphosphoric acid ...... Pyrophosphoric acid ....... Phosphorus trifluoride ..... Phosphorus pentafiuoride .. Phosphorus oxyfiuoride .... Phosphorus trichloride ..... Phosphorus pentachloride .. Phosphorus oxychloride .... Phosphorus tribromide .... Phosphorus penta bromide .. Phosphorus oxybromide ...

P

.......... . .......... . ...........

P2 P. P.06 P.O' O PH, HPO, H.P 2 O, PF, PF, POF, -PCla PCI, POCI, PBr, PBr, POBr,

30.974

••

D

•••

••

....... ....... ...... .

. ....... . .......

61. 948 123.895 219.892 283.889 33.998 79.980 177.975 87.969 125.966 103.968 137.333 208.239 153:332 270.701 430.494 286.700

34.5 14.08

..0 .....

....... . ....... . 1.3

....... .

........

. ....... . .......

. .......

0.0 -4.2 -9.4 -1.8

........ ........

-392.0 -713.2

........ ........

-226.7 -535.6

-219.6 -381.4 -289.5

........

-76.4

-89.6

........

-33.3

-142.7 -44.1

........ ....... .

........

•••••••

•••

0

0 .••••

-106.0 -64.5 -109.6

HEATS OF FORMATION AND !tElATS OF COMBUSTlON TABLE

41-2.

4-321

HEATS OF FORMATION OF ELEMENTS AND

INORGANIC COMPOUNDS

(Continued) !:J.HjO, kcal mole- 1 at 25°C

Substance name

I w~~tt I--G-a-s--'-L-iq-u-i-d---;--S-o-l-id--

Formula

Phosphorus (Cont.) Phosphorus triiodide. . . . .. Ammonium dihydrogen phosphate .......... , Ammonium hydrogen ... . phosphate ...... " ... , Ammonium phosphate.....

PI,

411.687

-10.9

NH4H,PO,

115.026

-345.94

(NH,),HP04 (NH,)aPO,

132.057 149.087

-347.50 -399.6

Boron Boron ................... amorphous ............ . Diborane ............... . Boric acid ............... . Boron trichloride. : ....... .

B

0.0 0.9

10.811

B,H 6 H,BO, BCla

27.670 61.833 117.170

8.5 -261.55 -96.50

-102.1

Silicon Silicon. . . . . . . . . . . . . . . . . .. amorphous ............ . Silicon .................. . Silicon monoxide ......... . Silane ................... . Disilane ................. . Metasilic acid ........... . Orthosilic acid ........... . Silicon tetrachloride ...... . Silicon tetrabromide ...... . Silicon tetraiodide ........ . Tetramethylsilane ........ . Hexamethyldisiloxane .... .

Si Si, SiO SiH, Si,H 6 H,SiO, H,Si04 SiC14 SiBr4 SiI, Si(CH,), [(CH,)aSij,O

28.086 56.172 44.085 32.118 62.220 78.100 96.116 169.898 347.722 535.704 88.226 162.382

0.0 1.0 142 -23.8 8.2 19.2 -284.1 -354.0 -157.03 -99.3

-164.2 -109.3

-57.15 -185.88

-63 -194.8

-45.3

Beryllium, Sodium, Potassium Beryllium ............... Beryllium oxide .......... Beryllium fluoride ........ Beryllium chloride ....... Sodium ................. Sodium oxide ............ Sodium hydride .......... Sodium hydroxide ........ Sodium fluoride .......... Sodium chloride ......... Sodium carbonate ........ Sodium formate .......... Sodium acetate .......... Potassium ............... Potassium oxide ......... Potassium hydride ....... Potassium hydroxide. Potassium fluoride ....... Potassium chloride .... ...

. . . . . . . . . . . . . . . .

Be BeO BeF, BeClz Na Na,O NaH NaOH NaF NaCl Na,CO, NaCHO, NaC,H,O, K

K,O KH KOH

. KF . KCI

9.012 25.012 47.009 79.918 22.990 61. 979 23.998 39.997 41.988 58.443 105.989 68.008 82.035 39.102 94.203 40.110 56.109 58.100 74.555

78.0 30.2 -186.1 -85.7 25.9 29.88 -70.1 -43.7

21.52 30.0 -78.2 -.51.0

0.0 -145.0 -245.3 -117.2 0.0 -99.4 -13.7 -101.72 -136.6 -98.5 -269.8 -155.03 -169.8 0.0 -86.4 -15.6 -101.52 -134.4 -104.1

4-322

HEAT TABLE

41-3.

HEATS OF FORMATION AND"HEATS OF

COMBUSTION OF COMPOUNDS OF CARBON

Rcal mole- 1 at 25°C

Substance name

Formula and state

Mol. weight

-!!.Hco !!.Hfo

Gross

I

Net

Carbon and Carbon-Oxygen Carbon ..................... graphite ................... diamond .................. Carbon ..................... Carbon monoxide ............ Carbon suboxide .............

C,g C,e C,e C2,g CO,g C30"g C30,,1

12.011

. ...... .0.·.0.

24.021 28.011 68.032

.......

171.29 0.0 0.45 199.03 -26.42 -22.20 -28.03

265.34 94.05 94.50 387.13 67.64 259.95 254.12

265.34 94.05 94.50 387.13 67.64 259.95 254.12

-17.88 -20.23 -24.81 -28.69 -30.14 -35.31 -32.14 -36.88 -34.98 -41.37 -36.90 -42.92 -39.66 -45.00 -39.92 -47.50 -41.62 -48.80 -40.99 -48.26 -44.32 -50.99 -42.46 -49.46 -44.85 -53.61 -46.57 -54.91 -45.92 -54.32 -45.29 -53.75 -49.25 -57.03 -46.78 -55.79 -48.26 -56.15

212.80 372.82 530.60. 526.72 687.64. 682.47 685.64 680.89 845.16 838.78 843.24. 837.22 840.49 835.14 1,002.59 995.01 1,000.89 993.71 1,001.52 994.25 998.19 991.52 1,000.06 993.05. 1,160.02 1,151.27 1,158.31 1,149.97 1,158.96 1,150.55 1,159.59 1,151.13 1,155.63 1,147.85 1,158.10 1;149.09 1,156.62 1,148.73

191. 76 341.26 :488.52 484.64 635.04 629.87 633.04 628.30 782.. 05 775.,66 780:13 774.11 777.38 772.03 928.95 921.38 927.26 920.08 927.89 920.62 924.56 917.89 926.42 919.42 1,075.87 1,067.12 1,074.16 1,065.82 1,074.80 1,066.40 1,075.44 1,066.98 1,071.48 1,063.70 1,073.95 1,064.94 1,072.47 1,064.58

Carbon-Hydrogen, Alkanes Methane .................... Ethane ..................... Propane ..................... n-Butane .................... 2-Methylpropane (isobutane) .. n-Pentane ................... 2-Methylbutane (isopentane) .. 2,2-Dimethylpropane (neopentane) n-Hexane ................... 2-Methylpentane ............. 3-Methylpentane ............. 2,2-Dimethylbutane .......... 2,3-Dimethylbutane .......... n-Heptane .................. 2-Methylhexane .............. 3-Methylhexane .............. 3 -Ethylpentane ..............

2,2-DimethylpentaIle ......... 2 ;3-Dimethylpentane ......... ~,4-Dimethylpentane .........

CH.,g C 2 H"g C 3H a,g C3HaJ C.H,o,g C.H,o,l C.H,o,g C.H,o,l C 5H",g C,H",l C,H",g C,H",l C,H 12,g C,H",l C.Hu,g C,Hu,l C 6H 14 ,g C,Hu,l C,H,.,g C,Hu,l C,Hu,g C.Hu,l C.H 14,g C.H,.,l C7H",g C7Hu,l C,H16,g C,H16,l C7H16,g C7H16,l C7H16,g C7H",l C7H",g C,H16,l C7H16,g C,H",l C,H16,g C7H,~,l

16.043 30.070 44.097 .0

•••••

58.124 ••••

0

••

58.124 ••••••

0

72.151

.......

72.151 .0

••

0

••

72.151

,

....... 86.178 ....... 86.178 ....... 86.178 ....... 86.178 ....... 86.178

,

.0 . . . 0.

100.206

.......

100.206 ••••

0

••

100.206 .0 . . . . .

100.206 ••••

;

••

!

0

••

100.206 0.0

••

100.206 ••••

0

••

100.206

.......

4-323

HEATS OF FORMATION AND HEATS OF COMBUSTION TABLE

41-3.

HEATS OF FORMATION AND HEATS OF

COMBUSTION OF COMPOUNDS OF CARBON

(Continued) Kcal mole-' at 25°0

Formula and state

Substance name

Mol. weight

-t:..Hco t:..Hr Gross

I

Net

Carbon-Hydrogen, Alkanes (Cont.) 3,3-Dimethylpentane ......... 2,2,3-Trimethylbutane ........ n-Octane .................... 2-Methylheptane ............. 3-Methylheptane ............. 4-Methylheptane ............. 3-Ethylhexane ............... 2,2-Dimethylhexane .......... 2,3-Dimethylhexane .......... 2, 4-Dimethylhexane .......... 2,5-Dimethylhexane .......... 3,3-Dimethylhexane .......... 3,4-Dimethylhexane .......... 2-Methyl-3-ethylpentane ...... 3-Methyl-3-ethylpentane ...... 2,2,3-Trimethylpentane ....... 2,2,4-Trimethylpentane ....... 2,3,3-Trimethylpentane ....... 2,3,4-Trimethylpentane ....... 2,2,3,3-Tetramethylbutane .... n-Nonane ................... 2,2-Dimethylheptane ......... 2,2,3-Trimethylhexane ........ 2,2,4-Trimethylhexane ........

C7H 16,g C7H16,1 C7H16,g C 7H16,1 CsH,s,g CsHlS,1 CSHlS,g CsHlS,1 CSHlS,g CsHlS,1 CSHlS,g CsHlS,1 CSHlS,g CsHlS,1 CSHlS,g CSHlS,1 CSHlS,g CSHlS,l CSHlS,g CSHlS,l CSHlS,g CSHlS,l CSHlS,g CSHlS,l CSHlS,g CSHlS,l CSHlS,g CSHlS,l CSHlS,g CsHlS,1 CSHlS,g CsHlS,1 CSHlS,g CSHlS,l CSHlS,g CsHlS,1 CSHlS,g CsHlS,1 CSHlS,g CsHlS,1 C,H.o,g C,H.o,l C.H.o,g C.H.o,l C.H.o,g C.H.o,l C.H.o,g C.H.o,l

100.206 100.206

....... 114.233

.......

114.233 00

•••••

114.233

....... 114.233

....... 114.233

....... 114.233

......... 114.233

....... 114.233

....... 114.233

....... 114.233 "

.....

114.233

....... 114.233

...... , 114.233

.......

114.233

....... 114.233

.......

114.233

....... 114.233

....... 114.233 .0 • . . . .

128.260

....... 128.260

....... 128.260

....... 128.260

.......

-48.12 -56.05 -48.92 -56.61 -49.79 -59.71 -51.47 -60.96 -50.80 -60.32 -50.66 -60.15 -50.37 -59.85 -53.68 -62.60 -51.10 -60.38 -52.41 -61.44 -53.19 -62.24 -52.58 -61.56 -50.88 -60.20 -50.45 -59.66 -51.36 -60.44 -52.58 -61.41 -53.55 -61.95 -51.70 ...,.60.60 -51.94 -60.96 -53.97 -64.21 -54.56 -65.66 -58.74 -68.85 -57.59 -67.56 -57.85 -67.58

1,156.75 1,148.83 1,155.96 1,148.27 1,317.45 1,307.53 1,315.77 1,306.28 1,316.45 1,306.92 1,316.58 1,307.09 1,316.87 1,307.39 1,313,56 1,304.64 1,316.14 1,306.86 1,314.83 1,305.80 1,314.06 1,305.00 1,314.66 1,305.68 1,316.36 1,307.04 1,316.79 1,307.58 1,315.88 1,306.80 1,314.66 1,305.83 1,313.69 1,305.29 1,315.54 1,306.64 1,315.30 1,306.28 1,313.28 1,303.03 1,475.05 1,463.95 1,470.87 1,460.76 1,472.02 1,462.05 1,471.76 1,462.03

1,072.60 1,064.68 1,071.81 1,064.12 1,222.78 1,212.86 1,221.10 1,211.61 1,221.78 1,212.25 1,221.91 1,212.42 1,222.20 1,212.72 1,218.89 1,209.97 1,221.47 1,212.19 1,220.16 1,211.13 1,219.38 1,210.33 1,219.99 1,211.01 1,221.69 1,212.37 1,222.12 1,212.91 1,221.21 1,212.13 1,219.99 1,211.16 1,219.02 1,210.62 1,220.87 1,211.97 1,220.63 1,211.61 1,218.61 1,208.36 1,369.86 1,358.76 1,365.68 1,355.57 1,366.83 1,356.86 1,366.57 1,356.84

4-324

HEAT TABLE

41-3. HEATS OF FORMATION AND HEATS OF

COMBUSTION OF COMPOUNDS OF CARBON

(Continued) Kcal mole- I at 25°C

Formula and state

Substance name

Mol. weight

-!:J.Hco !:J.Hr

Gross

Net

I Carbon-Hydrogen, Alkanes (Cont.) 2,2,5-Trimethylhexane ........ 2,3,3-Trimethylhexane ........ 2,3,5-Trimethylhexane ........ 2,4,4-Trimethylhexane ........ 3,3,4-Trimethylhexane ........ 2,2-Dimethyl-3-ethylpentane .. 2,4-Dimethyl-3-ethylpentane .. n-Decane ....................

C,H",g C,H,o,l C,H 20 ,g C,H,o,l C,H,o,g C,H,o,l C,H,o,g C,H,o,l C,H,o,g C,H,o,l C,H 20,g C,H·,o,l C,H,o,g C,H,o,l

128.260

. . . . . ..

128.260

.......

128.260 .......

128.260

.......

128.260

.......

128.260 ...

,

...

128.260

.......

Cl0H2~,g

142.287

C,oH 22 ,l

.......

-60.36 -69.97 -57.13 -67.18 -57.91 -67.81 -57.06 -66.87 -56.20 -66.33 -55.21 -65.17 -54.30 -64.42 -59.64 -71.92

1,469.24 1,459.64 1,472.48 1,462.43 1,471.70 1,461. 80 1,472.55 1,462.74 1,473 .41 1,463.28 1,474.40 1,464.44 1,475.31 1,465.19 1,632.34 1,620.06

1,364.06 1,354.45 1,367.29 1,357.24 1,366.51 1,356.61 1,367.36 1,357.55 1,368.22 1,358.09 1,369.21 1,359.25 1,370.12 1,360.00 1,516.63 1,504.35

499.85 655.78 650.22 793.42 786.55 948.86 941.28 1,106.23 1,097.50 1,103.54 1,095.44 1,105.62 1,097.06 1,103.92 1,095.64 1,104.12 1,095.90 1,104.66 1,096.39 1,263.56 1,253.74 1,421.10 1,410.10 1,578.54 1,566.36 1,735.99 1,722.63 1,893.43 1,878.89

468.29 613.70 608.14 740.83 733.96 885.75 878.17 1,032.60 1,023.87 1,029.91 1,021.81 1,031. 99 1,023.43 1,030.29 1,022.01 1,030.49 1,022.27 1,031.03 1,022.76 1,179.41 1,169.59 1,326.43 1,315.43 1,473.35 1,461.17 1,620.28 1,606.92 1,767.20 1,752.66

Carbon-Hydrogen, Cycloalkanes Cyclopropane ................ Cyclobutane ................. Cyclopentane ................ Methylcyclopentane .......... Ethylcyclopentane ........... 1,1-Dimethylcyclopentane .... 1-cis-2-Dimethylcyclopentane.. 1-lrans-2-Dimethylcyclopentane l-cis-3-Dimethylcyclopentane ..

I-trans-3-Dimethylcyclopentane n-Propylcyclopentane ......... n-Butylcyclopentane ......... n- Pentylcyclopentane .........

n-Hexylcyclopentane ......... n-Heptylcyclopentane ........

C,H 6 ,g C 4H s,g C4Hs,1 C 5H' lo ,g CoH,O,l C 6H ,2,g C 6H",l C,H ,4,g C,H ,4,1 C,H 14 ,g C,H 14 ,l C,H ' 4,g C,H, .,! C,H ,4,g C,H ,.,! C 7H14,g C,H ,4,1 C,H 14,g C 7 H 14,l C SH ,6 ,g CSHIO,l C,HIs,g C,H",l C,oH 20,g C,oH2o,1 Cl1H 22 ,g CllH",l C 12 H 24,g C 12 H,.,!

42.081 56.108

.......

70.135

.......

84.163

.......

98.190

.......

98.190 .......

98.190

.......

98.190

....... 98.190 ....... 98.190 .......

112.217 ..... -. 126.244 .......

140.271

..... ..

154.298

.......

168.325 ......

.

12.75 6.32 0.76 -18.41 -25.28 -25.34 -32.92 -30.33 -39.06 -33.02 -41.12 -30.94 -39.50 -32.64 -40.92 -32.44 -40.66 -31.90 -40.17 -35.37 -45.19 -40.19 -51.19 -45.12 -57.30 -50.04 -63.40 -54.96 -69.50

4-325

HEATS OF FORMATION AND HEATS OF COMBUSTION TABLE

41-3.

HEATS OF FORMATION AND HEATS OF

COMBUSTION OF COMPOUNDS OF CARBON

(Continued) Kcal mole- l at 25°C

Formula and state

Substance name

Mol. weight

-t1Hco t1HJO

Gross

I

Net

Carbon-Hydrogen, Cycloalkanes (Cont.) n-Octylcyclopentilne ..........

n-N onylcyclopentane ......... n-DecYlcyclopentane .... , .... Cyc1ohexane ................. Methylcyclohexane ........... Ethylcyclohexane ............ 1,1-Dimethylcyclohexane ..... 1-cis-2-Dimethylcyclohexane .. 1-trans-2-Dimethylcyclohexane 1-cis-3-Dimethylcyclohexane ..

1-trans-3-Dimethylcyclohexane 1-cis-4-Dimethylcyclohexane .. 1-trans-4-Dimethylcyclohexane Cycloheptane ................ Cyclooctane ................. Cyclotetradecane .............

C13H26,g CuH.6,l C14H",g CuH",1 C15H,o,g CuH,o,1 C6H12,g C6Hu,! C,Hu,g C,H",! C 8H",g CsH16,1 C SH l6,g C8Hle.l C SH ,6,g C SH l6,l C,H,6,g CsH",! C SH16,g CSHI.,1 C SH I6,g C 8H 16,1 C sH 16,g CSH,.,1 C sHl. 1,929.47 1,912.77 2,076.69 2,058.81 2,223.30 2,204.24 1,018.82 1,008.43 1,461.11 1,441.61 1,207.83 1,190.46 1,337.00 1,334.29

182.62 173.55 337.02 326.85 494.13 484.75

161.58 152.51 305.46 295.29 452.06 440.68

Carbon-Oxygen-Hydrogen, Alkanols Methanol (methyl alcohol) ....

CH,O,g CH40,1 Ethanol (ethyl alcohol) ....... C2H,O,g C2H,O,1 I-Propanol (n-propyl alcohol) .. C3H80,g C 3H sO,l

32.042

.......

46.070

.......

60.097 ••

•

•••

0

-48.06 -57.13 -56.03 -66.20 -61.28 -72.66

HEAT

4-332 TABLE

41-3.

HEATS 0]' FORMATION AND HEATS OF

COMBUSTION OF COMPOUNDS OF CARBON

(Continued) Kcal mole- 1 at 25°C

Substance name

Formula and state

Mol. weight

-I:!.Hco I:!.HjD

Gross

-

I

Net

Carbon-Oxygen-Hydrogen, Alkanols (Cont.) 2-Propanol (isop:ropyl alcohol) . I-Butanol. .................. 2-Butanol

.

. .... . ..

2-Methyl-l-propanol

... . . . . ..

2"Methyl-2-propanol ......... 1-Pentanol. ................. 2-Pentanol. ................. 3-Pentanol. ................. 2-Methyl-1-butanol .......... 3-Methyl-1-butanol .......... 2-Methyl-2-butanol. .......... 3-Methyl-2-butanol. .......... 1-Hexanol. .......

•••


c', and a > a'

Small sphere of radius a midway between planes a distance 2c apart 1 C .." 1.1128?: 10-10 ( Ii

1 In 2 )-' - c

Sphere of radius b on axis of infinite cylinder of radius a C = b[1.11285 - 0.9277r - 0.114r' - 0.1955r 3

where r = bja.

+ 1.8858r(1

The error is less than 1 part in 4,000 for 0

-

r)-0.S463]

a, placed in a field that would be uniform and of strength E except for the spheroid is T

=

27rE v (K - 1)2b 2aE2(3P - 2) sin 2" 3[(K - 1)2P2 (K - 1)(2 - K)P - 2Kj

+

where P = A[(l + A2) cot- 1 A - A], A = a(b 2 - a 2 )-!, and K = If the -above obb_t.e spheroi.rt is eoncillet.ing, the torque is T

=

E' v- 1 •

27rfvb 2aE2(3P - 2) sin 2a 3P(P - 1)

The torque tending to increase the angle a between the field and the major axis of a prolate dielectric spheroid of capacitivity • with semiaxes a and b where b < a placed in a field that would be uniform and of strength E except for the spheroid is 27rEv(K - 1)2b 2aE2(2 - 3Q) sin 2a + (K - 1)(2 - K)Q - 2K]

T = 3[(K - 1)2Q2 where Q = C[(l - C2) coth- 1 C

+ C].

C = a(a 2 - b2)-! and K =

E'v- 1•

If the above prolate spheroid is conducting, the torque becomes!

T

=

27rEv b 2aE2(2 - 3Q) sin 2a 3Q(Q - 1)

The axis of rotational symmetry of a right circular solid conducting cylinder of radius a and length 2b makes an angle e with a field that would be uniform and of strength E except for the cylinder. The torque tending to align the axis with the field is

T

= 7rEa 2bE2

sin 20(al - at)

I For torque on general ellipsoid, see Stratton, "Electromagnetic Theory," p. 215, McGraw-Hill Book Company, New York, 1941.

ELECTRICITY AND MAGNETISM

where

Gt G)

+ 2.1444 (~rB2B + 0.7171 2 + 0.84883 + 0.369

a1 = 1 at =

G)

0.548

6752

tanho.6 (~) 0.712

The torque vanishes at (a/b) = 1.1958. The errors in these formulas are less tlmu i part in 4,000 for 0.25
a c satisfy the scalar Helmholtz equation, (5b-76) with k 2 = W 2P.E. By choosing a appropriately, one may also apply'Eqs. (5b-74) and (5b-75) to the case of an inhomogeneous medium.2 Basic Wave Types

1. Transverse electromagnetic waves (TEM waves)-containing neither an electric nor a magnetic field component in the direction of propagation. 2. Transverse magnetic waves (TM or E waves)-containing an electric field component but not a magnetic field cotnponent in the direction of· propagation. 1 The vector aef> may be identified as the electric Hertz ~ector and, the vector a'lr may be identified as the magnetic Hertz vector. a in this case is a constant vector. 2 c. Yeh, Phys. Rev. 131, 2350 (1963).

6-46

ELECTRICITY AND MAGNETISM

3. Transverse electric waves (TE or H waves)~ontaining a magnetic ·field.component but not an electric field component in the direction of propagation. 4. Hybrid waves (HE waves)-containing all components of electric and magnetic fields. These hybrid waves are obtainable by linear superposition of TE and TM waves. Formal Solutions for the Time-harmonic Vector Wave Equation. INTEGRAL REPRESENTATIONS. Upon direct integration of the wave equation in homogeneous isotropic medium, integral solutions in terms of the sources can be obtained.. Aharmonic time dependence of e'w, is assumed and suppressed in this section. Direct integration of Eqs. (5b-68) to (5b-7I) gives p, ~.

A. = -4 'It"

eiklr-r'l

-I-'-'1 dv'

(5b-77)

V· J(r') r - r

r -3 H

Resistance R,ohms/m

Internal illductance Lo, henrys/m (for high frequency)

~: (~+~) ~

2R. [

-;a:

sid Vi (s/d) 2

] -

+ 1 + 2p2 (1 4p4 SR •• [1 +q + "D q

2R.2 [1 "d

1

2

R

'"

_ 4

2 _

q

2)]

1+Sp44p2 ]

~ 2R.

b

6-:-60

ELECTRICITY AND. MAGNETISM

two differe,nt media. Special features of surface-wave modes ~ having the ilsual propagation constant ei"{z along the axis and the structure are given in the following: 1. The field is characterized by an exponential decay away from the surface of the structure. 2. In most cases in which dEo, p./p.o > 1, the phase velocities of the propagating surface-wave modes are less than the velocity of light in vacuum. 3. Below the cutoff frequency, a mode simply does not exist. In other words unlike the bounded waveguide' case no evanescent mode exists. 4. The finite number of discrete surface-wave modes does riot represent a complete set of solutions. In addition to.the eigenfunction solutipns there exists solutions with a continuous spectrum. (This property is in direct contrast to the mode property in bounded waveguides.) 5. Only TE, TJJ1, or HE modes may exist on a surface-wave structure. Detailed formulas are given for the circular dielectric waveguide as a representative surface-wave structure. It is understood that all fields vary as ei"{z-iwt. The dielectric rod of radiusa~having· E1 and P.o as its permittivity and permeability" is assumed ,to be embedded in another dielectric medium with E = EO and p. = p.o. F;urthermore E1

>

EO·

FIELD COMPONENTS

1. HEnm modes with n

~ 0: Ez = A nJ n(slr) cos ncf> = B"Kn(sorYcos ncf> Hz = C"Jn(Slr) sin ncf> = DnKn(sor) sin ncf>

r:$a r;:::a r:$a r;:::a

(5b-148) (5b-149) (5b~150)

(5b-151)

2. TMom modes:

E. =A.Jo(slr)

r :$ a

= BoKo(sor) Hz = 0

r;:::a for all r

Ez = 0 Ez = C OJ O(slr)

for all r r:$a r;:::a

3. TEom modes: = DoKo(sor)

All other transverse field components may be found from Eqs. (5b-120) to (5b-123) with E = E1, P. = P.o for r :$ a and E = EO, P. = P.o for r ;::: a. An, Bn, Cn, Dn are amplitude coefficients. I n and Kn are respectively the Bessel and modified Bessel functions. PROPAGATION CONSTANT. The propagation constant 'Y is obtained by solving the following equations: 1. HEnm modes (n ~ 0):

(5b"152) (5b~153)

(5b-154) 2. TMom modes: (EdEO)J~(sla)

slaJO(sla) with Eqs •. (5b-153) and (5b-154)

K~(soa) + soaKo(soa) .

= 0

(5b-155)

TABLE

5b-4.

SEVERAL IMPORTAWI FORMULAS FOR SOME COMMON TRANSMISSION LINES*

Quantity

General lire

Pr pagation constant I' = '" - i{3.

Ideal line

VCR - iwL)(G - iwC)

Ph 'se constant {3 .....................

1m (1')

At enuation constant", ...............

Re (1')

w

-iw VLC w 271'

VLC

= -

v

=-

- iwl~ Ch ,racteristic impedance Z o........... G - iwe: Inp ut impedance Z; . ................. Z (ZL cosh 1'1 -t- Zo sinh 1'1) o Z 0 cosh 1'1 ZL sinh 1'1

Zo

Zo tanh 1'1

-iZo tan {31

1m Jedance of open line ...............

Zo coth 1'1

+iZo cot {31

"'I)

Zo'

(ZL sinh al Z:' cosh Zo sinh al Z1. cosh al (ZL cosh al Zo sinh cd) 1m Jedance of half-wave line ........... Zo Zo cosh al Z}; sinh al Vo tage along line V(z) ..... ........... Vi cosh 'YZ - IiZO sinh 'Yz I i COSh 'YZ - Zo Vi mn . h 'Yz Cu rent along line I(z) . ...............

Re lection coefficient KR . .............

ZL - Zo ZL Zo

Sta llding-wave ratio ..................

1 + IKEI 1 - IKRI

per unit length l = length of line Subscript i denotes input end quantities.

ZL ZL

(3 below)

G'

R' )

+ 8w'C' + 8w'L' GZo

2

~~[l-i(~-J!:...-)J C 2wC 2wL

("'Icoscos

(31 - i sin (31) {31 - ial sin {31 Z (cos {31 - ial sin (31 ) o al cos {31 - i sin {31 Z

o

Zo (Zo + ZLal) ZL + Zoal Zo (ZL + Zoal) Zo + ZLal

I:;j

o

~

cj t"< ~

UJ.

ZL - Zo ZL + Zo

1 + IKRI 1 - IRRI z

= distance along line from input end

A

=

wavelength measured along line

v = phase velocity of line equals velocity of light in dielectric of line for an ideal line

Subscript L denotes load end quantities.

* Ramo and Whinnery,

(See", and RG 1 - 4w'LC J!:...- + 2Zo

Vi cos {3z + iliZo sin (3z .Vi. Ii cos {3z + t Zo sm {3z

+

R. L, G, C = distributed resistance, inductance, conductance, capacitance

VLC

(ZL cos (31 - iZo sin (31) Zo cos {31 - iZL sin {31

1m Jedance of shorted line .............

+ + + +

w

~~

+

Zo

A

(

0

~R

1m Jedance of quarter-wave line ........

Approximate results for low-loss lines

"Fields and Waves in Modern Radio," 2d ed., John Wiley & Sons, Inc., New York, 1953.

en J

~

5-58

ELECTRICITY AND MAGNETISM

Conventional TEM Transmission Lines. For a two-conductor uniform line supporting the TEM waves, the differential equations for the voltage V and current I are aV = -L aI _ RI az at

(5b-140)

'!!

(5b-141)

az

-C aV - GV at

=

where L, C, R, and G are the inductance, capacitance, resistance, and conductance, respectively, all per unit length of the line. If steady-state sinusoidal conditions of the form e-iwt are considered, then the equations become aV -(R - iwL)I (5b-142) az aI -(G - iwC)V (5b-143) az Combining the above equations gives (5b-144) where the propagation constant

x

=

VCR - iwL)(G - iwC)

=

a

+ i(3

(5b-145)

The solution for Eq. (5b-144) is V '= Ae-xz I =

+ Bexz

-.l (Ae-Xz Zo

(5b-146)

_ BeXz)

(5b-147)

where Zo = VCR - iwL)/(G - iwC) and is called the characteristic impedance. A and B are constants to be determined according to the input and termination conditions. TaLles 5L-3 aud 5b-4 summarize constants for some common lines and some important formulas for transmission lines. Another kind of quasi-TEM microwave transmission line is the strip line 1 which basically consists of two (or more) parallel metallic strips of generally different width separated by a dielectric medium. This structure cannot support a TEM wave although the dominant mode closely resembles the TEM wave of a simplified microstrip with dielectric material uniformly filling the entire region. Under this TEM wave approximation, the problem is essentially one of finding the electrostatic potential iI>(x,Y) which satisfies the Laplace's equation V 2