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CHEN 354 – Chemical Engineering Thermodynamics II Due Wednesday, February 10th, 2016

Spring 2016

Partial Molar Properties Problem 1. The molar volume (cm3mol-1) of a binary liquid mixture at T and P is given by: 𝑉 = 120𝑥1 + 70𝑥2 + 𝑥1 𝑥2 (15𝑥1 + 8𝑥2 ) a) Find expressions for the partial molar volume of species 1 and 2 at T and P. ∞ b) Plot values of V, 𝑉�1 and 𝑉�2 vs x1. Identify the following points in the graph: V1, V2, 𝑉�1 , and ∞ 𝑉�2 with their respective values. � 𝑑𝑉

c) Demonstrate that �𝑑𝑥1 � 1

𝑥1 =1

� 𝑑𝑉

= �𝑑𝑥2 � 1

𝑥1 =0

=0

Problem 2. The molar enthalpy (J.mol-1) of a binary liquid mixture of species 1 and 2 at T and P is given by: 𝐻 = 375𝑥1 + 273𝑥2 + 𝑥1 𝑥2 (45𝑥1 + 36𝑥2 ) a) Find expressions for the partial molar enthalpy of species 1 and 2 at T and P. b) Use the expressions found in numeral a) to demonstrate the summability relation, which states ��1� + 𝑥2 �𝐻��2� that 𝐻 = 𝑥1 �𝐻 ��1� and �𝐻��2� vs x1. Identify the following points in the graph: H1, H2, 𝐻 ���1�∞ , and c) Plot values of H, �𝐻 ∞ �� 𝐻��2 with their respective values.

Problem 3. The partial molar volume of species a in a binary mixture (a + b) at constant T and P is given by: 𝑣̅𝑎 = 𝑣𝑎0 − 0.2𝑥𝑏2 0 Where 𝑣𝑎 is the molar volume of pure species a, and xb the mol fraction of species b. If the molar volume of pure species b is 𝑣𝑏0 , derive an expression for 𝑣̅𝑏 . Draw a sketch showing 𝑣̅𝑎 , 𝑣̅𝑏 and xb.

Problem 4. The Gibbs-Duhem equation can be deduced by combining the total derivative of property M (P, T, xi) in terms of P, T and x: 𝜕𝜕 𝜕𝜕 �𝑖 𝑑𝑥𝑖 𝑑𝑑 = � � 𝑑𝑑 + � � 𝑑𝑑 + � 𝑀 𝜕𝜕 𝑇,𝑥 𝜕𝜕 𝑃,𝑥 𝑖 �: and the total derivative of M in terms of partial molar 𝑀 �𝑖 � 𝑑 �𝑛𝑛 = � 𝑛𝑖 𝑀 𝑖

a) Use these two equations to derive the Gibbs-Duhem equation: 𝜕𝜕 𝜕𝜕 �𝑖 = 0 � � 𝑑𝑑 + � � 𝑑𝑑 − � 𝑥𝑖 𝑑𝑀 𝜕𝜕 𝑇,𝑥 𝜕𝜕 𝑃,𝑥 𝑖 b) Deduce an expression for Gibbs-Duhem equation for M = G (Gibbs free energy) of a binary mixture at constant T and P. Give your answer in terms of chemical potential. c) Briefly comment on the importance and applications of the Gibbs-Duhem equation. d) Demonstrate that the expressions for the molar volume and molar enthalpy given in problems 1 and 2 respectively, satisfy the Gibbs-Duhem equation.