Chapter 10, Problem 10.32P

Interpretation Introduction

Interpretation:

We have to prove excess properties of binary gas mixtures in given terms.

Concept introduction:

The gas mixture fugacity coefficients are partial molar properties of the excess Gibbs free energy, hence partial molar properties of the total fugacity can be written as follows:

  $\ln \varphi ={\left[\frac{\partial \left(G^E/RT\right)}{\partial n_i}\right]}_{P,T,n_j}$

Excess properties are the difference between real gas property to ideal gas property.

Equation (3.36) and (10.62) mentioned in question are:

  $\begin{array}{l}Z=1+\frac{BP}{RT}\\ B={\displaystyle \sum _i{\displaystyle \sum _j{y}_i{y}_j{B}_{i,j}}}\end{array}$

All excess properties in terms of excess Gibbs free energy given in Table 10.1 in the book are:

  $\begin{array}{l}{V}^E={\left(\frac{\partial \left(G^E\right)}{\partial P}\right)}_{T,y}\\ S^E=-{\left(\frac{\partial \left(G^E\right)}{\partial T}\right)}_{P,y}\\ H^E=G^E+T{S}^E\\ H^E=G^E+T\left[-{\left(\frac{\partial \left(G^E\right)}{\partial T}\right)}_{P,y}\right]\\ H^E=-R{T}^2{\left(\frac{\partial \left(G^E/RT\right)}{\partial T}\right)}_{P,y}\\ C_P^E={\left(\frac{\partial H^E}{\partial T}\right)}_{P,y}\\ C_P^E=-T{\left(\frac{{\partial }^2{G}^E}{\partial T^2}\right)}_{P,y}\end{array}$