OUESTION $4.$ Thermodynamics. The chemical potential of an ideal gas is given by

${}_{i,IG}(T,P)={}_{i,IG}(T,p_i)+RTln(\frac{P}{p_i})$

The chemical potential of component $i$ in an ideal gas mixture at temperature $T$ and pressure $P$ is:

${}_{i,\text{I}\text{G}\text{M}}(T,P)={}_{i,IG}(T,p_i)$

Where $p_i=y_{i}P$ is the partial pressure of the ideal gas component in the ideal gas mixture.

$($a$)$ Use these equations as starting point and define the fugacity hat$(f{)}_i$ and fugacity coefficient hat$({)}_i$ of a

component in a real gas mixture.

$($b$)$ Show that RTlnhat$({)}_i$ is a partial molar quantity and write down an equation that will allow the

determination of $lnhat({)}_i$ from $ln.$

$($c$)$ Start from the relationship constant, $P$ and show that the fugacity coefficient of

a mixture can be evaluated from $ln={\displaystyle \underset{0}{\overset{P}{}}}(z-1)\frac{dP}{P}$

$($d$)$ Show that along any isotherm

$\frac{dP}{P}=\frac{dz}{z}-\frac{dV}{V}$

$($e$)$ Now show that $ln={\displaystyle \underset{V}{\overset{}{}}}(z-1)\frac{dV}{V}+z-1-lnz$

$($f$)$ Consider a binary mixture described by this modified van der Waals EOS with $a=0.$ Let the mixing rule

for the $b$ parameter be given by: $b=(y_1\sqrt[\phantom{2}]{b_1}+y_2\sqrt[\phantom{2}]{b_2}{)}^2.$ Use the expressions above to show that the

fugacity coefficient for component $1$ can then be expressed as:

RTlnhat$({)}_1=P(C_1+C_2{y}_2^2)$

Determine the constants $C_1$ and $C_2.$