a$)$ Given the fugacity coefficient, $=\frac{f}{P}=$exp$(\frac{1}{RT}{\displaystyle \underset{0}{\overset{P}{}}}((\frac{?}{\text{b}\text{a}\text{r}}(V))\frac{-}{b}ar(V^{id}))dP),$ a real gas following the equation of state below: $P(\frac{?}{\text{b}\text{a}\text{r}}(V)-b)e^{\frac{a}{RT(\frac{?}{\text{b}\text{a}\text{r}}(V))}}=RT$

Prove that $f=$Pexp$[\frac{P}{RT}(-\frac{a}{RT}+b)]$

Hint: Use Taylor serial expansion and only keep the first order correction. The final expression of fugacity contains $P,R,T,a,$ and $b.$

b$)$ Calculate the fugacity of this real gas at $100$ bar and $400K.$ a $=1\mathrm{.}5L^2.$ bar. $mo{l}^{-2}$ and b $=0\mathrm{.}04L*mo{l}^{-1}.$

c$)$ Calculate the difference in chemical potential of this real gas and an ideal gas.

d$)$ Calculate the Helmholtz energy $\frac{?}{\text{b}\text{a}\text{r}}(A)$ of this gas at $100$ bar and $400K$ given the standard molar Gibbs energy $\frac{?}{\text{b}\text{a}\text{r}}(G{)}^o$ is $-100k\frac{J}{m}ol.$