PLEASE SHOW UR WORK ON HOW THEY GOT THESE EQUATIONS Problem $2.$ Given the equation of state $P(\frac{V}{n}-b)=RT$ and the following relations

$(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T=T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V-P$

$(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{P}}{)}_T=-T(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P+V$

$($a$)$ evaluate $(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V,(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P,(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T,$ and $(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{P}}{)}_T$

$($b$)$ evaluate $U$ and $H$ for an isothermal process from $P_1,V_1$ to $P_2,V_2$Problem $2.$ Given the equation of state $P(\frac{V}{n}-b)=RT$ and the following relations

$(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T=T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V-P$

$(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{P}}{)}_T=-T(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P+V$

evaluate $(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V,(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P,(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T,$ and $(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{P}}{)}_T$

$($a$)$

$(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V=\frac{nR}{V-nb}$

$(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P=\frac{nR}{P}$

From the EOS, $P=R\frac{T}{\frac{V}{n}-b}$

$(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_T=T(\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{T}}{)}_V-P=T(\frac{nR}{V-nb})-\frac{RT}{(\frac{V}{n}-b)}=T(\frac{nR}{V-nb})-\frac{nRT}{(V-bn)}=0$

From the EOS, $V=\frac{nRT}{P}+nb$

$(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{P}}{)}_T=T(\frac{\text{d}\text{e}\text{l}\text{V}}{\text{d}\text{e}\text{l}\text{T}}{)}_P+V=-T(\frac{nR}{P})+V=-T(\frac{nR}{P})+\frac{nRT}{P}+nb=nb$

$($b$)$

$U={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(\frac{\text{d}\text{e}\text{l}\text{U}}{\text{d}\text{e}\text{l}\text{V}}{)}_{T}dV={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}0dV=0$

$H={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(\frac{\text{d}\text{e}\text{l}\text{H}}{\text{d}\text{e}\text{l}\text{P}}{)}_{T}dP={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}nbdP=nb(P_2-P_1)$