C$)$ The enthalpy departure in terms of volume$-$explicit $v=v(T,P)$ volumetric EOS is given by Equation $(4C)$ below:

$h_{(T,P)}^{\text{d}\text{e}\text{p}}={\displaystyle \underset{P=0}{\overset{P=P}{}}}[v_{-}-T(\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{T}}{)}_P]dP$

i$)$ Transform $($showing your calculations clearly$)$ Eq$.(4$C$)$ to in terms of the compressibility facior $(Z)$ and pressure$-$explicit $P=P(T,V)$ volumetric EOS

ii$)$ Use your answer in Q$4$ C $($i$)$ above and the van der Waals $($VdW$)$ EOS, calculate the molar enthalpy departure for $0\mathrm{.}0016m^3$ of butane at $2$MPa and $433K[10$ marks$]$ Data

$VdW*EOS$:$P=\frac{R_{0}T}{v-b}-\frac{a}{w^2}aa=1\mathrm{.}4m^{6}P$