QUESTION $4$

By the departure function methodology, the changes in thermodynamic properties of real gases can be estimated from ideal gas heat capacities and real gas volumetric Equation Of State $($EOS$).$

A$)$ Explain why ideal gas heat capacities are used for calculation of property changes for real gases under the departure function methodology

$[4$ marks$]$

B$)$ The computed departure function of a property is a positive numerical value. $($Y$$s or No$)?$ Why?

$[4$ marks$]$

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

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

Eq$.(4$C$)$

i$)$ Transform $($showing your calculations clearly$)$ Eq$.(4$C$)$ to $h_{(T,p)}^{\text{d}\text{e}\text{p}}$ in terms of the compressibility factor $(Z)$ and pressure$-$explicit $P=P(T,v)$ volumetric EOS $[7$ marks$]$

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

$VdW*EOS$:$P=\frac{R_{0}T}{v-b}-\frac{a}{v^2}$;$a=1\mathrm{.}4m^{6}P\frac{a}{m}o{l}^2$;$b=0\mathrm{.}00012\frac{m^3}{m}ol$;$R_0=8\mathrm{.}314\frac{J}{m}$olK;$Z=0\mathrm{.}81$