(Solved): (a) Derive an equation relating the Joule-Thomson coefficient (JT) to the volume expansion coe ...

(a) Derive an equation relating the Joule-Thomson coefficient $\left({?}_{JT}\right)$ to the volume expansion coefficient $(?)$, the molar heat capacity at constant pressure $\left(c_p\right)$ and the molar volume $(v)$. Note the definitions $\begin{array}{c}{?}_{JT}={\left(\frac{?T}{?P}\right)}_h\\ ?=\frac{1}{V}{\left(\frac{?V}{?T}\right)}_P\end{array}$ $[40\%]$ (b) A certain fluid has the equation of state $P(\mathrm{v}?\mathrm{b})=RT?\frac{AP}{T^{3.33}}$ where $\mathrm{A}$ and $\mathrm{b}$ are constants $\begin{array}{l}A=2.00\times 10^3{\mathrm{m}}^3{\mathrm{K}}^{3.33}{\mathrm{mol}}^{?1}\\ b=0.18{\mathrm{m}}^3{\mathrm{mol}}^{?1}.\end{array}$ Calculate the inversion temperature for the fluid i.e., the temperature at which the Joule-Thomson coefficient is zero.