A virial gas changes its Joule-Thomson sign at twice the characteristic temperature

A gas obeys the low-pressure molar-volume model \[ V(T,P)=\frac{RT}{P}+b\left(1-\frac{T_*}{T}\right), \] where $b>0$ and $T_*>0$. Over each small pressure step below, take the molar heat capacity $C_P$ and $b$ as constant. A throttle is steady, adiabatic, has negligible changes in kinetic and potential energy, and therefore has $dH=0$. 1. Starting from \[ dH=C_P\,dT+\left[V-T\left(\frac{\partial V}{\partial T}\right)_P\right]dP, \] derive \[ \mu_{JT}=\left(\frac{\partial T}{\partial P}\right)_H =\frac{T(\partial V/\partial T)_P-V}{C_P}. \] 2. Insert the volume model and show that \[ \mu_{JT}=\frac b{C_P}\left(\frac{2T_*}{T}-1\right). \] Find the inversion temperature and state which side cools under a pressure drop. 3. Let $T_*=300\ \mathrm{K}$ and $b/C_P=1.00\times10^{-6}\ \mathrm{K\,Pa^{-1}}$. For a small pressure change $\Delta P=-1.00\times10^5\ \mathrm{Pa}$, estimate $\Delta T\approx\mu_{JT}\Delta P$ first at $T=300\ \mathrm{K}$ and then at $T=1200\ \mathrm{K}$. 4. Explain why these are local linear estimates and why the ideal term $RT/P$ contributes zero to the numerator. Do not replace the specified virial model by a van der Waals spinodal calculation.

At fixed $P$, $(\partial V/\partial T)_P=R/P+bT_*/T^2$.