Derive the Maxwell relation

$$\left(\frac{\partial S}{\partial p}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{p}$$

The diagram below illustrates the Joule-Thomson throttling process for a porous barrier. A gas of volume $V_{1}$, initially on the left-hand side of a thermally insulated pipe, is forced by a piston to go through the barrier using constant pressure $p_{1}$. As a result the gas flows to the right-hand side, resisted by a piston which applies a constant pressure $p_{2}$ (with $p_{2}<p_{1}$ ). Eventually all of the gas occupies a volume $V_{2}$ on the right-hand side. Show that this process conserves enthalpy.

The Joule-Thomson coefficient $\mu_{\mathrm{JT}}$ is the change in temperature with respect to a change in pressure during a process that conserves enthalpy $H$. Express the JouleThomson coefficient, $\mu_{\mathrm{JT}} \equiv\left(\frac{\partial T}{\partial p}\right)_{H}$, in terms of $T, V$, the heat capacity at constant pressure $C_{p}$, and the volume coefficient of expansion $\alpha \equiv \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}$.

What is $\mu_{\mathrm{JT}}$ for an ideal gas?

If one wishes to use the Joule-Thomson process to cool a real (non-ideal) gas, what must the $\operatorname{sign}$ of $\mu_{\mathrm{JT}}$ be?