(a) A sample of gas has pressure $p$, volume $V$, temperature $T$ and entropy $S$.

(i) Use the first law of thermodynamics to derive the Maxwell relation

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

(ii) Define the heat capacity at constant pressure $C_{p}$ and the enthalpy $H$ and show that $C_{p}=(\partial H / \partial T)_{p}$.

(b) Consider a perfectly insulated pipe with a throttle valve, as shown.

Gas initially occupying volume $V_{1}$ on the left is forced slowly through the valve at constant pressure $p_{1}$. A constant pressure $p_{2}$ is maintained on the right and the final volume occupied by the gas after passing through the valve is $V_{2}$.

(i) Show that the enthalpy $H$ of the gas is unchanged by this process.

(ii) The Joule-Thomson coefficient is defined to be $\mu=(\partial T / \partial p)_{H}$. Show that

$$\mu=\frac{V}{C_{p}}\left[\frac{T}{V}\left(\frac{\partial V}{\partial T}\right)_{p}-1\right]$$

[You may assume the identity $(\partial y / \partial x)_{u}=-(\partial u / \partial x)_{y} /(\partial u / \partial y)_{x} \cdot$ ]

(iii) Suppose that the gas obeys an equation of state

$$p=k_{B} T\left[\frac{N}{V}+B_{2}(T) \frac{N^{2}}{V^{2}}\right]$$

where $N$ is the number of particles. Calculate $\mu$ to first order in $N / V$ and hence derive a condition on $\frac{d}{d T}\left(\frac{B_{2}(T)}{T}\right)$ for obtaining a positive Joule-Thomson coefficient.