Using the Gibbs free energy $G(T, P)=E-T S+P V$, derive the Maxwell relation

$$\left.\frac{\partial S}{\partial P}\right|_{T}=-\left.\frac{\partial V}{\partial T}\right|_{P}$$

Define the notions of heat capacity at constant volume, $C_{V}$, and heat capacity at constant pressure, $C_{P}$. Show that

$$C_{P}-C_{V}=\left.\left.T \frac{\partial V}{\partial T}\right|_{P} \frac{\partial P}{\partial T}\right|_{V}$$

Derive the Clausius-Clapeyron relation for $\frac{d P}{d T}$ along the first-order phase transition curve between a liquid and a gas. Find the simplified form of this relation, assuming the gas has much larger volume than the liquid and that the gas is ideal. Assuming further that the latent heat is a constant, determine the form of $P$ as a function of $T$ along the phase transition curve. [You may assume there is no discontinuity in the Gibbs free energy across the phase transition curve.]