Derive the following two relations:

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

and

$$T d S=C_{V} d T+\left.T \frac{\partial p}{\partial T}\right|_{V} d V$$

[You may use any standard Maxwell relation without proving it.]

Experimentalists very seldom measure $C_{V}$ directly; they measure $C_{p}$ and use thermodynamics to extract $C_{V}$. Use your results from the first part of this question to find a formula for $C_{p}-C_{V}$ in terms of the easily measured quantities

$$\alpha=\left.\frac{1}{V} \frac{\partial V}{\partial T}\right|_{p}$$

(the volume coefficient of expansion) and

$$\kappa=-\left.\frac{1}{V} \frac{\partial V}{\partial p}\right|_{T}$$

(the isothermal compressibility).