(a) Show that for a gas obeying the van der Waals equation of state (10.3.9),

\[C_{P}-C_{V}=N k\left\{1-\frac{2 a}{k T v^{3}}(v-b)^{2}\right\}^{-1}\]

(b) Also show that, for a van der Waals gas with constant specific heat \(C_{V}\), an adiabatic process conforms to the equation

\[(v-b) T^{C_{V} / N k}=\text { const.; }\]

compare with equation (1.4.30).

(c) Further show that the temperature change resulting from an expansion of the gas (into vacuum) from volume \(V_{1}\) to volume \(V_{2}\) is given by

\[T_{2}-T_{1}=\frac{N^{2} a}{C_{V}}\left(\frac{1}{V_{2}}-\frac{1}{V_{1}}\right)\]