For a van der Waals gas that obeys the state equation

\[p=\frac{R T}{v-\mathrm{b}}-\frac{\mathrm{a}}{v^{2}}\]

shows that the coefficient of thermal expansion, \(\beta\), is given by \[\beta=\frac{R v^{2}(v-\mathrm{b})}{R T v^{3}-2 \mathrm{a}(v-\mathrm{b})^{2}}\]
and the isothermal compressibility, \(k\), is given by \[k=\frac{v^{2}(v-\mathrm{b})^{2}}{R T v^{3}-2 \mathrm{a}(v-\mathrm{b})^{2}}\]
Also evaluate \(T\left(\frac{\partial p}{\partial v}\right)_{T}\left(\frac{\partial v}{\partial T}\right)_{p}^{2}\) for the van der Waals gas. From this find \(c_{p}-c_{v}\) for an ideal gas, stating any assumptions made in arriving at the solution.