Show, for a gas obeying the state equation

\[p v=(1+\alpha) \Re T\]

where \(\alpha\) is a function of temperature alone, that the specific heat at constant pressure is given by

\[c_{p}=-\Re T \frac{\mathrm{d}^{2}(\alpha T)}{\mathrm{d} T^{2}} \ln p+c_{p_{0}}\]

where \(c_{p_{0}}\) is the specific heat at unit pressure.

\(\left[c_{p}=-\Re T \frac{\mathrm{d}^{2}(\alpha T)}{\mathrm{d} T^{2}} \ln p+c_{p_{0}}\right]\)