The equation of state for a certain gas is

\[\frac{p v}{\Re T}=1+p \mathrm{e}^{-\mathrm{A} T}\]

where A is constant.

Show that if the specific heat at constant pressure at some datum pressure \(p_{0}\) is \(c_{p_{0}}\), then the value of the specific heat at constant pressure at the state \((T, p)\) is given by \[c_{p}-c_{p_{0}}=\Re T \mathrm{~A} e^{-\mathrm{A} T}(2-\mathrm{A} T)\left(p-p_{o}\right) .\]