How can the equation of state in the form of a relationship between pressure, volume and temperature be used to extend limited data on the entropy of a substance.

A certain gas, A, has the equation of state

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

where \(\alpha\) is a function of temperature alone. Show that

\[\left(\frac{\delta s}{\delta p}\right)_{T}=-\Re\left(\frac{1}{p}+\alpha+T \frac{\mathrm{d} \alpha}{\mathrm{d} T}\right)\]

Another gas B behaves as an ideal gas. If the molar entropy of gas A is equal to that of gas B when both are at pressure \(p_{0}\) and the same temperature \(T\), show that if the pressure is increased to \(p\) with the temperature maintained constant at \(T\) the molar entropy of gas B exceeds that of gas A by an amount

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