Assuming that entropy is a continuous function, \(s=s(T, v)\), derive the expression for entropy change

\[\mathrm{d} s=c_{v} \frac{\mathrm{d} T}{T}+\left(\frac{\partial p}{\partial T}\right)_{v} \mathrm{~d} v\]

and similarly, for \(s=s(T, p)\) derive

\[\mathrm{d} s=c_{p} \frac{\mathrm{d} T}{T}-\left(\frac{\partial v}{\partial T}\right)_{p} \mathrm{~d} p\]

Apply these relationships to a gas obeying van der Waals equation

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

and derive an equation for the change of entropy during a process in terms of the basic properties and gas parameters. Also derive an expression for the change of internal energy as such a gas undergoes a process.

\[\left[s_{2}-s_{1}=c_{p} \ln \frac{T_{2}}{T_{1}}-R \ln \left(\frac{p_{2}+\mathrm{a} / v^{2}-2 \mathrm{a}(v-\mathrm{b}) / v^{3}}{p_{1}+\mathrm{a} / v^{2}-2 \mathrm{a}(v-\mathrm{b}) / v^{3}}\right) ; u_{2}-u_{1}=\int_{1}^{2} c_{v} \mathrm{~d} T-\mathrm{a}\left(\frac{v_{1}-v_{2}}{v_{1} v_{2}}\right)\right]\]