A monoatomic ideal gas goes through a process $$p = {p_0} - \alpha V$$   where $${p_0}$$ and $$\alpha $$ are positive constants and $$V$$ is its volume. At what volume will the entropy of gas be maximum?

Solution :
$$\eqalign{ & ds = n{C_v}dT + PdV = 0 \cr & nR\frac{{dT}}{{dV}} + \left( {{p_0} - \alpha V} \right) = 0 \cr & pV = nRT \cr & {p_0}V - \alpha {V^2} = nRT \cr & {p_0} - 2\alpha V = nR\frac{{dT}}{{dV}} \cr & - \left( {{p_0} - \alpha V} \right)\left( {\gamma - 1} \right) = {p_0} - 2\alpha V \cr & - {p_0}\left( {\gamma - 1} \right) + \alpha \left( {\gamma - 1} \right)V = {p_0} - 2\alpha V \cr & {p_0}V = \alpha V\left( {\gamma + 1} \right) \cr & V = \frac{{{p_0}\gamma }}{{\alpha (\gamma + 1)}} \cr & V = \frac{{{p_0} \times \frac{5}{3}}}{{\alpha \left( {\frac{5}{3} + 1} \right)}} = \frac{{5{p_0}}}{{8\alpha }} \cr} $$