 Solution

ID: past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2014/paper-1/1/b/solution

Using

C_{V} = T (\partial S / \partial T)_{V}

and the stated Maxwell relation, interchange derivatives within a smooth single phase:

(\frac{\partial C _{V}}{\partial V})_{T} = T \frac{\partial}{\partial T} (\frac{\partial S}{\partial V})_{T} = T (\frac{\partial ^{2} p}{\partial T ^{2}})_{V} = 0.

Therefore $C_{V}$ depends only on temperature for the given temperature-linear equation of state. It need not be constant in temperature.

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