Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section B / B2 / c / Solution

At fixed particle number, $dU=TdS-PdV$. The stated Maxwell relation givesFor the intended classical monatomic Van der Waals gas, equipartition supplies $f(T)=3N{k}_{B}T/2$: the attractive mean-field term is temperature independent and there are three translational quadratic degrees of freedom. Consequently,The caloric closure of the van der Waals equation is essential. Caloric assumption: the pressure equation alone determines the volume dependence, but not the function $f(T)$. The monatomic kinetic-energy model is the additional assumption behind the claimed formula.^[ref]