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 / d / Solution

Equating the initial and final internal energies, with the same additive constant,soThis is cooling in free expansion of a van der Waals gas. The gas cools because separating attractive particles raises their potential energy at the expense of kinetic energy. The excluded-volume coefficient $b$ cancels from this caloric model. The result assumes a homogeneous equilibrium gas with $V_0/3>Nb$ and positive $T_f$; a two-phase final state would require a coexistence calculation rather than this homogeneous formula. The ideal-gas limit $a=0$ has no temperature change.