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

The Van der Waals equation replaces the freely available volume by $V-Nb$ to represent short-range repulsion and excluded volume. The attractive mean field reduces the pressure by $a{N}^2/V^2$: particles near the boundary are pulled back into the gas, lowering their momentum transfer to the walls. Both are effective corrections, rather than an exact microscopic interaction law.

The total internal energy and particle number are conserved. The outer container is rigid and isolated, so there is no external work or heat transfer. Its total volume stays $V_0$, while the volume accessible to the gas increases from $V_0/3$ to $V_0$. Neither the gas temperature nor its entropy is generally conserved: this is an irreversible free expansion, with increasing entropy.

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]

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.