Cooling in free expansion of a van der Waals gas

For the classical monatomic caloric model, energy conservation in expansion from $V_i$ to $V_f$ gives $T_f-T_i=2aN(1/V_f-1/V_i)/(3k_B)$. Attractions make the gas cool. The formula assumes homogeneous equilibrium states, not a two-phase coexistence state.