Caloric closure of the van der Waals equation

The thermodynamic identity $(\partial U/\partial V{)}_T=T(\partial P/\partial T{)}_V-P$ fixes the volume-dependent term $-a{N}^2/V$ but leaves an arbitrary function of temperature. Classical monatomic mean-field modelling adds $3N{k}_{B}T/2$, giving $U=3N{k}_{B}T/2-a{N}^2/V+C$.