Solution
= Solution
For van der Waals, the identity gives $(\partial E/\partial V)_T=aN^2/V^2$, so $E(T,V)=f(T)-aN^2/V$. The dilute monatomic <limit> fixes $f(T)=3NkT/2$, hence $E=3NkT/2-aN^2/V$ and $C_V=3Nk/2$ at every volume.
= Solution
For van der Waals, the identity gives $(\partial E/\partial V)_T=aN^2/V^2$, so $E(T,V)=f(T)-aN^2/V$. The dilute monatomic <limit> fixes $f(T)=3NkT/2$, hence $E=3NkT/2-aN^2/V$ and $C_V=3Nk/2$ at every volume.