Critical-isochore compressibility of a van der Waals gas (${\kappa }_T=4b/[3R(T-T_c)]$)

For molar equation $p=RT/(v-b)-a/v^2$, the critical values are $v_c=3b$ and $T_c=8a/(27Rb)$. At $v=v_c$, $(\partial p/\partial v{)}_T=-R(T-T_c)/(4b^2)$. Thus ${\kappa }_T=-v_c^{-1}(\partial v/\partial p{)}_T=4b/[3R(T-T_c)]$ on the stable branch above $T_c$. The mean-field compressibility exponent is one; the continuation below the critical temperature is unstable rather than an equilibrium negative compressibility.