 Spinodal densities of a cubic gas

For the cubic equation of state $p=n{k}_{B}T-b{n}^2/2+c{n}^3/6$, with $b,c>0$, a spinodal occurs where the isothermal density derivative vanishes:Using the critical point of a cubic density equation of state givesBelow $T_c$, the interval between these densities has negative isothermal compressibility and is mechanically unstable. At $T_c$ the two roots merge; above it there are no real spinodal densities. The spinodal is a local stability boundary, not the liquid–vapor coexistence curve: the latter requires equal pressure and chemical potential in the two phases.