= Spinodal densities of a cubic gas
For the cubic <equation of state> $p=nk_BT-bn^2/2+cn^3/6$, with $b,c>0$, a <spinodal> occurs where the isothermal density derivative vanishes:
$$
\left(\frac{\partial p}{\partial n}\right)_T=k_BT-bn+\frac c2n^2=0.
$$
Using the <critical point of a cubic density equation of state> gives
$$
n_\pm=\frac bc\left(1\pm\sqrt{1-T/T_c}\right),\qquad k_BT_c=\frac{b^2}{2c}.
$$
Below $T_c$, 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.
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