= Solution
At every $T>0$, $V>b$,
$$
\left(\frac{\partial p}{\partial V}\right)_T=-\frac{RT}{(V-b)^2}<0,
\qquad
\left(\frac{\partial^2p}{\partial V^2}\right)_T=\frac{2RT}{(V-b)^3}>0.
$$
There is no constant-pressure coexistence interval, unstable loop or finite-temperature critical inflection. The <isothermal compressibility> is positive:
$$
\kappa_T=-\frac1V\left(\frac{\partial V}{\partial p}\right)_T
=\frac{V-b}{pV}>0.
$$
Thus \b[this <equation of state> predicts no liquid-gas <phase transition> in its physical domain]. Its <divergence> at the excluded-volume boundary is a repulsive <constraint>, not a liquid-gas coexistence mechanism. This conclusion concerns the stated model and cannot rule out an ordering transition requiring additional physics or an additional thermodynamic variable. Extrapolation to <absolute zero> is also outside the classical <gas> model's reliable regime.
Back to article page
