How to deal with cubic terms in Gibbs free energy?

Question

Consider as a toy example a system with Gibbs free energy given by

$$G(p,T) = p^3 - 2p^2 - p + 2.$$

If I was to find the Helmholtz free energy, I would try Legendre transforming $F = G - pV$,

$$V = \frac{\partial G}{\partial p} = 3p^2 - 4p - 1$$

Solving for $p$,

$$p = \frac{2}{3} \pm \sqrt{\frac{7}{9} + \frac{V}{3}}$$

So

$$F(V,T) = G-pV = -2p^3 - 2p^2 +2 = -2\left(\frac{2}{3} \pm \sqrt{\frac{7}{9} + \frac{V}{3}} \right)^3 - 2\left(\frac{2}{3} \pm \sqrt{\frac{7}{9} + \frac{V}{3}} \right)^2 + 2$$

$$F(V,T) = -2\left(\frac{2}{3} \pm \sqrt{\frac{7}{9} + \frac{V}{3}} \right)^3 - 2\left(\frac{2}{3} \pm \sqrt{\frac{7}{9} + \frac{V}{3}} \right)^2 + 2$$

and apparently the free energy $F(V,T)$ is not even single-valued. How do I actually find the free energy as a function of volume, and how do I make sense of this result?

Answer 1

Your equation is probably not convex, so the Legendre transform is not expected to be single valued. This what happens when you get mixed phases. You will need a Maxwell construction.

Answer 2

It is enough to look at the second derivative of your toy Gibbs free energy to discover that is a concave function of $p$ only in the interval [$0,2/3$] for values of $p>\frac{2}{3}$ it is a convex function. This fact implies negative isothermal compressibility, i.e., a mechanically unstable system.

Moreover, a Maxwell construction cannot cure this drawback for two reasons.

The first is that from a mathematical point of view, it would require a limited interval of convexity between two intervals where the function is concave.

The second, more important in my opinion, is that a Maxwell construction, i.e., the construction of a concave hull of the original free energy, would result in an interval where the function would be linear. Physically, it would correspond to an interval of increasing pressures where the volume would remain constant. That would be a highly unphysical behavior.

If you want to play with a toy Gibbs free energy, I would advise starting with functions satisfying the fundamental requirements of concavity as a function of $p$ and $T$. First-order phase transitions should appear as a jump discontinuity of the first derivatives.