4.3 Phase transitions and coexistence

A substance can exist in several phases — solid, liquid, vapour — and which one is stable at a given temperature and pressure is decided by whichever has the lowest Gibbs free energy per particle. A phase transition is the point where two phases cross in free energy and the stable state switches.

First-order transitions and latent heat

At a transition the two phases have equal Gibbs free energy per unit mass, $g_1 = g_2$ — the equal-chemical-potential condition. The free energy itself is continuous across the transition, but its first derivatives need not be. Recall $s = -(\partial g/\partial T)_p$ and $v = (\partial g/\partial p)_T$ : a jump in the slope of $g$ is a jump in entropy and volume. A transition where these first derivatives are discontinuous is called first order, and the entropy jump is what we observe as latent heat:

L \;=\; T\,\Delta s \;=\; T\,(s_2 - s_1).

Melting and boiling are first order. Heat poured into ice at $0^\circ$ C does not raise its temperature; it pays the latent heat $L = T\Delta s$ to convert the low-entropy solid into the higher-entropy liquid at constant $T$ . The accompanying volume jump $\Delta v$ — water famously contracts on melting, most substances expand — is the discontinuity in the other first derivative.

A transition where the first derivatives are continuous but the second derivatives (the heat capacity, the compressibility) diverge is called continuous, or second order; there is no latent heat, and the two phases merge smoothly. The liquid–vapour transition becomes continuous exactly at the critical point.

The phase diagram

Plotting which phase has the lowest $g$ across the $p$ – $T$ plane produces the phase diagram: regions of solid, liquid, and vapour, separated by coexistence curves along which two phases have equal free energy and so coexist. Three features organise it:

The triple point, where all three coexistence curves meet and solid, liquid, and vapour are simultaneously in equilibrium — a fixed point of temperature and pressure for a given substance.
The critical point, where the liquid–vapour curve ends: beyond it the distinction between liquid and gas disappears and one can pass continuously from one to the other.
The slopes of the coexistence curves, $dp/dT$ , fixed by the latent heat and the volume change through the Clausius–Clapeyron relation of the next lesson.

The phase diagram is the global picture; the chemical potentials of the previous lesson are its local rule. Everywhere in a single-phase region one phase wins; on a curve two tie; at the triple point three do.