Thermodynamics

Phase Equilibrium

What fixes the pressure and temperature at which two phases sit side by side.

Updated May 2026

Water at one atmosphere boils at exactly 373.15 K. Not approximately. The coexistence condition is sharp, and it comes from a single requirement.

The condition

Two phases coexist when a molecule is indifferent about which one it is in. In thermodynamic terms, the chemical potentials are equal:

μ_{liquid} (T, p) = μ_{gas} (T, p)

That is one equation in two unknowns, so its solutions form a curve in the $(T, p)$ plane, the phase boundary. Fix the pressure and the temperature is determined.

Fig. 1

Three coexistence curves meeting at the triple point, where all three phases are simultaneously in equilibrium. The vaporisation curve ends at the critical point; the fusion curve does not end at all.

The triple point is not particular to water. It is a counting result. Gibbs' phase rule says the number of free variables is $F = C - P + 2$ for $C$ components and $P$ phases. One component in three phases gives $F = 0$ : no freedom at all, so the triple point is a single point, fixed by the substance. That is precisely why the triple point of water was used to define the kelvin until 2019.

Clausius–Clapeyron

Differentiate the coexistence condition along the boundary and the slope emerges:

\frac{d p}{d T} = \frac{L}{T Δ v}

with $L$ the latent heat and $Δ v$ the volume change per particle.

Consequence

Why ice is strange

For almost every substance $Δ v > 0$ on melting and the boundary slopes forward: squeeze it and it freezes. Water expands on freezing, so $Δ v < 0$ and the melting curve leans backwards, so pressure melts ice. It is why a glacier slides on a thin film of its own meltwater, and it is a direct consequence of the hydrogen bond holding the crystal open.

The critical point

Follow the liquid–gas boundary upward and the two phases become progressively more alike: the liquid thins, the vapour densifies. At the critical point the distinction disappears entirely and the boundary simply stops.

Above it there is no transition to cross. You can take a liquid to a gas continuously by going around the critical point, never boiling, never seeing a meniscus.

Order of a transition

Ehrenfest's classification is by which derivative of the free energy first jumps.

First order. $S$ and $V$ are discontinuous. There is latent heat, the phases coexist, and you can watch a boundary between them. Boiling and melting.
Second order. The first derivatives are continuous, the second ones (heat capacity, compressibility) diverge. No latent heat, no coexistence. The Curie point, the superfluid transition, the critical point itself.

Second-order transitions are where the universality of the previous section lives: at a continuous transition the correlation length diverges, the microscopic structure stops setting the scale, and unrelated materials share the same exponents.