Phase coexistence and the Clapeyron equation

Consider one substance in two bulk phases alpha and beta, with negligible interface curvature, surface energy, gravity, or external-field differences. Thermal equilibrium requires equal temperature, mechanical equilibrium equal pressure, and chemical equilibrium equal chemical potential. To see the last condition, transfer a small mole amount from alpha to beta at fixed common temperature and pressure. The Gibbs change is $(\mu_\beta-\mu_\alpha)d\nu$. If the potentials differ, one transfer direction lowers G and is thermodynamically favored.

For a pure extensive phase the molar Gibbs free energy equals chemical potential. Thus coexistence equates two molar free-energy branches. It does not require equal molar volumes, entropies, or internal energies; their discontinuities are the characteristic first-order phase changes. The amount in each phase depends on the total constraints. At a fixed coexistence temperature, changing average volume between the two phase volumes can change the phase fraction while pressure remains on the coexistence value.

Move a small distance along the coexistence curve, keeping the two chemical potentials equal. Their differentials must remain equal, so subtracting the two molar Gibbs differentials gives $\Delta v_m\,dp=\Delta s_m\,dT$. Here every delta means beta minus alpha. Reversible latent heat per mole for alpha-to-beta conversion is $L_m=T\Delta s_m$. Substitution gives the Clapeyron equation. Its sign depends jointly on the entropy and volume jumps, not just on the fact that a transition occurs.

For ordinary vaporization, molar entropy and molar volume both increase, giving positive slope. Melting usually increases entropy, but if the liquid has smaller molar volume than the solid, the slope is negative. Increasing pressure then favors the smaller-volume liquid and lowers the equilibrium melting temperature locally. These conclusions are about bulk equilibrium boundaries; hysteresis and delayed nucleation can make a measured transient transformation occur away from the equilibrium coexistence curve.

The general Clapeyron equation does not assume an ideal gas. To obtain the common exponential vapor-pressure formula, neglect the condensed-phase molar volume compared with vapor volume and replace the latter by $RT/p$. Then the slope is approximately $L_mp/(RT^2)$. Treating latent heat as constant allows direct integration between two temperatures. Every approximation has a physical domain, usually a restricted vaporization interval away from strong nonideality and critical behavior.

Near a critical endpoint, entropy and volume jumps both approach zero, so substituting zero into their ratio is not a valid way to find a slope. Along a broad coexistence curve, latent heat can also vary substantially. Separate questions about which phase is stable from questions about how quickly it appears: a metastable phase can persist behind a nucleation barrier even when another phase has lower Gibbs free energy. The Clapeyron slope describes equilibrium geometry and provides no nucleation rate or transformation timescale.