Clausius–Clapeyron relation

The relation states that the pressure–temperature slope of a phase boundary equals the entropy change divided by the volume change, or equivalently the transition enthalpy divided by temperature times the volume change: dP/dT = ΔS/ΔV = ΔH/(TΔV).¹ Here, ΔH and ΔV refer to the molar enthalpy and volume changes from one phase to the other. The equation applies along a line where the two phases coexist in equilibrium, such as liquid and vapor on a vapor-pressure curve.

Émile Clapeyron first formulated the phase-boundary relation in 1834 using the emerging theory of heat engines. Rudolf Clausius later recast it in terms of entropy and developed the simplified vaporization form. The result links measurable equilibrium behavior to microscopic energy and volume changes without requiring a complete molecular model.

The general equation follows from equality of the chemical potentials of the two coexisting phases. Differentiating that equality with respect to temperature and pressure gives the Gibbs–Duhem relation for phase equilibrium, while the entropy and volume differences supply the slope.³ For liquid–vapor equilibrium, the vapor volume usually dominates the liquid volume, and an ideal-gas approximation gives ΔV ≈ RT/P. The equation then becomes d ln P/dT = ΔHvap/(RT²).

If the enthalpy of vaporization is treated as constant over a limited temperature interval, integration yields ln(P₂/P₁) = −ΔHvap/R(1/T₂ − 1/T₁). This form estimates an unknown vapor pressure from a reference pressure and temperature, but its accuracy decreases when the enthalpy changes substantially or the vapor is nonideal.

The relation predicts how strongly a phase boundary moves when temperature changes. A large transition enthalpy or a small volume change produces a steep pressure–temperature slope, whereas a large volume change produces a shallower one. For ordinary vaporization, ΔH is positive and the vapor has a much larger volume than the liquid, so vapor pressure increases rapidly with temperature.²

In atmospheric science, the equation helps quantify how saturation vapor pressure changes with temperature, a foundation for interpreting humidity, condensation, cloud formation, and latent-heat effects. Engineers use it in distillation, evaporation, heat-pump design, and refrigerant calculations. Geophysicists apply the same principle to melting curves and volatile transport in planetary interiors. Experimental vapor-pressure data, such as those compiled by NIST, can be differentiated or fitted to obtain thermodynamic quantities.²

The simplified Clausius form is not the full relation and should not be applied indiscriminately. Near a critical point, the distinction between liquid and vapor disappears, nonideal behavior becomes important, and both ΔH and ΔV approach limiting values that require more careful treatment. For solid–solid transitions, the same general equation applies, but the volume and entropy changes may be small, making the phase-boundary slope especially sensitive to measurement uncertainty.

The sign of the slope need not be positive. Ice melting under ordinary pressures is a familiar exception because liquid water occupies less volume than ice; its melting curve therefore slopes toward lower pressure with increasing temperature. The equation also describes sublimation curves and solid–vapor equilibria. In practice, tabulated property data and equations of state are preferred when broad temperature ranges, high pressures, or mixtures make the idealized assumptions unreliable.⁴

Phase boundary

A curve or surface in a phase diagram along which two phases coexist at equilibrium.

Enthalpy of transition

The heat absorbed or released per mole during a phase transition at constant pressure.

Coexistence curve

The set of pressure–temperature conditions at which two specified phases are in equilibrium.

Vapor pressure

The equilibrium pressure of a vapor in contact with its condensed phase at a given temperature.