Clausius–Clapeyron relation

The Clausius–Clapeyron relation gives the slope of a phase-coexistence curve on the pressure–temperature plane: dP/dT = L/(TΔV), where L is the latent heat absorbed in crossing the boundary and ΔV the volume change across it. It follows from a single requirement — that two phases in equilibrium have equal molar Gibbs free energies, so that stepping along the coexistence curve changes both by the same amount. Substituting dG = −S dT + V dP for each phase and equating gives dP/dT = ΔS/ΔV, which becomes L/(TΔV) once the latent heat is written as L = TΔS.

Its power lies in what it does not assume. The derivation never refers to what the phases are, what the substance is made of, or how its molecules interact; it applies to melting, boiling, sublimation, one crystal structure transforming into another, and to transitions well outside conventional chemistry. The shape of every line on a phase diagram is therefore fixed by two quantities a laboratory can measure with a calorimeter and a balance.

The relation's most celebrated consequence is the sign of water's melting line. Because ice is less dense than liquid water, melting has ΔV < 0 and so dP/dT < 0: pressure lowers the melting point, a behaviour almost unique among common substances. Applied to liquid–vapour coexistence, where ΔV is dominated by the gas volume RT/P, the relation integrates to ln P ≈ const − L/RT — the exponential rise of vapour pressure with temperature that governs boiling points on mountains, pressure cookers and evaporation rates.

Written down by Benoît Clapeyron in 1834 as a practical formula in his memoir reviving Carnot's work, and re-derived from the second law by Rudolf Clausius around 1850, once entropy gave the argument a foundation Clapeyron did not have.