The Clausius–Clapeyron relation, named after Rudolf Clausius^[1] and Benoît Paul Émile Clapeyron,^[2] is a way of characterizing a discontinuous phase transition between two phases of matter of a single constituent.

On a pressure–temperature (P–T) diagram, the line separating the two phases is known as the coexistence curve. The Clausius–Clapeyron relation gives the slope of the tangents to this curve. Mathematically,

The Clausius–Clapeyron relation characterizes behavior of a closed system during a phase change, during which temperature and pressure are constant by definition. Therefore,^{[3] []}

Using the appropriate Maxwell relation gives^{[3] []}

Given constant pressure and temperature (during a phase change), we obtain^{[3] []}

Furthermore, along the coexistence curve,

One may therefore use the Gibbs–Duhem relation

Rearrangement gives

from which the derivation of the Clapeyron equation continues as in the previous section.

at low temperatures. If pressure is also low, the gas may be approximated by the ideal gas law, so that

we can obtain the Clausius–Clapeyron equation^{[3] []}

or^{[5] []}

These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to the latent heat of the phase change, without requiring specific volume data.

For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as

Atmospheric water vapor drives many important meteorologic phenomena (notably precipitation), motivating interest in its dynamics. The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions (near standard temperature and pressure) is

where:

(This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate.)^[9] But see also this discussion of the accuracy of different approximating formulae for saturation vapour pressure of water.

and substituting in

we obtain

To provide a rough example of how much pressure this is, to melt ice at −7 °C (the temperature many ice skating rinks are set at) would require balancing a small car (mass = 1000 kg^[11]) on a thimble (area = 1 cm²).

While the Clausius–Clapeyron relation gives the slope of the coexistence curve, it does not provide any information about its curvature or second derivative. The second derivative of the coexistence curve of phases 1 and 2 is given by ^[12]