Clausius-Clapeyron Equation

The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, 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,

\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{L}{T\,\Delta v}=\frac{\Delta s}{\Delta v},

where \mathrm{d}P/\mathrm{d}T is the slope of the tangent to the coexistence curve at any point, L is the specific latent heat, T is the temperature, \Delta v is the specific volume change of the phase transition and \Delta s is the entropy change of the phase transition.

Derivation from state postulate

Using the state postulate, take the specific entropy s for a homogeneous substance to be a function of specific volume v and temperature T.


The Clausius–Clapeyron relation characterizes behavior of a closed system during a phase change, during which temperature and pressureare constant by definition. Therefore,[3]:508


Using the appropriate Maxwell relation gives


where P is the pressure. Since pressure and temperature are constant, by definition the derivative of pressure with respect to temperature does not change.[4][5]:57, 62 & 671 Therefore the partial derivative of specific entropy may be changed into atotal derivative


and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase \alphato a final phase \beta, to obtain


where \Delta s\equiv s_{\beta}-s_{\alpha} and \Delta v\equiv v_{\beta}-v_{\alpha} are respectively the change in specific entropy and specific volume. Given that a phase change is an internally reversible process, and that our system is closed, the first law of thermodynamics holds


where u is the internal energy of the system. Given constant pressure and temperature (during a phase change) and the definition of specific enthalpy h, we obtain

d h = T\;\mathrm{d}s
\mathrm{d}s = \frac {\mathrm{d} h}{T}

Given constant pressure and temperature (during a phase change), we obtain

\Delta s = \frac {\Delta h}{T}

Substituting the definition of specific latent heat L = \Delta h gives

\Delta s = \frac{L}{T}

Substituting this result into the pressure derivative given above (\mathrm{d}P/\mathrm{d}T = \mathrm{\Delta s}/\mathrm{\Delta v}), we obtain

\frac{\mathrm{d} P}{\mathrm{d} T} = \frac {L}{T \Delta v}.

This result (also known as the Clapeyron equation) equates the slope of the tangent to the coexistence curve \mathrm{d}P/\mathrm{d}T, at any given point on the curve, to the function {L}/{T {\Delta v}} of the specific latent heat L, the temperature T, and the change in specific volume \Delta v .

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