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.

clausius-clapeyron

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

clausius-clapeyron

Using the appropriate Maxwell relation gives

clausius-clapeyron

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

clausius-clapeyron

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

clausius-clapeyron

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

clausius-clapeyron

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

clausius-clapeyron
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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