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# 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 $\alpha$to 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 $\Delta v$.