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1.0.s: The Clapeyron Equation

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  • The analysis in the two previous sections can be repeated for any phase change of a pure substance. Let \(\alpha\) and \(\beta\) denote the two phases that are at equilibrium.

    \[\alpha \rightleftharpoons \beta\]

    Let \(\overline{G}_{\alpha }\), \(\overline{S}_{\alpha }\), and \({\overline{V}}_{\alpha }\) represent the Gibbs free energy, the entropy, and the volume of one mole of pure phase \(\alpha\) at pressure \(P\) and temperature \(T\). Let \(\overline{G}_{\beta }\), \(\overline{S}_{\beta }\), and \(\overline{V}_{\beta }\) represent the corresponding properties of one mole of pure phase \(\beta\). The equations

    \[d\overline{G}\left(\alpha \right)=\overline{V}_\alpha dP-\overline{S}_{\alpha }dT\]


    \[d\overline{G}\left(\beta \right)= \overline{V}_{\beta } dP- \overline{S}_{\beta }dT\]

    describe the changes in the Gibbs free energy of a mole of \(\alpha\) and a mole of \(\beta\) when they go from one \(\alpha-\beta\)-equilibrium state at \(P\) and \(T\) to a second \(\alpha-\beta\)-equilibrium state at \(P+dP\) and \(T+dT\). Since these Gibbs free energy changes must be equal, we have

    \[ \begin{align*} d\overline{G}\left(\beta \right)-d\overline{G}\left(\alpha \right) &=\left({\overline{V}}_{\beta }-{\overline{V}}_{\alpha }\right)dP-\left({\overline{S}}_{\beta }-\overline{S}_{\alpha }\right)dT \\[4pt] &=\Delta \overline{V}dP-\Delta \overline{S}dT \\[4pt] &=0 \end{align*}\]


    \[\frac{dP}{dT}=\frac{\Delta \overline{S}}{\Delta \overline{V}}\]

    where \(\Delta \overline{S}\) and \(\Delta \overline{V}\) are the entropy and volume changes that occur when one mole of the substance goes from phase \(\alpha\) to phase \(\beta\). Since \(\Delta \overline{S}=\Delta \overline{H}/T\), the condition for equilibrium between phases \(\alpha\) and \(\beta\) becomes

    \[\frac{dP}{dT}=\frac{\Delta \overline{H}}{T\ \Delta \overline{V}} \label{Clap1}\]

    Equation \ref{Clap1} is known as the Clapeyron equation.