1.14.9: Clausius - Clapeyron Equation
A give closed system contains chemical substance j present in both liquid and gas phases. The system is at equilibrium. In terms of the Phase Rule, the following parameters are defined; \(\mathrm{P} = 2\), \(\mathrm{C} = 1\) and hence \(\mathrm{F} = 1\). Hence, if the temperature is fixed by the observer, the equilibrium pressure \(\mathrm{p}^{\mathrm{eq}}\) is defined. The equilibrium can be described in terms of an equality of chemical potentials of pure liquid and pure gas.
\[\mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\mu_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \label{a} \]
Both chemical potentials in Equation \ref{a} are functions of both \(\mathrm{T}\) and \(\mathrm{p}\). In general terms,
\[\mathrm{d} \mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=\left(\frac{\partial \mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})}{\partial \mathrm{T}}\right)_{\mathrm{p}} \, \mathrm{dT}+\left(\frac{\partial \mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})}{\partial \mathrm{p}}\right)_{\mathrm{T}} \, \mathrm{dp} \label{b} \]
or
\[ \mathrm{d} \mu_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})=-\mathrm{S}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dp} \label{c} \]
Similarly
\[\mathrm{d} \mu_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p})=-\mathrm{S}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dp} \label{d} \]
The condition in Equation \ref{a} applies at all \(\mathrm{T}\) and \(\mathrm{p}\).
\[\begin{aligned}
&\text { Then, }-\mathrm{S}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dp} \\
&=-\mathrm{S}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p}) \, \mathrm{dp}
\end{aligned} \nonumber \]
or [1],
\[\left(\frac{\mathrm{dp}}{\mathrm{dT}}\right)^{e q}=\frac{\mathrm{S}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p})-\mathrm{S}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})}{\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p})-\mathrm{V}_{\mathrm{j}}^{*}(\ell ; \mathrm{T} ; \mathrm{p})} \nonumber \]
\[\left(\frac{\mathrm{dp}}{\mathrm{dT}}\right)^{\mathrm{eq}}=\frac{\Delta_{\mathrm{vap}} \mathrm{S}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}{\Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})} \nonumber \]
But at equilibrium,
\[\Delta_{\text {vap }} \mathrm{G}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})=\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})-\mathrm{T} \, \Delta_{\text {vap }} \mathrm{S}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})=0 \nonumber \]
\[\left(\frac{\mathrm{dp}}{\mathrm{dT}}\right)^{\text {eq }}=\frac{\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}{\mathrm{T} \, \Delta_{\mathrm{vap}} \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})} \nonumber \]
The latter is the Clausius-Clapeyron Equation [2]. In a modern development, equation (i) was exactly integrated [3]. Equation(i) does not have the form of an exact differential in the independent variables \(\mathrm{p}\) and \(\mathrm{T}\).[3] The corresponding integrating factor is \(\mathrm{T}^{-1} \, \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*}\).
Thus
\[\mathrm{T}^{-1} \, \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*} \, \mathrm{dp}-\mathrm{T}^{-2} \, \Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*} \, \mathrm{dT}=0 \nonumber \]
or,
\[\mathrm{T}^{-1} \, \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*} \, \mathrm{dp}+\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*} \, \mathrm{dT}^{-1}=0 \nonumber \]
The latter equation is an exact differential as a consequence of equation (\(\ell\)) [4].
\[\left(\frac{\partial\left(\mathrm{T}^{-1} \, \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*}\right)}{\partial \mathrm{T}^{-1}}\right)_{\mathrm{p}}=\left(\frac{\partial \Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
A mathematical solution is known for differential equations having the form of equation (k) [3]. A comprehensive set of equations have been derived describing first order transitions for pure substances [5] and hence the phase equilibrium curves For liquid-vapour equilibria, both \(\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p}) \text { and } \Delta_{\text {vap }} \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p}) \text { are }>0\). Therefore the equilibrium vapor pressure of a liquid increases with increase in temperature. A useful approximation assumes that gas \(j\) is a perfect gas; i.e. \(\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp})=\mathrm{R} \, \mathrm{T}\) and \(\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{gp} ; \mathrm{T} ; \mathrm{p})>>\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{l} ; \mathrm{T} ; \mathrm{p})\).
\[\left(\frac{\mathrm{d} \ln (\mathrm{p})}{\mathrm{dT}}\right)^{\text {eq }}=\frac{\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}{\mathrm{R} \, \mathrm{T}^{2}} \nonumber \]
\[\left(\frac{\mathrm{d} \ln (\mathrm{p})}{\mathrm{d}\left(\mathrm{T}^{-1}\right)}\right)^{\mathrm{eq}}=-\frac{\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T} ; \mathrm{p})}{\mathrm{R}} \nonumber \]
Within the limits of the approximations outlined above, \(\ln \left(p^{e q}\right)\) is a linear function of \(\mathrm{T}^{-1}\).
Exactly integrated equations have also been established for other first-order transitions (\(\mathrm{p}{\mathrm{eq}}\), \(\mathrm{T}{\mathrm{eq}}\)) curves of pure substances [5].
Footnotes
[1] \(\frac{\left[\mathrm{N} \mathrm{m}^{-2}\right]}{[\mathrm{K}]}=\frac{\left[\mathrm{J} \mathrm{mol}^{-1} \mathrm{~K}^{-1}\right]}{\left[\mathrm{m}^{3} \mathrm{~mol}^{-1}\right]}=\frac{\left[\mathrm{J} \mathrm{m}^{-3}\right]}{[\mathrm{K}]}=\frac{\left[\mathrm{N} \mathrm{m}^{-2}\right]}{[\mathrm{K}]}\)
[2] We have derived the equation for vapor-liquid equilibrium which is the generally quoted form. An equivalent form expresses \(\left(\frac{\mathrm{dp}}{\mathrm{dT}}\right)^{e q}\) for the equilibrium for chemical substance \(j\) in two phases \(\alpha\) and \(\beta\).
[3] C. Mosselman, W. H. van Vugt and H. Vos, J. Chem. Eng. Data 1982, 27 ,246.
[4] From \(\mathrm{dH}=\mathrm{T} \, \mathrm{dS}+\mathrm{V} \, \mathrm{dp}\)
\(\begin{aligned}
&\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{T} \, \left(\frac{\partial \mathrm{S}}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{V} \\
&\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=-\mathrm{T} \, \left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}}\right)_{\mathrm{T}}+\mathrm{V} \\
&\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{T}^{-1} \, \left(\frac{\partial \mathrm{V}}{\partial \mathrm{T}^{-1}}\right)_{\mathrm{T}}+\mathrm{V} \\
&\left(\frac{\partial \mathrm{H}}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\left(\frac{\partial\left(\mathrm{T}^{-1} \, \mathrm{V}\right)}{\partial \mathrm{T}^{-1}}\right)_{\mathrm{T}}
\end{aligned}\)
[5] L. Q. Lobo and A. G. M. Ferreira, J. Chem. Thermodyn., 2001, 33 ,1597.