1.14.54: Poynting Relation
A given closed system comprises chemical substance j in two homogeneous subsystems which are separated by an appropriate semipermeable diaphragm and which are at the same temperature but different pressures. The subsystems I and II are in thermodynamic equilibrium. Thus (cf. Topic 690),
\[\mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)=\mu_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right) \nonumber \]
For subsystem I,
\[\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)=\left(\frac{\partial \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)}{\partial \mathrm{T}}\right)_{\mathrm{p}} \, \mathrm{dT}+\left(\frac{\partial \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \, \mathrm{dp}_{1} \nonumber \]
Or,
\[\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)=-\mathrm{S}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right) \, \mathrm{dp}_{1} \nonumber \]
Here \(\mathrm{S}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)\) and \(\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{\mathrm{l}}\right)\) are molar properties of chemical substance \(j\). Similarly,
\[\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right)=-\mathrm{S}_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right) \, \mathrm{dT}+\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right) \, \mathrm{dp}_{2} \nonumber \]
The equality expressed in equation (a) is valid at all \(\mathrm{T}\) and \(\mathrm{p}\). Clearly this condition can only be satisfied if the following equation is satisfied.
\[\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)=\mathrm{d} \mu_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right) \nonumber \]
Then at constant temperature,
\[V_{j}^{*}\left(I, T, p_{1}\right) \, d p_{1}=V_{j}^{*}\left(I I, T, p_{2}\right) \, d_{2} \nonumber \]
Hence,
\[\frac{\mathrm{dp}_{1}}{\mathrm{dp}_{2}}=\frac{\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}_{1}, \mathrm{p}_{2}\right)}{\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)} \nonumber \]
The latter is the Poynting Equation [1]. An interesting application of this equation concerns the case where system II is the vapor phase and system I is the liquid phase. The vapor phase is described as an ideal gas using equation (h) for one mole of chemical substance \(j\).
\[\mathrm{p}_{2} \, \mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{II}, \mathrm{T}, \mathrm{p}_{2}\right)=\mathrm{R} \, \mathrm{T} \nonumber \]
The liquid phase comprises one mole of liquid j for which \(\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)\) is the molar volume which is assumed to be a constant, independent of pressure.
Hence from equations (g) and (h),
\[\frac{\mathrm{dp}_{1}}{\mathrm{dp}_{2}}=\frac{1}{\mathrm{~V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)} \, \frac{\mathrm{R} \, \mathrm{T}}{\mathrm{p}_{2}} \nonumber \]
Or,
\[\mathrm{R} \, \mathrm{T} \, \mathrm{d} \ln \left(\mathrm{p}_{2}\right)=\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right) \, d \mathrm{p}_{1} \nonumber \]
The assumption is made that, phase I being a liquid, \(\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right)\) is independent of pressure. Then equation (j) is integrated between pressure limits \(\mathrm{p}_{2}\) and \({\mathrm{p}_{2}}^{\prime}\) and between \(\mathrm{p}_{1}\) and \({\mathrm{p}_{1}}^{\prime}\). Hence,
\[\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{p}_{2}^{\prime} / \mathrm{p}_{2}\right)=\mathrm{V}_{\mathrm{j}}^{*}\left(\mathrm{I}, \mathrm{T}, \mathrm{p}_{1}\right) \,\left[\mathrm{p}_{1}^{\prime}-\mathrm{p}_{1}\right] \nonumber \]
An interesting application of equation (k) concerns the impact of an increase in pressure from \(\mathrm{p}_{1}\) and \({\mathrm{p}_{1}}^{\prime}\) on liquid \(j\). This increase might be produced for example by an increase in confining pressure of an inert gas insoluble in liquid \(j\). Equation (k) describes the increase in vapor pressure from \(\mathrm{p}_{2}\) to \({\mathrm{p}_{2}}^{\prime}\) of liquid \(j\). This pattern might seem intuitively somewhat unexpected.
Footnotes
[1] J. J. Vanderslice, H. W. Schamp Jr and E. A. Mason, Thermodynamics, Prentice Hall,Englewood Cliffs, N.J., 1966, page 106.
[2] Poynting, Phil. Mag.,1881,[4], 12 ,32.