1.14.10: Gibbs - Helmholtz Equation
The Gibbs energy and enthalpy of a closed system are related;
\[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S} \nonumber \]
The two properties \(\mathrm{G}\) and \(\mathrm{H}\) are also related by the Gibbs - Helmholtz equation through the dependence of \(\mathrm{G}\) on temperature at fixed pressure. We envisage a situation in which a closed system at equilibrium having Gibbs energy \(\mathrm{G}\) is displaced to a neighbouring equilibrium state by a change in temperature at constant pressure. We are interested in the partial derivative, \(\left[\frac{\partial(\mathrm{G} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \mathrm{A}=0}\). In general terms we consider the isobaric differential dependence of \((\mathrm{G} / \mathrm{T})\) on temperature.
\[\frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}}=\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{p}-\frac{\mathrm{G}}{\mathrm{T}^{2}} \nonumber \]
\[\mathrm{T}^{2} \, \frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}}=\mathrm{T} \,\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}}-\mathrm{G} \nonumber \]
But
\[\mathrm{S}=-\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
For an equilibrium change, equations (b) and (c) yield equation (e).
\[\mathrm{T}^{2} \, \frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}}=-(\mathrm{G}+\mathrm{T} \, \mathrm{S}) \nonumber \]
But \(\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}\). Then,
\[\mathrm{H}=-\mathrm{T}^{2} \, \frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}} \nonumber \]
For an equilibrium change,
\[\Delta \mathrm{H}(\mathrm{A}=0)=-\mathrm{T}^{2} \, \frac{\mathrm{d}}{\mathrm{dT}}\left(\frac{\Delta \mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p} ; \mathrm{A}=0} \nonumber \]
or,
\[\Delta \mathrm{H}(\mathrm{A}=0)=\frac{\mathrm{d}}{\mathrm{dT}^{-1}}\left(\frac{\Delta \mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p} ; \mathrm{A}=0} \nonumber \]
In a similar manner we obtain the Gibbs -Helmholtz equation for a system perturbed at constant composition [1].
\[\Delta \mathrm{H}(\text { fixed } \xi)=\frac{\mathrm{d}}{\mathrm{dT}^{-1}}\left(\frac{\Delta \mathrm{G}}{\mathrm{T}}\right)_{\mathrm{p}, \bar{\xi},} \nonumber \]
Equation (f) is the starting point for the development of another important equation. Thus,
\[\mathrm{H}=-\mathrm{T}^{2} \,\left[-\frac{\mathrm{G}}{\mathrm{T}^{2}}+\frac{1}{\mathrm{~T}} \, \frac{\mathrm{dG}}{\mathrm{dT}}\right] \nonumber \]
Hence,
\[\mathrm{H}=\mathrm{G}-\mathrm{T} \,\left[\frac{\mathrm{dG}}{\mathrm{dT}}\right] \nonumber \]
Equation (k) is differentiated with respect to temperature at constant pressure and at ‘\(\mathrm{A}=0\)’.
\[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}-\mathrm{T} \,\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{T}^{2}}\right)_{p, A=0}-\left(\frac{\partial \mathrm{G}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0} \nonumber \]
Hence,
\[\left(\frac{\partial \mathrm{H}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}=0}=-\mathrm{T} \,\left(\frac{\partial^{2} \mathrm{G}}{\partial \mathrm{T}^{2}}\right)_{\mathrm{p}, \mathrm{A}=0} \nonumber \]
But
\[\left(\frac{\partial^{2} G}{\partial T^{2}}\right)_{p, A=0}=\frac{\partial}{\partial T}\left(\frac{\partial G}{\partial T}\right)=-\left(\frac{\partial S}{\partial T}\right)_{p, A=0} \nonumber \]
Also the equilibrium isobaric heat capacity,
\[C_{p}(A=0)=\left(\frac{\partial H}{\partial T}\right)_{p, A=0} \nonumber \]
Equations (m), (n) and (o) yield equation (p).
\[\left(\frac{\partial S}{\partial T}\right)_{p, A=0}=\frac{C_{p}(A=0)}{T} \nonumber \]
Equation (p) relates the isobaric equilibrium dependence of entropy of a closed system on temperature to the isobaric heat capacity. Also starting from, \(\mathrm{H}=\mathrm{G}+\mathrm{T} \, \mathrm{S}\), then
\[(\partial \mathrm{H} / \partial \mathrm{p})_{\mathrm{T}}=(\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}}+\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{p})_{\mathrm{T}} \nonumber \]
Using a Maxwell Equation,
\[(\partial H / \partial p)_{T}=\mathrm{V}-\mathrm{T} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}} \nonumber \]
Similarly,
\[(\partial \mathrm{U} / \partial \mathrm{T})_{\mathrm{V}}=\mathrm{C}_{\mathrm{V}}=\mathrm{T} \,(\partial \mathrm{S} / \partial \mathrm{T})_{\mathrm{V}} \nonumber \]
And
\[(\partial \mathrm{U} / \partial \mathrm{V})_{\mathrm{T}}=-\mathrm{p}-\mathrm{T} \,(\partial \mathrm{V} / \partial \mathrm{T})_{\mathrm{p}} \,(\partial \mathrm{p} / \partial \mathrm{V})_{\mathrm{T}} \nonumber \]
Footnote
[1] There are many thermodynamic equations which are of the GibbsHelmholtz type. As a common feature they conform to the following calculus property.
Given
\[\mathrm{f}=\mathrm{f}(\mathrm{x}, \mathrm{y}) \nonumber \]
Then
\[\left(\frac{\partial(f / x)}{\partial(1 / x)}\right)_{y}=-x^{2} \,\left(\frac{\partial(f / x)}{\partial x}\right)_{y}=f-x \,\left(\frac{\partial f}{\partial x}\right)_{y} \nonumber \]
Similarly,
\[\left(\frac{\partial(f / y)}{\partial(1 / y)}\right)_{x}=-y^{2} \,\left(\frac{\partial(f / x)}{\partial y}\right)_{x}=f-y \,\left(\frac{\partial f}{\partial y}\right)_{x} \nonumber \]
Normally \(\mathrm{f}\) stands for a thermodynamic potential and \(x\) and \(y\ for its natural variables. Thus a total of 8 equations of the Gibbs - Helmholtz type holding for closed systems can be constructed from \(\mathrm{U}=\mathrm{U}(\mathrm{S}, \mathrm{V}), \mathrm{F}=\mathrm{F}(\mathrm{T}, \mathrm{V}), \mathrm{H}=\mathrm{H}(\mathrm{S}, \mathrm{p}) \text { and } \mathrm{G}=\mathrm{G}(\mathrm{T}, \mathrm{p})\).