1.13.1: Equilibrium and Frozen Properties
The Gibbs energy \(\mathrm{G}\) of a given closed system is characterised by the independent variables temperature \(\mathrm{T}\), pressure \(\mathrm{p}\) and composition \(\xi\).
\[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi] \label{a} \]
In the state defined by Equation \ref{a} the affinity for spontaneous change is \(\mathrm{A}\). Starting with the system in the state defined by equation (a) it is possible to change the pressure (at fixed temperature) and thereby perturb the system to neighbouring states where the affinity \(\mathrm{A}\) is the same. The differential dependence of \(\mathrm{G}\) on pressure along this path is given by the partial differential \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \mathrm{A}}\). Returning to the state defined by Equation \ref{a} we envisage a perturbation by a change in pressure (at fixed temperature) along a path such that the extent of chemical reaction \(\xi\) remains constant; the corresponding differential dependence of \(\mathrm{G}\) is given by \((\partial \mathrm{G} / \partial \mathrm{p})_{\mathrm{T}, \xi}\).The two partial derivatives are related by equation (b) for a system at constant temperature.
\[\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\mathrm{A}}=\left[\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right]_{\xi}-\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\xi} \,\left[\frac{\partial \xi}{\partial \mathrm{A}}\right]_{\mathrm{p}} \,\left[\frac{\partial \mathrm{G}}{\partial \xi}\right]_{\mathrm{p}} \label{b} \]
The important result which emerges from this equation concerns the properties of a system at chemical equilibrium where the affinity for spontaneous change is zero, the rate of change \(\mathrm{d} \xi / \mathrm{dt}\) is zero, the Gibbs energy is a minimum and, significantly, \((\partial G / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero. Hence
\[V=\left[\frac{\partial G}{\partial p}\right]_{T, A=0}=\left[\frac{\partial G}{\partial p}\right]_{T, \xi(e q)} \nonumber \]
Thus we confirm that the volume \(\mathrm{V}\) of a system is a strong state variable, the dependence of \(\mathrm{G}\) on pressure (at constant \(\mathrm{T}\)) at constant ‘\(\mathrm{A}=0\)’ and at constant composition, \(\xi^{\mathrm{eq}}\) are identical. However if we turn our attention on to expansibilities and compressibilities we find that it is important to distinguish between two sets of properties, equilibrium and frozen.