1.14.65: Spontaneous Change: Isothermal and Isobaric
By definition,
\[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S} \nonumber \]
\(\mathrm{G}, \mathrm{~H} \text { and } \mathrm{S}\) are extensive functions of state. At fixed \(\mathrm{T}\) and \(\mathrm{p}\), the dependences of these variables on extent of reaction, \(\xi\) are related.
\[(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}=(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}-\mathrm{T} \,(\partial \mathrm{S} / \partial \xi)_{\mathrm{T}, \mathrm{p}} \nonumber \]
For a spontaneous change \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\) where the affinity for spontaneous change \(\mathrm{A}\left[=-(\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\right]\) is positive. This can arise under two limiting circumstances.
- The spontaneous process is exothermic; [i.e.\(\left[\text { i.e. }(\partial \mathrm{H} / \partial \xi)_{\mathrm{T}, \mathrm{p}}<0\right]\) such that \(\left.\left|\mathrm{T} \,(\partial \mathrm{S} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\right|<\mid \partial \mathrm{H} / \partial \xi\right)_{\mathrm{T}, \mathrm{p}} \mid\). The decrease in \(\mathrm{G}\) is enthalpy driven.
- The spontaneous process is endothermic; \(\left[\text { i.e. }(\partial H / \partial \xi)_{\mathrm{T}, \mathrm{p}}>0\right]\) such that \(\left.\left|\mathrm{T} \,(\partial \mathrm{S} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\right|>\mid \partial \mathrm{H} / \partial \xi\right)_{\mathrm{T}, \mathrm{p}} \mid\). The decrease in \(\mathrm{G}\) is entropy driven.
If for a given possible process, \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}>0\), then the process is not spontaneous; there is no affinity for spontaneous change. If \((\partial \mathrm{G} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is zero at defined \(\mathrm{T}\) and \(\mathrm{p}\), the system is at equilibrium with the surroundings; the affinity for spontaneous change is zero. The chemical equilibrium is stable if \((\partial \mathrm{A} / \partial \xi)_{\mathrm{T}, \mathrm{p}}\) is negative.