1.2.5: Affinity for Spontaneous Reaction- Dependence on Temperature
Using the definition of the Gibbs energy \(\mathrm{G} [=\mathrm{U}+\mathrm{p} \, \mathrm{V}-\mathrm{T} \, \mathrm{S}=\mathrm{H}-\mathrm{T} \, \mathrm{S}]\), we form an equation for the entropy of a closed system. Thus \(\mathrm{T} \, \mathrm{S}=-\mathrm{G}+\mathrm{H}\). The entropy of the closed system is perturbed by a change in composition/organisation, \(\xi\) at fixed \(\mathrm{T}\) and \(p\).
\[\text { Then, } T \,\left(\frac{\partial S}{\partial \xi}\right)_{T, p}=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p}+\left(\frac{\partial H}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
\[\text { But the affinity for spontaneous reaction, } A=-\left(\frac{\partial G}{\partial \xi}\right)_{T, p} \nonumber \]
A Maxwell equation requires that, \(\left(\frac{\partial \mathrm{S}}{\partial \xi}\right)_{\mathrm{T}_{, \mathrm{p}}}=\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\).
\[\text { Hence, } \mathrm{T} \,\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}=\mathrm{A}+\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
Equation (c) is rearranged to yield the following interesting equation.
\[\mathrm{A}-\mathrm{T} \,\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}=-\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}]
The affinity for spontaneous change and its dependence on temperature are simply related to the enthalpy of reaction at fixed \(\mathrm{T}\) and \(p\). We exploit this link by considering the derivative \(\mathrm{d}(\mathrm{A} / \mathrm{T}) / \mathrm{dT}\) (at fixed \(p\) and fixed \(\xi\)).
\[\mathrm{d}(\mathrm{A} / \mathrm{T}) / \mathrm{dT}=(1 / \mathrm{T}) \,(\mathrm{dA} / \mathrm{dT})-\mathrm{A} / \mathrm{T}^{2} \nonumber \]
\[\text { Hence }\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}=-\frac{1}{\mathrm{~T}^{2}} \,\left[\mathrm{A}-\mathrm{T} \,\left(\frac{\partial \mathrm{A}}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}\right] \nonumber \]
\[\text { Using equation }(\mathrm{d}),\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi}=\frac{1}{\mathrm{~T}^{2}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
The latter equation is an analogue of the Gibbs-Helmholtz Equation relating the change in Gibbs energy to the enthalpy of reaction, \(\left(\frac{\partial H}{\partial \xi}\right)_{T, \mathrm{p}}\). The background to equation (g) is the definition of the dependent variable (\(\mathrm{A} / \mathrm{T}\)) in terms of independent variables, \(\mathrm{T}\), \(p\) and \(\xi\).
\[\text { Thus } \quad(\mathrm{A} / \mathrm{T})=(\mathrm{A} / \mathrm{T})[\mathrm{T}, \mathrm{p}, \xi] \nonumber \]
The general differential of the latter equation has the following form.
\[\mathrm{d}(\mathrm{A} / \mathrm{T})=\left[\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right]_{\mathrm{p}, \xi} \, \mathrm{dT}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi} \, \mathrm{dp}+\frac{1}{\mathrm{~T}} \,\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \nonumber \]
But, from equation (e)
\[\mathrm{d}(\mathrm{A} / \mathrm{T})=-\left(\mathrm{A} / \mathrm{T}^{2}\right) \, \mathrm{dT}+(1 / \mathrm{T}) \, \mathrm{dA} \nonumber \]
\[\text { Or, } \mathrm{dA}=\mathrm{T} \, \mathrm{d}(\mathrm{A} / \mathrm{T})+(\mathrm{A} / \mathrm{T}) \, \mathrm{dT} \nonumber \]
We incorporate equation (i) for the term (\(\mathrm{A} / \mathrm{T}\)). Thus
\[\mathrm{dA}=\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial(\mathrm{A} / \mathrm{T})}{\partial \mathrm{T}}\right)_{\mathrm{p}, \xi}+\frac{\mathrm{A}}{\mathrm{T}}\right] \, \mathrm{dT}+\left[\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right]_{\mathrm{T}, \xi} \, \mathrm{dp}+\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \nonumber \]
Then using equation (g),
\[\mathrm{dA}=\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right] \, \mathrm{dT}-\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{dp}+\left[\frac{\partial \mathrm{A}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \, \mathrm{d} \xi \nonumber \]
The latter is a general equation for the change in affinity. We rearrange this equation as an equation for a change in extent of reaction.
\[\begin{aligned}
\mathrm{d} \xi=-\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left[\frac{1}{\mathrm{~T}}\right.&\left.\,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right] \, \mathrm{dT}+\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dp} \\
&+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA}
\end{aligned} \nonumber \]
The latter equation has the form of a general differential for the extent of reaction written as,
\[\xi=\xi[T, \mathrm{p}, \mathrm{A}] \nonumber \]
\[\text { Or, } \mathrm{d} \xi=\left(\frac{\partial \xi}{\partial \mathrm{T}}\right)_{\mathrm{p}, \mathrm{A}} \, \mathrm{dT}+\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}} \, \mathrm{dp}+\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \, \mathrm{dA} \nonumber \]
Hence from equation (n),
\[\left(\frac{\partial \xi}{\partial T}\right)_{\mathrm{p}, \mathrm{A}}=-\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \,\left[\frac{1}{\mathrm{~T}} \,\left(\frac{\partial \mathrm{H}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}+\frac{\mathrm{A}}{\mathrm{T}}\right] \nonumber \]
Equation (q) describes the dependence of extent of reaction on temperature at fixed pressure and affinity for spontaneous reaction. Then from equation (n),
\[\left(\frac{\partial \xi}{\partial \mathrm{p}}\right)_{\mathrm{T}, \mathrm{A}}=+\left[\frac{\partial \mathrm{V}}{\partial \xi}\right]_{\mathrm{T}, \mathrm{p}} \,\left(\frac{\partial \xi}{\partial \mathrm{A}}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]
Equation (r) describes the dependence of extent of reaction at fixed temperature and fixed affinity for spontaneous change.