1.2.6: Affinity for Spontaneous Reaction - Dependence on Pressure

The Gibbs energy of a given closed system is defined by equation (a) where \(\xi\) describes the chemical composition.

\[\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi] \nonumber \]

We consider the dependence of Gibbs energy on pressure and extent of reaction at fixed temperature \(\mathrm{T}\).

\[\frac{\partial}{\partial p}\left(\frac{\partial \mathrm{G}}{\partial \xi}\right)=\frac{\partial}{\partial \xi}\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right) \nonumber \]

But volume \(\mathrm{V}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}\) and affinity \(A=-\left(\frac{\partial G}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\).

Volume \(\mathrm{V}\) and affinity \(\mathrm{A}\) are given by first differentials of the Gibbs energy, \(\mathrm{G}\).

\[\text { Then }-\left(\frac{\partial \mathrm{A}}{\partial \mathrm{p}}\right)_{\mathrm{T}, \xi}=\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}} \nonumber \]

Here \(\left(\frac{\partial \mathrm{V}}{\partial \xi}\right)_{\mathrm{T}, \mathrm{p}}\) is the volume of reaction, being the increase volume accompanying unit increase in extent of reaction, \(\xi\).

---