1.2.2: Affinity for Spontaneous Reaction- Chemical Potentials
For a closed system containing \(\mathrm{k}\) − chemical substances, the differential dependence of Gibbs energy on temperature, pressure and chemical composition, is given by the following equation.
\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}}\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}} \nonumber \]
The condition at constant \(n(i \neq j)\) indicates that the amounts of each \(i\) chemical substance except chemical substance \(j\) is constant. The Gibbs energy of a closed system is a thermodynamic potential function; equation (b).
\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi \nonumber \]
Here \(\mathrm{A}\) is the affinity for spontaneous chemical reaction producing a change in extent of reaction, \(\mathrm{d} \xi\), in this case a change in composition. Further the chemical potential of chemical substance \(j\),
\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial \mathrm{n}_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \nonumber \]
Comparison of equations (a) and (b) yields equation(d).
\[-A \, d \xi=\sum_{j=1}^{j=k} \mu_{j} \, d n_{j} \nonumber \]
The stoichiometry in a chemical reaction for chemical substance \(j\), \(ν_{j}\) is defined such that \(ν_{j}\) is positive for products and negative for reactants; a mnemonic is ‘P for P’.
\[v_{j}=d n_{j} / d \xi \nonumber \]
Hence the affinity for spontaneous change,
\[A=-\sum_{j=1}^{j=k} v_{j} \, \mu_{j} \nonumber \]
But at equilibrium, the affinity for spontaneous change \(\mathrm{A}\) is zero.
\[\text { Then, } \sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{k}} \mathrm{v}_{\mathrm{j}} \, \mu_{\mathrm{j}}^{\mathrm{eq}}=0 \nonumber \]
Equation (g) in terms of its simplicity is misleading. Chemists are experts at assaying a system at equilibrium in order to determine the chemical substances present and their amounts. For example, an assay of a given system yields (for defined temperature and pressure) the amounts of un-dissociated acid \(\mathrm{CH}_{3}\mathrm{COOH}(\mathrm{aq})\), and the conjugate base \(\mathrm{CH}_{3}\mathrm{COO}^{−} (\mathrm{aq})\) and hydrogen ions at equilibrium. We write equation (g) as follows.
\[-\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COOH} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COO}^{-} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)=0 \nonumber \]
Or, representing a balance of chemical potentials, (a useful approach)
\[\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COOH} ; \mathrm{aq}\right)=\mu^{\mathrm{eq}}\left(\mathrm{CH}_{3} \mathrm{COO}^{-} ; \mathrm{aq}\right)+\mu^{\mathrm{eq}}\left(\mathrm{H}^{+} ; \mathrm{aq}\right) \nonumber \]
The concept of a balance of equilibrium chemical potentials at thermodynamic equilibrium is often the starting point for a description of the properties of closed systems.