1.14.5: Extent of Reaction
For chemists, chemical reaction is the key thermodynamic process. By definition chemical reaction produces a change in composition of a closed system. The extent of chemical reaction is measured by a quantity \(\mathrm{d}\xi\), where the chemical composition is described by the symbol \(\xi\). An example makes the point.
An aqueous solution is prepared at temperature \(\mathrm{T}\) and pressure \(\mathrm{p}\) contains solute \(\mathrm{X}\). The latter undergoes spontaneous chemical reaction to form chemical substance \(\mathrm{Y}\).
Thus
| \(\mathrm{X}(\mathrm{aq})\) | \(\rightarrow\) | \(\mathrm{Y}(\mathrm{aq})\) | |
| At \(t = 0\) | \(\mathrm{n}_{\mathrm{X}}^{0}\) | \(0 \mathrm{~mol}\) | |
| At time \(t\), | \(\mathrm{n}_{\mathrm{X}}^{0}-\xi\) | \(\xi \mathrm{~mol}\) | |
| Rate of reaction | \(= \mathrm{d}\xi / \mathrm{dt}\) |
[Time is a legitimate thermodynamic property.]
A key concept states that spontaneous chemical reaction is driven by the affinity for spontaneous change, \(\mathrm{A}\). Then by definition equilibrium corresponds to the state where \(\mathrm{A} = 0\), and \(\mathrm{d}\xi / \mathrm{dt} = 0\).
General Terms
For a system containing \(\mathrm{i}\)-chemical substances, the chemical potential of chemical substance \(\mathrm{j}\) is given by equation (a).
\[\mu_{\mathrm{j}}=\left(\frac{\partial \mathrm{G}}{\partial n_{\mathrm{j}}}\right)_{\mathrm{T}, \mathrm{p}, \mathrm{n}(\mathrm{i} \neq \mathrm{j})} \nonumber \]
Then,
\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}+\sum_{\mathrm{j}=1}^{\mathrm{j}=\mathrm{i}} \mu_{\mathrm{j}} \, \mathrm{dn} \mathrm{j}_{\mathrm{j}} \nonumber \]
But,
\[\mathrm{dG}=-\mathrm{S} \, \mathrm{dT}+\mathrm{V} \, \mathrm{dp}-\mathrm{A} \, \mathrm{d} \xi \nonumber \]
By comparison,
\[A \, d \xi=-\sum_{j=1}^{j=i} \mu_{j} \, d n_{j} \nonumber \]
But \(\mathrm{dn}_{\mathrm{j}}=\mathrm{v}_{\mathrm{j}} \, \mathrm{d} \xi\) where \(\mathrm{ν}_{\mathrm{j}}\) is positive for products and negative for reactants. Hence,
\[A=-\sum_{j=1}^{j=i} v_{j} \, \mu_{j} \nonumber \]
This remarkable equation relates the affinity for chemical reaction \(A\) with the chemical potentials of the chemical substances involved in the chemical reaction. Moreover at equilibrium, \(A\) is zero. Hence,
\[\sum_{j=1}^{j=i} v_{j} \, \mu_{j}^{e q}=0 \nonumber \]
We have a condition describing chemical equilibrium in terms of the chemical potentials of reactants and products at equilibrium.