1.14.39: Kinetic Salt Effects
The chemical potential of a given solute \(\mathrm{j}\) in an aqueous solution is related to the concentration \(\mathrm{c}_{\mathrm{j}}\) using equation (a) where \(\mathrm{c}_{\mathrm{r}}\) is a reference concentration, \(1 \mathrm{~mol dm}^{-3}\), and yj is the solute activity coefficient.
\[\mu_{\mathrm{j}}(\mathrm{aq})=\mu_{\mathrm{j}}^{0}\left(\mathrm{c}_{\mathrm{j}}=1 \mathrm{~mol} \mathrm{dm}{ }^{-3} ; \mathrm{aq} ; \mathrm{id}\right)+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right) \nonumber \]
By definition, at all \(\mathrm{T}\) and \(\mathrm{p}\),
\[\operatorname{limit}\left(\mathrm{c}_{\mathrm{j}} \rightarrow 0\right) \mathrm{y}_{\mathrm{j}}=1.0 \nonumber \]
In the application of equation (a) to the rates of chemical reactions in solution, transition state theory [1] is used. In the case of a second order bimolecular reaction involving solutes \(\mathrm{X}(\mathrm{aq})\) and \(\mathrm{Y}(\mathrm{aq})\), the reaction proceeds as described by equation (c).
\[\mathrm{X}(\mathrm{aq})+\mathrm{Y}(\mathrm{aq}) \Leftarrow \Rightarrow \mathrm{TS}^{\neq} \rightarrow \text { products } \nonumber \]
An equilibrium between reactants and transition state, \(\mathrm{TS}^{\neq}\) is described by an equilibrium constant \(\mathrm{K}^{\neq}\). Hence,
\[\Delta^{\neq} \mathrm{G}^{0}=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}^{\neq}\right)=\mu_{\neq}^{0}(\mathrm{aq})-\mu_{\mathrm{X}}^{0}(\mathrm{aq})-\mu_{\mathrm{Y}}^{0}(\mathrm{aq}) \nonumber \]
At equilibrium,
\[\mu^{\mathrm{eq}}(\mathrm{X} ; \mathrm{aq})+\mu^{\mathrm{eq}}(\mathrm{Y} ; \mathrm{aq})=\mu^{\mathrm{eq}}(\mathrm{TS} ; \mathrm{aq}) \nonumber \]
Using equation (a),
\[\mathrm{K}^{\neq}=\frac{\mathrm{c}^{\neq}(\mathrm{aq}) \, \mathrm{y}^{\neq}(\mathrm{aq}) \, \mathrm{c}_{\mathrm{r}}}{\mathrm{c}_{\mathrm{x}}^{\mathrm{eq}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{X}}(\mathrm{aq}) \, \mathrm{c}_{\mathrm{Y}}^{\mathrm{eq}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{Y}}(\mathrm{aq})} \nonumber \]
According to \(\mathrm{TS}\) theory [1] rate constant \(\mathrm{k}\) is related to \(\mathrm{K}^{\neq}\) using equation (g) where \(\kappa\) is a transmission coefficient, customarily set to unity. Then,
\[\mathrm{k}=\mathrm{K} \,(\mathrm{k} \, \mathrm{T} / \mathrm{h}) \, \mathrm{K}^{\neq} \, \mathrm{y}_{\mathrm{X}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{Y}}(\mathrm{aq}) / \mathrm{y}_{\neq}(\mathrm{aq}) \nonumber \]
In the event that the thermodynamic properties of the aqueous solution are ideal, equation (g) simplifies to equation (h).
\[\mathrm{k}(\mathrm{id})=\kappa \,(\mathrm{k} \, \mathrm{T} / \mathrm{h}) \, \mathrm{K}^{\neq} \nonumber \]
For a real system,
\[\mathrm{k}=\mathrm{k}(\mathrm{id}) \, \mathrm{y}_{\mathrm{X}}(\mathrm{aq}) \, \mathrm{y}_{\mathrm{Y}}(\mathrm{aq}) / \mathrm{y}_{\neq}(\mathrm{aq}) \nonumber \]
The Bronsted-Bjerrum analysis concerns rates of chemical reaction between ions having electric charges, \(\mathrm{z}_{\mathrm{x}} \, \mathrm{e}\) and \(\mathrm{z}_{\mathrm{y}} \, \mathrm{e}\) where the transition state has charge z ⋅ e ≠ ( z e z e) X Y = ⋅ + ⋅ .
In most applications, the activity coefficients are related to the ionic strength of the solution using the Debye - Huckel Limiting Law. For reactant \(\mathrm{j}\),
\[\ln \left(\mathrm{y}_{\mathrm{j}}\right)=-\mathrm{S}_{\mathrm{Y}} \, \mathrm{z}_{\mathrm{j}}^{2} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]
Then,
\[\ln (\mathrm{k})=\ln (\mathrm{k}(\mathrm{id}))+\ln \left(\mathrm{y}_{\mathrm{X}}\right)+\ln \left(\mathrm{y}_{\mathrm{Y}}\right)-\ln \left(\mathrm{y}_{z}\right) \nonumber \]
\[\ln (\mathrm{k})=\ln (\mathrm{k}(\mathrm{id}))-\mathrm{S}_{\mathrm{y}} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \,\left[\mathrm{z}_{\mathrm{X}}^{2}+\mathrm{z}_{\mathrm{Y}}^{2}-\left(\mathrm{z}_{\mathrm{X}}+\mathrm{z}_{\mathrm{Y}}\right)^{2}\right] \nonumber \]
Or,
\[\ln (\mathrm{k})-\ln (\mathrm{k}(\mathrm{id}))=\mathrm{S}_{\mathrm{y}} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \,\left[2 \, \mathrm{z}_{\mathrm{X}} \, \mathrm{Z}_{\mathrm{Y}}\right] \nonumber \]
Equation (m) forms the basis of the classic and oft-quoted plot of \([\ln (\mathrm{k})-\ln (\mathrm{k}(\mathrm{id}))]\) against \(\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2}\) in which the slope is determined by the product of charge numbers, \(\mathrm{z}_{\mathrm{x}} \, \mathrm{z}_{\mathrm{y}}\); [1;see Footnote (1), page 429].
An interesting feature was noted by Rosseinsky [2]. Equation (m) can be written in a quite general form for a reaction involving \(\mathrm{n}\) ions. Then,
\[\ln (\mathrm{k})-\ln (\mathrm{k}(\mathrm{id}))=\mathrm{S}_{\mathrm{y}} \,\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \, \sum_{\mathrm{i}}^{\mathrm{n}} \sum_{\mathrm{j}}^{\mathrm{n}} \mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}} \quad(\mathrm{i} \neq \mathrm{j}) \nonumber \]
For chemical reaction involving cations and anions , cases can arise where the double sum in equation(n) is zero. Hence the rate constant will be independent of ionic strength. Rosseinsky cites the following reaction as a case in point [3].
\[2 \mathrm{Mn}^{2+}(\mathrm{aq})+\mathrm{MnO}_{4}^{-} \text {(aq) } \rightarrow \mathrm{Mn}_{3} \mathrm{O}_{4}^{3+} \nonumber \]
Footnotes
[1] S. A. Glasstone, K. J. Laidler and H. Eyring, The Theory of Rate Processes, McGraw-Hill, New York, 1941, pp. 427-429.
[2] D. R. Rosseinsky, J. Chem. Phys.,1968, 48 , 4806.
[3] D. R. Rosseinsky and M. J. Nicol, Trans. Faraday Soc.,1965, 61 , 2718.