1.14.10: Extrathermodynamics - Solvent Effects in Chemical Kinetics

An enormous chemical literature describes the effects of solvents on rates of chemical reactions. C. K. Ingold [1] in his classic monograph actually uses the phrase 'solvent polarity' when commenting on the relative rates of reactions through a series of solvents of diminishing "polarity". One of the reactions discussed by Ingold concerns the solvolysis of 2-chloro-2-methyl -propane, \(\left(\mathrm{CH}_{3}\right)_{3}\mathrm{CCl}\). In 1948, Winstein and Grunwald [2] used this reaction as a basis for a quantitative treatment of solvent polarities leading to the definition of solvent Y-value. The basis of their analysis can be understood using an extrathermodynamic analysis [3]. We use a description based on substituent zone \(\mathrm{R}\) and reaction zone \(\mathrm{X}\) for a solute molecule \(\mathrm{RX}\). Here the interaction between the two zones is solvent dependent.

The starting point is kinetic data describing an (assumed) unimolecular first order solvolysis of a solute \(\mathrm{RX}\). The chemical reaction proceeds through a transition state \(\mathrm{RX}^{neq}\). For a given solvent medium \(\mathrm{M}\) at defined \(\mathrm{T}\) and \(\mathrm{p}\), transition state theory [9] describes the standard activation Gibbs energy as follows.

\[\Delta^{\neq} \mathrm{G}^{0}(\mathrm{RX} ; \mathrm{M})=\mu^{0}\left(\mathrm{RX} \mathrm{X}^{\neq} ; \mathrm{M}\right)-\mu^{0}(\mathrm{RX} ; \mathrm{M})\]

The basic postulate states that the reference chemical potential of solute \(\mathrm{RX}\) in solution, \(\mu^{0}(\mathrm{RX} ; \mathrm{sln})\) at defined \(\mathrm{T}\) and \(\mathrm{p}\) is given by the sum of contributions from the substituent zone, \(\mu^{0} (\mathrm{R})\) and reaction zone , \(\mu^{0} (\mathrm{X})\) together with terms describing the interaction of \(\mathrm{R}\) and \(\mathrm{X}\) with the solvent, \(\mathrm{I}(\mathrm{R}, \mathrm{M})\) and \(\mathrm{I}(\mathrm{X}, \mathrm{M})\) and the effect of solvent on this interaction \(\mathrm{II}(\mathrm{R}, \mathrm{X}, \mathrm{M})\).

\[\begin{aligned}
&\mu^{0}(\mathrm{RX} ; \text { in medium } \mathrm{M})= \\
&\qquad \mu^{0}(\mathrm{R})+\mu^{0}(\mathrm{X})+\mathrm{I}(\mathrm{R}, \mathrm{M})+\mathrm{I}(\mathrm{X}, \mathrm{M})+\mathrm{II}(\mathrm{R}, \mathrm{X}, \mathrm{M})
\end{aligned}\]

Thus the solvent \(\mathrm{M}\) contributes to the interaction between \(\mathrm{R}\) and \(\mathrm{X}\). A key postulate is advanced at this stage which states that the interactions terms can be factorised.

\[\begin{aligned}
&\mu^{0}(\mathrm{X} ; \text { in medium } \mathrm{M})= \\
&\mu^{0}(\mathrm{R})+\mu^{0}(\mathrm{X})+\mathrm{I}(\mathrm{R}) \, \mathrm{I}(\mathrm{M})+\mathrm{I}(\mathrm{X}) \, \mathrm{I}(\mathrm{M})+\mathrm{II}(\mathrm{R}) \, \mathrm{II}(\mathrm{X}) \, \mathrm{II}(\mathrm{M})
\end{aligned}\]

A similar equation is set down for the transition state.

\[\begin{aligned}
&\mu^{0}\left(\mathrm{RX}^{\neq} ; \text {in medium } \mathrm{M}\right)= \\
&\mu^{0}\left(\mathrm{R}^{\neq}\right)+\mu^{0}\left(\mathrm{X}^{\neq}\right)+\mathrm{I}\left(\mathrm{R}^{\neq}\right) \, \mathrm{I}(\mathrm{M})+\mathrm{I}\left(\mathrm{X}^{\neq}\right) \, \mathrm{I}(\mathrm{M})+\mathrm{II}\left(\mathrm{R}^{\neq}\right) \, \mathrm{II}\left(\mathrm{X}^{\neq}\right) \, \mathrm{II}(\mathrm{M})
\end{aligned}\]

Equations (c) and (d) are combined with equation (a). For reaction in solvent medium \(\mathrm{M}\),

\[\begin{aligned}
&\Delta^{\neq} \mathrm{G}^{0}(\mathrm{RX}, \mathrm{M})=\\
&\left.\left[\mu^{0}\left(R^{\neq}\right)-\mu^{0}(R)\right]+\left[\mu^{0}\left(X^{\neq}\right)-\mu^{0} X\right)\right]\\
&+\mathrm{I}(\mathrm{M}) \,\left[\mathrm{I}\left(\mathrm{R}^{\neq}\right)-\mathrm{I}(\mathrm{R})\right]+\mathrm{I}(\mathrm{M}) \,\left[\mathrm{I}\left(\mathrm{X}^{\neq}\right)-\mathrm{I}(\mathrm{X})\right]\\
&+\mathrm{II}(\mathrm{M}) \,\left[\mathrm{II}\left(\mathrm{R}^{*}\right) \, \mathrm{II}\left(\mathrm{X}^{\neq}\right)-\mathrm{II}(\mathrm{R}) \, \mathrm{II}(\mathrm{X})\right]
\end{aligned}\]

A second postulate state that \(\mathrm{II}(\mathrm{M})\) and \(\mathrm{I}(\mathrm{M})\) are simply related; i.e. equation (f).

\[\mathrm{II}(\mathrm{M})=\alpha \, \mathrm{I}(\mathrm{M})\]

If \(\Delta_{m} \Delta^{\neq} G^{0}(\mathrm{RX})\) describes the effect of solvent \(\mathrm{M}\) on \(\Delta^{\neq} \mathrm{G}^{0}(\mathrm{RX})\), \(\Delta_{\mathrm{m}} \Delta^{\neq} \mathrm{G}^{0}(\mathrm{RX})\) is given by the product of a (solvent operator) and a (substrate operator). By definition,

\[\Delta_{\mathrm{m}} \Delta^{\pm} \mathrm{G}^{0}(\mathrm{RX})=\Delta_{\mathrm{m}} \mathrm{Y} \,(\text { substrate operator })\]

Originally the substrate operator was set to unity for \(\left(\mathrm{CH}_{3}\right)_{3}\mathrm{CCl}\), and \(\mathrm{Y}\) was set to zero for an 80:20 ethanol + water mixture [4]. The outcome was a set of \(\mathrm{Y}\)-values for many solvents, particularly alcohol + water mixtures at \(298.5 \mathrm{~K}\) and ambient pressure

Footnotes

[1] C. K. Ingold, Structure and Mechanism in Organic Chemistry, G. Bell, London, 1953; see page 347.

[2] E. Grunwald and S. Winstein, J. Am. Chem. Soc., 1948,70, 841; 846.

[3] J. E. Leffler and E. Grunwald, Rates and Equilibria of Organic Reactions, Wiley, New York, 1963; Dover Publications , New York,1989.

[4] \(20 \mathrm{~cm}^{3}\) of ethanol(\(\ell\)) was poured from a volumetric flask containing \(1 \mathrm{~dm}^{3}\) of ethanol(\(\ell\)). The liquid in the flask was then ‘topped up’ with water(\(\ell\)). The mixture is a good solvent for both apolar and polar solutes. Unfortunately the exact composition of the mixture is unknown. As rarely stated, the volume of water required is slightly larger than \(20 \mathrm{~cm}^{3}\). Professor Ross E Robertson (University of Calgary) viewed with interest that so much information in the chemical literature describes rates of chemical reactions where the solvent is vodka.