1.14.8: Extrathermodynamics - Equilbrium - Acid Strength

In aqueous solution at ambient pressure and \(298.15 \mathrm{~K}\), benzoic acid exists in the form of a chemical equilibrium described in equation (a)

\[\mathrm{PhCOOH}(\mathrm{aq}) \Leftrightarrow=\mathrm{H}^{+}(\mathrm{aq})+\mathrm{PhCOO}^{-}(\mathrm{aq}) \nonumber \]

At defined \(\mathrm{T}\) and \(\mathrm{p}\),

\[\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhCOOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{PhCOO}^{-} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)-\mu^{0}(\mathrm{PhCOOH} ; \mathrm{aq}) \nonumber \]

In the case of a substituted benzoic acid, \(\mathrm{XC}_{6} \mathrm{H}_{4} \mathrm{COOH} \quad[=\mathrm{XPhCOOH}]\), the corresponding description of the chemical equilibrium takes the following form.

\[\begin{aligned}
&\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhCOOH} ; \mathrm{aq}) \\
&=\mu^{0}(\mathrm{XPhCOO} ; ; \mathrm{aq})+\mu^{0}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)-\mu^{0}(\mathrm{XPhCOOH} ; \mathrm{aq})
\end{aligned} \nonumber \]

In aqueous solution at ambient pressure and \(298.15 \mathrm{~K}\), the properties of an aqueous solution containing phenol can be described in terms of the following equilibrium.

\[\mathrm{PhOH}(\mathrm{aq}) \Leftrightarrow=\Longrightarrow \mathrm{H}^{+}(\mathrm{aq})+\mathrm{PhO}^{-}(\mathrm{aq}) \nonumber \]

Then, (cf. equation (b)),

\[\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{PhO}^{-} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)-\mu^{0}(\mathrm{PhOH} ; \mathrm{aq}) \nonumber \]

In the case of a substituted phenol \(\mathrm{XPhOH}\), the equation corresponding to equation (d) takes the following form.

\[\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{XPhO}^{-} ; \mathrm{aq}\right)+\mu^{0}\left(\mathrm{H}^{+} ; \mathrm{aq}\right)-\mu^{0}(\mathrm{XPhOH} ; \mathrm{aq}) \nonumber \]

In the following we compare situations where \(\mathrm{X}\) is common to the substituted phenol and benzoic acid including position in the aromatic ring. The interesting point to emerge is that for a range of substituents, \(\mathrm{X}\), the recorded dependence of \(\Delta_{\Delta_{r}} \mathrm{G}^{0}(\mathrm{XPhOH} ; \mathrm{aq})\) on \(\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhCOOH} ; \mathrm{aq})\) is linear. Such a pattern is not a requirement of thermodynamics [1]. The challenge is to suggest a set of minimum relationships which account for this pattern [2,3].

Zone Model POSTULATE---Single Interaction Mechanism

Consider the reference chemical potential for solute \(\mathrm{RX}\) in aqueous solution at fixed \(\mathrm{T}\) and \(\mathrm{p}\), \(\mu^{0}(\mathrm{RX} ; \mathrm{aq})\). As chemists we recognise that groups \(\mathrm{R}\) and \(\mathrm{X}\) do not make independent contributions to \(\mu^{0}(\mathrm{RX} ; \mathrm{aq})\) [4]. The postulate, Single Interaction Mechanism, recognises that the groups \(\mathrm{R}\) and \(\mathrm{X}\) interact such that \(\mu^{0}(\mathrm{RX} ; \mathrm{aq})\) is given by equation (g).

\[\mu^{0}(\mathrm{RX} ; \mathrm{aq})=\mu^{0}(\mathrm{R})+\mu^{0}(\mathrm{X})+\mathrm{I}(\mathrm{R}, \mathrm{X}) \nonumber \]

Here symbol \(\mathrm{R}\) identifies the substituent zone and \(\mathrm{X}\) identifies the reaction zone so that \(\mathrm{I}(\mathrm{R}, \mathrm{X})\) describes interaction between these two zones.

Separability Postulate

The interaction variable \(\mathrm{I}(\mathrm{R}, \mathrm{X})\) is a function of scalar variables. Then

\[\mu^{0}(\mathrm{RX} ; \mathrm{aq})=\mu^{0}(\mathrm{R})+\mu^{0}(\mathrm{X})+\mathrm{I}(\mathrm{R}) \, \mathrm{I}(\mathrm{X}) \nonumber \]

Hence for benzoic acid \(\mathrm{PhCOOH}(\mathrm{aq})\),

\[\mu^{0}(\mathrm{PhCOOH} ; \mathrm{aq})=\mu^{0}(\mathrm{Ph})+\mu^{0}(\mathrm{COOH})+\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}(\mathrm{COOH}) \nonumber \]

Similarly,

\[\mu^{0}(\mathrm{PhCOO} ; ; \mathrm{aq})=\mu^{0}(\mathrm{Ph})+\mu^{0}\left(\mathrm{COO}^{-}\right)+\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right) \nonumber \]

Hence,

\[\begin{gathered}
\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhCOOH} ; \mathrm{aq})=\mu^{0}(\mathrm{Ph})+\mu^{0}\left(\mathrm{COO}^{-}\right)+\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)+\mu^{0}\left(\mathrm{H}^{+}\right) \\
-\mu^{0}(\mathrm{Ph})-\mu^{0}(\mathrm{COOH})-\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}(\mathrm{COOH})
\end{gathered} \nonumber \]

Or,

\[\begin{gathered}
\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhCOOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{COO}^{-}\right)+\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)+\mu^{0}\left(\mathrm{H}^{+}\right) \\
-\mu^{0}(\mathrm{COOH})-\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}(\mathrm{COOH})
\end{gathered} \nonumber \]

A similar equation emerges describing the acid dissociation of the substituted acid. Thus,

\[\begin{gathered}
\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhCOOH} ; \mathrm{aq})=\mu^{0}\left(\mathrm{COO}^{-}\right)+\mathrm{I}(\mathrm{XPh}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)+\mu^{0}\left(\mathrm{H}^{+}\right) \\
-\mu^{0}(\mathrm{COOH})-\mathrm{I}(\mathrm{XPh}) \, \mathrm{I}(\mathrm{COOH})
\end{gathered} \nonumber \]

By definition,

\[\Delta_{\mathrm{r}} \mathrm{G}^{0}=\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{XPhCOOH} ; \mathrm{aq})-\Delta_{\mathrm{r}} \mathrm{G}^{0}(\mathrm{PhCOOH} ; \mathrm{aq}) \nonumber \]

Hence,

\[\begin{gathered}
\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0}(\text { acids })=\left[\mathrm{I}(\mathrm{XPh}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)-\mathrm{I}(\mathrm{XPh}) \, \mathrm{I}(\mathrm{COOH})\right] \\
-\left[\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}\left(\mathrm{COO}^{-}\right)-\mathrm{I}(\mathrm{Ph}) \, \mathrm{I}(\mathrm{COOH})\right]
\end{gathered} \nonumber \]

Or,

\[\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0}(\text { acids })=[\mathrm{I}(\mathrm{XPh})-\mathrm{I}(\mathrm{Ph})] \,\left[\mathrm{I}\left(\mathrm{COO}^{-}\right)-\mathrm{I}(\mathrm{COOH})\right] \nonumber \]

Thus \(\Delta_{r} G^{0}\) is given by the product of two terms;

1. a difference in substituent parameters, and
2. a difference in reaction zone parameters.

We turn our attention to the acid strength of phenol and susbstituted phenols in aqueous solution at the same \(\mathrm{T}\) and \(\mathrm{p}\). A similar analysis to that set out above yields the following equation.

\[G (phenols) [I(XPh) I(Ph)] [I(O ) I(OH)] 0 ∆∆r = − ⋅ − − (q)

Comparison of equations (p) and (q) yields equation (r).

\[\left.\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0}(\text { phenols })=\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0} \text { (acids }\right) \,\left\{\left[\mathrm{I}\left(\mathrm{O}^{-}\right)-\mathrm{I}(\mathrm{OH})\right] /\left[\mathrm{I}\left(\mathrm{COO}^{-}\right)-\mathrm{I}(\mathrm{COOH})\right]\right\} \nonumber \]

The analysis rationalises the observation that \(\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0} \text { (phenols) }\) is a linear function of \(\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0} \text { (acids) }\). In other words we have not proved that such a linear function exists. Rather we have identified the minimum hypothesis required to account for the observation. In these terms the extrathermodynamic analysis has pointed to a reason for the recorded dependences of \(\Delta \Delta_{\mathrm{r}} G^{0} \text { (phenols) }\) on \(\Delta \Delta_{\mathrm{r}} \mathrm{G}^{0} \text { (acids) }\). The pattern is not a requirement of thermodynamics.

Footnotes

[1] See for example,

1. D. T. Y. Chen and K. J. Laidler, Trans. Faraday Soc., 1962, 58, 486.; and
2. C. M. Judson and M. L. Kilpatrick, J. Am. Chem. Soc., 1949, 71, 3115.

[2] J. E. Leffler and E. Grunwald, Rates and Equilibria of Organic Reactions, Wiley, London, 1963.

[3] E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1997.

[4] The superscript ‘0’ is retained although the meaning here is somewhat obscure. It effectively reminds us that we are dealing with the properties of a solute in its solution reference state.