1.1.1: Acid Dissociation Constants- Weak Acids- Debye-Huckel Limiting Law

For a weak acid HA in aqueous solution at temperature T and pressure p (which is ambient pressure and so close to the standard pressure) the following chemical equilibrium is established.

\[\mathrm{HA}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq}) \nonumber \]

The r.h.s. of equation (1.1.3) describes a 1:1 ‘salt’ in aqueous solution. At equilibrium (i.e. at a minimum in Gibbs energy), the thermodynamic description of the solution takes the following form.

\[\mu^{e q}(\mathrm{HA} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p})+\mu^{\mathrm{eq}}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\mu^{\mathrm{eq}}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right) \nonumber \]

We express µ^eq(H₂O;aq;T;p) in terms of the practical osmotic coefficient φ for the solution.

\[\mu^{e q}\left(\mathrm{H}_{2} \mathrm{O} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)=\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda ; \mathrm{T} ; \mathrm{p}\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left[\mathrm{m}(\mathrm{HA})+2 \, \mathrm{m}_{\mathrm{j}}\right]^{e q} \nonumber \]

Here m_j is the molality of the ‘salt’ H₃O⁺A^-. The latter yields 2 moles of ions for each mole of H₃O⁺A^-. A full description of the solution takes the following form.

\[\begin{aligned}
\mu^{0}(\mathrm{HA} ; \mathrm{aq})+& \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}(\mathrm{HA}) \, \gamma(\mathrm{HA}) / \mathrm{m}^{0}\right]^{e q} \\
&+\mu^{*}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda\right)-\phi \, \mathrm{R} \, \mathrm{T} \, \mathrm{M}_{1} \,\left[\mathrm{m}(\mathrm{HA})+2 \, \mathrm{m}_{\mathrm{j}}\right]^{\mathrm{eq}} \\
&=\mu^{0}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-} ; \mathrm{aq}\right)+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left[\mathrm{m}_{\mathrm{j}} \, \gamma_{\pm}\right]^{\mathrm{eq}} \quad
\end{aligned} \nonumber \]

The practical osmotic coefficient φ describes the properties of solvent, water in the aqueous solution; γ_± is the mean ionic activity coefficient for the ‘salt’ H₃O⁺A^-. By definition, if ambient pressure p is close to the standard pressure p⁰, the standard Gibbs energy of acid dissociation,

\[\begin{gathered}
\Delta_{\mathrm{d}} \mathrm{G}^{0}=\mu^{0}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p}\right)-\mu^{0}(\mathrm{HA} ; \mathrm{aq} ; \mathrm{T} ; \mathrm{p})-\mu^{0}\left(\mathrm{H}_{2} \mathrm{O} ; \lambda ; \mathrm{T} ; \mathrm{p}\right) \\
=-\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{K}_{\mathrm{A}}^{0}\right)
\end{gathered} \nonumber \]

\(\mathrm{K}_{\mathrm{A}}^{0}\) is the acid dissociation constant. Combination of equations (1.1.4) and (1.1.5) yields equation (1.1.6).

\[\mathrm{K}_{\mathrm{A}}^{0} = \frac{\left[\mathrm{m}\left(\mathrm{H}_{3} \mathrm{O}^{+} \, \mathrm{A}^{-}\right)^{\mathrm{cq}} \, \gamma_{\pm}\left(\mathrm{H}_{3} \mathrm{O}^{+} \mathrm{A}^{-}\right)^{\mathrm{eq}} / \mathrm{m}^{0}\right]^{2} \, \exp \left[\phi \, \mathrm{M}_{1} \,\left(\mathrm{m}(\mathrm{HA})+2 \, \mathrm{m}_{\mathrm{j}}\right)\right]^{\mathrm{cq}}}{\left[\mathrm{m}(\mathrm{HA}) \, \gamma(\mathrm{HA}) / \mathrm{m}^{0}\right]^{\mathrm{eq}}} \nonumber \]

For dilute aqueous solutions, several approximations are valid. The exponential term and γ(HA)^eq are close to unity. There are advantages in defining a quantity \(\mathrm{K}_{\mathrm{A}}^{0}\) (app) . Further, γ_±(H₃O⁺A^-) is obtained using the Debye - Huckel Limiting Law, DHLL.

By definition,

\[\mathrm{K}_{\mathrm{A}}(\mathrm{app})=\left[\mathrm{m}\left(\mathrm{H}^{+} \mathrm{A}^{-}\right)^{\mathrm{eq}}\right]^{2} / \mathrm{m}(\mathrm{HA})^{\mathrm{eq}} \, \mathrm{m}^{0} \nonumber \]

Then

\[\ln \mathrm{K}_{\mathrm{A}}(\operatorname{app})=\ln \mathrm{K}_{\mathrm{A}}^{0}+2 \, \mathrm{S}_{\gamma}\left(\mathrm{I} / \mathrm{m}^{0}\right)^{1 / 2} \nonumber \]

In other words, with increase in ionic strength \(I\), \(\mathrm{K}_{\mathrm{A}}(\operatorname{app})\) increases as a consequence of ion - ion interactions which stabilize the dissociated form of the acid.