Relative Strengths of Acids and Bases

As we did with acids, we can measure the relative strengths of bases by measuring their base-ionization constant (K_b) in aqueous solutions. In solutions of the same concentration, stronger bases ionize to a greater extent, and so yield higher hydroxide ion concentrations than do weaker bases. A stronger base has a larger ionization constant than does a weaker base. For the reaction of a base, B:

\(\text{B}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\phantom{\rule{0.2em}{0ex}}⇌\phantom{\rule{0.2em}{0ex}}{\text{HB}}^{\text{+}}\left(aq\right)+{\text{OH}}^{\text{−}}\left(aq\right),\)

we write the equation for the ionization constant as:

\({K}_{\text{b}}=\phantom{\rule{0.2em}{0ex}}\cfrac{\left[{\text{HB}}^{\text{+}}\right]\left[{\text{OH}}^{\text{−}}\right]}{\left[\text{B}\right]}\)

where the concentrations are those at equilibrium. Again, we do not include [H₂O] in the equation because water is the solvent. The chemical reactions and ionization constants of the three bases shown are:

\({\text{NO}}_{2}{}^{\text{−}}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\phantom{\rule{0.2em}{0ex}}⇌\phantom{\rule{0.2em}{0ex}}{\text{HNO}}_{2}\left(aq\right)+{\text{OH}}^{\text{−}}\left(aq\right)\phantom{\rule{5em}{0ex}}{K}_{\text{b}}=2.22\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-11}\)

\({\text{CH}}_{3}{\text{CO}}_{2}{}^{\text{−}}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\phantom{\rule{0.2em}{0ex}}⇌\phantom{\rule{0.2em}{0ex}}{\text{CH}}_{3}{\text{CO}}_{2}\text{H}\left(aq\right)+{\text{OH}}^{\text{−}}\left(aq\right)\phantom{\rule{5em}{0ex}}{K}_{\text{b}}=5.6\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-10}\)

\({\text{NH}}_{3}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\phantom{\rule{0.2em}{0ex}}⇌\phantom{\rule{0.2em}{0ex}}{\text{NH}}_{4}{}^{\text{+}}\left(aq\right)+{\text{OH}}^{\text{−}}\left(aq\right)\phantom{\rule{5em}{0ex}}{K}_{\text{b}}=1.8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\)

A table of ionization constants of weak bases appears in this appendix (with a partial list in this table). As with acids, percent ionization can be measured for basic solutions, but will vary depending on the base ionization constant and the initial concentration of the solution.

Consider the ionization reactions for a conjugate acid-base pair, HA − A⁻:

\(\text{HA}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\phantom{\rule{0.2em}{0ex}}⇌\phantom{\rule{0.2em}{0ex}}{\text{H}}_{3}{\text{O}}^{\text{+}}\left(aq\right)+{\text{A}}^{\text{−}}\left(aq\right)\phantom{\rule{5em}{0ex}}{K}_{\text{a}}=\phantom{\rule{0.2em}{0ex}}\cfrac{\left[{\text{H}}_{3}{\text{O}}^{\text{+}}\right]\left[{\text{A}}^{\text{−}}\right]}{\left[\text{HA}\right]}\)

\({\text{A}}^{\text{−}}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)\phantom{\rule{0.2em}{0ex}}⇌\phantom{\rule{0.2em}{0ex}}{\text{OH}}^{\text{−}}\left(aq\right)+\text{HA}\left(aq\right)\phantom{\rule{5em}{0ex}}{K}_{\text{b}}=\phantom{\rule{0.2em}{0ex}}\cfrac{\left[\text{HA}\right]\left[\text{OH}\right]}{\left[{\text{A}}^{\text{−}}\right]}\)

Adding these two chemical equations yields the equation for the autoionization for water:

\(\require{cancel}\cancel{\text{HA}\left(aq\right)}+{\text{H}}_{2}\text{O}\left(l\right)+\require{cancel}\cancel{{\text{A}}^{\text{−}}\left(aq\right)}+{\text{H}}_{2}\text{O}\left(l\right)\phantom{\rule{0.2em}{0ex}}⇌\phantom{\rule{0.2em}{0ex}}{\text{H}}_{3}{\text{O}}^{\text{+}}\left(aq\right)+\require{cancel}\cancel{{\text{A}}^{\text{−}}\left(aq\right)}+{\text{OH}}^{\text{−}}\left(aq\right)+\require{cancel}\cancel{\text{HA}\left(aq\right)}\)

\({\text{2H}}_{2}\text{O}\left(l\right)\phantom{\rule{0.2em}{0ex}}⇌\phantom{\rule{0.2em}{0ex}}{\text{H}}_{3}{\text{O}}^{\text{+}}\left(aq\right)+{\text{OH}}^{\text{−}}\left(aq\right)\)

Because of the way mass-action expressions are defined, the K expression for any chemical equation that is derived by adding together two or more other equations is equal to the mathematical product of the other equations’ K expressions. Multiplying the mass-action expressions together and canceling common terms, we see that:

\({K}_{\text{a}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{K}_{\text{b}}=\phantom{\rule{0.2em}{0ex}}\cfrac{\left[{\text{H}}_{3}{\text{O}}^{\text{+}}\right]\left[{\text{A}}^{\text{−}}\right]}{\text{[HA]}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\cfrac{\text{[HA]}\left[{\text{OH}}^{\text{−}}\right]}{\left[{\text{A}}^{\text{−}}\right]}\phantom{\rule{0.2em}{0ex}}=\left[{\text{H}}_{3}{\text{O}}^{\text{+}}\right]\left[{\text{OH}}^{\text{−}}\right]={K}_{\text{w}}\)

For example, the acid ionization constant of acetic acid (CH₃COOH) is 1.8 \(×\) 10⁻⁵, and the base ionization constant of its conjugate base, acetate ion \(\left({\text{CH}}_{3}{\text{COO}}^{\text{−}}\right),\) is 5.6 \(×\) 10⁻¹⁰. The product of these two constants is indeed equal to K_w:

\({K}_{\text{a}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{K}_{\text{b}}=\left(1.8\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}\right)\phantom{\rule{0.4em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(5.6\phantom{\rule{0.4em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-10}\right)=1.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-14}={K}_{\text{w}}\)

The extent to which an acid, HA, donates protons to water molecules depends on the strength of the conjugate base, A⁻, of the acid. If A⁻ is a strong base, any protons that are donated to water molecules are recaptured by A⁻. Thus there is relatively little A⁻ and \({\text{H}}_{3}{\text{O}}^{\text{+}}\) in solution, and the acid, HA, is weak. If A⁻ is a weak base, water binds the protons more strongly, and the solution contains primarily A⁻ and H₃O⁺—the acid is strong. Strong acids form very weak conjugate bases, and weak acids form stronger conjugate bases (see the figure below).

This diagram shows the relative strengths of conjugate acid-base pairs, as indicated by their ionization constants in aqueous solution.

The figure below lists a series of acids and bases in order of the decreasing strengths of the acids and the corresponding increasing strengths of the bases. The acid and base in a given row are conjugate to each other.

The chart shows the relative strengths of conjugate acid-base pairs.

The first six acids in the figure above are the most common strong acids. These acids are completely dissociated in aqueous solution. The conjugate bases of these acids are weaker bases than water. When one of these acids dissolves in water, their protons are completely transferred to water, the stronger base.

Those acids that lie between the hydronium ion and water in the figure above form conjugate bases that can compete with water for possession of a proton. Both hydronium ions and nonionized acid molecules are present in equilibrium in a solution of one of these acids. Compounds that are weaker acids than water (those found below water in the column of acids) in the figure above exhibit no observable acidic behavior when dissolved in water. Their conjugate bases are stronger than the hydroxide ion; if any conjugate base were formed, it would react with water to form the hydroxide ion.

The extent to which a base forms hydroxide ion in aqueous solution depends on the strength of the base relative to that of the hydroxide ion, as shown in the last column in the figure above. A strong base, such as one of those lying below hydroxide ion, accepts protons from water to yield 100% of the conjugate acid and hydroxide ion. Those bases lying between water and hydroxide ion accept protons from water, but a mixture of the hydroxide ion and the base results. Bases that are weaker than water (those that lie above water in the column of bases) show no observable basic behavior in aqueous solution.