Acid/Base Conjugates

The equilibrium expression for the autoionization of water can be used to relate the strength of an acid to its conjugate base (and vice versa).

Take a generic acid, HA in water. It will dissociate and form its conjugate base A^- and the hydronium ion.

\[\rm{HA(aq) + H_2O(l) \rightleftharpoons A^-(aq) + H_3O^+(aq)}\]

The equilibrium constant for this reaction is

\[{\rm K_a = {{[H_3O^+][A^-]} \over {[HA]}}}\]

Conversely the conjugate base of HA is A^-. When A^- is placed in water it will accept a proton from water (protonate) and form HA and the hydroxide ion

\[\rm{A^-(aq) + H_2O(l) \rightleftharpoons HA(aq) + OH^-(aq)}\]

The equilibrium constant for this reaction is

\[{\rm K_b = {{[OH^-][HA]} \over {[A^-]}}}\]

If we add these two reactions together we can see that the overall reaction for the two combined reactions is simply the auto-ionization of water.

\[\rm{2H_2O(l) \rightleftharpoons OH^-(aq) + H_3O^+(aq)}\]

The equilibrium constant for the sum of two reactions is the product of the equilibrium constants for the individual reactions. Thus the equilibrium constant for the two reactions added together is K_b x K_a. And because the net reaction is the auto-ionization of water with the equilibrium constant K_w, we know that

\[{\rm K_w = {K_a \times K_b}}\]

We can use this relationship to convert between the K_a for an acid HA and the K_b for its conjugate base A^-. Or we can convert between the K_b for a base B and the Ka for its conjugate acid BH⁺.

Because the product of K_a and K_b is a constant, there is a natural relationship between the strengths of the acids and bases that form a conjugate pair. The acid strength depends on the magnitude of K_a. The larger the K_a the stronger the acid. The strength of the base depends similarly on the K_b. The larger the K_b the stronger the base.

For conjugated acid/base partners both K_a and K_b can't be large. Because their product is a constant, the larger the K_a of the acid the smaller the K_b of its conjugate base (and vice versa). Thus the stronger the acid, the weaker its conjugate base. Or the stronger the base, the weaker its conjugate acid.

For example, the K_a for acetic acid, CH₃COOH, is 1.8x10^-5. From this we can find the K_b of its conjugate base partner the acetate ion, CH₃COO^-. K_b = K_w/K_a = 1x10^-14/1.8x10^-5 = 5.6x10^-10. This can be compared to the acid/base strength of the conjugate pair for hydrofluoric acid, HF/F^-. The K_a for HF, is 7.2x10^-4. Using this and K_w we find the K_b for F^- is 1.4x10^-11. HF is a stronger acid than CH₃COOH (it has a larger K_a). However, its conjugate base F^- is a weaker base than acetic acid's conjugate base, the acetate ion, CH₃COO^- since K_b for F^-1 is smaller than K_b for CH₃COO^-. For all pairs, the stronger the acid the weaker the base; the stronger the base, the weaker the acid.

This relationship is typically used for weak acids and bases. However, the same is true for strong acids/bases. Because strong acids are assumed to be "infinitely strong (have essentially infinite K_a's), their conjugate bases have K_b that are zero. or "infinitely weak". This makes the conjugate partner for strong acids and bases infinitely weak. They are so weak that we do not even consider them to be acids/bases. Instead these are the ions we refer to as spectator ions.

© 2013 mccord/vandenbout/labrake

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