15.5: Relative Strengths of Conjugate Acid-Base Pairs

Brønsted-Lowry acid-base chemistry is the transfer of protons; thus, logic suggests a relation between the relative strengths of conjugate acid-base pairs. The strength of an acid or base is quantified in its ionization constant, K_a or K_b, which represents the extent of the acid or base ionization reaction. For the conjugate acid-base pair HA / A⁻, the ionization equilibrium equations and ionization constant expressions are

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

As previously discussed, the equilibrium constant for a summed reaction is equal to the mathematical product of the equilibrium constants for the added reactions, and so

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This equation states the relation between ionization constants for any conjugate acid-base pair, namely, their mathematical product is equal to the ion product of water, K_W. By rearranging this equation, a reciprocal relation between the strengths of a conjugate acid-base pair becomes evident:

The inverse proportional relation between K_a and K_b means the stronger the acid or base, the weaker its conjugate partner.

Taking the negative log of both sides of the equation, K_a × K_b = K_W yields

then

As pK_W is 14 at 25 °C, this equation can also be written as

The pK_a and pK_b also represent the strength of acids and bases, respectively. Like pH and pOH, the higher the value pK_a or pK_b, the weaker the acid or base, respectively.

The listing of conjugate acid–base pairs shown is arranged to show the relative strength of each species as compared with water. In the acid column, those species listed below water are weaker acids than water. These species do not undergo acid ionization in water; they are not Brønsted-Lowry acids. All the species listed above water are stronger acids, transferring protons to water to some extent when dissolved in an aqueous solution to generate hydronium ions. Species above water but below hydronium ion are weak acids, undergoing partial acid ionization, whereas those above hydronium ion are strong acids that are completely ionized in aqueous solution.

This text is adapted from Openstax, Chemistry 2e, Section 14.3: Relative Strengths of Acids and Bases.

The relationship between the dissociation constants of a conjugate acid-base pair can be expressed quantitatively.

For a weak acid, HA, its acid dissociation constant, K_a, is expressed as the hydronium ion concentration times the A ion concentration divided by the concentration of HA.

The base dissociation constant, K_b, for its conjugate base, A ion, is expressed as the HA concentration times the hydroxide ion concentration divided by the concentration of the A ion.

If the expressions for K_a and K_b are multiplied, the resulting equation is the equilibrium expression for K_w. This equation shows that the K_a for a weak acid and the K_b for its conjugate base are inversely proportional. A stronger acid with a higher K_a always has a proportionately weaker conjugate base with a lower K_b and vice versa.

As the value of K_W is constant, this equation can be used to calculate K_a or K_b of a conjugate acid-base pair if either one of them is known. For example, if the K_b for a base is 1 × 10⁻⁶, then the K_a for its conjugate acid can be calculated to be 1 × 10⁻⁸.

The relationship between K_a, K_b, and K_W can also be expressed in terms of their negative logarithms, pK_a, pK_b, and pK_W.

For the equation, K_a times K_b is equal to K_w, when the negative logarithm of both sides is taken, the resulting equation is pK_a plus pK_b is equal to pK_W, which is 14 at 25 °C.

pK_a and pK_b can also be used to compare the strengths of weak acids and bases. The lower the value of the pK_a, the stronger the acid. In the same manner, the lower the value of the pK_b, the stronger the base. For example, an acid with a pK_a of 2.1 is stronger than an acid with a pK_a of 4.6.