Ionization Constants of Weak Acids and Bases: $K_a$, $K_b$ and their Relationship

Every weak acid has a characteristic ionization constant, $K_a$, and every weak base its own

$K_b$, each defined exactly as for any equilibrium constant. For a weak acid

$\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-$, $K_a = [\text{H}^+][\text{A}^-]/[\text{HA}]$;

for a weak base $\text{BOH} \rightleftharpoons \text{B}^+ + \text{OH}^-$,

$K_b = [\text{B}^+][\text{OH}^-]/[\text{BOH}]$. A larger $K_a$ (or $K_b$) means the acid (or base)

ionizes more extensively and is therefore the stronger of two weak electrolytes being compared.

Finding the pH of a weak acid or base. Because only a small fraction of a weak electrolyte

ionizes, an ICE table is used exactly as in Ostwald's dilution law: starting from an initial

concentration $C$ and letting $x = [\text{H}^+]$ at equilibrium, $K_a = x^2/(C-x) \approx x^2/C$ (the

same small-$\alpha$ approximation as before), giving $[\text{H}^+] = \sqrt{K_a C}$ and hence

$\text{pH} = -\log\sqrt{K_a C}$. The identical approach, with $K_b$ and $[\text{OH}^-]$, gives the pOH

of a weak base, which is then converted to pH via $\text{pH} = 14 - \text{pOH}$.

Relationship between $K_a$ and $K_b$ for a conjugate pair. Every weak acid $\text{HA}$ has a

conjugate base $\text{A}^-$, and that conjugate base has its own basic-ionization equilibrium,

$\text{A}^- + \text{H}_2\text{O} \rightleftharpoons \text{HA} + \text{OH}^-$, with its own $K_b$.

Multiplying the two equilibrium expressions together,

so for any conjugate acid-base pair, $K_a \times K_b = K_w$. This is extremely useful: the ionization

constant of the conjugate base of a weak acid (or the conjugate acid of a weak base) can always be

found from $K_a$ (or $K_b$) alone, without needing a separate independent measurement — for instance,

acetic acid's conjugate base, the acetate ion, has …