Relationship between Ka and Kb

For conjugate-acid base pairs, the acid dissociation constant 
$K_\text{a}$K, start subscript, a, end subscript and base ionization constant $K_\text{b}$K, start subscript, b, end subscript are related by the following equations:

• $K_\text{a}\cdot K_\text{b}=K_\text{w}$K, start subscript, a, end subscript, dot, K, start subscript, b, end subscript, equals, K, start subscript, w, end subscript
  where $K_\text{w}$K, start subscript, w, end subscript is the autoionization constant

• $\text{p}K_\text{a}+\text{p}K_\text{b}=14~~\text{at }25\,^\circ\text C$

Introduction: Weak acid and bases ionize reversibly

Weak acids, generically abbreviated as $\greenD{\text {HA}}$start color greenD, H, A, end color greenD, donate $\text H^+$H, start superscript, plus, end superscript (or proton) to water to form the conjugate base $\greenD{\text A^-}$start color greenD, A, start superscript, minus, end superscript, end color greenD and $\text{H}_3\text{O}^+$H, start subscript, 3, end subscript, O, start superscript, plus, end superscript:

$\text{HA}(aq)+{\text{H}}_2\text{O}(l)\rightleftharpoons {\text{H}}_3{\text{O}}^{+}(aq)+{\text{A}}^{-}(aq)$

$~~\greenD{\text{acid}}~~~~~~~~~~\text{base}~~~~~~~~~~~~\text{acid}~~~~~~~~~~~~~~\greenD{\text{base}}$space, space, start color greenD, a, c, i, d, end color greenD, space, space, space, space, space, space, space, space, space, space, b, a, s, e, space, space, space, space, space, space, space, space, space, space, space, space, a, c, i, d, space, space, space, space, space, space, space, space, space, space, space, space, space, space, start color greenD, b, a, s, e, end color greenD

Similarly, a base (abbreviated as $\purpleC{\text{B}}$start color purpleC, B, end color purpleC) will accept a proton in water to form the conjugate acid, $\purpleC{\text{HB}^+}$start color purpleC, H, B, start superscript, plus, end superscript, end color purpleC, and $\text{OH}^-$O, H, start superscript, minus, end superscript:

$\text{B}(aq)+\text{H}_2\text{O}(l)\rightleftharpoons\text{HB}^+(aq)+\text{OH}^-(aq)$

$~~\purpleC{\text{base}}~~~~~~~~~\text{acid}~~~~~~~~~~\purpleC{\text{acid}}~~~~~~~~~~\text{base}$space, space, start color purpleC, b, a, s, e, end color purpleC, space, space, space, space, space, space, space, space, space, a, c, i, d, space, space, space, space, space, space, space, space, space, space, start color purpleC, a, c, i, d, end color purpleC, space, space, space, space, space, space, space, space, space, space, b, a, s, e

For a weak acid or base, the equilibrium constant for the ionization reaction quantifies the relative amounts of each species. In this article, we will discuss the relationship between the equilibrium constants $K_\text{a}$K, start subscript, a, end subscript and $K_\text{b}$K, start subscript, b, end subscript for a conjugate acid-base pair.

Summary

For conjugate-acid base pairs, the acid dissociation constant $K_\text{a}$K, start subscript, a, end subscript and base ionization constant $K_\text{b}$K, start subscript, b, end subscript are related by the following equations:

• $K_\text{w}=K_\text{a}\cdot K_\text{b}$K, start subscript, w, end subscript, equals, K, start subscript, a, end subscript, dot, K, start subscript, b, end subscript

• $\text{p}K_\text{a}+\text{p}K_\text{b}=14~~\text{at }25\,^\circ\text C$

$\text{HA}(aq)+\text{H}_2\text{O}(l)\rightleftharpoons\text{H}_3\text{O}^+(aq)+\text{A}^-(aq)HH$HA(aq)+H

$~~\greenD{\text{acid}}~~~~~~~~~~\text{base}~~~~~~~~~~~~\text{acid}~~~~~~~~~~~~~~\greenD{\text{base}}$