Conjugate Acid-Base Pair: The Relationship Between $ K_a $, $ K_b $, and $ K_w $
NoteIn this article we will recap on acid and base ionization constants which were covered in the previous article, as well as explore their relation to $K_w$.
Acid Ionization Constant ($ K_a $)
- When a weak acid $ HA $ dissociates in water, the equilibrium can be expressed as: $$HA(aq) \rightleftharpoons H^+(aq) + A^-(aq)$$
- The equilibrium constant for this reaction, $ K_a $, is defined as: $$K_a = \frac{[H^+][A^-]}{[HA]}$$ Here:
- $ [H^+] $: Concentration of hydrogen ions (protons)
- $ [A^-] $: Concentration of the conjugate base
- $ [HA] $: Concentration of the undissociated acid
- For strong acids, $ K_a $ is very large because the acid dissociates almost completely.
- For weak acids, $ K_a $ is small, indicating partial dissociation.
Base Ionization Constant ($ K_b $)
- The conjugate base $ A^- $ of the weak acid $ HA $ can react with water to form $ OH^- $ ions: $$A^-(aq) + H_2O(l) \leftrightharpoons OH^-(aq) + HA(aq)$$
- The equilibrium constant for this reaction, $ K_b $, is defined as: $$K_b = \frac{[OH^-][HA]}{[A^-]}$$
- The stronger the conjugate base, the larger the $ K_b $.
- Conversely, weak conjugate bases have smaller $ K_b $ values.
The Ionic Product of Water ($ K_w $)
- Water itself undergoes a small degree of ionization: $$H_2O(l) \leftrightharpoons H^+(aq) + OH^-(aq)$$
- The equilibrium constant for this reaction, known as the ionic product of water, is $$K_w = [H^+][OH^-]$$ at 298 K, $ K_w = 1.0 \times 10^{-14} $.
- Many students forget that $ K_w $ changes with temperature.
- Always check the temperature before using $ K_w = 1.0 \times 10^{-14} $.
Deriving $ K_a \times K_b = K_w $
- Let’s combine the ionization reactions for $ HA $ and $ A^- $:
- Acid ionization: $$ HA \leftrightharpoons H^+ + A^- $$
- Base ionization: $$ A^- + H_2O \leftrightharpoons OH^- + HA $$
- Adding these reactions cancels $ HA $ and $ A^- $, leaving: $$H_2O \leftrightharpoons H^+ + OH^-$$
