1. The Equilibrium Reactions:
* Acid Dissociation: Consider a weak acid, HA, that dissociates in water:
HA(aq) + H₂O(l) ⇌ H₃O⁺(aq) + A⁻(aq)
with an acid dissociation constant, Ka:
Ka = [H₃O⁺][A⁻] / [HA]
* Base Hydrolysis: The conjugate base, A⁻, undergoes hydrolysis:
A⁻(aq) + H₂O(l) ⇌ HA(aq) + OH⁻(aq)
with a base hydrolysis constant, Kb:
Kb = [HA][OH⁻] / [A⁻]
2. Combining the Equilibria:
* Multiplying the Ka and Kb expressions:
Ka * Kb = ([H₃O⁺][A⁻] / [HA]) * ([HA][OH⁻] / [A⁻])
* Simplifying: Notice that [HA] and [A⁻] cancel out.
Ka * Kb = [H₃O⁺][OH⁻]
3. The Ion Product of Water:
* Kw Definition: The ion product of water, Kw, is defined as:
Kw = [H₃O⁺][OH⁻]
4. The Conclusion:
* Combining the equations: Since we showed that Ka * Kb = [H₃O⁺][OH⁻] and Kw = [H₃O⁺][OH⁻], we can conclude:
Kw = Ka * Kb
Therefore, the ion product of water (Kw) is equal to the product of the acid dissociation constant (Ka) and the base hydrolysis constant (Kb) for a conjugate acid-base pair.
Important Note: This relationship highlights the interdependence of acid and base strengths within a conjugate pair. If one member of the pair is a strong acid (large Ka), its conjugate base will be a weak base (small Kb) and vice versa. This ensures that Kw remains constant at a given temperature.
