Conjugate Acid–Base Relationships and Ka × Kb = Kw

Core Concept

Every acid, once it donates a proton, becomes a conjugate base; every base, once it accepts one, becomes a conjugate acid. The two are locked together by the proton they exchange, and their strengths are inversely linked through water's ion-product constant: Ka × Kb = Kw = 1.0 × 10⁻¹⁴ at 25 °C. This single equation is the bridge between the "acid world" and the "base world," letting you convert the Ka of any acid into the Kb of its conjugate base and reason about which ions hydrolyze water.

Key Ideas

• Conjugate pair: two species differing by exactly one H⁺ (HCl/Cl⁻, NH₄⁺/NH₃, H₂O/OH⁻).
• Inverse strength: the stronger an acid, the weaker its conjugate base (and vice versa).
• Ka × Kb = Kw holds for any conjugate pair at a given temperature (25 °C: 1.0 × 10⁻¹⁴).
• Log form: pKa + pKb = pKw = 14.00 at 25 °C.
• Amphiprotic species (H₂O, HCO₃⁻, H₂PO₄⁻, HSO₄⁻) can donate or accept; which dominates is set by the relative sizes of their Ka and Kb.

Equations and Variables

• Conjugate pair: HA / A⁻ (differ by one proton)
• Ka(HA) × Kb(A⁻) = Kw
• Kb(A⁻) = Kw / Ka(HA)
• Ka(BH⁺) = Kw / Kb(B)
• pKa + pKb = 14.00 (25 °C)

How It Works

1. Identify the conjugate pair: remove one H⁺ from an acid to get its conjugate base (and subtract one from the charge); add one H⁺ to a base to get its conjugate acid (and add one to the charge).
2. Multiply the two equilibrium constants: Ka for the acid ionization and Kb for the reverse reaction (the conjugate base grabbing a proton from water). The H₃O⁺ and OH⁻ they produce multiply to give Kw, and everything else cancels.
3. Use Ka × Kb = Kw to find whichever constant you're missing. For example, acetic acid (Ka = 1.8 × 10⁻⁵) has an acetate conjugate base with Kb = 1.0 × 10⁻¹⁴ / 1.8 × 10⁻⁵ = 5.6 × 10⁻¹⁰.
4. To predict whether an ion is acidic or basic in water, compare its Ka and Kb: if Ka > Kb it's a net acid; if Kb > Ka it's a net base.

Worked Example

(a) Find Kb for CN⁻. (b) Find Ka for NH₄⁺.

Given Ka(HCN) = 6.2 × 10⁻¹⁰ and Kb(NH₃) = 1.8 × 10⁻⁵.

(a) CN⁻ is the conjugate base of HCN:

Kb(CN⁻) = Kw / Ka(HCN) = (1.0 × 10⁻¹⁴) / (6.2 × 10⁻¹⁰) = 1.6 × 10⁻⁵

(b) NH₄⁺ is the conjugate acid of NH₃:

Ka(NH₄⁺) = Kw / Kb(NH₃) = (1.0 × 10⁻¹⁴) / (1.8 × 10⁻⁵) = 5.6 × 10⁻¹⁰

Notice the symmetry: HCN (Ka 6.2 × 10⁻¹⁰) and NH₄⁺ (Ka 5.6 × 10⁻¹⁰) are comparably weak acids, and their conjugate bases CN⁻ (Kb 1.6 × 10⁻⁵) and NH₃ (Kb 1.8 × 10⁻⁵) are comparably weak bases.

How it works

1. Identify the conjugate pair: remove one H⁺ from an acid to get its conjugate base (and subtract one from the charge); add one H⁺ to a base to get its conjugate acid (and add one to the charge).
2. Multiply the two equilibrium constants: Ka for the acid ionization and Kb for the reverse reaction (the conjugate base grabbing a proton from water). The H₃O⁺ and OH⁻ they produce multiply to give Kw, and everything else cancels.
3. Use Ka × Kb = Kw to find whichever constant you're missing. For example, acetic acid (Ka = 1.8 × 10⁻⁵) has an acetate conjugate base with Kb = 1.0 × 10⁻¹⁴ / 1.8 × 10⁻⁵ = 5.6 × 10⁻¹⁰.
4. To predict whether an ion is acidic or basic in water, compare its Ka and Kb: if Ka > Kb it's a net acid; if Kb > Ka it's a net base.

Common confusions

• "Conjugate base of a weak acid is strong." — It is weak too, just with Kb = Kw/Ka; only the conjugate base of a strong acid is truly negligible.
• "Ka × Kb = Kw applies to any two species." — It applies only to a conjugate pair, not to unrelated acids and bases.
• "Adding H⁺ doesn't change the charge." — It does: adding a proton raises the charge by +1, so an acid and its conjugate base differ by one charge unit.
• "HCO₃⁻ is always a base." — It is amphiprotic; in pure water its Kb (≈2.3 × 10⁻⁸) exceeds its Ka (≈4.7 × 10⁻¹¹), so it makes a basic solution, but against a strong acid it acts as a base.

Quick review

• Conjugate pair = differs by one H⁺ (and one charge unit).
• Ka × Kb = Kw = 1.0 × 10⁻¹⁴; pKa + pKb = 14.00.
• Kb(A⁻) = Kw/Ka(HA); Ka(BH⁺) = Kw/Kb(B).
• Stronger acid ⇒ weaker conjugate base.
• Compare Ka vs Kb to decide net acid/base character of an amphiprotic ion.