Ka, Kb and Conjugate Strength

Introduction

The acid dissociation of HA and the base reaction of its conjugate A⁻ are not independent. Adding their equations leaves water autoionization, so their constants multiply to Kw. This relationship quantifies the inverse strength of a conjugate acid and base at a stated temperature.

Core explanation

Acid reaction: HA + H₂O ⇌ H₃O⁺ + A⁻, with Ka. Conjugate-base reaction: A⁻ + H₂O ⇌ HA + OH⁻, with Kb. Add them and cancel HA and A⁻ to obtain 2H₂O ⇌ H₃O⁺ + OH⁻. Reaction-constant multiplication gives Ka(HA)Kb(A⁻) = Kw under matching standard states and temperature.

At 25 °C in the familiar concentration approximation, Kw ≈ 1.0 × 10⁻¹⁴. If Ka = 1.0 × 10⁻⁵ for HA, then Kb of A⁻ is approximately 1.0 × 10⁻⁹. A stronger acid with larger Ka has a weaker conjugate base with smaller Kb, because their product remains fixed at that temperature.

Define pKa = −log₁₀Ka and pKb = −log₁₀Kb. Taking negative base-ten logarithms of KaKb = Kw gives pKa + pKb = pKw. At 25 °C, pKw is about 14.00. This does not mean pKa and pKb are always individually positive or always add to 14 at every temperature; use the correct temperature-specific pKw.

The Kb paired with Ka must be for the conjugate base of that acid . Ka of acetic acid should be multiplied by Kb of acetate, not by Kb of ammonia. Likewise Ka for NH₄⁺ pairs with Kb for NH₃. Wrong pair selection can produce a mathematically neat but chemically meaningless result.

For polyprotic acids, each dissociation step has its own conjugate pair. Ka for H₂CO₃/HCO₃⁻ connects with Kb for HCO₃⁻ acting as base to reform H₂CO₃; the next acid step HCO₃⁻/CO₃²⁻ connects with Kb of CO₃²⁻. A species can thus be both an acid in one pair and a base in another.

Step-by-step reasoning

1. Write the acid and its exact conjugate base. 2. Identify Ka for the acid reaction and Kb for that base reaction. 3. Use KaKb = Kw at the same temperature and convention. 4. Convert to pKa + pKb = pKw only after pairing correctly.

Visual explanation

Stack the acid and conjugate-base equations. Cross out HA and A⁻ on opposite sides, leaving water autoionization and the product KaKb.

Real-world analogy

Two reciprocal transfer tendencies share an overall constraint. If one direction becomes much more favorable, the other must become less favorable under the same reference conditions.

Real-world example

Knowing the Ka of ammonium allows estimation of ammonia's Kb at the same temperature. This can support a weak-base pH calculation without a separately tabulated Kb.

Why?

Why multiply to Kw? The two conjugate reactions sum to water's autoionization after their intermediate acid and base species cancel from the combined balanced equation.

Common misconception

“Any Ka times any Kb equals Kw.” The identity applies only to a matched conjugate acid-base pair at the same temperature and consistent standard states.

Worked example

At 25 °C, suppose Ka(BH⁺) = 2.0 × 10⁻⁹. Then Kb(B) = Kw/Ka ≈ (1.0 × 10⁻¹⁴)/(2.0 × 10⁻⁹) = 5.0 × 10⁻⁶. pKa ≈ 8.70 and pKb ≈ 5.30, whose sum is about 14.00. The base B is exactly the conjugate of BH⁺.

Quick check

1. If Ka increases for one acid at fixed T, what happens to its conjugate base Kb? Answer: It decreases, because their product equals the fixed Kw.

Exam focus

Write the conjugate pair before substituting numbers. Use pKw for the stated temperature rather than automatically writing 14.

Advanced insight

The relation is thermodynamic when constants are activity-based. Concentration approximations can deviate at high ionic strength because activity coefficients enter the acid, base and water equilibria.

Summary

For a matched conjugate pair, KaKb = Kw and pKa + pKb = pKw. The relationship expresses the inverse acid-base strength connection at fixed temperature under consistent standards.

Practice questions

1. At 25 °C, what is Kb if Ka = 1.0 × 10⁻⁴ for the conjugate acid? Answer: 1.0 × 10⁻¹⁰. 2. Which Kb pairs with Ka of NH₄⁺? Answer: Kb of NH₃, its conjugate base. 3. Does pKa + pKb always equal 14.00? Answer: Only approximately at 25 °C under the familiar Kw value and matching conventions.