Henderson-Hasselbalch Relation

Introduction

Buffer pH depends strongly on the ratio of conjugate base to weak acid. The Henderson-Hasselbalch relation makes that dependence visible. It follows from the weak-acid equilibrium expression, but its familiar concentration form is approximate, so every numerical use should be tied to the conditions of the mixture.

Core explanation

For HA + H₂O ⇌ H₃O⁺ + A⁻, the dilute concentration expression is Ka ≈ [H₃O⁺][A⁻]/[HA]. Rearranging gives [H₃O⁺] ≈ Ka[HA]/[A⁻]. Take the negative base-ten logarithm of both sides: pH ≈ pKa + log₁₀([A⁻]/[HA]). Equal acid and base concentrations give pH ≈ pKa. A tenfold increase in the base-to-acid ratio raises the predicted pH by about one unit; a tenfold decrease lowers it by about one unit.

The ratio is of conjugate partners, not arbitrary acid and base concentrations. For an acetate buffer, numerator is acetate and denominator is acetic acid. For ammonia/ammonium, writing the acid form as NH₄⁺ puts ammonia in the numerator and ammonium in the denominator, using the Ka of ammonium. It is also possible to use a pOH form with Kb, provided the equation and species labels are consistent.

When strong acid or strong base is added, first perform complete stoichiometric neutralization of the added reagent against one buffer component. Then use the remaining moles in the ratio if the components share essentially the same final volume. If appreciable volume changes or multiple equilibria are present, adjust the calculation accordingly. Writing “initial ratio” after adding acid is a common error.

The relation is most reliable when both components are appreciable, the strong reagent has not exhausted one, and autoionization of water and additional acid dissociation are comparatively small. At very low concentrations, extreme ratios, or high ionic strength, direct mass balance, charge balance, and activity-based equilibrium may be needed. The thermodynamic expression includes activity coefficients; the textbook concentration equation is a useful model, not a universal exact identity.

Step-by-step reasoning

1. Choose the acid form and its matching conjugate base. 2. Write Ka and solve algebraically for hydronium activity or concentration. 3. Take base-ten logarithms to obtain pH versus pKa. 4. Insert post-reaction amounts and assess the approximation.

Visual explanation

Plot pH vertically against log₁₀(base/acid) horizontally. The line crosses pH = pKa at log ratio zero and rises one pH unit for each unit increase in log ratio.

Real-world analogy

Think of a balance scale with acid on one pan and conjugate base on the other. Equal amounts place the reading near pKa; changing their ratio moves the reading predictably, not linearly in the raw amounts.

Real-world example

When preparing an acetate buffer, a chemist may choose an acetate-to-acetic-acid ratio to place the pH near a desired value. The formula gives a starting estimate before calibration with a pH meter.

Why?

Why does equal concentration imply pH near pKa? Equal numerator and denominator make their ratio one; log₁₀(1) is zero, leaving pH approximately equal to pKa.

Common misconception

“Doubling both buffer components changes pH.” Their ratio stays the same in the simple model, though the larger total amount increases how much strong reagent they can absorb.

Worked example

A buffer contains 0.040 mol A⁻ and 0.100 mol HA in the same final volume. If pKa = 4.76, then pH ≈ 4.76 + log₁₀(0.040/0.100). Since log₁₀(0.40) ≈ −0.398, pH ≈ 4.36. The answer is below pKa because acid exceeds conjugate base. Reporting pH above pKa would contradict the composition before any detailed calculation.

Quick check

1. At what approximate pH are HA and A⁻ equal in concentration? Answer: pH ≈ pKa under the concentration approximation.

Exam focus

Label numerator and denominator explicitly. After any added strong acid or base, update mole amounts by stoichiometry before using the logarithm.

Advanced insight

In terms of activities, the relation is pH = pKa + log₁₀(aA⁻/aHA) under consistent standard states. Replacing activities with molar concentrations silently assumes a suitable activity-coefficient approximation.

Summary

The buffer relation is a rearranged weak-acid equilibrium: pH ≈ pKa + log₁₀(base/acid). Equal partners give pH near pKa, while their ratio controls the direction and size of a pH change.

Practice questions

1. If base/acid = 10, what is pH relative to pKa? Answer: About one unit above pKa. 2. If base/acid = 0.1, what is pH relative to pKa? Answer: About one unit below pKa. 3. Which amounts belong in the ratio after adding HCl? Answer: The post-neutralization amounts of conjugate base and acid, provided neither is exhausted.