Starting from the equilibrium-constant expression for a weak acid $ ext{HA}$, derive the Henderson-Hasselbalch equation, $ ext{pH} = ext{p}K_a + \log \dfrac{[ ext{A}^-]}{[ ext{HA}]}$, and state the assumption it makes about the equilibrium concentrations of the acid and its

Question

Starting from the equilibrium-constant expression for a weak acid $	ext{HA}$, derive the Henderson-Hasselbalch equation, $	ext{pH} = 	ext{p}K_a + \log \dfrac{[	ext{A}^-]}{[	ext{HA}]}$, and state the assumption it makes about the equilibrium concentrations of the acid and its conjugate base.

Accepted Answer

Concept understanding — Buffer Solution pH

What is a Buffer Solution?

Imagine you're making lemonade. If you add a few drops of lemon juice to a glass of water, the pH drops sharply — it becomes very acidic. But if you add the same few drops to a glass of already acidic lemonade, the pH barely changes. Why? Because lemonade contains a buffer — a mixture that resists pH change when small amounts of acid or base are added.

A buffer solution is a mixture of a weak acid and its conjugate base (or a weak base and its conjugate acid). It "soaks up" added H⁺ or OH⁻ ions without letting the pH swing wildly.

Note

The key is that both components must be present in significant amounts. A weak acid alone won't buffer — you need its conjugate base partner too.

The Intuition: A Chemical Sponge

Think of a buffer as a two-way sponge:

If you add acid (H⁺): The conjugate base in the buffer grabs the extra H⁺, turning into the weak acid. The H⁺ is "absorbed" — pH barely drops.
If you add base (OH⁻): The weak acid donates an H⁺ to neutralise the OH⁻, turning into the conjugate base. The OH⁻ is "absorbed" — pH barely rises.

The buffer works best when the amounts of weak acid and conjugate base are roughly equal. That's when the sponge is most "spongy" — it can absorb shocks in either direction.

The Precise Statement: The Henderson–Hasselbalch Equation

For a buffer made from a weak acid $HA$ and its conjugate base $A^-$ , the pH is given by:

\text{pH} = \text{p}K_a + \log_{10} \left( \frac{[A^-]}{[HA]} \right)

Where:

$\text{p}K_a = -\log_{10} K_a$ (a measure of the weak acid's strength — lower pKa = stronger acid)
$[A^-]$ = concentration of the conjugate base
$[HA]$ = concentration of the weak acid

This equation tells you exactly how the pH depends on the ratio of base to acid, not their absolute amounts.

Tip

When $[A^-] = [HA]$ , the ratio is 1, $\log(1) = 0$ , so $\text{pH} = \text{p}K_a$ . This is the buffer's optimal pH — it resists change most strongly here.

Why This Works: A Quick Derivation

Start from the weak acid equilibrium:

HA \rightleftharpoons H^+ + A^-

The acid dissociation constant is:

K_a = \frac{[H^+][A^-]}{[HA]}

Take negative logs of both sides:

-\log K_a = -\log [H^+] - \log \frac{[A^-]}{[HA]}

Which gives:

\text{p}K_a = \text{pH} - \log \frac{[A^-]}{[HA]}

Rearrange:

\text{pH} = \text{p}K_a + \log \frac{[A^-]}{[HA]}

That's it. The derivation is just algebra on the definition of $K_a$ .

Watch out

The Henderson–Hasselbalch equation assumes that the concentrations $[HA]$ and $[A^-]$ are the initial concentrations you mixed. It works well when both are much larger than $[H^+]$ or $[OH^-]$ from dissociation — which is true for a properly made buffer.

Example: Making an Acetate Buffer

You mix 0.1 M acetic acid ( $\text{p}K_a = 4.76$ ) with 0.1 M sodium acetate. What's the pH?

$\text{pH} = 4.76 + \log \frac{0.1}{0.1} = 4.76 + \log 1 = 4.76$ …

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Starting from the weak acid's own ionization equilibrium, $\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-$ , the equilibrium constant is $K_a = [\text{H}^+][\text{A}^-]/[\text{HA}]$ . Rearranging for $[\text{H}^+]$ : $[\text{H}^+] = K_a \times [\text{HA}]/[\text{A}^-]$ . Taking $-\log_{10}$ of both sides: $-\log[\text{H}^+] = -\log K_a - \log([\text{HA}]/[\text{A}^-])$ , i.e. $\text{pH} = \text{p}K_a + \log([\text{A}^-]/[\text{HA}])$ (flipping the sign of the log inverts the fraction). This derivation assumes that the equilibrium concentrations $[\text{HA}]$ and $[\text{A}^-]$ are well approximated by the concentrations of acid and salt as originally mixed — valid because the deliberately-added salt supplies a large concentration of the common ion $\text{A}^-$ , which suppresses the weak acid's own (otherwise small) ionization almost to not …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set ANNUAL1 mark
$Q. The solutions which resist change in their pH on dilution are called ______ solution.$
›Reveal solutionSolution

Solutions that resist a change in pH on dilution (or on adding small amounts of acid/base) are called buffer solutions.

A buffer solution typically consists of a weak acid together with its conjugate base (e.g. CH3COOH + CH3COO- from CH3COONa), or a weak base together with its conjugate acid (e.g. NH4OH + NH4+ from NH4Cl). Because both the acidic and basic components are present in appreciable, comparable amounts, any small addition of H+ or OH- (or dilution, which shifts conce …

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CBSE 2026Set ANNUAL1 markMCQ
$Q. Equilibrium : Assertion (A) : A solution containing a mixture of ammonium chloride and ammonium hydroxide maintains a constant value of pH on addition of small amounts of acid or alkali. Reason (R) : A solution containing mixture of ammonium chloride and ammonium hydroxide act as a buffer solution around pH 9.25. (Choose the correct option for the Assertion-Reason pair.)$
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.
›Reveal solutionSolution
Both the assertion and the reason are true, and the reason correctly explains the assertion: NH4Cl + NH4OH is a basic buffer solution with pH close to 9.25.

A buffer solution resists changes in pH upon addition of small amounts of acid or base. A basic buffer is typically made of a weak base and its salt with a strong acid — exactly the NH4OH (weak base) + NH4Cl (its salt with strong acid HCl) combination described here.

How the buffering works: NH4OH partially ionises to give NH4+ and OH-, while NH4Cl fully dissociates to give a large reservoir of NH4+ ions and Cl- ions.

On adding a small amount of acid (H+), the excess H+ is neutralised by the NH4OH present (NH4OH + H+ → NH4+ + H2O), so pH barely changes.

On adding a small amount of base (OH-), the NH4+ from the salt neutralises it (NH4+ + OH- → NH4OH), again keeping pH nearly constant. …
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CBSE 2026Set ANNUAL1 mark
$Q. If Bronsted acid is a weak acid then its conjugate base is a ________.$
›Reveal solutionSolution
[!TLDR]

strong base

Method

The weaker the acid, the stronger its conjugate base (inverse relationship between Ka of acid …
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CBSE 2023Set ANNUAL1 mark
$Q. What is buffer solution?$
›Reveal solutionSolution
A buffer resists pH change on adding small amounts of acid/base; made of a weak acid + its conjugate base (or weak base + its conjugate acid).

A buffer solution is a solution whose pH changes very little when a small amount of a strong acid or strong base is added to it, or on dilution. It is typically prepared by mixing:

a weak acid with a salt of its conjugate base (an acidic buffer, e.g. CH3COOH + CH3COONa), or

a weak base with a salt of its conjugate acid (a basic buffer, e.g. NH4OH + NH4Cl). …
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CBSE 2023Set ANNUAL1 mark
$Q. Write the Henderson equation.$
›Reveal solutionSolution

The Henderson-Hasselbalch equation: pH = pKa + log([salt]/[acid]), used to calculate the pH of a buffer solution.

For a buffer made of a weak acid (HA) and its conjugate base/salt (A-, e.g. from a sodium salt), the equilibrium HA <-> H+ + A- gives Ka = [H+][A-]/[HA].

…

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CBSE 2017Set ANNUAL1 mark
$Q. Define the buffer solution.$
›Reveal solutionSolution
A buffer solution maintains an almost constant pH despite small additions of acid, base, or dilution, because it contains a weak acid/conjugate-base or weak base/conjugate-acid pair that can absorb added H+ or OH-.

A buffer solution resists changes in pH upon the addition of small quantities of an acid or a base, or upon dilution. There are two main types:

Acidic buffer: a mixture of a weak acid and its salt with a strong base (e.g., CH3COOH + CH3COONa). It resists pH changes near acidic pH values.

Basic buffer: a mixture of a weak base and its salt with a strong acid (e.g., NH4OH + NH4Cl). It resists pH changes near basic pH values. …
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