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Weak acids and bases $K_a$ for weak acids (A-level only)

What you'll learn

  • What “weak” means for acids and bases in aqueous solution.
  • How to write a KaK_aKa​ expression for a weak acid.
  • How to calculate pH from KaK_aKa​ and concentration, and calculate KaK_aKa​ from pH.
  • How to convert between KaK_aKa​ and pKapK_apKa​, including the half-neutralisation method.

The prerequisites: acids, bases and pH

A Brønsted–Lowry acid is a proton donor. A Brønsted–Lowry base is a proton acceptor. A proton is a hydrogen ion, written as H+\text{H}^+H+.

In aqueous solution, H+\text{H}^+H+ does not really float around alone — it bonds to water to form H3O+\text{H}_3\text{O}^+H3​O+. At A-Level, we usually write H+\text{H}^+H+ as a shorthand.

Definition

pH

pH is a logarithmic measure of hydrogen ion concentration:

pH=−log⁡10[H+]\text{pH}=-\log_{10}[\text{H}^+]pH=−log10​[H+]

In calculations, use the numerical value of [H+][\text{H}^+][H+] in mol dm⁻³.

Because pH is logarithmic, a change of 1 pH unit means a factor of 10 change in [H+][\text{H}^+][H+].

Example

Converting pH into hydrogen ion concentration

A solution has pH 3.25. Find [H+][\text{H}^+][H+].

  1. Rearrange the pH equation by reversing the log:

    [H+]=10−pH[\text{H}^+]=10^{-\text{pH}}[H+]=10−pH
  2. Substitute the pH value:

    [H+]=10−3.25[\text{H}^+]=10^{-3.25}[H+]=10−3.25
  3. Evaluate and include units:

    [H+]=5.62×10−4 mol dm−3[\text{H}^+]=5.62 \times 10^{-4}\ \text{mol dm}^{-3}[H+]=5.62×10−4 mol dm−3

Strong, weak, concentrated and dilute

Dissociation means a substance separates into ions in solution. An aqueous solution is a solution in water.

A strong acid dissociates completely in water. For example, hydrochloric acid effectively produces one mole of H+\text{H}^+H+ for every mole of HCl dissolved.

A weak acid dissociates only slightly in water. Most acid particles remain as undissociated molecules.

Definition

Weak acid

A weak acid is an acid that only partially dissociates in aqueous solution, setting up a reversible equilibrium between the acid and its ions.

A weak base also forms ions only slightly in aqueous solution. For example, ammonia reacts only partially with water:

NH3(aq)+H2O(l)⇌NH4+(aq)+OH−(aq)\text{NH}_3\text{(aq)}+\text{H}_2\text{O(l)} \rightleftharpoons \text{NH}_4^+\text{(aq)}+\text{OH}^-\text{(aq)}NH3​(aq)+H2​O(l)⇌NH4+​(aq)+OH−(aq)

This section focuses mainly on KaK_aKa​, the equilibrium constant for weak acids.

Common Mistake

Weak does not mean dilute

Weak describes the extent of dissociation. Dilute describes low concentration. You can have a concentrated weak acid, or a dilute strong acid.

The weak acid equilibrium

Most A-Level weak acid calculations use a general weak monoprotic acid, HA. Monoprotic means each acid molecule can donate one proton.

The equilibrium is:

HA(aq)⇌H+(aq)+A−(aq)\text{HA(aq)} \rightleftharpoons \text{H}^+\text{(aq)}+\text{A}^-\text{(aq)}HA(aq)⇌H+(aq)+A−(aq)

The ion A−\text{A}^-A− is the conjugate base of HA. A conjugate base is what remains after an acid has donated a proton.

An equilibrium is a dynamic situation where the forward and reverse reactions continue, but the concentrations of substances stay constant.

Key Idea

Weak acids are equilibrium systems

For a weak acid, HA, both undissociated acid molecules and ions are present at equilibrium. You must use equilibrium concentrations in KaK_aKa​ expressions.

The acid dissociation constant, KaK_aKa​

The acid dissociation constant, KaK_aKa​, measures how far a weak acid dissociates in water.

For:

HA(aq)⇌H+(aq)+A−(aq)\text{HA(aq)} \rightleftharpoons \text{H}^+\text{(aq)}+\text{A}^-\text{(aq)}HA(aq)⇌H+(aq)+A−(aq)

the expression is:

Ka=[H+][A−][HA]K_a=\frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}Ka​=[HA][H+][A−]​

Square brackets mean concentration in mol dm⁻³ at equilibrium.

For this type of expression, the units of KaK_aKa​ are:

(mol dm−3)(mol dm−3)mol dm−3=mol dm−3\frac{(\text{mol dm}^{-3})(\text{mol dm}^{-3})}{\text{mol dm}^{-3}}=\text{mol dm}^{-3}mol dm−3(mol dm−3)(mol dm−3)​=mol dm−3
Example

Constructing a Ka expression

Write the KaK_aKa​ expression for ethanoic acid.

  1. Write the dissociation equilibrium:

    CH3COOH(aq)⇌H+(aq)+CH3COO−(aq)\text{CH}_3\text{COOH(aq)} \rightleftharpoons \text{H}^+\text{(aq)}+\text{CH}_3\text{COO}^-\text{(aq)}CH3​COOH(aq)⇌H+(aq)+CH3​COO−(aq)
  2. Put the product concentrations on the top and the undissociated acid concentration on the bottom:

    Ka=[H+][CH3COO−][CH3COOH]K_a=\frac{[\text{H}^+][\text{CH}_3\text{COO}^-]}{[\text{CH}_3\text{COOH}]}Ka​=[CH3​COOH][H+][CH3​COO−]​
  3. Work out the units from the concentration terms:

    Ka=mol dm−3K_a=\text{mol dm}^{-3}Ka​=mol dm−3

Calculating the pH of a weak acid

Suppose the initial concentration of a weak acid HA is ccc mol dm⁻³.

At equilibrium, let the concentration that dissociates be xxx mol dm⁻³.

So:

  • [H+]=x[\text{H}^+]=x[H+]=x
  • [A−]=x[\text{A}^-]=x[A−]=x
  • [HA]=c−x[\text{HA}]=c-x[HA]=c−x

The exact expression is:

Ka=x2c−xK_a=\frac{x^2}{c-x}Ka​=c−xx2​

For a weak acid, only a small amount dissociates, so xxx is usually much smaller than ccc. This lets you use the approximation:

c−x≈cc-x \approx cc−x≈c

So:

Ka≈x2cK_a \approx \frac{x^2}{c}Ka​≈cx2​

and therefore:

[H+]≈Kac[\text{H}^+] \approx \sqrt{K_a c}[H+]≈Ka​c​
Key Idea

The square-root shortcut

For a weak monoprotic acid of concentration ccc, if dissociation is small:

[H+]≈Kac[\text{H}^+] \approx \sqrt{K_a c}[H+]≈Ka​c​

Then use pH=−log⁡10[H+]\text{pH}=-\log_{10}[\text{H}^+]pH=−log10​[H+].

Example

Calculating pH from Ka and concentration

Ethanoic acid has Ka=1.74×10−5 mol dm−3K_a=1.74 \times 10^{-5}\ \text{mol dm}^{-3}Ka​=1.74×10−5 mol dm−3. Calculate the pH of 0.100 mol dm⁻³ ethanoic acid.

  1. Use the weak acid approximation:

    [H+]≈Kac[\text{H}^+] \approx \sqrt{K_a c}[H+]≈Ka​c​
  2. Substitute the values:

    [H+]≈(1.74×10−5)(0.100)[\text{H}^+] \approx \sqrt{(1.74 \times 10^{-5})(0.100)}[H+]≈(1.74×10−5)(0.100)​
  3. Calculate the hydrogen ion concentration:

    [H+]≈1.74×10−6=1.32×10−3 mol dm−3[\text{H}^+] \approx \sqrt{1.74 \times 10^{-6}}=1.32 \times 10^{-3}\ \text{mol dm}^{-3}[H+]≈1.74×10−6​=1.32×10−3 mol dm−3
  4. Convert to pH:

    pH=−log⁡10(1.32×10−3)=2.88\text{pH}=-\log_{10}(1.32 \times 10^{-3})=2.88pH=−log10​(1.32×10−3)=2.88
  5. Check the approximation is sensible:

    1.32×10−30.100×100=1.32%\frac{1.32 \times 10^{-3}}{0.100} \times 100=1.32\%0.1001.32×10−3​×100=1.32%

    Only a small percentage dissociated, so the approximation is reasonable.

Common Mistake

Using the starting concentration as [H+]

For a weak acid, the initial acid concentration ccc is not the hydrogen ion concentration. Only a small fraction dissociates, so [H+][\text{H}^+][H+] must be found using KaK_aKa​.

Common Mistake

When the shortcut may not work

If the acid is not very weak, or the solution is extremely dilute, c−x≈cc-x \approx cc−x≈c may be a poor approximation. Then use the exact expression Ka=x2c−xK_a=\frac{x^2}{c-x}Ka​=c−xx2​ instead.

Calculating KaK_aKa​ from pH

If you know the pH and the initial concentration of a weak acid, you can work backwards.

First convert pH into [H+][\text{H}^+][H+]. For a weak monoprotic acid:

[A−]=[H+][\text{A}^-]=[\text{H}^+][A−]=[H+]

Then substitute into:

Ka=[H+][A−][HA]K_a=\frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}Ka​=[HA][H+][A−]​
Example

Calculating Ka from pH

A 0.0500 mol dm⁻³ weak monoprotic acid has pH 3.10. Calculate KaK_aKa​.

  1. Convert pH to hydrogen ion concentration:

    [H+]=10−3.10=7.94×10−4 mol dm−3[\text{H}^+]=10^{-3.10}=7.94 \times 10^{-4}\ \text{mol dm}^{-3}[H+]=10−3.10=7.94×10−4 mol dm−3
  2. Use the weak acid relationships:

    [A−]=7.94×10−4 mol dm−3[\text{A}^-]=7.94 \times 10^{-4}\ \text{mol dm}^{-3}[A−]=7.94×10−4 mol dm−3

    and, using the small dissociation approximation:

    [HA]≈0.0500 mol dm−3[\text{HA}] \approx 0.0500\ \text{mol dm}^{-3}[HA]≈0.0500 mol dm−3
  3. Substitute into the KaK_aKa​ expression:

    Ka=(7.94×10−4)(7.94×10−4)0.0500K_a=\frac{(7.94 \times 10^{-4})(7.94 \times 10^{-4})}{0.0500}Ka​=0.0500(7.94×10−4)(7.94×10−4)​
  4. Calculate:

    Ka=1.26×10−5 mol dm−3K_a=1.26 \times 10^{-5}\ \text{mol dm}^{-3}Ka​=1.26×10−5 mol dm−3

pKapK_apKa​

pKapK_apKa​ is another way of expressing acid strength. It is especially useful because many KaK_aKa​ values are very small numbers written in standard form.

Definition

pKa

pKapK_apKa​ is defined as:

pKa=−log⁡10KapK_a=-\log_{10}K_apKa​=−log10​Ka​

It has no units.

To convert back:

Ka=10−pKaK_a=10^{-pK_a}Ka​=10−pKa​

A larger KaK_aKa​ means a stronger weak acid. A smaller pKapK_apKa​ also means a stronger weak acid.

Tip

Comparing acid strength

Think: large KaK_aKa​, low pKapK_apKa​, strong acid.

Example

Converting between Ka and pKa

Convert between KaK_aKa​ and pKapK_apKa​.

  1. For Ka=6.3×10−5 mol dm−3K_a=6.3 \times 10^{-5}\ \text{mol dm}^{-3}Ka​=6.3×10−5 mol dm−3:

    pKa=−log⁡10(6.3×10−5)=4.20pK_a=-\log_{10}(6.3 \times 10^{-5})=4.20pKa​=−log10​(6.3×10−5)=4.20
  2. For pKa=3.50pK_a=3.50pKa​=3.50:

    Ka=10−3.50=3.16×10−4 mol dm−3K_a=10^{-3.50}=3.16 \times 10^{-4}\ \text{mol dm}^{-3}Ka​=10−3.50=3.16×10−4 mol dm−3
  3. Compare the two acids: the acid with pKa=3.50pK_a=3.50pKa​=3.50 is stronger because it has the larger KaK_aKa​.

Finding KaK_aKa​ from half-neutralisation

The half-neutralisation point is the point in a titration where exactly half the original acid has been neutralised by the added base.

For a weak acid HA titrated with a strong alkali such as NaOH:

HA+OH−→A−+H2O\text{HA}+\text{OH}^- \to \text{A}^-+\text{H}_2\text{O}HA+OH−→A−+H2​O

At half-neutralisation, half of the original HA has been converted into A−\text{A}^-A−. Therefore:

[HA]=[A−][\text{HA}]=[\text{A}^-][HA]=[A−]

Substitute this into the KaK_aKa​ expression:

Ka=[H+][A−][HA]K_a=\frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}Ka​=[HA][H+][A−]​

Since [A−]=[HA][\text{A}^-]=[\text{HA}][A−]=[HA], they cancel:

Ka=[H+]K_a=[\text{H}^+]Ka​=[H+]

So:

pH=pKa\text{pH}=pK_apH=pKa​

This is why measuring the pH at half-neutralisation is a practical method for finding KaK_aKa​ of a weak acid.

Weak acid strong base titration curve showing half-neutralisation where pH equals pKa

Example

Using half-neutralisation to find Ka

A weak monoprotic acid is titrated with NaOH. The equivalence point occurs after 30.0 cm³ of NaOH is added. At 15.0 cm³, the measured pH is 4.76. Calculate KaK_aKa​.

  1. Identify the half-neutralisation volume:

    30.02=15.0 cm3\frac{30.0}{2}=15.0\ \text{cm}^3230.0​=15.0 cm3
  2. At half-neutralisation, use the relationship:

    pH=pKa\text{pH}=pK_apH=pKa​

    so:

    pKa=4.76pK_a=4.76pKa​=4.76
  3. Convert pKapK_apKa​ into KaK_aKa​:

    Ka=10−4.76K_a=10^{-4.76}Ka​=10−4.76
  4. Calculate:

    Ka=1.74×10−5 mol dm−3K_a=1.74 \times 10^{-5}\ \text{mol dm}^{-3}Ka​=1.74×10−5 mol dm−3
Exam technique

In the exam

  1. Decide what the question gives you: KaK_aKa​ and concentration usually means find pH; pH and concentration usually means find KaK_aKa​; pKapK_apKa​ means use logs.
  2. For weak acid pH calculations, start from Ka=[H+][A−][HA]K_a=\frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}Ka​=[HA][H+][A−]​ and use [H+]=[A−][\text{H}^+]=[\text{A}^-][H+]=[A−] for a monoprotic acid.
  3. Give units carefully: KaK_aKa​ is usually mol dm⁻³ for HA calculations, while pH and pKapK_apKa​ have no units.
Self review

Check yourself

  • Why does 0.100 mol dm⁻³ ethanoic acid have a higher pH than 0.100 mol dm⁻³ hydrochloric acid?
  • A weak acid has concentration 0.200 mol dm⁻³ and Ka=4.0×10−6 mol dm−3K_a=4.0 \times 10^{-6}\ \text{mol dm}^{-3}Ka​=4.0×10−6 mol dm−3. How would you estimate its pH?
  • Why is pH equal to pKapK_apKa​ at half-neutralisation?
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A Brønsted–Lowry acid is a proton (H+\text{H}^+H+) donor. When an acid dissolves in water, it dissociates (splits) into ions. A strong acid, such as HCl\text{HCl}HCl, dissociates completely in water, meaning every molecule of acid releases a proton.

A weak acid only partially dissociates. Most of the acid molecules stay intact, creating a reversible equilibrium between the undissociated acid (HA\text{HA}HA) and its constituent ions:

HA(aq)⇌H+(aq)+A−(aq) \text{HA(aq)} \rightleftharpoons \text{H}^+\text{(aq)} + \text{A}^-\text{(aq)} HA(aq)⇌H+(aq)+A−(aq)

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What is the definition of a weak acid?

Weak acids and bases $K_a$ for weak acids (A-level only) Revision Guide

  1. A Level
  2. /Chemistry
  3. /Weak acids and bases $K_a$ for weak acids (A-level only)