IB Chemistry › R3.1 › R3.1.13

pH of Weak Acids

Calculating pH using Ka and the "small x" approximation.

Reactivity 3.1 HL Extension ⏱️ ~5 min revision
HL Extension

The Small-x Approximation Formula

For a weak monoprotic acid \(\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-\) with initial concentration \(c\):

  1. Assumption 1: \([\text{H}^+] \approx [\text{A}^-]\) (negligible contribution from water auto-ionisation).
  2. Assumption 2 (Small-\(x\) approximation): \([\text{HA}]_{\text{eq}} = c - [\text{H}^+] \approx c\) (degree of dissociation is < 5%).
\[K_{\text{a}} \approx \frac{[\text{H}^+]^2}{c} \implies [\text{H}^+] = \sqrt{K_{\text{a}} \times c}\] \[\text{pH} = -\log_{10}\sqrt{K_{\text{a}} \times c} = \frac{1}{2}(\text{p}K_{\text{a}} - \log_{10} c)\]

The Approximation

Worked Example

Worked Example

pH of 0.10 M Ethanoic Acid

Problem: Calculate the pH of \(0.10\text{ mol dm}^{-3}\) ethanoic acid (\(K_{\text{a}} = 1.8 \times 10^{-5}\text{ mol dm}^{-3}\)).

\[[\text{H}^+] = \sqrt{1.8 \times 10^{-5} \times 0.10} = \sqrt{1.8 \times 10^{-6}} = 1.34 \times 10^{-3}\text{ mol dm}^{-3}\] \[\text{pH} = -\log_{10}(1.34 \times 10^{-3}) = 2.87\]
Examiner Trap

Approximation Limits & Dilution Effect

  • Validity check: If \(\frac{[\text{H}^+]}{c} > 0.05\) (5% ionisation), the quadratic equation must strictly be used.
  • Dilution effect: Diluting a weak acid by 10-fold increases its percentage ionisation (Ostwald's dilution law), so pH increases by less than 1 full unit (\(\Delta\text{pH} \approx 0.5\)).
AQA GCSE & IB Chemistry

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