Meaning of a neutralisation curve
A neutralisation curve (neutralization curve) is the graph of \(\mathrm{pH}\) on the vertical axis versus the volume of titrant added on the horizontal axis during an acid–base titration. The curve shape reflects the dominant species controlling \([\mathrm{H_3O^+}]\) and \([\mathrm{OH^-}]\) as neutralization proceeds.
\[ \mathrm{H_3O^+(aq) + OH^-(aq) \rightarrow 2H_2O(l)} \]
Reference relationships for pH and pOH
At \(25^\circ\text{C}\), water autoionization is described by
\[ K_w = [\mathrm{H_3O^+}][\mathrm{OH^-}] = 1.0\times 10^{-14} \]
The pH and pOH definitions are
\[ \mathrm{pH} = -\log_{10}([\mathrm{H_3O^+}]),\qquad \mathrm{pOH} = -\log_{10}([\mathrm{OH^-}]) \]
\[ \mathrm{pH} + \mathrm{pOH} = 14.00 \quad (25^\circ\text{C}) \]
Strong acid–strong base neutralisation curve model
A standard general-chemistry model uses a strong monoprotic acid (such as \(\mathrm{HCl}\)) titrated by a strong base (such as \(\mathrm{NaOH}\)). Strong electrolytes are treated as fully dissociated in dilute aqueous solution.
The pH calculation is controlled by the excess strong species after the reaction stoichiometry is accounted for: excess \(\mathrm{H_3O^+}\) before equivalence and excess \(\mathrm{OH^-}\) after equivalence.
General pH expressions as a function of added volume
Let \(C_a\) and \(V_a\) be the initial acid concentration and volume, and let \(C_b\) and \(V_b\) be the base concentration and the added base volume. Moles of acid equivalents and base equivalents are
\[ n_a = C_aV_a,\qquad n_b = C_bV_b \]
Total solution volume (assuming volume additivity) is
\[ V_{\text{tot}} = V_a + V_b \]
Region before the equivalence point
Before equivalence, \(n_a > n_b\). Excess hydronium equivalents are \(n_a-n_b\), giving
\[ [\mathrm{H_3O^+}] = \frac{n_a-n_b}{V_{\text{tot}}} = \frac{C_aV_a - C_bV_b}{V_a+V_b} \]
\[ \mathrm{pH} = -\log_{10}\!\left(\frac{C_aV_a - C_bV_b}{V_a+V_b}\right) \]
Equivalence point
The equivalence point satisfies \(n_a=n_b\), so
\[ C_aV_a = C_bV_{b,\text{eq}} \quad\Rightarrow\quad V_{b,\text{eq}} = \frac{C_aV_a}{C_b} \]
For a strong acid–strong base titration at \(25^\circ\text{C}\), the solution at equivalence is dominated by spectator ions and water, so \(\mathrm{pH}\approx 7.00\).
Region after the equivalence point
After equivalence, \(n_b > n_a\). Excess hydroxide is \(n_b-n_a\), giving
\[ [\mathrm{OH^-}] = \frac{n_b-n_a}{V_{\text{tot}}} = \frac{C_bV_b - C_aV_a}{V_a+V_b} \]
\[ \mathrm{pOH} = -\log_{10}\!\left(\frac{C_bV_b - C_aV_a}{V_a+V_b}\right),\qquad \mathrm{pH} = 14.00 - \mathrm{pOH} \]
Worked values for a typical neutralisation curve
Consider \(V_a = 25.00\ \text{mL}\) of \(C_a = 0.100\ \text{M}\ \mathrm{HCl}\) titrated by \(C_b = 0.100\ \text{M}\ \mathrm{NaOH}\) at \(25^\circ\text{C}\). The equivalence volume is
\[ V_{b,\text{eq}} = \frac{(0.100)(0.02500)}{0.100} = 0.02500\ \text{L} = 25.00\ \text{mL} \]
| Added base volume \(V_b\) (mL) | Dominant excess species | Key concentration | pH |
|---|---|---|---|
| 0.00 | excess \(\mathrm{H_3O^+}\) | \([\mathrm{H_3O^+}] = 0.100\ \text{M}\) | \(\mathrm{pH}= -\log_{10}(0.100)=1.00\) |
| 10.00 | excess \(\mathrm{H_3O^+}\) | \[ [\mathrm{H_3O^+}] = \frac{(0.100)(0.02500)-(0.100)(0.01000)}{0.02500+0.01000} = \frac{0.00150}{0.03500} = 4.29\times 10^{-2} \] | \(\mathrm{pH}\approx -\log_{10}(4.29\times 10^{-2})=1.37\) |
| 25.00 | equivalence | \(\mathrm{pH}\approx 7.00\) | 7.00 |
| 30.00 | excess \(\mathrm{OH^-}\) | \[ [\mathrm{OH^-}] = \frac{(0.100)(0.03000)-(0.100)(0.02500)}{0.02500+0.03000} = \frac{0.00050}{0.05500} = 9.09\times 10^{-3} \] | \[ \mathrm{pOH} = -\log_{10}(9.09\times 10^{-3}) = 2.04,\quad \mathrm{pH} = 14.00-2.04=11.96 \] |
Visualization of a neutralisation curve
Indicator choice and endpoint meaning
The endpoint is the observed signal change (often a color change) from an acid–base indicator. The equivalence point is the stoichiometric condition \(n_a=n_b\). For a strong acid–strong base neutralisation curve, the rapid pH change occurs near \(\mathrm{pH}\approx 7\), so many common indicators can work because the pH jump spans several pH units in a small volume range.
Comparison with other neutralisation curves
Curves involving weak acids or weak bases show buffer regions and equivalence-point pH values that differ from \(7.00\). The strong acid–strong base curve is distinguished by the absence of a buffer plateau and a near-neutral equivalence point at \(25^\circ\text{C}\).
| Titration type | Typical equivalence-point pH (25°C) | Distinct curve feature |
|---|---|---|
| strong acid + strong base | \(\approx 7\) | very steep jump centered near neutral pH |
| weak acid + strong base | \(> 7\) | buffer region; equivalence in basic range |
| strong acid + weak base | \(< 7\) | smaller jump; equivalence in acidic range |
| weak acid + weak base | depends on \(K_a\) and \(K_b\) | limited pH jump; potentiometric methods common |
Common pitfalls
Confusion between equivalence point and endpoint is common; the equivalence point is defined by stoichiometry, while the endpoint depends on the indicator transition range and experimental observation.
Dilution effects are significant away from the equivalence point; concentration expressions use \(V_{\text{tot}}=V_a+V_b\), not the initial volume alone.