pH Titration Curves and Indicators

AS · 13 min

If you follow the pH of an acid with a pH meter while adding alkali from a burette, you get a characteristic S-shaped curve whose shape depends on whether the acid and the alkali are strong or weak. This note shows how to sketch the four types of pH titration curve, what each feature (starting pH, gentle slope, steep section, equivalence point, final pH) tells you, and how to choose an indicator whose colour change falls inside the steep section. Sketching a curve and choosing an indicator from given data are standard Paper 2 questions, and they matter for every acid–base titration you carry out in Paper 3.

The equivalence point and the end point

The equivalence point is the point at which exactly enough alkali has been added to react with all the acid, in the ratio given by the equation. For $25.0\ \text{cm}^3$ of $0.100\ \text{mol dm}^{-3}$ acid (monoprotic) and $0.100\ \text{mol dm}^{-3}$ alkali, it is at $25.0\ \text{cm}^3$ of alkali added.

The end point is the point at which the indicator changes colour. A good indicator is chosen so that the end point coincides with the equivalence point: the colour change happens within the steep part of the curve, where one drop of alkali changes the pH by several units.

Sketching a pH titration curve

All four curves below show $0.100\ \text{mol dm}^{-3}$ alkali added from a burette to $25.0\ \text{cm}^3$ of $0.100\ \text{mol dm}^{-3}$ acid. The horizontal axis is the volume of alkali added (0 to $50\ \text{cm}^3$ ); the vertical axis is pH (0 to 14). Every curve has its equivalence point at $25.0\ \text{cm}^3$ , but the pH values differ.

Strong acid with strong alkali

For example, hydrochloric acid with sodium hydroxide.

y = 7 + 5.8tanh(0.8(x - 25)) + 0.008(x - 25)

Starts at pH 1: the strong acid is fully dissociated, so $[\ce{H+}] = 0.100\ \text{mol dm}^{-3}$ .
Rises very slowly at first: there is a large excess of $\ce{H+}$ , so adding a little $\ce{OH-}$ hardly changes $[\ce{H+}]$ .
Steep vertical section from about pH 3 to pH 11 around the equivalence point.
Equivalence point at pH 7: the salt (NaCl) is neutral.
Levels off at about pH 13, approaching the pH of the alkali.

Weak acid with strong alkali

For example, ethanoic acid with sodium hydroxide.

y = 8.7 + 4.3tanh(0.8(x - 25)) + 0.06(x - 25)(1 - tanh(0.8(x - 25)))/2

Starts at about pH 3: the weak acid is only partially dissociated.
Rises gently and steadily before the equivalence point (the mixture of ethanoic acid and ethanoate ions resists pH change).
Steep section from about pH 7 to pH 11: shorter than for strong–strong, and entirely above 7.
Equivalence point above 7 (about pH 8.7): the salt, sodium ethanoate, is slightly alkaline because ethanoate ions accept protons from water: $\ce{CH3COO- + H2O <=> CH3COOH + OH-}$ .
Levels off at about pH 13.

Strong acid with weak alkali

For example, hydrochloric acid with aqueous ammonia.

y = 5.5 + 4.3tanh(0.8(x - 25)) + 0.06(x - 25)(1 + tanh(0.8(x - 25)))/2 + 0.008(x - 25)(1 - tanh(0.8(x - 25)))/2

Starts at pH 1 (strong acid).
Steep section from about pH 3 to pH 7: entirely below 7.
Equivalence point below 7 (about pH 5.5): the salt, ammonium chloride, is slightly acidic because ammonium ions donate protons: $\ce{NH4+ <=> NH3 + H+}$ .
Levels off at only about pH 11, the pH of the weak alkali, after the equivalence point.

Weak acid with weak alkali

For example, ethanoic acid with aqueous ammonia.

y = 7 + 4tanh(0.12(x - 25))

Starts at about pH 3 and finishes at about pH 11.
No steep section: the pH changes gradually throughout, with only a slight increase in slope around the equivalence point (near pH 7 for these two).

Key result

acid + alkali	start pH	steep section	pH at equivalence	final pH
strong + strong	1	about 3 to 11	7	about 13
weak + strong	about 3	about 7 to 11	above 7 (about 9)	about 13
strong + weak	1	about 3 to 7	below 7 (about 5)	about 11
weak + weak	about 3	none	about 7 (varies)	about 11

Tip

If the acid is added to the alkali instead, the curve is reflected: it starts at high pH and falls, with the steep section in the same pH range. Always check which solution is in the flask and which is in the burette.

Indicators

An acid–base indicator is a weak acid whose undissociated form and conjugate base have different colours. It changes colour over a range of about 1.5 to 2 pH units.

Key result

indicator	colour in acid	colour in alkali	pH range of colour change
methyl orange	red	yellow	3.1 to 4.4
methyl red	red	yellow	4.2 to 6.3
bromothymol blue	yellow	blue	6.0 to 7.6
phenolphthalein	colourless	pink	8.3 to 10.0

These ranges are typical; in an exam, use the values given in the question.

Method

Choosing an indicator

Identify the type of titration (strong or weak acid; strong or weak alkali) and so the pH range of the steep section.
Choose an indicator whose whole colour-change range lies within the steep section.
If there is no steep section (weak acid with weak alkali), no indicator is suitable: use a pH meter and find the equivalence point from the graph.

titration	steep section	suitable indicator(s)
strong acid + strong alkali	3 to 11	methyl orange or phenolphthalein (or any in between)
weak acid + strong alkali	7 to 11	phenolphthalein (methyl orange would change far too early)
strong acid + weak alkali	3 to 7	methyl orange (phenolphthalein would change far too late)
weak acid + weak alkali	none	none suitable

Why does it matter? In a weak acid–strong alkali titration, methyl orange would change colour from red to yellow at about pH 4, while the pH is still rising slowly in the gentle region, long before the equivalence point. The titre would be far too small and would not even be reproducible, because the colour would change gradually over several $\text{cm}^3$ .

Worked examples

Routine: choosing an indicator

Choose a suitable indicator from the table above for each titration, giving a reason: (a) nitric acid with potassium hydroxide; (b) ethanoic acid with sodium hydroxide; (c) hydrochloric acid with ammonia.

Solution

(a) Strong acid and strong alkali: steep section about pH 3 to 11. Methyl orange or phenolphthalein: both ranges lie within the steep section.

(b) Weak acid and strong alkali: steep section about pH 7 to 11. Phenolphthalein (8.3 to 10.0) lies within it.

Explaining the pH at equivalence

Explain why the pH at the equivalence point is above 7 when ethanoic acid is titrated with sodium hydroxide.

Solution

At the equivalence point the solution contains sodium ethanoate, $\ce{CH3COONa}$ , and water. The ethanoate ion is the conjugate base of a weak acid, so it accepts protons from water molecules:

\ce{CH3COO-(aq) + H2O(l) <=> CH3COOH(aq) + OH-(aq)}

This produces a small excess of $\ce{OH-}$ ions, so the solution is slightly alkaline and the pH is above 7.

Sketching with different concentrations

$20.0\ \text{cm}^3$ of $0.150\ \text{mol dm}^{-3}$ hydrochloric acid is titrated with $0.100\ \text{mol dm}^{-3}$ sodium hydroxide. (a) Calculate the volume of alkali at the equivalence point. (b) Describe the curve you would sketch.

Solution

(a) $n(\ce{HCl}) = 0.0200 \times 0.150 = 3.00 \times 10^{-3}\ \text{mol}$ . Ratio $1 : 1$ , so $n(\ce{NaOH}) = 3.00 \times 10^{-3}\ \text{mol}$ . Volume $= \dfrac{3.00 \times 10^{-3}}{0.100} = 0.0300\ \text{dm}^3 = 30.0\ \text{cm}^3$ .

(b) Starting pH just below 1 (strong acid, $[\ce{H+}] = 0.150\ \text{mol dm}^{-3}$ ); pH rises slowly up to about $29\ \text{cm}^3$ ; vertical section from about pH 3 to pH 11 at $30.0\ \text{cm}^3$ , centred on pH 7; then the curve levels off towards about pH 13 as more alkali is added.

Reading an unfamiliar indicator table

The table shows some indicators and their pH ranges.

indicator	pH range
thymol blue (acid range)	1.2 to 2.8
bromocresol green	3.8 to 5.4
thymolphthalein	9.3 to 10.5
alizarin yellow	10.1 to 12.0

Which indicator is most suitable for titrating aqueous ammonia with hydrochloric acid, and which for propanoic acid (a weak acid) with sodium hydroxide? Explain.

Solution

Ammonia (weak alkali) with hydrochloric acid (strong acid): steep section about pH 3 to 7, equivalence below 7. Bromocresol green (3.8 to 5.4) lies within it. Thymol blue changes too early (too acidic); the others too late.

Propanoic acid (weak) with sodium hydroxide (strong): steep section about pH 7 to 11, equivalence above 7. Thymolphthalein (9.3 to 10.5) lies within it. Alizarin yellow's range extends to 12, beyond the steep section, so it would change gradually after the equivalence point.

Exam-hard: a titration with two equivalence points

When $0.100\ \text{mol dm}^{-3}$ hydrochloric acid is added to $25.0\ \text{cm}^3$ of $0.100\ \text{mol dm}^{-3}$ sodium carbonate, the pH curve has two steep sections: the first at about pH 8.5 after $25.0\ \text{cm}^3$ of acid, and the second at about pH 4 after $50.0\ \text{cm}^3$ .

(a) Write equations for the two stages. (b) Explain why there are two equivalence points and why the second is at twice the volume of the first. (c) Suggest an indicator for each stage.

Solution

(a) Stage 1: $\ce{CO3^2-(aq) + H+(aq) -> HCO3-(aq)}$ .

Stage 2: $\ce{HCO3-(aq) + H+(aq) -> H2O(l) + CO2(g)}$ .

(b) The carbonate ion accepts two protons, one at a time. The first proton converts all the carbonate into hydrogencarbonate; this needs $n(\ce{H+}) = n(\ce{CO3^2-}) = 2.50 \times 10^{-3}\ \text{mol}$ , i.e. $25.0\ \text{cm}^3$ of acid. The second proton converts the hydrogencarbonate into carbonic acid (water and $\ce{CO2}$ ), needing another $2.50 \times 10^{-3}\ \text{mol}$ : another $25.0\ \text{cm}^3$ , so a total of $50.0\ \text{cm}^3$ . At each equivalence point the pH changes sharply because one stage has just finished.

(c) First stage (around pH 8.5): phenolphthalein (pink to colourless as acid is added). Second stage (around pH 4): methyl orange (yellow to red/orange).

Plotting a pH titration curve

Method:

Calibrate a pH meter with buffer solutions of known pH (for example pH 4 and pH 7).
Pipette $25.0\ \text{cm}^3$ of the acid into a beaker with a magnetic stirrer; record the initial pH.
Add the alkali from a burette in $2\ \text{cm}^3$ portions, recording the pH after each addition. Near the expected equivalence point, add $0.1$ to $0.5\ \text{cm}^3$ portions, because the pH changes rapidly there.
Continue to about twice the equivalence volume.
Plot pH (vertical) against volume of alkali (horizontal) and draw a smooth curve. The equivalence point is the midpoint of the steep section.

Points examiners ask about: why smaller additions are made near the equivalence point; why the pH meter is calibrated; why the probe is rinsed with distilled water between solutions; how to read the equivalence volume from the graph; which indicator from a list would be suitable, judged against the graph you have drawn.

Watch out

Equivalence point always at pH 7. Only for strong acid with strong alkali. Weak acid–strong alkali is above 7; strong acid–weak alkali below 7.
Choosing an indicator because it changes "near 7". The indicator's range must lie within the steep section, not at pH 7.
Drawing the steep section at the wrong volume. The equivalence volume is set by the moles of acid and the concentration of alkali, not by the strengths. Calculate it first.
Starting a strong acid curve at pH 3 or a weak acid curve at pH 1. Strong acids start lower.
Using an indicator for weak acid–weak alkali. There is no sharp change, so no indicator gives a clear end point.

Exam tip

When asked to sketch a titration curve, label the axes (pH; volume of alkali added / $\text{cm}^3$ ), mark the equivalence volume on the horizontal axis, and make sure the start pH, steep section, pH at equivalence and final pH are realistic for the combination. These are usually the four marking points.
When asked to choose an indicator, quote the indicator's range and the pH range of the steep section, and say that one lies within the other.
The syllabus states that $\text{p}K_a$ values will not be used at AS: indicator questions give pH ranges.

Summary

Equivalence point: exactly enough alkali to react with all the acid. End point: indicator changes colour.
Strong–strong: starts pH 1, steep 3 to 11, equivalence pH 7.
Weak acid–strong alkali: starts about pH 3, steep 7 to 11, equivalence above 7.
Strong acid–weak alkali: starts pH 1, steep 3 to 7, equivalence below 7.
Weak–weak: no steep section; no suitable indicator.
An indicator is suitable if its whole colour-change range lies within the steep section. Methyl orange (3.1 to 4.4) for strong acid–weak alkali; phenolphthalein (8.3 to 10.0) for weak acid–strong alkali; either for strong–strong.

Practice

Question

Define the equivalence point of a titration and explain how it differs from the end point.
Sketch (or describe) the pH curve for adding $0.100\ \text{mol dm}^{-3}$ NaOH to $25.0\ \text{cm}^3$ of $0.100\ \text{mol dm}^{-3}$ HCl. Mark the equivalence point.
State the approximate pH at the equivalence point for (a) $\ce{HNO3}$ with KOH, (b) $\ce{CH3COOH}$ with KOH, (c) $\ce{H2SO4}$ with $\ce{NH3}$ .
Explain why methyl orange is not a suitable indicator for the titration of ethanoic acid with sodium hydroxide.
Explain why no indicator is suitable for the titration of ethanoic acid with aqueous ammonia, and suggest how the equivalence point could be found.
$25.0\ \text{cm}^3$ of $0.0500\ \text{mol dm}^{-3}$ sulfuric acid is titrated with $0.100\ \text{mol dm}^{-3}$ sodium hydroxide. Calculate the equivalence volume and describe the shape of the curve.
Explain why the pH at the equivalence point of a hydrochloric acid–ammonia titration is below 7.
Describe how the pH curve would differ if $0.100\ \text{mol dm}^{-3}$ hydrochloric acid were added from the burette to $25.0\ \text{cm}^3$ of $0.100\ \text{mol dm}^{-3}$ sodium hydroxide, instead of the other way round.
A $25.0\ \text{cm}^3$ sample of a weak monoprotic acid HA was titrated with $0.120\ \text{mol dm}^{-3}$ sodium hydroxide. The pH curve started at pH 3.1 and had its steep section between pH 7.2 and pH 11.0, centred on $18.75\ \text{cm}^3$ . Calculate the concentration of HA, and choose an indicator from: methyl orange (3.1 to 4.4), bromothymol blue (6.0 to 7.6), phenolphthalein (8.3 to 10.0).
A student titrates $25.0\ \text{cm}^3$ of $0.100\ \text{mol dm}^{-3}$ ethanoic acid with $0.100\ \text{mol dm}^{-3}$ sodium hydroxide but mistakenly uses methyl orange. Predict the volume of alkali at which the colour change is first seen, explain why the result is neither accurate nor precise, and calculate the percentage error if the student records $3.0\ \text{cm}^3$ .

Answers

The equivalence point is where the amounts of acid and alkali added have exactly reacted in the stoichiometric ratio. The end point is where the indicator changes colour. With a suitable indicator, the end point occurs at (or within one drop of) the equivalence point.
Starts at pH 1, rises slowly to about pH 2 at $24\ \text{cm}^3$ , rises steeply from about pH 3 to pH 11 at $25.0\ \text{cm}^3$ (equivalence point, pH 7), then levels off at about pH 13 by $50\ \text{cm}^3$ .
(a) 7. (b) Above 7, about 8 to 9. (c) Below 7, about 5.
In this titration the steep section lies between about pH 7 and 11. Methyl orange changes colour between pH 3.1 and 4.4, which is reached in the gently rising part of the curve, well before the equivalence point. The colour would change gradually and too early.
With a weak acid and weak alkali, the pH changes gradually throughout, with no steep section. No indicator gives a sharp colour change at the equivalence point. Use a pH meter to record the pH throughout (or a conductivity meter), plot the curve, and find the point of greatest gradient.
$n(\ce{H2SO4}) = 0.0250 \times 0.0500 = 1.25 \times 10^{-3}\ \text{mol}$ . $\ce{H2SO4 + 2NaOH -> Na2SO4 + 2H2O}$ , so $n(\ce{NaOH}) = 2.50 \times 10^{-3}\ \text{mol}$ ; volume $= 25.0\ \text{cm}^3$ . Strong acid–strong alkali shape: start about pH 1 ( $[\ce{H+}] = 0.100$ ), steep from 3 to 11 at $25.0\ \text{cm}^3$ , equivalence pH 7, finishing near pH 13.
At the equivalence point the solution contains ammonium chloride. Ammonium ions are the conjugate acid of the weak base ammonia and donate protons to water: $\ce{NH4+ + H2O <=> NH3 + H3O+}$ . This produces excess $\ce{H3O+}$ , so the solution is slightly acidic.
The curve is reflected: it starts at pH 13 and stays high, falling slowly at first; it drops steeply from about pH 11 to pH 3 at $25.0\ \text{cm}^3$ (equivalence at pH 7), then levels off near pH 1.
$n(\ce{NaOH}) = 0.01875 \times 0.120 = 2.25 \times 10^{-3}\ \text{mol} = n(\ce{HA})$ . $[\ce{HA}] = \dfrac{2.25 \times 10^{-3}}{0.0250} = 0.0900\ \text{mol dm}^{-3}$ . Indicator: phenolphthalein (8.3 to 10.0 lies within 7.2 to 11.0); bromothymol blue's range extends below the steep section, and methyl orange changes far too early.
Methyl orange changes between pH 3.1 and 4.4. Ethanoic acid starts at about pH 3, so the colour starts to change almost immediately and is complete after only a few $\text{cm}^3$ of alkali (a gradual change over several $\text{cm}^3$ , not a sharp end point). The titre is far too small (inaccurate) and different runs would give different readings because the change is gradual (imprecise). Percentage error $= \dfrac{25.0 - 3.0}{25.0} \times 100 = 88\%$ .

The equivalence point and the end point

Sketching a pH titration curve

Strong acid with strong alkali

Weak acid with strong alkali

Strong acid with weak alkali

Weak acid with weak alkali

Indicators

Worked examples

Practice

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