Back titrations and chemical analysis

Why and how back titrations are used, with step-by-step calculations for limestone, antacids, ammonium fertilisers and aspirin; analysing a household substance; and evaluating the role of acid–base titration in industry, quality control and research, including Australian examples and First Nations knowledge behind a modern drug.

Try titrating a lump of limestone with hydrochloric acid and you'll be waiting a long time. The solid fizzes slowly, the indicator keeps changing back, and you can never tell when it's finished. Chemists solve this with a neat trick: add a known excess of acid, let it react completely, then titrate whatever acid is left over. The difference tells you exactly how much the limestone used.

When to use a back titration

In a back titration, a known excess of a standard reagent is added to the sample. After the reaction is complete, the leftover reagent is titrated with a second standard solution. Use one when the substance being analysed:

is insoluble or only slightly soluble, such as CaCO₃ in limestone or eggshells, and Mg(OH)₂ or Al(OH)₃ in antacid tablets
reacts slowly, so the end point drifts, such as the hydrolysis of aspirin by NaOH
is volatile, such as ammonia, which would escape during a direct titration
gives no sharp end point with an available indicator.

The logic of a back titration: moles that reacted with the sample = total moles added − moles left over.

The calculation chain

Moles of reagent added in total: $n = cV$.
Moles of reagent left over, from the titration and its mole ratio.
Moles that reacted with the sample = total − left over.
Moles of sample, from the mole ratio of the first reaction.
Mass, concentration or percentage, as asked.

Worked example: limestone

A 2.00 g sample of crushed limestone was added to 50.0 mL of 1.00 mol/L HCl. After the fizzing stopped, the excess acid needed 26.8 mL of 0.500 mol/L NaOH. Calculate the percentage by mass of CaCO₃ in the limestone.

n(HCl) = 1.00 × 0.0500 = 0.0500 mol.

total HCl

n(NaOH) = 0.500 × 0.0268 = 0.0134 mol; HCl + NaOH is 1 : 1, so 0.0134 mol of HCl was left over.

excess HCl

n(HCl) reacted = 0.0500 − 0.0134 = 0.0366 mol.

reacted

CaCO₃ + 2HCl → CaCl₂ + H₂O + CO₂, so n(CaCO₃) = 0.0366 ÷ 2 = 0.0183 mol; m = 0.0183 × 100.09 = 1.83 g.

CaCO₃

1.83 ÷ 2.00 × 100 = 91.6% CaCO₃.

percentage

Worked example: ammonium sulfate fertiliser

1.250 g of a fertiliser containing (NH₄)₂SO₄ (M = 132.14 g/mol) was boiled with 50.00 mL of 0.5000 mol/L NaOH until all the ammonia had been driven off. The remaining NaOH needed 22.60 mL of 0.4800 mol/L HCl. Find the percentage of (NH₄)₂SO₄ in the fertiliser.

NH₄⁺(aq) + OH⁻(aq) → NH₃(g) + H₂O(l); boiling drives off the NH₃.

the reaction

n(NaOH) added = 0.5000 × 0.05000 = 0.02500 mol. Excess = n(HCl) = 0.4800 × 0.02260 = 0.01085 mol.

total and excess

n(OH⁻) reacted = 0.02500 − 0.01085 = 0.01415 mol = n(NH₄⁺).

reacted

Each (NH₄)₂SO₄ contains 2 NH₄⁺: n = 0.01415 ÷ 2 = 7.075 × 10⁻³ mol; m = 7.075 × 10⁻³ × 132.14 = 0.9349 g.

salt

0.9349 ÷ 1.250 × 100 = 74.79%.

percentage

Worked example: aspirin

Aspirin (C₉H₈O₄, M = 180.16 g/mol) reacts with NaOH in two steps: its carboxylic acid group is neutralised, and on warming its ester group is hydrolysed, so overall 1 mol of aspirin uses 2 mol of NaOH. A 0.400 g tablet was warmed with 50.00 mL of 0.1000 mol/L NaOH, and the excess NaOH needed 15.80 mL of 0.1000 mol/L HCl. Find the mass of aspirin in the tablet.

0.1000 × 0.05000 = 5.000 × 10⁻³ mol.

total NaOH

n(HCl) = 0.1000 × 0.01580 = 1.580 × 10⁻³ mol of NaOH left over.

excess

5.000 × 10⁻³ − 1.580 × 10⁻³ = 3.420 × 10⁻³ mol NaOH reacted.

reacted

n(aspirin) = 3.420 × 10⁻³ ÷ 2 = 1.710 × 10⁻³ mol; m = 1.710 × 10⁻³ × 180.16 = 0.308 g (308 mg; the rest of the tablet is binder and filler).

aspirin

InteractiveBack titration calculator

Try this

The sim opens on a limestone sample. Press "Next step" five times, predicting each line before you see it. What percentage of CaCO₃ does it find for the first sample (2.048 g, titre 37.05 mL)?
Watch the mole bars. Which bar is measured by the titration, and which bar is found by subtraction?
Switch to the antacid tablet. Why does the sim say a direct titration won't work, and what mole ratio is used in step 4?
Switch to aspirin. Which reagent is in excess this time, and what is the titrant?
Press "New sample" a few times. The percentage changes a lot even though the titres differ by only a few mL. Use the note at the end to explain why.

Investigation: analysing a household substance (antacid tablet)

Aim: to find the mass of base in an antacid tablet and compare it with the label.

Method: Weigh a tablet, crush it with a mortar and pestle, and transfer it all to a conical flask. Pipette in 50.00 mL of standardised 0.5 mol/L HCl (an excess) and swirl or warm gently until the solid stops reacting (some insoluble binder may remain). Add 2–3 drops of methyl orange or bromothymol blue. Titrate the excess HCl with standardised 0.2 mol/L NaOH until the colour changes. Repeat with two more tablets.

Processing: follow the five-step chain above. Antacids may contain CaCO₃, Mg(OH)₂ or Al(OH)₃ (each 1 : 2 or 1 : 3 with HCl); the label tells you which. You can also report the result as moles of HCl neutralised per tablet, which is what matters to a patient.

Other household analyses: the same skills apply to the ethanoic acid in vinegar (direct titration, 6H), citric acid in juice or soft drink (triprotic, phenolphthalein, after removing CO₂ by warming), the acidity of wine, or the ammonia in cloudy ammonia (titrated with HCl, methyl orange). Strongly coloured samples should be diluted or followed with a pH probe.

Validity and safety: make sure all the solid has reacted before titrating, and use the same indicator each time. Wear safety glasses; 0.5 mol/L HCl is an irritant.

Titration in industry, research and quality control

Automatic titrators and digital pH and conductivity probes now do much of this work, but the chemistry is exactly what you do by hand.

Field	What is titrated, and why
Wine making	Australian winemakers titrate juice and wine with NaOH to find the titratable acidity (reported as g/L of tartaric acid), which controls taste, colour stability and microbial spoilage. Sulfur dioxide preservative is also measured by titration.
Food and drink	Acidity of fruit juices and soft drinks; the acid number of cooking oils, which rises as fats break down; the ripeness of fruit (sugar to acid ratio).
Pharmaceuticals	Pharmacopoeia assays of drugs such as aspirin, and of alkaloid bases, by acid–base titration. These official methods set the purity a medicine must meet before it is sold.
Environment	Alkalinity (buffering capacity) of rivers and dams; monitoring acid mine drainage and acid sulfate soils; how much lime is needed to treat acidic soil or water.
Energy and manufacturing	Acid number of biodiesel and lubricating oils; the strength of acids and alkalis made and sold by chemical plants.

Evaluating titration as a technique: it is cheap, quick, very accurate and precise when done well, and it is an absolute method: the answer comes straight from stoichiometry, without needing calibration against samples of known composition. Its limitations are that it measures total acid or base without identifying which substances are present; it needs a known, fast, complete reaction and a clear end point; it can be affected by coloured or cloudy samples; and it consumes the sample. The instrumental methods in Module 8 complement it when you need to know which substance is there, or when concentrations are very small.

Going further: First Nations science: from corkwood to the operating theatre

Aboriginal peoples of eastern and central Australia used corkwood trees (Duboisia species) long before European arrival. The leaves of D. myoporoides were used to stun fish in waterholes, and the related pituri plant (6E) was prepared as a stimulant. During the Second World War, when supplies from Europe were cut off, Australian scientists followed this knowledge and found that corkwood leaves are rich in the alkaloids hyoscine (scopolamine) and hyoscyamine. Queensland plantations of hybrid Duboisia now supply a large share of the world's hyoscine, used to prevent motion sickness and to dilate pupils for eye surgery. These alkaloids are bases: they are extracted by converting them between their protonated (water-soluble) and free-base (organic-soluble) forms, and their purity is checked by acid–base titration. It's a reminder that this knowledge belongs to its original custodians, and that benefit-sharing is now an expected part of research built on it.

Glossy green leaves and small round green fruits of a corkwood tree — Leaves and fruit of corkwood, Duboisia myoporoides, which grows in the forests of eastern Australia.Photo: John Moss, Public domain

Test yourself

A back titration is most appropriate for determining
0.0400 mol of HCl was added to an antacid containing Mg(OH)₂. The excess HCl needed 0.0150 mol of NaOH. How many moles of Mg(OH)₂ were in the sample?
In a back titration, some of the excess acid is lost as spray when the sample fizzes vigorously. The calculated amount of the sample will be
Which is a limitation of acid–base titration as an analytical technique?

Complete the back titration.

Put the steps of the back titration calculation in order.

In a back titration, a known of a standard reagent is added to the sample. The amount left over is found by with a second standard solution. Back titrations suit samples that are in water or that are , like ammonia. The moles that reacted with the sample equal the total moles added minus the moles .

7 marks

To find the purity of a sample of chalk (impure CaCO₃), a student added 1.500 g of chalk to 100.0 mL of 0.5000 mol/L HCl. When the reaction stopped, the solution was transferred to a 250.0 mL volumetric flask and made up to the mark. 25.00 mL aliquots of this solution needed an average of 24.50 mL of 0.1000 mol/L NaOH. (a) Explain why a back titration was used. (b) Calculate the percentage by mass of CaCO₃ in the chalk. (c) Identify one assumption made in the calculation.

Your answer