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Identifying a compound is only half the analytical chemist's job. The other half — usually the half that pays — is saying how much of it is there. A forensic toxicologist needs to know not just that a blood sample contains paracetamol, but whether the concentration is therapeutic, toxic, or fatal. A pharmaceutical company needs to know not just that a tablet contains the right active ingredient, but that it contains 500 ± 5 mg of it per tablet. A water-quality scientist needs to know not just that a river contains nitrate, but whether the level breaches the 50 mg dm⁻³ EU drinking-water limit. This lesson covers the quantitative toolkit that delivers those numbers: percentage purity from titration, percentage yield from synthesis, percentage uncertainty and how it propagates through multi-step calculations, the Beer–Lambert law that converts absorbance into concentration, and calibration curves that turn raw instrument responses into reportable results. The skills knit every Required Practical into a single coherent quantitative framework.
Spec mapping (AQA 7405): This lesson maps to §3.3.15 (chromatography — quantitative interpretation of peak areas) and §3.1.2 (amount of substance — concentration, percentage yield, percentage atom economy, percentage uncertainty). It cross-references the atomic-structure course lesson on titration (Required Practical 1, where percentage uncertainty in volumetric apparatus first appears), the L4 chromatography lesson in this course (quantitative GC and HPLC), and every Required Practical in the AS and A2 specification — the percentage-uncertainty toolkit applies to all of them. Refer to the official AQA specification document for the exact wording of each section.
Assessment objectives: Definitions of percentage purity, percentage yield, percentage uncertainty, and the Beer–Lambert law are AO1 recall items. AO2 dominates this topic: computing percentage purity from titration data, combining individual percentage uncertainties into a result uncertainty, and applying A = εcl or a calibration curve to convert an instrument reading into a concentration. AO3 reasoning is tested when candidates must evaluate the precision of a technique, choose between titrimetric and spectroscopic methods for a given sample, or design a quantitative analysis (e.g. propose how to determine the iron content of a vitamin tablet to within 2%).
The percentage purity of a sample is the mass of the desired pure substance, expressed as a percentage of the total mass of the sample:
Percentage purity = (mass of pure substance / total mass of sample) × 100%
A pharmaceutical-grade reagent might be quoted as ">99.5% pure"; a laboratory-grade salt as ">98%". The "impurity" balance includes water of crystallisation, decomposition products, residual starting materials, and trace contaminants from manufacture.
We cannot weigh "the pure substance" directly inside a mixture — that's the whole problem. Instead, we determine the amount of the pure substance by a reaction whose stoichiometry is known (most commonly an acid–base or redox titration), convert that amount to a mass, and divide by the total sample mass.
A 2.50 g sample of impure calcium carbonate (CaCO₃) is reacted with 50.0 cm³ of 1.00 mol dm⁻³ HCl — a known excess. The excess acid is then back-titrated with 0.500 mol dm⁻³ NaOH; the mean titre is 21.40 cm³. Calculate the percentage purity of the CaCO₃.
Equations:
Step 1 — Total moles of HCl added. n(HCl, total) = c × V = 1.00 × (50.0/1000) = 5.00 × 10⁻² mol = 0.0500 mol.
Step 2 — Moles of HCl unreacted (titrated by NaOH). n(NaOH) = 0.500 × (21.40/1000) = 1.070 × 10⁻² mol. HCl reacts 1:1 with NaOH, so n(HCl, excess) = 1.070 × 10⁻² mol.
Step 3 — Moles of HCl that reacted with CaCO₃. n(HCl, reacted) = 0.0500 − 1.070 × 10⁻² = 3.93 × 10⁻² mol.
Step 4 — Moles of CaCO₃. From CaCO₃ + 2HCl, n(CaCO₃) = n(HCl, reacted) / 2 = 3.93 × 10⁻² / 2 = 1.965 × 10⁻² mol.
Step 5 — Mass of pure CaCO₃. M(CaCO₃) = 40.1 + 12.0 + 3 × 16.0 = 100.1 g mol⁻¹. m(pure CaCO₃) = 1.965 × 10⁻² × 100.1 = 1.967 g.
Step 6 — Percentage purity. Percentage purity = (1.967 / 2.50) × 100 = 78.7% (3 s.f.).
The 21% impurity might be calcium oxide (from partial decomposition), water of crystallisation, or insoluble silicate from the geological source.
Exam Tip: Back-titration is the standard route when the analyte is a sparingly-soluble solid or reacts slowly. Direct titration of a suspension is unreliable because the rate-limiting step is dissolution, not the acid–base reaction.
The percentage yield of a synthesis is the actual mass of product obtained, as a percentage of the maximum mass predicted by stoichiometry (the theoretical yield):
Percentage yield = (actual yield / theoretical yield) × 100%
(Recap from the atomic-structure course, lesson 6. Percentage yield and percentage purity are different quantities: yield measures how much of the reaction's potential product survived isolation; purity measures what fraction of the isolated material is the target compound. A reaction can give a 90% yield of 70%-pure crude product, or a 30% yield of 99.9%-pure recrystallised product.)
Industrial chemists also report atom economy = (Mᵣ of desired product / sum of Mᵣ of all products) × 100%, a measure of how much of the reactant atoms end up in the product rather than as waste. High atom economy plus high yield is the green-chemistry ideal.
Every measurement carries an uncertainty — the range within which the true value lies. The uncertainty is typically half the smallest division on the instrument (an analogue burette reading to ±0.05 cm³) or the manufacturer's quoted tolerance for digital instruments (a balance reading to ±0.001 g, a Class A pipette delivering 25.00 ± 0.06 cm³).
The percentage uncertainty in a single measurement is:
Percentage uncertainty = (absolute uncertainty / measured value) × 100%
A 24.80 cm³ titre with a burette uncertainty of ±0.10 cm³ (two readings, each ±0.05 cm³, combined) has a percentage uncertainty of (0.10 / 24.80) × 100 = 0.40%.
When a calculated result combines several measured quantities, the uncertainties propagate according to the operations involved:
| Operation | Rule |
|---|---|
| Addition (x = a + b) | Add absolute uncertainties: Δx = Δa + Δb |
| Subtraction (x = a − b) | Add absolute uncertainties: Δx = Δa + Δb |
| Multiplication (x = a × b) | Add percentage uncertainties: %Δx = %Δa + %Δb |
| Division (x = a / b) | Add percentage uncertainties: %Δx = %Δa + %Δb |
| Power (x = aⁿ) | Multiply percentage uncertainty by the power: %Δx = n × %Δa |
The subtraction rule has a hidden trap: if you subtract two similar numbers, the absolute uncertainty stays the same but the result shrinks — so the percentage uncertainty in the difference can explode. A burette titre of (24.80 − 0.50) cm³ has an absolute uncertainty of ±0.10 cm³ (same as adding ±0.05 twice), but a percentage uncertainty of 0.10/24.30 × 100 = 0.41%; in a titration where two near-equal readings are subtracted to yield 0.20 cm³, the percentage uncertainty would jump to 50%. This is why titres are designed to be 20–30 cm³.
A student titrates a 25.00 ± 0.06 cm³ aliquot (Class A pipette) against a 0.100 ± 0.001 mol dm⁻³ standard NaOH solution. The mean titre is 24.80 ± 0.10 cm³. The calculated concentration of the analyte (assumed 1:1 stoichiometry) is c = (0.100 × 24.80) / 25.00 = 0.0992 mol dm⁻³. Find the percentage uncertainty in c.
Step 1 — Percentage uncertainty in each measurement.
Step 2 — The calculation involves only multiplication and division. Add percentage uncertainties:
Total percentage uncertainty = 0.24 + 0.40 + 1.00 = 1.64% (or 1.6% to 2 s.f.).
Step 3 — Absolute uncertainty in c. Δc = 1.64% × 0.0992 = 0.0016 mol dm⁻³.
Final result: c = 0.099 ± 0.002 mol dm⁻³ (or 0.0992 ± 0.0016 mol dm⁻³).
Notice that the standard solution's concentration uncertainty (1.00%) dominates the result. To reduce the overall uncertainty, the student should buy a more accurately certified standard (e.g. 0.1000 ± 0.0002 mol dm⁻³), not re-do the titration with greater care — extra titration precision has marginal impact when one input dominates.
Exam Tip: When asked to reduce the uncertainty in a result, identify which input contributes the largest percentage uncertainty and target that one. Improving a 0.2% measurement to 0.1% is wasted effort if another input is at 1%.
For dilute solutions, the absorbance A of light at a chosen wavelength is directly proportional to the concentration c of the absorbing species and the path length l of the light through the solution:
A = εcl
where:
Absorbance is defined as A = log₁₀(I₀ / I), where I₀ is the incident light intensity and I is the transmitted intensity. A solution with A = 1.0 transmits 10% of the incident light; A = 2.0 transmits 1%; A = 3.0 transmits 0.1%.
At low concentrations, each absorbing molecule contributes additively to the absorbance, so doubling c doubles A. The graph of A vs c is a straight line through the origin with gradient εl.
At high concentrations the linearity breaks down. The dominant causes are:
The working range of A = εcl is roughly A ≤ 1.5 for routine work; above this, dilute the sample.
A solution of [Cu(H₂O)₆]²⁺ has ε = 14.5 dm³ mol⁻¹ cm⁻¹ at 800 nm. A sample in a 1.00 cm cuvette gives an absorbance of 0.435. Calculate the concentration.
A = εcl 0.435 = 14.5 × c × 1.00 c = 0.435 / 14.5 = 3.00 × 10⁻² mol dm⁻³ (i.e. 30.0 mmol dm⁻³).
Note that copper(II) aqua complex absorbs only weakly (ε ~ 15) — it is a forbidden d–d transition. Strong charge-transfer absorbers like KMnO₄ have ε ~ 2500 and can be quantified down to micromolar levels.
A calibration curve converts a raw instrument response (absorbance, peak area, fluorescence intensity) into a concentration without needing to know ε independently. The procedure:
A calibration curve is more robust than a single-point calculation because:
Standard solutions of an iron(II)–phenanthroline complex are prepared and absorbances measured at 510 nm in a 1.00 cm cuvette:
| c / mol dm⁻³ | A |
|---|---|
| 0.00 | 0.000 |
| 2.00 × 10⁻⁵ | 0.224 |
| 4.00 × 10⁻⁵ | 0.448 |
| 6.00 × 10⁻⁵ | 0.671 |
| 8.00 × 10⁻⁵ | 0.896 |
| 1.00 × 10⁻⁴ | 1.120 |
An unknown gives A = 0.535. Find its concentration and the molar absorption coefficient.
Gradient (εl): Using the (1.00 × 10⁻⁴, 1.120) point, gradient = 1.120 / 1.00 × 10⁻⁴ = 1.12 × 10⁴ dm³ mol⁻¹.
ε: with l = 1.00 cm, ε = 1.12 × 10⁴ dm³ mol⁻¹ cm⁻¹ — consistent with a strong allowed charge-transfer absorption.
Unknown: c = A / (εl) = 0.535 / (1.12 × 10⁴ × 1.00) = 4.78 × 10⁻⁵ mol dm⁻³ (or read directly from the graph at A = 0.535).
Exam Tip: When reading from a calibration graph, draw clean horizontal and vertical construction lines from the y-axis intercept of the measurement, through the calibration line, to the x-axis. Quote the concentration to a sensible number of significant figures (usually 3 s.f. — matching the graph's resolution).
Errors fall into two categories with very different implications:
Systematic errors shift every measurement in the same direction. Common sources:
Systematic errors cannot be reduced by repetition — averaging ten biased readings gives the same biased mean. They are addressed by calibration against certified reference materials and by procedural controls (e.g. running a blank, using a standard reference solution to verify the spectrometer).
Random errors scatter measurements symmetrically about the true value. Sources include reading the burette meniscus, electrical noise in the detector, fluctuations in lamp output, and small temperature changes affecting solution density.
Random errors are reduced by repetition: the standard error of the mean falls as 1/√n. Three concordant titres are the minimum; the published mean is reported with the standard deviation as the uncertainty.
These two words are often used interchangeably in everyday speech but mean distinct things in analytical chemistry:
A bullseye analogy: three darts tightly clustered in one corner of the board are precise but not accurate; three darts scattered around the bullseye are accurate but not precise; three darts clustered on the bullseye are both. Quantitative chemistry aims for both — but the routes are different. Accuracy is improved by calibration; precision is improved by replication and instrument quality.
A common reporting convention: a result is quoted as mean ± standard deviation (precision) with a separate note of any calibration check against a certified standard (accuracy). Bench scientists routinely report both.
Practical-skills box — minimising uncertainty.
- Maximise the titre. A 24 cm³ titre has half the percentage uncertainty of a 12 cm³ titre. If your titres are too small, dilute the standard or take a larger aliquot.
- Use Class A glassware. A Class B pipette has 2× the tolerance of a Class A — and the cost difference is trivial.
- Replicate. Three concordant titres (within 0.10 cm³) is the standard; for higher-stakes work, five or more.
- Run a blank. Subtracts background absorbance, reagent impurities, and instrument zero offset.
- Use a certified reference standard. Available from NIST (USA) and LGC (UK) for most common analytes; ties your measurement chain back to a primary calibration.
- Match conditions — temperature, ionic strength, solvent — between standards and unknown. Beer–Lambert sensitivity drifts ~1% per 10 °C for many transition-metal complexes.
Quantitative analysis is the connective tissue that turns every other Required Practical into a numerical result:
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