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Every measurement has an uncertainty. The question isn’t whether your result is exact (it never is), but how confident you can be in it. These sixteen problems cover the uncertainty calculations that appear in practical assessments and exams, from reading a single instrument to combining uncertainties in a full titration or enthalpy calculation.
For the underlying ideas, see calculating uncertainty, propagating uncertainty and volumetric glassware tolerances.
Key rules
- Absolute uncertainty: the ± value, in the same units as the measurement.
- Percentage uncertainty = (absolute uncertainty ÷ measured value) × 100.
- Adding or subtracting quantities: add the absolute uncertainties.
- Multiplying or dividing: add the percentage uncertainties.
- Raising to a power: multiply the percentage uncertainty by the power.
(These simple rules give a maximum, “worst-case” uncertainty and are the ones most courses use. More advanced treatments combine uncertainties in quadrature.)
Part A: Single readings
1. A balance reads to 0.01 g. What is the uncertainty of a single reading?
Answer: Commonly taken as ±0.005 g (half the smallest division) for a single reading on a digital instrument, although many courses and manufacturers quote ±0.01 g. Follow the convention of your course, and state it.
2. A mass is found by difference: container + solid = 25.46 g; empty container = 23.12 g. Each reading has an uncertainty of ±0.005 g. Find the mass and its uncertainty.
Answer: Mass = 2.34 g. Two readings are subtracted, so add the absolute uncertainties: ±0.005 + 0.005 = ±0.01 g. Result: 2.34 ± 0.01 g.
3. A burette can be read to ±0.05 cm³. What is the uncertainty in a titre?
Answer: A titre is the difference between two readings (final − initial), so the uncertainty is ±0.05 + 0.05 = ±0.10 cm³. (Some courses also add ±0.05 cm³ for judging the end-point drop.)
4. Calculate the percentage uncertainty in a titre of 24.20 cm³ (uncertainty ±0.10 cm³).
Answer: 0.10 ÷ 24.20 × 100 = 0.41%
Part B: Choosing equipment
5. Compare the percentage uncertainty of measuring 25 cm³ with (a) a 25.0 cm³ pipette (±0.06 cm³) and (b) a 50 cm³ measuring cylinder (±0.5 cm³).
Answer: (a) 0.06 ÷ 25.0 × 100 = 0.24% (b) 0.5 ÷ 25 × 100 = 2% The pipette is almost ten times more precise, which is why it’s used for accurate volumes. See measuring volume accurately.
6. A student weighs 0.25 g of solid on a balance with uncertainty ±0.01 g. Calculate the percentage uncertainty and suggest how to reduce it.
Answer: 0.01 ÷ 0.25 × 100 = 4%. Reduce it by using a larger mass (e.g. 2.50 g gives 0.4%) and scaling up the rest of the experiment, or by using a more precise balance (±0.001 g gives 0.4% for 0.25 g). See using a balance.
7. A titration gives a titre of only 8.50 cm³. Why is this a problem, and what could be changed?
Answer: Uncertainty ±0.10 cm³ gives 0.10 ÷ 8.50 × 100 = 1.2%, about three times worse than for a 25 cm³ titre. Use a more dilute titrant or a larger volume of analyte so the titre is closer to 20–30 cm³.
Part C: Combining uncertainties
8. A concentration is calculated from c = n ÷ V. n has an uncertainty of 0.5% and V has an uncertainty of 0.24%. Find the percentage uncertainty in c.
Answer: Division, so add percentages: 0.5 + 0.24 = 0.74%
9. A temperature rises from 21.0 °C to 34.5 °C, each reading ±0.5 °C. Find ΔT and its percentage uncertainty.
Answer: ΔT = 13.5 °C. Uncertainty = ±0.5 + 0.5 = ±1.0 °C. Percentage = 1.0 ÷ 13.5 × 100 = 7.4% Small temperature changes carry large percentage uncertainties.
10. The area of a square sample is found from its side length, 2.00 ± 0.02 cm. Find the percentage uncertainty in the area.
Answer: Percentage uncertainty in length = 0.02 ÷ 2.00 × 100 = 1%. Area = length², so multiply by 2: 2%.
Part D: Full calculations
11. In a titration, 25.00 cm³ of NaOH (pipette ±0.06 cm³) needs 23.45 cm³ of 0.1000 mol/dm³ HCl (titre ±0.10 cm³). The HCl concentration has an uncertainty of ±0.2%. Calculate [NaOH] and its absolute uncertainty.
Answer: [NaOH] = (0.1000 × 23.45) ÷ 25.00 = 0.09380 mol/dm³ Percentage uncertainties:
- titre: 0.10 ÷ 23.45 × 100 = 0.43%
- pipette: 0.06 ÷ 25.00 × 100 = 0.24%
- HCl concentration: 0.2% Total = 0.43 + 0.24 + 0.2 = 0.87% Absolute = 0.0087 × 0.09380 = ±0.00082 Result: 0.0938 ± 0.0008 mol/dm³ See titration calculations.
12. In an enthalpy of neutralisation experiment, q = mcΔT with m = 50.0 g (±0.5%), c = 4.18 J g⁻¹ K⁻¹ (taken as exact) and ΔT = 6.8 °C (±1.0 °C). Find q and its percentage uncertainty.
Answer: q = 50.0 × 4.18 × 6.8 = 1421 J (≈ 1.42 kJ) Percentage uncertainty in ΔT = 1.0 ÷ 6.8 × 100 = 14.7% Total = 0.5 + 14.7 = 15.2%, dominated by the temperature measurement. See enthalpy of neutralisation.
13. In question 12, which single change would most improve the result?
Answer: Reduce the uncertainty in ΔT: use a thermometer or probe reading to ±0.1 °C, or increase ΔT by using more concentrated solutions. Improving the mass measurement would make almost no difference, because it contributes only 0.5%.
Part E: Evaluating results
14. A student’s result for the enthalpy of neutralisation is −52.1 kJ/mol, with a total uncertainty of ±4%. A data book gives −57.1 kJ/mol. Can the difference be explained by measurement uncertainty alone?
Answer: 4% of 52.1 = ±2.1 kJ/mol, so the result range is −50.0 to −54.2 kJ/mol. The literature value (−57.1) lies outside this range, so measurement uncertainty alone doesn’t explain it. There must be a systematic error, most likely heat loss to the surroundings, which makes the measured temperature rise too small.
15. Calculate the percentage difference between the result and the literature value in question 14.
Answer: (57.1 − 52.1) ÷ 57.1 × 100 = 8.8%
16. A result is 0.0938 ± 0.0008 mol/dm³. How many significant figures should be quoted, and why?
Answer: The uncertainty is in the fourth decimal place, so the result is quoted to the same place: 0.0938 (3 significant figures). Quoting 0.093802 would suggest far more precision than the measurement has. See significant figures.
Common mistakes
- Forgetting that a titre or mass by difference involves two readings.
- Adding absolute uncertainties when quantities are multiplied (use percentages instead).
- Quoting results to more decimal places than the uncertainty supports.
- Blaming “human error” instead of naming a specific random or systematic error.
- Assuming a large difference from the literature value is always due to measurement uncertainty. Compare the difference with the calculated uncertainty first.
Key takeaways
- Percentage uncertainty = absolute ÷ value × 100; bigger measurements mean smaller percentage uncertainties.
- Add absolute uncertainties for sums and differences; add percentages for products and quotients.
- The largest percentage uncertainty usually dominates; improve that measurement first.
- If the literature value lies outside your uncertainty range, look for a systematic error.
- Quote results to a precision that matches their uncertainty.
- Uncertainty is not a mistake; it’s an honest statement of how well a quantity is known.
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