Topic 8 · page 22 of 29
Measurement uncertainty in volumetric analysis
Every reading has an uncertainty set by the apparatus. Express each as a percentage of the value measured, add them to estimate the overall uncertainty in the result, and reduce it by making measurements large and using precise apparatus.
Read first: Core Practical 4: a standard solution from a solid acid
Uncertainty in one measurement 8.22
No measurement is exact. Every instrument has an uncertainty: the range within which the true value is likely to lie. A burette reading of 12.35 cm³ with an uncertainty of ±0.05 cm³ means the true volume is between 12.30 and 12.40 cm³.
The values to use are printed on the apparatus, or given in the question. Typical ones are:
| Apparatus | Typical uncertainty |
|---|---|
| Burette (50 cm³), each reading | ±0.05 cm³ |
| Pipette (25.0 cm³) | ±0.06 cm³ |
| Volumetric flask (250 cm³) | ±0.3 cm³ |
| Balance reading to 3 d.p. | ±0.0005 g each reading |
| Balance reading to 2 d.p. | ±0.005 g each reading |
Two points that cost marks:
- A burette titre is the difference of two readings, an initial and a final, each with an uncertainty of ±0.05 cm³. The titre's uncertainty is ±0.10 cm³: the burette uncertainty is doubled.
- A mass by difference needs two readings of the balance, so its uncertainty is also doubled.
Worked exampleUncertainties from one titration
Estimate the overall uncertainty in the concentration of sodium hydroxide found from a standard solution of acid (as in Core Practical 4): mean titre 24.55 cm³, pipette 25.0 cm³, flask 250 cm³, acid mass 3.162 g by difference on a 3 d.p. balance. The values are illustrative.
- Burette: two readings, ±0.05 each, so ±0.10 cm³. 0.10 ÷ 24.55 × 100 = 0.41 %
- Pipette: 0.06 ÷ 25.0 × 100 = 0.24 %
- Volumetric flask: 0.3 ÷ 250 × 100 = 0.12 %
- Balance: two readings, ±0.0005 each, so ±0.001 g. 0.001 ÷ 3.162 × 100 = 0.03 %
- Overall: 0.41 + 0.24 + 0.12 + 0.03 = 0.80 % (0.7989 before rounding)
- In a concentration of 0.0982 mol dm⁻³: 0.80 % of 0.0982 = 0.00079, so the result is 0.0982 ± 0.0008 mol dm⁻³.
Reducing the uncertainty 8.22
The percentage uncertainty falls when the measured quantity is larger, or the apparatus more precise.
| Change | Effect | Example |
|---|---|---|
| Use a larger titre | ±0.10 cm³ is a smaller fraction | titre 12.00 cm³: 0.83 %; titre 24.55 cm³: 0.41 % |
| Weigh a larger mass | ±0.001 g is a smaller fraction | 0.500 g: 0.2 %; 3.162 g: 0.03 % |
| Use a balance with more decimal places | smaller absolute uncertainty | 2 d.p. balance on 3.162 g: ±0.01 g = 0.32 %; 3 d.p.: 0.03 % |
| Use a pipette rather than a measuring cylinder | much smaller uncertainty | pipette ±0.06 in 25.0 = 0.24 %; cylinder ±0.5 in 25 = 2.0 % |
| Weigh by difference | the whole sample is counted | no solid left on the container is missed |
| Choose a concentration that gives a titre of about 20–25 cm³ | large titre, no refilling | a titre of 20–25 cm³ is typical |
Comparing with the expected value 8.22
If the result is compared with an accepted value, calculate the percentage difference and compare it with the uncertainty.
Worked exampleIs the difference explained by the apparatus?
The label on the bottle of acid says 0.100 mol dm⁻³. A titration gives 0.0982 ± 0.0008 mol dm⁻³. Does the apparatus uncertainty explain the difference?
- Difference = 0.100 − 0.0982 = 0.0018 mol dm⁻³.
- As a percentage of 0.100: 0.0018 ÷ 0.100 × 100 = 1.8 %.
- The uncertainty of the result is 0.8 %, which is smaller than 1.8 %.
- So the difference is greater than the uncertainty: it is not explained by the apparatus. A systematic error (for example the acid label being wrong, or the sodium hydroxide being partly carbonate) is likely.
Check yourself
- A 25.0 cm³ pipette has an uncertainty of ±0.06 cm³. Calculate the percentage uncertainty.
- A burette is read before and after a titration: 0.00 and 18.40 cm³, each ±0.05 cm³. Calculate the percentage uncertainty in the titre.
- A student weighs 0.50 g of solid by difference on a balance reading to 2 d.p. (±0.005 g per reading). Calculate the percentage uncertainty, and say how it changes with a 3 d.p. balance (±0.0005 g per reading).
- Calculate the overall percentage uncertainty from a burette titre of 18.40 cm³ (±0.10), a 25.0 cm³ pipette (±0.06) and a 250 cm³ flask (±0.3).
- Suggest two ways to reduce the percentage uncertainty in a titration whose titre is only 10.00 cm³.
- Explain why using the mean of three concordant titres does not remove the uncertainty of the burette.
Answers cover these until you have tried the questions
- 0.06 ÷ 25.0 × 100 = 0.24 %
- The titre is 18.40 cm³ with an uncertainty of 2 × 0.05 = ±0.10 cm³. 0.10 ÷ 18.40 × 100 = 0.54 %
- Uncertainty = 2 × 0.005 = ±0.01 g. 0.01 ÷ 0.50 × 100 = 2.0 %. On a 3 d.p. balance: 2 × 0.0005 = ±0.001 g, 0.001 ÷ 0.50 × 100 = 0.2 %, ten times smaller.
- 0.10 ÷ 18.40 = 0.543 %; pipette 0.24 %; flask 0.3 ÷ 250 = 0.12 %. Total = 0.543 + 0.24 + 0.12 = 0.90 %.
- Use a larger titre: a more dilute titrant, or a larger amount of sample, so the titre is about 25 cm³ (the percentage falls from 1.0 % to about 0.4 %). Use more precise apparatus, for example a burette with a smaller tolerance.
- The mean averages out the random scatter between runs. But every reading is taken from the same burette, with the same ±0.05 cm³ tolerance on each reading, so the uncertainty of each titre is the same. Only more precise apparatus, or a larger volume, reduces it.