Topic 16 · page 17 of 19
Uncertainty and validity
Every measurement has an uncertainty. Turn each into a percentage, add them, and compare the total with how far your result is from the expected value to decide whether the result is valid.
Read first: Redox titration calculations
Percentage uncertainty 16.16
Every piece of apparatus has an uncertainty: the amount by which a reading could be wrong. A balance reading to two decimal places, a burette, a pipette and a volumetric flask all have one. The same uncertainty matters more when the quantity measured is small.
A burette is read twice, at the start and at the end, so its uncertainty counts twice. A balance reading to 0.01 g is uncertain by ± 0.005 g in each reading, and weighing by difference uses two readings, so the uncertainty is ± 0.01 g.
Worked exampleTotal uncertainty in the brass experiment
Typical uncertainties (use the values printed on your own glassware, which depend on its class):
| Apparatus | Uncertainty | Measured value | Percentage uncertainty |
|---|---|---|---|
| balance, by difference | ± 0.01 g | 1.50 g | 0.67% |
| volumetric flask | ± 0.3 cm³ | 250.0 cm³ | 0.12% |
| pipette | ± 0.06 cm³ | 25.0 cm³ | 0.24% |
| burette, two readings | ± 0.10 cm³ | 16.40 cm³ | 0.61% |
| Total | 1.6% |
- Add the percentage uncertainties: 0.67 + 0.12 + 0.24 + 0.61 = 1.64, which is 1.6%.
- Apply it to the result: 1.6% of 69.4 = 1.1.
- The copper content is 69.4 ± 1.1%.
Is the result valid? 16.16
A result is valid if it is close enough to the true value that the difference can be explained by the uncertainties in the apparatus. Compare two numbers:
- the percentage difference between your result and the expected value;
- the total percentage uncertainty from the apparatus.
If the percentage difference is smaller than the total percentage uncertainty, the result agrees with the expected value within the limits of the measurements. If it is larger, the apparatus cannot explain the difference, so there must also be systematic errors.
Worked exampleJudging the brass result
The brass is sold as 70.0% copper. The experiment gave 69.4% with a total uncertainty of 1.6%.
- Percentage difference = (70.0 − 69.4) ÷ 70.0 × 100 = 0.9%.
- 0.9% is smaller than 1.6%.
- The result agrees with the stated value within the uncertainty of the apparatus. There is no evidence of a systematic error.
Suppose a student had obtained 66.0%. The percentage difference would be (70.0 − 66.0) ÷ 70.0 × 100 = 5.7%, which is larger than 1.6%. Systematic errors must be present. Likely ones in these titrations are:
- iodine lost by evaporation before it is titrated, so the titre is too small (iodine–thiosulfate);
- some of the brass not dissolved, so too little copper is in the solution (iodine–thiosulfate);
- Fe²⁺ solution oxidised by air before the titration, so the titre is too small (manganate(VII));
- hydrochloric acid used with manganate(VII), so the titre is too large;
- overshooting the end point, so the titre is too large;
- a pipette or burette not rinsed with the right solution, so the concentration is wrong.
Reducing the uncertainty 16.16
To reduce the percentage uncertainty in the burette, make the titre larger. The uncertainty of each reading stays the same, so a larger volume makes it a smaller fraction.
Worked exampleHalving the concentration of the titrant
The titre is 16.40 cm³ and the uncertainty is ± 0.10 cm³, which is 0.61%. If the titrant is made half as concentrated, the titre doubles to 32.80 cm³.
- New percentage uncertainty = 0.10 ÷ 32.80 × 100 = 0.30%.
- The percentage uncertainty has halved.
Check yourself
- A titre of 22.40 cm³ is read with a burette that has an uncertainty of ± 0.05 cm³ for each reading. Calculate the percentage uncertainty.
- A 25.0 cm³ pipette has an uncertainty of ± 0.06 cm³. Calculate the percentage uncertainty.
- In the iron tablet experiment the tolerances are: balance ± 0.01 g on 0.350 g, 100.0 cm³ flask ± 0.2 cm³, pipette ± 0.06 cm³ on 25.0 cm³, and burette ± 0.10 cm³ on 11.60 cm³. Find the total percentage uncertainty and give the result 18.5% with its uncertainty.
- State how the percentage uncertainty from a burette can be reduced.
- A student obtains 66.0% copper in brass labelled 70.0%. The total percentage uncertainty is 1.6%. Is the result valid? Suggest a likely cause.
- Three titres agree within 0.05 cm³, but the final concentration is 8% too high. Explain how this can happen.
Answers cover these until you have tried the questions
- Two readings: ± 0.10 cm³. 0.10 ÷ 22.40 × 100 = 0.45%.
- 0.06 ÷ 25.0 × 100 = 0.24%.
- Balance: 0.01 ÷ 0.350 × 100 = 2.86%. Flask: 0.2 ÷ 100.0 × 100 = 0.20%. Pipette: 0.24%. Burette: 0.10 ÷ 11.60 × 100 = 0.86%. Total = 2.86 + 0.20 + 0.24 + 0.86 = 4.2%. 4.2% of 18.5 = 0.8, so the result is 18.5 ± 0.8%.
- Make the titre larger, for example with a more dilute titrant. Repeating the titration does not reduce the uncertainty of each reading.
- Percentage difference = (70.0 − 66.0) ÷ 70.0 × 100 = 5.7%. This is larger than 1.6%, so the result is not valid (it is not explained by the apparatus alone): there must be systematic errors. A result that is too low could be caused by some of the brass not dissolving, or by iodine lost by evaporation before it was titrated.
- Concordant titres show precision, not accuracy. A systematic error (for example, a wrongly standardised solution or an overshot end point each time) shifts every titre by the same amount, so the readings agree with each other but the result is still wrong.