5.12-5.13 - Measurement Uncertainty And Error

What Makes A Result Trustworthy

A student titrates an acid and writes the concentration as $0.103 \mathrm{mol\,dm^{-3}}$. The number looks precise, but an examiner can ask a harder question: how much confidence should we have in it?

A chemical result is only useful when the measurement route is clear. The student has to show what was measured, how uncertain those measurements are, and whether the final result is close to an accepted or expected value.

The two formulae to keep separate are:

If the direction matters, keep the sign in words as well as doing the percentage calculation. For example, a measured enthalpy change that is less exothermic than the accepted value may suggest heat loss, even if the percentage error calculation uses the size of the difference.

Common-error contrast

"More decimal places" does not automatically mean "more accurate". Writing $0.103000 \mathrm{mol\,dm^{-3}}$ cannot remove an uncertain titre, a poorly rinsed pipette, a balance zero error, gas leakage, or heat loss. The final number should reflect the weakest measurement in the route.

Calculating Measurement Uncertainty

Most uncertainty calculations begin with the apparatus reading. Use the uncertainty given in the question whenever it is supplied. If it is not supplied, use the stated resolution or the conventional reading uncertainty for that apparatus in the context of the question.

[DIAGRAM: asset_name: Edexcel A Level Chemistry L027 Burette Titre Uncertainty; asset_slug: edexcel_a_level_chemistry_l027_burette_titre_uncertainty; recommended_method: deterministic_code; description: Initial 0.10 and final 23.70 cm3 readings give a 23.60 cm3 titre with maximum uncertainty ±0.10 cm3. Pale 1600 by 900 diagram rendered with Pillow; technical geometry and Unicode chemical labels inspected.]

Core rules

For a single measured value, use the percentage-uncertainty formula above. For a value found by subtraction, add the absolute uncertainties. This is why a burette titre has two reading uncertainties: one from the initial reading and one from the final reading.

These are maximum-uncertainty estimates for this course. A fixed balance zero offset can cancel in a mass difference, whereas independent reading uncertainties still contribute. Use the stated uncertainty model rather than automatically doubling every quoted instrument uncertainty.

For products and quotients, add percentage uncertainties. This is useful when a calculated result depends on several measured quantities, such as concentration from a measured volume and amount.

For repeated direct readings of the same quantity, a question may ask you to use the spread of values. A simple estimate of random uncertainty is often:

Use this only when the data and wording make it appropriate. In many titration contexts, the intended calculation is the apparatus uncertainty in the titre from the two burette readings.

Worked example: burette uncertainty

A titre is found from a burette with readings that each have an uncertainty of $\pm 0.05 \mathrm{cm^{3}}$.

Initial reading $= 0.10\,\mathrm{cm^{3}}$; final reading $= 23.70\,\mathrm{cm^{3}}$; titre $= 23.70 - 0.10 = 23.60\,\mathrm{cm^{3}}$.

The titre uses two readings, so the absolute uncertainty is:

The percentage uncertainty in the titre is:

To a sensible number of significant figures, this is $0.42\%$.

Chemical interpretation: the absolute uncertainty is small, but it is not zero. A titre near $24 \mathrm{cm^{3}}$ gives a much smaller percentage uncertainty than a very small titre because the same $0.10 \mathrm{cm^{3}}$ uncertainty is divided by a larger measured volume.

Worked example: combining measured volumes

A concentration calculation uses:

• a $25.0 \mathrm{cm^{3}}$ pipette with an uncertainty of $\pm 0.06 \mathrm{cm^{3}}$
• a $23.60 \mathrm{cm^{3}}$ titre with an uncertainty of $\pm 0.10 \mathrm{cm^{3}}$

The percentage uncertainties are:

Pipette: $0.06 / 25.0 \times 100 = 0.24\%$

Titre: $0.10 / 23.60 \times 100 = 0.42\%$

If the final concentration is calculated by multiplying or dividing by these measured volumes, the approximate combined percentage uncertainty from these two measurements is:

$0.24\% + 0.42\% = 0.66\%$

Chemical interpretation: the titre contributes more uncertainty than the pipetted volume. If you wanted to improve this method, the evidence points first toward getting a larger, reliable titre or reducing burette reading uncertainty.

Guided practice

The result should not be reported with excessive precision just because the calculator shows many digits. A sensible statement is $3.164\,\mathrm{g} \pm 0.002\,\mathrm{g}$, with a percentage uncertainty of about $0.063\%$.

Commenting On Sources Of Error

A source of error is not just "something went wrong". A strong chemistry comment says what caused the error, whether it is random or systematic, how it affects the measured result, and how the method could be improved.