Part 6: Uncertainty, yield and atom economy

Quantify measurement limits, separate precision from accuracy, and distinguish how much product was collected from how efficiently a reaction uses atoms.

Uncertainty is not the same as a known error

Uncertainty expresses the range associated with a measurement; error is the difference between a measured value and a reference value. Random variation causes scatter, while a systematic effect shifts results in a consistent direction. A faulty balance zero is systematic; inconsistent judgement of a faint end point can introduce random variation. Repeats help assess scatter but do not automatically correct calibration or gas-loss errors.

Use the uncertainty stated on the apparatus or in the question. Do not memorise one universal uncertainty for every pipette, balance or cylinder. For an analogue reading, half the smallest scale division may be an appropriate reading estimate when no other specification is given; resolution and calibration tolerance are not identical quantities.

Count the measurements and use the right denominator

A titre uses two burette readings. If each has uncertainty ±0.05 cm³, the simple maximum absolute uncertainty in their difference is ±0.10 cm³. For a 24.60 cm³ titre, percentage uncertainty = (0.10/24.60) × 100 = 0.407%. Using only ±0.05 cm³ omits one reading; dividing by the 50 cm³ burette capacity ignores the actual measured quantity.

A mass found by difference similarly uses two balance readings. If each is ±0.001 g, the transferred mass uncertainty is ±0.002 g. For 0.200 g this is 1.0%; for 2.000 g it is 0.10%. A larger measured amount reduces relative uncertainty only if the method still works safely and completely.

For a stated maximum-uncertainty estimate of a result formed by multiplication or division, add the contributing percentage uncertainties. For c = m/(MV), treating M as exact, 0.08% in m and 0.24% in V gives 0.32% in c. This school-level worst-case convention is not the same as a statistical standard uncertainty; follow the question's stated method.

Choose an improvement that fixes the actual limitation

A 10.00 cm³ titre with ±0.10 cm³ uncertainty has 1.00% uncertainty; a 25.00 cm³ titre has 0.400%. Adjusting concentration or aliquot size to obtain a larger sensible titre can help, provided the burette capacity and reaction conditions permit it. Using a volumetric pipette rather than a coarse measuring cylinder improves delivery precision for a fixed aliquot.

To compare measurements with a reference, suppose measured Vm = 23.1 dm³ mol⁻¹ and the suitable reference is 24.0 dm³ mol⁻¹. Percentage error magnitude = (0.9/24.0) × 100 = 3.75%. If the estimated measurement uncertainty is only 0.8%, the discrepancy suggests an unaccounted effect such as gas loss; agreement within an uncertainty interval would be consistent with the reference, not proof that every systematic effect is absent.

'Repeat and average' is appropriate for random scatter. 'Use a lid and insulation' targets heat exchange; 'seal before mixing' targets initial gas loss; 'calibrate the instrument' targets offset. The explanation must connect the change to a measurable effect and, where possible, its direction.

Percentage yield compares actual and theoretical product

The theoretical yield comes from the limiting reactant and the balanced equation. Percentage yield = actual amount of desired product / theoretical amount × 100%, with consistent quantities and units. A reaction can have less than 100% yield because it is incomplete, reversible, competes with side reactions or loses product during transfer, filtration and purification.

Example: heating 8.40 g NaHCO₃ gives 4.00 g dry Na₂CO₃. From 2NaHCO₃ → Na₂CO₃ + CO₂ + H₂O, n(NaHCO₃) = 8.40/84.0 = 0.100 mol and theoretical n(Na₂CO₃) = 0.0500 mol. Theoretical mass = 0.0500 × 106.0 = 5.30 g. Percentage yield = 4.00/5.30 × 100 = 75.5%. Dividing 4.00 g product by 8.40 g reactant would ignore gaseous products and the reaction ratio.

An apparent yield over 100% does not show creation of matter. Product may be wet, contaminated or incorrectly identified, or the starting amount may be underestimated. Drying to constant mass can address retained solvent, while purity testing addresses impurities; repeating the same flawed isolation is not sufficient.

Atom economy follows the equation, not the measured yield

Atom economy measures the fraction of reactant mass that the balanced reaction puts into the desired product. Include stoichiometric coefficients. By mass conservation, the denominator may be the sum of coefficient × molar mass for all reactants or for all products. Solvents and catalysts are not part of the ideal overall stoichiometric equation, although their real environmental costs still matter.

For 2NaHCO₃ → Na₂CO₃ + CO₂ + H₂O with Na₂CO₃ desired, atom economy = 106.0/(106.0 + 44.0 + 18.0) × 100 = 63.1%. This remains 63.1% whether the measured percentage yield is 75.5% or 95%. Changing isolation technique can improve yield without changing the equation's atom economy.

An addition reaction with a single product can have 100% atom economy. That does not guarantee a sustainable process: energy use, hazards, solvent, catalyst recovery, feedstock source, yield and waste treatment also matter. Selling a useful by-product improves economics but does not change the atom economy calculated for one specified desired product.

Quick checks

Original Finesse questions. Reveal the indicative worked solutions after attempting each question; these are not official Edexcel mark allocations.

Sources