A 2.4% Standardization Drift and a Two-Tonne Rejection
A government analytical laboratory assays citric acid monohydrate (declared 99.5–100.5%; reject if below 99.0%) by titration with approximately 0.1 M sodium hydroxide using phenolphthalein. The NaOH stock has stood for five weeks in a high-density polyethylene carboy fitted with a soda-lime guard tube; when fresh it standardized against potassium hydrogen phthalate (KHP) at a factor of 1.000. A new analyst now standardizes the same carboy against KHP that has been stored open on the bench for three weeks (it was dried at 105 °C on receipt, and not since) and obtains a factor of 0.976. Using that factor she assays the batch and reports 97.9% citric acid, and recommends rejection of a 2-tonne consignment. The supplier has invoked a contractual penalty clause. A second analyst, using freshly prepared NaOH and freshly dried KHP, obtains a factor of 1.001 and an assay of 99.8%.
1. Rank the plausible causes of the 2.4% low factor and identify the single diagnostic that would discriminate between the two leading candidates.
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Use the direction of each bias first - it eliminates candidates faster than any test.
| Candidate | Direction of effect on the calculated factor |
|---|---|
| CO₂ uptake by the NaOH (carbonate formation) | lowers it |
| KHP wet from bench storage | raises it |
| Endpoint overshoot, or a bubble/drop not delivered so recorded volume exceeds delivered volume | lowers it |
| KHP overheated at drying | raises it (less titratable acid per gram than assumed) |
Because wet KHP pushes the factor up, it cannot explain a factor that fell. On the contrary, it means the downward cause must be larger than the observed 2.4% - the two errors are partly cancelling. With a five-week-old carboy the leading candidate is carbonate: each mole of CO₂ consumes two moles of NaOH but the resulting carbonate still supplies one titratable equivalent at the phenolphthalein endpoint, so half the affected equivalents are lost as a systematic low bias.
Discriminating diagnostic. Titrate an aliquot of the NaOH itself with standard acid to the phenolphthalein endpoint, then continue to a bromocresol green or methyl orange endpoint. For pure NaOH the two volumes are equal; carbonate makes the first stage smaller than the total, and the carbonate content follows directly from the ratio. A barium chloride precipitation check is a quick cross-check. If no carbonate is found, weigh the water delivered by the burette over 20 mL and check for a tip bubble before blaming chemistry.
2. What should the laboratory report on the consignment, and how should it handle the penalty clause and the first analyst's result?
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The first result cannot be rescued arithmetically. The assay is directly proportional to the standardization factor, so a 2.4% factor error is a 2.4% assay error. Applying a blanket correction to a number produced with an uncharacterised factor is not a correction, it is a guess.
Report
- Withdraw the 97.9% report and issue a formal correction stating 99.8% against a specification of 99.0–101.0%.
- Re-run the assay in duplicate, with both analysts and the fresh standard, so the corrected figure rests on fresh data rather than on re-scaling the old titration.
- State the reason plainly: the standardization factor was biased low by carbonate in the NaOH stock, and the confirming standardization is 1.001.
Penalty clause. Write to the supplier before the penalty is enforced, admit the laboratory error without hedging, and offer a referee analysis of the sealed retained sample at an independent accredited laboratory at the laboratory's cost. Defending a known-biased figure to protect institutional reputation converts a technical error into an integrity problem.
Corrective action. Prepare carbonate-free NaOH from a settled 50% w/v stock diluted with CO₂-free water; check and replace the guard tube; restandardize on each day of use and after any standing; dry KHP at 105 °C for two hours and cool it in a desiccator; require duplicate standardization by two analysts on a control chart, verified against a second primary standard.
3. The laboratory is accredited to ISO/IEC 17025. Explain how the standardization uncertainty should enter the reported assay and how you would treat a residual 0.3% difference between the two analysts after correction.
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Propagation. The assay is a product of factor, titre volume, stoichiometric constant and mass, so relative uncertainties combine in quadrature:
u(assay)/assay = √[ (u(f)/f)² + (u(V)/V)² + (u(m)/m)² + (u_rep)² ]
Typical realistic magnitudes: factor 0.10%, volume 0.05%, mass 0.02%, repeatability 0.15% - combined about 0.19%, expanded (k = 2) about 0.4%. That statement belongs on the report. What must not be included is the 2.4% bias: a known systematic error is corrected, not absorbed into an uncertainty budget, and including it would launder a defect into a wide but respectable-looking interval.
The residual 0.3% between analysts. Compare it with the uncertainty of the standardization step, not with the final assay uncertainty. If duplicate standardizations agree within about 0.3%, record both, use the mean, and plot on the control chart. Investigate if the difference exceeds roughly 0.5%, or if the control chart shows a trend rather than scatter - a trend points to carbonate build-up or a drifting burette, not to analyst variability. Document the decision rule in the SOP so the choice is not made case by case after the result is known.