SCIENCE · INVESTIGATE AND EXPLAIN

Design a dilution with final-volume and uncertainty accounts

Conserve amount while propagating transfer and calibration bounds.

Learning outcome: Compare10/100 and 20/200 nominal volume pairs, then inspect the second-stage bounds. Retain endpoint formulas and distinguish bounded intervals from statistical confidence.

Starting recap

Review amount dimensions, chemical formulas and charge, signed differences, ratio bounds and the distinction between a theoretical constraint and measured conversion.

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Try

Transferring10 mL of .200 M stock to 100 mL final solution gives .0200 M; adding100 mL solvent is not the same specified final-volume operation.

Open the experiment

Predict

Does a second dilution stage remove uncertainty simply because its nominal factor is convenient?

Your prediction and reason

Experiment

Compare10/100 and 20/200 nominal volume pairs, then inspect the second-stage bounds. Retain endpoint formulas and distinguish bounded intervals from statistical confidence.

Propagate transfer and final-volume bounds through dilution

Worked teaching model · synthetic values · Stock concentration0.200±0.002 mol/L; aliquot bound±0.05 mL and final solution-volume bound±0.20 mL. Independent worst-case intervals, not confidence limits. Optional second stage transfers10.00±0.05 mL to100.00±0.20 mL. No loss/reaction assumed. Final volume is not added solvent volume.

Stock aliquot: 10 mL Final solution volume: 100 mL

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Calculated evidence under the stated assumptions
Quantity	Value	Unit
One-stage concentration	0.02	mol/L
One-stage bounded minimum	0.01966168	mol/L
One-stage bounded maximum	0.02034168	mol/L
Second-stage nominal concentration	0.002	mol/L
Second-stage bounded minimum	0.001952432	mol/L
Second-stage bounded maximum	0.002048436	mol/L
Stock amount transferred	2	mmol

Conserve solute amount while distinguishing final solution volume from added solvent.

Intervals use worst-case monotonic combinations, not a statistical confidence claim. A second transfer can access a smaller concentration but adds measurement bounds. No loss, reaction or volume-temperature mismatch is included; the UI is a virtual design ledger, not evidence of pipetting technique.

Retain the reaction/measurement boundary with every number. This mathematical packet is not a chemical procedure or evidence of actual laboratory technique.

Conserve solute amount while distinguishing final solution volume from added solvent.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Begin with amount conservation

Molarity is amount of a specified solute per final solution volume. With no reaction or loss, an aliquot contains n=Cstock Valiquot and dilution gives Cfinal=n/Vfinal. Consistent volume units cancel in the ratio . The final volume is the volume of the entire solution, not the amount of solvent added; real volumes need not add exactly.

Compare equivalent nominal designs

A 10 mL aliquot of .200 mol/L stock contains 2.00 mmol solute. At 100 mL final volume its nominal concentration is .0200 mol/L. A 20 mL aliquot brought to 200 mL has the same nominal concentration but different relative effects from the stated absolute volume bounds. Equal nominal ratios do not imply identical uncertainty budgets.

Propagate the declared bounds

The stock interval is .198–.202 M, aliquot bound±.05 mL and final volume bound±.20 mL. For positive inputs, the smallest concentration uses the smallest stock/aliquot and largest final volume; the largest uses the opposite combination. The10/100 packet therefore uses .198×9.95/100.20 and .202×10.05/99.80 as its endpoints. These are worst-case bounds, not a confidence interval or a standard error.

Audit a serial step

The optional second stage transfers10.00±.05 mL into100.00±.20 mL final volume. Its nominal concentration is one tenth of the first stage, but its interval multiplies the first interval by the corresponding lower and upper volume ratios. It adds an operation and additional bounds rather than making the original stock uncertainty disappear.

Plan beyond the arithmetic

A genuine analytical design would also consider contamination, transfer losses, temperature-dependent calibration and possible correlated volume errors. The workbench does not claim those were checked. It is a virtual comparison with explicit assumptions, not a pipetting protocol or certification. A changed-case report should show the conserved amount, final-volume interpretation, endpoint calculations and at least one omitted error source that an independent measurement could test.

Sources and evidence boundary

Conceptual sources checked 30 October 2026: Molarity and dilution. All prose, cases, code and diagrams are original. The supplied models are not measured structures, a performed laboratory investigation or certification of independent mastery. No chemical handling, heating, pressure vessel or cryogenic experiment is requested. Retain predictions, contrasting evidence and a changed-case explanation.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which unmodeled loss, temperature or calibration correlation could change the inference?

Your explanation and limitations

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Previous: Infer an empirical ratio without assigning oxidant oxygen to the sample Next: Infer concentration from blank-corrected equivalents

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