Serial dilutions: calculations, execution and error control

The equation can be correct while the solution is wrong: using diluent volume instead of final volume, skipping a mixing step or drawing from the wrong preceding vessel is enough. Serial dilution needs a clear quantitative model and a reconstructable sequence. Additional steps make some large dilution ratios practical, but they also introduce further opportunities for error.

Start with concentration and units

State what concentration means: analyte mass per solution volume, amount of substance per volume or mass fraction. mg/mL and µg/mL can be converted; mol/L also requires the appropriate chemical identity and molar mass. Do not treat mass/mass, mass/volume and volume/volume percentages as interchangeable, or convert a mass fraction using volumes alone without the necessary data, such as density.

Make units consistent before substituting values. For example, 1 mg/mL equals 1000 µg/mL and 1 mL equals 1000 µL. Keep the stock solution’s assigned concentration separate from subsequent factors. If the method requires corrections for assigned content, purity or another reporting basis, apply them at the specified point and make them reviewable, avoiding double application.

Use C₁V₁ = C₂V₂ with its assumptions

For dilution without loss or formation of analyte, C₁ × V₁ = C₂ × V₂. C₁ is the concentration of the solution sampled; V₁ is the transferred aliquot volume; C₂ is the resulting concentration; V₂ is the new solution’s final volume. V₁ need not be the entire contents of the original vessel. The relationship expresses a balance, also illustrated in the Rose-Hulman Institute of Technology teaching material on dilutions.

The model assumes a representative aliquot, quantitative transfer and preservation of analyte in the measured form. Adsorption, degradation or precipitation may invalidate the simple application. The diluent must not introduce an unaccounted quantity of analyte. The volumetric equation does not automatically apply to mass percentages: these require a mass-based balance.

Making 2 mL up to 100 mL does not mean adding 100 mL of solvent. Nor is final volume always the sum of the initial volumes: NISTIR 8342 demonstrates this for ethanol–water mixtures. Follow the specified approach for obtaining final volume, considering relevant thermal conditions.

Multiply the factors in the sequence

Here the factor is defined as F = V_final/V_aliquot, greater than 1 for a dilution. Concentration is divided by F. Some resources use its reciprocal: stating the convention avoids ambiguity. “1:10” alone is less clear than “take 1 mL and make up to a final volume of 10 mL”.

Simulated numerical example. Stock solution S nominally contains 1000 µg/mL. Assume homogeneous, stable solutions, no losses and correctly achieved final volumes. These are teaching values, not a validated procedure or a statement of uncertainty.

F_total = 50 × 10 × 2 = 1000; C_C = 1000/1000 = 1 µg/mL. The factors need not be equal. In a series, each step uses the specified preceding solution; preparing all levels directly from the stock is a different design and requires separate calculations.

Select volumes that can actually be delivered

Design the steps within demonstrated instrument operating ranges, considering liquid, tips, vessel and the amount needed for analysis and controls. A single dilution may require an aliquot unsuited to the available system; numerous steps can increase transfers, consumption and uncertainty contributions. Compare alternatives against the purpose, without assuming that more steps are always better.

For pipettes, consider technique appropriate to the liquid and the pipette–tip system. A volumetric flask defines a contained volume; it is not automatically a device for delivering that volume. ISO 1042:1998 concerns one-mark volumetric flasks; referencing it does not replace identification, status and suitability of the actual glassware.

Mix, transfer and identify every solution

Prepare unambiguous identifiers for the stock and derived solutions. Verify source and destination before withdrawal, perform the specified mixing before the next aliquot and record quantities and sequence during the work. Define arrangements compatible with the matrix and vessel: foam, sedimentation or evaporation can compromise sampling.

Retain the link between solution, operator, instrument, diluent and critical times. The available quantity decreases after withdrawal, but the concentration of the remaining homogeneous solution does not change from that withdrawal alone. Do not confuse residual volume with dilution factor. Protect preparations under conditions and for periods demonstrated by the method.

Distinguish errors from shared contributions

An identified execution error is not handled simply by widening uncertainty. For a correctly performed sequence, the model instead includes stock concentration and volumetric ratios. If the same pipette or reference contributes to several steps, some effects may be shared: these are not automatically independent observations.

The law of propagation of uncertainty described by NIST includes covariances. Adding only squared relative contributions is a simplification that requires appropriate assumptions. The sign and magnitude of an effect depend on the model; a common cause can affect several levels together. Repeated readings of the final solution do not make the preparations independent.

Retain sufficient digits in intermediate calculations and round according to method rules. Many decimal places do not demonstrate accuracy. Documented, specified metrological corrections differ from invented factors used to make a result fit.

Check backwards and reconstruct an error

The reverse check starts from C: 1 × (20/10) = 2 µg/mL in B; 2 × (50/5) = 20 in A; 20 × (100/2) = 1000 in the stock. Check units, data origins and transcription too. EU GMP Chapter 6 requires critical examination of calculations; rereading the same copied equation is not an effective independent check.

Simulated case. At step B, 4.00 mL of A is transferred instead of 5.00 mL, still to a final volume of 50.0 mL. Under the example’s assumptions, B contains 20 × 4/50 = 1.6 µg/mL; C, prepared as specified from B, contains 0.8 µg/mL, 20% below target. A is outside this particular error unless other evidence indicates otherwise. Further solutions derived from B are affected.

Place the affected preparations on hold, retain original data and document the event. The calculation clarifies theoretical impact but does not authorise relabelling, additions or use outside the method. The responsible person determines the investigation and repeat-preparation route. Incorporate these controls into the sample preparation workflow in the PL-03 area.

For the related steps, see: Analytical weighing: technique, static electricity and sample transfer.

Sources and example limitations

Sources consulted on 30 September 2026: Rose-Hulman teaching material, dilution section; NIST TN 1297 (1994), Appendix A; NISTIR 8342 (2021), volume non-additivity; EU GMP Chapter 6 (2014). ISO 1042:1998: public description, published edition; ISO/DIS 1042 remains under development. No tolerance is inferred from the unconsulted full ISO text. Table and case are original and arithmetically checked, without presenting them as experimental results.