Dilution calculator: C₁V₁ = C₂V₂

This dilution calculator applies C₁V₁ = C₂V₂ to prepare a less concentrated solution from a stock solution or to check a laboratory calculation. Enter any three of the initial concentration C₁, aliquot volume V₁, final concentration C₂ and final volume V₂; the tool solves the fourth, converts compatible units and reports the C₁/C₂ dilution factor. The relationship follows from conservation of the solute during dilution: solvent is added, but the amount of solute contained in the measured aliquot is not created or removed.

The general dilution equation is C₁V₁ = C₂V₂. The left side represents the amount of solute in the volume V₁ taken from the stock solution, while the right side represents that same amount of solute after adjustment to the final volume V₂. OpenStax derives the relation from n = CV: because the amount n of solute is unchanged by adding solvent, the two concentration × volume products are equal.

Any one quantity can be isolated algebraically. The stock volume is V₁ = C₂V₂/C₁. The final concentration is C₂ = C₁V₁/V₂. The final volume is V₂ = C₁V₁/C₂, and the stock concentration can be recovered from C₁ = C₂V₂/V₁. The calculator displays the relevant rearrangement and numerical substitution instead of returning an unexplained number.

Concentrations can be expressed as amount concentration — mol/L, mmol/L or µmol/L — or mass concentration — g/L, mg/L or µg/L. Conversions are allowed within one family, for example 500 mmol/L to 0.5 mol/L. The calculator deliberately refuses to convert mol/L to g/L because that conversion requires the molar mass of the solute, which is information outside the dilution equation.

Volumes may be entered in µL, mL or L. Internally they are converted to litres and then displayed in the unit selected for each field. This removes a frequent source of mistakes, such as multiplying a value in millilitres by another volume that was already expressed in litres.

The dilution factor is F = C₁/C₂ and, from the dilution equation, also F = V₂/V₁. If a 1.0 mol/L stock is diluted to 0.10 mol/L, F = 10: the final volume must be ten times the aliquot volume. The factor is therefore a useful reasonableness check before reading the detailed calculation.

In laboratory practice, V₂ means final solution volume, not necessarily the measured volume of solvent added. To prepare 100 mL, an aliquot V₁ is transferred and solvent is added until the total solution reaches the 100 mL calibration mark. It is not always correct to measure exactly 100 − V₁ mL of solvent because solution volumes are not universally additive.