Dilution arithmetic: C1V1 = C2V2

When a solution is diluted, the amount of solute does not change; only the volume it is spread through does. Amount equals concentration times volume, so:

C1 × V1 = C2 × V2

C1 and V1 are the concentration and volume of the starting solution; C2 and V2 are the concentration and volume after dilution. Know any three and the fourth follows. The only rule is that C1 and C2 must be in the same unit, and V1 and V2 must be in the same unit. Which unit does not matter, as long as they match.

Using deliberately abstract numbers:

• How much stock do I take? A stock at 10 mg/mL; a target of 1 mg/mL in a final volume of 5 mL. V1 = (C2 × V2) / C1 = (1 × 5) / 10 = 0.5 mL of stock, made up to 5 mL.
• What concentration do I end with? 0.2 mL of a 10 mg/mL stock made up to 2 mL. C2 = (C1 × V1) / V2 = (10 × 0.2) / 2 = 1 mg/mL.
• What final volume do I need? 1 mL of a 10 mg/mL stock, target 2 mg/mL. V2 = (C1 × V1) / C2 = (10 × 1) / 2 = 5 mL.

Note that V2 is the final volume, not the volume of diluent added. Diluent added is V2 minus V1. Confusing those two is the most common dilution mistake in practice.

The dilution factor is V2 divided by V1 — how many times more dilute the result is than the start. A dilution factor of 10 means C2 is one tenth of C1. Dilution factors multiply across steps: a 10-fold dilution followed by another 10-fold dilution is a 100-fold dilution overall. Serial dilution is simply this arithmetic repeated, and it is preferred over a single very large dilution because it keeps every pipetted volume within the range where a pipette is accurate — see pipetting accuracy basics.

The equation is exact. The result is not, and the difference comes from three places:

• Volume measurement. A pipette used at the bottom of its range, or a graduated cylinder read at eye level from the wrong angle, introduces a percentage error that carries straight into C2.
• The starting concentration. If C1 is wrong, everything downstream is wrong by the same proportion. C1 usually rests on a mass, a volume and a unit conversion — see concentration units.
• Mixing. A solution that is not homogeneous before the aliquot is taken is not at C1 where the pipette tip is.

None of these appear in the equation, which is why a calculation can be perfect and the solution still wrong.

Record C1, V1, C2, V2, the dilution factor and the calculation itself, with units on every term. A future reader — often you — needs to be able to reconstruct what was done from the page alone. Chain of custody for the material starts with the lot number; the calculation is the next link. See a lab notebook and chain of custody for reference materials.

Before trusting any dilution result, check it against the direction and the magnitude you expected. A dilution always lowers concentration: if C2 came out higher than C1, two terms have been swapped. The dilution factor should match the ratio of the volumes: taking 1 part into a final 10 should give a tenth, and if the answer is a hundredth or a fifth, a unit has slipped. Round numbers help — set the calculation up with a factor of 10 or 100 where the protocol allows, because a thousand-fold error in a round-number calculation is obvious and the same error in an awkward one is not. Record the check alongside the calculation. The material's own measured purity is reported on its certificate, described on the quality page; the arithmetic in this post is about volumes and concentrations of a solution, which are the laboratory's measurements, not the supplier's. Questions about what a certificate reports go to support.

This post is arithmetic that applies to any solution of anything. It is not a preparation, dilution or usage instruction for any material sold on this site, and we publish none; we cannot advise on them. Every material is supplied strictly for in-vitro laboratory research, and handling is the responsibility of the qualified researcher — see research use only.