How to Dilute a Solution with C1V1 = C2V2

The formula

C1 × V1 = C2 × V2

• C1 is the concentration of the stock solution you have.
• V1 is the volume of stock to use. This is usually what you want to find.
• C2 is the concentration you want.
• V2 is the final volume you want.

To find the volume of stock, rearrange it: V1 = C2 × V2 ÷ C1. Then add water (or another solvent) until the total reaches V2.

It works because diluting doesn’t change the amount of dissolved substance. It only spreads it through more liquid.

Example 1: a 10× buffer

You need 1 liter of 1× buffer and have a 10× stock.

V1 = 1 × 1,000 mL ÷ 10 = 100 mL. Mix 100 mL of stock with 900 mL of water.

Example 2: a molar stock

You need 200 mL of a 50 mM solution and have a 1 M stock. First, put both concentrations in the same unit: 50 mM is 0.05 M.

V1 = 0.05 × 200 mL ÷ 1 = 10 mL. Mix 10 mL of stock with 190 mL of water.

Example 3: 70% alcohol from 95%

You need 500 mL of 70% alcohol and have 95% alcohol. Both are percentages by volume.

V1 = 70 × 500 mL ÷ 95 ≈ 368 mL. Pour 368 mL of 95% alcohol into a measuring cylinder, then add water up to the 500 mL mark. Alcohol and water shrink slightly when mixed, so filling up to the mark is more accurate than adding exactly 132 mL of water.

Dilution factor and ratios

The dilution factor is C1 ÷ C2, or V2 ÷ V1. Diluting a 10× stock to 1× is a dilution factor of 10.

Be careful with ratios such as “1:10”. In labs, a 1:10 dilution usually means 1 part in 10 parts total (1 mL of stock plus 9 mL of water). On product labels and in recipes, “1:10” often means 1 part plus 10 parts of water. Check which one is meant.

For a serial dilution, repeat the same step. Three 1:10 dilutions in a row make a 1:1,000 dilution.

Common mistakes

• Mixing units. Both concentrations must be the same kind: both %, both ppm or both molar. The concentration converter switches between %, ppm and mg/L. To go between percent and molar, you also need the substance’s molar mass.
• Treating V2 as the water to add. V2 is the final total volume. The water to add is V2 − V1.
• Adding water to concentrated acid. Always add acid to water, slowly, never the other way round.
• Using it for reactions. C1V1 = C2V2 only works for simple dilution, when nothing reacts.

The dilution calculator does the arithmetic for you. Enter any three values and it works out the fourth, with mixing instructions.