How do you use C1V1 = C2V2?

Put the stock's concentration and volume on the left and the final concentration and volume on the right, then solve for the unknown. To make 500 mL of 0.1 M from a 2 M stock: V₁ = 0.1 × 500 ÷ 2 = 25 mL of stock, made up to 500 mL with 475 mL of water. Pick what to solve for above.

How much water do I add to dilute a solution?

Final volume minus the stock volume you started with. Diluting 100 mL of 1 M to 0.25 M needs a final volume of 1 × 100 ÷ 0.25 = 400 mL, so 300 mL of water. The page shows that line whenever it solves for a volume.

What is a dilution factor?

The ratio of final volume to stock volume, or equivalently stock concentration to final concentration. A 1 : 10 dilution has a factor of 10 — 1 part stock in 10 parts total. Note that '1 : 10' in biology usually means 1 part in 10 total, not 1 part stock to 10 parts water.

Can I use this for percent solutions or ppm?

Yes — the equation only needs the same concentration unit on both sides. Diluting 10% bleach to 0.5% is a factor of 20, so 50 mL of bleach in 1 L of solution.

Dilution Calculator (C₁V₁ = C₂V₂)

Solve C₁V₁ = C₂V₂ for the final volume, the final concentration or the stock volume, with the solvent to add and the dilution factor.

Inputs

Solve for

Stock concentration C₁(mol/L)

0 mol/L1,000,000 mol/L

Stock volume V₁(L)

0 L1,000,000 L

Target concentration C₂(mol/L)

0 mol/L1,000,000 mol/L

Final volume V₂(L)

0 L1,000,000 L

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Final concentration (C₂)

2.3611 mol/L

Working: C₂ = C₁V₁ ÷ V₂ = 5 × 0.85 ÷ 1.8 = 2.3611 mol/L
Moles of solute (C₁V₁ = C₂V₂): 4.25 mol
Dilution factor: 2.1176×

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What each Dilution input means

Solve for: The unknown: the final volume, the final concentration, or the volume of stock to measure out.
Stock concentration C₁: The stock's concentration, in mol/L. Any unit works if C₂ uses the same one — %, g/L or ppm.
Stock volume V₁: The volume of stock used, in litres.
Target concentration C₂: The concentration you want after diluting, in the same unit as C₁.
Final volume V₂: The total volume after diluting, in litres — stock plus solvent.

Dilution formula

C₁ × V₁ = C₂ × V₂ (the moles of solute do not change) · V₂ = C₁V₁ ÷ C₂ · C₂ = C₁V₁ ÷ V₂ · V₁ = C₂V₂ ÷ C₁ · solvent to add = V₂ − V₁

C_1 V_1 = C_2 V_2

Adding solvent to a solution spreads the same amount of solute through a larger volume, so the concentration falls in proportion. Because concentration × volume is the amount of solute, and that amount is unchanged, C₁V₁ before equals C₂V₂ after. Knowing any three of the four gives the fourth: how much stock to measure out to make a given volume at a target concentration, what concentration a stock becomes when made up to a volume, or what final volume a portion of stock must reach. The units just have to match on both sides — mL with mL, mol/L with mol/L, or even percent with percent — because they cancel. The page also gives the solvent to add (final minus stock volume) and the dilution factor.

Dilution worked example

OpenStax's Example 3.19: 0.850 L of 5.00 M copper nitrate is diluted to 1.80 L. C₂ = C₁V₁ ÷ V₂ = 5.00 × 0.850 ÷ 1.80 = 2.36 M, a dilution factor of 1.80 ÷ 0.850 = 2.12. The same solute, 4.25 mol, is in both. Asked the other way — how much 5.00 M stock makes 1.80 L of 2.36 M — V₁ = 2.36 × 1.80 ÷ 5.00 = 0.850 L, and 0.95 L of water is added.

More Dilution examples

500 mL of 1 M hydrochloric acid from 12 M concentrated stock: V₁ = 1 × 0.5 ÷ 12 = 0.0417 L — about 41.7 mL of acid, made up to 500 mL by adding it to about 458 mL of water (acid into water, never the reverse).

25 mL of a 2 M solution diluted to 250 mL in a volumetric flask: C₂ = 2 × 0.025 ÷ 0.25 = 0.2 mol/L, a tenfold dilution.

Where the Dilution calculator can give the wrong answer

Solvent to add is final volume minus stock volume only when volumes are additive, which they nearly are for dilute aqueous solutions. Concentrated acids and alcohol–water mixes shrink on mixing; make the final volume up to the mark rather than adding a calculated amount.
Add acid to water, never water to concentrated acid. The arithmetic is the same; the order is a safety rule.
A target concentration above the stock's cannot be reached by dilution — the page says so instead of returning a negative solvent volume.
Any consistent units work, but mixing them does not: a stock in mol/L and a target in mg/mL need converting first.
Serial dilutions multiply: two 1 : 10 steps are 1 : 100. Run the page once per step, feeding each C₂ in as the next C₁.

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Dilution reference table

Making 1 litre from 12 M stock
Target concentration	Stock to measure	Water to make it up to 1 L (about)
6 M	500 mL	500 mL
3 M	250 mL	750 mL
1 M	83.3 mL	917 mL
0.5 M	41.7 mL	958 mL
0.1 M	8.3 mL	992 mL

Source: Arithmetic: V₁ = C₂V₂ ÷ C₁ (OpenStax Chemistry 2e, §3.3); volumes do not always add exactly, so make up to the mark rather than adding a fixed volume of water

Dilution FAQ

: Put the stock's concentration and volume on the left and the final concentration and volume on the right, then solve for the unknown. To make 500 mL of 0.1 M from a 2 M stock: V₁ = 0.1 × 500 ÷ 2 = 25 mL of stock, made up to 500 mL with 475 mL of water. Pick what to solve for above.
: Final volume minus the stock volume you started with. Diluting 100 mL of 1 M to 0.25 M needs a final volume of 1 × 100 ÷ 0.25 = 400 mL, so 300 mL of water. The page shows that line whenever it solves for a volume.
: The ratio of final volume to stock volume, or equivalently stock concentration to final concentration. A 1 : 10 dilution has a factor of 10 — 1 part stock in 10 parts total. Note that '1 : 10' in biology usually means 1 part in 10 total, not 1 part stock to 10 parts water.
: Yes — the equation only needs the same concentration unit on both sides. Diluting 10% bleach to 0.5% is a factor of 20, so 50 mL of bleach in 1 L of solution.