Dilution calculator: C1V1 = C2V2 with worked examples

Diluting a solution adds solvent but not solute, so the amount of solute stays the same: C₁V₁ = C₂V₂, where C₁ and V₁ are the stock solution's concentration and volume, and C₂ and V₂ are the diluted solution's. To make 250 mL of 0.40 M solution from 2.0 M stock, you need 50 mL of stock made up to 250 mL. This dilution calculator solves C₁V₁ = C₂V₂ for any one value, says how much solvent to add, and plans serial dilutions tube by tube.

Open the full dilution calculator over this page to try your own numbers.

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Dilution Calculator

Use C1V1 = C2V2 to plan dilutions and serial dilutions

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How to use the dilution calculator

1. Under Calculate, choose C1V1 = C2V2 for a single dilution or Serial dilution for a series of tubes.
2. For a single dilution, choose what to find: C₂, V₁, V₂ or C₁. Each one opens with an example already filled in.
3. Type the three values you know. Each box has its own unit: volumes in mL, cm³, L, dm³ or m³, and concentrations in mol/L, mol/dm³, mmol/dm³, µmol/dm³ or mol/m³. Mix units if you need to, and the calculator converts.
4. Read the answer, then the extra results: the solvent to add and the dilution factor.
5. The steps show the rearranged formula with your numbers, and the graph shows the concentration falling as the final volume grows.
6. To practise, press Random problem in the Practice section.

mol/L is the same as molarity (M), and mol/dm³ is the UK name for the same unit. A cm³ is a mL and a dm³ is a litre. The calculator refuses a final concentration above the stock, or a final volume below the stock volume, because adding solvent can't do either.

The dilution formula: C1V1 = C2V2

Concentration times volume is the amount of solute (in moles, for molar concentrations). Dilution doesn't change that amount, so the two products are equal. US textbooks often write it as M₁V₁ = M₂V₂, with M for molarity. Rearranged for each letter:

Any units work as long as both sides match: both volumes in mL, or both in L. The OpenStax Chemistry textbook makes the same point. Two more results follow:

• Solvent to add = V₂ − V₁. This is the water you pour in, assuming the volumes simply add, which is close enough for dilute solutions.
• Dilution factor = C₁ ÷ C₂ = V₂ ÷ V₁: how many times weaker the new solution is.

Worked examples you can check in the calculator

Example 1: how much stock do I need?

You need 250 mL of 0.40 M solution and have a 2.0 M stock. Choose V₁, with C₁ = 2.0 mol/L, C₂ = 0.40 mol/L and V₂ = 250 mL:

Measure out 50 mL of stock and make it up to 250 mL, which means adding 200 mL of water. The dilution factor is 2.0 ÷ 0.40 = 5. These are the numbers the V₁ mode opens on, written there as 0.25 dm³ and 0.05 dm³.

Example 2: the new concentration

10 mL of 6.0 M hydrochloric acid is diluted to 150 mL. What is its new concentration?

The diluted acid is 0.40 M, fifteen times weaker. When diluting a strong acid in the lab, add the acid to the water, not water to the acid.

Example 3: what volume to make it up to

You have 25 cm³ of 1.5 mol/dm³ sodium chloride solution and want it at 0.30 mol/dm³. Choose V₂:

Make it up to 125 cm³, which means adding 100 cm³ of water.

Example 4: working back to the stock

20 mL of an unknown stock was diluted to 500 mL, and the diluted solution measured 0.080 M. How concentrated was the stock? Choose C₁:

The stock was 2.0 M.

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Serial dilution calculator

A serial dilution repeats the same dilution several times, each tube made from the one before. It reaches very low concentrations accurately without measuring tiny volumes. If each step dilutes by a factor f, after n steps:

Choose Serial dilution and enter the stock concentration, the dilution factor per step and the number of steps. You can also give the volume moved each step, and the calculator works out how much solvent goes in each tube. It lists the concentration of up to eight tubes and plots them on a log scale, where each step is the same drop.

Example 5: a tenfold serial dilution

Start with a 1.0 M stock, dilute tenfold in each of 5 steps, and move 1 mL each time. Each tube holds 1 mL from the tube before plus 9 mL of water.

The total dilution factor is 10⁵ = 100,000, so the last tube is 1.0 × 10⁻⁵ M. These are the numbers the Serial dilution mode opens on.

Common mistakes

• Mixing units. Put both volumes in the same unit, or let the calculator convert them for you.
• Adding V₂ of water. V₂ is the final volume, stock included. The water to add is V₂ − V₁.
• Swapping stock and final values. The stock is always the stronger solution and the smaller volume.
• Reading a ratio as a factor. "1 : 4" can mean 1 part stock to 4 parts water (a dilution factor of 5) or 1 part in 4 total (a factor of 4). Check which your method means.
• Using C₁V₁ = C₂V₂ for a reaction. It only works when the amount of solute stays the same. For a titration with a 1 : 2 ratio, use the mole ratio from the equation instead.

Read next

Chemistry · Moles and quantitiesSerial dilution calculator: tube by tube concentrationsPlan a serial dilution: the concentration of every tube, the total dilution factor and the solvent to add, from the stock and the step factor.