Dilution calculator
Enter three of C₁, V₁, C₂, V₂, get the fourth.
How do I use C1V1 = C2V2?
The amount of solute stays the same when you add water, so starting concentration × starting volume = final concentration × final volume. To make 500 mL of 1 M from a 12 M stock: V₁ = 1 × 500 ÷ 12 = 41.7 mL.
How much water do I add?
The final volume minus the stock volume: 500 − 41.7 = 458.3 mL, added up to the 500 mL mark.
Can I use percent or mg/mL?
Yes — any concentration unit works as long as C₁ and C₂ use the same one, and V₁ and V₂ the same volume unit.
What is a dilution factor?
The final volume ÷ the starting volume — 12 in the example above, a 1:12 dilution.
Is it safe to add water to acid?
For strong acids, add the acid slowly to the water, not water to the acid, and follow your lab's safety rules.
Why does the answer say “add water up to” the final volume?
Because V₂ in the equation is the volume of the finished solution, not the volume of water you add. Put the measured stock volume V₁ in a flask, then add water until the total reaches V₂. In the 12 M to 1 M example that means 41.7 mL of stock and about 458.3 mL of water; pouring 500 mL of water onto 41.7 mL of stock would give a solution weaker than intended.
Can I use C₁V₁ = C₂V₂ to mix two solutions?
No. The equation assumes the moles of solute stay the same, which is true when you only add solvent (OpenStax Chemistry 2e, section 3.3). If you combine two solutions that both contain the solute, add their moles instead of using one equation for both.
Does it work for concentrating a solution?
The algebra is the same, but the calculator tells you when C₂ is larger than C₁, because adding solvent cannot do that. You would have to evaporate solvent or add more solute.
Where the equation comes from
Adding solvent to a stock solution lowers its concentration but leaves the moles of solute unchanged. Since moles = concentration × volume, the amount before (n₁ = C₁V₁) equals the amount after (n₂ = C₂V₂), which gives C₁V₁ = C₂V₂ (OpenStax Chemistry 2e, section 3.3). The units only have to cancel, so molarity, percent or mg/mL all work, as long as C₁ and C₂ share one unit and V₁ and V₂ share another. This calculator labels the volumes in milliliters and leaves the concentration unit to you. The dilution factor is the same ratio either way: C₁ ÷ C₂ = V₂ ÷ V₁.
Worked examples
Finding a concentration (OpenStax Example 3.19): 0.850 L of 5.00 M copper(II) nitrate is diluted to 1.80 L. C₂ = 5.00 × 0.850 ÷ 1.80 = 2.36 M. Finding a volume (Example 3.20): how much 0.45 M hydrobromic acid must be diluted to 0.12 M, starting from 11 mL? V₂ = 0.45 × 11 ÷ 0.12 = 41 mL, so add about 30 mL of water. Choose “Final volume V₂” and enter 0.45, 11 and 0.12 to see it.
Common mistakes
Mixing volume units is the usual one: V₁ in milliliters with V₂ in liters is off by a factor of 1000, so convert both to the same unit first. Treating V₂ as the water to add, rather than the total, comes second. Third is using the equation after the solute amount has changed, for example when two solutions are mixed or when something precipitates or evaporates. Last, C₁ and C₂ must be the same kind of concentration: a stock labeled in mass percent cannot be paired with a target in molarity until you convert it (see the percent to molarity calculator).
Sources: OpenStax Chemistry 2e, section 3.3 “Molarity” (dilution equation and Examples 3.19 to 3.21). Related: molarity calculator, serial dilution calculator, percent to molarity calculator and ppm calculator.