C1V1 = C2V2 is the most used equation in any wet lab. It is also the one people stop thinking about, which is how a 10X buffer ends up at 0.1X.
The idea in one line
When you dilute, the amount of solute does not change. Only the volume does. The amount before (C₁ × V₁) equals the amount after (C₂ × V₂).
C₁ is your stock concentration. V₁ is how much stock you take. C₂ is the concentration you want. V₂ is the final total volume. Three of these you know. The fourth you solve for, and nine times out of ten it is V₁.
Four steps, every time
- Write down C₁, C₂ and V₂ with units.
- Put C₁ and C₂ in the same unit. Put V₂ in the unit you want V₁ in.
- Solve V₁ = (C₂ × V₂) / C₁.
- Diluent volume = V₂ − V₁. Add stock to diluent, not the other way round, when the stock is an acid or a concentrated organic.
Want more options, saved recipes and warnings? Open the full C1V1 dilution calculator in BenchMate.
Worked examples
10X TBE to 1 L of 1X
V₁ = (1X × 1000 mL) / 10X = 100 mL of 10X stock plus 900 mL water. The "X" works as a unit as long as both sides use it.
1 M Tris to 50 mL of 50 mM
Convert first: 50 mM = 0.05 M. V₁ = (0.05 × 50) / 1 = 2.5 mL stock plus 47.5 mL diluent.
10 mM drug stock to 200 µL of 25 µM
10 mM = 10,000 µM. V₁ = (25 × 200) / 10,000 = 0.5 µL. That is below what most pipettes deliver accurately. Do an intermediate dilution first, for example 1:10 to 1 mM, then take 5 µL of that into 195 µL.
Rule of thumb: if V₁ comes out under about 1 µL, or under 1% of a P20's range, do an intermediate dilution. The error in a 0.5 µL pipetting step can be larger than the effect you are measuring.
The unit trap
The equation does not care which units you use. It only cares that they match. Mixing mM on one side and µM on the other is the classic 1000-fold error. So is mg/mL against µg/µL. These last two are actually equal (1 mg/mL = 1 µg/µL), and people convert anyway and get it wrong.
| Unit | Equals |
|---|---|
| 1 M | 1000 mM = 10⁶ µM |
| 1 mM | 1000 µM = 10⁶ nM |
| 1 mg/mL | 1 µg/µL = 1 g/L |
| 1 L | 1000 mL = 10⁶ µL |
When C1V1 does not work
- Mixing two different solutions that both contain the solute. Add the amounts separately: C₁V₁ + C₂V₂ = C₃V₃.
- Solvents that change volume on mixing. Ethanol and water do not add linearly. 50 mL ethanol plus 50 mL water gives a bit less than 100 mL. For exact % v/v, make up to final volume.
- Percent w/w solutions such as concentrated HCl or ammonia. You need density to convert to molarity first.
BenchMate solves C1V1 in either direction, handles the unit conversion, and warns you when V₁ is too small to pipette.
Frequently asked questions
What does C1V1 = C2V2 mean?
C₁ is the stock concentration, V₁ the volume of stock you take, C₂ the final concentration and V₂ the final volume. The amount of solute stays the same, so C₁V₁ equals C₂V₂.
How do I dilute 10X buffer to 1X?
Take one tenth of the final volume as 10X stock and make up the rest with water. For 1 L of 1X, use 100 mL of 10X and 900 mL of water.
Can I use different units for C1 and C2?
No. C₁ and C₂ must be in the same unit, and V₁ comes out in whatever unit you used for V₂.
What if the stock volume is too small to pipette?
Make an intermediate dilution first, for example 1:10 or 1:100, then dilute that to the final concentration.
22 core bench techniques, from molarity to sterilisation. Volume 1 of The Biotechnology Techniques Bible.