Most working solutions are made by diluting a concentrated stock. The arithmetic rests on one idea: adding solvent changes the volume, but not the amount of solute.
Where C1V1 = C2V2 comes from
The amount of solute is concentration × volume. Before dilution that is C1V1. After dilution it is C2V2. Nothing is added or removed, so the two are equal:
C1 × V1 = C2 × V2
- C1: stock concentration
- V1: volume of stock to transfer
- C2: final concentration
- V2: final total volume. This is not the volume of diluent:
diluent = V2 − V1
The equation works with any units, as long as both concentrations use the same unit and both volumes use the same unit. It also works for mass concentrations (mg/mL) as well as molar ones.
Worked example
Make 1 mL of 100 µM from a 10 mM stock. Convert the stock to the same unit first: 10 mM = 10 000 µM.
V1 = C2V2 / C1 = (100 µM × 1 mL) ÷ 10 000 µM = 0.01 mL = 10 µL.
Add 10 µL of stock to 990 µL of diluent. This is a 100-fold dilution.
What “1:10” means
Dilution ratios are written two ways, and the difference is a factor of about 10%:
- 1:10 as a dilution factor (usual in biology): 1 part stock in 10 parts total, for example 100 µL stock + 900 µL diluent. The concentration falls 10-fold.
- 1:10 as a mixing ratio (common in chemistry recipes): 1 part stock + 10 parts diluent, 11 parts total. The concentration falls 11-fold.
The calculators on this site always use the first meaning: 1 : F means a total dilution of F-fold. When following a protocol, check which convention its author used.
Serial dilutions
A serial dilution repeats the same step, each tube made from the one before. With a factor F per step:
Ck = C0 / Fk
For a final volume V in each tube, transfer V / F from the previous tube into V − V / F of diluent. A 2-fold series with 1 mL tubes moves 500 µL into 500 µL each time.
Serial dilutions reach very low concentrations without pipetting tiny volumes, but errors compound. A 2% pipetting error repeated over ten steps can add up to about 20%. When accuracy at the low end matters, make each tube directly from the stock or from an intermediate (see planning a concentration series).
Common mistakes
- Mixed units. Putting 10 mM and 100 µM into the equation without converting gives an answer 1000-fold out.
- Adding V2 of diluent. Adding 1 mL of diluent to 10 µL of stock gives 1.01 mL and a slightly lower concentration.
- Volumes below the pipette’s range. Volumes under about 2 µL are hard to deliver accurately. Use an intermediate dilution instead.
- Not mixing between steps. In a serial dilution, mix each tube thoroughly before taking the next transfer.