C1V1 = C2V2 Dilution Calculator
C1V1 = C2V2 Dilution Calculator
Work out how much stock solution to take for a given final concentration and volume — and, just as important, how much diluent to add to it.
Stock volume needed
C1V1 = C2V2 → stock volumeA 100 mmol/L stock, 50 mL of a 10 mmol/L working solution required
Formula
V₁ = (C₂ × V₂) ÷ C₁
diluent = V₂ − V₁
- C₁, C₂
- stock and final concentration — these must be in the SAME unit; the calculation does not convert between mol/L, mg/mL and per cent
- V₂
- final volume, in any unit you like; V₁ comes back in that same unit
- V₁
- the volume of stock to take — not the volume to add to the tube
- V₂ − V₁
- the volume of diluent, which is what you actually pipette alongside the stock
Worked example
A 100 mmol/L stock, 50 mL of a 10 mmol/L working solution required
V₁ = (10 × 50) ÷ 100 = 500 ÷ 100 = 5.000 mL of stock
Diluent = 50 − 5 = 45.000 mL
Take 5 mL of stock and make up to 50 mL — do not add 50 mL of diluent to it
Worked dilutions, with the diluent volume spelled out
| Stock (C₁) | Final (C₂) | Final volume (V₂) | Stock to take (V₁) | Diluent to add |
|---|---|---|---|---|
| 100 mmol/L | 10 mmol/L | 50 mL | 5.000 mL | 45.000 mL |
| 1 mol/L | 0.1 mol/L | 100 mL | 10.000 mL | 90.000 mL |
| 10 mg/mL | 0.25 mg/mL | 20 mL | 0.500 mL | 19.500 mL |
| 5,000 U/mL | 50 U/mL | 10 mL | 0.100 mL | 9.900 mL |
How far to go in a single step
| Single-step dilution | Verdict |
|---|---|
| Up to 1 in 20 | Comfortable. Both volumes are usually within the reliable range of a single pipette. |
| 1 in 20 to 1 in 100 | Acceptable if the small volume is still well inside the pipette’s calibrated range — not at the very bottom of it. |
| > 1 in 100 | Poor practice. The stock volume becomes small enough that pipetting error dominates. Use a serial dilution instead. |
Getting the diluent volume right
C₁V₁ = C₂V₂ is a statement of mass conservation: the amount of solute you take out of the stock is the amount that ends up in the working solution, so concentration multiplied by volume is the same on both sides. Two unit rules follow. C₁ and C₂ must be expressed in the same concentration unit, because the calculation divides one by the other and a mismatch between milligrams per millilitre and micrograms per millilitre passes through silently as a thousandfold error. V₁ comes back in whatever unit you used for V₂, so if the final volume was in millilitres, so is the answer.
The step people get wrong is not V₁ but the diluent. The volume of diluent is V₂ − V₁, not V₂: taking 5 mL of stock and adding 50 mL of buffer gives 55 mL of a solution that is about 9 mmol/L, not the 10 mmol/L intended. Write the instruction as make up to rather than add, and prepare the solution that way too — stock into the flask first, then diluent to the mark — so the arithmetic and the technique agree.
The relationship assumes volumes are additive, which is a good approximation for dilute aqueous solutions and a poor one for concentrated alcohols and some concentrated acids, where mixing produces a measurable volume contraction. Ethanol and water are the classic demonstration: the mixture occupies less space than the sum of the parts. Making up to a mark in a volumetric flask sidesteps the problem entirely, because the final volume is then measured rather than assumed.
There is also a practical ceiling on how far a single step should go. Beyond roughly 1 in 100, the stock volume becomes small enough that the relative error of the pipette dominates the result, and any bias in that one delivery propagates undiluted into the final concentration. A serial dilution splits the same total dilution across several larger, more accurately pipetted transfers. The other guard against small-volume error is to stay well inside a pipette’s calibrated range rather than at the very bottom of it, where its specified accuracy is worst.
Frequently asked questions
What does C1V1 = C2V2 actually mean?
It states that the amount of solute is unchanged by dilution: the concentration times the volume of the stock you take equals the concentration times the volume of the solution you end up with. Rearranged, V1 = C2V2 ÷ C1.
How much diluent do I add?
V2 − V1, not V2. If you need 5 mL of stock for a 50 mL final volume, add 45 mL of diluent — or better, put the 5 mL into a volumetric flask and make up to the 50 mL mark.
Can C1 and C2 be in different units?
No. The two concentrations must share a unit, because the calculation divides one by the other. Convert first — 250 µg/mL becomes 0.25 mg/mL — or the answer will be wrong by whatever factor separates the units.
What unit does the answer come out in?
The same volume unit you entered for V2. Enter the final volume in millilitres and the stock volume is in millilitres; enter it in litres and so is the answer.
When should I use a serial dilution instead?
When a single step would be more dilute than about 1 in 100. Below that the stock volume is small enough that pipetting error dominates, and splitting the dilution over several larger transfers is more accurate.
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References
- Sambrook J, Green MR. Molecular Cloning: A Laboratory Manual. 4th ed. Cold Spring Harbor Laboratory Press; 2012.
- Rifai N, Horvath AR, Wittwer CT, eds. Tietz Textbook of Clinical Chemistry and Molecular Diagnostics. Elsevier — sections on solution preparation and volumetric technique.