Dilution Calculator (C₁V₁ = C₂V₂)
Work out how much stock to take, how much solvent to add, or what concentration you will end up at when diluting a solution. Solves any of the four variables in C₁V₁ = C₂V₂.
Stock volume to take (V₁)
0.0208 L (20.83 mL)
- Stock concentration C₁
- 6 mol/L
- Stock volume V₁
- 0.0208 L (20.83 mL)
- Final concentration C₂
- 0.5 mol/L
- Final volume V₂
- 0.25 L (250 mL)
- Solvent to add
- 0.2292 L (229.17 mL)
- Dilution factor (V₂/V₁)
- 12×
Dilution conserves moles: n = C·V is the same before and after, so C₁V₁ = C₂V₂. Rearrange for whichever quantity you are missing. Solvent added is V₂ − V₁; the dilution factor V₂/V₁ equals C₁/C₂.
How to use this calculator
Pick what you are solving for at the top — most lab work asks "how much stock do I take?", which is V₁. Fill in the other three quantities. Concentrations are in mol/L (M); volumes are in litres. If you have volumes in millilitres, divide by 1000 first — 250 mL is 0.250 L, 20 mL is 0.020 L. The calculator returns the missing quantity as the primary result and the full breakdown of all four variables, plus the volume of solvent you need to add to reach V₂ and the dilution factor V₂/V₁. Note the calculator rejects impossible dilutions: you cannot make C₂ bigger than C₁ or V₂ smaller than V₁ by adding solvent. If you need to concentrate a solution, evaporate solvent or start with a stronger stock instead.
How the calculation works
Dilution is the process of adding solvent to a stock solution to lower the concentration. The amount of solute, in moles, does not change — only the volume of solvent it is spread through. Moles are concentration times volume (n = C·V), so n₁ = n₂ gives C₁V₁ = C₂V₂. That is the entire formula. Rearranging: V₁ = C₂V₂/C₁ (volume of stock to take), C₂ = C₁V₁/V₂ (concentration after diluting), V₂ = C₁V₁/C₂ (final volume needed for a target concentration), and C₁ = C₂V₂/V₁ (the back-calculation, used when checking a stock you suspect has drifted). The dilution factor DF = V₂/V₁ equals C₁/C₂ and is a useful shorthand — "a 10× dilution" means V₂ = 10·V₁ and C₂ = C₁/10. In practice you draw V₁ from the stock with a pipette, transfer it to a volumetric flask of size V₂, and top up with solvent to the calibration mark. The flask is calibrated to the final volume, so the solvent added is V₂ − V₁ rather than exactly V₂ — that small distinction is what volumetric glassware is built around.
Worked example
How much 3.00 M HCl stock do you need to make 250 mL of 0.500 M HCl? Set the solver to V₁, enter C₁ = 3.00, C₂ = 0.500, V₂ = 0.250 L. The calculator returns V₁ = (0.500 × 0.250) / 3.00 = 0.04167 L, i.e. 41.67 mL of stock. Solvent to add is V₂ − V₁ = 250 − 41.67 = 208.33 mL, and the dilution factor is 250/41.67 = 6.00× (matching C₁/C₂ = 3.00/0.500 = 6.00). This is exactly the worked example in Harris, Quantitative Chemical Analysis 9e, problem 1-19. In a real prep you would pipette 41.67 mL of the concentrated HCl into a 250 mL volumetric flask, add distilled water to the calibration mark, and stopper-and-invert to mix.
Frequently asked questions
What does C₁V₁ = C₂V₂ actually mean?
It is a statement that moles of solute are conserved during dilution. Concentration times volume gives moles (n = C·V), so if you take V₁ of a stock at C₁ you have n = C₁·V₁ moles in that aliquot. Topping up to V₂ with pure solvent does not change how much solute is there — it only changes the volume it is distributed in. So n is still C₁V₁, and the new concentration is C₂ = n/V₂ = C₁V₁/V₂. Rearrange to taste. The equation works for any concentration unit that scales linearly with volume — molarity (mol/L), mass/volume (g/L), and so on. It fails for molality (mol/kg) because that is per-mass not per-volume, and for mole fraction or percent-by-mass for the same reason.
How do I make a 1:10 dilution?
A 1:10 dilution means one part stock to nine parts solvent, giving a dilution factor of 10× (V₂/V₁ = 10) and C₂ = C₁/10. To make 10 mL of 1:10 diluted solution, you take 1 mL of stock and add 9 mL of solvent — final volume is 10 mL. To make 100 mL, take 10 mL of stock and add 90 mL of solvent. Note "1:10" is sometimes written ambiguously: in chemistry it usually means one part stock in ten parts total (V₁/V₂ = 1/10), but some biology and clinical contexts use "1:10" to mean one part stock to ten parts solvent (V₁/(V₁+solvent) = 1/11). Check the convention in your protocol; this calculator works in unambiguous V₁ and V₂.
Does the equation work for serial dilutions?
Yes, applied step by step. A serial dilution is just a chain of single dilutions where each step's output is the next step's input. For a 1:10 serial across four tubes starting from 1 M stock: tube 1 is 0.1 M (DF 10), tube 2 is 0.01 M, tube 3 is 0.001 M, tube 4 is 0.0001 M. The cumulative dilution factor multiplies — four 10× steps give 10⁴ = 10 000× overall. Serial dilutions are how you reach very low concentrations (nM, pM ranges) accurately, because each single step uses easy-to-pipette volumes. Pipetting 1 µL out of 1 mL directly is hard to do reproducibly; ten 10× steps each pipetting 100 µL into 900 µL is straightforward.
What if my volumes are in millilitres or microlitres?
The formula does not care about the volume unit as long as both sides use the same one — C₁V₁ = C₂V₂ holds whether you express volumes in L, mL, µL, gallons or teaspoons. This calculator uses litres on the form to keep the moles-conserved math obvious (since molarity is mol/L). If your protocol gives volumes in millilitres, divide by 1000 before entering: 250 mL becomes 0.250 L, 20 µL becomes 0.000020 L (or 2.0 × 10⁻⁵). The results panel always shows volumes in both L and mL so you can read off whichever you need at the bench.
Why does the calculator refuse my values?
A dilution must obey two physical constraints: the final concentration cannot be higher than the stock (C₂ ≤ C₁) and the final volume cannot be smaller than the stock volume drawn (V₂ ≥ V₁). If you enter C₂ > C₁ or V₂ < V₁, you have described concentrating a solution rather than diluting it, and the equation no longer fits — you would need to evaporate solvent or add more solute, not just add solvent. The calculator returns an error and asks you to revise. The other common failure mode is entering zero or negative values for one of the three required inputs; all four quantities are strictly positive in a real dilution.
Does this work for percent-by-volume (% v/v) or mass/volume (g/L) concentrations?
Yes for any volume-scaled unit. The dilution equation C₁V₁ = C₂V₂ applies whenever concentration is mass-or-moles per volume, including % v/v (volume fraction × 100), % w/v (g solute per 100 mL solution × 100), ppm by mass/volume, and mg/mL. The form on this page labels the C inputs as mol/L because molarity is the dominant lab unit, but you can use it with any consistent C unit — enter 6 M as 6, or 6 g/L as 6, or 6 % w/v as 6, and the answer comes out in the same units. The one trap is mass/mass percent (% w/w) and molality (mol/kg solvent), which are mass-scaled rather than volume-scaled — for those, neither side of the equation simplifies to "moles" and you cannot use C₁V₁ = C₂V₂.