Quick answer: the dilution equation is C₁V₁ = C₂V₂ — stock concentration × stock volume equals final concentration × final volume. To find the stock volume you need: V₁ = C₂V₂ ÷ C₁. To make 100 mL of 1 M from 10 M stock, take 10 mL of stock and top up to 100 mL.
Why it works
Adding solvent to a solution does not change how much solute is in it — you are just spreading the same amount through more liquid. The quantity of solute is concentration × volume, and since that quantity is unchanged, the product C × V before dilution equals C × V after. That single conservation idea is the whole of the formula.
Solve for stock volume: V₁ = (C₂ × V₂) ÷ C₁
Solve for final concentration: C₂ = (C₁ × V₁) ÷ V₂
Worked example — 100 mL of 1 M solution from a 10 M stock:
- Stock volume: V₁ = (1 × 100) ÷ 10 = 10 mL of stock
- Solvent to add: 100 − 10 = 90 mL (or bring to 100 mL in a volumetric flask)
- Dilution factor: 100 ÷ 10 = 10-fold (1:10)
Units and safety
Any concentration unit works — molarity, percent, mg/mL, ppm — provided C₁ and C₂ use the same one. Likewise V₁ and V₂ must share a unit. Concentration and volume units are independent of each other.
One practical rule that the arithmetic will not tell you: when diluting concentrated acids, always add the acid to the water, never water to the acid. The reaction is exothermic, and adding water to concentrated acid can cause it to boil and spit. Work out the volumes here, then add carefully in the right order.
Frequently asked questions
What is the dilution formula?
C1V1 = C2V2, where C1 and V1 are the concentration and volume of the stock, and C2 and V2 are the concentration and volume of the diluted solution. It expresses that the amount of solute does not change when you add solvent — you are only spreading it through a larger volume.
How do I dilute a stock solution?
To make a target volume at a target concentration, the stock volume needed is V1 = C2 × V2 ÷ C1. To make 100 mL of 1 M solution from a 10 M stock, you need 1 × 100 ÷ 10 = 10 mL of stock, then add solvent to 100 mL total. Always add stock to solvent, not the other way around, for concentrated acids.
How much solvent do I add?
Solvent to add = final volume − stock volume. In the example above, 100 mL final minus 10 mL of stock means adding 90 mL of solvent. In practice you bring the solution up to the final volume in a volumetric flask rather than measuring the solvent separately, which is more accurate.
Do the concentration units have to match?
The two concentrations must be in the same unit as each other, and the two volumes must be in the same unit as each other — but concentration and volume units are independent. Molarity, percentage, mg/mL or any concentration unit works, as long as C1 and C2 use the same one.
What is a dilution factor?
The dilution factor is the ratio of final volume to stock volume, or equivalently the ratio of stock concentration to final concentration. Diluting 10 M to 1 M is a 10-fold or 1:10 dilution. A series of tenfold dilutions is called a serial dilution and is standard for preparing calibration standards.