Dilution Math: How C1V1 = C2V2 Actually Works

Every lab bench ritual — the volumetric flask, the pipette, the careful topping-up to the mark — is secretly one equation: C1V1 = C2V2. Four numbers, one rule, and the rule is not really about concentration at all. It is a conservation statement: dilution adds solvent, never solute, so whatever you pipette is all there ever is. That is why any three of the four numbers nail down the fourth.

The amount of solute never changes

Concentration times volume equals amount. Dissolve 0.5 moles of salt into water and the solution holds 0.5 moles no matter how dilute it becomes; pour it into a bigger beaker and it still holds 0.5 moles. In symbols:

amount = C × V
before = after
C1 × V1 = C2 × V2

Read the left side as "what you take from the stock bottle" and the right side as "what ends up in the flask". The act of dilution — adding solvent — changes V and spreads the same amount thinner, which is exactly what "concentration" means.

This is also why the equation is so forgiving of units. If C1 and C2 are both molar, fine; if both are mg/mL, equally fine. The units cancel as long as each pair speaks the same language. Mixing units across a pair (a molar stock against a milligram-per-millilitre target) is the single most common way this calculation goes wrong in practice.

Solving for any missing value

Algebra does the rest. To find the volume of stock you need, isolate V1:

V1 = (C2 × V2) / C1
C1 = (C2 × V2) / V1
C2 = (C1 × V1) / V2
V2 = (C1 × V1) / C2

A concrete example: you need 500 mL of 0.5 M working solution and the stock bottle says 10 M. Then V1 = (0.5 × 500)/10 = 25 mL. Measure 25 mL of stock into a volumetric flask and top up to the 500 mL mark. The same three numbers also tell you the strength ratio, called the dilution factor: 10/0.5 = 20, so this is a 20× dilution — one part stock in twenty parts of final solution.

The dilution factor is worth internalising because it is how biologists actually talk. "Make a 1:100 dilution" means a factor of 100, done in one step or as ten-fold steps in series. Two sequential 10× dilutions multiply to 100×; a six-step ten-fold series spans a million-fold range while every pipetting step stays in the accurate middle of the pipette's range. Big single-step dilutions are surprisingly inaccurate because a tiny pipetted volume carries a large relative error.

"Dilute to" versus "add" — the mark matters

Recipes say "dilute to 500 mL", not "add 500 mL of water", and the distinction is chemistry rather than pedantry. Dissolved solute occupies volume, so mixing 25 mL of stock with exactly 475 mL of water does not guarantee a final volume of 500 mL — the error is small for dilute solutions but grows with concentration. Volumetric flasks exist precisely so you can fill to a calibrated mark and let the final volume be what it should be. The spirit of C1V1 = C2V2 is that V2 is the final volume you end up with, not the solvent you poured.

Where the equation quietly fails

C1V1 = C2V2 assumes the solute stays put and nothing reacts. It holds beautifully for salts, sugars, buffers and most diluted stocks. It drifts when the "solute" changes identity or amount during dilution: weak acids shift their dissociation as they are diluted, volatile solvents evaporate, and concentrated sulfuric acid releases heat that changes volumes. None of these break the arithmetic of the solute you are tracking — but they remind you that the equation tracks the amount you put in, not every consequence of the water you add.

Significant figures deserve a closing word. A dilution factor computed as 20 is rarely better than two significant figures, because pipettes, flasks and concentrations all carry their own uncertainty. Reporting "25 mL" is honest; "24.99972 mL" is not.The Sig-Fig Calculator helps keep the decimals honest, and the molar mass guide covers how to get from grams on the balance to the molar stock you start from.

Try it yourself

The Dilution Calculator solves C1V1 = C2V2 in any direction: enter any three of stock concentration, stock volume, target concentration and final volume, leave the unknown blank, and it returns the missing value with the dilution factor and a recipe sentence. To work out how much solid to weigh for a stock in the first place, the Molar Mass Calculator converts formulas like CuSO4·5H2O into grams per mole, and the Ideal Gas Law Calculator applies the same "any three, solve the fourth" pattern to PV = nRT.