How to Calculate a Dilution with C1V1 = C2V2

What the equation actually conserves

C1V1 = C2V2 is a statement about the solute, not about the liquid. Concentration times volume is an amount of substance, so the left side is the moles you pipette out of the stock bottle and the right side is the moles sitting in the flask when you have finished. Diluting adds solvent and nothing else, so that amount cannot change, and the equation is simply the two ways of writing it set equal. Reading it that way makes the direction obvious: if the concentration falls by a factor, the volume must rise by the same factor, because their product is fixed.

That reading also tells you where the equation stops working. Because both sides are a concentration multiplied by a volume, any unit that is already an amount per unit of volume cancels cleanly: mol/L, g/L, percent weight per volume, ppm expressed per litre. Units built on mass do not. Molality is moles per kilogram of solvent, and percent weight per weight is grams per gram of solution, so multiplying either by a volume gives you nothing meaningful. For those you need the mass analogue, m1w1 = m2w2 with mass fractions and total masses, and you weigh rather than pipette. The tell is in the name of the unit: if the denominator is not a volume, the volume form of the equation does not apply.

The subtraction: add V2 minus V1, and make up to the mark

V2 is the total volume of the finished solution, stock included. It is not the volume of solvent, and treating it as such is the error that quietly ruins a preparation. Take 25 mL of stock and add 500 mL of water and you have 525 mL, not 500, so the concentration is 25/525 of the stock rather than 25/500 — 0.0952 mol/L instead of 0.1000, about 5 percent low. At a dilution factor of 20 the mistake is small enough to survive a casual glance and large enough to matter; at a factor of 2 it is a third of the way off. The safe phrasing to write on the protocol is bring up to V2, never add V2.

There is a second reason to write it that way. Volumes are not strictly additive: mixing ethanol and water produces less total volume than the sum of the two, and concentrated acids behave similarly, so V2 − V1 is a good working figure rather than an exact one. The rigorous procedure is therefore to measure V1 accurately, add most of the solvent, mix, and only then top up to the graduation of a volumetric flask. For a dilute aqueous dilution the difference between the two approaches is far below the uncertainty of the pipette; for alcohols and concentrated acids it is not. One safety note belongs here too, because the order of addition is not free: when diluting a concentrated acid, add the acid to the water and never the reverse, since the heat released is dumped into a large volume rather than a small one.

Dilution factors, and the two conventions behind 1:10

The dilution factor is the ratio V2 / V1, which equals C1 / C2. It is a single number that captures the whole operation, and it is worth computing because it gives you an independent check: if the factor you wanted was 20 and your two volumes are not in a ratio of 20, something is wrong before you touch a pipette. The notation, unfortunately, is ambiguous. Written here, 1:10 means one part stock brought up to ten parts total, a factor of 10. Elsewhere the same string means one part stock added to ten parts solvent, eleven parts total and a factor of 11. Both conventions are in current use, and neither is wrong — they are simply different. The way out is to write the factor as a number, or to state the two volumes, and to read someone else's 1:10 as a question rather than an instruction.

Serial dilutions follow from the same ratio, with one rule: the factors multiply, they do not add. Three successive steps of 1:10 give an overall factor of 10 × 10 × 10 = 1000, not 30, which is exactly why the technique exists. Reaching a factor of 1000 in one step would mean pipetting 0.5 mL into 500 mL, and the relative error on that half millilitre dominates the whole result; three steps of 5 mL into 50 mL each carry a far smaller relative error, though they do accumulate across the chain. Mix thoroughly between steps and use a fresh pipette tip for each one, because carryover on a tip is a systematic error that pushes every downstream concentration in the same direction.

Frequently asked questions

How much solvent do I actually add?

V2 − V1, not V2. If the calculation says take 25 mL of stock and end up with 500 mL, the solvent is 475 mL. In practice do not measure that 475 mL separately: put the 25 mL of stock into the flask and top the contents up to the 500 mL graduation, which absorbs both the arithmetic and the fact that volumes do not add exactly. The distinction is largest when the dilution factor is small — going from 1.0 mol/L to 0.5 mol/L, V1 is half of V2, so adding V2 instead of V2 − V1 would leave you a third under target rather than on it.

Does 1:10 mean one part in ten, or one part to ten?

Both readings circulate, which is why the notation on its own is not a reliable instruction. This article and the calculator use one part in ten: one volume of stock brought up to ten volumes total, a dilution factor of 10. The other convention reads it as one volume of stock plus ten volumes of solvent, eleven total, a factor of 11 — a 10 percent difference in the final concentration. Analytical and clinical protocols usually mean the first; some formulation and cleaning-product literature means the second. When you write a protocol, give the factor as a number or spell out both volumes, and when you read one, check whether the totals add up before you trust the colon.

What if the calculator returns a final volume smaller than the stock volume?

It means you asked to concentrate, not to dilute, and no amount of solvent will get you there. C1V1 = C2V2 is symmetric and will happily return V2 < V1 whenever the target C2 is above the stock C1, but the operation it describes then is removing solvent — evaporation, or a different, stronger stock. Check the two concentrations first: C2 must be lower than C1 for a dilution to exist at all, and the ratio C1 / C2 tells you immediately how much larger the final volume has to be. If your target really is more concentrated than anything you have, the answer is a new stock, not a new calculation.