Dilution Calculator (C1V1 = C2V2)

Background.

A dilution calculator solves the single most-used equation in wet-lab chemistry — the dilution law, C1 × V1 = C2 × V2 — for whichever of the four quantities you do not already know: the concentration of your stock solution (C1), the volume of stock you take (V1), the concentration you want after dilution (C2), or the total final volume you need (V2). Pick a 'solve for' mode, type in the other three values, and the calculator returns the missing one along with two practical extras: the volume of diluent (water, buffer, saline, solvent) you need to add, and the dilution factor, the single number lab technicians write on tube labels as a shorthand for how many times a stock was diluted.

The dilution law itself is nothing more than a statement of conservation. When you take V1 litres from a stock solution at concentration C1, you remove a fixed amount of solute — n = C1 × V1 units of it, whether those units are moles, milligrams, or colony-forming units. Adding solvent to bring the total volume up to V2 does not create or destroy any of that solute; it only spreads the same amount through a larger volume. The new concentration is therefore C2 = n / V2 = (C1 × V1) / V2, which rearranges into the familiar symmetric form C1V1 = C2V2. Because the law is a statement about a conserved quantity divided by a volume, it holds for any consistent pair of concentration and volume units — molarity in mol/L, percent solutions in % w/v, drug concentrations in mg/mL, environmental contaminant levels in parts per million, and bacterial or viral titres in colony-forming units or plaque-forming units per millilitre. This calculator is deliberately built to be unit-agnostic for exactly that reason: it is as useful to a microbiology student running a ten-fold serial dilution series as it is to an analytical chemist preparing a 0.01 M working standard from a 1 M stock.

Dilution calculations show up constantly outside the research bench, too. A pharmacist compounding a paediatric liquid dose from a concentrated adult formulation is solving a dilution problem. A pool-maintenance technician diluting concentrated chlorine stock to a target parts-per-million reading is solving a dilution problem. A cleaning-products manufacturer specifying '1 part concentrate to 40 parts water' on a bottle label has already solved the dilution problem for the end user and expressed the answer as a dilution factor. Home brewers diluting a hop extract, hydroponics growers diluting a nutrient concentrate to a target electrical-conductivity reading, and photography darkroom technicians diluting stock developer to a working strength are all running the identical C1V1 = C2V2 calculation, just with different instruments measuring the concentration axis.

Most general-purpose molarity calculators — including the molarity calculator elsewhere on this site — bundle a single dilution submode into a broader tool that also solves M = n/V, and that submode typically only answers one direction of the question: given a stock concentration, a stock volume, and a target concentration, what is the final volume? That is a common question, but it is not the only one. Equally common in practice is the inverse: given a stock concentration, a target concentration, and a target final volume, how much stock do you actually need to pipette? This calculator treats all four directions as equally first-class, because in a real protocol you often know the target concentration and target volume (dictated by the assay or recipe) and need to solve backward for how much stock to measure out — precisely the calculation a fixed single-direction tool cannot do without manual algebra.

Below the calculator you will find the full derivation of the dilution law from conservation of solute, a worked example preparing a 0.500 mol/L sodium hydroxide working solution from a 10.00 mol/L stock, guidance on choosing which of the four modes matches the data you actually have, an explanation of dilution factor and how serial dilutions chain multiple dilution factors multiplicatively, and safety notes for diluting concentrated acids and bases. The solver itself runs on Decimal.js arbitrary-precision arithmetic and is registered at dilution.calculate in the Quanta engine, with every mode validated against division-by-zero and non-positive concentration or volume inputs.

What is dilution calculator?

Dilution is the process of reducing the concentration of a solute in a solution, almost always by adding more solvent while holding the amount of solute fixed. The dilution law, C1V1 = C2V2, is the mathematical expression of that process: C1 and V1 are the concentration and volume of the solution before dilution (the 'stock' or 'initial' state), and C2 and V2 are the concentration and total volume after dilution (the 'working' or 'final' state). The law follows directly from the definition of concentration as amount of solute per unit volume: amount of solute n = C × V is fixed by the act of taking an aliquot from the stock, and diluting changes only the volume it occupies, not the amount itself, so C1V1 (the amount taken from stock) must equal C2V2 (the same amount now expressed at the new, larger volume). Two closely related quantities appear throughout dilution work. The volume of diluent to add is simply V2 − V1 — the amount of solvent you pour in after measuring out your V1 aliquot of stock. The dilution factor, commonly written DF, is V2 / V1 (equivalently C1 / C2): a dilution factor of 10 means the final volume is ten times the aliquot volume, so the concentration has dropped to one-tenth of the stock value. Dilution factors are typically reported as a ratio, '1:10' or '1-in-10', meaning one part stock combined with enough diluent to reach ten total parts — not one part stock to ten parts diluent, a common point of confusion. Serial dilutions chain this idea: taking a 1-in-10 dilution and diluting it 1-in-10 again produces an overall 1-in-100 dilution, because dilution factors multiply along the chain (10 × 10 = 100), which is exactly how microbiologists prepare the wide concentration range needed to get a countable number of colonies on an agar plate from an unknown, highly concentrated bacterial culture.

How to use this calculator.

1. 01Pick the one quantity you don't know from the 'Solve for' menu: initial (stock) volume, final volume, final concentration, or initial (stock) concentration.
2. 02Fill in the other three fields. The calculator ignores whichever field you are solving for — you can leave it at its default.
3. 03Use any consistent unit for the two concentration fields (mol/L, %, mg/mL, ppm, CFU/mL) and any consistent unit for the two volume fields — the calculator's math is unit-agnostic, but C1 and C2 must share a unit, and V1 and V2 must share a unit (litres, by default, on this page; convert mL to L by dividing by 1000).
4. 04Read the primary result (the value you solved for) plus the full echoed quartet of C1, V1, C2, V2 underneath it.
5. 05Check 'Diluent to add' (V2 − V1) for the practical answer to 'how much water do I pour in?' and 'Dilution factor' (V2 / V1) for the lab-notebook shorthand.
6. 06For serial dilutions, chain the calculator: use the final volume of one dilution step as the starting point for planning the next, and multiply the individual dilution factors to get the overall dilution across the whole series.
7. 07Safety: when diluting concentrated acids or bases, always add the concentrated reagent to the larger volume of water, never the reverse — the dissolution/mixing is exothermic and can spatter if water is added to a concentrated acid.

The formula.

The dilution law is a direct consequence of two definitions: concentration is amount of solute per unit volume, and diluting a solution does not add or remove solute. Start from C = n / V, so the amount of solute in a stock aliquot is n = C1 × V1. When that aliquot is diluted to a total volume V2, the amount of solute is unchanged — it is still n — but it is now expressed in a larger volume, so the new concentration is C2 = n / V2. Substituting n = C1V1 into C2 = n/V2 gives C2 = C1V1/V2, which rearranges into the symmetric form C1V1 = C2V2. Every one of the calculator's four modes is this single identity solved for a different unknown: V1 = C2V2/C1, V2 = C1V1/C2, C2 = C1V1/V2, and C1 = C2V2/V1.

Two secondary quantities follow immediately. The volume of diluent to add is V2 − V1, because the total final volume is made up of the original stock aliquot plus whatever solvent you pour in afterward. The dilution factor is V2 / V1 — equivalently, by the dilution law, C1 / C2 — a single dimensionless number describing how many times more dilute the final solution is relative to the stock. Because the dilution law is homogeneous in the concentration units (C1 and C2 always appear as a ratio or together in a product with matching volumes), the equation is exactly as valid for mass-percent concentrations, ppm, or colony-forming-unit titres as it is for molarity — nothing in the derivation assumes moles or litres specifically, only that whatever unit you choose for concentration is used consistently on both sides and whatever unit you choose for volume is used consistently on both sides.

Serial dilutions extend the same logic across multiple steps. If a first dilution has factor DF1 = V2,1/V1,1 and the resulting solution is itself diluted again with factor DF2 = V2,2/V1,2, the overall dilution factor relative to the original stock is DF1 × DF2, because each step divides the concentration by its own factor in sequence: C_final = C_original / (DF1 × DF2). This multiplicative chaining is why microbiologists can span nine or ten orders of magnitude in concentration with only nine or ten pipetting steps, each an easy-to-execute 1-in-10 dilution.

A worked example.

Frequently asked questions.

References& sources.

1. [1]IUPAC Gold Book, 'amount-of-substance concentration, c' — the formal IUPAC definition of concentration, c = n/V, underlying the dilution law's conservation-of-solute derivation. International Union of Pure and Applied Chemistry.
2. [2]Harvey, D.T. Analytical Chemistry 2.1, §2.5 'Preparing Solutions' — derives the dilution law C_o × V_o = C_d × V_d from conservation of the amount of solute and gives worked dilution examples. LibreTexts / DePauw University.
3. [3]NIST Chemistry WebBook — searchable reference database of molar masses, concentrations, and physical property data used to cross-check dilution calculations for real compounds. National Institute of Standards and Technology, Standard Reference Database 69.
4. [4]Skoog, D.A., West, D.M., Holler, F.J. & Crouch, S.R. (2013). Fundamentals of Analytical Chemistry, 9th ed. Cengage. Chapter 4 covers concentration units, dilution calculations, and standard laboratory dilution practice. ISBN 978-0495558286.
5. [5]Atkins, P.W. & de Paula, J. (2014). Atkins' Physical Chemistry, 10th ed. Oxford University Press. Foundations section on amount of substance and molar concentration, the basis for the conservation argument behind C1V1 = C2V2.