MathBench

Dilution Calculator (C₁V₁ = C₂V₂)

Pick the variable you need (usually how much stock to measure), enter the other three with their units, and get the answer plus the dilution recipe in plain words.

Result

—stock volume needed
—dilution factor

How this calculator works

The dilution equation says the amount of dissolved substance doesn't change when you add solvent: concentration times volume is the same before and after. Choose which of the four quantities you need (almost always the stock volume V₁), enter the other three with their units, and the calculator solves the equation and writes out the recipe: how much stock to measure and what final volume to bring it to.

Units convert automatically within a family (M/mM/µM/nM, or %(w/v)/mg/mL/µg/mL, or X-fold), so a molar stock can be diluted to a millimolar target without hand conversion. If the two concentrations are from different families, the tool says so rather than guessing: crossing between molar and mass units requires the compound's molecular weight.

The formula

C₁ × V₁ = C₂ × V₂

V₁ = C₂V₂ ÷ C₁     (stock volume to measure)
Dilution factor = C₁ ÷ C₂

This is conservation of solute: moles (or grams, or "parts") in equals moles out. Any consistent unit pair works, because both sides scale identically. That's also why the equation alone can't convert between unit families.

Worked example

Say you need 500 mL of 50 mM buffer from a 1 M stock:

  1. Match units: 1 M = 1000 mM
  2. V₁ = C₂V₂ ÷ C₁ = (50 × 500) ÷ 1000 = 25 mL
  3. Check the factor: 1000 ÷ 50 = 20× dilution

Recipe: measure 25 mL of the 1 M stock into your vessel, then add water to the 500 mL mark. (Not 25 mL + 500 mL of water. That would give 525 mL and a concentration 5% low.)

Assumptions & tips

  • Bring to volume. Don't add volumes. Transfer the stock first, then fill to the final mark. The difference is small on one dilution and compounding on a serial series.
  • Keep transfers in your pipette's sweet spot. Accuracy suffers below ~10% of a pipette's range. If the answer comes out to 3 µL with a P1000, switch to a smaller pipette or do a serial dilution instead.
  • Acid into water, always. When the "stock" is a concentrated acid, add it to most of the final water, mix, then top to volume. Never pour water into concentrated acid.
  • Label with both concentrations. "50 mM Tris (from 1 M, 2026-08-13)" on the bottle saves the next person, and future you, from re-deriving what's inside.
  • Making the stock itself from powder? That's a molarity problem, not a dilution problem. Use the molarity calculator to get from grams and molecular weight to a stock concentration first.

Frequently asked questions

What does C1V1 = C2V2 assume?

That you are only adding solvent: no reaction, no evaporation, and the amount of dissolved substance stays constant. The moles you start with equal the moles you end with. Only the volume changes. It also treats volumes as additive, which is very slightly untrue for some solvent mixtures (ethanol and water famously contract), but the error is negligible for routine lab dilutions.

Do I add the calculated volume of solvent, or dilute to the final volume?

Dilute to the final volume. The two are not quite the same thing. Proper technique is to measure the stock volume, transfer it, then add solvent up to the final volume mark ("q.s. to volume"), ideally in a volumetric flask. Adding (V2 − V1) of solvent assumes perfectly additive volumes and an accurately delivered stock. Bringing to volume forgives both.

Can I mix units, like a molar stock and a mg/mL target?

Not directly, C1V1 = C2V2 is a ratio, so both concentrations must be in the same family: molar with molar, mass-per-volume with mass-per-volume, fold (X) with fold. Converting between molar and mg/mL requires the molecular weight, which this equation does not know. Do that conversion first with the molarity calculator, then dilute.

How do serial dilutions work?

A serial dilution repeats the same modest dilution step to reach a large total factor: three 1:10 steps give 1:1000. It exists because one giant dilution is inaccurate: pipetting 1 µL into a liter multiplies any pipetting error dramatically, while three 1:10 steps keep every transfer in the accurate range of your pipettes. The total factor is the product of the steps.

Sources

  1. Chemistry 2e, §3.3 — Molarity. OpenStax, Rice University. openstax.orgDerives the dilution equation, given there as M₁V₁ = M₂V₂ and in general form as C₁V₁ = C₂V₂, from the fact that the amount of solute does not change when solvent is added: the single relationship this calculator rearranges for each of the four variables.
  2. Quantities, Units and Symbols in Physical Chemistry (the IUPAC "Green Book"). International Union of Pure and Applied Chemistry / RSC Publishing. iupac.orgThe definitions of amount concentration (mol per unit volume) and mass concentration (mass per unit volume) that make M/mM/µM and %(w/v)/mg/mL separate unit families: the reason the calculator refuses to mix them without a molecular weight.
  3. The Calibration of Small Volumetric Laboratory Glassware (NBSIR 74-461). Josephine Lembeck, National Bureau of Standards (now NIST), December 1974. nvlpubs.nist.govMeniscus reading, filling and draining technique for volumetric flasks and pipets, and the contained-versus-delivered distinction behind the "bring to the final volume, don't add volumes" instruction in the recipe and tips.
  4. ISO 8655-2:2022, Piston-operated volumetric apparatus — Part 2: Pipettes. International Organization for Standardization, 2022. iso.orgSets the maximum permissible errors for air-displacement and positive-displacement pipettes across their working volume: the basis for the tip about keeping transfers out of the low end of a pipette's range and serial-diluting instead. The link is the ISO catalogue record; the standard itself is sold, not published openly.
  5. ISO 6887-1:2017, Microbiology of the food chain — Preparation of test samples, initial suspension and decimal dilutions for microbiological examination — Part 1: General rules for the preparation of the initial suspension and decimal dilutions. International Organization for Standardization, 2017. iso.orgCodifies the decimal (1:10) dilution series described in the serial-dilution FAQ, where repeated modest steps replace one large, error-prone transfer. The link is the ISO catalogue record; the standard itself is sold, not published openly.

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