Dilution Calculator

Enter a stock concentration, a target concentration, and a final volume to find how much stock solution and solvent to combine.

Worked example — Dilution Calculator

Stock volume (V₁)	50
Solvent to add	200
Dilution factor	5x

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Dilution Calculator

A dilution is a recipe, not a reaction: you are choosing how much concentrated stock to take and how much solvent to put with it. The calculator answers the practical question rather than the chemical one — given the strength you have and the strength and volume you want, how much of the bottle do you pour?

What each symbol means

C₁: the stock concentration, the strength of what is on the shelf. It sets how little you need: at 100 against a target of 0.5, a full litre of working solution takes 5 units of stock.
V₁: the stock volume to measure out — the number you actually pipette, and the one row the calculation exists to produce.
C₂: the target concentration you want to end up with. Together with C₁ it fixes the dilution factor, independently of how much you make.
V₂: the final volume, which is what you end up holding — not what you add. The difference between the two is the row below it.

Where the formula comes from

Rearranging the conservation statement for the quantity you need gives V₁ = C₂V₂ / C₁. Every figure on the right is something you already decided, which is why there is nothing to solve.
The solvent to add is then V₂ − V₁, and this is the step that gets skipped at the bench. For 250 of a 1 solution from a 5 stock the calculator asks for 50 of stock and 200 of solvent — you top up to the final volume, you do not add the final volume.
The dilution factor is just C₁/C₂, so it is fixed before any volume is chosen. A 2x dilution is therefore always half stock and half solvent: 12 down to 6 in a final 500 asks for 250 and 250.

How to read what you see

Read the middle row against the bottom one. A 200x dilution — 100 stock down to 0.5 — needs 5 of stock and 995 of solvent, so the solvent row is almost the whole final volume and the stock row is a rounding error. At 2x the two rows are equal. The factor tells you which of those two situations you are in before you measure anything.

Assumes: That volumes add: that 50 of stock plus 200 of solvent occupies 250. It also assumes one dilution step and one solute, with nothing reacting and nothing already in the solvent.
Breaks when: Volumes are not strictly additive, and concentrated acids are where it shows. Mixing equal volumes of concentrated sulphuric acid and water gives measurably less than their sum, because the molecules pack differently together than apart — and it releases enough heat to matter besides. The calculator’s 2x answer of 250 and 250 is a good instruction and a poor prediction of the level in the flask, which is why real protocols say to make up to the mark rather than to add a measured volume of water.

It will happily ask you for 0.05 mL 🖖

C₁V₁ = C₂V₂ gives you the volumes and has no opinion about whether you can measure them. The trace preset asks for 5 mL of stock in 1000 — a 200× dilution, and comfortable. Set the target concentration to 0.005 instead and the panel answers 0.05 mL with exactly the same confidence, for a factor of 20,000. Nobody pipettes fifty microlitres into a litre flask and calls the result accurate. That is what serial dilution is for, and it settles the challenge above: 200× followed by 100× is 20,000×, on volumes you can actually deliver. The chemistry is indifferent to the route. The measurement is not.

Practice

Check yourself

Predict the answer first, then use the controls above to find out. Reveal only after you have committed to a guess — that is what makes it practice.

Ask for the impossible: stock 1, target 5, final volume 250. The page will warn you — but what does it put in the two volume rows first?

Show answer

1250 of stock and 0 of solvent, at a dilution factor of 0.2x. The arithmetic is untouched: V₁ = C₂V₂/C₁ is 5 × 250 / 1, which genuinely is 1250. What it describes is a recipe calling for five times as much stock as the volume you are making. The factor is the tell — it has dropped below 1, and a dilution factor under one is a concentration, which no amount of pouring will achieve. The warning is doing the work the formula cannot.
Now make stock and target equal: 10 and 10, final volume 100. No warning this time. What do the rows say?

Show answer

100 of stock, 0 of solvent, factor 1x. It is a correct answer to a question nobody asks — pour out the volume you want and add nothing. But it is the boundary: below it the solvent row is positive and the factor is above 1, above it the factor falls under 1 and the warning appears, and this single point is the last one where every row is still honest.

Problem solved in full

250 mL of 1 M solution from a 5 M bottle 5 steps
You need 250 mL of 1 M solution and the bottle on the shelf is 5 M. Work out the two volumes — and be clear about which one you measure first, because the order matters in the lab.
1. Nothing is created by adding water. The number of moles of solute in the pipetted stock is the number of moles in the final flask, and that single sentence is the whole equation.
  
  $n = C V is conserved ⟹ C_{1} V_{1} = C_{2} V_{2}$
2. Writing moles as concentration times volume on both sides and solving gives 50 mL of stock.
  
  $V_{1} = \frac{C _{2} V _{2}}{C _{1}} = \frac{1 \times 250}{5} = 50 mL$
3. The solvent is the difference, not the final volume: 200 mL of water added to 50 mL of stock.
  
  $V_{solvent} = 250 - 50 = 200 mL$
4. Check it by going back to moles. 5 × 50 and 1 × 250 are both 250 millimoles, which they must be.
  
  $check : 5 \times 50 = 250 = 1 \times 250 ✓$
5. The dilution factor can be read two ways — the concentration ratio or the volume ratio — and they agree at 5, because the two are reciprocals by construction.
  
  $dilution factor = \frac{C _{1}}{C _{2}} = \frac{V _{2}}{V _{1}} = 5 \times$
Answer

The tool prints V₁ = 50, 200 of solvent and a dilution factor of 5×. The laboratory caveat is the one the formula cannot express: you make this up to 250 mL in a volumetric flask, you do not add 200 mL to 50 mL in a beaker. Volumes are not strictly additive when a solute is present, and for concentrated acids the difference is measurable. C₁V₁ = C₂V₂ tells you how much stock to take; the glassware tells you how to finish.

$C_{1} V_{1} = C_{2} V_{2} is conservation of moles, not a rule to memorise$

References (2)

The quantity C₁ and C₂ actually are, defined: IUPAC, “amount concentration.” The IUPAC Compendium of Chemical Terminology (the Gold Book), International Union of Pure and Applied Chemistry, 2014.
Why “volumes add” is an approximation — the concentrative-properties tables let you check the contraction for yourself: John R. Rumble (ed.), CRC Handbook of Chemistry and Physics, “Concentrative Properties of Aqueous Solutions: Density, Refractive Index, Freezing Point Depression, and Viscosity.” CRC Press / Taylor & Francis, 2021. ISBN 978-1-032-12171-0.

By the 42Math team · Updated 2026-05-21

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Lab buffer - 5 down to 1 in a final 250: measure 50 of stock and add 200 of solvent, a 5x dilution
Trace dilution - 100 down to 0.5 in 1000 is 200x — only 5 of stock against 995 of solvent
Concentrated acid - 12 down to 6 in 500 is the 2x case, where the stock volume and the solvent to add are both 250