The C1V1 = C2V2 Equation

C1V1 = C2V2 is the dilution equation: the concentration of your stock times the volume you take equals the concentration of the final solution times its total volume. It works because diluting adds solvent but not solute, so the amount of solute is the same on both sides. This guide explains each term, rearranges the equation all four ways, states the unit rule that catches most errors, and translates the “10x stock” language you see on reagent bottles into concrete ratios. The C1V1 calculator solves any one unknown and lets C1 and C2 use different units.

What does each term mean?

• C1 — the concentration of the stock (the concentrated solution you start with).
• V1 — the volume of stock you pipette into the new solution.
• C2 — the concentration you want in the final (diluted) solution.
• V2 — the total final volume after diluent is added.

The key relationship hiding in the equation is that C1 × V1 = amount of solute, and that amount does not change when you add solvent. So whatever combination of C and V you choose on the left must equal the combination on the right.

How do I rearrange C1V1 = C2V2?

Divide both sides to isolate the one value you need:

1. Stock volume: V1 = (C2 × V2) / C1 — how much concentrate to pipette
2. Final concentration: C2 = (C1 × V1) / V2 — what you end up with
3. Stock concentration needed: C1 = (C2 × V2) / V1 — how strong a stock to make
4. Final volume: V2 = (C1 × V1) / C2 — how much you can make from a fixed aliquot

The diluent is always V2 − V1. That subtraction is easy to forget: the equation gives you the stock volume, not the water volume, and adding V1 of stock to V2 of water overshoots the target.

Worked example: diluting a 10x buffer to 1x

A “10x” stock is ten times the working concentration. To make 500 mL of 1x from a 10x stock:

• C1 = 10x, C2 = 1x, V2 = 500 mL
• V1 = (1 × 500) / 10 = 50 mL of stock
• Diluent = 500 − 50 = 450 mL of water
• This is a 1:10 dilution (50 mL in 500 mL total)

Pipette 50 mL of the 10x concentrate and bring the total to 500 mL. The “x” notation is just a concentration ratio, so C1V1 = C2V2 handles it directly with no unit conversion.

How does 10x map to a 1:10 or 1:100 dilution?

People describe the same dilution three ways — as a fold-concentration, a ratio, and a dilution factor. They are interchangeable, and this table pins the relationships down:

Stock	Working	Ratio (1:n)	Dilution factor	Stock per 10 mL
10x	1x	1:10	10	1.0 mL
20x	1x	1:20	20	0.5 mL
50x	1x	1:50	50	0.2 mL
100x	1x	1:100	100	0.1 mL

The rule: fold-concentration = dilution factor = n in 1:n. A 100x stock diluted to 1x is a 1:100 dilution with DF = 100. Stock per final volume is always V_total ÷ DF, which is just V1 = (C2 × V2)/C1 with C2/C1 = 1/n.

The unit-matching rule

C1 and C2 must be in the same concentration unit, and V1 and V2 must be in the same volume unit — but the two pairs do not have to match each other. You can use M for concentration and mL for volume, or mg/mL with L; the equation only needs internal consistency on each side.

The trap is mixed scales. Dividing 100 mM by 1 M without converting gives a factor of 100 instead of 10. Convert first:

• 1 M = 1000 mM = 1000 mmol/L
• 1 mL = 0.001 L; 1 L = 1000 mL
• mg/mL, g/L and ppm (for dilute aqueous) are numerically equal

The C1V1 calculator accepts different units on each side and normalises them internally, so you can type 100 mM for C1 and 1 M for the stock and it still returns the right volumes.

When does C1V1 = C2V2 not apply?

The equation assumes the solute is conserved and volumes are additive. It breaks or needs care when:

• Volumes are not additive. Mixing ethanol and water gives a final volume slightly less than the sum, so a strict % v/v dilution is approximate.
• A reaction consumes or releases solute. Diluting a concentrated acid is a dilution, but neutralisation is not — C1V1 = C2V2 is not a titration equation.
• You are diluting by mass, not volume. For % w/w you need masses and a density; the volume equation does not transfer.

For ordinary aqueous dilutions of salts, buffers, sugars and most reagents, the additivity assumption holds well and the equation is exact enough for bench work.

Worked example: making a standard from a concentrated stock

You have a 1000 ppm chloride standard and need 100 mL of a 25 ppm working standard:

• V1 = (25 ppm × 100 mL) / 1000 ppm = 2.5 mL
• Diluent = 100 − 2.5 = 97.5 mL
• DF = 1000 / 25 = 40 (a 1:40 dilution)

Because ppm, mg/L and µg/mL are interchangeable for dilute solutions, the same equation works whether the label is in ppm or mg/L. The PPM calculator handles the unit conversions if you need the answer in a different form.

Common mistakes

• Using diluent volume as V2. V2 is the total, not the water you add.
• Forgetting to subtract. The equation yields V1; the diluent is V2 − V1.
• Mixed concentration scales. mM vs M, ppm vs %, mg/mL vs g/L.
• Assuming additivity for non-aqueous mixes. Alcohol-water volumes contract.
• Pipetting V1 too small. If V1 is under ~2 µL, switch to a serial dilution.

If C2 comes out higher than C1, or V1 larger than V2, a term has been swapped between the two sides — a good signal to re-check which is stock and which is final.

Where does this fit in the wider workflow?

C1V1 = C2V2 is the algebra underneath the dilution calculator, which presents the same relation in dilution-factor language, and the serial dilution calculator, which chains several C1V1 steps to reach large factors. When you need to prepare the stock itself rather than dilute one, the stock solution calculator starts from a target concentration and mass. All of these are collected in the laboratory calculators, and the concept of dilution factor is covered in how to calculate a dilution.