What does C1V1 = C2V2 mean?

C1V1 = C2V2 is the dilution equation, where C1 is the initial (stock) concentration, V1 is the initial volume you take from the stock, C2 is the final (desired) concentration after dilution, and V2 is the total final volume of the diluted solution. The equation comes from the conservation of moles: the number of moles of solute stays the same before and after adding the diluent (usually water). Rearranging the formula lets you solve for any one of the four variables given the other three.

How do I make a 10-fold dilution?

A 10-fold dilution (1:10) means the final solution is one-tenth the concentration of the original. To make it, take 1 part of your stock solution and add 9 parts diluent, giving a total of 10 parts. For example, take 1 mL of stock and add 9 mL of water to get 10 mL at one-tenth the original concentration. Using C1V1 = C2V2: if C1 = 1 mol/L and V2 = 10 mL, then C2 = 0.1 mol/L and V1 = 1 mL.

What units do I use in the dilution formula?

The units for concentration (C1 and C2) must match each other, and the units for volume (V1 and V2) must match each other. Common concentration units include mol/L (molarity), mg/mL, g/L, and percent (%). Common volume units include microlitres (uL), millilitres (mL), and litres (L). As long as both concentrations use the same unit and both volumes use the same unit, the equation works correctly. The result will be in whichever unit you are solving for.

Dilution Calculator (C1V1 = C2V2)

This calculator solves the dilution equation C1V1 = C2V2, the standard formula used whenever you dilute a concentrated stock solution down to a target working concentration. Choose which of the four variables you need to find, either the final volume (V2), the final concentration (C2), the initial volume taken from stock (V1), or the initial stock concentration (C1), then pick matching concentration units (mol/L, mmol/L, mg/mL, ug/mL, percent, or fold) and volume units (uL, mL, L). Enter the three values you already know and the calculator works out the fourth automatically as you type. The results bar shows all four values side by side, with the one you solved for highlighted, and a step-by-step working panel sets out the rearranged formula, each known value, the calculation itself, the dilution factor (C1/C2), and the volume of diluent you need to add to reach the final volume. A worked example using the default figures shows a 1 mol/L stock diluted from 10 mL to 100 mL, giving a 10-fold dilution made by adding 90 mL of diluent. Use it to prepare lab reagents, serial dilutions, or working solutions from concentrated stocks, and double check that both concentration values and both volume values use the same units before you rely on the result, since mixing units will give a wrong answer.

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Standard formula C1V1 = C2V2 is a fundamental chemistry relationship derived from conservation of moles of solute.

1. What to Solve For

Solve for

Concentration unit

Volume unit

2. Known Values

C1 Initial (stock) concentration

V1 Volume taken from stock

C2 Final (desired) concentration

V2 Total final volume after dilution

Please enter valid positive numbers for all three known values.

Dilution Result

C1 (Initial Conc.)

mol/L

V1 (Volume Taken)

V2 (Final Volume)
100
mL

C2 (Final Conc.)

0.1

mol/L

Step-by-Step Working

EquationC1 × V1 = C2 × V2

Rearranged to solve forV2 = (C1 × V1) / C2

Known: C11 mol/L

Known: V110 mL

Known: C20.1 mol/L

Known: V2(solving)

CalculationV2 = (C1 × V1) / C2 = (1 × 10) / 0.1

V2 =100 mL

Dilution factor10× (1:10 dilution)

Volume of diluent to add90 mL (i.e. V2 − V1)

Worked Example (default values)

Stock solution: C1 = 1 mol/L, take V1 = 10 mL. Desired final concentration: C2 = 0.1 mol/L.

C1 × V1 = C2 × V2 → V2 = (C1 × V1) / C2

V2 = (1 mol/L × 10 mL) / 0.1 mol/L = 100 mL

This is a 10-fold dilution (dilution factor = 10). Add 10 mL of stock to 90 mL of diluent to reach 100 mL total.

The Dilution Equation Explained

The formula C1V1 = C2V2 expresses conservation of solute. When you dilute a solution, you are not adding or removing solute molecules; you are simply increasing the amount of solvent. The number of moles of solute is the same in the stock aliquot and in the final diluted solution.

Since moles = concentration x volume, the product C1V1 (moles in the aliquot you take) equals C2V2 (moles in the final solution). Rearranging lets you solve for any one of the four variables:

Unknown	Formula	When to use
V2 (final volume)	V2 = (C1 × V1) / C2	You know how much stock to take and want to know the total volume to make up
C2 (final concentration)	C2 = (C1 × V1) / V2	You know how much stock and how much final volume; find what concentration results
V1 (initial volume)	V1 = (C2 × V2) / C1	You know the target concentration and volume; find how much stock to take
C1 (initial concentration)	C1 = (C2 × V2) / V1	You know how much you took and what you made; back-calculate the stock concentration

Units and How to Use Them

The formula works for any unit system as long as you are consistent. Both concentrations must use the same unit, and both volumes must use the same unit. Common concentration units in laboratory work include:

mol/L (M, molarity): moles of solute per litre of solution. Used for most chemical reactions.
mmol/L (mM): millimolar. Used for biochemistry assays, physiological solutions.
mg/mL (= g/L): mass per volume. Common for protein and drug concentrations.
ug/mL (= mg/L): micrograms per millilitre. Used in trace analysis and ELISA assays.
% (w/v or v/v): grams per 100 mL, or volume per 100 mL. Common in microbiology media and SDS-PAGE gels.
Fold (x): stock solutions are often described as a multiple of working concentration (for example, 10x PBS, 5x loading buffer).

Dilution Factor

The dilution factor is the ratio by which you reduce the concentration: DF = C1 / C2 = V2 / V1. A 1:10 dilution has DF = 10. A 1:100 dilution has DF = 100. Serial dilutions multiply the dilution factor at each step, so two 1:10 serial dilutions give an overall DF of 100.

Volume of Diluent to Add

Note that V2 is the total final volume, not the volume of diluent added. To prepare the dilution, take V1 of stock and add enough diluent to bring the total volume up to V2. The volume of diluent added is (V2 - V1). For example, to make 100 mL from 10 mL of stock, add 90 mL of water.

Common Laboratory Applications

Preparing working solutions from concentrated stocks (10x PBS, 5x TAE, 100x antibiotics)
Titrating samples to fall within a standard curve range
Making serial dilution plates for cell counting or MIC assays
Diluting reagents to the manufacturer's recommended working concentration
Back-calculating the original concentration of an unknown sample

Related Calculators

Science and Engineering Calculators: full list of science tools.
Moles Calculator (mass to moles): convert between mass, moles, and molar mass.
Charles's Law Calculator: gas volume and temperature relationships.
Avogadro Number Calculator: number of particles from moles.

Sources and method: The dilution formula C1V1 = C2V2 is derived from conservation of moles of solute (n = cV, so c1V1 = c2V2). It is covered in standard general chemistry textbooks including Zumdahl and Zumdahl, "Chemistry" (10th ed.), and Chang and Goldsby, "Chemistry" (13th ed.).

This calculator is for educational and laboratory planning purposes. Always verify calculations before preparing solutions used in clinical, pharmaceutical, or safety-critical applications. Concentrations and volumes must use consistent units for the result to be valid.