C1V1 = C2V2 Equation: Dilution Formula Explained

The equation C₁V₁ = C₂V₂ is the dilution equation, also called the dilution formula or conservation-of-solute equation. It calculates how much of a concentrated starting solution is needed to prepare a more dilute solution of a specified concentration and final volume. Enter the starting and target concentrations and the target volume to calculate the required starting-solution volume.

Formula

The dilution equation is C₁V₁ = C₂V₂. Rearranged to find the starting-solution volume, it is V₁ = C₂V₂ / C₁.

• C₁ is the starting solution’s concentration, in mol/L (M).
• V₁ is the volume of starting solution required, reported here in mL.
• C₂ is the target solution’s concentration, in mol/L (M).
• V₂ is the final solution volume, entered in mL.

The concentration units must match. The formula uses the same volume unit for V₁ and V₂; this calculator converts the entered final volume from millilitres to litres by dividing by 1000, calculates V₁ in litres, then multiplies by 1000 to display millilitres. The solvent estimate is V₂ − V₁. It is an arithmetic estimate, not a guarantee that separately measured liquid volumes add exactly to the stated final volume.

Worked Example

Using the calculator’s defaults, find the volume of a 1 mol/L starting solution needed to make 100 mL of a 0.1 mol/L target solution.

1. Convert the final volume: 100 mL / 1000 = 0.10000 L.
2. Calculate the target amount of solute: 0.1 mol/L × 0.10000 L = 0.010000 mol.
3. Divide by the starting concentration: V₁ = 0.010000 mol / 1 mol/L = 0.010000 L.
4. Convert to millilitres: 0.010000 L × 1000 = 10.000 mL. Rounded to the calculator’s three decimal places, the result is 10.000 mL.
5. The estimated solvent volume is 100 mL − 10 mL = 90.000 mL.

For an actual preparation, the final volume is the important target: add the measured starting solution and bring the mixture to the specified final volume according to the applicable lab method, rather than assuming the solvent volume calculation produces an exact final volume. Follow your lab method, the safety data sheet (SDS) and your supervisor.

Assumptions

• The solute is conserved during dilution: no reaction, precipitation, evaporation or loss changes the amount being measured.
• Starting and target concentrations refer to the same solute and use the same concentration units.
• V₂ means the total final solution volume, not the volume of solvent added.
• The solvent output is only the volume difference V₂ − V₁; real solution volumes may not be additive.

What the Equation Means

The equation expresses conservation of solute. The amount of solute in a solution is concentration multiplied by volume. If a portion of a starting solution is diluted without adding or removing solute, the amount of solute in that portion before dilution equals the amount in the finished solution: C₁V₁ = C₂V₂.

For example, a tenfold dilution means the target concentration is one tenth of the starting concentration. To make 100 mL at one tenth the starting concentration, the equation gives 10 mL of starting solution; the remaining volume is made up to 100 mL. A lower target concentration generally requires a smaller starting-solution volume for the same final volume.

The equation is not a separate chemical law with a unique named discoverer. It is a rearrangement of the basic concentration relationship, amount of solute = concentration × volume. In chemistry classes and laboratory instructions, “dilution equation” is the usual name; “C₁V₁ equals C₂V₂” is also commonly used as a spoken description.

When to Use It

Use the equation when preparing a diluted solution from a known stock solution, provided both concentrations describe the same solute and the amount of solute stays unchanged. It is useful for laboratory planning, solution-concentration exercises and scaling a stock solution to a chosen final volume.

• Identify the stock concentration as C₁.
• Set the desired final concentration as C₂ and the total final volume as V₂.
• Calculate V₁ = C₂V₂/C₁, keeping concentration units consistent.
• Check that V₁ is no greater than V₂. If it is, the requested target cannot be made by diluting that stock to that final volume.

The calculator’s input limits prevent negative concentrations and volumes, but they do not validate whether a chosen solution is chemically compatible or whether a real preparation is appropriate. A zero starting concentration cannot supply solute; the equation’s division by zero makes that case undefined. A target concentration greater than the starting concentration is not achieved by dilution alone.

Common Mistakes and Limits

Confusing final volume with solvent volume: V₂ is the total volume after dilution. If a task asks for a 100 mL final solution, putting 100 mL of solvent into the starting solution usually makes the total exceed 100 mL. Use the calculated starting-solution volume, then make up to the specified final volume.

Mixing units: If one concentration is in mol/L and the other is in mmol/L, convert them before using the equation. Likewise, use the same units for V₁ and V₂ in the equation. This calculator handles the mL-to-L conversion explicitly.

Rounding too soon: Keep extra digits while calculating and round only the final value to match the precision of the measurements and the required method. A result with many decimal places does not mean the physical measurement is that precise.

Assuming it covers every concentration problem: The formula does not calculate a stock concentration from a chemical’s mass unless the amount, molar mass and volume are known and converted separately. It also does not account for chemical reactions, changes in solute form, nonideal behavior or measurement uncertainty. For serial dilutions, apply the equation to each step using that step’s starting and target concentrations; do not confuse the concentration of an intermediate dilution with the original stock concentration.

Frequently Asked Questions

What is C₁V₁ = C₂V₂ called?
It is called the dilution equation or dilution formula. It relates a starting solution to a diluted solution when the amount of solute remains constant.

What do C₁, V₁, C₂ and V₂ mean?
C₁ and V₁ are the starting concentration and the portion of starting solution used. C₂ and V₂ are the desired final concentration and total final volume.

Does V₂ mean the amount of water to add?
No. V₂ is the total final solution volume. The solvent amount is not necessarily exactly V₂ − V₁ in practice, so preparations are generally brought to the stated final volume.

Can I use C₁V₁ = C₂V₂ for a concentration increase?
Not by dilution alone. If C₂ is greater than C₁, the calculated V₁ exceeds V₂; concentrating the solution or using a stronger stock would be needed instead.

Can the equation be used for serial dilution?
Yes. Use it separately for each step, treating the previous step’s solution as the next step’s starting solution. Track each intermediate concentration carefully.

Related

• C1V1 = C2V2 Calculator for Dilution
• Dilution Factor Formula and How to Use It
• C1V1 = C2V2 Calculator for Dilution

in c1 | Starting concentration | mol/L | 1 | 0
in c2 | Target concentration | mol/L | 0.1 | 0
in v2 | Final solution volume | mL | 100 | 0
out v1 | Starting-solution volume | mL | c2 * (v2 / 1000) / c1 * 1000 | 3
out solvent | Approximate added solvent volume | mL | v2 - v1 | 3