C1V1 = C2V2 Calculator for Dilution

The C₁V₁ = C₂V₂ calculator finds how much of a starting solution is needed to make a target volume at a lower concentration. Enter the starting concentration, starting volume, and final volume to calculate the resulting concentration; or use the equation to solve for the volume of stock required.

Formula

The dilution equation is C₁V₁ = C₂V₂. Solving for the final concentration gives C₂ = C₁ × V₁ / V₂.

• C₁ is the starting, or stock, concentration in mol/L.
• V₁ is the volume of stock solution used, in mL.
• C₂ is the concentration after dilution, in mol/L.
• V₂ is the total final solution volume, in mL.

Because both volumes use the same unit, their units cancel in the ratio V₁/V₂. If the volumes are in different units, convert them before using the equation. For example, 1 L is 1000 mL. The second calculator result is C₂ converted from mol/L to µmol/L by multiplying by 1,000,000.

Worked Example

Using the calculator defaults, the starting solution is 1 mol/L, the stock volume is 10 mL, and the final volume is 100 mL.

C₂ = C₁ × V₁ / V₂
C₂ = 1 mol/L × 10 mL / 100 mL
C₂ = 0.1 mol/L

Converted to micromoles per liter: 0.1 mol/L × 1,000,000 = 100,000 µmol/L. The calculator displays the primary result to four decimal places (0.1000 mol/L) and the converted result to two decimal places (100000.00 µmol/L).

Assumptions

• The amount of dissolved solute is conserved during dilution: no solute is added, removed, or lost.
• V₂ means the final total solution volume, not the volume of solvent added.
• Concentrations are expressed on the same basis, here mol/L.
• The calculation assumes a homogeneous solution and does not account for chemical reactions, precipitation, evaporation, or volume changes caused by mixing.

How to Use the Calculator

Enter the values in the matching fields. The calculator above treats C₁, V₁, and V₂ as inputs, so its direct output is C₂—the final concentration. This is useful when you know the stock concentration and the two volumes and want to check the result of a dilution.

If your practical question is “How much stock do I need?”, rearrange the same relationship instead:

V₁ = C₂ × V₂ / C₁

Insert the desired final concentration, desired final volume, and stock concentration. The output is the stock volume required, as long as concentration and volume units are used consistently. For example, a 1 mol/L stock used to prepare 100 mL at 0.1 mol/L requires 10 mL of stock. The final solution is then brought to 100 mL total volume; it is not made by adding 100 mL of solvent to the stock.

The calculator is a general concentration arithmetic tool for study and lab planning. For bench work, follow your lab method, the safety data sheet (SDS), and your supervisor.

What C₁V₁ = C₂V₂ Means

The equation expresses conservation of solute during a simple dilution. Before dilution, the amount of solute is represented by concentration times volume, C₁V₁. After adding solvent and bringing the mixture to a larger final volume, the same solute amount is represented by C₂V₂. The concentration falls because the same amount is distributed through more solution.

For instance, if you take one part of a stock solution and make the total volume five parts, the final concentration is one-fifth of the stock concentration. “Fivefold dilution” describes that final-to-stock volume relationship, not five parts of solvent added to one part stock.

Common Mistakes and Limits

• Using solvent volume instead of final volume: V₂ is the volume after the stock and solvent are combined. Adding 90 mL of solvent to 10 mL of stock gives an approximate total of 100 mL, not 90 mL.
• Mixing concentration units: Convert values to a common unit first. For example, 1 mM is 0.001 mol/L; it is not 1 mol/L.
• Mixing volume units: The calculator uses the same volume unit for V₁ and V₂, so mL-to-mL works directly. If one value is in liters and the other in milliliters, convert one of them.
• Rounding too early: Keep extra digits during intermediate calculations and round the reported result to a precision justified by the measurements.
• Assuming every mixture is a simple dilution: The equation does not handle reactions, changing solute amounts, nonhomogeneous mixtures, or cases where mixing changes volume appreciably.

A zero or negative volume is not physically meaningful for this calculation, and the stock concentration must be positive to use the rearranged equation for V₁. A calculated result also cannot be more precise than the inputs. If a stock label reports only two significant figures, displaying many calculator decimals does not create additional measurement accuracy.

Frequently Asked Questions

What does C₁V₁ = C₂V₂ mean?
It states that concentration multiplied by volume is unchanged when a fixed amount of solute is diluted. C₁ and V₁ describe the starting solution; C₂ and V₂ describe the final solution.

How do I calculate C₂?
Rearrange the equation to C₂ = C₁ × V₁ / V₂. Use matching units for the two volumes and the same concentration units throughout.

How do I find the stock volume V₁?
Use V₁ = C₂ × V₂ / C₁. This gives the stock volume needed for a specified final concentration and total volume.

Is V₂ the amount of water or solvent I add?
No. V₂ is the final total solution volume. The solvent amount is not generally equal to V₂ because the stock solution already contributes volume.

Can I use C₁V₁ = C₂V₂ for any concentration units?
Yes, if C₁ and C₂ use the same concentration unit and V₁ and V₂ use the same volume unit. Convert mixed units first; for example, 1 L = 1000 mL and 1 mol/L = 1000 mmol/L.

Related

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

in c1 | Starting concentration C1 | mol/L | 1 | 0
in v1 | Starting volume V1 | mL | 10 | 0
in v2 | Final solution volume V2 | mL | 100 | 0
out c2 | Final concentration C2 | mol/L | c1 * v1 / v2 | 4
out um | Final concentration | µmol/L | c2 * 1000000 | 2