C1V1 = C2V2: Calculate Dilutions Step by Step

The equation C₁V₁ = C₂V₂ calculates how much of a concentrated solution you need to make a more dilute solution at a chosen concentration and final volume. Enter the stock and target concentrations and the final volume to find the stock volume required.

Formula

The dilution equation is C₁V₁ = C₂V₂. Solving for the volume of stock solution gives V₁ = (C₂ × V₂) / C₁.

• C₁ is the stock solution concentration, in mol/L.
• V₁ is the volume of stock solution required, in mL.
• C₂ is the desired final concentration, in mol/L.
• V₂ is the desired final solution volume, in mL.

The equation states that the amount of dissolved substance taken from the stock is the same as the amount present in the diluted solution. Since both volumes in the calculation use mL, those units cancel consistently. Concentrations must use the same unit, here mol/L. The calculator’s second result, V₂ − V₁, is an approximate diluent volume; in practice, make the solution up to the specified final volume rather than assuming volumes always add exactly.

Worked Example

Using the calculator’s default values, the stock concentration is 1 mol/L, the target concentration is 0.1 mol/L, and the final volume is 100 mL.

1. Write the rearranged equation: V₁ = (C₂ × V₂) / C₁.
2. Substitute the values: V₁ = (0.1 mol/L × 100 mL) / 1 mol/L.
3. Cancel the matching concentration units and calculate: V₁ = 10 mL.
4. Find the approximate diluent amount: 100 mL − 10 mL = 90 mL.

The calculator reports 10.00 mL of stock and an approximate 90.00 mL of diluent. For a reliable final concentration, add diluent until the total solution volume is 100 mL.

Assumptions

• The stock concentration is known and the stock is correctly prepared.
• The dissolved substance does not react, evaporate, or precipitate during dilution.
• Volume is measured at the relevant laboratory temperature; the calculation treats the stated final volume as the total solution volume.
• The formula is a concentration calculation, not a substitute for a validated laboratory method.

What You’ll Need

• The stock concentration, C₁, and desired final concentration, C₂, in matching units.
• The desired final volume, V₂, with its unit clearly identified.
• A calculator, or the calculator above.
• For bench work: suitable measuring equipment, a container, and the appropriate diluent. Follow your lab method, the safety data sheet (SDS) and your supervisor.

Step-by-Step

1. Check that this is a dilution. The stock concentration should be greater than or equal to the target concentration. If the target is more concentrated than the stock, dilution cannot achieve it; use a suitable more concentrated stock or another validated approach.
2. Write down the known values. Record C₁, C₂, and V₂, including each unit. For example, note whether a volume is mL or L and whether a concentration is mol/L, mmol/L, or another unit.
3. Make the concentration units match. Convert before calculating if needed. For example, 100 mmol/L equals 0.100 mol/L, because 1 mmol/L is 0.001 mol/L. Do not enter 100 as if it were 100 mol/L when the label says 100 mmol/L.
4. Choose one volume unit. Use mL for both V₁ and V₂, or convert both to L. If the stock volume is wanted in mL and the final volume is in L, convert the final volume using 1 L = 1000 mL.
5. Rearrange the equation. Starting from C₁V₁ = C₂V₂, divide both sides by C₁ to get V₁ = C₂V₂/C₁. This is the quantity the calculator displays as its main answer.
6. Calculate the stock volume. Multiply the target concentration by the desired final volume, then divide by the stock concentration. Check that the result is positive and no greater than the final volume.
7. Plan for the final volume. In a simple estimate, subtract V₁ from V₂ to estimate the diluent amount. For accurate preparation, measure the stock, add some diluent, then bring the mixture to the specified final volume using the appropriate calibrated technique.
8. Label and verify. Record the target concentration, final volume, date, and any other information your lab method requires. If the concentration is critical, use the method’s specified verification rather than relying on arithmetic alone.

Common Mistakes

• Mixing concentration units: A value in mmol/L is not numerically interchangeable with one in mol/L. Convert both concentrations to the same unit first.
• Mixing volume units: Using V₂ in liters while expecting V₁ in milliliters introduces a factor-of-1000 error. Keep both volumes in the same unit during the calculation.
• Using the diluent volume as V₂: V₂ is the final total solution volume, not the amount of water or other diluent added.
• Reversing the concentrations: The stock concentration C₁ belongs in the denominator. Reversing C₁ and C₂ can produce a stock volume larger than the final volume.
• Rounding too early: Keep unrounded values through the calculation and round the measured volume only as appropriate for the equipment and method.
• Assuming volumes add exactly: Adding a measured stock volume to the same volume of diluent does not always produce precisely the sum as a final volume. Make up to the final-volume mark when accuracy matters.

Tips

• Write units beside every number before substituting into the equation; this often reveals a unit mismatch immediately.
• Use a consistent concentration basis, such as mol/L for both solutions, even if labels originally use different units.
• Do a quick sense check: if the target is one-tenth the stock concentration, the stock should make up about one-tenth of the final volume.
• For a serial dilution, calculate each step from the concentration actually produced in the previous step, and account for any transfer volume removed.
• When the calculated stock volume is too small to measure reliably, consider a suitable intermediate dilution if your approved method allows it.

Troubleshooting

The calculator says the result cannot be computed or gives an impossible volume: Check that C₁ is above zero and that C₂, C₁, and V₂ use matching units. If V₁ exceeds V₂, the target is more concentrated than the stock or a value or unit has been entered incorrectly.

The answer differs by a factor of 1000: Look for a mix-up between mol/L and mmol/L, or between L and mL. Convert explicitly before calculating.

The required stock amount is below the equipment’s reliable measurement range: Do not round it to zero or guess. Ask whether a permitted intermediate dilution or a different preparation scale is appropriate.

The final concentration appears wrong despite correct arithmetic: Check stock labeling, solution identity, measurement technique, final-volume adjustment, and whether the substance remains fully dissolved. Follow the method’s verification steps and consult your supervisor if the discrepancy could affect safety or results.

Frequently Asked Questions

What does C₁V₁ = C₂V₂ mean?
C₁V₁ represents the amount of solute in the portion of stock used; C₂V₂ represents the amount in the finished diluted solution. The equation assumes that amount is conserved during dilution.

How do I calculate V₁?
Rearrange the equation to V₁ = C₂V₂/C₁. Use matching units for the two concentrations and the two volumes.

Can I use the formula if my concentrations are in different units?
Yes, after converting them to the same concentration unit. For example, convert mmol/L to mol/L by dividing by 1000.

Does V₂ mean the amount of water I add?
No. V₂ is the total final solution volume. The diluent amount is only approximately V₂ − V₁; accurate preparation means bringing the mixture to V₂.

What if C₂ is greater than C₁?
A dilution cannot raise concentration above that of the stock. You need a more concentrated stock or a different preparation method.

Does the equation work for every solution?
It is a standard dilution calculation when solute amount is conserved and concentrations are defined consistently. Reactions, precipitation, evaporation, or nonstandard concentration behavior may require a different method.

Related

• Dilution Formula Calculator: C1V1 = C2V2 for Molarity
• Dilution Factor Calculator: Final Molarity From Volumes
• Serial Dilution Calculator: Molar Concentration After n Steps

in c1 | Stock concentration | mol/L | 1 | 0
in c2 | Target concentration | mol/L | 0.1 | 0
in v2 | Final solution volume | mL | 100 | 0
out v1 | Stock solution needed | mL | c2 * v2 / c1 | 2
out vd | Diluent to add, approximately | mL | v2 - v1 | 2