Solution Dilution

What the C1·V1 = C2·V2 equation calculates

The equation C1·V1 = C2·V2 describes a simple dilution: a volume V1 of a solution with concentration C1 is made up with solvent to the final volume V2, and the concentration drops to C2. The amount of solute does not change, only the volume in which it is distributed.

The four symbols mean:

• C1: initial concentration, of the stock solution or the sample.
• V1: initial volume, the aliquot you pipette.
• C2: desired or obtained final concentration.
• V2: total final volume, with the diluent already added.

In the clinical laboratory, the equation is used to prepare standards and working reagents, to dilute a sample that exceeded the method's linearity limit and to check the factor used to correct the result. For serial dilutions, in which the factors multiply, the scientific calculator helps with powers of 10.

How to calculate the dilution in this tool, step by step

The tool calculates any of the four terms from the other three. The procedure:

1. In the Which value do you want to calculate? field, choose the unknown term: Initial Concentration (C1), Initial Volume (V1), Final Concentration (C2) or Final Volume (V2). The default is V2.
2. The chosen field becomes disabled. Fill in the other three.
3. The result appears as soon as three fields have values, with three decimal places.
4. Use Copy Result to take the value to your record, or Clear to reset the fields (the choice of term is kept).

The fields have no fixed unit. The tool accepts any concentration unit (mol/L, g/L, mg/dL, %) and volume unit (L, mL, µL), as long as C1 and C2 are in the same unit, and V1 and V2 as well. The result comes out in the unit you used.

If any divisor is zero, an invalid values message appears followed by the term; if the calculation does not give a finite number, an invalid result message (NaN or Infinity) appears.

Dilution formula and its four forms

The tool starts from C1 × V1 = C2 × V2 and isolates the chosen term. The table shows the expression used in each case.

Two derived quantities do not appear on screen, but come from the same inputs:

• Dilution factor = V2 ÷ V1 = C1 ÷ C2.
• Diluent volume = V2 − V1.

The concentration and volume units follow the International System, described in the SI Brochure (BIPM): amount-of-substance concentration (mol/L) or mass concentration (g/L). For monovalent ions, such as sodium, mEq/L is equivalent to mmol/L, as in the corrected sodium calculator.

Worked example: preparing 500 mL from a stock 10 times more concentrated

To prepare 500 mL of a working solution with concentration 1 from a stock with concentration 10 (in the same unit), you need to pipette 50 mL of the stock.

1. Choose Initial Volume (V1) in "Which value do you want to calculate?".
2. Enter C1 = 10, C2 = 1 and V2 = 500.
3. The tool calculates V1 = (1 × 500) ÷ 10 = 50.000.
4. Diluent volume, calculated separately: 500 − 50 = 450 mL.
5. Dilution factor: 500 ÷ 50 = 10.

The reverse path checks the calculation: with C1 = 10, V1 = 100 and C2 = 1, choosing V2, the tool returns 1,000.000, that is, 100 mL of stock made up to 1,000 mL.

In practice, transfer volume V1 into a volumetric flask of volume V2 and fill to the mark, instead of adding 450 mL measured in a graduated cylinder. That way the final volume is exact even when the volumes do not add up perfectly.

Common errors, units and verification checklist

Almost every dilution error comes from mixed units or from confusing final volume with diluent volume. The tool detects neither.

• Same units: C1 and C2 in the same unit (for example, both in mg/dL); V1 and V2 as well (both in mL). Mixing mL with µL causes a 1,000-fold error.
• V2 is the total volume: if you want 10 mL in the end, V2 = 10, not 10 plus the aliquot.
• Type of percentage: % w/v, % v/v and % w/w are not interchangeable.
• Factor notation: confirm whether "1:10" in your procedure means 1 part in 10 total parts or 1 part plus 10 of diluent; both conventions occur in practice.
• Signs: the tool does not block negative numbers; check the inputs.
• Disabled field: if it still holds a value typed earlier, the tool counts it among the three filled in. Clear before changing the calculated term.
• Result correction: when diluting a sample, multiply the reading by the dilution factor.

Glucose in mg/dL divided by 18 gives mmol/L, a conversion used in the calculated serum osmolality.

When C1·V1 = C2·V2 does not apply

The equation holds for a simple dilution, in which only solvent is added and nothing reacts. It no longer describes the process in four common situations:

• Mixing two solutions that contain solute: when both parts have the same solute at different concentrations, the calculation becomes C1·V1 + C2·V2 = Cf·(V1 + V2).
• Volumes that do not add up: mixtures such as ethanol and water undergo volume contraction; for that reason, fill to the flask mark instead of adding volumes.
• Chemical reaction or precipitation: if the solute reacts, the initial amount is not conserved.
• Serial dilutions: the tool calculates one step at a time; the total factor is the product of the factors of each step (three 1 in 10 dilutions give 1 in 1,000).

Real accuracy also depends on calibrated glassware and pipettes: the calculation is exact, the execution is not. When an analyte result goes into a derived index, such as the fractional excretion of sodium, check that the dilution correction was applied before the calculation.

Frequently asked questions about solution dilution with the C1·V1 = C2·V2 equation

References

1. Bureau International des Poids et Mesures. The International System of Units (SI). 9th ed. Sèvres: BIPM; 2019. View
2. Thompson A, Taylor BN. Guide for the Use of the International System of Units (SI). NIST Special Publication 811. Gaithersburg: National Institute of Standards and Technology; 2008. View