TAHA Engineering Toolsby TAHA Knowledge Lab

Chemical Engineering

Solution Dilution Calculator (C₁V₁ = C₂V₂)

Solve C₁V₁ = C₂V₂ for any value, the diluent volume to add, or the result of a serial dilution.

Inputs

Calculate from
A small beaker labelled C1 V1 and a jug of diluent both pour into a larger beaker labelled C2 V2.C₁, V₁ (stock)diluentC₂, V₂ (final)C₁V₁ = C₂V₂
A volume V₁ of stock at concentration C₁ is made up with diluent to a total volume V₂ at concentration C₂. The amount of solute stays the same: C₁V₁ = C₂V₂. Diagram not to scale.

Any concentration unit (mol/L, g/L, %, ppm…) — use the same unit for C₁ and C₂.

Result

Stock concentration C₁
1
Diluent to add (V₂ − V₁)
90
Dilution factor C₁ ÷ C₂
10
Working
C₁ × V₁ = C₂ × V₂ → 1 × 10 mL = 0.1 × 100 mL Diluent to add = V₂ − V₁ = 90 mL

Take the stock and make up to the final volume; this needs about 90 mL of diluent.

  • The diluent volume assumes the volumes simply add. This is close for most dilute water solutions, but not exact for mixtures such as alcohol and water or concentrated acids: make up to the final volume instead.

Educational estimate — check the assumptions below.

About this calculator

Diluting a solution keeps the amount of solute the same while the volume grows, so C₁ × V₁ = C₂ × V₂. Enter any three of the four values to find the fourth, and the volume of diluent to add. The serial-dilution mode repeats the same dilution several times, as used for calibration standards and microbiology.

Concentration can be in any unit (mol/L, g/L, %, ppm…) as long as C₁ and C₂ use the same one. This is a calculation tool only, not chemical handling advice.

Formula

C₁ V₁ = C₂ V₂    Vdiluent = V₂ − V₁    DF = C₁ ÷ C₂ = V₂ ÷ V₁

Serial dilution: Cn = C₀ ÷ DFn

C₁, V₁ = concentration and volume of the stock taken; C₂, V₂ = final concentration and total volume; DF = dilution factor; n = number of steps.

Worked example

  • To make 100 mL of 0.1 mol/L from a 1 mol/L stock: V₁ = 0.1 × 100 ÷ 1 = 10 mL of stock, made up with about 90 mL of diluent.
  • Three 1 : 10 serial dilutions of a 1 mol/L solution: 1 ÷ 10³ = 0.001 mol/L (total dilution 1000 times).

Assumptions and limitations

  • The amount of solute does not change (no reaction, no evaporation).
  • The diluent volume assumes the volumes add up exactly; for concentrated or non-ideal mixtures, make up to the final volume instead of adding a calculated volume.
  • C₁ and C₂ must be in the same unit. Mass-based units (% w/w, ppm by mass) combine with volumes only when the densities are about equal.
  • A calculation tool only, not chemical handling advice.

References

  • IUPAC Green Book (amount and mass concentration); dilution as conservation of the amount of solute.