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Mixing two solutions

Concentration of a mixture, proportions to reach a target concentration (alligation), or volume to add to an existing solution.

Parameters

Any units (%, g/L, mol/L…), the same everywhere; nothing is converted.

Result

Volumes of solution 1 and 2

400 + 100

Calculation
share₁ = (12 − 20) / (10 − 20) = 0.8
Mixture concentration
12
Volume of solution 1
400
Volume of solution 2
100
Total volume
500
020040005101520volumeconcentrationtarget 12V₁ = 400V₂ = 100
Note

Mix 400 of solution 1 and 100 of solution 2 to reach the target.

Calculation, assumptions and sources

OverviewMass balance of a mixture of two solutions: the concentration obtained, the proportions to reach a target (alligation, mixing cross), or the volume to add to an existing solution.

Mass balance of a mixture of two solutions: the concentration obtained, the proportions to reach a target (alligation, mixing cross), or the volume to add to an existing solution. The target must lie between the two concentrations; aiming at 25% with 10 and 20 is impossible.

Method & formulasC_f = (C₁V₁ + C₂V₂) / (V₁ + V₂)
Cf = (C₁V₁ + C₂V₂) / (V₁ + V₂)share₁ = (Cf − C₂) / (C₁ − C₂)V₂ = V₁ × (C₁ − Cf) / (Cf − C₂)

The amounts of solute add up, and so do the volumes (within the domain of validity); the final concentration is their quotient, a volume-weighted average. The two inverse formulas solve the same equality.

Mass balance
the amount of solute in the mixture is the sum of the amounts brought in.
Alligation (mixing cross)
a graphical method in which each solution’s share is proportional to the other one’s gap to the target.
Mixture range
the interval between the two concentrations, bounds included; no mixture leaves it.
Mixing versus dilution
the special case where solution 2 is pure solvent (C₂ = 0), handled by the C₁V₁ = C₂V₂ dilution tool.
Validity domainVolumes are assumed additive and the solute conserved, which holds for dilute aqueous solutions.

Volumes are assumed additive and the solute conserved, which holds for dilute aqueous solutions. Concentrated mixtures contract (up to 4% of the expected volume for ethanol-water) and distort per-volume concentrations; masses are used in that case. Reactive mixtures (strong acid and strong base) are excluded.

The mixing crossThe target goes at the centre, the two concentrations at the left corners and, diagonally, the gaps, which give the shares.

The target goes at the centre, the two concentrations at the left corners and, diagonally, the gaps, which give the shares. To aim at 12 between 10 and 20, write 8 opposite the 10 and 2 opposite the 20, that is 4:1 in volumes of weak and strong.