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Mixing Solutions Calculator

Find the concentration after combining solutions of the same solute, or the volumes of a stronger and a weaker solution that mix to a target (alligation).

Formula

Cmix=C1V1+C2V2+⋯V1+V2+⋯C_{\text{mix}} = \dfrac{C_1V_1 + C_2V_2 + \cdots}{V_1 + V_2 + \cdots}
Vstrong=Vtotal Ctarget−CweakCstrong−CweakV_{\text{strong}} = V_{\text{total}}\,\dfrac{C_{\text{target}} - C_{\text{weak}}}{C_{\text{strong}} - C_{\text{weak}}}
Vweak=Vtotal−VstrongV_{\text{weak}} = V_{\text{total}} - V_{\text{strong}}
CmixC_{\text{mix}}
concentration of the combined solution
Ci, ViC_i,\ V_i
concentration and volume of each solution being combined
Cstrong, CweakC_{\text{strong}},\ C_{\text{weak}}
the two solutions available, in the same unit as the target

How it works

Mixing does not create or destroy solute. The amount in the mixture is the sum of the amounts in each part (each concentration times its volume), and it is spread through the sum of the volumes. The result is a volume-weighted average, which always lies between the weakest and the strongest solution.

Running the same balance backwards gives the volumes of two solutions needed for a target in between, a method pharmacy texts call alligation. With a weaker solution of zero it reduces to an ordinary dilution.

Worked example

100 mL of 2 M sodium chloride is combined with 300 mL of 0.5 M sodium chloride.

  1. Amount = 2 mol/L × 0.1 L + 0.5 mol/L × 0.3 L = 0.35 mol.
  2. Volume = 0.1 L + 0.3 L = 0.4 L.
  3. C mix = 0.35 mol / 0.4 L = 0.875 mol/L.

The mixture is 400 mL of 0.875 M sodium chloride.

These are the values the calculator opens with, so you can check its output against this example.

Assumptions

  • Every solution contains the same solute, and all concentrations use one unit.
  • Volumes add on mixing. This fails for mixtures such as ethanol and water and for concentrated acids.
  • The solutions do not react with each other.

Common mistakes

  • Averaging the concentrations without weighting by volume.
  • Setting a target outside the two solutions. Mixing cannot exceed the stronger or fall below the weaker.
  • Using it for a w/w percentage with volumes. Mass-based concentrations need masses.

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