Solution Dilution & Concentration (C₁V₁ = C₂V₂)
Solves the dilution equation for whichever of the four quantities you don’t know: how much stock to take, how far to dilute it, or what concentration you end up with. Lab chemists, technicians and process operators use it to make up standards, reagents and cleaning solutions [1].
Equations
C1·V1 = C2·V2
C1 = 10·ρ·w/M
C1 and V1 are the concentration and volume of the stock you take. C2 and V2 are the concentration and total volume of the diluted solution. The amount of solute is the same before and after, which is all the first equation says [1].
The second equation gives the molarity of a concentrated reagent, in mol/L. ρ is its density in g/mL [3], w its assay in mass percent [4], and M the molar mass of the solute in g/mol, from its formula [2].
Inputs
- Choose the unknown in Solve For. Its field is then hidden, and you enter the other three.
- Choosing a Stock Solution fills in C1 from the reagent’s typical assay and density. Check them against the bottle label or the certificate, since they vary from lot to lot.
- The equation works with any concentration unit, as long as C1 and C2 use the same one. Volumes can be entered in any unit; the Solution gives them in liters.
Limits and common mistakes
- V2 is the final volume, not the water you add. Make the solution up to the mark in a volumetric flask. Volumes are not always additive, so V2 − V1 of water is only close.
- Don’t mix mass and volume units. A mass percent diluted by volume needs the densities too.
- Always add acid to water, never water to acid. Concentrated sulfuric acid gives off a lot of heat as it dilutes. Let the solution cool before making it up to the mark.
- The page can’t concentrate a solution. A final concentration above the stock is rejected.
Worked example
A lab needs 1 L of 0.5 M sulfuric acid for a titration, made from the 98 % reagent on the shelf. How much concentrated acid should go into the flask?
- Choose Solve For Stock Volume (V1).
- Choose Stock Solution Sulfuric acid, 98 % H₂SO₄ (1.84 g/mL). C1 fills in from the reagent.
- Enter Final Concentration 0.5 M and Final Volume 1 L, and press Solve. The Solution shows:
Stock Concentration (C1): 18.3865 M
Stock Volume (V1): 0.0271939 L
Final Concentration (C2): 0.5 M
Final Volume (V2): 1 L
So measure about 27.2 mL of the concentrated acid. Add it slowly to about 700 mL of water, let it cool, and make it up to 1 L.
Check by hand: M = 98.07 g/mol, so C1 = 10 × 1.84 × 98/98.07 = 18.39 M. Then V1 = 0.5 × 1/18.39 = 0.0272 L. The widely quoted strength of concentrated sulfuric acid is about 18 M, which agrees.