Diluting a solution does just one thing: it spreads the same amount of solute through a larger volume of solvent. Nothing is added or removed except solvent, so the quantity of solute — which is concentration times volume — stays exactly the same. That is the whole meaning of C₁V₁ = C₂V₂: the solute in the stock you take equals the solute in the final solution. Rearrange it to find how much concentrated stock to measure out, then top up with solvent to the target volume.
Reviewed: June 19, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: conservation of solute (C₁V₁ = C₂V₂).
The dilution equations
Because the units cancel on each side, the two concentrations only need to match each other (both molarity, both %), and likewise the two volumes — you can freely mix molarity with millilitres. After finding V₁, the rest of the final volume is solvent: V₂ − V₁. The dilution factor tells you how many "times" more dilute the result is; a 10× dilution turns a 5 M stock into 0.5 M.
Worked example — making a working solution
Scenario: You have a 5 M stock and need 100 mL of a 0.5 M working solution.
Measure 10 mL of the 5 M stock into a flask and add solvent up to the 100 mL mark — that is 90 mL of water — to get 0.5 M. It is a tenfold (1:10) dilution. If you instead needed it twice as dilute, 0.25 M, you would take only 5 mL of stock and add 95 mL of water. Always make up to the final volume mark rather than adding a fixed solvent volume, since mixing can change the total slightly.
Frequently Asked Questions
C₁V₁ = C₂V₂. Stock to take: V₁ = C₂V₂/C₁. 5 M → 100 mL of 0.5 M = 10 mL stock + 90 mL water.
The dilution equation — solute is conserved, so concentration × volume is equal before and after.
solvent = V₂ − V₁. After V₁ = 10 mL for a 100 mL final, add 90 mL. Acid into water, never the reverse.
DF = C₁/C₂ = V₂/V₁. 5 M to 0.5 M is 10× (1:10). Check if a protocol means total parts or added parts.
Both concentrations same unit, both volumes same unit — they cancel each side. Don't mix concentration units.