Dilution Calculator

The dilution equation C1V1 = C2V2 is a statement of conservation: adding solvent to a solution changes its volume but not the amount of solute it contains. Concentration multiplied by volume gives that amount, so the product must be identical before and after. Enter any three of the four quantities and the fourth follows. The relationship holds for any concentration unit, molarity in mol/L, mass concentration in mg/mL, or percent weight per volume, provided the same unit appears on both sides of the equation and both volumes are expressed in the same unit as each other. Two practical points cause most laboratory errors. V2 is the total final volume, not the volume of diluent added; the diluent required is V2 minus V1, which is why a 10 mol/L stock taken to 2 mol/L in a 100 mL final volume needs 20 mL of stock and 80 mL of solvent. The equation also assumes volumes are additive, a good approximation for dilute aqueous solutions but one that fails for concentrated mixtures where solute and solvent interact; mixing ethanol and water, for example, yields less total volume than the sum of the parts. It cannot describe a solute that reacts, precipitates or changes protonation state on dilution. And for very large dilution factors a single step demands impractically small pipetted volumes, so serial dilution in stages of ten or one hundred is used instead.

The equation is a consequence of how concentration itself came to be defined. Quantitative volumetric analysis was developed in France in the early nineteenth century, with Joseph Louis Gay-Lussac performing systematic titrations in the 1820s and 1830s and introducing standard solutions of known strength for assaying silver and other materials. Karl Friedrich Mohr's textbook of titrimetric analysis, published in Germany in 1855, codified the apparatus and arithmetic of the field, including the burette and pipette in recognisably modern form, and made volume-based calculation the routine language of the analytical laboratory. The specific form C1V1 = C2V2 requires a concentration unit proportional to amount of substance per unit volume. That came with the atomic and molecular framework built on Amedeo Avogadro's 1811 hypothesis and settled at the Karlsruhe congress of 1860, and with Wilhelm Ostwald's promotion of the mole as a unit of amount of substance around the turn of the twentieth century. Molarity, moles of solute per litre of solution, became standard laboratory usage from there, and the mole was formally adopted as an SI base unit in 1971. Once concentration was defined in that way, the dilution relation was simply the arithmetic that followed from it.