Dilution Equation (C1V1 = C2V2)

Adding solvent to a solution spreads the same solute through a larger volume — the moles do not change, only their crowding. Since moles equal concentration times volume, $C_1V_1$ must equal $C_2V_2$ before and after any dilution. To prepare 250 mL of 1.0 M hydrochloric acid from a 12.1 M concentrated stock, solve for $V_1$: (1.0 × 250)/12.1 ≈ 20.7 mL of stock, made up to the 250 mL mark with water. (And always add acid to water, never the reverse.)

The law is not limited to molarity — any concentration measure proportional to moles per volume works, as long as both sides use the same one. Water-treatment operators lean on it daily when dosing inhibitor or biocide from concentrated drums into recirculating loops, and biologists use the identical arithmetic for serial dilutions, where each step divides concentration by a fixed factor.