Dilution (C₁V₁ = C₂V₂)
Leave any one of the four blank — stock, dose, or final volume solved.
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The engineering
Conservation of the dissolved stuff: what leaves the stock bottle is what ends up in the flask. Concentration can be in any unit — molar, mg/L, percent — as long as both C fields use the same one; the volumes likewise just need to match each other. The 'diluent to add' row assumes volumes are additive, which is a very good approximation for dilute aqueous work and slightly off for strong mixtures (ethanol–water famously shrinks).
Practice notes the formula cannot encode: add concentrated acid to water, never the reverse, and make up to the final volume in a volumetric flask rather than adding a calculated diluent when accuracy matters — 'to volume' beats 'plus volume' because of that non-additivity.
Where this math comes from
Quantitative 'how much is in there' chemistry starts with Lavoisier's balance-sheet chemistry in the 1780s, but dilution arithmetic became a daily tool with volumetric analysis: Gay-Lussac's burette work in the 1820s — he coined 'burette' and 'pipette' — let chemists titrate and dilute by volume with real precision.
The equation itself is older than any name attached to it; what changed is what C means. Ostwald and Arrhenius's 1880s physical chemistry made molarity the lingua franca, and C₁V₁ = C₂V₂ became the one formula every chemist, nurse, brewer, and pool owner shares.
- 1789Antoine LavoisierConservation of mass — the bookkeeping behind dilution.
- 1824Joseph Gay-LussacVolumetric analysis; the burette and pipette named (circa).
- 1887Svante ArrheniusIonic dissociation theory — concentration becomes physical chemistry.
See the full timeline of the math behind every calculator →
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