Use the C₁V₁ = C₂V₂ dilution equation to solve for any unknown. Calculate stock volume, final volume, or concentration for any dilution.
The dilution equation C₁V₁ = C₂V₂ is arguably the most-used formula in laboratory science. It expresses the conservation of solute: the amount of solute in the stock aliquot (C₁ × V₁) equals the amount in the final diluted solution (C₂ × V₂). This simple relationship lets you solve for any one of the four variables when the other three are known.
Every scientist — from first-year chemistry students to senior researchers — uses this equation regularly for preparing dilutions from stock solutions. Whether diluting concentrated acid for a titration, making working solutions from a protein stock, diluting a PCR primer to working concentration, or preparing drug solutions for cell culture, C₁V₁ = C₂V₂ is the essential calculation.
This calculator solves for any of the four variables (C₁, V₁, C₂, or V₂), shows the dilution factor, the volume of diluent to add, and provides a visual diagram of the dilution process. It supports multiple concentration and volume units, making it instantly useful regardless of your field or the units in your protocol.
The C₁V₁ = C₂V₂ equation is simple, but unit mix-ups and arithmetic errors are common under time pressure. This calculator eliminates those mistakes and shows the diluent volume directly. This solution dilution calculator helps you compare outcomes quickly and reduce avoidable mistakes when making day-to-day care decisions. Use the estimate as a planning baseline and confirm final decisions with a qualified professional when risk is high.
C₁V₁ = C₂V₂. Solve for: V₁ = C₂V₂/C₁, V₂ = C₁V₁/C₂, C₁ = C₂V₂/V₁, C₂ = C₁V₁/V₂. Diluent Volume = V₂ − V₁. Dilution Factor = C₁/C₂.
Result: V₁ = 83.33 mL of stock
V₁ = (1 × 1000) / 12 = 83.33 mL. Take 83.33 mL of 12 M HCl and add 916.67 mL of water to make 1000 mL of 1 M HCl. Dilution factor = 12.
The dilution equation is a statement of mass conservation. The number of moles of solute doesn't change during dilution — you're only adding solvent. Since moles = concentration × volume, the product C × V before dilution must equal C × V after. This holds for ideal solutions where mixing is perfectly additive.
In molecular biology, C₁V₁ = C₂V₂ is used to dilute primers from 100 µM stock to 10 µM working solutions, dilute antibodies for Western blots, prepare agarose gel solutions, and set up PCR reaction mixes. In analytical chemistry, it's used to prepare calibration standards from a certified reference material stock. In pharmacology, it generates dose-response curves from a drug stock.
The biggest source of dilution error is measuring very small volumes of stock. If V₁ comes out to 0.5 µL, consider using a two-step dilution instead: first dilute to an intermediate concentration, then dilute again to the final target. This keeps all pipetting volumes in the accurate range of your equipment (typically >1 µL for micropipettes, >1 mL for glass pipettes).
Yes, as long as both concentrations use the same unit. Molarity, mg/mL, %, ppm — any consistent unit works because the equation is based on conservation of solute mass.
Yes. Both V₁ and V₂ must be in the same unit (both mL, both L, etc.). The units cancel in the equation, but they must be consistent.
C₁V₁ = C₂V₂ assumes volume additivity (V_total = V₁ + V_diluent). For solutions where mixing causes significant volume contraction (like ethanol + water), the final volume may differ slightly from the expected V₂.
You can't dilute to a higher concentration. If C₂ > C₁, you need a more concentrated stock or a different approach (like dissolving additional solute).
Always add concentrated acid to water, never the reverse. Adding water to concentrated acid causes violent boiling and spattering because the exothermic dissolution superheats the small water volume.
Accuracy depends on your volume measurement tools. Micropipettes (±0.5–1%), volumetric flasks (±0.05%), and graduated cylinders (±1–3%) each contribute uncertainty.