Solution Dilution Calculator (C₁V₁ = C₂V₂)
Enter any three of C₁, V₁, C₂, V₂ and the dilution equation C₁V₁ = C₂V₂ instantly returns the fourth — together with the volume of diluent to add, the dilution factor and a step-by-step recipe. Built for everyday wet-lab, molecular biology, pharmacy and cell-culture work.
Enter three positive numbers (any three of C₁, V₁, C₂, V₂); C₁ and C₂ must be in the same unit family.
Result
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—
Equation
C₁ × V₁ = C₂ × V₂
C₁ and C₂ must share the same concentration unit (both M, or both mg/mL, etc.); V₁ and V₂ can be in different volume units.
Formula
C₁ × V₁ = C₂ × V₂ ⇒ V₁ = (C₂ × V₂) / C₁; diluent volume = V₂ − V₁
- · Underlying assumption: dilution adds or removes no solute, so concentration × volume is conserved across the step (Atkins, Physical Chemistry 11e §5.2).
- · C₁ and C₂ must share the same concentration unit (both M, both mg/mL, both % w/v, …) — otherwise the units do not cancel. This widget converts between molar units automatically when both sides are molar.
- · V₁ and V₂ can use different volume units (mL, L, µL); they are converted internally before solving.
- · Dilution factor = V₂ / V₁ = C₁ / C₂. A "1:10 dilution" means 1 part stock + 9 parts diluent — not 1 part stock + 10 parts diluent.
- · Serial dilution = repeating the same fold dilution n times; total factor = (per-step factor)^n. Useful for standard curves and assay titration.
- · For concentrated acids (concentrated HCl, H₂SO₄, etc.) always add acid to water, never the reverse, to avoid localised boiling and splashing.
- · The tool assumes volume additivity (V_total ≈ V_stock + V_diluent). At high organic-solvent concentrations (≥ 10% v/v ethanol, etc.) this introduces a small error — use a volumetric flask and bring up to the mark for precision work.
Frequently asked
I have a 1 M NaCl stock and want to make 250 mL of a 0.15 M working solution — how do I do it?
Use C₁V₁ = C₂V₂. With C₁ = 1 M, C₂ = 0.15 M, V₂ = 250 mL, solve V₁ = (0.15 × 250) / 1 = 37.5 mL. So pipette 37.5 mL of the 1 M stock and add 250 − 37.5 = 212.5 mL of deionised water (or your buffer) to bring the total volume to 250 mL — a 0.15 M solution. In this tool: pick "solve for V₁", set C₁ = 1 M, C₂ = 0.15 M, V₂ = 250 mL, and the answer and recipe appear immediately.
Does a "1:10 dilution" mean 1 part stock + 9 parts water, or 1 part stock + 10 parts water?
In biology and chemistry, "1:N" means 1 part stock to N parts total volume — so 1:10 = 1 part stock + 9 parts diluent (10 parts total) = a 10× dilution = new concentration is one-tenth the old. Writing it as "1+9" makes this unambiguous. Some industrial, pharmacy and continental-European usages instead read 1:10 as "1 part stock + 10 parts diluent" (an 11× dilution), so verify with the source. This tool always uses the international "final volume / stock volume" convention.
Why must C₁ and C₂ use the same unit? Can I mix M with mg/mL?
The dilution equation works because concentration units cancel on both sides — they must match. Molar units (M, mM, µM) interconvert freely (they're all mol/L), but to swap between molar and mg/mL you must know the molecular weight: c (M) = c (g/L) / MW. This widget auto-converts between molar units when both sides are molar; for mg/mL ↔ M you have to convert separately first, then run the dilution.
How does serial dilution work?
Serial dilution repeats the same fold dilution n times. With 10× dilutions five times you reach 10⁵ = 100,000× total. Example: starting from 1 mg/mL and taking 100 µL into 900 µL of PBS each time, tube #5 has 1 mg/mL ÷ 10⁵ = 10 ng/mL. Run it one step at a time in this widget: set C₁ = previous tube's C₂, V₁ = 100 µL, V₂ = 1000 µL, and repeat — the last C₂ is your final concentration.
The tool assumes that total volume equals stock plus diluent. Is that accurate?
For dilute aqueous samples (water, buffer, salt solutions, almost all cell-culture media) volume additivity holds to better than ~1% — no issue. For high-concentration organic solvents (concentrated ethanol, DMSO, glycerol) or concentrated acids, you see partial-molar-volume contraction on mixing: 50 mL pure ethanol + 50 mL water ≈ 96 mL, not 100 mL. For rigorous preparation use a volumetric flask and top up to the calibration mark rather than adding a fixed diluent volume.
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