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Dilution Calculator

Enter any three values and leave one empty — the calculator will solve for the missing value using C₁V₁ = C₂V₂.

The dilution equation

Diluting a solution means adding solvent without adding more solute. The number of moles of solute is the same before and after — they’re just spread through a larger volume. Since (concentration) × (volume) equals moles of solute, that product is conserved:

C₁ × V₁ = C₂ × V₂

  • C₁ and V₁: concentration and volume of the stock solution you start with
  • C₂ and V₂: concentration and final total volume after dilution

The calculator takes three of the four values and solves for the fourth. It detects which field is empty and runs the corresponding rearrangement: V₁ = C₂V₂/C₁ if you need to know how much stock to pipette, C₂ = C₁V₁/V₂ if you want to predict the final concentration, and so on.

The non-obvious twist is unit consistency. Because C₁ and C₂ appear in a ratio (and so do V₁ and V₂), the equation is unit-agnostic — molarity, mass percent, mg/mL, ppm, and any other concentration measure all work. But the units have to match across each pair. C₁ in molarity and C₂ in mg/mL gives nonsense. The equation won’t warn you; it’ll just produce a number that’s confidently wrong. Pick a unit before plugging in.

The other thing that trips people up is what V₂ actually means. V₂ is the final total volume — stock plus added solvent — not the amount of solvent you added. If you need 500 mL of 0.1 M HCl from 12 M stock, V₁ = 4.17 mL of stock and V₂ = 500 mL of final solution, meaning you add about 496 mL of water (or, more precisely, dilute to the 500 mL mark in a volumetric flask).

The equation only handles dilutions of a single solution. Mixing two different solutes or two stocks of the same solute at different concentrations needs a mass-balance calculation across both components.

Worked examples

How much stock? 500 mL of 0.1 M HCl from 12 M stock. V₁ = (0.1 × 500)/12 = 4.17 mL of 12 M HCl, then dilute to 500 mL total.

Final concentration? 25 mL of 6.0 M NaOH diluted to 250 mL. C₂ = (6.0 × 25)/250 = 0.60 M.

Final volume? Take 10 mL of 5.0 M H₂SO₄ and dilute to 0.25 M. V₂ = (5.0 × 10)/0.25 = 200 mL total.

Percent dilution. 50 mL of 30% H₂O₂ diluted to 3%. V₂ = (30 × 50)/3 = 500 mL — same equation, different units.

Frequently Asked Questions

What is the dilution equation?
C₁V₁ = C₂V₂. C₁ is the stock concentration, V₁ is the volume of stock you take, C₂ is the diluted concentration, and V₂ is the final total volume after adding solvent. The equation works because moles of solute (which equals concentration times volume) doesn't change when you only add solvent. It rearranges to solve for any one of the four when the other three are known.
How do you solve for any variable in C₁V₁ = C₂V₂?
Algebraically rearrange to isolate the unknown. To find stock volume needed: V₁ = C₂V₂/C₁. To find final concentration: C₂ = C₁V₁/V₂. To find final volume: V₂ = C₁V₁/C₂. To find required stock concentration: C₁ = C₂V₂/V₁. The calculator detects which of the four input fields you left blank and runs the matching rearrangement.
What concentration units can I use?
Any units, as long as C₁ and C₂ match. Molarity, mass percent, mg/mL, ppm, weight/volume — all work because the units cancel out of the ratio C₁V₁/V₂. Volumes also need to match each other (mL with mL or L with L). The equation is unit-agnostic but unit-strict: mismatched units give wrong answers without raising any obvious flag.
Why does C₁V₁ = C₂V₂ work?
Diluting a solution adds solvent but doesn't change the moles of solute already in the flask. Concentration times volume equals moles of solute, so C × V is conserved across the dilution. C₁V₁ is moles of solute before dilution, C₂V₂ is moles of solute after, and they're the same number. This only holds when you're not adding more solute or removing any — pure solvent in, that's it.
How do I prepare a serial dilution?
A serial dilution is a chain of dilutions, each one a fixed factor below the last. For a 1:10 series, take 1 mL of the previous tube and add 9 mL of solvent (final volume 10 mL). Each step gives a tenfold drop, so five tubes span 10⁻¹ to 10⁻⁵ of the starting concentration. Apply C₁V₁ = C₂V₂ at each step, or just multiply by the dilution factor.