C1V1 = C2V2: How to Calculate Any Dilution (with Serial Dilutions)

The dilution equation C₁V₁ = C₂V₂ says the amount of solute stays the same when you dilute — so to find how much stock you need, rearrange to V₁ = C₂V₂ / C₁. To make 100 mL of a 1× solution from a 10× stock, that’s (1 × 100) / 10 = 10 mL of stock plus 90 mL of diluent. Solve for any unknown, or plan a full serial dilution, with the dilution calculator; here’s how it works.

The equation in one picture

Why it works: solute is conserved

When you dilute a solution, you add solvent but not more solute — so the amount of solute (grams, moles, whatever) is the same before and after. Amount = concentration × volume, so:

C₁ × V₁ = C₂ × V₂

where 1 is the concentrated stock and 2 is the diluted final solution. That’s the whole idea. To find any one value, rearrange for it — most often you know the stock concentration (C₁), the concentration you want (C₂) and the final volume (V₂), and you’re solving for how much stock to use:

V₁ = C₂ × V₂ / C₁

A worked example

Make 100 mL of 1× working solution from a 10× stock:

• C₁ = 10×, C₂ = 1×, V₂ = 100 mL
• V₁ = (1 × 100) / 10 = 10 mL of stock
• Diluent = 100 − 10 = 90 mL

So: measure 10 mL of the 10× stock, add 90 mL of diluent, and you have 100 mL of 1×. The dilution calculator solves for whichever value you leave blank, so you never have to rearrange by hand.

Why the units don’t matter (as long as they’re consistent)

Because concentration appears on both sides, the units cancel. Use molarity, mg/mL, µg/mL or % — the equation doesn’t care, provided both concentrations share a unit and both volumes share a unit. That’s also why this is M₁V₁ = M₂V₂ when the concentration happens to be molarity: same equation, one specific unit. And it’s why the tool needs no compound database and never goes stale — the maths is exact for whatever units you use.

Serial dilutions and the dilution factor

Sometimes one dilution can’t reach the concentration you need accurately (pipetting 1 µL of stock into 10 L isn’t practical). A serial dilution solves this by repeating a smaller, fixed fold-dilution step by step:

• 1:10, then 1:10 again → 1:100, then again → 1:1000, and so on.

Each step’s dilution factor is final volume ÷ sample volume, and the cumulative factor is the product of the steps. The dilution calculator’s serial-dilution planner lays out the transfer volume, diluent volume and cumulative fold-dilution for every tube — the recipe you actually pipette. Once you have a target concentration, the molarity calculator tells you how much to weigh out for the stock in the first place.

Quick summary

C₁V₁ = C₂V₂ conserves solute, so V₁ = C₂V₂/C₁ tells you how much stock to use — 10 mL stock + 90 mL diluent gives 100 mL of 1× from a 10× stock. Units cancel, so it works for any consistent concentration unit (that’s the same equation as M₁V₁ = M₂V₂). For very dilute targets, a serial dilution repeats a fold-step for an exponential series. Solve any dilution, or plan a serial one tube-by-tube, with the dilution calculator.

Sources: conservation of mass (amount of solute = concentration × volume) — standard analytical chemistry; general laboratory dilution practice. Educational information.