Dilution Calculator —
M₁V₁ = M₂V₂ solution calculator.

Field guide

The conservation of solute.

Dilution is one of the most common operations in a chemistry or biology lab, and the math behind it comes from a single fact: when you add solvent to a solution, the amount of solute in the container doesn’t change. Only the total volume — and hence the concentration — does.

Amount of solute = concentration × volume. Set that product equal before and after dilution and you have the formula every chemistry student memorises:

Subscript 1 is the stock you start from; subscript 2 is the diluted solution you finish with. Given any three of the four, the other is forced. The four rearrangements:

And the question every bench scientist actually wants the answer to — “how much solvent do I add?” — is just:

Molarity versus dilution.

These are often introduced together and confused. Molarity is a property of a single solution: how many moles of solute per litre. The Molarity Calculator is what you reach for to prepare a solution from a solid, where you need molecular weight to convert grams to moles.

Dilution is a property of two solutions — a relationship. You don’t need molecular weight here; the dilution equation doesn’t care what unit you put concentration in, as long as you use the same unit on both sides. M, mM, %w/v, mg/mL, ppm — all work, because the unit cancels.

How to prepare a dilution at the bench.

1. Decide your target. What concentration C₂ and final volume V₂ do you need?
2. Solve for V₁ from your stock: V₁ = (C₂ × V₂) ÷ C₁. The calculator does this and tells you the solvent volume directly.
3. Measure V₁ of the stock with the most accurate pipette or graduated cylinder available for that volume. For very small V₁ values, consider a serial dilution instead — repeatedly measuring 1 μL is less accurate than two 1:10 steps from a more concentrated solution.
4. Transfer to the receiving vessel and bring the total volume to V₂ with solvent (typically distilled or deionised water for aqueous work). Mix thoroughly — gentle inversion for most aqueous solutions; vortexing for small volumes.
5. Label everything — the new concentration, the date, the solvent, and your initials. Future-you will thank present-you.

Serial dilutions and why you need them.

Suppose you have a 1 M stock and you need a final solution at 10 μM. A single dilution would require V₁ = (10⁻⁵ × V₂) ÷ 1 — a 100,000-fold dilution. Even with a 1 L final volume, you’d be pipetting 10 μL of stock and praying for accuracy. The real answer is to do it in steps.

A serial dilution performs equal dilutions in sequence, each made from the previous step rather than from the original stock. With fold factor F, the concentration after k steps is:

Five 1:10 steps from a 1 M stock land at 1 × 10⁻⁵ M — the same 100,000-fold dilution, but built from five accurately-measurable 1:10 transfers. The calculator’s serial mode generates the full schedule, including the cumulative fold at each step, so you can pick the exact tube you need.

Common pitfalls.

• Unit symmetry on C. You can use any concentration unit, but C₁ and C₂ must be in the same unit. Mixing M and mM without converting first will give you an answer that’s off by a factor of 1000.
• V₁ > V₂ means you’re trying to concentrate. The dilution equation only adds solvent. If your calculation demands you remove solvent, you need a fresh, more concentrated stock — or evaporation.
• Don’t pipette tiny volumes from a tiny stock. If V₁ comes out under a few microlitres, switch to a serial dilution; the measurement error of a single small transfer can swamp the result.
• Mix after each step. In serial dilutions, an under-mixed tube poisons every step downstream.

Related calculators

For single-solution prep, use the Molarity Calculator; for molar masses, the Molecular Weight Calculator; for reaction yields, the Stoichiometry Calculator; and for acid–base solutions, the pH Calculator.

Disclaimer: This dilution calculator follows IUPAC 2021 recommendations for concentration units and uses the CODATA / NIST values where applicable. It is provided for educational reference only and is not a substitute for professional laboratory training, safety review, or institutional protocols. Always verify critical preparations against your method documentation and your facility’s safety data sheets.