C₁V₁=C₂V₂ explained

§ 1The variables

• C₁ · concentration of the stock solution (M, mM, %, mg/ml).
• V₁ · volume of stock you pipette out (ml, µl).
• C₂ · target concentration of the working solution.
• V₂ · final total volume of the working solution.

§ 2Rearranged forms

• Solve for V₁: V₁ = (C₂ × V₂) ÷ C₁
• Solve for V₂: V₂ = (C₁ × V₁) ÷ C₂
• Solve for C₁: C₁ = (C₂ × V₂) ÷ V₁
• Solve for C₂: C₂ = (C₁ × V₁) ÷ V₂

§ 3Worked example

1. You hold a 100 µM (0.1 mM) tetracycline stock. You need 10 ml of 5 µM working solution for a plate assay.
2. Convert to common units. 100 µM and 5 µM are already compatible.
3. V₁ = (5 × 10 000 µl) ÷ 100 = 500 µl of stock.
4. Pipette 500 µl of stock into a 10 ml vessel. Top up with diluent to 10 ml. Mix.
5. Sanity check. 100 µM × 0.5 ml = 50 nmol. 5 µM × 10 ml = 50 nmol. Moles match.

§ 4When it stops working

Real solutions deviate from ideal when solute and solvent have markedly different densities, when there is partial molar volume contraction (ethanol-water), or when the solute is a hydrate whose formula mass already includes water. For routine bench work, the formula holds to better than 1 % below 30 % w/v.^[14]

§ 5Related

• C₁V₁ calculator
• Molarity mode
• Molarity from stock
• Serial dilution