Dilution Formula (M1 × V1 = M2 × V2): How to Calculate, Examples & Calculator

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The dilution formula is M1 × V1 = M2 × V2. The amount of solute (in moles) is the same before and after dilution; only the volume of solvent changes. M1 and V1 are the concentration and volume of the initial (stock) solution; M2 and V2 are the concentration and volume of the final (diluted) solution.

M1 × V1 = M2 × V2
Symbol Meaning Common unit
M1 Initial (stock) concentration M (mol/L), N, or %
V1 Volume of stock solution taken mL or L
M2 Final (diluted) concentration same as M1
V2 Final volume same as V1

Both sides must use the same units. The same relation holds for normality (N1 × V1 = N2 × V2) and for percent (%1 × V1 = %2 × V2) as long as the solutions are dilute enough for density changes to be negligible.

Dilution calculator
Enter three values and leave one blank. V1 and V2 must use the same unit (e.g. mL), and so must M1 and M2 (M, N, or %).

Need molarity, the mass to prepare a solution, normality, or a solution from a concentrated reagent (37% HCl, 98% H2SO4)? Use the full chemistry calculator.

How to dilute a solution correctly

  1. Calculate V1 with the dilution formula: V1 = (M2 × V2) / M1.
  2. Take V1 of the stock solution with a volumetric pipette or burette.
  3. Transfer it to a volumetric flask of volume V2 that already contains some distilled water.
  4. Fill to the mark with distilled water, stopper, and invert to mix.
  5. Label it: substance, concentration, date, and name.

Safety: when diluting concentrated acids, always add acid to water slowly, never the reverse, because the heat of dilution can make the liquid spatter. Work in a fume hood and wear eye protection. See also hazardous chemical symbols.

Worked examples

1. Finding the stock volume (V1)

Prepare 100 mL of 0.1 M HCl from 1 M HCl. V1 = (0.1 × 100) / 1 = 10 mL.

2. Finding the final concentration (M2)

50 mL of 2 M NaOH is diluted to 250 mL. M2 = (2 × 50) / 250 = 0.4 M.

3. From a concentrated reagent in percent

Concentrated 37% HCl has a density of about 1.19 g/mL and a molar mass of 36.46 g/mol. Its molarity is M = (10 × % × density) / molar mass = (10 × 37 × 1.19) / 36.46 ≈ 12.08 M. For 100 mL of 0.1 M HCl: V1 = (0.1 × 100) / 12.08 ≈ 0.83 mL. Use the exact assay and density from your bottle's label and CoA.

4. Serial dilution

For large dilution factors (e.g. 1:1000) it is more accurate to dilute in steps: 1 mL to 10 mL (1:10), then 1 mL of that to 10 mL, and so on. The total factor is the product of the step factors (10 × 10 × 10 = 1000).

Common mistakes

  • Mixing units for V1 and V2 (mL with L).
  • Treating V2 as the volume of water added. V2 is the total final volume; water added ≈ V2 − V1.
  • Using a measuring cylinder for standard solutions; use a volumetric flask and pipette for accuracy.

Equipment and chemicals

Class A volumetric flasks and pipettes (e.g. Iwaki), distilled water, and analytical-grade reagents with lot CoA are available from AMI Scientific.

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