How to use
Find the missing value in M₁V₁ = M₂V₂ with mixed units, dilution checks, significant figures and careful final-volume preparation guidance.
Enter exactly three positive values and leave the fourth numeric field blank. Choose units separately for each quantity. Results update as you type; the blank field determines what is calculated. Select Auto or 2, 3 or 4 significant figures, and optionally change the calculated result’s display unit.
Reset restores 2.0 M stock, a blank stock volume, 0.1 M target and 100 mL final volume. Swap concentrations exchanges only the stock and target concentration values and their units; volumes stay as entered. Copy Result exports the known values, calculated result, dilution factor and preparation or warnings. Show calculation opens six algebra steps. No values are saved or sent for calculation.
Formula and molarity
Molarity is the amount of solute in moles divided by the total solution volume in liters: M = n / V. Here M is molarity, n is amount of solute in moles and V is solution volume in liters. Molarity is commonly used for amount concentration expressed in mol/L. This tool calculates straightforward dilutions, rather than molar masses or amounts from chemical formulas.
The dilution equation M₁V₁ = M₂V₂ expresses conservation of the amount of solute. The amount before dilution equals the amount afterward.
- Stock concentration (M₁)
- The initial concentration of the solution being diluted.
- Stock volume (V₁)
- The volume of stock solution used.
- Target concentration (M₂)
- The final concentration after dilution.
- Final volume (V₂)
- The final total solution volume, including stock and solvent.
- M₁ = M₂V₂ / V₁
- V₁ = M₂V₂ / M₁
- M₂ = M₁V₁ / V₂
- V₂ = M₁V₁ / M₂
Units
Every concentration converts to mol/L and every volume converts to liters before calculation. Decimal SI scaling factors are exact and do not reduce significant figures. Mixed concentration and volume units are supported.
| Unit | Normalized value |
|---|---|
| M | 1 mol/L |
| mM | 10⁻³ M |
| µM | 10⁻⁶ M |
| nM | 10⁻⁹ M |
| L | 1 L |
| mL | 10⁻³ L |
| µL | 10⁻⁶ L |
The result defaults to the largest supported unit giving a value of at least 1. If none does, it uses the smallest supported unit. Thus 0.005 L becomes 5 mL, 0.000025 L becomes 25 µL and 0.005 M becomes 5 mM. A display-unit dropdown lets you choose another supported unit without changing the underlying quantity.
Dilution factor
For a valid dilution, dilution factor = M₁ / M₂ = V₂ / V₁. A 2 M stock diluted to 0.1 M has a factor of 20×. Equal concentrations give 1×, meaning no concentration change. A target higher than the stock is a concentration increase, so the calculator does not label its ratio as a valid dilution factor.
How to prepare a dilution
Measure the calculated stock volume and dilute to a final total volume. For example, measure 5.00 mL of stock and dilute to a final total volume of 100 mL. The 100 mL is the volume of the finished solution, not the amount of solvent added.
Do not universally substitute “add 95 mL of solvent.” Mixture volumes are not guaranteed to add exactly, and this calculator does not model contraction or expansion. Follow the appropriate laboratory procedure and use suitable volumetric equipment to reach the final total volume.
When solving a concentration, the result explains the calculated stock or resulting target concentration. It does not invent a stock-volume recipe. Concentration-increase scenarios show warnings instead of preparation instructions.
Common mistakes
- Leave exactly one numeric field blank. All four known values overdetermine this interface; multiple blanks do not define a unique result.
- Zero, negative, non-finite and malformed values are rejected.
- A target concentration greater than the stock cannot be made by simple dilution.
- A required stock volume greater than the final total volume indicates the same invalid dilution scenario.
- Choose concentration and volume units carefully. V₂ means final total volume, not solvent volume.
A concentration increase still has an algebraic solution to the equation, but the result is flagged and no dilution preparation instruction is supplied.
Precision and significant figures
Auto uses the fewest significant figures in the three known written values, capped at 6. Leading zeros do not count; interior zeros and decimal trailing zeros do. In an ordinary integer without a decimal point, trailing zeros are treated as ambiguous and do not count. Therefore 100 has 1 significant figure under this convention; write 100. or 1.00e2 to explicitly state 3. The default 2.0, 0.1 and 100 inputs produce 5 mL with 1 significant figure.
The explicit 2, 3 and 4 options format the result to that many significant figures. These settings add no measurement accuracy. Calculations and SI scaling use exact fractions; display rounding uses round half to even. Scientific notation keeps intended significant zeros unambiguous. Known values retain their entered text in preparation wording. Non-integer dilution factors use the selected result precision; integer ratios are shown without unnecessary decimal zeros.
The worked examples below choose stated reporting precision for illustration. Follow the precision warranted by your measurements and laboratory protocol.
You can check significant figures in the dilution arithmetic as a separate reporting check. Rounded significant figures do not add measurement accuracy, change the exact-fraction calculation or replace laboratory protocol and safety guidance.
Worked examples
Find stock volume
Stock concentration 2.00 M, target 0.100 M, final volume 100 mL. Find V₁.
- V₁ = M₂V₂ / M₁.
- V₁ = (0.100 × 100) / 2.00 = 5.00 mL, shown with the 3-significant-figure setting.
- Dilution factor = 2.00 / 0.100 = 20×.
Measure 5.00 mL of stock solution and dilute to a final total volume of 100 mL. Plain 100 is ambiguous for measurement precision; this example explicitly chooses 3 figures for the calculated result.
Mixed concentration units
Stock 2.5 M, target 50 mM, final volume 250 mL. Convert 50 mM = 0.050 M. V₁ = (0.050 × 250) / 2.5 = 5 mL. Dilution factor = 50×.
Find target concentration
M₁ = 1.0 M, V₁ = 10 mL and V₂ = 100 mL. M₂ = M₁V₁ / V₂ = (1.0 × 10) / 100 = 0.10 M, shown with 2 significant figures. This is 100 mM (1.0 × 10² mM at the same stated precision).
Assumptions and limitations
M₁V₁ = M₂V₂ assumes the amount of solute is conserved during dilution. It applies to straightforward dilution with a consistent solute and amount-concentration definition. It does not imply all real solution systems behave ideally.
This calculator does not model chemical reactions, solution contraction or expansion, density effects, precipitation, changing chemical species, acid/base equilibrium, serial dilution chains or mixing multiple stock solutions. It has no chemical or molecular-weight database, pH, reagent recipes or inventory.
Inputs allow positive decimals or e notation, at most 64 characters and exponents from −100 to 100. Both known and calculated quantities must be finite and positive, with concentration at most 10¹² M and volume at most 10¹⁵ L after normalization. These are implementation safety bounds, not physically realistic chemistry limits. Input remains in page memory; refresh clears it.
Dilution molarity FAQ
What does M₁V₁ = M₂V₂ mean?
It means the amount of solute in the stock volume equals the amount in the final total solution volume when solute is conserved.
What is molarity?
Molarity is amount of solute in moles divided by solution volume in liters, commonly expressed as mol/L or M.
Can I solve for any missing variable?
Yes. Leave exactly one of M₁, V₁, M₂ or V₂ blank and enter the other three.
Can I mix M and mM in the same calculation?
Yes. The tool normalizes each concentration to M. It also supports µM and nM.
Can I mix liters, milliliters, and microliters?
Yes. Every volume normalizes to liters before the equation is solved.
What is a dilution factor?
For a valid dilution it is stock concentration divided by target concentration, equal to final volume divided by stock volume.
Why does the calculator say “dilute to final volume” instead of “add solvent”?
The final volume is the finished solution volume. Volumes may not add exactly, so an assumed solvent-volume difference can be misleading.
What if my target concentration is higher than my stock concentration?
Simple dilution cannot increase concentration. The calculator flags this case and suppresses dilution preparation instructions.
How many significant figures should I use?
Use the precision supported by your measured inputs. Auto follows the fewest written figures, capped at 6; ordinary integer trailing zeros are ambiguous. Explicit settings change display precision, not accuracy.
When should M₁V₁ = M₂V₂ not be used?
Do not use it as a model for reactions, changing species, precipitation, equilibria, density or volume effects, serial chains or mixtures of multiple stock solutions.
Sources
Reviewed October 1, 2026. IUPAC supports the amount-concentration terminology, NIST explains SI prefixes, and OpenStax covers molarity and dilution. Auto precision and safety bounds are documented product conventions. These sources do not endorse BuzCalculator.