The M1V1 = M2V2 equation
In general chemistry the dilution equation is usually written with molarity:
M1 × V1 = M2 × V2
M1 is the molarity of the concentrated solution and V1 the volume of it you take; M2 is the molarity after diluting and V2 the final volume. Molarity is moles of solute per liter, so M × V is simply the number of moles. Adding water does not change the number of moles, which is why the two products are equal. OpenStax Chemistry 2e, section 3.3 derives the equation this way and uses it in three worked examples, all of which this M1V1 = M2V2 calculator reproduces.
Solving M1V1 = M2V2 for each variable
| Unknown | Rearranged equation |
|---|---|
| M1 | M1 = M2 × V2 ÷ V1 |
| V1 | V1 = M2 × V2 ÷ M1 |
| M2 | M2 = M1 × V1 ÷ V2 |
| V2 | V2 = M1 × V1 ÷ M2 |
Volumes only need to share a unit, because the unit cancels. The calculator accepts L, mL and µL on either side and M, mM, µM, nM and pM for the molarities, converting everything before solving.
Worked examples
Finding the volume of stock (V1). How much 1.59 M KOH is needed to make 5.00 L of 0.100 M KOH? V1 = 0.100 M × 5.00 L ÷ 1.59 M = 0.314 L (OpenStax Example 3.21). This is the default problem loaded in the calculator above.
Finding the final molarity (M2). If 0.850 L of 5.00 M copper(II) nitrate is diluted to 1.80 L, M2 = 5.00 M × 0.850 L ÷ 1.80 L = 2.36 M (OpenStax Example 3.19).
Finding the final volume (V2). What volume of 0.12 M HBr can be made from 11 mL of 0.45 M HBr? V2 = 0.45 M × 0.011 L ÷ 0.12 M = 0.041 L, or 41 mL (OpenStax Example 3.20).
A large dilution factor. The LibreTexts introductory chemistry chapter on solution dilution makes 8.00 L of 0.50 M nitric acid from 16 M stock: V1 = 0.50 × 8.00 ÷ 16 = 0.25 L, a dilution by a factor of 32.
Common mistakes with M1V1 = M2V2
- Treating V2 as the water added. V2 is the total volume. In the KOH example you measure 0.314 L of stock and add water until the total reaches 5.00 L, which is about 4.69 L of water.
- Mixing millimolar and molar. 50 mM is 0.050 M. If M1 is in M and M2 in mM, convert first, or select the units in the calculator and let it convert.
- Expecting M2 to be larger than M1. A dilution can only lower the concentration. The calculator warns you if the numbers imply the opposite, which usually means two values were swapped.
Beyond molarity
The same relationship works for any concentration unit, as long as both sides use the same kind: mg/mL, percent, or fold (10X to 1X). For those, use the general dilution calculator on the home page, or the percent solution dilution calculator for % w/v and % v/v. If you have to prepare the concentrated stock from a solid first, start with the stock solution calculator.