TL;DR:
Molarity is the amount of solute per litre of solution:
M = mol / L.Diluting adds solvent but not solute, so the moles stay the same:
M₁V₁ = M₂V₂.To make 250 mL of 0.5 M from a 2 M stock, take 62.5 mL of stock and add water up to 250 mL.
A lab shelf holds a bottle of 2 M solution, but your experiment needs 0.5 M. You cannot just pour some in and hope. Two ideas solve it: what molarity means, and the fact that adding water does not change the amount of dissolved substance.
What molarity measures
Molarity (symbol M) is concentration expressed as moles of solute per litre of solution: M = n / V. A 2 M solution has 2 mol of dissolved substance in every litre. The unit M is read as molar, and mol/L means the same thing.
Notice that the volume is that of the finished solution, not of the water you started with. Dissolving a solid can change the total volume slightly, which is why chemists use a volumetric flask and fill to a marked line, instead of adding a measured amount of water.
Rearranged, n = M × V gives the moles in any volume. For example, 0.100 M solution in 500 mL (0.500 L) contains 0.100 × 0.500 = 0.0500 mol. See the mole, explained with a dozen if you want a refresher on moles.
Why M₁V₁ = M₂V₂ works
When you dilute, you add solvent only. The amount of solute does not change, so the moles before equal the moles after. Moles equal M × V, so M₁V₁ = M₂V₂, where 1 is the concentrated stock and 2 is the diluted solution.
The two volumes can use any matching unit, millilitres or litres, as long as both are the same, since the unit cancels. Concentrations must also match each other.
Worked example: 2 M down to 0.5 M
You need 250 mL of 0.5 M solution, and the stock is 2 M. How much stock do you take?
Write the knowns:
M₁ = 2 M,M₂ = 0.5 M,V₂ = 250 mL,V₁ = ?.Rearrange:
V₁ = M₂V₂ / M₁.Substitute:
V₁ = (0.5 × 250) / 2 = 125 / 2 = 62.5 mL.So take 62.5 mL of stock and add water until the total volume is 250 mL.
Check by counting moles. Stock taken: 2 × 0.0625 L = 0.125 mol. Final solution: 0.5 × 0.250 L = 0.125 mol. The same, as it must be.
A second check is the dilution factor: 2 ÷ 0.5 = 4, and 250 ÷ 62.5 = 4. The solution was made four times more dilute, so the final volume is four times the stock volume. Add roughly 250 − 62.5 = 187.5 mL of water, but top up to the mark rather than relying on that sum.
Making a solution from a solid
Say you want 500 mL of 0.100 M sodium chloride. Moles needed: 0.100 × 0.500 = 0.0500 mol. The molar mass of NaCl is 22.99 + 35.45 = 58.44 g/mol. Mass: 0.0500 × 58.44 = 2.92 g. You dissolve 2.92 g in some water in a 500 mL volumetric flask, then fill to the mark.
Lab steps for a dilution
Calculate first: Work out the volume of stock and write it down, with units, before you touch anything.
Put some water in the flask: Add roughly half of the final volume of water to the volumetric flask first.
Add the measured stock: Use a pipette or a burette to add the stock. Always add the concentrated solution to the water.
Fill to the mark: Add water until the bottom of the curved surface (the meniscus) sits on the line at eye level, adding the last bit with a dropper.
Mix and label: Stopper the flask and invert it several times. Label with the substance, concentration, date and your name.
Add acid to water: Wear splash goggles, gloves and a lab coat. When diluting a concentrated acid, always add the acid slowly to the water, never the other way round, because the mixing releases heat and can spit. Do these steps only in a supervised lab and never pipette by mouth.
How to check your answer
Mole check. Moles in the stock portion must equal moles in the final solution.
Direction. The diluted solution has a lower concentration and a larger volume than the portion of stock.
Dilution factor. The ratio of concentrations equals the ratio of volumes.
Units. Convert mL to L when you compute moles.
Mistakes that cost marks
Calling the volume of water added the final volume. The final volume is the whole solution.
Using
M₁V₁ = M₂V₂for mixing two different solutions, which it does not describe.Mixing mL for one volume and L for another.
Forgetting that M is per litre of solution, not per kilogram of solvent.
Pouring water into concentrated acid.
Where you will use it next
Molarity turns volumes into moles, which lets you use stoichiometry and limiting reagents with solutions. It also gives the concentrations behind pH and the logarithmic scale.
Frequently asked questions
What is the difference between molarity and molality?
Molarity is moles per litre of solution. Molality is moles per kilogram of solvent. Molality does not change with temperature, while molarity can, since volume changes.
Can I use M₁V₁ = M₂V₂ with percentages?
Yes, if both concentrations use the same units, because the amount of solute stays fixed. With molarity it is the most common use.
Get started
