🧫 Molarity and Dilution

Written and maintained by the PhDino author · Last reviewed 21 September 2026 · Checked against 1 independent reference calculation · how PhDino checks its numbers

Combining a concentrated stock solution with water to reach a target, more dilute concentration.

Concentrated stock solutions are common in a lab because they're more practical to prepare and store than every working concentration a procedure might call for — one bottle of 1 M stock can become dozens of different working dilutions as needed, instead of weighing out fresh solid every time.

Diluting a stock rests on one simple fact: the number of moles of solute doesn't change when you add water — only the volume it's dissolved in changes, so the concentration drops in exact proportion to how much the volume increased. That's the entire content of the dilution equation, expressed as concentration times volume being conserved before and after.

Key formula

M₁V₁ = M₂V₂
Volume of stock needed = (target concentration × target volume) ÷ stock concentration

Variables

M₁, V₁

concentration and volume of the concentrated stock

M₂, V₂

concentration and volume of the final diluted solution

How to use the Molarity / Dilution calculator

Use this to plan a dilution: how much concentrated stock to measure out, and how much water to bring the total to the volume you need, for a target concentration. It is the everyday calculation in a teaching lab, a home brewery or a cleaning-solution recipe, all built on C₁V₁ = C₂V₂.

Enter the stock concentration, the concentration you want and the final volume. The target cannot be higher than the stock: water can only dilute, never concentrate.

Stock Concentration (M)

The molarity of the solution you start from, in moles per litre, from the bottle label. Concentrated hydrochloric acid is about 12 M, sulfuric about 18 M and nitric about 16 M. A bottle labelled in percent needs converting to molarity first.

Target Concentration (M)

The molarity you want after dilution, in the same units as the stock.

Target Volume (mL)

The final volume of the diluted solution in millilitres, for example the size of the volumetric flask you will use.

Worked example: a dilute hydrochloric acid solution

A lab needs 500 mL of 0.25 M hydrochloric acid, and the stock on the shelf is 12 M concentrated acid. How much stock is needed, and how much water?

Worked by hand:

1. Moles of solute needed. 0.25 mol/L × 500 mL ÷ 1,000 = 0.125 mol of HCl in the final solution.
2. Volume of stock that holds that many moles. V₁ = C₂V₂ ÷ C₁ = 0.25 × 500 ÷ 12 = 10.42 mL.
3. Water to make up the volume. 500 − 10.42 = 489.6 mL.
4. Dilution factor. 12 ÷ 0.25 = 48, so the stock is diluted 48-fold.

Measure 10.4 mL of the concentrated acid and dilute to 500 mL. Use safe technique: add the acid to the water, never the water to the acid, with eye protection, because concentrated acid releases a lot of heat when it meets water. If the only stock were 6 M, the same recipe would need 20.8 mL.

Reading the result: accuracy at small volumes

The arithmetic is exact, but the pipette is not. Measuring 10.4 mL of stock is easy with a burette or a graduated pipette; measuring 0.3 mL is much less reliable, and any error is multiplied by the dilution factor. When the stock volume comes out below a millilitre or so, dilute in two stages instead of one.

• In practice, put the measured stock into a volumetric flask and add water up to the mark, rather than adding the calculated volume of water, because the volumes of concentrated solutions do not add exactly.
• Always add concentrated acid to water slowly while stirring, never the reverse, and use appropriate protective equipment.
• A more concentrated stock means a larger dilution factor and a larger error from any given measurement, so choose a stock as close to your target as is convenient.
• Label the finished solution with its concentration, the date and the hazards.

Notes & limitations

• This assumes the volumes are simply additive (stock volume plus water volume equals final volume), which is a good approximation for dilute aqueous solutions but not exact for concentrated solutions where mixing changes total volume slightly — precise work uses a volumetric flask filled to the mark with water rather than adding a calculated water volume.
• Always add stock solution to a partial volume of water and then dilute to the final mark, rather than adding water to a concentrated stock — for most solutions this is just good practice, but for some (concentrated acids especially) adding water to concentrate can cause dangerous spattering from the heat released.

Common mistakes

• Mixing units, such as a stock in molarity and a target in percent, or a volume in litres against millilitres. Convert everything before you enter it.
• Adding water to acid. The heat released can boil the water and spatter concentrated acid.
• Adding the calculated water volume rather than diluting to the mark, which makes the final volume and concentration slightly wrong for strong solutions.
• Using a target concentration above the stock concentration, which is not a dilution and is rejected by the calculator.
• Trusting a very small pipetted volume. If the stock volume is under about a millilitre, dilute in steps.

Frequently asked questions