Molarity & dilution: practice problems with step-by-step solutions

Free Molarity & dilution practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: molarity and mass, dilution, mixing solutions.

Free practice problemEasy
200 mLc = 0.1 mol/L

What mass m of NaOH (M = 40 g/mol) do you need to make 200 mL of a 0.1 mol/L solution?

V = 200 mLc = 0.1 mol/LM = 40 g/mol
g
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What you practise

  1. Easymolarity and mass
  2. Mediumdilution
  3. Hardmixing solutions

How to solve Molarity & dilution problems

  1. Convert mass to moles with n = m/M.
  2. Convert the volume to litres.
  3. Molarity c = n/V in mol/L.
  4. For a dilution, the moles stay the same: c₁V₁ = c₂V₂.

Common mistakes

  • Leaving the volume in mL.
  • Giving the amount in moles instead of the concentration.
  • Confusing the volume of stock solution with the volume of water to add.

Key formulas

Molarity
c = n/V (mol/L)
Moles
n = m/M
Dilution
c1V1 = c2V2
Mixing
c = (c1V1 + c2V2)/(V1 + V2)

Learn Molarity & dilution from scratch

Before this: Stoichiometry

1What concentration is

A solution is a substance, the solute, dissolved in a liquid, the solvent. Salt water is salt dissolved in water.

A spoon of sugar in a cup tastes sweeter than the same spoon in a bucket. What matters is how much solute there is per volume: the concentration.

Chemists count the solute in moles, so they use molarity: the number of moles of solute per litre of solution.

c = n / V (mol/L)

A 1 mol/L solution contains 1 mol of solute in every litre. Half a litre of it contains 0.5 mol.

The unit mol/L is often written as M, so “0.5 M” means 0.5 mol/L.

Self-check 0.2 mol of glucose is dissolved to make 0.5 L of solution. What is the molarity?
  1. 0.1 mol/L
  2. 0.2 mol/L
  3. 0.4 mol/L
  4. 2.5 mol/L

Answer: 0.4 mol/L. c = n/V = 0.2/0.5 = 0.4 mol/L. Multiplying (0.1) or dividing the other way (2.5) uses the formula wrongly, and 0.2 is the amount, not the concentration.

2From grams to molarity

In the lab you weigh the solute in grams and measure the volume in millilitres. Two conversions bring you to mol/L.

  • Grams to moles with the molar mass: n = m / M.
  • Millilitres to litres: divide by 1000. So 250 mL = 0.250 L.

Together:

c = m / (M · V)

You can also go the other way: the mass you need for a solution is m = c · V · M.

Example: 100 mL of a 0.5 mol/L KCl solution (M = 74.55 g/mol) needs 0.5 · 0.100 · 74.55 = 3.73 g of KCl.

Self-check 5.85 g of NaCl (M = 58.5 g/mol) is dissolved to make 250 mL of solution. What is the molarity?
  1. 0.0004 mol/L
  2. 0.1 mol/L
  3. 0.4 mol/L
  4. 25 mol/L

Answer: 0.4 mol/L. n = 5.85/58.5 = 0.1 mol and V = 0.250 L, so c = 0.1/0.250 = 0.4 mol/L. 0.0004 keeps the volume in mL, and 0.1 is the amount in moles.

3Dilution and mixing

To dilute a solution you add solvent. The volume grows, but the number of moles of solute stays the same.

c1​V1​+ H₂Oc2​V2​

Before: n = c₁ · V₁. After: n = c₂ · V₂. The moles are equal, so:

c₁ · V₁ = c₂ · V₂

Here both volumes have the same unit, so you can keep them in mL. The water to add is V₂ − V₁.

When you mix two solutions of the same substance, add up the moles and divide by the total volume:

c = (c₁V₁ + c₂V₂) / (V₁ + V₂)

The result always lies between the two concentrations, closer to the one with the larger volume.

Self-check 100 mL of a 0.4 mol/L solution is mixed with 200 mL of a 0.6 mol/L solution of the same substance. What is the concentration of the mixture?
  1. 0.16 mol/L
  2. 0.5 mol/L
  3. 0.53 mol/L
  4. 1.0 mol/L

Answer: 0.53 mol/L. Moles: 0.4 · 100 + 0.6 · 200 = 160 mmol in 300 mL, so 160/300 = 0.53 mol/L. 0.5 is the plain average, which ignores that there is twice as much of the second solution.

4The method step by step

  1. Write down what is given and what is asked, each with its unit.
  2. For molarity from a mass: convert grams to moles with n = m/M, and mL to L.
  3. Calculate c = n/V, or m = c · V · M for the mass you need.
  4. For a dilution: use c₁V₁ = c₂V₂ and solve for the unknown. The water to add is V₂ − V₁.
  5. For mixing: add the moles of both solutions and divide by the total volume.
  6. Check that the answer makes sense: a diluted solution is weaker than the stock, a mixture lies between its parts.

5A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

You need 200 mL of a 0.25 mol/L solution and have a 2 mol/L stock solution. What volume V₁ of stock solution do you need, and how much water do you add?

1

List the values

Stock: c₁ = 2 mol/L, V₁ is unknown. Wanted: c₂ = 0.25 mol/L and V₂ = 200 mL.

2

Write the dilution rule

The moles of solute do not change when you add water.

c₁ · V₁ = c₂ · V₂
3

Solve for V₁

V₁ = c₂ · V₂ / c₁ = 0.25 · 200 / 2 = 25 mL

Both volumes are in mL, so no conversion is needed.

4

Find the water

Vwater = V₂ − V₁ = 200 − 25 = 175 mL
c1​ = 2 MV1​ = 25 mL+ H₂Oc2​ = 0.25 MV2​ = 200 mL
5

Check the answer

The stock is 2 / 0.25 = 8 times stronger, so it must be diluted 8 times. 25 mL × 8 = 200 mL. The numbers fit.

Answer

Take V₁ = 25 mL of stock and add 175 mL of water.

6Common mistakes

  • Leaving the volume in mL in c = n/V. The answer comes out 1000 times too small.
  • Giving the number of moles when the concentration is asked.
  • Confusing the volume of stock solution (V₁) with the volume of water to add (V₂ − V₁).
  • Swapping c₁ and c₂, which gives a stock volume larger than the final volume.
  • Averaging two concentrations when mixing, instead of adding the moles.
Self-check You need 100 mL of 0.4 mol/L solution from a 1 mol/L stock. How much water do you add to the stock?
  1. 100 mL
  2. 250 mL
  3. 40 mL
  4. 60 mL

Answer: 60 mL. V₁ = 0.4 · 100 / 1 = 40 mL of stock, so the water is 100 − 40 = 60 mL. 40 mL is the stock volume, not the water. 250 mL swaps c₁ and c₂.

7How exam questions look

  • Easy: molarity and mass. Find the molarity from a mass and a volume, or the mass needed for a solution.
  • Medium: dilution. Find the volume of stock solution needed, or the volume of water to add.
  • Hard: mixing solutions. Two volumes of the same substance at different concentrations are mixed; find the new concentration.

The molar mass is given in the question. Assume volumes add up when solutions are mixed.

Learn it step by step in the appLearn it step by step in the app

Worked examples

Example 1Easy
5 g KMnO4​V = 1000 mL

5 g of KMnO4 (M = 158.03 g/mol) is dissolved in water to make 1000 mL of solution. Find the molarity c.

m = 5 gM = 158.03 g/molV = 1000 mL

Find c (mol/L)

Show hints
  1. Convert the mass to moles: n = m/M.
  2. Molarity is moles per litre, so convert mL to L.
  3. n = 0.03164 mol.
Show the answer and the step-by-step solution

Molarity: concentration c = n/V in mol/L, with n = m/M. Diluting adds solvent but not solute, so the number of moles stays the same: c₁·V₁ = c₂·V₂. When two solutions of the same solute are mixed, the moles add up and so do the volumes.

Amount of solute

n = m/M = 5/158.03 = 0.03164 mol

Concentration

c = n/V = 0.03164/1 = 0.03164 mol/L

Answer

c = 0.03164 mol/L
Example 2Medium
2 mol/LV1​ = ?0.5 mol/L1000 mL

You need 1000 mL of 0.5 mol/L solution and have a 2 mol/L stock solution. What volume V1 of stock solution do you need?

c1 = 2 mol/Lc2 = 0.5 mol/LV2 = 1000 mL

Find V1 (mL)

Show hints
  1. Dilution keeps the number of moles: c₁·V₁ = c₂·V₂.
  2. Solve for V₁ = c₂·V₂/c₁. You can keep the volumes in mL.
  3. c₂·V₂ = 500 mmol.
Show the answer and the step-by-step solution

Molarity: concentration c = n/V in mol/L, with n = m/M. Diluting adds solvent but not solute, so the number of moles stays the same: c₁·V₁ = c₂·V₂. When two solutions of the same solute are mixed, the moles add up and so do the volumes.

Volume of stock solution

V1 = c2·V2/c1 = 0.5·1000/2 = 250 mL

Answer

V1 = 250 mL
Example 3Hard
1.5 mol/L250 mL0.6 mol/L500 mLc = ?750 mL

250 mL of a 1.5 mol/L solution is mixed with 500 mL of a 0.6 mol/L solution of the same substance. Find the concentration c of the mixture (volumes are additive).

c1 = 1.5 mol/LV1 = 250 mLc2 = 0.6 mol/LV2 = 500 mL

Find c (mol/L)

Show hints
  1. Find the moles in each solution: n = c·V.
  2. Add the moles and add the volumes.
  3. n = 375 + 300 mmol.
Show the answer and the step-by-step solution

Molarity: concentration c = n/V in mol/L, with n = m/M. Diluting adds solvent but not solute, so the number of moles stays the same: c₁·V₁ = c₂·V₂. When two solutions of the same solute are mixed, the moles add up and so do the volumes.

Total amount

n = 1.5·250 + 0.6·500 = 675 mmol

Concentration of the mixture

c = 675/750 = 0.9 mol/L

Answer

c = 0.9 mol/L

Questions and answers

What is Molarity & dilution?

A solution is a substance, the solute, dissolved in a liquid, the solvent. Salt water is salt dissolved in water.

A spoon of sugar in a cup tastes sweeter than the same spoon in a bucket. What matters is how much solute there is per volume: the concentration.

How do I solve Molarity & dilution problems step by step?
  1. Convert mass to moles with n = m/M.
  2. Convert the volume to litres.
  3. Molarity c = n/V in mol/L.
  4. For a dilution, the moles stay the same: c₁V₁ = c₂V₂.
What are common mistakes in Molarity & dilution?
  • Leaving the volume in mL.
  • Giving the amount in moles instead of the concentration.
  • Confusing the volume of stock solution with the volume of water to add.
What do the Molarity & dilution exercises cover?
  • Easy: molarity and mass
  • Medium: dilution
  • Hard: mixing solutions

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

Topics before and after this one

Learn first

Stoichiometry

Continue with

Chemical equilibrium

Calculators

Molarity calculatorDilution calculator (C1V1 = C2V2)

Courses with this topic

General Chemistry

More topics in Natural sciences

StoichiometryChemical equilibriumCalorimetry & enthalpypH & buffersGeneticsCell biology

Printable worksheet

Molarity & dilution worksheet with answers
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