Ways of expressing concentration

Many labels for one idea

Look at four labels. A bottle of vinegar says 5 % acetic acid. A hospital drip says 0.9 g of salt in 100 mL. A report on well water says arsenic 8 ppb. A bottle in the laboratory says hydrochloric acid, 12 mol/L.

Each label tells you how much solute is in a solution. Why are there so many ways to say it?

Because each job needs a different answer. A cook wants a percentage. A chemist who must count particles wants moles. An engineer who heats the liquid wants a number that does not change with temperature. A health officer who finds a trace of poison wants a very small unit.

The measures you know

You met four measures in Class 11. Call the solute A and the solvent B.

• Mass percent: $\dfrac{\text{mass of A}}{\text{mass of solution}} \times 100$. It has no unit.
• Mole fraction: $x_A = \dfrac{n_A}{n_A + n_B}$. It has no unit. The mole fractions of all the parts add up to 1.
• Molarity: $M = \dfrac{n_A}{\text{volume of solution in litres}}$. Its unit is $\text{mol/L}$.
• Molality: $m = \dfrac{n_A}{\text{mass of solvent in kg}}$. Its unit is $\text{mol/kg}$.

The picture shows the difference that causes most mistakes. Molarity needs the whole solution made up to a volume. Molality needs only the solvent weighed on a balance, and the solution can take any volume.

Three more measures

Volume percent is used when a liquid is dissolved in a liquid. It is $\dfrac{\text{volume of A}}{\text{volume of solution}} \times 100$. Rubbing alcohol that is 70 % ethanol by volume has 70 mL of ethanol in every 100 mL of solution. When ethanol and water mix, the volume shrinks a little, so the volume of the solution is measured. It is not found by adding.

Mass by volume percent is the grams of solute in 100 mL of solution. A drip that says 0.9 g in 100 mL is 0.9 % (mass/volume). Take care: this is not a mass percent, because the 100 mL is a volume.

Parts per million (ppm) is for traces. It is a mass fraction multiplied by a million:

A mass percent is parts per hundred, so $1\ \% = 10\,000\ \text{ppm}$. For parts per billion, multiply by a billion. In water, $1\ \text{ppm} = 1\ \text{mg}$ of solute in $1\ \text{kg}$ of solution, and $1\ \text{ppb} = 1\ \mu\text{g}$ in $1\ \text{kg}$.

In a very dilute water solution, $1\ \text{L}$ has a mass of about $1\ \text{kg}$. So $1\ \text{ppm}$ is also about $1\ \text{mg/L}$. The World Health Organization guideline for fluoride in drinking water is $1.5\ \text{ppm}$, and for arsenic it is $10\ \text{ppb}$.

Try it yourself: a trace in water

What changes with temperature?

Warm a solution and it expands. Its volume grows, the mass stays the same, and the number of moles stays the same.

• Molarity divides by a volume, so it falls a little. So do volume percent and mass by volume percent.
• Mass percent, ppm, mole fraction and molality use only masses or moles. They do not change.

That is why careful work over a range of temperatures uses molality or mole fraction.

Changing from one measure to another

Do not memorise a formula for every change. Use one method.

1. Pick a starting amount that makes the sum easy. Take $1\ \text{L}$ of solution if you know the molarity. Take $100\ \text{g}$ of solution if you know a mass percent. Take $1\ \text{kg}$ of solvent if you know the molality.
2. Fill in everything. Find the moles of solute, the mass of solute, the mass of solvent, the moles of solvent, the mass of solution and the volume of solution.
3. Use the density to join mass and volume. Mass of solution $=$ volume $\times$ density.
4. Write the measure you want.

Example 1: from molarity. A solution of sodium carbonate, $\text{Na}_2\text{CO}_3$ ($106\ \text{g/mol}$), is $2.00\ \text{mol/L}$ and its density is $1.19\ \text{g/mL}$. Find the mass percent, the molality and the mole fraction.

• Take $1\ \text{L}$. The solute is $2.00\ \text{mol}$, which is $2.00 \times 106 = 212\ \text{g}$.
• The solution has a mass of $1000 \times 1.19 = 1190\ \text{g}$. So the water is $1190 - 212 = 978\ \text{g}$, which is $54.3\ \text{mol}$.
• Mass percent: $\dfrac{212}{1190} \times 100 = 17.8\ \%$.
• Molality: $\dfrac{2.00\ \text{mol}}{0.978\ \text{kg}} = 2.04\ \text{mol/kg}$.
• Mole fraction: $x = \dfrac{2.00}{2.00 + 54.3} = 0.0355$.

Example 2: from molality. A solution of ammonium chloride, $\text{NH}_4\text{Cl}$ ($53.5\ \text{g/mol}$), is $1.50\ \text{mol/kg}$ and its density is $1.02\ \text{g/mL}$. Find the molarity.

• Take $1\ \text{kg}$ of water. The solute is $1.50\ \text{mol}$, which is $1.50 \times 53.5 = 80.3\ \text{g}$.
• The solution has a mass of $1000 + 80.3 = 1080.3\ \text{g}$. Its volume is $\dfrac{1080.3}{1.02} = 1059\ \text{mL} = 1.059\ \text{L}$.
• Molarity: $\dfrac{1.50\ \text{mol}}{1.059\ \text{L}} = 1.42\ \text{mol/L}$.

Check the answer. The molarity is a little less than the molality. One litre of this solution holds less than one kilogram of water, so that is what we expect.

Try it yourself: from a label

When do molarity and molality agree?

In a very dilute water solution, $1\ \text{L}$ of solution holds about $1\ \text{kg}$ of water. So the molarity and the molality are almost equal. As the solution gets stronger, the solute adds mass and volume, and the two measures drift apart. Move the slider in the sim and watch the curves.

The sim uses the measured density of sucrose solutions, which rises from $1.000\ \text{g/mL}$ for pure water as sucrose is added. At every setting, the molality is above the molarity. One litre of this solution holds less than one kilogram of water, and the molality divides by the water alone.

Diluting, mixing and ions

When you add water to a solution, the moles of solute do not change. So the moles before equal the moles after:

When you mix two solutions of the same solute, the moles add, and so do the volumes (for weak water solutions):

A salt that splits into ions gives each ion a concentration. Count the ions in the formula. A solution of aluminium sulphate, $\text{Al}_2(\text{SO}_4)_3$, at $0.16\ \text{mol/L}$ has $0.32\ \text{mol/L}$ of $\text{Al}^{3+}$ and $0.48\ \text{mol/L}$ of $\text{SO}_4^{2-}$.

Try it yourself: mixing

Try it in Python

The program turns three labels into molarity, molality and mole fraction. All three use 100 g of solution as the starting amount.

Common mistakes

• Using the volume of the solvent in a molarity. Molarity divides by the volume of the whole solution.
• Using the mass of the solution in a molality. Molality divides by the mass of the solvent only.
• Adding the volumes of two strong solutions without care. For strong solutions, the volumes do not add exactly. The question will tell you what to assume.
• Averaging two molarities. Add the moles, then divide by the total volume.
• Forgetting that one ppm is one milligram per kilogram. Do not change it to grams.

Remember these

• Mass percent, ppm, mole fraction and molality do not change with temperature. Molarity does.
• $1\ \%= 10\,000\ \text{ppm}$. In water, $1\ \text{ppm} \approx 1\ \text{mg/L}$.
• To change a measure, choose a starting amount, fill in every quantity, and use the density to join mass and volume.
• $M_1V_1 = M_2V_2$ for dilution. For mixing, add the moles and divide by the total volume.
• Ion concentration is the salt concentration times the number of that ion in the formula.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A bottle of rubbing alcohol says 70 % ethanol by volume. What does this mean, and is it a mass percent?

2. A solution is warmed from $20\ ^\circ\text{C}$ to $50\ ^\circ\text{C}$. Which of these changes: mass percent, molarity, molality, mole fraction?

3. A sample of river water has a nitrate concentration of $5\ \text{ppm}$. How many milligrams of nitrate are in $3\ \text{kg}$ of the water?

4. Why are the molarity and the molality of a $0.001\ \text{mol/L}$ solution in water almost equal?

5. A solution of calcium chloride, $\text{CaCl}_2$, has a concentration of $0.38\ \text{mol/L}$. What is the concentration of chloride ions?