Concentration of solutions

Solute, solvent and solution

In a glass of sugar water, the sugar is the solute and the water is the solvent. Together they make the solution. A strong solution has much solute in it, and a weak one has little. Chemists use four ways to say exactly how much.

Mass percent

• $\text{mass percent} = \dfrac{\text{mass of solute}}{\text{mass of solution}} \times 100$

The mass of the solution is the solute and the solvent together. $12\ \text{g}$ of sugar in $188\ \text{g}$ of water makes $200\ \text{g}$ of solution, so the sugar is $\dfrac{12}{200} \times 100 = 6\%$.

Mole fraction

The mole fraction of a substance is its share of all the moles in the solution. It has no unit.

• $x_A = \dfrac{n_A}{n_A + n_B}$, and $x_A + x_B = 1$.

$46\ \text{g}$ of ethanol is $1\ \text{mol}$, and $90\ \text{g}$ of water is $5\ \text{mol}$. In this mixture, $x_{\text{ethanol}} = \dfrac{1}{6} = 0.167$ and $x_{\text{water}} = \dfrac{5}{6} = 0.833$.

Molarity

Molarity, $M$, is the number of moles of solute in one litre of solution. Its unit is $\text{mol L}^{-1}$, often written $\text{M}$.

• $M = \dfrac{n}{V}$, with $V$ in litres.

$4.0\ \text{g}$ of sodium hydroxide ($40\ \text{g mol}^{-1}$) is $0.1\ \text{mol}$. Made up to $250\ \text{mL}$ of solution, it gives $\dfrac{0.1}{0.250} = 0.40\ \text{mol L}^{-1}$.

To make this solution, you weigh the solid, dissolve it, and then add water up to a mark on a volumetric flask. You do not simply add $250\ \text{mL}$ of water, because the solute takes up space too.

Molality

Molality, $m$, is the number of moles of solute in one kilogram of solvent. Its unit is $\text{mol kg}^{-1}$.

• $m = \dfrac{n}{\text{mass of solvent in kg}}$

$4.0\ \text{g}$ of sodium hydroxide in $500\ \text{g}$ of water gives $\dfrac{0.1}{0.500} = 0.20\ \text{mol kg}^{-1}$.

Molarity or molality?

When a solution warms up, it expands. Its volume grows, so its molarity falls a little. The masses do not change, so its molality stays the same. That is why careful work over a range of temperatures uses molality.

Practise with new numbers

Try it in Python

The program gives all four measures for one solution of glucose.

Common mistakes

• Millilitres in place of litres. $250\ \text{mL}$ is $0.250\ \text{L}$. Change it before you divide.
• The solution in place of the solvent. Molality uses the mass of the solvent alone.
• The solvent in place of the solution. Molarity uses the volume of the whole solution.
• Mole fractions that do not add up. The mole fractions of all the parts of a solution add up to $1$.

Remember these

• Mass percent $=$ mass of solute $\div$ mass of solution $\times 100$.
• Mole fraction: $x_A = \dfrac{n_A}{n_A + n_B}$. It has no unit, and all the fractions add up to $1$.
• Molarity: moles per litre of solution, in $\text{mol L}^{-1}$. It changes a little with temperature.
• Molality: moles per kilogram of solvent, in $\text{mol kg}^{-1}$. It does not change with temperature.

Check yourself

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

1. A syrup has $36\ \text{g}$ of sugar in $204\ \text{g}$ of water. What is the mass percent of sugar?

2. Which measure uses the volume of the solution, molarity or molality?

3. What mass of sodium hydroxide ($40\ \text{g mol}^{-1}$) is in $2\ \text{L}$ of a $0.3\ \text{M}$ solution?

4. Why is a solution of $1\ \text{mol L}^{-1}$ not the same as one of $1\ \text{mol kg}^{-1}$?

5. Why do we add water up to the mark of the flask, instead of adding $250\ \text{mL}$ of water?