The mole and molar mass

Counting by weighing

A bank does not count a bag of one-rupee coins one by one. It weighs the bag. If one coin weighs $3.76$ g and the bag weighs $3.76$ kg, the bag holds $1000$ coins.

Chemists count atoms in the same way. Atoms are far too small and too many to count. But if we know the mass of one atom, a balance can count them for us. The mole is the unit that makes this easy.

The mole

One mole is the amount of a substance that contains exactly $6.02214076 \times 10^{23}$ particles.

This number is the Avogadro constant, $N_A = 6.022 \times 10^{23}\ \text{mol}^{-1}$. The particles can be atoms, molecules, ions or electrons; we always say which. One mole of water is $6.022 \times 10^{23}$ water molecules. One mole of oxygen atoms is a different amount from one mole of oxygen molecules, $\text{O}_2$.

Why such an odd number? It was chosen so that one mole of carbon-12 atoms has a mass of exactly $12$ grams. Since 2019, the number itself is fixed by definition.

How big is it? If you counted one particle every second, a mole would take about $1.9 \times 10^{16}$ years, more than a million times the age of the universe. Yet a mole of water is only $18$ mL, about a tablespoonful. Molecules are that small.

Molar mass

The molar mass, $M$, is the mass of one mole of a substance, in grams per mole.

The molar mass in g/mol has the same number as the molecular mass (or formula mass) in u.

This works because one u is $\dfrac{1}{N_A}$ of a gram. The same number for one particle in u and for one mole in grams is the whole point of the mole.

Three conversions

Everything in this chapter comes down to two small formulas.

Here $n$ is the amount in moles, $m$ the mass in grams, $M$ the molar mass in grams per mole, and $N$ the number of particles.

• From mass to moles: divide by the molar mass.
• From moles to particles: multiply by the Avogadro constant.
• From molecules to atoms: multiply by the number of those atoms in one molecule.

Keep the mass at $27$ g and raise the molar mass. The same mass holds fewer moles of a heavier substance, so the line gets flatter.

A worked example

How many oxygen atoms are in $8.80$ g of carbon dioxide? Take $\text{C} = 12.01$ u and $\text{O} = 16.00$ u.

1. Molar mass. $M = 12.01 + 2 \times 16.00 = 44.01\ \text{g mol}^{-1}$.
2. Moles of $\text{CO}_2$. $n = \dfrac{8.80}{44.01} = 0.2000\ \text{mol}$.
3. Moles of oxygen atoms. Each molecule has $2$ oxygen atoms, so $2 \times 0.2000 = 0.4000\ \text{mol}$.
4. Number of atoms. $0.4000 \times 6.022 \times 10^{23} = 2.409 \times 10^{23}$ oxygen atoms.

The mass of one molecule follows too: $\dfrac{44.01\ \text{g}}{6.022 \times 10^{23}} = 7.308 \times 10^{-23}\ \text{g}$.

Practise with new numbers

Try it in Python

The program takes $25.0$ g of three substances and finds the moles, the molecules and the atoms in each.

Common mistakes

• Forgetting which particle. One mole of $\text{O}_2$ has $2$ moles of oxygen atoms. Always say molecules or atoms.
• Using the atomic mass for a gas of molecules. Oxygen gas is $\text{O}_2$, so its molar mass is $32.00\ \text{g mol}^{-1}$, not $16.00$.
• Dividing the wrong way. Moles are mass divided by molar mass. A check: a larger sample must give more moles.
• Losing the power of ten. Write $6.022 \times 10^{23}$ in full, and keep the power through each step.
• Mixing up mole and molar mass. The mole is an amount; the molar mass is grams for each mole.

Remember these

• One mole is $6.022 \times 10^{23}$ particles: the Avogadro constant, $N_A$.
• Molar mass in g/mol has the same number as the molecular or formula mass in u.
• $n = \dfrac{m}{M}$ and $N = n \times N_A$.
• Multiply by the atoms in one molecule to count atoms.
• The mass of one particle is $\dfrac{M}{N_A}$.

Check yourself

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

1. How many moles of molecules are in $54.06$ g of water, $M = 18.02\ \text{g mol}^{-1}$?

2. Which is more: one mole of oxygen atoms or one mole of oxygen molecules, by mass?

3. How many molecules are in $0.0250$ mol of a gas?

4. Why does one mole of carbon-12 atoms have a mass of exactly $12$ g?

5. Equal masses of hydrogen gas and nitrogen gas are taken. Which has more molecules, and how many times as many?