Lesson 4 · 30 min

The mole: counting atoms by weighing

The Avogadro constant has been exact since 2019; compute molar masses from formulas and move freely between mass, moles and particles.

In this lesson

State the definition of the mole and the Avogadro constant (exact since 2019)
Compute molar masses from formulas, including brackets and water of crystallisation
Convert between mass, amount in moles and number of particles
Explain why relative atomic masses are not whole numbers

How many water molecules are in a teaspoon of water (about 5.0 g)? Roughly $1.7\times10^{23}$ — counting one per second would take about $5\times10^{15}$ years, hundreds of thousands of times the age of the Universe. Atoms are too small and too many to count, yet reactions happen by number: two hydrogen molecules for every oxygen molecule. Chemists do what a bank does with coins: weigh instead of count. If you know how much each particle weighs, weighing the whole is the same as counting. The mole is a unit that counts particles but is used by weighing.

Predict first

Which contains more atoms: 12.0 g of carbon or 12.0 g of magnesium?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

Carbon Magnesium The same — the masses are equal Cannot be compared

The mole and the Avogadro constant

Since 20 May 2019 one mole (mol) contains exactly $6.022\,140\,76\times10^{23}$ elementary entities — atoms, molecules, ions or formula units, and you must say which. That number is the Avogadro constant, $N_\mathrm{A} = 6.022\,140\,76\times10^{23}\ \text{mol}^{-1}$ , a defined value with no uncertainty.

The mass of one mole of a substance is its molar mass $M$ , in g/mol. It is numerically equal to the relative atomic or molecular mass: water’s relative molecular mass is 18.015, so $M(\ce{H2O}) = 18.015$ g/mol.

n = \frac{m}{M} = \frac{N}{N_\mathrm{A}} \qquad M = \sum_i \nu_i\,A_\mathrm{r}(i)\ \text{g/mol}

The amount of substance links the mass you weigh to the number of particles.

$n$: amount of substance, in mol
$m$: mass, in g
$M$: molar mass, in g/mol
$N$: number of particles (a pure number)
$N_\mathrm{A}$: Avogadro constant, 6.022 140 76 × 10²³ mol⁻¹ (exact)
$\nu_i\,A_\mathrm{r}(i)$: number of atoms of element i in the formula times its relative atomic mass

Formula → atoms → molar mass → mass ⇄ moles ⇄ particles

Molar-mass calculator

Formula

Examples

\ce{CuSO4*5H2O}

Molar mass M 249.68g/mol

Mass of one formula unit: 249.677 u = 4.146 × 10⁻²² g

Element	Atoms	Atomic weight	Subtotal / (g/mol)	Mass %
Cu Copper	1	63.546	63.546	25.45%
S Sulfur	1	32.06	32.060	12.84%
O Oxygen	9	15.999	143.991	57.67%
H Hydrogen	10	1.008	10.080	4.04%
Σ	21		249.677	100.00%

Mass ⇄ moles ⇄ particles

Mass m / g Amount n / mol Particles N

Converted values keep the significant figures of the number you typed (3).

Atoms of each element in it: Cu: 6.03 × 10²¹S: 6.03 × 10²¹O: 5.43 × 10²²H: 6.03 × 10²²

Look at the breakdown for copper sulfate pentahydrate: the water of crystallisation is 36% of the mass. Then try Ca3(PO4)2, K4[Fe(CN)6] and a deliberately wrong co2.

Worked example

How many molecules and hydrogen atoms in 36.0 g of water?

Find the amount, the number of molecules and the number of hydrogen atoms in 36.0 g of water.

Concept check

How many oxygen atoms are in 1.00 mol of $\ce{CO2}$ ?

6.02\times10^{23}

1.20\times10^{24}

1.81\times10^{24}

3.01\times10^{23}

Try the calculation

What is the mass of 0.250 mol of sodium chloride (NaCl)?

Key takeaways

One mole is exactly $6.022\,140\,76\times10^{23}$ specified particles — the Avogadro constant.
Molar mass in g/mol equals the relative atomic or molecular mass, summed over the atoms in the formula.
$n = m/M = N/N_\mathrm{A}$ connects mass, amount and number of particles.
Always name the entity: 1 mol of O and 1 mol of O₂ differ in mass by a factor of two.

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Data and references

BIPM, SI Brochure, 9th ed. (2019): definition of the mole.
IUPAC Commission on Isotopic Abundances and Atomic Weights, “Standard atomic weights of the elements 2021”, Pure Appl. Chem. 94, 573 (2022) — abridged values used in the element table.
E. Tiesinga et al., CODATA 2018, Rev. Mod. Phys. 93, 025010 (2021): molar mass of carbon-12.