Physics 9702 · Paper 4

Moles & the Avogadro Constant

Gas questions constantly switch between counting molecules one by one and counting them in bulk. The mole is the bridge, and Paper 4 tests the crossing every year.

Counting in moles

One of any substance contains $6.02 \times 10^{23}$ particles. That number is the $N_A$ , defined as the number of particles per unit amount of substance (9702/41/O/N/24 Q3(a)).

n = \dfrac{N}{N_A}

Symbols

Watch the letters: capital $N$ counts molecules, small $n$ counts moles. Mixing them is the single most common slip in this topic, and the equation above converts between them in one line.
Mass connects through the molar mass: $m = n M$ . One mole of carbon-12 has a mass of exactly 12 g, and one molecule of mass 28 u (nitrogen, N₂) gives a molar mass of 28 g.

Worked example

Smallest case: molecules to moles

A container holds $3.2 \times 10^{24}$ molecules of oxygen. How many moles is that?

$n = \dfrac{N}{N_A} = \dfrac{3.2 \times 10^{24}}{6.02 \times 10^{23}} = 5.3\ \text{mol}$ .

Answer

n \approx 5.3\ \text{mol}

N_A links the two gas constants

The molar gas constant

R = 8.31\ \text{J K}^{-1}\text{mol}^{-1}

and the Boltzmann constant

k = 1.38 \times 10^{-23}\ \text{J K}^{-1}

say the same thing at two scales — per mole and per molecule. So

N_A = R/k

: divide and you get

6.02 \times 10^{23}\ \text{mol}^{-1}

. A 2024 paper asked for exactly this link (9702/41/O/N/24 Q3(a)).