Moles and the Avogadro Constant

(Higher Tier only) Everything on this page — moles, the Avogadro constant and the mole–mass conversion — is assessed only at Higher Tier.

Atoms and molecules are so small that a spoonful of any substance contains an astronomical number of them. Counting particles one by one is impossible, so chemists group them into a standard package called the mole, in the same way a shop counts eggs in dozens.

The mole (mol) is the unit for amount of substance. One mole of any substance contains 6.02 × 10²³ particles. This number is the Avogadro constant.

The particles can be atoms, molecules, ions or formula units, depending on the substance. One mole of carbon atoms, one mole of water molecules and one mole of sodium ions all contain 6.02 × 10²³ particles.

The mole is powerful because it links the invisible world of particles to masses you can actually weigh on a balance. That link is the relative formula mass, which the next slide sets out.

The definition that makes moles usable in the lab is this: the mass of one mole of a substance in grams is numerically equal to its relative formula mass.

The mass of one mole of a substance, in grams, equals its relative formula mass (Mr). This mass is called the molar mass.

So if you know the Mr, you immediately know the mass of one mole.

Substance	Formula	Mr	Mass of 1 mole
Carbon	C	12	12 g
Water	H₂O	18	18 g
Carbon dioxide	CO₂	44	44 g
Sodium hydroxide	NaOH	40	40 g
Calcium carbonate	CaCO₃	100	100 g

Each of these different masses contains the very same number of particles, 6.02 × 10²³. One mole of carbon (12 g) and one mole of calcium carbonate (100 g) hold identical numbers of particles; the masses differ only because the particles themselves have different masses.

The central equation of quantitative chemistry connects moles, mass and Mr.

$\text{moles}=\frac{\text{mass (g)}}{M_r}$

Worked example — how many moles are in 20 g of sodium hydroxide, NaOH?

First the Mr: $23+16+1=40$.

$\text{moles}=\frac{20}{40}=0.5\text{mol}$

Worked example — how many moles are in 88 g of carbon dioxide, CO₂?

Mr of CO₂ = $12+(2\times 16)=44$.

$\text{moles}=\frac{88}{44}=2\text{mol}$

Always work out the Mr first and write it down. Dividing the mass by the wrong Mr is the single most common error in these questions.

Rearranging the same equation lets you find a mass from a number of moles.

$\text{mass (g)}=M_r\times \text{moles}$

Worked example — what is the mass of 0.25 mol of calcium carbonate, CaCO₃?

Mr of CaCO₃ = $40+12+(3\times 16)=40+12+48=100$.

$\text{mass}=100\times 0.25=25\text{g}$

Worked example — what is the mass of 3 mol of water, H₂O?

Mr of H₂O = 18.

$\text{mass}=18\times 3=54\text{g}$

A formula triangle helps you rearrange reliably: put mass on top, with Mr and moles side by side underneath. Cover the quantity you want and the triangle shows whether to multiply or divide.

Once you know the number of moles, you can find the actual number of particles by multiplying by the Avogadro constant.

$\text{number of particles}=\text{moles}\times (6.02\times 10^{23})$

Worked example — how many molecules are in 0.5 mol of carbon dioxide?

$0.5\times (6.02\times 10^{23})=3.01\times 10^{23}\text{molecules}$

Worked example — how many atoms are in 4.6 g of sodium, Na (Ar = 23)?

First find the moles: $\text{moles}=\frac{4.6}{23}=0.2\text{mol}$.

Then multiply by the Avogadro constant:

$0.2\times (6.02\times 10^{23})=1.204\times 10^{23}\text{atoms}$

Handle the standard-form arithmetic carefully: multiply the ordinary numbers, then attach the power of ten. Here 0.2 × 6.02 = 1.204, giving 1.204 × 10²³.

Harder questions chain the conversions: mass to moles, moles to particles, or the reverse. Break every question into single steps and never try to jump straight from grams to particles.

Worked example — a sample contains 1.204 × 10²⁴ molecules of water. What is its mass?

Step 1, find the moles by dividing by the Avogadro constant:

$\text{moles}=\frac{1.204\times 10^{24}}{6.02\times 10^{23}}=2\text{mol}$

Step 2, convert moles to mass using mass = Mr × moles, with Mr of H₂O = 18:

$\text{mass}=18\times 2=36\text{g}$

So 1.204 × 10²⁴ water molecules have a mass of 36 g. Laying out each step, with its own equation, keeps the standard-form and unit handling under control.

1. Confusing the mole and the Avogadro constant

The mole is the unit of amount; the Avogadro constant, 6.02 × 10²³, is the number of particles in one mole. You only multiply by 6.02 × 10²³ when the question asks for a number of actual particles.

2. Forgetting to calculate the Mr first

moles = mass ÷ Mr only works once you have the correct Mr. Work it out and write it down before dividing.

3. Rearranging the formula the wrong way

To find mass from moles you multiply (mass = Mr × moles); to find moles from mass you divide. A formula triangle prevents mixing these up.

4. Mishandling standard form

When multiplying by 6.02 × 10²³, deal with the ordinary numbers and the power of ten separately. 0.2 × 6.02 × 10²³ is 1.204 × 10²³, not 1.204 × 10²².

5. Using grams where moles are needed

Balanced equations work in moles, not grams. Convert every mass to moles before comparing amounts of different substances.