IMAT Chemistry · Stoichiometry and Reactions

The Mole, Molar Mass and Avogadro's Number

A mole is a fixed count of particles, 6.02 × 10²³ of them, and the molar mass turns a mass in grams into that count.

Why this matters

Counting particles in a given mass appears in 2012, 2013, 2017 and 2024. The 2024 ministry question asked how many atoms are in a small mass of a gas made of two-atom molecules, so the trap was the factor of two.

Concept 1 of 3: Relative atomic mass, relative formula mass and molar mass

Atoms are far too light to weigh in grams, so chemists compare them with a standard: one twelfth of a carbon-12 atom. A formula tells you which atoms are in one unit of a substance, so adding their relative masses gives the mass of that unit. The same number, in grams, is the mass of one mole.

Definition

Relative atomic mass $A_r$ : the average mass of an atom of an element (over its isotopes) compared with one twelfth of the mass of a carbon-12 atom. It has no unit; in atomic mass units it is written in u.
Relative formula mass $M_r$ : the sum of the $A_r$ values of all the atoms in the formula. For molecules it is also called the relative molecular mass.
Molar mass $M$ : the mass of one mole of the substance. It has the same number as $M_r$ , with the unit g/mol.
A number in front of a bracket, or after a bracket, multiplies everything inside it. In a hydrate such as $\mathrm{CuSO_4 \cdot 5H_2O}$ , the water counts too.

Relative formula mass

M_r = \sum (\text{number of atoms} \times A_r)

$M_r$ relative formula mass (no unit)
$A_r$ relative atomic mass of each element

Worked example

Find the molar mass of blue copper(II) sulfate crystals,

\mathrm{CuSO_4 \cdot 5H_2O}

. Use

A_r

: Cu = 63.5, S = 32, O = 16, H = 1.

Practice this conceptself-check · 4 quick reps

A bracket multiplies everything inside it

\mathrm{Ca(NO_3)_2}

there are 2 nitrogen atoms and 6 oxygen atoms, not 1 and 3. Forgetting the number outside the bracket gives a formula mass that is too small, and that wrong value is usually among the options.

Concept 2 of 3: Amount of substance: moles from a mass

Equations count particles, but a balance measures grams. The molar mass is the exchange rate between the two: one mole of any substance has a mass in grams equal to its formula mass. Divide a mass by the molar mass and you know how many moles, and so how many particles, you have.

Definition

The mole (mol) is the SI unit of amount of substance: one mole contains exactly $6.022 \times 10^{23}$ particles.
Moles from mass: divide the mass in grams by the molar mass in g/mol.
Mass from moles: multiply the moles by the molar mass.
Convert kilograms and milligrams to grams first: $1\ \text{kg} = 1000\ \text{g}$ , $1\ \text{mg} = 10^{-3}\ \text{g}$ .
The sample with the largest mass does not always have the most moles: a light molecule gives many moles per gram.

Moles from mass

n = \frac{m}{M}

$n$ amount of substance, in mol
$m$ mass, in g
$M$ molar mass, in g/mol

Worked example

(a) How many moles are in 4.9 g of sulfuric acid,

\mathrm{H_2SO_4}

(

M = 98\ \text{g/mol}

)? (b) What is the mass of 0.25 mol of calcium carbonate (

M = 100\ \text{g/mol}

Practice this conceptself-check · 3 quick reps

Kilograms must become grams before you divide

Molar masses are in grams per mole. Dividing a mass in kilograms by a molar mass gives an answer 1000 times too small. Convert first: 3.2 kg is 3200 g.

Concept 3 of 3: Avogadro's number: counting molecules and atoms

Once you have moles, the number of particles is just moles times Avogadro's number. The question then is which particles are being counted. One mole of water molecules contains two moles of hydrogen atoms, so counting atoms means one more multiplication by the number of those atoms in the formula.

Definition

Avogadro's number (the Avogadro constant) $N_A = 6.02 \times 10^{23}\ \text{mol}^{-1}$ : the number of particles in one mole.
Number of formula units or molecules: $N = n \times N_A$ .
Number of atoms of one element: multiply again by how many of those atoms are in the formula.
Many elements exist as diatomic molecules: $\mathrm{H_2}$ , $\mathrm{N_2}$ , $\mathrm{O_2}$ , $\mathrm{F_2}$ , $\mathrm{Cl_2}$ , $\mathrm{Br_2}$ , $\mathrm{I_2}$ . Noble gases exist as single atoms.
Equal moles of any gases at the same temperature and pressure take up equal volumes (Avogadro's law; see States of Matter and Gas Laws).

Number of particles

N = n \times N_A = \frac{m}{M} \times N_A

$N$ number of particles (molecules or formula units)
$N_A$ Avogadro's number, $6.02 \times 10^{23}\ \text{mol}^{-1}$

Worked example

How many oxygen atoms are there in 8.8 g of carbon dioxide? (C = 12, O = 16,

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

)

Practice this conceptself-check · 4 quick reps

Nitrogen gas is made of two-atom molecules

A mass of

\mathrm{N_2}

divided by 28 gives moles of molecules, and the number of nitrogen atoms is twice that. Dividing by 28 and stopping gives half the right number of atoms, and that answer is usually an option. (Dividing the mass by 14 gives moles of atoms directly.) Decide first whether the question counts atoms or molecules.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Relative atomic mass, relative formula mass and molar mass
Relative formula mass
$M_r = \sum (\text{number of atoms} \times A_r)$
Amount of substance: moles from a mass
Moles from mass
$n = \frac{m}{M}$
Avogadro's number: counting molecules and atoms
Number of particles
$N = n \times N_A = \frac{m}{M} \times N_A$

Watch out for (3)

A bracket multiplies everything inside it
→ Relative atomic mass, relative formula mass and molar mass
Kilograms must become grams before you divide
→ Amount of substance: moles from a mass
Nitrogen gas is made of two-atom molecules
→ Avogadro's number: counting molecules and atoms

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