SCH3U The Mole Concept: Notes and Practice

Atoms and molecules are far too small to see, let alone count one at a time, and any real sample you could hold in your hand — a glass of water, a pinch of salt — contains an absurdly large number of them. You can't weigh out "500 water molecules" on a lab balance; the balance isn't sensitive enough, and even if it were, nobody could count that fast. Chemists solve this the same way you solve the problem of counting eggs: instead of counting one at a time, you count in a convenient group. A dozen is a group of 12 items, useful for eggs. A mole is a group of $6.022 \times 1 0^{23}$ items, useful for atoms — a number chosen specifically because it's the number of atoms it takes to make the mass in grams equal to the atomic mass in amu.

How big is Avogadro's number, really?

One mole of standard soft drink cans would cover the entire surface of Earth (land and ocean) to a depth of hundreds of kilometres. Yet a single drop of water (about

0.05 g

) holds roughly

1.7 \times 1 0^{21}

water molecules. Atoms are so small that even huge numbers of them fit into a sample you can hold.

Fig 1The mole as the counting unit that bridges particles and measurable mass.

Avogadro's Number

Avogadro's number, $N_{A} = 6.022 \times 1 0^{23} /mol$ , is the number of particles — atoms, molecules, ions, or formula units — in exactly one mole of anything. It plays the exact same role as "12" plays for a dozen: it's the conversion factor between "how many groups" (moles) and "how many individual items" (particles). Just as 3 dozen eggs is $3 \times 12 = 36$ eggs, $3 mol$ of atoms is $3 \times 6.022 \times 1 0^{23}$ atoms. And just as 60 eggs is $60 \div 12 = 5$ dozen, $1.2044 \times 1 0^{24}$ atoms is $1.2044 \times 1 0^{24} \div 6.022 \times 1 0^{23} = 2.000$ mol. Notice that a dozen of anything is 12 of them whether they are eggs or elephants; in the same way, a mole of hydrogen atoms and a mole of lead atoms contain exactly the same number of atoms, even though their masses are very different.

n = \frac{N}{N _{A}} N = n \times N_{A}

n = amount in moles, N = number of particles, NA = 6.022 × 10²³ per mole.

The mole ( $mol$ ) is the SI base unit for amount of substance.
One mole always contains $6.022 \times 1 0^{23}$ particles, regardless of what the particles are.
"Particles" means atoms for an element, molecules for a molecular compound, and formula units for an ionic compound.

Counting unit	Group size	Typically used for
pair	2	shoes, socks
dozen	12	eggs, doughnuts
ream	500	sheets of paper
mole	$6.022 \times 1 0^{23}$	atoms, molecules, ions, formula units

The mole is a counting unit like any other. Only the size of the group is unusual.

Why This Particular Number?

The size of the mole was chosen to connect the atomic scale to the lab scale. The atomic mass unit ( $u$ , also written amu) is defined as $\frac{1}{12}$ of the mass of one carbon-12 atom, which is $1.6605 \times 1 0^{- 24} g$ . Multiply that by Avogadro's number and you get exactly one gram. So if one atom of an element has a mass of $x u$ , one mole of those atoms has a mass of $x g$ . That is why a carbon atom is $12.01 u$ and a mole of carbon is $12.01 g$ , and it's the reason the periodic table can be read directly in grams per mole (next lesson). Since 2019 the SI defines the mole by fixing $N_{A}$ at exactly $6.02214076 \times 1 0^{23}$ ; the 4-sig-fig value $6.022 \times 1 0^{23}$ is plenty for this course.

6.022 \times 1 0^{23} \times 1.6605 \times 1 0^{- 24} g = 1.000 g

Avogadro's number of atomic mass units is one gram. This is what makes molar mass work.

Counting the Right Particle

Substance	One particle is	What 1 mol contains
Fe (element)	one Fe atom	$6.022 \times 1 0^{23}$ Fe atoms
$O_{2}$ (diatomic element)	one $O_{2}$ molecule	$6.022 \times 1 0^{23}$ molecules, but $2 mol$ = $1.204 \times 1 0^{24}$ O atoms
$H_{2} O$ (molecular)	one $H_{2} O$ molecule	$2 mol$ H atoms and $1 mol$ O atoms
NaCl (ionic)	one formula unit	$1 mol Na^{+}$ and $1 mol Cl^{-}$ ions
$CaCl_{2}$ (ionic)	one formula unit	$1 mol Ca^{2 +}$ and $2 mol Cl^{-}$ : $3 mol$ of ions

Subscripts inside the formula tell you how many moles of each atom or ion one mole of the substance contains.

Particles vs. atoms

Be careful what's being counted. A mole of

O_{2}

molecules contains

6.022 \times 1 0^{23}

molecules, but

2 \times 6.022 \times 1 0^{23}

individual oxygen atoms, since each molecule has 2 atoms. When a question asks for atoms or ions inside a compound, find moles of the compound first, multiply by the subscript, and only then multiply by

N_{A}

(or the reverse order — the result is the same).

AMoles to particles: multiply by NA

$0.50 mol \times 6.022 \times 1 0^{23} /mol = 3.0 \times 1 0^{23}$
The answer should be a huge number
Enter $N_{A}$ with the EE or EXP key (6.022E23) so it acts as one number

BParticles to moles: divide by NA

$\frac{1.2 \times 1 0 ^{22}}{6.022 \times 1 0 ^{23} /mol} = 0.020 mol$
Lab-sized samples usually give 0.001 to 100 mol
A tiny particle count like $1 0^{- 24}$ means you divided when you should have multiplied

Why bother with a counting unit at all, when chemists could just quote enormous particle counts? Because reactions happen in whole-number ratios of particles, and the mole keeps those ratios intact at a human scale. The equation $2 H_{2} + O_{2} \to 2 H_{2} O$ means two molecules of hydrogen for every one of oxygen; multiplying every count by $N_{A}$ turns it into two moles of hydrogen for every one mole of oxygen, the same ratio in amounts you can actually measure. Every calculation in the rest of this unit (molar mass, percent composition, empirical formulas, stoichiometry, limiting reagents and yield) is built on that one idea: count particles by counting moles.

The Mole Concept

Key things to remember

Avogadro's Number

Why This Particular Number?

Counting the Right Particle

Worked examples

Where marks get lost

Flashcards 9

Practice