S1.4.1 The mole and Avogadro constant

The Mole: A Gateway to Quantifying the Atomic World

You are holding a single grain of sand in your hand. Now imagine trying to count every grain of sand on a beach. The task seems impossible, doesn’t it?

Similarly, when dealing with atoms, molecules, or ions—particles so small that billions of them fit into a single drop of water—counting them individually is impractical.

The Mole: The Chemist’s Counting Unit

Definition

Mole

The mole (mol) is the SI unit for the amount of substance, defined as $6.02 \times 10^{23}$ particles (atoms, molecules, or ions).

Think of it as the "chemist's dozen," but instead of 12, one mole contains 6.02 × 10²³ elementary entities. This number is known as the Avogadro constant ( $N_{A}$ ).

Why 6.02 × 10²³?

The Avogadro constant wasn’t chosen randomly.
It was specifically defined to connect the microscopic world of atoms and molecules to measurable quantities in the macroscopic world.

Example

One mole of carbon-12 atoms has a mass of exactly 12 grams.
This precise relationship between the number of particles and their mass makes the mole an invaluable tool in chemistry.

Analogy

Imagine you’re buying apples by weight at a market. The mole is like the scale that tells you how many apples you’re getting without counting each one individually—it bridges the gap between quantity and weight.

Elementary Entities: What Are We Counting?

When using the mole, it’s essential to specify what you’re counting. The term elementary entities refers to the type of particle involved, which could include:

Atoms (e.g., one mole of helium atoms)
Molecules (e.g., one mole of water molecules)
Ions (e.g., one mole of sodium ions)
Electrons (e.g., one mole of electrons)
Other specified groups of particles (e.g., formula units in ionic compounds like NaCl)

Note

Always clarify the type of elementary entity in a calculation. For example, one mole of water (H₂O) contains one mole of molecules, but it also contains two moles of hydrogen atoms and one mole of oxygen atoms.

Using the Avogadro Constant for Conversions

The Avogadro constant ( $N_{A}$ ) serves as a bridge between the amount of substance (n) in moles and the number of entities (N) in a sample. The relationship is expressed mathematically as:

$N = n \times N_{A}$

Where:

$N$ = number of entities
$n$ = amount of substance in moles
$N_{A} = 6.022 \times 10^{23} {mol}^{- 1}$

Rearranging the Formula

To calculate the amount of substance ( $n$ ) when the number of entities ( $N$ ) is known, rearrange the equation:

$n = \frac{N}{N_{A}}$

Example question

Counting Atoms in a Sample

How many atoms are in 2.5 moles of copper (Cu)?

Solution

Identify the formula to use: $N = n \times N_{A}$
Substitute the given values: $n = 2.5 mol$ , $N_{A} = 6.02 \times 10^{23} {mol}^{- 1}$
Perform the calculation: $N = 2.5 mol \times 6.02 \times 10^{23} {mol}^{- 1} = 1.51 \times 10^{24} atoms .$

Tip

Always check your significant figures! Match the precision of your answer to the least precise value in the data.

Example question

Determining Moles from Entities

A sample contains $4.5 \times 10^{22}$ water molecules. How many moles of water does the sample contain?

Solution

Identify the formula to use: $n = \frac{N}{N_{A}}$ .
Substitute the given values: $N = 4.5 \times 10^{22}$ , $N_{A} = 6.02 \times 10^{23} {mol}^{- 1}$ .
Perform the calculation: $n = \frac{4.5 \times 10^{22}}{6.02 \times 10^{23}} = 0.075 mol .$

Common Mistake

Forgetting to divide by the Avogadro constant when converting from entities to moles is a common mistake. Always double-check your formula!

Example question

Calculating Number of Particles from Mass

How many molecules are in a 10.0 g sample of carbon dioxide $(C O_{2})$ ?

Solution

Step 1: Convert mass to moles using the formula:

$n = \frac{m}{M}$

where:

$m = 10.0 g$
$M = (12.01) + (2 \times 16.00) = 44.01 g/mol$

Calculation:

$n = \frac{10.0}{44.01} = 0.227 mol$

Step 2: Convert moles to number of molecules using Avogadro's constant:

$N = n \times N_{A}$

where $N_{A} = 6.02 \times 10^{23} {mol}^{- 1}$ .

Calculation:

$N = 0.227 \times 6.02 \times 10^{23} = 1.37 \times 10^{23} molecules$

Why Is the Mole Important?

The mole is more than just a counting unit—it’s a tool that connects the microscopic and macroscopic worlds. Here’s why it’s essential:

Practical Measurements: The mole allows us to measure incredibly small particles (like atoms and molecules) using macroscopic quantities like grams.
Chemical Calculations: It simplifies stoichiometric calculations, enabling chemists to predict the outcomes of reactions.
Universal Understanding: The mole provides a standardized way for scientists around the world to communicate quantities.

Analogy

Think of the mole as a bridge. Just as a bridge connects two separate landmasses, the mole connects the invisible world of atoms to the tangible world of grams and liters.

Reflection and Practice

Self review

How would you calculate the number of molecules in 0.15 moles of oxygen gas (O₂)? What about the number of oxygen atoms?

Theory of Knowledge

In 2018, the definition of the mole was revised to be based on a fundamental physical constant rather than a specific substance like carbon-12.

How does this reflect the evolving nature of scientific knowledge?
What challenges might arise when redefining fundamental units?

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