Avogadro's Number

In chemistry, atoms and molecules are far too small to count individually. Even a tiny sample of a substance contains an enormous number of particles.

Chemists solve this problem using the mole.

One mole of any substance contains a specific number of particles known as Avogadro's number.

Avogadro's number is:

6.022 × 10²³ particles per mole

This means:

1 mol = 6.022 × 10²³ particles

The symbol commonly used for Avogadro's constant is Nₐ.

Why Is Avogadro's Number So Large?

Atoms, molecules, and ions are incredibly small.

A sample that looks small to us can contain trillions upon trillions of particles.

For example:

1 mole of carbon atoms contains: 6.022 × 10²³ carbon atoms

1 mole of water molecules contains: 6.022 × 10²³ water molecules

1 mole of sodium ions contains: 6.022 × 10²³ sodium ions

The type of particle changes, but the number of particles in one mole does not.

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The Mole as a Counting Unit

We use counting words for groups of objects.

For example:

1 pair = 2 objects

1 dozen = 12 objects

1 mole = 6.022 × 10²³ objects

The difference is that a mole represents an enormously larger group.

The mole is therefore a counting unit.

However, instead of counting eggs or pencils, chemists use moles to count microscopic particles.

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What Counts as a Particle?

The word particle can refer to different things depending on the substance.

It may refer to:

• atoms
• molecules
• ions
• formula units

You must identify what type of particle the question is asking about.

For example:

1 mol He = 6.022 × 10²³ helium atoms

1 mol H₂O = 6.022 × 10²³ water molecules

1 mol Na⁺ = 6.022 × 10²³ sodium ions

1 mol NaCl = 6.022 × 10²³ formula units of sodium chloride

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Atoms

Elements such as helium, iron, copper, and carbon can be counted in atoms.

For example:

1 mol Fe = 6.022 × 10²³ Fe atoms

2 mol Fe = 2 × 6.022 × 10²³ atoms

= 1.2044 × 10²⁴ Fe atoms

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Molecules

Covalent substances often exist as molecules.

Examples include:

• H₂O
• CO₂
• O₂
• NH₃
• CH₄

One mole of any molecular substance contains Avogadro's number of molecules.

For example:

1 mol CO₂ = 6.022 × 10²³ CO₂ molecules

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Ions

Ions are charged particles.

Examples include:

Na⁺

Cl⁻

Mg²⁺

O²⁻

One mole of ions contains:

6.022 × 10²³ ions

For example:

0.5 mol Cl⁻ contains:

0.5 × 6.022 × 10²³

= 3.011 × 10²³ chloride ions

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Formula Units

Ionic compounds do not normally exist as separate molecules.

Instead, they form giant ionic lattices.

For ionic compounds, we therefore use the term formula unit.

For example:

1 mol NaCl contains:

6.022 × 10²³ formula units of NaCl

Each formula unit contains:

• 1 Na⁺ ion
• 1 Cl⁻ ion

So 1 mol NaCl contains:

6.022 × 10²³ Na⁺ ions

and:

6.022 × 10²³ Cl⁻ ions

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Converting Moles to Particles

The most important Avogadro's number calculation is:

Number of particles = number of moles × Avogadro's number

Using symbols:

N = n × Nₐ

where:

N = number of particles

n = number of moles

Nₐ = 6.022 × 10²³ mol⁻¹

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Worked Example: Moles to Atoms

How many atoms are present in 2.0 mol of helium?

Use:

Number of particles = moles × Avogadro's number

N = 2.0 × 6.022 × 10²³

N = 1.2044 × 10²⁴

Therefore:

2.0 mol He contains 1.2044 × 10²⁴ helium atoms.

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Worked Example: Moles to Molecules

How many molecules are present in 0.50 mol of water?

N = n × Nₐ

N = 0.50 × 6.022 × 10²³

N = 3.011 × 10²³

Therefore:

0.50 mol H₂O contains 3.011 × 10²³ water molecules.

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Worked Example: Moles to Ions

How many Mg²⁺ ions are present in 3.0 mol Mg²⁺?

N = 3.0 × 6.022 × 10²³

N = 1.8066 × 10²⁴

Therefore:

3.0 mol Mg²⁺ contains 1.8066 × 10²⁴ Mg²⁺ ions.

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Converting Particles to Moles

Sometimes we know the number of particles and need to determine the number of moles.

Use:

Number of moles = number of particles ÷ Avogadro's number

In symbols:

n = N ÷ Nₐ

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Worked Example: Atoms to Moles

A sample contains 1.2044 × 10²⁴ carbon atoms.

How many moles of carbon are present?

n = N ÷ Nₐ

n = (1.2044 × 10²⁴) ÷ (6.022 × 10²³)

n = 2.0

Therefore:

2.0 mol of carbon atoms are present.

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Worked Example: Molecules to Moles

A sample contains 3.011 × 10²³ oxygen molecules.

How many moles of O₂ are present?

n = (3.011 × 10²³) ÷ (6.022 × 10²³)

n = 0.50

Therefore:

0.50 mol O₂

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The Two Main Calculations

The relationship can be summarized simply.

To go from:

moles → particles

multiply by 6.022 × 10²³.

To go from:

particles → moles

divide by 6.022 × 10²³.

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Using Chemical Formulae

Sometimes a question asks for the number of atoms inside molecules.

This requires an extra step.

Consider water:

H₂O

Each water molecule contains:

• 2 hydrogen atoms
• 1 oxygen atom

Therefore, 1 mole of H₂O contains:

• 2 mol H atoms
• 1 mol O atoms

This means 1 mole of water contains a total of:

3 mol of atoms

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Worked Example: Atoms in Water

How many hydrogen atoms are present in 1.0 mol H₂O?

First determine the number of water molecules:

1.0 × 6.022 × 10²³ = 6.022 × 10²³ H₂O molecules

Each molecule contains 2 hydrogen atoms.

Therefore:

2 × 6.022 × 10²³ = 1.2044 × 10²⁴ hydrogen atoms

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Worked Example: Oxygen Atoms in Carbon Dioxide

How many oxygen atoms are present in 2.0 mol CO₂?

Each CO₂ molecule contains 2 oxygen atoms.

Therefore:

2.0 mol CO₂ contains:

2 × 2.0 = 4.0 mol O atoms

Now convert to atoms:

N = 4.0 × 6.022 × 10²³

N = 2.4088 × 10²⁴

Therefore:

2.4088 × 10²⁴ oxygen atoms

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Worked Example: Total Atoms in Ammonia

How many total atoms are present in 0.25 mol NH₃?

Each NH₃ molecule contains:

1 nitrogen atom + 3 hydrogen atoms = 4 atoms

First determine the number of molecules:

N = 0.25 × 6.022 × 10²³

N = 1.5055 × 10²³ molecules

Each molecule contains 4 atoms.

Total atoms:

4 × 1.5055 × 10²³

= 6.022 × 10²³ atoms

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Counting Ions in Ionic Compounds

Chemical formulae can also tell us how many ions are present.

Consider magnesium chloride:

MgCl₂

Each formula unit contains:

• 1 Mg²⁺ ion
• 2 Cl⁻ ions

Therefore:

1 mol MgCl₂ corresponds to:

• 1 mol Mg²⁺ ions
• 2 mol Cl⁻ ions

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Worked Example: Ions in Magnesium Chloride

How many chloride ions correspond to 0.50 mol MgCl₂?

Each formula unit contains 2 chloride ions.

Therefore:

0.50 mol MgCl₂ corresponds to:

0.50 × 2 = 1.0 mol Cl⁻

Now convert to ions:

N = 1.0 × 6.022 × 10²³

Therefore:

6.022 × 10²³ chloride ions

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A More Challenging Example

How many total ions are represented by 2.0 mol CaCl₂?

Each formula unit contains:

1 Ca²⁺ + 2 Cl⁻

Total = 3 ions

Therefore:

2.0 mol CaCl₂ corresponds to:

2.0 × 3 = 6.0 mol ions

Number of ions:

N = 6.0 × 6.022 × 10²³

N = 3.6132 × 10²⁴ ions

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Scientific Notation

Avogadro's number is written in scientific notation because it is extremely large.

6.022 × 10²³ means:

602,200,000,000,000,000,000,000

Writing:

6.022 × 10²³

is much easier and reduces errors.

Scientific notation is especially important when performing mole calculations.

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Using a Calculator

Suppose you need to calculate:

0.35 × 6.022 × 10²³

On a scientific calculator, you may enter something similar to:

6.022 EXP 23 × 0.35

or:

6.022 EE 23 × 0.35

depending on the calculator.

The answer is:

2.1077 × 10²³

Be careful not to enter the exponent incorrectly.

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Why Avogadro's Number Matters

Avogadro's number creates a connection between the microscopic and macroscopic worlds.

Chemists cannot practically count individual atoms.

However, they can:

• measure mass
• calculate moles
• use Avogadro's number
• determine the number of particles

This makes quantitative chemistry possible.

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Moles, Mass and Particles

Later calculations often connect three important quantities:

mass ↔ moles ↔ particles

Mass and moles are connected using molar mass.

Moles and particles are connected using Avogadro's number.

For example:

mass → moles → number of molecules

This allows chemists to determine how many microscopic particles are present in a measurable sample.

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Avogadro's Number and Chemical Reactions

Chemical equations describe particles reacting in specific ratios.

For example:

2H₂ + O₂ → 2H₂O

At the particle level:

2 molecules H₂ react with 1 molecule O₂.

At the mole level:

2 mol H₂ react with 1 mol O₂.

Because every mole contains the same number of particles, mole ratios allow chemists to scale reactions up from individual molecules to laboratory quantities.

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Worked Example: Multi-Step Problem

A sample contains 1.5 mol of CH₄.

How many hydrogen atoms are present?

Step 1: Identify the number of H atoms in each molecule.

CH₄ contains 4 H atoms.

Step 2: Determine moles of H atoms.

1.5 × 4 = 6.0 mol H atoms

Step 3: Convert moles to atoms.

N = 6.0 × 6.022 × 10²³

N = 3.6132 × 10²⁴

Therefore:

1.5 mol CH₄ contains 3.6132 × 10²⁴ hydrogen atoms.

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Worked Example: Working Backwards

A sample contains 9.033 × 10²³ molecules of CO₂.

How many moles of CO₂ are present?

Use:

n = N ÷ Nₐ

n = (9.033 × 10²³) ÷ (6.022 × 10²³)

n = 1.5

Therefore:

1.5 mol CO₂

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A Useful Problem-Solving Strategy

When solving Avogadro's number problems:

1. Identify what you are given.
2. Identify what the question asks for.
3. Decide whether you need to multiply or divide by Avogadro's number.
4. Check the chemical formula if atoms or ions inside a compound are being counted.
5. Perform the calculation.
6. Include the correct type of particle in your answer.

For example:

Given: moles

Wanted: molecules

Use:

moles × 6.022 × 10²³

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Common Misconceptions

Avogadro's number is the mass of one mole.

Incorrect. Avogadro's number tells us the number of particles in one mole.

One mole of every substance has the same mass.

Incorrect. One mole always contains the same number of particles, but different substances have different molar masses.

One mole of H₂O contains 6.022 × 10²³ atoms.

Incorrect. It contains 6.022 × 10²³ water molecules. Each molecule contains three atoms.

Ionic compounds contain molecules.

It is generally more accurate to describe ionic compounds using formula units rather than molecules.

To convert particles to moles, multiply by Avogadro's number.

Incorrect.

Particles → moles means divide by Avogadro's number.

0.5 mol contains 0.5 particles.

Incorrect. Even a fraction of a mole contains an enormous number of particles.

Did You Know?

Avogadro's number is so large that it is difficult to imagine.

If you could count particles at a rate of one billion particles every second, it would still take millions of years to count all the particles in just one mole.

The mole allows chemists to work with this enormous microscopic population using quantities that can actually be measured in a laboratory.

Key Terms

Avogadro's number – The number of particles in one mole: 6.022 × 10²³.

Avogadro constant (Nₐ) – The constant 6.022 × 10²³ mol⁻¹.

Mole (mol) – The amount of substance containing 6.022 × 10²³ specified entities.

Particle – A general term that may refer to an atom, molecule, ion, or formula unit.

Atom – The smallest particle of an element that retains its chemical identity.

Molecule – A group of atoms joined by covalent bonds.

Ion – A charged particle formed when electrons are gained or lost.

Formula unit – The simplest whole-number ratio of ions represented by the formula of an ionic compound.

Scientific notation – A method for expressing very large or very small numbers using powers of ten.

Key Takeaways

• Avogadro's number is 6.022 × 10²³.
• One mole contains 6.022 × 10²³ particles.
• The particles may be atoms, molecules, ions, or formula units.
• Avogadro's number connects microscopic particles with measurable amounts of substances.
• To convert moles to particles, multiply by 6.022 × 10²³.
• To convert particles to moles, divide by 6.022 × 10²³.
• Use N = n × Nₐ to calculate the number of particles.
• Chemical formulae must be considered when counting individual atoms within molecules.
• One mole of H₂O contains one mole of water molecules but two moles of H atoms and one mole of O atoms.
• Ionic formulae can be used to determine the number of individual ions.
• Avogadro's number is fundamental to quantitative chemistry and stoichiometry.