Unit 1 · Atomic Structure and Properties

AP Chemistry Avogadro's Number

What the mole is, what 6.022 × 10²³ counts, and how to count atoms, molecules, and ions in a sample, in four worked examples.

Chemical reactions happen between individual particles, but laboratory measurements are in grams. The mole is the bridge, and Avogadro's number is the exchange rate between particles and moles. This page explains the idea and the particle-counting calculations that use it. The full grams-to-moles-to-particles route is in mole conversions, and the mass side comes from the molar mass.

What the Mole and Avogadro's Number Mean

• The mole is a counting unit, like a dozen, for very large numbers of particles.
• Avogadro's number is the number of particles in one mole: 6.022 × 10²³ mol⁻¹ (defined as exactly 6.02214076 × 10²³).
• Molar mass is the mass of one mole in grams. It is numerically equal to the mass of one particle in atomic mass units, which is why the mole connects the particle scale to the gram scale.

The particles counted can be atoms, molecules, ions, or formula units, so always say what you are counting. The chart shows one mole of six substances: each holds the same number of particles, and the masses differ because the particles do.

Worked Example 1: Finding the Mass of a Single Atom or Molecule

Find the mass in grams of one water molecule and of one copper atom.

Water: molar mass 18.02 g/mol, so (18.02 g/mol) ÷ (6.022 × 10²³ mol⁻¹) = 2.99 × 10⁻²³ g per molecule.
Copper: 63.55 g/mol ÷ 6.022 × 10²³ = 1.06 × 10⁻²² g per atom.

The numbers are tiny, which is why chemists count in moles. Dividing the molar mass by Avogadro's number is the reverse of the usual conversion, and it gives the mass of one particle. The molar masses used here are calculated in molar mass.

Worked Example 2: Counting the Atoms in a Mass of Iron

How many iron atoms are in 5.00 g of iron?

Moles: 5.00 g ÷ 55.85 g/mol = 0.0895 mol.
Atoms: 0.0895 mol × 6.022 × 10²³ atoms/mol = 5.39 × 10²² atoms.

The route is grams → moles → particles, using molar mass first and Avogadro's number second. Keep an extra digit in the moles, and report the answer with three significant figures, matching the 5.00 g. See significant figures for the rule.

Worked Example 3: Counting Atoms of Each Element in a Sample of Ethanol

How many molecules, how many hydrogen atoms, and how many atoms in total are in 0.500 mol of ethanol, C₂H₅OH?

Molecules: 0.500 mol × 6.022 × 10²³ = 3.01 × 10²³ molecules.
Atoms per molecule: 2 C + 6 H + 1 O = 9 atoms.
Hydrogen atoms: 3.01 × 10²³ × 6 = 1.81 × 10²⁴.
Total atoms: 3.01 × 10²³ × 9 = 2.71 × 10²⁴.

The subscripts in the formula are the number of atoms per molecule, so they multiply the number of molecules. The same idea applies to the ions in a formula unit of an ionic compound, as in ionic bonds.

Worked Example 4: Comparing the Number of Molecules in Equal Masses

Which has more molecules, 10.0 g of water or 10.0 g of carbon dioxide? By what factor?

Water: 10.0 ÷ 18.02 = 0.555 mol, or 3.34 × 10²³ molecules.
Carbon dioxide: 10.0 ÷ 44.01 = 0.227 mol, or 1.37 × 10²³ molecules.
Ratio: 0.555 ÷ 0.227 ≈ 2.4.

Water has about 2.4 times as many molecules because its molar mass is smaller, so each gram holds more particles. Equal masses do not mean equal numbers of particles; equal moles do. This reasoning is the basis for deciding the limiting reactant in limiting reactant and percent yield.

Five Avogadro's Number Practice Questions With Answers

1. How many aluminum atoms are in 0.250 mol of aluminum?
   Show answer

   0.250 × 6.022 × 10²³ = 1.51 × 10²³ atoms.
2. What mass of copper contains 2.50 × 10²² atoms?
   Show answer

   (2.50 × 10²² ÷ 6.022 × 10²³) = 0.0415 mol, and 0.0415 × 63.55 g/mol = 2.64 g.
3. How many atoms are in 0.100 mol of NH₃?
   Show answer

   0.100 × 6.022 × 10²³ = 6.02 × 10²² molecules, and each has 4 atoms, so 2.41 × 10²³ atoms.
4. Which has more molecules, 1.00 g of H₂ or 1.00 g of O₂, and about how many times as many?
   Show answer

   H₂. Its molar mass is 2.016 against 32.00, so it has about 32.00/2.016 ≈ 15.9 times as many molecules.
5. What is the mass of 3.01 × 10²³ molecules of CO₂?
   Show answer

   3.01 × 10²³ ÷ 6.022 × 10²³ = 0.500 mol, and 0.500 × 44.01 g/mol = 22.0 g.

Common Avogadro's Number Mistakes

• Multiplying when you should divide. Moles to particles multiplies by 6.022 × 10²³; particles to moles divides.
• Counting molecules instead of atoms. Multiply the number of molecules by the atoms per molecule when asked for atoms.
• Assuming equal masses mean equal numbers of particles. Equal moles do.
• Skipping the molar mass step when starting from grams.
• Losing the exponent or writing it as 6.022 × 10⁻²³; the units are particles per mole.
• Not saying what the particles are. Atoms, molecules, ions, and formula units are all counted, and the answer depends on which.

Particle counting is short but shows up inside many longer problems. To see how a stronger Unit 1 would shift your overall estimate, the AP Chem Score Calculator turns section scores into a 1–5 range.

Related Resources

• AP Chem Score Calculator
• Unit 1 Review: Atomic Structure and Properties
• AP Chemistry Mole Conversions
• AP Chemistry Molar Mass
• Molar Mass Calculator
• AP Chemistry Stoichiometry
• AP Chemistry Limiting Reactant & Percent Yield
• AP Chemistry Significant Figures Rules
• AP Chemistry Scientific Notation
• Scientific Notation Calculator
• AP Chemistry Study Guide

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