How can we count particles too numerous to count one by one?

Support map

Model diagram

Use only paper, a calculator and supplied data. Do not carry out reactions, generate gases, heat substances or mix household chemicals.

Situation and question

A report says “we used 1 mole”. That is incomplete: 1 mole of water molecules and 1 mole of hydrogen atoms describe different sets. How can a short quantity describe an enormous count? Work with numbers and paper cards, without samples or balances.

New terms, fully explained

Amount of substance n measures the number of specified entities. Its unit, the mole (mol), contains exactly 6.02214076 × 10²³ specified entities. The Avogadro constant Nₐ is exactly 6.02214076 × 10²³ mol⁻¹. N is the dimensionless entity count. The relation is N = nNₐ. Approximate calculations here use Nₐ ≈ 6.02 × 10²³ mol⁻¹.

Predict first

Compare 0.5 mol of H₂O molecules with 0.5 mol of CO₂ molecules: which has more molecules? Then compare the total atom counts. Explain why equal molecule counts need not mean equal masses or equal numbers of atoms of each element.

Investigate or analyse given data

Build a table: 1 mol of H₂O molecules contains 1 mol of molecules, 2 mol of H atoms and 1 mol of O atoms, or 3 mol of atoms in total. For NaCl, count formula units describing the crystal composition rather than separate NaCl molecules. When counting ions, name their species and charge.

Worked example

For 0.5 mol of H₂O molecules, N ≈ 0.5 × 6.02 × 10²³ = 3.01 × 10²³ molecules. There are twice as many H atoms, 6.02 × 10²³, and as many O atoms as molecules. These are estimates using a rounded constant; the definition of the mole remains exact.

A common mistake

Mistake: “a mole is a mass in grams”, or “water has as many H atoms as molecules”. The mole measures amount; the gram measures mass. The subscript 2 in H₂O multiplies the H atom count. The number 6.02 × 10²³ is not a particle mass.

Transfer to a new case

Transfer: a set contains 0.25 mol of O₂ molecules. Find the amount of O atoms and the approximate molecule count. Also find how many moles of Cl⁻ ions correspond to the composition of 1 mol of CaCl₂ formula units. No actual dissolution is needed.

Answer and recall

Check: 0.50 mol of O atoms; approximately 1.51 × 10²³ O₂ molecules; 2 mol of Cl⁻ ions in the CaCl₂ composition. Reconstruct “name the entity → find n → multiply by Nₐ → account for subscripts”. Check units and round only the final result.

Exercise

How many mol of H atoms are in 0.10 mol H₂O? How many mol of Cl⁻ correspond to 0.30 mol CaCl₂?

Answers

When to revisit

Close the page and, after 10 minutes, reconstruct the map, define the terms in your own words and solve the new case. Repeat without hints after a day and a week. If an error persists, return to the worked example and write down the missing step.

Sources and limits

• BIPM: SI definition of the mole
• OpenStax: formula mass and the mole

The salt case: how do we check a transformation without losing matter?

Why do equal mole amounts have different masses?