D. Quantities in Chemical Reactions · Quantities in Chemical ReactionsThe Mole and Avogadro's Constant
D3.2 D2.1 D2.3 A1.12 A1.13About 70 minutes
Learning goals- I can explain why chemists count particles in moles.
- I can state Avogadro's constant and what it means.
- I can convert between amount in moles and number of particles.
- I can find the number of atoms or ions inside a given amount of a compound.
- I can write very large and very small numbers in scientific notation with the right number of significant figures.
Words to know
mole (mol)
The SI unit for amount of substance. One mole is exactly 6.02214076 × 10²³ elementary entities (atoms, molecules, ions or formula units).
Avogadro's constant (N_A)
The number of entities per mole: 6.02 × 10²³ mol⁻¹ (to three significant figures).
amount of substance (n)
How many particles a sample has, measured in moles.
elementary entity
The particle being counted: an atom (Fe), a molecule (H₂O), an ion (Na⁺) or a formula unit (NaCl).
formula unit
The smallest repeating ratio of ions in an ionic compound, such as one Na⁺ with one Cl⁻ in NaCl.
scientific notation
Writing a number as a × 10ⁿ with 1 ≤ a < 10, for example 6.02 × 10²³.
significant figures
The digits in a measured or calculated value that carry meaning about its precision.
Too many to count
A single drop of water holds more molecules than there are grains of sand on a large beach. Atoms are far too small and far too many to count one at a time. Yet a chemist who makes a medicine needs to know how many molecules of each reactant to mix, because reactions happen particle by particle.
The answer is the same trick people use with eggs, pencils and paper. We do not count eggs one by one at the store; we buy them by the dozen. Chemists use a counting unit too. It is called the mole, and it is a very, very big "dozen".
Remember from Grade 10
In Grade 10 (Chemical Reactions) you wrote balanced chemical equations and learned the law of conservation of mass. A coefficient such as the 2 in 2H₂O told you how many molecules react. In this unit you will see that the same coefficients also tell you how many moles react, and that turns an equation into a recipe you can weigh out on a balance.
The mole is a counting unit
A mole (symbol mol) is a number of particles, just as a dozen is 12 things:
- 1 dozen = 12
- 1 mole = 6.02 × 10²³
That number is called Avogadro's constant, N_A. Written out in full it is about 602 000 000 000 000 000 000 000. Its units are particles per mole (mol⁻¹), because it tells you how many particles are in each mole.
Since 2019 the mole has had an exact definition in the International System of Units (SI): one mole contains exactly 6.02214076 × 10²³ elementary entities. In this course we round it to 6.02 × 10²³, which has three significant figures.
What are we counting?
The mole can count any kind of particle, so you must always say which particle you mean. These are called elementary entities:
- atoms for elements such as iron, Fe, or helium, He
- molecules for molecular substances such as water, H₂O, carbon dioxide, CO₂, or oxygen gas, O₂
- formula units for ionic compounds such as sodium chloride, NaCl (an ionic crystal has no separate molecules, so we count the smallest ratio of ions)
- ions such as Na⁺ or Cl⁻
"One mole of oxygen" is unclear. One mole of oxygen atoms (O) is 6.02 × 10²³ atoms. One mole of oxygen molecules (O₂) is 6.02 × 10²³ molecules, which is 1.20 × 10²⁴ atoms.
Converting between moles and particles
The symbol n stands for the amount in moles, and N for the number of particles. They are linked by Avogadro's constant:
N = n × N_A and n = N ÷ N_A
Think of it like dozens. Three dozen eggs is 3 × 12 = 36 eggs. Forty-eight eggs is 48 ÷ 12 = 4 dozen. Moles work the same way, with 6.02 × 10²³ in place of 12.
Scientific notation and significant figures
Scientific notation writes a number as a × 10ⁿ, where 1 ≤ a < 10. For example 45 000 = 4.5 × 10⁴ and 0.000 32 = 3.2 × 10⁻⁴.
On most scientific calculators you enter 6.02 × 10²³ as 6.02 EE 23 (the key may say EXP or ×10ˣ). Do not type × 10 ^ 23 with an extra multiplication sign inside a longer calculation, or the calculator may group it wrongly.
Significant figures (sig figs): all non-zero digits count; zeros between non-zero digits count (205 has 3); leading zeros do not count (0.0045 has 2); trailing zeros after a decimal point count (2.50 has 3). When you multiply or divide, round the answer to the fewest significant figures in the measured values you used. In this unit most data have three significant figures, so most answers have three as well. Keep extra digits in your calculator and round only at the end.
Worked exampleExample 1: Moles to molecules
How many molecules are in 2.50 mol of carbon dioxide, CO₂?
- We know n = 2.50 mol and want N, so multiply by N_A.
- N = n × N_A = 2.50 mol × 6.02 × 10²³ molecules/mol
- = 1.505 × 10²⁴ molecules. The units mol cancel.
- Both values have 3 significant figures, so round to 3: 1.51 × 10²⁴ molecules.
2.50 mol of CO₂ contains 1.51 × 10²⁴ molecules.
Worked exampleExample 2: Atoms to moles
A copper wire contains 3.01 × 10²² copper atoms. What amount of copper is this, in moles?
- We know N and want n, so divide by N_A.
- n = N ÷ N_A = (3.01 × 10²² atoms) ÷ (6.02 × 10²³ atoms/mol)
- = 0.0500 mol. (On the calculator: 3.01 EE 22 ÷ 6.02 EE 23 = 0.05.)
- Three significant figures: 0.0500 mol (or 5.00 × 10⁻² mol).
The wire contains 0.0500 mol of copper atoms.
Atoms inside molecules
A chemical formula is a particle recipe. One molecule of water, H₂O, contains 2 hydrogen atoms and 1 oxygen atom. If you take a mole of water molecules, you have a mole of those recipes, so you have 2 mol of H atoms and 1 mol of O atoms.
The subscripts in a formula give the ratio of atoms, and the ratio of moles of atoms.
Worked exampleExample 3: Counting atoms in a compound
How many atoms in total are in 0.250 mol of water, H₂O?
- Molecules: N = 0.250 mol × 6.02 × 10²³ mol⁻¹ = 1.505 × 10²³ molecules.
- Each H₂O molecule has 2 + 1 = 3 atoms.
- Atoms = 3 × 1.505 × 10²³ = 4.515 × 10²³.
- Another way: 0.250 mol × 3 = 0.750 mol of atoms, and 0.750 × 6.02 × 10²³ = 4.515 × 10²³.
- Round to 3 significant figures: 4.52 × 10²³ atoms. (The 3 atoms per molecule is an exact count, so it does not limit the significant figures.)
0.250 mol of water contains 4.52 × 10²³ atoms.
| Sample | Entity counted | Amount | Number of entities |
|---|
| 1 mol of helium, He | atoms | 1 mol | 6.02 × 10²³ atoms |
| 1 mol of oxygen gas, O₂ | molecules | 1 mol | 6.02 × 10²³ molecules (1.20 × 10²⁴ O atoms) |
| 1 mol of NaCl | formula units | 1 mol | 6.02 × 10²³ formula units (6.02 × 10²³ Na⁺ and 6.02 × 10²³ Cl⁻) |
| 1 mol of CaCl₂ | formula units | 1 mol | 6.02 × 10²³ Ca²⁺ ions and 1.20 × 10²⁴ Cl⁻ ions |
Why such a strange number?
Avogadro's constant was chosen so that the mole connects neatly to masses we can weigh. For many years the mole was defined as the number of atoms in exactly 12 g of carbon-12. That choice means the mass of one mole of an element, in grams, has the same number as its atomic mass on the periodic table. You will use that link in the next lesson, Molar Mass.
The constant is named after the Italian scientist Amedeo Avogadro, who suggested in 1811 that equal volumes of gases at the same temperature and pressure contain equal numbers of particles. He never measured the number himself; later scientists such as the French physicist Jean Perrin measured it and proposed naming it in Avogadro's honour.
Just for funChemistry students celebrate Mole Day on October 23, from 6:02 a.m. to 6:02 p.m., because the date and time look like 6.02 × 10²³. In 2026, Mole Day falls on a Friday.
Hands-on activityCount by weighing
This activity shows how the mole works: you count huge numbers of tiny things by weighing them.
You need: a kitchen scale (grams), a bag of uncooked rice or dried lentils, a small bowl, a measuring cup and a sheet of paper.
- Count out exactly 100 grains onto the paper. Pour them into the bowl and weigh them. (If the scale cannot read such a small mass, count 200 or 300.)
- Work out the mass of one grain: mass ÷ 100.
- Fill the measuring cup with grains, weigh the grains (subtract the bowl's mass), and predict how many grains are in the cup: total mass ÷ mass of one grain.
- Check part of your prediction: weigh out a mass that should contain 200 grains and count them.
- Write two sentences: How is a "package of 100 grains" like a mole? Why do chemists weigh instead of counting?
What to look for: a cup of rice usually holds several thousand grains, and your check count should be within a few percent of your prediction. Grains are not all the same size, so there is some error; atoms of the same isotope are identical, which is one reason weighing works so well in chemistry. Your sentences should say that a mole is a fixed number of particles, and that weighing lets you count particles you could never count one by one.
Your turn
1How many iron atoms are in 4.00 mol of iron, Fe?
N = 4.00 × 6.02 × 10²³ = 2.41 × 10²⁴ atoms.
2A glass holds 1.20 × 10²⁴ water molecules. What amount of water is this?
n = 1.20 × 10²⁴ ÷ 6.02 × 10²³ = 1.993… = 1.99 mol.
3How many formula units are in 0.0350 mol of sodium chloride?
N = 0.0350 × 6.02 × 10²³ = 2.11 × 10²² formula units.
4How many moles of oxygen atoms are in 1 mol of CO₂? How many oxygen atoms is that?
2 mol of O atoms, which is 2 × 6.02 × 10²³ = 1.20 × 10²⁴ O atoms.
5How many carbon atoms are in 0.500 mol of glucose, C₆H₁₂O₆?
Moles of C = 6 × 0.500 = 3.00 mol. Atoms = 3.00 × 6.02 × 10²³ = 1.81 × 10²⁴ C atoms.
6Which contains more molecules: 1 mol of hydrogen gas, H₂, or 1 mol of table sugar, C₁₂H₂₂O₁₁? Explain.
They contain the same number, 6.02 × 10²³ molecules. A mole is a number of particles, no matter how big each particle is. (The sugar sample is much heavier, though.)
7Write 0.000 000 602 in scientific notation. How many significant figures does it have?
6.02 × 10⁻⁷, with 3 significant figures (the leading zeros do not count).
More practice: warm-up
1What is Avogadro's constant, to three significant figures?
6.02 × 10²³ particles per mole (mol⁻¹).
2How many molecules are in 2.00 mol of methane, CH₄?
2.00 × 6.02 × 10²³ = 1.20 × 10²⁴ molecules.
3Write 45 000 in scientific notation.
4How many significant figures are in 0.00250 g?
3 (the 2, the 5 and the trailing zero after the decimal point).
5Finish the sentence: a dozen is to 12 as a mole is to ____.
More practice: on your own
1A gold ring contains 5.00 × 10²⁰ gold atoms. What amount of gold is this?
n = 5.00 × 10²⁰ ÷ 6.02 × 10²³ = 8.31 × 10⁻⁴ mol. The answer is much less than 1 mol because 5 × 10²⁰ is much less than 6 × 10²³.
2How many hydrogen atoms are in 2.50 mol of ammonia, NH₃?
Each NH₃ has 3 H atoms, so mol H = 3 × 2.50 = 7.50 mol. Atoms = 7.50 × 6.02 × 10²³ = 4.52 × 10²⁴ H atoms.
3A sample of methane, CH₄, contains 3.01 × 10²³ hydrogen atoms. What amount of methane is in the sample?
mol H = 3.01 × 10²³ ÷ 6.02 × 10²³ = 0.500 mol. Each CH₄ has 4 H, so mol CH₄ = 0.500 ÷ 4 = 0.125 mol.
4How many ions in total are in 0.200 mol of calcium chloride, CaCl₂?
Each formula unit gives 1 Ca²⁺ + 2 Cl⁻ = 3 ions. mol ions = 3 × 0.200 = 0.600 mol. Ions = 0.600 × 6.02 × 10²³ = 3.61 × 10²³ ions.
5Explain to a friend why chemists use the mole instead of counting atoms directly.
Sample: Atoms are so small that even a tiny sample has trillions of trillions of them, far too many to count. The mole is a counting unit, like a dozen, that bundles 6.02 × 10²³ particles together, and because it is linked to mass, chemists can count particles by weighing a sample.
Challenge
1Suppose you could count one atom every second, day and night, without stopping. About how many years would it take to count one mole of atoms? (Use 1 year ≈ 3.16 × 10⁷ s.)
Time = 6.02 × 10²³ s ÷ 3.16 × 10⁷ s/year ≈ 1.9 × 10¹⁶ years. That is about a million times longer than the age of the universe, which is why counting by weighing matters.
2Which sample contains more oxygen atoms: 1.00 mol of oxygen gas, O₂, or 0.700 mol of ozone, O₃? Show your reasoning.
O₂: 2 × 1.00 = 2.00 mol O atoms. O₃: 3 × 0.700 = 2.10 mol O atoms. The ozone sample has more oxygen atoms, even though it has fewer molecules.
3A sample of glucose, C₆H₁₂O₆, contains 1.00 × 10²⁴ atoms in total. What amount of glucose is in the sample?
Each molecule has 6 + 12 + 6 = 24 atoms. Molecules = 1.00 × 10²⁴ ÷ 24 = 4.167 × 10²² molecules. n = 4.167 × 10²² ÷ 6.02 × 10²³ = 0.0692 mol of glucose.
Quiz
1. What is Avogadro's constant, to three significant figures?
2. How many molecules are in 0.500 mol of water?
3. How many oxygen atoms are in 1.00 mol of CO₂?
4. A sample contains 1.204 × 10²⁴ atoms of neon. What amount is this?
5. Which statement about 1 mol of H₂ and 1 mol of sucrose, C₁₂H₂₂O₁₁, is true?
6. How many significant figures are in 0.0450 mol?
7. Why is NaCl counted in formula units rather than molecules?
Big idea: The mole is a counting unit, like a dozen, but for 6.02 × 10²³ particles. Expectation D3.2 asks students to describe how Avogadro's constant, the mole and molar mass are related; this lesson builds the first two and the next lesson adds molar mass. D2.1 is the vocabulary, D2.3 the calculations, and A1.13 the precision and significant-figure skills used all unit.
How to help: Use the dozen analogy often: "Three dozen eggs is how many eggs? So three moles of atoms is how many atoms?" Do the rice activity together; it is a model of how chemists count by weighing. Make sure your teen can use the EE (or EXP) key on their calculator; many errors in this unit are typing errors, not chemistry errors.
Common difficulties:
- Multiplying when they should divide. A quick check helps: a number of particles is enormous; an amount in moles is usually an ordinary number such as 0.25 or 3.0.
- Not saying which particle is counted. "A mole of oxygen" could mean O atoms or O₂ molecules, which differ by a factor of 2.
- Forgetting to multiply by the subscript when counting atoms in a compound.
- Typing 6.02 × 10 ^ 23 inside a longer calculation; using EE avoids order-of-operation mistakes.
- Rounding too early. Keep all digits until the final answer, then round to three significant figures.