Counting Atoms in Formulae

Introduction

Reading H₂O is easy, but what about Ca(OH)₂ or 3Al₂(SO₄)₃? Formulae often contain brackets, and in equations they are often written with a large number in front. Both multiply the atoms, and both follow simple rules. Counting atoms correctly is an essential skill: it is needed for balancing equations, working out relative formula masses and checking that mass is conserved in reactions.

Core explanation

Rule 1: a subscript counts the symbol before it. In Na₂CO₃, there are 2 Na, 1 C and 3 O.

Rule 2: a subscript after a bracket multiplies everything inside the bracket. Brackets group atoms that stay together, usually a polyatomic ion. In Ca(OH)₂: - inside the bracket: 1 O and 1 H; - the 2 outside multiplies both: 2 O and 2 H; - plus 1 Ca. Total: 1 Ca, 2 O, 2 H, which is 5 atoms.

If a subscript is inside the bracket as well, multiply the two numbers. In Al₂(SO₄)₃, the group SO₄ contains 1 S and 4 O; the 3 outside gives 3 S and 3 × 4 = 12 O. With 2 Al, the total is 2 + 3 + 12 = 17 atoms.

Rule 3: a coefficient multiplies the whole formula. A large number written in front of a formula is a coefficient . It tells you how many molecules or formula units there are. 3H₂O means three water molecules: 3 × 2 = 6 H and 3 × 1 = 3 O. The coefficient multiplies every atom in the formula, including those in brackets.

Combining the rules. For 2Mg(NO₃)₂: - one formula unit: 1 Mg, 2 × 1 = 2 N, 2 × 3 = 6 O; - multiply by the coefficient 2: 2 Mg, 4 N, 12 O; - total atoms: 2 + 4 + 12 = 18.

Elements appearing more than once. Some formulae mention an element in two places. In CH₃COOH (ethanoic acid), hydrogen appears as H₃ and as H: 3 + 1 = 4 H. Carbon appears twice: 2 C. Oxygen appears twice: 2 O. Always add every appearance.

Why brackets are used. Writing Ca(OH)₂ instead of CaO₂H₂ shows that the substance contains two hydroxide ions, OH⁻. The brackets keep the chemistry visible.

Useful table.

Expression Atoms of each element Total atoms --- --- --- NH₄NO₃ 2 N, 4 H, 3 O 9 (NH₄)₂SO₄ 2 N, 8 H, 1 S, 4 O 15 4CO₂ 4 C, 8 O 12 3Ca(OH)₂ 3 Ca, 6 O, 6 H 15

Step-by-step reasoning

To count atoms in any expression:

1. Ignore the coefficient at first and work on one formula. 2. Multiply atoms inside each bracket by the subscript after the bracket. 3. Add up each element wherever it appears. 4. Multiply every count by the coefficient. 5. Add the counts to find the total, if needed.

Visual explanation

Draw Ca(OH)₂ as one calcium sphere with two separate O–H pairs attached, each pair circled. The two circles show why the bracketed group is doubled. For 3Ca(OH)₂, draw three copies of the whole picture side by side and count the spheres of each colour.

Real-world analogy

A bracket is like a pack: "2 × (1 sandwich + 1 apple)" means two sandwiches and two apples. A coefficient is like buying several identical lunch boxes: three lunch boxes, each with two packs, gives six sandwiches and six apples.

Real-world example

Fertiliser bags list compounds such as ammonium sulfate, (NH₄)₂SO₄, and ammonium nitrate, NH₄NO₃. Counting the nitrogen atoms in each formula helps farmers and chemists compare how much nitrogen, an essential plant nutrient, each fertiliser supplies.

Why?

Why does a coefficient multiply everything, while a subscript multiplies only one symbol or bracket? A coefficient counts whole molecules or formula units, so each copy brings all of its atoms. A subscript describes the make-up of one formula unit, so it applies only to the part it follows.

Common misconception

"In Ca(OH)₂ there are 2 oxygen atoms but only 1 hydrogen." The subscript after a bracket multiplies every atom inside it, so there are 2 hydrogen atoms too. Another common slip is applying a coefficient to the first element only: 2H₂O has 4 H and 2 O.

Worked example

Question: How many atoms of each element are in 2(NH₄)₃PO₄?

Reasoning: One formula unit: N = 1 × 3 = 3; H = 4 × 3 = 12; P = 1; O = 4. Multiply by the coefficient 2: N = 6, H = 24, P = 2, O = 8. Total = 6 + 24 + 2 + 8 = 40.

Answer: 6 N, 24 H, 2 P and 8 O: 40 atoms in total.

Quick check

1. How many hydrogen atoms are in 3H₂SO₄? Answer: Six, because each molecule has 2 hydrogen atoms and there are 3 molecules.

Exam focus

Show your working clearly, element by element. Examiners award marks for correct counts in bracketed formulae and for applying coefficients to every element. These counts are the basis of balancing equations and calculating relative formula mass, so accuracy here prevents errors later.

Advanced insight

When balancing equations, you may change only coefficients, never subscripts or brackets. Changing 2H₂O to H₂O₂ would change water into hydrogen peroxide. Coefficients change how many particles take part; subscripts define what the particles are. Atom counting on both sides then confirms that atoms are conserved.

Summary

A subscript counts the atom before it; a subscript after a bracket multiplies everything inside; a coefficient in front multiplies the whole formula. Add every appearance of an element. Work through one formula unit first, then multiply by the coefficient, and total the counts.

Practice questions

1. Count the atoms of each element in Mg(OH)₂. Answer: 1 Mg, 2 O and 2 H: 5 atoms in total. 2. How many oxygen atoms are in 2Al₂(SO₄)₃? Answer: One formula unit has 4 × 3 = 12 O, so 2 formula units have 24 oxygen atoms. 3. How many carbon and hydrogen atoms are in 5CH₄? Answer: 5 carbon atoms and 5 × 4 = 20 hydrogen atoms. 4. A student says (NH₄)₂CO₃ contains 4 hydrogen atoms. Correct the error. Answer: The 2 after the bracket multiplies the NH₄ group, so there are 2 × 4 = 8 hydrogen atoms (and 2 nitrogen atoms). 5. Find the total number of atoms in CH₃COOH. Answer: 2 C, 4 H and 2 O, giving 8 atoms in total.