Masses of Formulae with Brackets

Introduction

Some formulae contain brackets, such as Ca(OH)₂ for calcium hydroxide (slaked lime) and Mg(NO₃)₂ for magnesium nitrate. Brackets are a compact way of saying "this group of atoms appears more than once". They are also the single biggest source of mistakes in mass calculations. Once you know exactly what the small number after a bracket multiplies, these formulae become no harder than H₂O or CO₂.

Core explanation

Why brackets are used. Many ionic compounds contain polyatomic ions — small groups of atoms that stay together and carry an overall charge. Examples are hydroxide (OH⁻), nitrate (NO₃⁻), sulfate (SO₄²⁻), carbonate (CO₃²⁻) and ammonium (NH₄⁺). When a formula needs two or more of the same polyatomic ion, the group is placed in brackets and the number is written after the closing bracket. Calcium ions carry a 2+ charge and hydroxide ions a 1− charge, so two hydroxide ions are needed: Ca(OH)₂.

What the subscript multiplies. A subscript written immediately after a bracket multiplies everything inside the bracket. A subscript inside the bracket applies only to the atom just before it.

- In Ca(OH)₂ there is 1 Ca, 2 × 1 = 2 O and 2 × 1 = 2 H. - In Mg(NO₃)₂ there is 1 Mg, 2 × 1 = 2 N and 2 × 3 = 6 O.

Two good methods. You can calculate the mass either by first counting all the atoms, or by working out the mass of the bracketed group and then multiplying. Both give the same answer, and using one to check the other is excellent practice.

Using Ar values Ca = 40, O = 16, H = 1, Mg = 24, N = 14:

Method 1 — count atoms first (Ca(OH)₂): Mr = 40 + (2 × 16) + (2 × 1) = 40 + 32 + 2 = 74

Method 2 — mass of the group first (Ca(OH)₂): mass of OH = 16 + 1 = 17; Mr = 40 + (2 × 17) = 40 + 34 = 74

Magnesium nitrate, Mg(NO₃)₂: mass of NO₃ = 14 + (3 × 16) = 14 + 48 = 62 Mr = 24 + (2 × 62) = 24 + 124 = 148

Relative formula mass has no units. Because it compares masses with one-twelfth of the mass of a carbon-12 atom, Mr is a pure number. Later, when you meet moles, the same number will appear with the unit g/mol.

Formulae

Mr = sum of (number of each atom × its Ar). For a bracket: mass of group = sum of Ar values inside; contribution = mass of group × subscript outside.

Step-by-step reasoning

To find Mr for Al₂(SO₄)₃ (Al = 27, S = 32, O = 16):

1. Find the mass of the bracketed group: SO₄ = 32 + (4 × 16) = 96. 2. Multiply by the subscript outside the bracket: 3 × 96 = 288. 3. Add the atoms outside the bracket: Al₂ = 2 × 27 = 54. 4. Total: Mr = 54 + 288 = 342. 5. Check by counting atoms: 2 Al, 3 S, 12 O → 54 + 96 + 192 = 342.

Visual explanation

Imagine each bracket as a sealed bag of atoms. Ca(OH)₂ is one calcium ball beside two identical bags, each holding one red oxygen ball and one small white hydrogen ball. Mg(NO₃)₂ is one magnesium ball beside two bags, each holding one blue nitrogen ball and three red oxygen balls. The number after the bracket tells you how many bags there are.

Real-world analogy

A shopping list says "2 × (bread, milk, eggs)". You do not buy two loaves and one bottle of milk and one box of eggs — you buy two of everything in the bracket. Chemical brackets work in exactly the same way.

Real-world example

Calcium hydroxide is spread on acidic fields and used in mortar and plaster, and magnesium nitrate appears in some fertilisers. Manufacturers and farmers need their formula masses to work out what fraction of a product is the useful element, such as the nitrogen in a fertiliser.

Why?

Why not simply write CaO₂H₂ instead of Ca(OH)₂? Writing the bracket shows that the compound contains hydroxide ions, each an oxygen bonded to a hydrogen. The bracketed form tells chemists about structure and how the substance will behave, not just which atoms are present.

Common misconception

"In Ca(OH)₂ the 2 applies only to the hydrogen." This gives 40 + 16 + 2 = 58, which is wrong. The subscript after a bracket multiplies every atom inside it, so there are two oxygen atoms as well as two hydrogen atoms, and the correct Mr is 74.

Worked example

Question: Calculate the relative formula mass of ammonium sulfate, (NH₄)₂SO₄ (N = 14, H = 1, S = 32, O = 16).

Reasoning: Mass of NH₄ = 14 + (4 × 1) = 18. Two ammonium groups: 2 × 18 = 36. Mass of SO₄ = 32 + (4 × 16) = 96. Total = 36 + 96 = 132. Check by counting: 2 N, 8 H, 1 S, 4 O → 28 + 8 + 32 + 64 = 132.

Answer: Mr = 132.

Quick check

1. How many oxygen atoms are shown in the formula Mg(NO₃)₂? Answer: Six, because the 2 after the bracket multiplies the 3 oxygen atoms in each nitrate group.

Exam focus

Show your working line by line, writing the mass of the bracketed group before multiplying. Examiners award method marks for correct steps even if an arithmetic slip occurs. Always check that the subscript outside the bracket has been applied to every atom inside.

Advanced insight

Brackets also appear in the formulae of complex ions, such as [Cu(H₂O)₆]²⁺, the hydrated copper ion that gives many copper salt solutions their blue colour. Square brackets in these complex ions group the atoms that surround a central metal ion. The rule is the same at every level: a subscript multiplies everything in the bracket directly before it, and nested brackets are multiplied from the inside outwards.

Summary

Brackets group atoms, usually a polyatomic ion, that appear more than once in a formula. A subscript after a bracket multiplies every atom inside it. To find Mr, either count all atoms first or calculate the group mass and multiply. Ca(OH)₂ has Mr = 74 and Mg(NO₃)₂ has Mr = 148; Mr has no units.

Practice questions

1. Calculate Mr for magnesium hydroxide, Mg(OH)₂ (Mg = 24, O = 16, H = 1). Answer: 24 + 2 × (16 + 1) = 24 + 34 = 58. 2. Calculate Mr for calcium nitrate, Ca(NO₃)₂ (Ca = 40, N = 14, O = 16). Answer: 40 + 2 × (14 + 48) = 40 + 124 = 164. 3. List the number of each type of atom in Fe(OH)₃ and calculate its Mr (Fe = 56). Answer: 1 Fe, 3 O, 3 H; Mr = 56 + 3 × 17 = 56 + 51 = 107. 4. A student calculates Mr of Ca(OH)₂ as 58. Explain the error. Answer: The student multiplied only the hydrogen by 2; the oxygen must also be doubled, giving 40 + 32 + 2 = 74. 5. Calculate Mr for calcium phosphate, Ca₃(PO₄)₂ (Ca = 40, P = 31, O = 16). Answer: 3 × 40 + 2 × (31 + 64) = 120 + 190 = 310.