Relative Formula Mass of Ionic Compounds

Introduction

Ionic compounds such as calcium carbonate and copper sulfate have no molecules, so the phrase "relative molecular mass" does not really fit them. Instead, we use the relative formula mass of one formula unit. The arithmetic is the same addition as for molecules, but ionic formulae often contain brackets around polyatomic ions, and misreading those brackets is one of the most common errors in chemistry calculations.

Core explanation

Mᵣ of a formula unit. For an ionic compound, Mᵣ is the sum of the Aᵣ values of all the atoms in one formula unit. For sodium chloride, NaCl: 23.0 + 35.5 = 58.5. For magnesium oxide, MgO: 24.3 + 16.0 = 40.3.

Charges make no difference. An ion is formed by gaining or losing electrons. An electron's mass is only about 1/1800 of the mass of a proton, so a few electrons make no measurable difference at this precision. We therefore use ordinary Aᵣ values for ions: Na⁺ counts as 23.0 and Cl⁻ as 35.5.

Brackets multiply everything inside. In calcium hydroxide, Ca(OH)₂, the subscript 2 applies to the whole OH group. The formula contains 1 Ca, 2 O and 2 H:

Mᵣ = 40.1 + 2 × (16.0 + 1.0) = 40.1 + 34.0 = 74.1

It is often quickest to find the mass of the group first, then multiply. Useful group masses (with the values used in this unit):

Group Mass --- --- OH 17.0 NO₃ 62.0 SO₄ 96.1 CO₃ 60.0 NH₄ 18.0 PO₄ 95.0

More examples.

Compound Working Mᵣ --- --- --- CaCO₃ 40.1 + 12.0 + 3 × 16.0 100.1 MgCl₂ 24.3 + 2 × 35.5 95.3 Mg(NO₃)₂ 24.3 + 2 × 62.0 148.3 (NH₄)₂SO₄ 2 × 18.0 + 96.1 132.1 Fe₂O₃ 2 × 55.8 + 3 × 16.0 159.6 Al₂(SO₄)₃ 2 × 27.0 + 3 × 96.1 342.3

Subscripts inside and outside. In Al₂(SO₄)₃, the 4 is inside the bracket and belongs to oxygen; the 3 outside multiplies the whole sulfate group. So there are 3 sulfur atoms and 3 × 4 = 12 oxygen atoms.

Step-by-step reasoning

1. Identify any brackets and the number outside each. 2. Find the mass of each bracketed group by adding its atoms. 3. Multiply each group mass by the number outside the bracket. 4. Add the Aᵣ values of the atoms outside brackets, multiplied by their subscripts. 5. Add everything together and check by counting atoms in full.

Visual explanation

Picture ammonium sulfate as a small parcel: two identical tetrahedral NH₄⁺ ions and one tetrahedral SO₄²⁻ ion. Weighing one NH₄⁺ (18.0) and doubling it, then adding the sulfate (96.1), gives the mass of the parcel, 132.1.

Real-world analogy

A café orders "3 × (sandwich + drink)". The bracket means three sandwiches and three drinks, not one sandwich and three drinks. Chemical brackets work in exactly the same way.

Real-world example

Fertiliser bags list compounds such as ammonium nitrate, NH₄NO₃ (Mᵣ 80.0), and ammonium sulfate, (NH₄)₂SO₄ (Mᵣ 132.1). Farmers and manufacturers use these Mᵣ values to compare how much nitrogen each fertiliser supplies per kilogram.

Why?

Why do ionic compounds use formula units rather than molecules? In a lattice, each ion is surrounded by many oppositely charged ions and there is no separate molecule to weigh. The formula unit is the smallest repeating ratio, so its mass is the sensible quantity to use.

Common misconception

"Ca(OH)₂ means one oxygen and two hydrogens." The bracket multiplies both atoms inside it, giving two oxygen atoms and two hydrogen atoms. Leaving out the bracket effect gives Mᵣ = 58.1 instead of the correct 74.1.

Worked example

Question: Calculate the Mᵣ of calcium phosphate, Ca₃(PO₄)₂.

Reasoning: Ca: 3 × 40.1 = 120.3. PO₄ group: 31.0 + 4 × 16.0 = 95.0; two groups = 190.0. Total = 120.3 + 190.0 = 310.3. Check by atoms: 3 Ca, 2 P, 8 O.

Answer: Mᵣ(Ca₃(PO₄)₂) = 310.3.

Quick check

1. Calculate the Mᵣ of sodium carbonate, Na₂CO₃. Answer: 2 × 23.0 + 12.0 + 3 × 16.0 = 106.0.

Exam focus

Examiners love formulae with brackets, such as Mg(OH)₂, Ca(NO₃)₂ and Al₂(SO₄)₃. Write out the atom count in full or work out the group mass first. Never include the charges of ions in the calculation, and do not give Mᵣ units.

Advanced insight

In very precise work, the mass of an ion does differ slightly from that of the neutral atom. A Cl⁻ ion is heavier than a Cl atom by the mass of one electron, about 0.0005 u. This matters in high-resolution mass spectrometry, but it is far below the precision of ordinary chemical calculations.

Summary

For ionic compounds, Mᵣ is the sum of the Aᵣ values of all atoms in one formula unit. Ionic charges do not change it, because electrons have almost no mass. A number outside a bracket multiplies every atom inside, so find each group's mass and multiply, or count all atoms in full.

Practice questions

1. Calculate the Mᵣ of magnesium hydroxide, Mg(OH)₂. Answer: 24.3 + 2 × 17.0 = 58.3. 2. Calculate the Mᵣ of calcium nitrate, Ca(NO₃)₂. Answer: 40.1 + 2 × 62.0 = 164.1. 3. How many oxygen atoms are in one formula unit of Al₂(SO₄)₃, and what is its Mᵣ? Answer: 3 × 4 = 12 oxygen atoms; Mᵣ = 2 × 27.0 + 3 × 96.1 = 342.3. 4. Explain why the Mᵣ of potassium chloride, KCl, is the same whether you treat it as ions or atoms. Answer: The ions differ from the atoms only by one electron each, and an electron's mass is negligible, so Mᵣ = 39.1 + 35.5 = 74.6 either way.