Relative Atomic Mass and Relative Formula Mass

Quantitative chemistry relies on comparing the masses of atoms and molecules using a standardised scale. Because individual atoms are far too small to weigh directly — a single carbon atom has a mass of about 2 × 10⁻²³ g — chemists use relative masses. These are dimensionless numbers that tell us how heavy one atom or molecule is compared to a universally agreed standard. This lesson covers relative atomic mass (Aᵣ) and relative formula mass (Mᵣ), which are the foundation of every mole calculation you will encounter at GCSE and beyond.

Relative Atomic Mass (Aᵣ)

The relative atomic mass (Aᵣ) of an element is defined as the weighted mean mass of an atom of that element compared to one-twelfth of the mass of a carbon-12 atom.

In simpler terms:

The carbon-12 isotope is chosen as the reference standard and is defined as having a mass of exactly 12
Every other element's Aᵣ is measured relative to this standard
Aᵣ has no units — it is a pure ratio (a dimensionless number)

The Aᵣ values for all elements are given on the periodic table. You should know some of the most common ones from memory:

Element	Symbol	Aᵣ	Element	Symbol	Aᵣ
Hydrogen	H	1	Sulfur	S	32
Carbon	C	12	Chlorine	Cl	35.5
Nitrogen	N	14	Potassium	K	39
Oxygen	O	16	Calcium	Ca	40
Sodium	Na	23	Iron	Fe	56
Magnesium	Mg	24	Copper	Cu	64
Aluminium	Al	27	Zinc	Zn	65

Notice that chlorine has an Aᵣ of 35.5, which is not a whole number. This is because chlorine exists naturally as a mixture of two isotopes: chlorine-35 (approximately 75% abundance) and chlorine-37 (approximately 25% abundance). The weighted mean of these gives 35.5.

Why "Weighted Mean"?

The phrase weighted mean is critical and appears frequently in exam mark schemes. A weighted mean accounts for the fact that different isotopes of an element exist in different proportions. It is not a simple average — more abundant isotopes contribute more to the final Aᵣ value.

Formula for calculating Aᵣ from isotopic data:

Aᵣ = Σ (isotope mass × percentage abundance) / 100

Or equivalently: Aᵣ = Σ (isotope mass × fractional abundance)

graph TD
    A["Calculating Aᵣ"] --> B["Step 1: Multiply each isotope\nmass by its % abundance"]
    B --> C["Step 2: Add all the products"]
    C --> D["Step 3: Divide by 100"]
    D --> E["Result: Weighted mean\n= Aᵣ"]

Worked Example 1: Calculating Aᵣ from Isotopic Data

Question: Copper exists as two isotopes: copper-63 (69.2%) and copper-65 (30.8%). Calculate the relative atomic mass of copper.

Step 1: Multiply each isotope mass by its percentage abundance.

63 × 69.2 = 4,359.6
65 × 30.8 = 2,002.0

Step 2: Add the products.

4,359.6 + 2,002.0 = 6,361.6

Step 3: Divide by 100.

6,361.6 / 100 = 63.6

Answer: The relative atomic mass of copper is 63.6 (this matches the periodic table value to 1 decimal place).

Worked Example 2: Aᵣ from Mass Spectrum Data

Question: A mass spectrum of boron shows two peaks: mass 10 with relative intensity 20, and mass 11 with relative intensity 80. Calculate the Aᵣ of boron.

Step 1: The relative intensities give us the percentage abundances (20% and 80%).

Step 2: Aᵣ = (10 × 20 + 11 × 80) / 100 = (200 + 880) / 100 = 1,080 / 100 = 10.8

Answer: The relative atomic mass of boron is 10.8.

Worked Example 3: Three Isotopes

Question: Silicon has three isotopes: Si-28 (92.2%), Si-29 (4.7%), and Si-30 (3.1%). Calculate the Aᵣ.

Aᵣ = (28 × 92.2 + 29 × 4.7 + 30 × 3.1) / 100 = (2,581.6 + 136.3 + 93.0) / 100 = 2,810.9 / 100 = 28.1

Relative Formula Mass (Mᵣ)

The relative formula mass (Mᵣ) of a compound is the sum of the relative atomic masses of all the atoms shown in its chemical formula. Like Aᵣ, it has no units.

Method: Write out the formula, count the number of each type of atom (being very careful with brackets and subscripts), multiply each count by the relevant Aᵣ, and add them all together.

Worked Example 4: Simple Mᵣ Calculation

Question: Calculate the relative formula mass of water (H₂O).

Atom	Number in formula	Aᵣ	Contribution
H	2	1	2
O	1	16	16
Total			18

Answer: Mᵣ of H₂O = 18

Worked Example 5: Mᵣ of an Ionic Compound

Question: Calculate the relative formula mass of calcium carbonate (CaCO₃).

Atom	Number in formula	Aᵣ	Contribution
Ca	1	40	40
C	1	12	12
O	3	16	48
Total			100

Answer: Mᵣ of CaCO₃ = 100

Worked Example 6: Mᵣ with Brackets

Question: Calculate the relative formula mass of magnesium hydroxide, Mg(OH)₂.

The bracket and subscript 2 means there are two OH groups, which gives two O atoms and two H atoms in total.

Atom	Number in formula	Aᵣ	Contribution
Mg	1	24	24
O	2	16	32
H	2	1	2
Total			58

Answer: Mᵣ of Mg(OH)₂ = 58

Common GCSE exam mistake: Forgetting to multiply the atoms INSIDE the bracket by the subscript outside. In Mg(OH)₂, there are 2 oxygen atoms and 2 hydrogen atoms, not 1 of each. This is the single most common arithmetic error in Mᵣ calculations and cascades through every subsequent calculation.

Worked Example 7: Mᵣ of a Hydrated Salt

Question: Calculate the Mᵣ of hydrated copper sulfate, CuSO₄·5H₂O.

The "·5H₂O" means 5 water molecules are included in each formula unit (called "water of crystallisation").

Component	Atoms	Aᵣ calculation	Contribution
Cu	1 Cu	1 × 64	64
S	1 S	1 × 32	32
O (in sulfate)	4 O	4 × 16	64
5H₂O	10 H + 5 O	(10 × 1) + (5 × 16)	10 + 80 = 90
Total			250

Answer: Mᵣ of CuSO₄·5H₂O = 250

Percentage Composition by Mass

You can calculate the percentage of a particular element in a compound using:

% by mass = (total Aᵣ of that element in the formula / Mᵣ of the compound) × 100

Worked Example 8: Percentage Composition

Question: Calculate the percentage by mass of iron in iron(III) oxide, Fe₂O₃. (Aᵣ: Fe = 56, O = 16)

Step 1: Mᵣ of Fe₂O₃ = (2 × 56) + (3 × 16) = 112 + 48 = 160

Step 2: % of Fe = (112 / 160) × 100 = 70.0%

Answer: Iron makes up 70.0% of the mass of iron(III) oxide.

Quick Reference: Common Mᵣ Values

Compound	Formula	Mᵣ
Water	H₂O	18
Carbon dioxide	CO₂	44
Hydrochloric acid	HCl	36.5
Sodium hydroxide	NaOH	40
Sulfuric acid	H₂SO₄	98
Nitric acid	HNO₃	63
Calcium carbonate	CaCO₃	100
Sodium chloride	NaCl	58.5
Magnesium oxide	MgO	40
Copper sulfate (anhydrous)	CuSO₄	160
Ammonium nitrate	NH₄NO₃	80

Memorising these common values helps you spot calculation errors quickly and saves time in exams.

Key Exam Tips

Aᵣ and Mᵣ have no units — they are dimensionless ratios. Never write "g" or "amu" after them at GCSE level
Always use the Aᵣ values given in the question or on the periodic table — do not round them unless specifically told to
Watch out for brackets in formulae — multiply ALL atoms inside the bracket by the subscript outside
For hydrated salts (like CuSO₄·5H₂O), include ALL the water molecules in the Mᵣ calculation
Show your working clearly in a table format: list each atom, its count, and its contribution. This earns method marks even if your final answer is wrong
Double-check your arithmetic — a Mᵣ error at this stage will cascade through every subsequent mole, mass, and concentration calculation in multi-step problems
When calculating Aᵣ from isotopic data, always use the phrase "weighted mean" — this exact wording is required for the definition mark