Last updated: 17 August 2026
Chemical calculations eventually require us to connect the tiny world of atoms with quantities that can be measured in the laboratory. Before we can calculate moles, reacting masses or concentrations, we therefore need a sensible way to compare the masses of atoms and molecules.
Writing the actual mass of an individual atom in grams would produce extremely small numbers. Instead, chemists ask a much more useful question: how heavy is this atom compared with an agreed reference? That idea gives us the relative mass scale.
The important part of this chapter is not simply remembering symbols such as Aᵣ and Mᵣ. You need to see the distinction between the mass of one isotope, the weighted mean mass of an element and the total relative mass represented by a chemical formula.
Diagnostic Quiz
Try these five questions before reading the chapter. They check the isotope, formula-reading and averaging ideas that relative-mass calculations build upon.
- 5 quick prerequisite questions
- Focus on isotopes, formulae and averages
- Every answer option is explained
The carbon-12 relative mass scale
To compare masses, chemists first need an agreed reference. The standard used is the carbon-12 isotope.
One carbon-12 atom is assigned a relative isotopic mass of exactly 12. The reference against which other atomic-scale masses are compared is therefore:
You can imagine this as defining one small reference portion of the mass of a carbon-12 atom. Other atoms are then described according to how many times heavier they are than this reference.
Whether we are discussing the mass of an isotope, the average mass of an element or the mass represented by a molecular formula, the comparison is ultimately made with 1/12 of the mass of one carbon-12 atom.
Why use a relative scale?
The actual mass of one atom is extraordinarily small. A relative scale replaces inconvenient numbers with values such as approximately 1 for hydrogen, 12 for carbon and 16 for oxygen.
These numbers are much easier to compare and use in chemical calculations.
Why do relative masses have no units?
A relative mass is a ratio between two masses:
Since both quantities are masses, their units cancel. Relative atomic mass and relative molecular mass are therefore written as numbers without units.
Incorrect: Aᵣ(Cl) = 35.5 g
Correct: Aᵣ(Cl) = 35.5
The value is a relative comparison, not the mass of one chlorine atom measured in grams.
Relative isotopic mass
Most elements can exist as more than one isotope. Each isotope has its own characteristic mass, so we first need a way of describing the mass of one particular isotope.
The relative isotopic mass of an isotope is the mass of one atom of that isotope compared with 1/12 of the mass of one carbon-12 atom.
For example, if an isotope has a relative isotopic mass of about 35, one atom of that isotope is about 35 times as massive as the carbon-12 reference mass.
Relative isotopic mass and nucleon number are not the same idea
At first, relative isotopic mass may look almost identical to the isotope’s nucleon number. However, the meanings are different.
| Quantity | Meaning | Must it be a whole number? |
|---|---|---|
| Nucleon number | Total number of protons and neutrons in the nucleus | Yes |
| Relative isotopic mass | Mass of one isotope atom relative to the carbon-12 standard | No |
A nucleon number is a count of particles and therefore has to be an integer. A relative isotopic mass is a measured mass comparison and can differ slightly from the nucleon number.
An isotope described as chlorine-35 has a nucleon number of exactly 35, but its experimentally determined relative isotopic mass need not be exactly 35.000.
Relative atomic mass, Aᵣ
A sample of an element in nature may contain atoms of several isotopes. If those isotopes have different masses, we need one value that represents the element as a whole.
The relative atomic mass, Aᵣ, of an element is the weighted mean mass of an atom of that element compared with 1/12 of the mass of one carbon-12 atom.
The phrase weighted mean is crucial. An isotope that is more abundant contributes more strongly to Aᵣ than an isotope that is less abundant.
When abundances are given as percentages
Percentage abundances add to 100, so:
Worked example: chlorine
Suppose a sample contains approximately 75% chlorine-35 and 25% chlorine-37. Using 35 and 37 as the isotope masses:
Notice that 35.5 lies between 35 and 37, but it is closer to 35 because chlorine-35 is more abundant.
A weighted mean is pulled towards the value that occurs more often.
Why a simple mean gives the wrong answer
If we simply averaged 35 and 37:
This would be correct only if the two isotopes were present in equal amounts. Chlorine-35 is more common, so it must make a larger contribution to the average.
Simple mean: assumes that all isotope masses contribute equally.
Weighted mean: allows a more abundant isotope to contribute more.
Aᵣ is therefore calculated using a weighted mean.
Calculating Aᵣ from isotope abundances
Example 1: percentage abundances
Element X contains two isotopes:
- X-24: relative isotopic mass 24.0, abundance 80.0%;
- X-26: relative isotopic mass 26.0, abundance 20.0%.
The result passes two useful checks:
- 24.4 lies between 24.0 and 26.0;
- 24.4 lies closer to 24.0 because isotope-24 is more abundant.
Example 2: abundances given as a ratio
Isotope abundances do not have to be expressed as percentages. Suppose element Y has isotopes of masses 10 and 11 in the abundance ratio 1 : 4.
There is no need to convert the ratio to percentages first. The total abundance is simply five ratio parts.
Divide by 100 when percentages are used. If the abundances are given as a ratio, divide by the total number of ratio parts.
Working backwards to find an abundance
The weighted-mean relationship can also be used in reverse. Suppose an element contains isotopes of relative isotopic masses 35 and 37 and has Aᵣ = 35.5.
Let the percentage abundance of isotope-37 be x. The abundance of isotope-35 must then be 100 − x.
Isotope-37 therefore has an abundance of 25%, while isotope-35 has an abundance of 75%.
What does a decimal Aᵣ actually mean?
Students sometimes see Aᵣ(Cl) ≈ 35.5 and imagine that a chlorine atom somehow contains “35.5 nucleons”. It does not.
Individual atoms belong to particular isotopes. A chlorine atom might be chlorine-35 or chlorine-37. The value 35.5 describes the weighted mean mass of a large collection of naturally occurring chlorine atoms.
One isotope: has a particular nucleon number and relative isotopic mass.
The element: has an Aᵣ that reflects all its naturally occurring isotopes and their abundances.
A useful reasonableness check
A weighted mean must lie between the smallest and largest values being averaged.
If an element contains only isotopes of masses 24 and 26, its Aᵣ cannot be 23 or 27. It must lie somewhere between 24 and 26.
The position of Aᵣ also tells you something about abundance. If Aᵣ is much closer to 24 than to 26, isotope-24 must make the larger contribution.
Relative molecular mass, Mᵣ
Once the relative atomic masses of the elements are known, we can extend the same idea to whole molecules.
The relative molecular mass, Mᵣ, is the mean mass of one molecule compared with 1/12 of the mass of one carbon-12 atom.
In calculations, Mᵣ is found by adding the Aᵣ values of every atom shown in the molecular formula.
Example: water
A molecule of H₂O contains two hydrogen atoms and one oxygen atom.
Example: carbon dioxide
CO₂ contains one carbon atom and two oxygen atoms.
Example: sulfuric acid
H₂SO₄ contains two H atoms, one S atom and four O atoms.
Most Mᵣ errors are not caused by difficult arithmetic. They occur because the wrong number of atoms has been counted from the formula.
Relative formula mass
The term relative molecular mass is appropriate only when the substance consists of discrete molecules.
Ionic compounds such as NaCl and MgO form giant ionic lattices. There is no separate NaCl molecule or MgO molecule within the solid. For these substances, we use the term relative formula mass.
The calculation itself is performed in exactly the same way: add the Aᵣ values of all the atoms represented by the formula.
| Substance | Structure represented | Appropriate term | Calculation |
|---|---|---|---|
| H₂O | Discrete molecules | Relative molecular mass | (2 × 1.0) + 16.0 = 18.0 |
| CO₂ | Discrete molecules | Relative molecular mass | 12.0 + (2 × 16.0) = 44.0 |
| NaCl | Ionic lattice | Relative formula mass | 23.0 + 35.5 = 58.5 |
| MgO | Ionic lattice | Relative formula mass | 24.3 + 16.0 = 40.3 |
Relative molecular mass refers to a molecule. Relative formula mass is used when the chemical formula represents a ratio of particles in a giant structure rather than a discrete molecule.
Handling brackets when calculating Mᵣ
Brackets are a common source of mistakes because the subscript outside a bracket applies to every atom inside the bracket.
Example: magnesium hydroxide
Mg(OH)₂ contains:
- 1 Mg atom;
- 2 O atoms;
- 2 H atoms.
Incorrect counting: Mg₁O₁H₂
Correct counting: Mg₁O₂H₂
The 2 outside (OH) multiplies both oxygen and hydrogen.
Example: calcium nitrate
Ca(NO₃)₂ contains one Ca atom, two N atoms and six O atoms.
Worked examples with relative masses
Example 1: choosing the more abundant isotope
Element X contains isotopes of relative isotopic masses 63 and 65. Its Aᵣ is 63.4. Which isotope is more abundant?
The weighted mean, 63.4, lies much closer to 63 than to 65. The isotope of mass 63 must therefore contribute more strongly to the average.
Answer: isotope-63 is more abundant.
Example 2: calculating Aᵣ from a ratio
Element Q contains isotope-24 and isotope-25 in the abundance ratio 3 : 2.
Since isotope-24 is more abundant, an answer closer to 24 than to 25 is sensible.
Example 3: calculating the relative molecular mass of ammonia
NH₃ contains one nitrogen atom and three hydrogen atoms.
Example 4: calculating the relative formula mass of magnesium hydroxide
Example 5: spotting an impossible Aᵣ
A student calculates Aᵣ = 82.3 for an element containing only isotopes of relative isotopic masses 79 and 81.
This cannot be correct. A weighted mean of 79 and 81 must lie between 79 and 81.
A weighted mean cannot be smaller than every isotope mass or larger than every isotope mass used to calculate it.
Common mistakes with relative masses
| Common mistake | Why it is wrong | Correct approach |
|---|---|---|
| Using the whole carbon-12 atom as the reference mass | The relative mass scale uses one-twelfth of its mass | Compare with 1/12 of the mass of one carbon-12 atom |
| Giving Aᵣ or Mᵣ units such as g | Relative masses are ratios and therefore have no units | Write Aᵣ and Mᵣ as numbers only |
| Confusing nucleon number with relative isotopic mass | Nucleon number counts particles, while relative isotopic mass compares masses | Remember that nucleon number must be an integer |
| Taking a simple average of isotope masses | The isotopes may not occur in equal amounts | Use a weighted mean based on isotopic abundance |
| Dividing by 100 when abundances are given as a ratio | The ratio may not contain 100 total parts | Divide by the sum of the ratio parts |
| Thinking Aᵣ = 35.5 means each atom has 35.5 nucleons | Aᵣ represents a weighted mean for many atoms | Distinguish individual isotopes from the element’s average |
| Using “relative molecular mass” for NaCl | NaCl forms an ionic lattice rather than discrete molecules | Use the term relative formula mass |
| Ignoring a subscript outside brackets | The subscript multiplies every atom inside the bracket | Expand the bracket mentally before adding Aᵣ values |
| Accepting an Aᵣ outside the isotope-mass range | A weighted mean must lie between the values being averaged | Check that the answer lies between the lightest and heaviest isotopes |
Relative-mass checklist
Identify the quantity
- One particular isotope → relative isotopic mass.
- Average for an element → Aᵣ.
- Discrete molecule → relative molecular mass.
- Formula unit or giant structure → relative formula mass.
Calculate Aᵣ
- Multiply each isotope mass by its abundance.
- Add the weighted contributions.
- Divide by 100 for percentage abundances.
- Divide by the total parts for an abundance ratio.
Calculate Mᵣ
- Read the chemical formula carefully.
- Multiply each Aᵣ by the number of atoms present.
- Apply outside subscripts to everything inside brackets.
- Add all contributions together.
Check the answer
- Do not give Aᵣ or Mᵣ a unit.
- An Aᵣ must lie within the isotope-mass range.
- Aᵣ should lie closer to the more abundant isotope.
- Check that every atom in the formula has been counted.
Mastery Quiz
Now apply the chapter to isotope data, weighted averages and unfamiliar chemical formulae. These questions focus on reasoning rather than definition recall alone.
- 10 application-focused questions
- Includes calculations and misconception testing
- Every answer option is explained
Frequently asked questions
Questions students often ask about relative masses
These questions bring together the distinctions between isotope mass, atomic mass and the relative mass represented by a chemical formula.
Why is carbon-12 assigned a relative isotopic mass of exactly 12?
A relative scale needs an agreed reference. Carbon-12 is defined to have a relative isotopic mass of exactly 12, so one-twelfth of the mass of one carbon-12 atom becomes the reference against which other atomic-scale masses are compared.
Why does Aᵣ often contain a decimal even though atoms contain whole numbers of protons and neutrons?
Aᵣ is not the nucleon number of one atom. It is a weighted mean that takes into account the relative isotopic masses and abundances of the different isotopes present in a sample of the element.
Can an element with isotopes of masses 20 and 22 have Aᵣ = 23?
No. A weighted mean must lie between the values being averaged. If 20 and 22 are the only isotope masses present, Aᵣ must lie between 20 and 22.
Does the isotope with the greatest mass always determine Aᵣ?
No. Both mass and abundance matter. A lighter isotope can dominate the value of Aᵣ if it is much more abundant than the heavier isotope.
Do I always divide by 100 when calculating Aᵣ?
Only when the abundances are given as percentages. If the abundances are given as a ratio, divide by the sum of the ratio parts instead.
Why is relative molecular mass written Mᵣ even though it has no unit?
The symbol Mᵣ represents a relative mass ratio. The subscript r reminds us that the quantity is relative to the carbon-12 reference rather than being an actual mass measured in grams.
What is the difference between relative molecular mass and relative formula mass?
Relative molecular mass is used when the substance consists of discrete molecules, such as H₂O or CO₂. Relative formula mass is more appropriate for substances such as NaCl that form giant structures rather than separate molecules.
Both are calculated by adding the relevant Aᵣ values in the chemical formula.
Why does the 2 in Mg(OH)₂ multiply both O and H?
The subscript is written outside the bracket, so it applies to the complete bracketed group. There are two OH groups, giving two oxygen atoms and two hydrogen atoms.
Should I use isotope mass numbers or periodic-table Aᵣ values when calculating Mᵣ?
For normal Mᵣ calculations, use the Aᵣ values supplied in the question or periodic table. These values represent the appropriate weighted mean atomic masses of the elements.
The relative mass scale gives chemists a practical way to compare particles that are far too small to weigh individually. Relative isotopic mass describes one isotope, while Aᵣ combines isotope masses and abundances into a weighted mean. Adding Aᵣ values then gives the relative molecular or formula mass represented by a chemical formula. With these distinctions secure, the relative masses needed for later mole and stoichiometric calculations become much easier to interpret correctly.