Relative Masses of Atoms and Molecules

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Why Do We Need a Relative Mass Scale?

Atoms are extremely small. A single hydrogen atom has a diameter of approximately 10⁻¹⁰ metres and weighs only 1.67 × 10⁻²⁷ kilograms. These numbers are so tiny that they are very difficult to work with in everyday calculations. Imagine trying to measure out 1.67 × 10⁻²⁷ kg of hydrogen on a balance — it would be impossible!

Because atoms are so incredibly light, scientists developed a relative mass scale. Instead of measuring the actual mass of atoms in kilograms, we compare the masses of atoms to a standard atom. This makes the numbers much easier to handle.

The Unified Atomic Mass Unit

The standard we use is the carbon-12 atom. Scientists chose carbon-12 because it is:

• Common and stable
• Easy to measure accurately in experiments
• A convenient reference point

Definition of the unified atomic mass unit:

The unified atomic mass unit (symbol: u, sometimes called a Dalton, symbol: Da) is defined as one-twelfth of the mass of a carbon-12 atom.

In other words:

• We take one atom of carbon-12
• We divide its mass by 12
• That gives us 1 unified atomic mass unit

By definition, one carbon-12 atom has a mass of exactly 12 u.

If we convert this to actual mass in kilograms: 1 u = 1.66 × 10⁻²⁷ kg

But in chemistry, we almost always use the relative scale (in units of u) rather than actual masses in kg, because it's much simpler.

Relative Isotopic Mass

Before we can understand relative atomic mass, we need to know about isotopes.

Isotopes are atoms of the same element that have:

• The same number of protons (same atomic number)
• Different numbers of neutrons (different mass numbers)

For example, chlorine has two main isotopes:

• Chlorine-35: 17 protons, 18 neutrons
• Chlorine-37: 17 protons, 20 neutrons

Definition of relative isotopic mass:

The relative isotopic mass is the mass of one atom of a specific isotope of an element compared to one-twelfth of the mass of one carbon-12 atom.

In simpler terms: it's the mass of a particular isotope measured on the unified atomic mass unit scale.

Important note: Relative isotopic mass has no units because it is a ratio (a comparison of two masses). The units cancel out.

Examples:

• The relative isotopic mass of carbon-12 is exactly 12 (by definition)
• The relative isotopic mass of chlorine-35 is approximately 35
• The relative isotopic mass of chlorine-37 is approximately 37

For most isotopes, the relative isotopic mass is very close to the mass number (the total number of protons and neutrons). However, they are not always exactly the same because of small differences in nuclear binding energy.

Relative Atomic Mass (Ar)

In nature, most elements exist as a mixture of different isotopes. For example, a sample of chlorine from nature contains about 75% chlorine-35 and 25% chlorine-37.

Because of this mixture, we need to calculate an average mass for the element.

Definition of relative atomic mass (Ar):

The relative atomic mass (Ar) is the weighted average mass of all the naturally occurring isotopes of an element compared to one-twelfth of the mass of one carbon-12 atom.

"Weighted average" means we take into account:

• The mass of each isotope
• How common (abundant) each isotope is

Important note: Relative atomic mass has no units because it is a ratio.

How to Calculate Relative Atomic Mass

If we know the isotopes of an element and their percentage abundances, we can calculate Ar using this formula:

Formula:

Or in words:

1. Multiply each isotope's mass by its percentage abundance
2. Add up all these values
3. Divide by 100

Worked Example 1: Chlorine

Chlorine has two isotopes:

• Chlorine-35 with a relative isotopic mass of 35 (75.5% abundance)
• Chlorine-37 with a relative isotopic mass of 37 (24.5% abundance)

Calculate the relative atomic mass of chlorine.

Solution:

$A_r = \frac{(35 \times 75.5) + (37 \times 24.5)}{100}$

$A_r = \frac{2642.5 + 906.5}{100}$

$A_r = \frac{3549}{100} = 35.49$

The relative atomic mass of chlorine is 35.5 (rounded to 1 decimal place).

Notice that the Ar (35.5) is closer to 35 than to 37. This makes sense because chlorine-35 is more abundant (75.5%).

Worked Example 2: Magnesium

Magnesium has three isotopes:

• Magnesium-24: 78.60% abundance
• Magnesium-25: 10.11% abundance
• Magnesium-26: 11.29% abundance

Calculate the relative atomic mass of magnesium to 2 decimal places.

Solution:

$A_r = \frac{(24 \times 78.60) + (25 \times 10.11) + (26 \times 11.29)}{100}$

$A_r = \frac{1886.4 + 252.75 + 293.54}{100}$

$A_r = \frac{2432.69}{100} = 24.33$

The relative atomic mass of magnesium is 24.33.

Relative Molecular Mass (Mr)

Many substances exist as molecules — groups of atoms bonded together. For example:

• Water (H₂O) is a molecule containing 2 hydrogen atoms and 1 oxygen atom
• Carbon dioxide (CO₂) is a molecule containing 1 carbon atom and 2 oxygen atoms

Definition of relative molecular mass (Mr):

The relative molecular mass (Mr) is the weighted average mass of one molecule of a substance compared to one-twelfth of the mass of one carbon-12 atom.

In simpler terms: it's the sum of the relative atomic masses of all the atoms in one molecule.

Important note: Relative molecular mass has no units because it is a ratio.

How to Calculate Relative Molecular Mass

To find Mr, simply:

1. Write down the molecular formula
2. Look up the Ar values for each element
3. Add them up, counting each atom

Worked Example 3: Water (H₂O)

Given: Ar(H) = 1.0, Ar(O) = 16.0

$M_r = (2 \times 1.0) + (1 \times 16.0)$

$M_r = 2.0 + 16.0 = 18.0$

The relative molecular mass of water is 18.0.

Worked Example 4: Potassium carbonate (K₂CO₃)

Given: Ar(K) = 39.1, Ar(C) = 12.0, Ar(O) = 16.0

$M_r = (2 \times 39.1) + (1 \times 12.0) + (3 \times 16.0)$

$M_r = 78.2 + 12.0 + 48.0 = 138.2$

The relative molecular mass of potassium carbonate is 138.2.

Worked Example 5: Ammonium sulfate ((NH₄)₂SO₄)

Given: Ar(N) = 14.0, Ar(H) = 1.0, Ar(S) = 32.1, Ar(O) = 16.0

First, work out what atoms are present:

• 2 nitrogen atoms (the subscript 2 outside the brackets means everything inside is multiplied by 2)
• 8 hydrogen atoms (4 in each NH₄, and there are 2 of them)
• 1 sulfur atom
• 4 oxygen atoms

$M_r = (2 \times 14.0) + (8 \times 1.0) + (1 \times 32.1) + (4 \times 16.0)$

$M_r = 28.0 + 8.0 + 32.1 + 64.0 = 132.1$

The relative molecular mass of ammonium sulfate is 132.1.

Relative Formula Mass

Some substances do not exist as simple molecules. Ionic compounds (like sodium chloride, NaCl) exist as giant structures where ions are arranged in a regular pattern called a lattice. There are no separate "molecules" of NaCl.

For these ionic compounds, we use the term relative formula mass instead of relative molecular mass. However, it is calculated in exactly the same way, and we still use the symbol Mr.

Definition of relative formula mass:

The relative formula mass is the sum of the relative atomic masses of all the atoms shown in the formula of an ionic compound, compared to one-twelfth of the mass of one carbon-12 atom.

Important note: Relative formula mass has no units because it is a ratio.

Worked Example 6: Sodium chloride (NaCl)

Given: Ar(Na) = 23.0, Ar(Cl) = 35.5

$M_r = (1 \times 23.0) + (1 \times 35.5) = 58.5$

The relative formula mass of sodium chloride is 58.5.

Worked Example 7: Calcium hydroxide (Ca(OH)₂)

Given: Ar(Ca) = 40.1, Ar(O) = 16.0, Ar(H) = 1.0

Atoms present: 1 Ca, 2 O (the subscript 2 means 2 OH groups), 2 H

$M_r = (1 \times 40.1) + (2 \times 16.0) + (2 \times 1.0)$

$M_r = 40.1 + 32.0 + 2.0 = 74.1$

The relative formula mass of calcium hydroxide is 74.1.

Summary of Key Differences

Important points to remember:

1. All relative masses are compared to 1/12 of a carbon-12 atom
2. All relative masses have no units (they are ratios)
3. Relative atomic mass (Ar) is a weighted average based on isotopic abundances
4. Relative molecular mass (Mr) is the sum of all Ar values in a molecule
5. Relative formula mass is calculated the same way as Mr, but is used for ionic compounds