Average atomic mass

Why is the atomic mass of chlorine 35.5 u?

Every proton and neutron has a mass of about $1\ \text{u}$, so the mass of any single atom is almost a whole number. Yet the atomic mass of chlorine is $35.5\ \text{u}$. How can this be?

The reason is that a sample of chlorine has two kinds of atom, the isotopes chlorine-35 and chlorine-37. The atomic mass in the table is the average for the mixture of them.

No single chlorine atom has a mass of $35.5\ \text{u}$. Every atom has a mass of about $35\ \text{u}$ or $37\ \text{u}$. The number $35.5\ \text{u}$ describes the mixture, as the average height of a class need not be the height of any one student.

Weighted average

In nature, out of every $100$ atoms of chlorine, about $75$ are chlorine-35 and about $25$ are chlorine-37. We cannot just take the middle of $35$ and $37$, because there are more atoms of the lighter one. We find the average by weight.

In general,

Here is an easy way to see why it is "weighted". Suppose that your marks are $80$ in a test that counts for $3$ parts and $60$ in a test that counts for $1$ part. The average is not $70$. It is $\dfrac{80 \times 3 + 60 \times 1}{4} = 75$. The test with the bigger weight pulls the average closer to its own value. The more common isotope does the same.

Another example: boron

Boron has two isotopes. Boron-10 is $20\ \%$ of the atoms, and boron-11 is $80\ \%$.

The answer is nearer to $11\ \text{u}$, because boron-11 is the more common isotope.

A third example: magnesium

Magnesium has three isotopes: magnesium-24 ($79\ \%$), magnesium-25 ($10\ \%$) and magnesium-26 ($11\ \%$).

The result, $24.3\ \text{u}$, is the value in the periodic table.

You should find that the average always lies between the two mass numbers, closer to the more common isotope.

Working backwards

Sometimes the average is known, and the percentages are not. Copper has two isotopes, copper-63 and copper-65, and the average atomic mass is $63.5\ \text{u}$. What is the percentage of each?

Let $x$ be the percentage of copper-63. Then copper-65 is $(100 - x)$.

So copper-63 is $75\ \%$ and copper-65 is $25\ \%$. Check: $\dfrac{63 \times 75 + 65 \times 25}{100} = \dfrac{4725 + 1625}{100} = 63.5$. ✓.

Practise with new numbers

Try it in Python

This program finds the average atomic mass of chlorine from its isotopes.

Common mistakes

• Taking a plain average. $\dfrac{35 + 37}{2} = 36$ is wrong for chlorine. You must weight by the percentages.
• Forgetting to divide by $100$ when the percentages add up to $100$.
• Using the atomic number instead of the mass number.
• Expecting a whole number. The average of whole numbers can be a decimal, as in $35.5\ \text{u}$.

Remember these

• The atomic mass in the table is the weighted average of the masses of the isotopes.
• Chlorine: $75\ \%$ of $35$ and $25\ \%$ of $37$ gives $35.5\ \text{u}$.
• The average is nearer to the mass of the more common isotope.
• It is the sum of (mass number times percentage), divided by $100$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Neon is $90\ \%$ neon-20 and $10\ \%$ neon-22. What is its average atomic mass?

2. Why is the atomic mass of an element usually not a whole number?

3. An element is $60\ \%$ of the isotope with the mass number $28$ and $40\ \%$ of the isotope with the mass number $30$. What is its average atomic mass?

4. Without working it out, is the atomic mass of chlorine nearer to $35\ \text{u}$ or to $37\ \text{u}$? Why?