B3.2 Relate isotopic abundance to relative atomic mass

Ontario Grade 11 Chemistry

Matter, Chemical Trends, and Chemical Bonding

How the proportions of isotopes affect an element’s average mass

A sample of an element contains many atoms. Atoms of the same element can have different masses because they can have different numbers of neutrons. These forms are called isotopes. The periodic table gives one relative atomic mass for an element, even when the element has more than one isotope. That value depends on both the masses of the isotopes and how common each isotope is. It is a weighted average: common isotopes contribute more than rare ones.

What you will learn

Define isotope, isotopic abundance, and relative atomic mass.
Explain how an isotope’s abundance affects relative atomic mass.
Calculate relative atomic mass from isotope masses and abundances.

1. Review: atoms and isotopes

An atom has a nucleus containing protons and neutrons. The number of protons identifies the element. Every carbon atom, for example, has six protons. Atoms of the same element can have different numbers of neutrons.

An isotope is one form of an element’s atoms, with a particular number of neutrons. Isotopes of the same element have the same number of protons but different numbers of neutrons. Since neutrons have mass, different isotopes have different masses.

The mass number is the total number of protons and neutrons in one nucleus. Carbon-12 and carbon-13 are carbon isotopes with mass numbers 12 and 13. A mass number describes one isotope. It is not the relative atomic mass shown for the element on the periodic table.

mass number=number of protons+number of neutrons

The number of protons determines which element an atom is.
Isotopes have the same number of protons but different numbers of neutrons.
Mass number counts the protons and neutrons in one nucleus.

2. Abundance: how common each isotope is

A sample of an element can contain several isotopes. Isotopic abundance is the fraction or percentage of the sample’s atoms that are a particular isotope. The abundances for all isotopes in a complete mixture add to 100%.

Imagine a collection of tokens that come in different masses. If one mass type makes up most of the collection, it has a larger effect on the collection’s average mass. Isotope mixtures work in the same way. Each atom has the mass of one isotope, but an individual atom does not have the average mass listed on the periodic table.

Relative atomic mass is often not a whole number because it represents an average for an isotope mixture. A more abundant isotope has a greater influence on that average.

\sum \text{isotopic abundances}=100\%

Isotopic abundance tells how much of a sample is a particular isotope.
The isotope abundances in a complete mixture add to 100%.
A more abundant isotope has a greater influence on the average.

3. Calculate relative atomic mass

A weighted average is an average in which each value counts according to how common it is. To calculate relative atomic mass, multiply each isotope’s mass by its fractional abundance, then add the contributions. Fractional abundance means the decimal form of a percentage. Divide a percentage by 100 to convert it to a decimal.

For example, 75% becomes 0.75. The fractional abundances in a complete mixture add to 1. Use isotope masses supplied in a question. If only mass numbers are supplied, use them as approximate isotope masses for that calculation.

Isotope masses may be given in atomic mass units, written as

\mathrm{u}

. The contributions have units of

\mathrm{u}

while they are being added. Relative atomic mass is a comparison value and is reported without a unit. Keep extra digits during the calculation, then round the final answer based on the precision of the supplied data.

A_r=\sum(\text{isotope mass}\times\text{fractional abundance})

Convert each percentage abundance to a decimal before multiplying.
Multiply each isotope mass by its own fractional abundance.
Add the contributions and report relative atomic mass without a unit.

4. Check what the result means

The calculated relative atomic mass should lie between the isotope masses used. For a two-isotope mixture, the result is closer to the mass of the more abundant isotope. This comparison is not a reliable way to identify the most abundant isotope when a mixture contains more than two isotopes.

The relative atomic mass is not the mass number of one particular atom. It describes the weighted average for the isotope mixture. Before calculating, check that the given percentage abundances add to 100%, or that the fractional abundances add to 1. If they do not, review the information rather than silently changing it.

Check that the abundances total 100% or, as fractions, 1.
The result should fall between the isotope masses.
For two isotopes, the result is closer to the mass of the more abundant isotope.

Worked example

Calculate a relative atomic mass

An element has two isotopes. Their masses are

34.97\ \mathrm{u}

and

36.97\ \mathrm{u}

. Their abundances are 75.8% and 24.2%, respectively. Calculate the relative atomic mass.

Check the abundance data
The isotope percentages should describe the full mixture. Their sum is 100.0%, so the data account for all the isotopes in this example.
$75.8\%+24.2\%=100.0\%$
Convert percentages to fractions
Divide each percentage by 100 to get its fractional abundance. These fractions are the weights used in the average.
$75.8\%=0.758,\quad 24.2\%=0.242$
Calculate the contributions
Multiply each isotope mass by its own fractional abundance. Each product is that isotope’s contribution to the average, and the products retain the mass unit $\mathrm{u}$ .
$(34.97\ \mathrm{u})(0.758)=26.50726\ \mathrm{u},\quad (36.97\ \mathrm{u})(0.242)=8.94674\ \mathrm{u}$
Add and round
Add the contributions without rounding them first. Both abundances are given to three significant figures, so report the final result to three significant figures. Remove the unit when reporting relative atomic mass.
$26.50726\ \mathrm{u}+8.94674\ \mathrm{u}=35.454\ \mathrm{u}\approx 35.5$

Answer: The relative atomic mass is 35.5.

Check: The result lies between

34.97

and

36.97

. Because there are two isotopes, its position closer to

34.97

is consistent with the greater abundance of the isotope with mass

34.97\ \mathrm{u}

.

Common mistakes and how to avoid them

Taking the ordinary average of isotope masses without considering abundance.

Correction: Use a weighted average: multiply each isotope mass by its fractional abundance, then add the contributions.

Multiplying a mass by a percentage such as 75.8 rather than its decimal form.

Correction: Convert the percentage to a fraction by dividing by 100. For example, 75.8% is 0.758.

Treating relative atomic mass as the mass number of one isotope.

Correction: Mass number belongs to one isotope and counts its protons and neutrons. Relative atomic mass describes the weighted average for the isotope mixture.

Giving relative atomic mass a unit.

Correction: Isotope masses may be expressed in

\mathrm{u}

during the calculation, but relative atomic mass is reported without a unit.

Lesson summary

Isotopes are atoms of the same element with different numbers of neutrons and different masses.
Isotopic abundance tells what fraction of the atoms are a particular isotope.
Relative atomic mass is found by multiplying each isotope mass by its fractional abundance and adding the contributions.
Check the abundance total, the result’s position between isotope masses, and the final rounding. For a mixture with more than two isotopes, do not use closeness to one isotope’s mass alone to decide which isotope is most abundant.

Check your understanding

Question 1

An element has isotopes with masses

10.0\ \mathrm{u}

and

11.0\ \mathrm{u}

. Their abundances are 20% and 80%, respectively. What is the relative atomic mass?

$10.2$
$10.8$
$10.5$
$11.0$

Show answer and explanation

10.8

Convert the abundances to 0.20 and 0.80. The weighted average is

(10.0\times0.20)+(11.0\times0.80)=10.8

. The calculation gives the relative atomic mass directly.

Question 2

Two isotopes have masses

20.0\ \mathrm{u}

and

22.0\ \mathrm{u}

. Which isotope is more abundant if the relative atomic mass is

20.4

?

The isotope with mass $20.0\ \mathrm{u}$
The isotope with mass $22.0\ \mathrm{u}$
They must be equally abundant
The relative atomic mass does not indicate which is more abundant

Show answer and explanation

The isotope with mass

20.0\ \mathrm{u}

For a two-isotope mixture, the weighted average is closer to the more abundant isotope’s mass. The value 20.4 is closer to 20.0 than to 22.0, so the isotope with mass

20.0\ \mathrm{u}

is more abundant.

Key terms

Isotope: One form of an element’s atoms, with a particular number of neutrons.
Mass number: The total number of protons and neutrons in one atom’s nucleus.
Isotopic abundance: The fraction or percentage of an element’s atoms that are a particular isotope.
Fractional abundance: The decimal form of an isotope’s percentage abundance.
Weighted average: An average in which each value counts according to its share of the total.
Relative atomic mass: The weighted average of an element’s isotope masses, reported without a unit.

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B3.2 · Relate isotopic abundance to relative atomic mass