Chemistry · General chemistry I · Worked example

Find an atomic mass from isotope abundances

Copper has two stable isotopes. Copper-63 has a mass of $62.930 u$ and a natural abundance of $69.15 %$ . Copper-65 has a mass of $64.928 u$ and a natural abundance of $30.85 %$ . Find the atomic mass of copper.

Turn the percentages into fractions

Divide each percentage by $100$ to get $0.6915$ and $0.3085$ . They add to $1$ , as the fractions of all the isotopes must.

0.6915 + 0.3085 = 1

Weight each mass

Multiply each isotope’s mass by its fraction. Each product is that isotope’s share of the average.

Each isotope’s contribution to copper’s atomic mass
Isotope	Mass $(u)$	Fraction	Contribution $(u)$
Copper-63	$62.930$	$0.6915$	$43.516$
Copper-65	$64.928$	$0.3085$	$20.030$

Add the contributions

$43.516 + 20.030$ $= 63.546 u$ , which is $63.55 u$ to the four significant figures of the abundances.

\begin{array}{l} m = 0.6915 \cdot 62.930 + 0.3085 \cdot 64.928 \\ m \approx 63.55 \end{array}

\begin{array}{c} m = 0.6915 \cdot 62.930 \\ + 0.3085 \cdot 64.928 \\ m \approx 63.55 \end{array}

Check against the periodic table

The periodic table lists copper as $63.55$ , the same value. The answer lies between the two isotope masses and nearer to copper-63, the more abundant isotope, which is a quick check that the weighting went the right way.

Result

The atomic mass of copper is $63.55 u$ .

Your turn

1. One like the example

Bromine is $50.65 %$ bromine-79 $(78.918 u)$ and $49.35 %$ bromine-81 $(80.916 u)$ . Find its atomic mass.

Show the answer and explanation

$79.90 u$ .

$0.5065 \times 78.918$ $+ 0.4935 \times 80.916$ $= 39.972 + 39.932$ $= 79.904$ , or $79.90 u$ . The two isotopes are almost equally common, so the average falls almost midway between them.

\begin{array}{l} m = 0.5065 \cdot 78.918 + 0.4935 \cdot 80.916 \\ m \approx 79.90 \end{array}

\begin{array}{c} m = 0.5065 \cdot 78.918 \\ + 0.4935 \cdot 80.916 \\ m \approx 79.90 \end{array}

2. A variation

Magnesium has three stable isotopes: magnesium-24 $(23.985 u, 78.99 %)$ , magnesium-25 $(24.986 u, 10.00 %)$ and magnesium-26 $(25.983 u, 11.01 %)$ . Find its atomic mass.

Show the answer and explanation

$24.31 u$ .

Three isotopes mean three products. $0.7899 \times 23.985 = 18.946$ , $0.1000 \times 24.986 = 2.499$ and $0.1101 \times 25.983 = 2.861$ . Their sum is $24.305$ , which is $24.31 u$ . The fractions add to $1.0000$ , and the answer sits close to magnesium-24, which is nearly four atoms in five.

\begin{array}{l} m = 0.7899 \cdot 23.985 + 0.1000 \cdot 24.986 + 0.1101 \cdot 25.983 \\ m \approx 24.31 \end{array}

\begin{array}{c} m = 0.7899 \cdot 23.985 \\ + 0.1000 \cdot 24.986 \\ + 0.1101 \cdot 25.983 \\ m \approx 24.31 \end{array}

3. Somewhere new

A mass spectrum of silver shows two peaks, of height $100$ at $106.905 u$ and of height $92.9$ at $108.905 u$ . Find silver’s atomic mass.

Show the answer and explanation

$107.87 u$ .

Peak heights are relative, so divide each by their sum, $192.9$ . The fractions are $0.5184$ and $0.4816$ . Then $0.5184 \times 106.905$ $+ 0.4816 \times 108.905$ $= 107.868$ , or $107.87 u$ , the value on the periodic table.

\begin{array}{l} \frac{100}{100 + 92.9} \approx 0.5184 \\ \frac{92.9}{100 + 92.9} \approx 0.4816 \\ m = 0.5184 \cdot 106.905 + 0.4816 \cdot 108.905 \\ m \approx 107.87 \end{array}

\begin{array}{c} \frac{100}{100 + 92.9} \approx 0.5184 \\ \frac{92.9}{100 + 92.9} \approx 0.4816 \\ m = 0.5184 \cdot 106.905 \\ + 0.4816 \cdot 108.905 \\ m \approx 107.87 \end{array}

Keep exploring

Open the rows in Math and use the percentages as they stand, $69.15$ and $30.85$ , in place of the fractions. The answer row turns to ✗: “That equality is false: 6354.64 is not 63.55.”

Return to the concept →

Sources and scope

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After reading

Study it, then try it

Practice the key terms as flashcards, then open the example in its tool, change a value and keep a useful result on your board.

Study the 7 key terms Check the weighted average in Math Open worked example on a board Periodic table: atomic masses and isotopes

Your existing work stays on this device. Examples open as editable copies.

Turn the percentages into fractions

Weight each mass

Add the contributions

Check against the periodic table

Result

Your turn

1. One like the example

79.90 u79.90 u.

2. A variation

24.31 u24.31 u.

3. Somewhere new

107.87 u107.87 u.

Keep exploring

Study it, then try it

$79.90 u$ .

$24.31 u$ .

$107.87 u$ .