Chemistry · General chemistry I · Concept

Isotopes and average atomic mass

Atoms of one element all have the same number of protons, but they can differ in their number of neutrons. Those versions are isotopes, told apart by mass number. The atomic mass on the periodic table is the average of the isotope masses, each weighted by how common that isotope is in nature. That average, not the mass of any single atom, is what molar mass is built from.

A concrete case

One hundred chlorine atoms

Picture $100$ chlorine atoms taken from ordinary salt. About $76$ of them have a mass of $34.969 u$ , and about $24$ have a mass of $36.966 u$ . The u is the atomic mass unit, a mass about the size of one proton.

The $76$ lighter atoms weigh $76 \times 34.969 = 2657.64 u$ together, and the $24$ heavier ones weigh $24 \times 36.966 = 887.18 u$ . All $100$ weigh $3544.83 u$ , so the average atom weighs $3544.83 \div 100 = 35.45 u$ .

That is the $35.45$ printed for chlorine on the periodic table. No single chlorine atom has that mass. Every one is about $35 u$ or about $37 u$ , and $35.45$ is the average over the mixture.

\frac{76 (34.969) + 24 (36.966)}{100} = 35.45

\begin{array}{c} \frac{76 (34.969) + 24 (36.966)}{100} \\ = 35.45 \end{array}

The mass of $100$ chlorine atoms
Kind of atom	Number of atoms	Mass of each $(u)$	Total mass $(u)$
Chlorine-35	$76$	$34.969$	$2657.64$
Chlorine-37	$24$	$36.966$	$887.18$
All	$100$	$35.45$ on average	$3544.83$

The rule

Same element, different mass

The atomic number $Z$ counts protons, and it decides the element. Every carbon atom has $6$ . The mass number $A$ counts protons and neutrons together, so the number of neutrons is $N = A - Z$ . Atoms with the same $Z$ and a different $A$ are isotopes. Carbon-12 has $6$ neutrons, carbon-13 has $7$ and carbon-14 has $8$ , and all three are carbon.

An isotope is named by its element and mass number, as in carbon-13. As a symbol, the mass number goes at the upper left, ¹³C, and the atomic number may go at the lower left. The atomic number is often left out, because the symbol C already says $6$ .

The mass number is a count, so it is a whole number. The mass of an isotope is a measurement in atomic mass units, close to its mass number but not equal to it. Chlorine-35 has a mass of $34.969 u$ .

N = A - Z 6_{13} C 13 C

Atomic mass is a weighted average

The natural abundance of an isotope is the percentage of an element’s atoms, in a natural sample, that are that isotope. The abundances of one element add to $100 %$ .

The atomic mass of an element is the average mass of its atoms, with each isotope counted in proportion to its abundance. Turn each percentage into a fraction by dividing by $100$ . Multiply each isotope’s mass by its fraction, and add the products. In the formula, $f_{i}$ is the fraction of isotope $i$ and $m_{i}$ is its mass.

\bar{m} = \sum_{i} f_{i} m_{i}, \sum_{i} f_{i} = 1

Chlorine’s two isotopes
Isotope	Protons	Neutrons	Mass $(u)$	Natural abundance
Chlorine-35	$17$	$18$	$34.969$	$75.77 %$
Chlorine-37	$17$	$20$	$36.966$	$24.23 %$

Why it works

Why the fractions do the weighting

An average is a total divided by a count. In the hundred-atom picture, the total is $76 \times 34.969 + 24 \times 36.966$ and the count is $100$ . Dividing each term by $100$ gives $0.76 \times 34.969 + 0.24 \times 36.966$ , the same number written with fractions.

A fraction is the share of the atoms that are one isotope, so each product is that isotope’s share of the average. The more common isotope pulls the average toward its own mass. With the real abundances, $0.7577 \times 34.969$ $+ 0.2423 \times 36.966$ $= 35.45 u$ .

The average does not depend on how many atoms you picture. A mole of chlorine holds both isotopes in the same proportions, so its atoms average $35.45 u$ too.

\frac{76 (34.969) + 24 (36.966)}{100} = 0.76 (34.969) + 0.24 (36.966)

\begin{array}{c} \frac{76 (34.969) + 24 (36.966)}{100} \\ = 0.76 (34.969) \\ + 0.24 (36.966) \end{array}

The method

From abundances to an atomic mass

The table works through copper, the first example below. Keep the isotope masses at the precision the problem gives and round only the final sum.

Finding copper’s atomic mass
Step	What to do	For copper
1	Turn each percentage into a fraction	$69.15 %$ $\to 0.6915$ , $30.85 %$ $\to 0.3085$
2	Check that the fractions add to $1$	$0.6915 + 0.3085$ $= 1$
3	Multiply each isotope’s mass by its fraction	$0.6915$ $\times 62.930$ $= 43.516$ and $0.3085$ $\times 64.928$ $= 20.030$
4	Add the products	$43.516 + 20.030$ $= 63.546$
5	Round, then check the size	$63.55 u$ lies between $62.930$ and $64.928$ , nearer the more common copper-63

From an atomic mass back to abundances

The weighted average can be run backward. With two isotopes, call the fraction of one of them $x$ . The other is then $1 - x$ , because the two fractions add to $1$ . Set the weighted average equal to the atomic mass and solve the linear equation for $x$ .

For chlorine, $34.969 x + 36.966 (1 - x)$ $= 35.45$ . Expanding gives $36.966 - 1.997 x = 35.45$ , so $1.997 x = 1.516$ and $x = 0.76$ . About $76 %$ of chlorine atoms are chlorine-35.

With three isotopes, one atomic mass cannot fix two unknown fractions. You need one more abundance from the problem.

\begin{array}{l} 34.969 x + 36.966 (1 - x) = 35.45 \\ 1.997 x = 1.516 \\ x \approx 0.76 \end{array}

Reading a mass spectrum

A mass spectrometer turns atoms into ions, separates them by mass, and counts them. The spectrum has one peak for each isotope, at that isotope’s mass, and the height of a peak shows how common it is.

Peak heights are often scaled so the tallest is $100$ . Divide each height by the sum of all of them to get the fractions. Silver’s spectrum has peaks of height $100$ at $106.905 u$ and $92.9$ at $108.905 u$ . The fractions are $100 \div 192.9 = 0.5184$ and $92.9 \div 192.9 = 0.4816$ .

Where it breaks

The average describes a natural mixture, not any atom. No chlorine atom has a mass of $35.45 u$ , and the mass number of the most common isotope, $35$ , is not the atomic mass either.

The average holds only for a sample with the natural mix. A sample made with one chosen isotope, as when chemists put carbon-13 at one position of a molecule to follow it through a reaction, has a different mass. Two molecules that differ only in their isotopes are isotopologues. Their bonds are the same, but their masses differ, and so can their reaction rates, noticeably for hydrogen.

Natural mixtures are not exactly fixed. Boron, lithium and magnesium vary a little from one source to another, so IUPAC gives their atomic weights as ranges. Boron’s is $10.806$ to $10.821$ . The periodic table prints one conventional value, $10.81$ .

An element with no stable isotope has no natural mixture to average. Its entry gives the mass number of a long-lived isotope in brackets, such as [98] for technetium.

Check your answer

Look at where the answer falls. A weighted average always lies between the lightest and the heaviest isotope masses, and nearer the most abundant one. If copper’s answer were $64.5 u$ , the weighting would be backward.

Add the fractions. They must total $1$ , or $100 %$ before dividing. If they do not, an isotope is missing or a percentage was copied wrong.

Compare with the periodic table. A computed atomic mass should match the printed one to the places the data support.

To have Chalk check the arithmetic, use a Math box. Write the weighted average on the first row as $m = 0.6915 \cdot 62.930$ $+ 0.3085 \cdot 64.928$ , and the result below it as $m \approx 63.55$ . The Checker marks the result ✓. With the percentages left as $69.15$ and $30.85$ , the result row reads ✗ with “That equality is false: 6354.64 is not 63.55.”

For the backward problem, write the equation and each step of its solution on its own row. Each step is checked against the one above. The calculator value and the rounded one, $x \approx 0.1998$ and $x \approx 0.20$ , both read ✓ in either order.

Why isotope masses are not whole numbers

The atomic mass unit is defined as one twelfth of the mass of a carbon-12 atom, so carbon-12 is exactly $12 u$ . Every other isotope comes out near a whole number but not on it. A proton and a neutron each weigh a little more than $1 u$ .

A nucleus also weighs less than its separate protons and neutrons. The missing mass, the mass defect, became the energy that binds the nucleus, by $E = m c^{2}$ . That is why chlorine-35, with $17$ protons and $18$ neutrons, weighs $34.969 u$ rather than about $35.3 u$ .

The small differences between isotopes are measurable in nature. The ratio of oxygen-18 to oxygen-16 in old ice records the temperature when the snow fell. The ratio of carbon-13 to carbon-12 shows how a plant took up its carbon. These are the same mass spectrometers, read with more care.

Common mistakes

Averaging the isotope masses without weighting them. $(34.969 + 36.966) / 2$ $= 35.97$ , not $35.45$ . A plain average counts both isotopes equally, but chlorine-35 is three times as common.
Using the percentages as they stand. $69.15 \times 62.930$ $+ 30.85 \times 64.928$ $= 6354.64$ , a hundred times too large. Each percentage must become a fraction first, $0.6915$ and $0.3085$ .
Confusing mass number with isotope mass. $35$ is a count of protons and neutrons, while $34.969 u$ is a measured mass. Using $35$ and $37$ for chlorine gives $35.48 u$ , not $35.45 u$ .
Expecting some atom to have the average mass. No chlorine atom weighs $35.45 u$ . The average describes the mixture, and every atom is either about $35 u$ or about $37 u$ .
Changing the number of protons to make an isotope. Adding a proton to carbon makes nitrogen. Isotopes of one element differ only in their neutrons, so their atomic number stays the same.
Solving for the wrong isotope. In $10.013 x + 11.009 (1 - x)$ $= 10.81$ , $x$ is the fraction of boron-10, and it comes out $0.20$ . Reporting $0.80$ for boron-10 swaps the two isotopes.

Key terms

Isotope: Atoms of the same element with different neutron counts. They have the same number of protons, but their masses, and sometimes their nuclear behavior, differ.
Mass number: The total number of protons and neutrons in a particular nucleus. It is an integer count, not an isotope’s measured mass.
Atomic number: The number of protons in an atomic nucleus, symbol $Z$ . It identifies the element: every atom with $Z = 6$ is carbon.
Natural abundance: The percentage of an element’s atoms in a natural sample that belong to a given isotope. Weighting isotope masses by these fractions gives the atomic mass.
Standard atomic weight: The average mass of an element’s atoms, weighted by how common each isotope is in nature, divided by the atomic mass unit. It is a ratio with no unit, so chlorine’s is $35.45$ , matching an average atomic mass of $35.45 u$ . It differs from the mass number, which counts the protons and neutrons in one particular isotope.
Atomic mass unit: A mass unit defined as one twelfth of the mass of a carbon-12 atom. Its symbol is $u$ , and it is also called the unified atomic mass unit.
Isotopologue: Molecules with the same chemical structure that differ only in which isotopes their atoms are, such as $H_{2} O$ , $HDO$ and $D_{2} O$ . They have different molar masses.

Work through an example

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.

Find an atomic mass from isotope abundances →

Find isotope abundances from an atomic mass →

Read an isotope label in a structure →

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