Isotopes and Calculating Average Atomic Mass
Isotopes are atoms of the same element that have the same number of protons but different numbers of neutrons, and the average atomic mass of an element is the weighted average of the masses of its naturally occurring isotopes. In brief: chlorine has two main isotopes, chlorine-35 (75.76%) and chlorine-37 (24.24%), so its average atomic mass is 0.7576 × 34.969 + 0.2424 × 36.966 = 35.45 u, which is the value shown on the periodic table.
The atomic masses on the periodic table are rarely whole numbers, because most elements are mixtures of isotopes. This article explains what isotopes are, why they behave the same chemically but differ in mass, how to calculate an average atomic mass and how to find isotope abundances from the average, with tables and worked examples, and describes the discovery of isotopes and some of their uses.
What are isotopes?
All atoms of an element have the same atomic number, but they can have different numbers of neutrons, so different mass numbers. These variants are called isotopes. Hydrogen has three well-known isotopes: protium (1H, no neutrons), deuterium (2H, one neutron) and tritium (3H, two neutrons). Carbon has isotopes 12C, 13C and 14C. The word comes from the Greek for “same place”, meaning the same place in the periodic table.
How do isotopes differ and how are they alike?
Isotopes of an element have the same number of electrons and the same electron arrangement, so their chemical properties are almost identical. They differ in mass, and so in physical properties that depend on mass, such as density, rate of diffusion and, for some, in nuclear stability. Some isotopes are stable, and some are radioactive. Heavy water (D2O) has a density of 1.107 g/cm3 and boils at 101.4 °C, compared with 1.000 g/cm3 and 100 °C for ordinary water.
How do you calculate the average atomic mass?
The relative atomic mass of an element is the weighted average of the isotope masses:
average atomic mass = Σ (fractional abundance × isotope mass)
- Convert each percent abundance to a fraction by dividing by 100.
- Multiply each fraction by the mass of that isotope.
- Add the results.
The abundances must add up to 100% (or fractions to 1). If you use mass numbers (whole numbers) instead of exact isotope masses, you get an approximate answer.
Isotope data for some elements
| Element | Isotope | Mass (u) | Abundance (%) | Average atomic mass (u) |
|---|---|---|---|---|
| Chlorine | 35Cl | 34.969 | 75.76 | 35.45 |
| 37Cl | 36.966 | 24.24 | ||
| Copper | 63Cu | 62.930 | 69.15 | 63.55 |
| 65Cu | 64.928 | 30.85 | ||
| Boron | 10B | 10.013 | 19.9 | 10.81 |
| 11B | 11.009 | 80.1 |
Worked examples
- Chlorine. 0.7576 × 34.969 = 26.493 and 0.2424 × 36.966 = 8.960. The sum is 35.453, which is rounded to 35.45 u.
- Copper. 0.6915 × 62.930 = 43.516 and 0.3085 × 64.928 = 20.030. The sum is 63.546, which is 63.55 u.
- Magnesium with three isotopes. 24Mg: 78.99% at 23.985 u; 25Mg: 10.00% at 24.986 u; 26Mg: 11.01% at 25.983 u. Contributions: 0.7899 × 23.985 = 18.946; 0.1000 × 24.986 = 2.499; 0.1101 × 25.983 = 2.861. The sum is 24.306, or 24.31 u.
- Finding an abundance from the average. Gallium has two isotopes: 69Ga (68.926 u) and 71Ga (70.925 u). The average atomic mass is 69.723 u. Let x be the fraction of gallium-69, so 1 − x is the fraction of gallium-71. Then 68.926x + 70.925(1 − x) = 69.723, which gives 70.925 − 1.999x = 69.723, so x = 1.202 ÷ 1.999 = 0.601. The abundances are 60.1% gallium-69 and 39.9% gallium-71.
- Number of neutrons. 37Cl has 37 − 17 = 20 neutrons, and 35Cl has 18.
- Check the sense of the result. The average atomic mass must lie between the isotope masses and closer to the more abundant isotope. 35.45 lies between 34.969 and 36.966 and closer to 34.969, as expected.
How are isotopes measured?
The mass and abundance of isotopes are measured by mass spectrometry. In a mass spectrometer, atoms are ionized, accelerated through an electric field and deflected by a magnetic field. Lighter ions are deflected more than heavier ions of the same charge, so they arrive at different places on the detector. The height of each peak is proportional to the abundance of the isotope.
What are isotopes used for?
- Radiocarbon dating: the radioactive isotope carbon-14, with a half-life of about 5,730 years, is used to date organic remains up to about 50,000 years old.
- Medicine: technetium-99m is used in imaging, iodine-131 treats thyroid disease and cobalt-60 is a source for radiotherapy.
- Energy: uranium-235 is the fissile isotope used in most nuclear reactors, and deuterium and tritium are fuels for fusion research.
- Tracers: isotopes follow the path of a substance in the body, in plants and in the environment.
- Geology and climate: ratios of oxygen isotopes in ice cores and shells record past temperatures.
How were isotopes discovered?
In the early 1900s, chemists found that radioactive decay products with different masses were chemically identical. Frederick Soddy proposed the term isotope in 1913 and received the Nobel Prize in Chemistry in 1921. J. J. Thomson found in 1912 that neon has two kinds of atoms of different mass, and Francis Aston developed the mass spectrograph in 1919 and showed that many elements are mixtures of isotopes. He received the Nobel Prize in Chemistry in 1922.
Which mistakes are common?
Mistake 1: Using a simple average of the isotope masses
The average must be weighted by abundance. The simple average of 35 and 37 is 36, which is wrong for chlorine.
Mistake 2: Thinking that isotopes have different chemical properties
Their chemistry is nearly identical, since they have the same electron arrangement.
Mistake 3: Forgetting that percentages must add up to 100
Check the sum of abundances before calculating.
Mistake 4: Assuming that all isotopes are radioactive
Many are stable. Chlorine-35 and chlorine-37, for example, are both stable.