The average atomic mass of an element is calculated by taking a weighted average of the masses of its naturally occurring isotopes, using the relative abundance of each isotope to determine how much it contributes to the overall value. An isotope is a form of an element whose atoms have the same number of protons but different numbers of neutrons.
This calculation explains why the atomic mass listed on the periodic table is usually not a whole number. Although individual isotopes have different masses, the value on the periodic table reflects the combined contribution of the element’s isotopes according to how common each one is in nature.
What is average atomic mass?
Every atom of a particular element has the same number of protons in its nucleus. However, atoms of that element can contain different numbers of neutrons. These variations are called isotopes.
For example, carbon has two common stable isotopes: carbon-12 and carbon-13. Carbon-12 contains six protons and six neutrons, while carbon-13 contains six protons and seven neutrons. Both are carbon because they have six protons, but their masses differ.
An element found in nature generally contains a mixture of its isotopes. Some isotopes are much more abundant than others, so they contribute more to the element’s average atomic mass.
Average atomic mass, also called the relative isotopic mass average in the context of isotope calculations, is expressed in atomic mass units (u). One atomic mass unit is defined as one-twelfth of the mass of a neutral carbon-12 atom in its ground state.
The average atomic mass is not necessarily the mass of any single atom. Instead, it represents a weighted average of the masses of the element’s isotopes.
The formula for calculating average atomic mass
To calculate average atomic mass, multiply the mass of each isotope by its fractional abundance, then add the resulting values.
The formula is:
Average atomic mass = Σ (isotope mass × fractional abundance)
The symbol Σ, called sigma, means to add the contributions from all the isotopes.
Fractional abundance is the proportion of the element’s atoms represented by a particular isotope. It is obtained by dividing the isotope’s percentage abundance by 100.
For example, if an isotope accounts for 75% of an element’s atoms, its fractional abundance is 0.75. If another isotope accounts for 25%, its fractional abundance is 0.25. The fractional abundances of all isotopes included in the calculation should add up to 1, or 100%.
Three pieces of information are therefore needed:
- Isotope mass: The measured mass of each isotope, usually expressed in atomic mass units.
- Isotopic abundance: The percentage of the element’s atoms represented by each isotope in the sample.
- Fractional abundance: The percentage abundance converted into a decimal for use in the calculation.
Once these values are known, the calculation is a straightforward weighted average.
How to calculate average atomic mass step by step
Consider a hypothetical element with two isotopes. One isotope has a mass of 10 u and makes up 80% of the element’s atoms. The other has a mass of 11 u and makes up the remaining 20%.
First, convert the percentage abundances into decimals:
- Isotope A: 80% ÷ 100 = 0.80
- Isotope B: 20% ÷ 100 = 0.20
Next, multiply each isotope’s mass by its fractional abundance:
- Isotope A: 10 u × 0.80 = 8.0 u
- Isotope B: 11 u × 0.20 = 2.2 u
Finally, add the contributions:
Average atomic mass = 8.0 u + 2.2 u = 10.2 u
The average atomic mass of this hypothetical element is 10.2 u.
Notice that the answer is closer to 10 u than to 11 u. This happens because isotope A is four times as abundant as isotope B. Its mass therefore has a greater influence on the average.
This example illustrates the central principle: the more abundant an isotope is, the more strongly its mass influences the average atomic mass.
A real example: calculating the average atomic mass of chlorine
Chlorine provides a useful real-world example because naturally occurring chlorine consists primarily of two stable isotopes: chlorine-35 and chlorine-37.
For an illustrative calculation, use approximate isotope masses and abundances:
| Isotope | Approximate isotope mass | Approximate natural abundance |
|---|---|---|
| Chlorine-35 | 34.969 u | 75.78% |
| Chlorine-37 | 36.966 u | 24.22% |
Convert the percentages into fractional abundances:
- Chlorine-35: 0.7578
- Chlorine-37: 0.2422
Multiply each isotope’s mass by its fractional abundance:
- Chlorine-35: 34.969 u × 0.7578 ≈ 26.49 u
- Chlorine-37: 36.966 u × 0.2422 ≈ 8.95 u
Add the contributions:
Average atomic mass ≈ 26.49 u + 8.95 u
Average atomic mass ≈ 35.45 u
This result explains why chlorine’s atomic mass on the periodic table is approximately 35.45 rather than exactly 35 or 37. The lighter isotope is more abundant, but the heavier isotope also contributes to the average.
The values above are rounded, so the final result is approximate. More precise calculations use measured isotope masses and abundances at greater precision.
Why average atomic mass is usually not a whole number
The mass number of an isotope is a whole number equal to the total number of protons and neutrons in its nucleus. For example, chlorine-35 has a mass number of 35, and chlorine-37 has a mass number of 37.
However, mass number and atomic mass are not the same thing. The actual mass of an isotope, expressed in atomic mass units, is generally not exactly equal to its mass number. This is partly because some nuclear mass is converted into binding energy when protons and neutrons form a nucleus, as described by Einstein’s mass-energy relationship.
The average atomic mass introduces another reason for a non-whole-number value: it combines the masses of different isotopes according to their abundances.
For chlorine, both effects matter. The actual isotope masses are not exactly 35 u and 37 u, and the natural mixture contains both isotopes in unequal proportions. The resulting average is approximately 35.45 u.
Some elements have only one naturally occurring stable isotope, so their standard atomic weights may be close to that isotope’s mass. Even then, the isotope’s actual mass need not be a whole number.
How scientists determine isotope masses and abundances
Scientists commonly use mass spectrometry to measure isotope masses and determine their relative abundances. In a mass spectrometer, atoms or molecules are converted into ions, which are then separated according to their mass-to-charge ratios. Detectors measure the signals from the different isotopes, allowing scientists to identify them and estimate their relative proportions.
The measured isotope masses and abundances can then be used in the weighted-average formula. Accurate measurements matter because even small differences in abundance or isotope mass can affect the calculated result.
The abundance values depend on the sample being measured. Most ordinary terrestrial samples of an element have characteristic isotopic compositions, but some elements show natural variations because of geological, biological, or other processes. For elements with significant natural isotopic variation, the average atomic mass can differ slightly from one sample to another.
This is why periodic tables may give a single standard atomic weight for some elements but an interval for others. A single value is useful for routine calculations, while an interval can better represent natural variation in isotopic composition.
Common mistakes when calculating average atomic mass
One frequent mistake is treating all isotopes as equally abundant. A simple arithmetic mean would give each isotope the same influence, regardless of how common it is. That is incorrect unless the isotopes actually occur in equal proportions.
Another common error is using percentage abundances directly without converting them into decimals. If abundances are given as percentages, either divide each percentage by 100 before multiplying or divide the final sum by 100 after using the percentages consistently.
For example, multiplying isotope masses by 80 and 20 rather than 0.80 and 0.20 produces a result 100 times too large unless the sum is subsequently divided by 100.
It is also important to distinguish an isotope’s mass number from its measured isotopic mass. The mass number is a whole-number count of protons and neutrons, whereas the measured isotope mass is a physical quantity expressed in atomic mass units. When accurate isotope masses are provided, use those values rather than substituting mass numbers.
Finally, check that the abundances add up to approximately 100%, or that the fractional abundances add up to approximately 1. Small discrepancies can arise from rounding. If the listed abundances do not add up to the expected total, verify the data before completing the calculation.
Why average atomic mass matters in chemistry
Average atomic mass connects the properties of individual atoms to the quantities used in laboratory calculations. Chemists use atomic masses to calculate molar masses, convert between grams and moles, and determine the amounts of substances involved in chemical reactions.
For example, the molar mass of chlorine atoms is numerically close to chlorine’s standard atomic weight, expressed in grams per mole. This relationship makes it possible to estimate how many atoms are present in a measured sample without weighing individual atoms.
The same principle applies throughout the periodic table. The atomic masses shown on periodic tables account for isotopic composition, helping chemists work with naturally occurring elements as they are commonly encountered.
Ultimately, average atomic mass is calculated by combining each isotope’s measured mass with its fractional abundance. Understanding this weighted average explains why periodic-table atomic masses often contain decimals and shows how the properties of individual isotopes determine the average mass of an element’s atoms.
