Average Atomic Mass: Formula and Worked Example

This calculator finds an element’s average atomic mass from the masses and natural abundances of its isotopes. Enter each isotope’s mass in atomic mass units (u) and its abundance as a percent; the result is a weighted average in u. The example uses chlorine’s two common isotopes.

Formula

Multiply each isotope’s mass by its fractional abundance, then add those products:

Average atomic mass = Σ(isotope mass × fractional abundance)

When abundances are entered as percentages, use:

Average atomic mass = Σ(isotope mass × percentage abundance) ÷ 100

Here, isotope mass is measured in atomic mass units (u), and fractional abundance is a proportion from 0 to 1. Percentage abundance is the same proportion expressed from 0% to 100%. The abundance values for all naturally occurring isotopes included in the calculation should add to 100% (or 1 as fractions). The answer is in u.

Worked Example

Use the calculator’s default values for chlorine:

• Isotope 1 mass: 34.96885 u; abundance: 75.78%
• Isotope 2 mass: 36.96590 u; abundance: 24.22%

1. Multiply the first mass by its percentage: 34.96885 × 75.78 = 2,650.939833.
2. Multiply the second mass by its percentage: 36.96590 × 24.22 = 895.914098.
3. Add the products: 2,650.939833 + 895.914098 = 3,546.853931.
4. Divide by 100 to convert the percentages into fractions: 3,546.853931 ÷ 100 = 35.46853931 u.
5. Round to five decimal places, as the calculator does: 35.46854 u.

This is a weighted average, not the ordinary midpoint of the isotope masses. The more abundant isotope contributes more to the final value.

What You’ll Need

• The mass of each isotope, preferably from a reliable periodic table, textbook, or isotope data source.
• The natural abundance of each isotope, reported as a percentage or decimal fraction.
• A calculator, spreadsheet, or the calculator fields above.
• A consistent set of isotopes and abundances. Make sure the data refer to the same element and, if relevant, the same sample or stated abundance conditions.

Step-by-Step

1. List the element’s isotopes. An isotope is a version of an element with a particular number of neutrons. Include every isotope for which the supplied abundance is nonzero. Do not confuse an isotope’s mass with its mass number: the mass number is a whole-number count of protons and neutrons, while measured isotope masses are usually decimal values.
2. Record each isotope’s mass. Use the isotope mass in u, not the element’s average atomic mass from a periodic table. Keep enough digits during the calculation to avoid unnecessary rounding error.
3. Record its abundance. Check whether the data are written as percentages, such as 75.78%, or fractions, such as 0.7578. These are equivalent, but the formula’s conversion must match the format you use.
4. Check the abundances add up correctly. Add the percentages; they should total about 100%. Small discrepancies can result from rounding in the source. If the total is substantially different, check for a missing isotope, a transcription error, or a mixture of percentages and fractions.
5. Convert percentages to fractions if needed. Divide each percentage by 100. For example, 75.78% becomes 0.7578. If you instead keep abundances as percentages, calculate the weighted sum and divide the total by 100.
6. Calculate each weighted contribution. Multiply an isotope’s mass by its fractional abundance. Repeat for every isotope. Each product represents that isotope’s contribution to the average.
7. Add the contributions. The sum of all the products is the average atomic mass. If using the percentage form, divide the sum by 100 before reporting the answer.
8. Round only the final result. Follow the precision requested by your assignment or the precision supported by the source data. Include the unit u, and do not report more meaningful precision than the isotope masses and abundances justify.

Common Mistakes

• Using mass numbers instead of isotope masses: mass numbers are useful approximations, but they may not give the expected result. Use measured isotope masses when available.
• Forgetting to divide by 100: if you multiply by percentage values such as 75.78 and 24.22, divide the resulting sum by 100. If you use 0.7578 and 0.2422, do not divide again.
• Entering the element’s average as an isotope mass: use the individual isotope masses, not the periodic-table average you are trying to calculate.
• Using incomplete or mismatched abundances: make sure each mass is paired with the abundance of the same isotope. Omitting an isotope or mixing incompatible datasets changes the weighted average.
• Rounding too early: rounding every product can shift the final digits. Keep the original precision through the calculation, then round once at the end.
• Assuming the result must be a whole number: average atomic mass is generally decimal because it combines isotope masses according to their relative abundances.

Tips

• Make a table with columns for isotope, mass, abundance, and mass × fractional abundance. It makes pairings and arithmetic easier to check.
• For three or more isotopes, use the same process for every isotope; do not average the isotope masses equally unless their abundances really are equal.
• As a quick reasonableness check, the answer should lie between the smallest and largest isotope masses used. It should usually be closer to the mass of the most abundant isotope.
• If using a spreadsheet, keep one row per isotope and calculate each product in a separate column. Verify the abundance total before summing the products.
• When a textbook gives rounded abundances that total slightly above or below 100%, use the instructions provided. If none are given, note the discrepancy rather than silently changing the data.

Troubleshooting

The result is far too large: check whether you entered percentages but forgot to divide by 100. A result around 3,500 u for chlorine instead of about 35.5 u is a typical sign of this error.

The result is far too small: check whether you entered decimal fractions and then divided by 100 again. Also confirm that masses are in u and that you did not enter a percentage such as 75.78 as 0.007578.

The abundances do not total 100%: confirm that all listed isotopes are included and that every value is in the same format. If the source rounds values, a small difference may be expected; if the difference is large, revisit the data before calculating.

Your answer differs slightly from a reference: compare the isotope masses, abundance values, and rounding rules used by both calculations. Different data precision or abundance sources can produce a small difference. Recalculate with unrounded values if available.

Your answer falls outside the isotope masses: check for a wrong mass-abundance pairing, a decimal-point error, or an incorrect formula. A weighted average with valid nonnegative abundances should not fall outside the range of the values being averaged.

Frequently Asked Questions

What is average atomic mass?
It is the abundance-weighted mean of an element’s isotope masses in a stated sample or natural-abundance dataset. It is commonly listed on periodic tables in u.

Why is average atomic mass usually not a whole number?
It combines isotope masses, which are not necessarily whole numbers, and weights them by their abundances. The result is therefore usually decimal.

Can I use percent abundances directly?
Yes. Multiply each isotope mass by its percentage, add the products, and divide by 100. Alternatively, convert percentages to fractions first and do not divide the final sum by 100.

What if an element has more than two isotopes?
Include every isotope in the dataset. Multiply each mass by its fractional abundance, then add all the products.

Is average atomic mass the same as mass number?
No. A mass number is the whole-number total of protons and neutrons in one isotope. Average atomic mass is a weighted average across isotopes and is usually decimal.

Why might my answer differ from the periodic table?
The table may use more precise isotope data, different rounding, or a different stated natural-abundance dataset. Sample composition can also affect measured isotope proportions.

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in isotope1_mass | Isotope 1 mass | u | 34.96885 | 0
in isotope1_abundance | Isotope 1 abundance | % | 75.78 | 0
in isotope2_mass | Isotope 2 mass | u | 36.96590 | 0
in isotope2_abundance | Isotope 2 abundance | % | 24.22 | 0
out average | Average atomic mass | u | (isotope1_mass * isotope1_abundance + isotope2_mass * isotope2_abundance) / 100 | 5