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Chemistry · Ch 1 — Some Basic Concepts of Chemistry

Average Atomic Mass

1.7.2

Average Atomic Mass

The Concept of Average Atomic Mass

Most elements in nature are not composed of atoms that are all identical. Instead, they exist as a mixture of isotopes — atoms of the same element that have the same number of protons but different numbers of neutrons. Because neutrons contribute to mass, each isotope has a distinct atomic mass. To talk about the mass of an element in a practical, real-world sense, we cannot simply use the mass of one isotope; we must account for how much of each isotope is actually present.

This is where the average atomic mass comes in. It is a weighted average of the atomic masses of all the naturally occurring isotopes of an element, where the weighting factor is the relative abundance (the percentage occurrence) of each isotope. The result is the mass you see listed for each element in the periodic table.

Note

The term "average" here is a weighted mean, not a simple arithmetic mean. A simple average would treat all isotopes equally, but nature does not — some isotopes are far more common than others.

Calculating the Average Atomic Mass: The Example of Carbon

The textbook uses carbon to illustrate this calculation perfectly. Carbon has three naturally occurring isotopes:

Table unnumbered-table-1.7.2-carbon-isotopesRelative abundance and atomic mass of the three naturally occurring isotopes of carbon: carbon-12, carbon-13, and carbon-14.
IsotopeRelative Abundance (%)Atomic Mass (amu)
12C^{12}\text{C}98.89212
13C^{13}\text{C}1.10813.00335

To find the average atomic mass, we convert each percentage to a decimal fraction and multiply it by the mass of that isotope. Then, we sum these products.

The calculation proceeds as follows:

  1. Convert percentages to decimal fractions:

    • 12C^{12}\text{C}: 98.892%=98.892100=0.9889298.892\% = \frac{98.892}{100} = 0.98892
    • 13C^{13}\text{C}: 1.108%=1.108100=0.011081.108\% = \frac{1.108}{100} = 0.01108
    • 14C^{14}\text{C}: 2×10−10%=2×10−10100=2×10−122 \times 10^{-10}\% = \frac{2 \times 10^{-10}}{100} = 2 \times 10^{-12}

  2. Multiply each isotope's mass by its fractional abundance:

    • Contribution of 12C^{12}\text{C}: (0.98892)×(12 u)(0.98892) \times (12 \text{ u})
    • Contribution of 13C^{13}\text{C}: (0.01108)×(13.00335 u)(0.01108) \times (13.00335 \text{ u})
    • Contribution of 14C^{14}\text{C}: (2×10−12)×(14.00317 u)(2 \times 10^{-12}) \times (14.00317 \text{ u})

  3. Sum the contributions to get the average atomic mass:

Average atomic mass of carbon=(0.98892)(12 u)+(0.01108)(13.00335 u)+(2×10−12)(14.00317 u)\text{Average atomic mass of carbon} = (0.98892)(12 \text{ u}) + (0.01108)(13.00335 \text{ u}) + (2 \times 10^{-12})(14.00317 \text{ u})

Evaluating this:

=11.86704 u+0.144077 u+2.800634×10−11 u= 11.86704 \text{ u} + 0.144077 \text{ u} + 2.800634 \times 10^{-11} \text{ u}

The contribution from 14C^{14}\text{C} is so vanishingly small that it is negligible for all practical purposes. The sum is:

Average atomic mass of carbon≈12.011 u\text{Average atomic mass of carbon} \approx 12.011 \text{ u}

Average atomic mass=∑i(fractional abundance of isotopei×mass of isotopei)\text{Average atomic mass} = \sum_{i} \left( \text{fractional abundance of isotope}_i \times \text{mass of isotope}_i \right)

The Meaning of Atomic Mass in the Periodic Table

The result of this calculation — 12.011 u for carbon — is the number you will find in the periodic table. This is a critical point: the atomic masses listed for elements in the periodic table are not the masses of any single atom. They are the weighted average masses of all the naturally occurring isotopes of that element. …

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