# Average Atomic Mass

As we said in the discussion of atomic mass, atoms come in isotopes of varying mass and all natural samples of an element will have a mixture of those isotopes.

Since any sample we take will have a mixture of isotopes, it is not terribly helpful to know the individual masses of the separate isotopes. What is needed is an average.

## Finding average mass

To show how this average is found we will use the example of carbon. We know that carbon has three isotopes: 12C with a mass of exactly 12 amu, 13C with a mass of 13.00335 amu, and 14C with a mass of 14.00324 amu. It would be tempting to average these three numbers as they are and decide that the average mass of carbon is 13.00220 amu. However that answer is misleading.

Here is a way to understand the problem with the math above. Imagine that you go to a small party (only 10 people are there). At the party you realize that 9 of the people there weigh 100 lbs, while the other person weigh 200 lbs. Hopefully you recognize that the average weight of a person at the party is not 150 (the average of 100 and 200). In order to find out the average you would add up all of the weights 100+100+100+100+100+100+100+100+100+200 and then divide by 10. The average weight at the party would work out to be 110 lbs.

However, what if the party was very large, so that you didn't know how many people were there. If the host told you that 90% of the people there weighed 100 lbs and the other 10% weighed 200 lbs could you still figure out the average weight? It's actually pretty easy:

Or, if you prefer not to deal with fractions, you could do it this way:

Note in this case, we have converted the percentages to decimal equivalents and then we don't need to divide by 100%.

## The Average Mass of Carbon

Finding the Correct average mass of carbon is simply a matter of knowing the masses of the isotopes and thier percent abundances (what percent of a natural sample of carbon they are). The data that you need is this:

Isotope | Pecent abundance | mass (amu) |
---|---|---|

^{12}C |
98.9% |
12 |

^{13}C |
1.1% |
13.00335 |

^{14}C |
trace (an amount too small to bother us here) |
14.00324 |

Out math, then, would look like this:

or

This number (12.011) just happens to be the atomic mass listed on the periodic table for carbon. In the same way, **ALL** of the masses listed on the periodic table are actually average masses of the isotopes for that element.