Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
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Average lifetimeYou might think that the halflife would also be the average lifetime of an atom, since half the atoms' lives are shorter and half longer. But the half whose lives are longer include some that survive for many halflives, and these rare longlived atoms skew the average. We can calculate the average lifetime as follows:
Using the convenient basee form again, we have
This integral is of a form that can either be attacked with integration by parts or by looking it up in a table. The result is
and the first term can be ignored for our purposes because it equals zero at both limits of integration. We end up with
which is, as expected, longer than one halflife.


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