Simple Nature - Light and Matter

Average lifetime example 6You might think that the half-life would also be the average lifetimeof an atom, since half the atoms’ lives are shorter and halflonger. But the half whose lives are longer include some that survivefor many half-lives, and these rare long-lived atoms skew theaverage. We can calculate the average lifetime as follows:(average lifetime) =0∫ ∞0t D(t) dtUsing the convenient base-e form again, we have(average lifetime) = ln 2 ∫ ∞(t exp − ln 2 )t dt .t 1/2 t 1/2This integral is of a form that can either be attacked with integration∫by parts or by looking it up in a table. The result isxe cx dx = x c ecx − 1 e cx , and the first term can be ignored forc 2our purposes because it equals zero at both limits of integration.We end up with(average lifetime) = ln 2 ( ) 2 t1/2t 1/2 ln 2= t 1/2ln 2= 1.443 t 1/2 ,which is, as expected, longer than one half-life.Section 13.1 Rules of Randomness 835