Basic Nuclear and Atomic Physics-Radioactivity and Radiation

We keep throwing around the term “half-life.” And now is as good of a time as any to put
some math to it. Looking at half-life, as you might expect,
we see that one half-life cuts things in half. Two half-lives cut things in fourths. Hmm, I think I got the picture. How about n over n naught is equal to one
half to the tau, where N is the number of atoms you have now, n sub-naught is the number
you started out with and tau is the number of half-lives. Next let’s change tau to an expression that
explicitly shows the number of half-lives. I have n over n naught is equal to one half
of t divided by t one half, where t is the time and t one half is the half-life. Well this is true, but it’s awkward. It’s usually changed to n over n naught is
equal to e to the minus 0.693t divided by the half-life, or e to the minus lambda t,
and finally, the form you see it on textbooks, n equals naught times e to the minus lambda
t, where lambda equals the decay constant which is zero point six nine three divided
by the half-life. In words, lambda is the probability that an
atom will decay in a second and has the units of inverse seconds. Now, let’s work a problem. Starting with 500 atoms that have a half-life
of 55 seconds, how many are left after 5 minutes? Get your pencil and paper, and do it again. I get 11.41 atoms and, just between you and
me, I think it is kind of hard to have .41 atoms. Well, what’s wrong with our equation? Nothing is wrong with our equation. In fact, our equation is better than reality. What the equation says is if we repeat the
experiment many, many times, on average you will get 11.41 atoms left after the five minutes. This is somewhat the same as the average couple
in the United States having 2.13 kids or 2.42 neutrons emitted during fission. The equation describes the expected value,
to use the statistical term. There is another wrinkle point because it
is statistical. Our formula gives the average value. In real life, if you have a small number of
atoms, the odds that you will get exactly the answer given by the formula, even if it
is a whole number, is small. How can this be? Let’s think of flipping coins for a minute. If you flip a fair coin 500 times, how many
heads do you expect? 250 right? The odds that you will get exactly 250 heads
when you flip a coin 500 times are only 3.566 percent. To be clear, if you repeated this experiment
many times, your average number of heads would get closer and closer to 250, about half the
trials higher, and half the trials lower, just as we predicted. You don’t get 250 each time. Leaving statistics behind for a minute, let’s
return to decay, radioactivity, and radiation, and I’ll show you some nifty things we can
do with our decay constant.

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