HALF-LIVES AND PENNIES

I have had trouble with the concept of radioactive half-life. It seemed to me that the more of the radioactive isotope you had the faster it decayed. If I had a kilogram of some radioactive element with a half-life of one hour, half a kilogram would decay in that hour. If I had five kilograms, 2½ kilograms would decay in that same hour. How did the radioactive isotope “know” there was more and more “needed” to decay?

If I had five 1 kilogram piles, widely separated, a half kilogram of each would decay in the first hour which would be a total of 2½ kilograms for the overall total. Of course that is only the first hour, for the second hour a quarter kilogram would decay. Intellectually I could understand the concept and deal with it. However I had trouble getting a “feel” for it, to visualize it. Until yesterday anyway.

Half-life of a radioisotope means that there is a 50-50 chance that any one atom will decay at some point within that period of time. It either decays or it doesn’t. If it decays it is “removed.” If it doesn’t decay then there is a 50-50 chance at some point within the next interval of time.

Suppose you have 1000 pennies and you flip each one of them once. There is a 50-50 chance of a heads. So you will probably have 500 heads and 500 tails. If you say that each penny represents one radioisotopic atom, heads equals “decay,” tails “not decayed,” and you define a “half-life” as one day. That is once each day you toss the coins to see if it “decays,” comes up heads or not. If the coin is heads you remove it (spend it or whatever) and keep only the coins that were tails for the next day when you repeat the trial over again, you will have an analogy to the radioactive decay process.

Starting with 1000 pennies, at the end of the first day, the first half-life, you would have 500 pennies. After the second day you will have 250, all the ones that came up tails both tosses. The coins do not “remember” what happened on any previous toss. The results of each toss is independent of the earlier ones and independent of the results of the other coins.

You remove the heads and only toss the ones that have again come up tails. The odds never change, but each “half-life” fewer coins (“radioisotopes”) come up heads (“decay”). Eventually you will wind up with one coin that hasn’t yet come up heads (if you start with 1024 coins, to make it work out in whole numbers, it will be the 10^{th} half-life, ten consecutive times that coin has come up tails, 2^{10}=1:1024). Start with 1,060,176 and 20 half-lives you will have one left (2^{>20}=1:1,060,176). That’s the odds that a particular coin picked at the beginning will be the one left. The coin you pick will most likely not be the last one left, but one will and it will still be a 50-50 chance that it will come up tails.

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