Half Lives

Clov: Do you believe in the life to come?

Hamm: Mine was always that.

– Samuel Beckett, Endgame

As each exponentially growing function has a fixed doubling time, each exponentially decaying function has a fixed halving time called its “half life”, a name that comes from chemistry. It is best explained with an example.

All living plants and animals contain carbon, which itself comes in several different forms, including the chemically unstable radiocarbon: Over time, radiocarbon “decays” into nitrogen. While a plant or animal lives, its ratio of radiocarbon to ordinary carbon remains constant. But once it dies, and thus ceases to take in new radiocarbon (through breathing or respiration), this ratio begins to decrease, for the radiocarbon in its tissues decays while the ordinary carbon remains chemically stable. Indeed, this ratio decreases by a fixed percentage per year, which means that it decays exponentially, and thus has a fixed half life. The half life of radiocarbon happens to be about 5600 years. $ ^{*} $ Thus, a piece of wood from a tree that died 5600 years ago, or a bone from a man or animal that died 5600 years ago, would have only half as much radiocarbon (proportionately speaking) as their living counterparts.

Radiocarbon dating is a method for determining the ratio of radiocarbon to ordinary carbon in any object made from organic material, and then using this ratio to determine the object’s approximate age. For instance, if we are studying a tool carved by a prehistoric man from a bone or a horn and we find that its radiocarbon-to-ordinary-carbon ratio is only 1/4 of the usual ratio, then, realizing that 1/4 is half of a half, we can conclude that the animal of which it was once part died approximately two “half lives” ago; that is, it must have died about $ 2(5600) = 11,200 $ years ago. On the other hand, if we find that the tool has, say, 40% of the usual radiocarbon-to-ordinary-carbon ratio, then it’s not immediately obvious how many half-lives have passed since the animal died. But… it’s easy to find out. After all, we just need to solve the equation $ (1/2)^{t} = .4 $ . Solving this, we find, as you should verify, that $ t $ . Hence, the animal must have died around $ 1.32(5600) $ years ago.

Radiocarbon dating was developed in the late 1940’s by Willard Libby, who was awarded a Nobel Prize for this work.