Half-life - Half-life in Non-exponential Decay

Half-life in Non-exponential Decay

The decay of many physical quantities is not exponential—for example, the evaporation of water from a puddle, or (often) the chemical reaction of a molecule. In such cases, the half-life is defined the same way as before: as the time elapsed before half of the original quantity has decayed. However, unlike in an exponential decay, the half-life depends on the initial quantity, and the prospective half-life will change over time as the quantity decays.

As an example, the radioactive decay of carbon-14 is exponential with a half-life of 5730 years. A quantity of carbon-14 will decay to half of its original amount (on average) after 5730 years, regardless of how big or small the original quantity was. After another 5730 years, one-quarter of the original will remain. On the other hand, the time it will take a puddle to half-evaporate depends on how deep the puddle is. Perhaps a puddle of a certain size will evaporate down to half its original volume in one day. But on the second day, there is no reason to expect that one-quarter of the puddle will remain; in fact, it will probably be much less than that. This is an example where the half-life reduces as time goes on. (In other non-exponential decays, it can increase instead.)

The decay of a mixture of two or more materials which each decay exponentially, but with different half-lives, is not exponential. Mathematically, the sum of two exponential functions is not a single exponential function. A common example of such a situation is the waste of nuclear power stations, which is a mix of substances with vastly different half-lives. Consider a sample containing a rapidly decaying element A, with a half-life of 1 second, and a slowly decaying element B, with a half-life of one year. After a few seconds, almost all atoms of the element A have decayed after repeated halving of the initial total number of atoms; but very few of the atoms of element B will have decayed yet as only a tiny fraction of a half-life has elapsed. Thus, the mixture taken as a whole does not decay by halves.