There are several quantitative aspects of radioactivity that are important for managing nuclear waste materials, including how much radioactivity a given substance emits, how concentrated the substance is in its surrounding medium, how dangerous its emissions are to human and other biological populations, and how long the substance goes on being radioactive. In this example we will address only the last of these issues, the time that a given substance remains dangerous.

A decaying exponential function  A(t)  takes a constant time to reach half of the original amount, no matter what the original amount is. (This is similar to a growing exponential with a constant doubling time.) The constant time for halving the amount is called the half-life of the substance, and is denoted by  T.

There is a simple – and delicious – experiment you can carry out with a supply of M&M's to demonstrate the concept of half-life. Go to the Fermi National Accelerator Lab for the details.

For purposes of studying danger time durations, it is very important to know the half-lives of the radioactive materials invovled. Some nuclides decay to stable substances sufficiently fast (half-lives ranging from fractions of a second to a matter of days) that they are not a serious threat to the environment. On the other hand, some of the isotopes of plutonium have half-lives of many millions of years. The U.S. has vast quantities of highly radioactive plutonium wastes from both weapons production and nuclear power generation.