In both cases, the notation means X is the chemical symbol for the element, A is the number of nucleons (protons and neutrons), and Z is the number of protons. Example (a) indicates that uranium-238 releases an alpha particle (i.e., a helium nucleus) to produce thorium-234. This is called alpha-decay. Example (b) shows that thorium-234 releases a beta particle (i.e., an electron) to produce protactinium-234. This is called beta-decay. Unstable nuclides, such as uranium-238, start series of disintegrations that continue until a stable nucleus results. Mass is lost in both alpha and beta decay processes. This mass is converted into energy: radiation.

For more details on radioactive decay processes, click here. To return to this page from pages outside the CCP site, use your browser's Back button.

In 1903, Rutherford and Soddy, in a paper entitled Radioactive Change, proposed the law of radioactive change. They observed, in every case they investigated, that the rate of decay of radioactive matter was proportional to the amount present. If we write (as chemists do) [A] for the concentration of the substance A, then the law of radioactive change becomes

The value of k is called the decay constant for A.

This law can be justified in the following way: For any fixed time interval, there is a certain probability (a number r between 0 and 1) that each atom of the radioactive substance will disintegrate. Thus, during this time interval, we should expect that the number of atoms that actually do disintegrate is r times the number present at the start of the interval. That's equivalent to saying that the rate of change of [A] is proportional to the amount present.

The functions that have rate of change proportional to the amount present are called exponential functions, and they have the form

for some base b, where y₀ is the starting amount. Thus, we would expect that a concentration function for a radioactive substance has the form

where b is a number between 0 and 1 (in order to have decay rather than growth), and [A]₀ is the starting concentration.

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 module we will address only the last of these issues, the time that a given substance remains dangerous.

• On the Contents page you downloaded a worksheet file to start your computer algebra system. Explain in the worksheet why the function [A] must have a constant time to reach half the original concentration, no matter what the starting concentration is. (This is similar to showing that a growing exponential has a constant doubling time.) The constant time for halving concentration is called the half-life of the substance, and is denoted T_1/2.

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. Click here 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. And we have vast quantities of highly radioactive plutonium wastes from both weapon production and nuclear power generation.

Here are some sources for reading about the environmental impact of nuclear waste management:

• Classifications of radioactive waste (high-level, low-level, etc.)
• Civilian low-level and Defense high-level wastes
• Nuclear waste disposal issues
• An environmentalist perspective
• A government nuclear lab perspective