Example: Radioactive Waste

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This example fits various forms of decaying exponential functions to data on the decay of radioactive elements. It reviews the concept of half-life and shows how it may be calculated from equations and data. An analysis is made of safe storage times for radioactive wastes.

Suppose we have a radioactive decay process of the general form

This means that each atom of the radioactive substance  X  breaks down into a collection of smaller atoms and/or subatomic particles. Everything from dating ancient artifacts to assessing the danger of nuclear wastes depends on knowing the rate of decay for such reactions.

Here are two examples of specific radioactive decay processes:

(a)  238 U 92  »»»  234 Th 90  +  4 H 2 (b)  234 Th 90  »»»  234 Pa 91  +  0 e –1

In both cases, the notation  ^Y X _Z  means  X  is the chemical symbol for the element,  Y  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 nuclear chemistry, click here.

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)  [X]  for the amount of the substance  X , then the law of radioactive change becomes

The constant  l  is called the decay constant or lambda value for  X .

This law can be understood 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  [X]  is proportional to the amount present.

The functions that have rates of change proportional to the amount present are the exponential functions:

for some base  b , where  y₀  is the starting amount. Thus, we would expect that the amount  A = [X]  of a radioactive substance  X  has the form

where  b  is a number between  0  and  1  (in order to have decay rather than growth), and  A₀  is the starting amount.