Radioactive Decay

Many radioactive materials disintegrate at a rate proportional to the amount present. For example, if X is the radioactive material and Q(t) is the amount present at time t, then the rate of change of Q(t) with respect to time t is given by

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where r is a positive constant (r>0). Let us call tex2html_wrap_inline46 the initial quantity of the material X, then we have

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Clearly, in order to determine Q(t) we need to find the constant r. This can be done using what is called the half-life T of the material X. The half-life is the time span needed to disintegrate half of the material. So, we have tex2html_wrap_inline60 . An easy calculation gives tex2html_wrap_inline62 . Therefore, if we know T, we can get r and vice-versa. Many chemistry text-books contain the half-life of some important radioactive materials. For example, the half-life of Carbon-14 is tex2html_wrap_inline68 . Therefore, the constant r associated with Carbon-14 is tex2html_wrap_inline72 . As a side note, Carbon-14 is an important tool in the archeological research known as radiocarbon dating.

Example: A radioactive isotope has a half-life of 16 days. You wish to have 30 g at the end of 30 days. How much radioisotope should you start with?


Solution: Since the half-life is given in days we will measure time in days. Let Q(t) be the amount present at time t and tex2html_wrap_inline78 the amount we are looking for (the initial amount). We know that

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where r is a constant. We use the half-life T to determine r. Indeed, we have

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Hence, since

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we get

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