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

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

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 . An easy calculation
gives . 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 . Therefore, the constant *r*
associated with Carbon-14 is . 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 the amount we are looking for (the initial amount). We
know that

,

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

Hence, since

,

we get

**
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