Page 90 - Radiochemistry and nuclear chemistry

Unstable Nuclei and Radioactive Decay 79

4.11. Kinetics of simple radioactive decay
Most radioactive isotopes which are found in the elements on earth must have existed for
at least as long as the earth. The nonexistence in nature of elements with atomic numbers
greater than 92 is explained by the fact that all the isotopes of these elements have life-times
considerably shorter than the age of the earth.
Radioactive decay is a random process. Among the atoms in a sample undergoing decay
it is not possible to identify which specific atom will be the next to decay. We denote the
decay rate by A. It is a measure of the number of disintegrations per unit time:

A = -dN/dt (4.39)

The decay rate is proportional to the number of radioactive atoms, N, present: A oc N. If
105 atoms show a decay rate of 5 atoms per second then 106 atoms show a decay rate of
50 atoms per second. If the number of radioactive nuclei and the number of decays per unit
time are sufficiently great to permit a statistical treatment, then

-dN/dt = X N (4.40a)
where h is the proportionality constant known as the decay constant. If the time of
observation At during which AN atoms decay is very small compared to tl,~ (usually <
1%), one may simply write

A = AN~At = X N (4.40b)

If the number of nuclei present at some original time t = 0 is designated as N 0, (4.40a)
upon integration becomes the general equation for simple radioactive decay:

N = N O e -ht (4.41a)

In Figure 4.8 the ratio of the number of nuclei at any time t to the original number at time
t = 0 (i.e. N/No) has been plotted on both a linear (left) and logarithmic (fight) scale as a
function of t. The linearity of the decay curve in the semi-logarithmic graph illustrates the
exponential nature of radioactive decay. Since A cx N, the equation can be rewritten as

A = A 0 e -ht (4.41b)

Commonly, log A is plotted as a function of t since it is simpler to determine the
disintegration rate than it is to determine the number of radioactive atoms in a sample.
Instead of the decay constant X, the average lifetime r is sometimes used:

r = 1/X (4.42)

Even more common is the use of the half-life, tt/~, which is the time needed to reduce the
amount of radioactive material by a factor of 2. Thus