Exponential Law of Decay

It is important to understand that radioactive decay is entirely a random and unpredictable process.  If we look at any one nucleus, it might decay in ten seconds or ten million years.  There is no way of telling when the decay will happen, and there is certainly no way of speeding up the process.  Remember that chemical reactions involve the outer shell electrons; radioactive decay involves the nucleus.

However when we have many millions of millions of nuclei, we can apply statistical models of probability.

The rate of disintegration of any nuclide at any time is directly proportional to the number of atoms left at the time:

This is calculus notation for rate of change in number.

The minus sign tells us that N decreases as time increases.

There is a constant of proportionality called the radioactive decay constant, which is given the physics code l (greek letter ‘l’, not to be confused with wavelength), and has the units s^-1.  So we can write:

The radioactive decay constant is defined as the fraction of the total number of nuclei present that decays per unit time, provided that the time interval is small.  For nuclides with relatively long half-lives, the decay constant will have units “per second”.  However some very unstable nuclides decay in microseconds, so the decay constant would have to be “per nanosecond”.

Example

The unit Becquerel (Bq) is often used.  1 Bq = 1 count per second.

Over longer periods of time, the relationship above does not hold.  It can be shown by calculus methods (which you are not expected to know), that the decay follows the relationship:

[N – no of nuclei; N₀ – original number of nuclei; e – exponential number, 2.718…; l - decay constant (s^-1); t – time (s)]

This relationship is described as an exponential decay and the graph looks like this:

We should note the following:

• The rate of decay represents the number of atoms remaining.  So we can use this graph for representation of the count rate, or ionisation current.

• The rate of decay is the number of disintegrations per second, or the activity.  It is measured in Becquerels  (Bq).

• 1 Bq = 1 disintegration per second.

Notice that the horizontal axis is calibrated in periods of time in which the decay goes from an initial value to half that value.  This period is called the half-life.  The formal definition of half-life is the time taken for the activity of a sample to decrease to half some initial value.  After 1 half life the activity is 50 %, between 1 and 2 half lives it falls from 50 % to 25 % and so on.  If we have a whole number of half-lives, we can do an iterative calculation of the fraction remaining, i.e. ½, ¼, 1/8, etc.

We can relate the half-life to the decay constant:

By definition

The formula is N = N₀e^-lt.  Rearranging gives us

T1/2 (pronounced "T-half") is the physics code for half-life.

This will work for any units of time, although we need to be consistent.

To work with these relationships, you must be familiar with the concept of logarithms (a way of expressing a number as a power of 10 or e).  You will see logarithms expressed as ‘lg’ or log₁₀, which means log to the base 10, or ‘ln’ or log_e, log to the base e.  The number e is the exponential number, 2.718... The latter is known as natural logarithms, which are at the heart of exponential functions.

Example

[Alternative method] If you are not so confident in the use of the natural logarithm, you can work out the number of half-lives a particular time gives.  In this case 3 days = 3/8 of a half life.

Activity = 1 ´ 10¹² ´ 1/2^3/8 = 1 ´ 10¹² ´ 1/1.30 = 7.7 ´ 10¹¹ Bq

This is a perfectly valid alternative approach.

The half-life has important implications for the storage of radioactive waste.  Radioactive waste is some of the nastiest muck known to man, so it has to be stored carefully.  Isotopes with a short half-life have intense activities, whereas those with a long half-life have lower activities, but it takes much longer for the activity to decay to a reasonable level.  Either way, it’s not very nice.

Even when there have been several half-lives, there remains quite a considerable activity.  If in our example above the activity had decayed to 1/1000^th of its original value, we would still have 10⁹ disintegrations per second, which is quite a lot.

Another use of the decay equations is in radioactive dating.  Isotopes used are carbon-14, rubidium-87, and hydrogen 3.