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As time passes the strength of a radioactive source gets weaker, its rate of decay (dN/dt) or activity gets smaller. A graph of activity against time is shown in Figure 1.

The rate at which an unstable nucleus decays depends only on what type of nucleus it is. This decay is a random process. This means that if we take a sample of unstable nuclei (for example 226Ra) we cannot know when any individual nucleus is going to decay. However we can measure what we call the half life (T - sometimes written as t1/2) for the element in the sample.


The half life (T) of an element is defined as:
(a) The average time taken for half the original number of nuclei in a sample of an element to decay.
(b) The average time taken for the activity of a radioactive source to fall to one half of its original value.

For example, the half life of 226Ra is about 1600 years. So if we start off with 1200 radium nuclei there will be 600 after 1600 years, 300 after 3200 years, 150 after 4800 years and so on. The number of radioactive radium nuclei remaining halves every half life, in this example every 1600 years.

Knowledge of the half lives of radioisotopes is vital in many situations, for example when they are used in medicine, in radioactive dating or when the environment has been contaminated in a nuclear accident.
Some values of the half lives of some well known radioisotopes are given in the following table:

Isotope Half life   Isotope Half life
Uranium 238 4.5x109 years   *Strontium 90 28 years
*Plutonium 239 2.4x104 years   *Cobalt 60 5.26 years
Carbon 14 5570 years   *Iodine 131 8.1 days
Radium 226 1622 years   Radon 220 55 s
*Caesium 137 30.2 years   Bismuth 214 1.6x10-4 s

* non -naturally occuring isotope.

The activity of a sample is directly proportional to the number of radioactive atoms in the sample and so we can use the half life to predict the activity of a sample at any time in the future.

Since the activity halves every half life it will fall to one half after one half life, a quarter after two, an eight after three and so on.

Example problem
The activity of a sample of radioactive material with a half life of 2 minutes is 48 Bq at the start. What will be the activity after:
(a) 2 mins      (b) 4 mins     (c) 8 mins     (d) 16 mins

(a) one half life          activity = 192
(b) two half lives        activity = 96
(c) four half lives        activity = 24
(d) eight half lives      activity = 1.5

A very useful formula for calculating the final activity for both times equal to complete half lives and also for other times is:

A = Ao/2n where n is the number of half lives that have passed.

The original activity is Ao and the activity after n half-lives have gone by is A. The formula works not only for simple cases where n is a whole number (i.e. for one half life, two half lives etc) but also when n is any number (i.e. 1.2 half lives, 4.3 half lives and so on).

Example problem
A scientist measures the activity due to caesium 137 contamination in a field near to the Chernobyl power station just after the accident in 1986 and finds it to be 10 kBqm-2.
If the half life of caesium 137 is 30 years what will be the activity in:
(a) 1996     (b) 2006     (c) 2016

(a) N = 10/30      A = 10 000/2[10/30] = 7937 Bqm-2
(b) N = 20/30      A = 10 000/2[20/30] = 6300 Bqm-2
(c) N = 30/30      A = 10 000/2[30/30] = 5000 Bqm-2
© Keith Gibbs 2011