# Half Life and Rate of Decay

The rate of decay is the number of radioactive atoms that emit nuclear radiation in one second. This is effectively the same as the activity of the sample. It is measured in becquerel (Bq).

It is sometimes referred to as the 'count' for a sample.

You should realize that the geiger counter only gives an indication of the activity as it does not detect ALL particles - only a proportion of them! Changing the setting on the meter changes the 'count' but not the activity it is measuring. When doing an experiment it is important not to change the settings on your counter - keep them the same! You can then make valid comparisons.

## The decay Constant ()

The rate of decay or activity (A) depends on the number of radioactive atoms present. It is proportional to the number that have not yet decayed in the sample. The constant of proportionality is called the decay constant and given the symbol (lamda).

A = n

The decay constant is characteristic to each radioactive isotope. It is the probability of a decay occurring. We can calculate the expected activity of a sample if we know its size and decay constant. The half life of a radioactive isotope is inversley proportional to the decay constant.

T1/2 = ln2/

## Half Life

The activity of radioisotopes decreases exponentially with time. After a given time period the amount that has yet to decay is halved. This is the case no matter when you start to measure the activity of the sample. The time taken for this 'halving' of activity is called the half-life.

Half-lives vary widely from microseconds to millions of years!

Uranium-238 has a half-life of 4.5 x 109 years (4,500,000,000 years) whereas Polonium-212 only has a half-life of 3 x 10-7 seconds (0.000 000 3 seconds). Also the tables of isotopes in the decay series section show a wide variation in half-lives.

Working out a half-life is best explained with an example:

Suppose you have a radioisotope producing a count rate of 640 Bq and after 14 minutes this had dropped to 5 Bq. What would be the half-life?

Well, it would take:

• one half-life for the activity to drop from 640 Bq to 320 Bq
• a second half life for the activity to drop to 160Bq
• a third for it to drop to 80Bq
• a fourth for it to drop to 40 Bq
• a fifth for it to drop to 20 Bq
• a sixth for it to drop to 10 Bq
• and a seventh for it to drop to 5 Bq

So it would take seven half lives for the activity to drop from 640 Bq to 5 Bq. The time for this to happen was 14 minutes so the half-life must be 2 minutes.

Sometimes questions include a value for the background rate. This is the rate of activity that is due to background radiation NOT the sample and must therefore be deducted before any calculations are done.

E.g. The background count for the laboratory was found to be 6 Bq. If it took 10 minutes for the count-rate to drop from 102 Bq to 12 Bq when a radioactive substance was being measured, what was the half-life of the sample?

Initial count-rate due only to the sample was (102-6) Bq = 96 Bq

Final count-rate due only to the sample was (12-6) Bq = 6 Bq

Well, it would take:

• one half-life for the activity to drop from 96 Bq to 48 Bq
• a second half life for the activity to drop to 24 Bq
• a third for it to drop to 12 Bq
• a fourth for it to drop to 6 Bq

So it would take four half lives for the activity of the sample alone to drop from 96 Bq to 6 Bq. The time for this to happen was 10 minutes so the half-life must be 2.5 minutes.

Questions of this type are not uncommon at GCSE

This topic requires the drawing and interpretation of graphs, so although it is suitable to use with spreadsheets on a computer, time needs to be spent ensuring that the work can be done by the pupils in an exam room too.

It is a useful rule of thumb to know that the activity of a sample drops to less than 1% of its value in seven half lives (see Tc99-m)

The activity of a sample can be measured with a Geiger-Müller tube connected to a rate-meter or by connecting it to a scaler and timing how long you allow the scaler to count for.

See site below for diagram of how a Geiger-counter works

http://www.atomicrocks.com/html/geiger.html

If the activity of a sample is plotted against time, an exponential curve is obtained.

(NB It must be the true activity - with the background count deducted from each reading. If you are given a 'corrected count rate' that has already been done for you!)

Examination questions often occur on this topic. When plotting a graph, examiners like to see candidates:

• Make maximum use of the graph paper (choose the best scale - have paper orientated the correct way so as to do this)
• Label axes with physical quantity and correct units
• Mark the points clearly. A neat cross is better than a 'blob'.. most computer programs go for 'blobs'.
• Draw the line of best fit. If the points indicate a curve, it should be smooth (not 'dot-to-dot' like in a puzzle book). If they indicate proportionality the line should be drawn with a ruler.
• Work should be neat (sharp pencil, long ruler, axes in ink etc.)
When analysing data from a graph (whether drawn by them or given to them) candidates must clearly show how the graph was used.

When dealing with radioacive materials inside a human body we have to look at the effective half life, rather than just the physical one.