Activity and Half-Life

The activity of a radioactive source is a measure of how many atoms decay radioactively every second:-

The unit for Activity is Becquerels ( Bq ) 

1 Bq  =  1 Radioactive decay per second

Note - As 1 Becquerel is a measure of one atomic nucleus decaying, this is a very small unit. It is much more common to record values of 1000s of Becquerels (kBq) or millions of Becquerels (MBq).

For Example - 

A Geiger counter records 4.5 million radioactive decays over a time of 3 minutes when placed near a radioactive source. What is the Activity of this source?

A  =  N / t

A  =  4.5x106 / ( 3 x 60 ) 

A  =  25000 Bq

A  =  25 kBq

Half-Life

In the above worked example, we assumed that the activity of a source was constant. This is not the case. When an unstable atom decays and emits radiation and forms a stable atom, it will not decay any further. 

This means that, over time, the Activity of a source will decrease. 

Although we cannot predict when an individual atom will decay, each Isotope will decay as a whole following a set time frame - the Half-Life, which is defined as :-

"1 Half-Life  =  The length of time required for the Activity of a Source to decrease to half the original value."

Each different isotope of a Radioactive Element will take a certain time for this process to occur, ranging from fractions of a second (7H has a half-life of 21x10-24 seconds) to longer than the universe has existed (128Te has a half-life of 2.2x1024 years or 159 billion times the age of the universe) 

Calculating Half-Life by graphical analysis

The Half-life of an isotope can be calculated by measuring the activity of a source over a long time period, then analysing the graph of Activity against time:-

The graph above shows how the activity of a radioactive source changes over 10 days. As can be seen from the graph, the Activity decreases over time. 

In order to calculate Half-Life, two corresponding values of activity are found, one half of the other (in the above case 40 and 20 Bq), and the graph is used to find the time between these points (in this case 4 - 2 = 2 days).

The time found through this method is equal to the Half-Life. 

Note - Any two values where 1 is double the other will give the same Half-Life value - check!

Calculating Half-Life Numerically 

The Half-life can also be found numerically by the following method:-

A sample of Technetium-99 has an initial activity of 256 Bq. 24 hours later this sample's activity is measured at 16 Bq. What is the Half-Life of Technetium-99? 

Initial Activity             256 Bq

After 1 Half-Lives      128 Bq

After 2 Half-Lives       64 Bq

After 3 Half-Lives       32 Bq

After 4 Half-Lives       16 Bq

In order for the activity to drop to 16 Bq, Four Half-Lives have passed.

As this occurred over 24 hours:-

Half-Life of Technetium-99    =  24 / 4  =  6 hours

The video below gives a summary of the concept of Half-Life:-