Half life

Radioactive decay is a random process. A block of radioactive material will contain many trillions of nuclei and not all nuclei are likely to decay at the same time so it is impossible to tell when a particular nucleus will decay.

It is not possible to say which particular nucleus will decay next, but given that there are so many of them, it is possible to say that a certain number will decay in a certain time. Scientists cannot tell when a particular nucleus will decay, but they can use statistical methods to tell when half the unstable nuclei in a sample will have decayed. This is called the half-life.

curriculum-key-fact
Half-life is the time it takes for half of the unstable nuclei in a sample to decay or for the activity of the sample to halve or for the count rate to halve. Count-rate is the number of decays recorded each second by a detector, such as the Geiger-Muller tube.

The illustration below shows how a radioactive sample is decaying over time.

Graph with time against activity in becquerels, with a downward sloping curve.

From the start of timing it takes two days for the count to halve from 80 down to 40. It takes another two days for the count rate to halve again, this time from 40 to 20.

Note that this second two days does not see the count drop to zero, only that it halves again. A third, two day period from four days to six days see the count rate halving again from 20 down to 10.

This process continues and although the count rate might get very small, it does not drop to zero completely.

The half-life of radioactive carbon-14 is 5,730 years. If a sample of a tree (for example) contains 64 grams (g) of radioactive carbon after 5,730 years it will contain 32 g, after another 5,730 years that will have halved again to 16 g.

Calculating the isotope remaining - Higher

It should also be possible to state how much of a sample remains or what the activity or count should become after a given length of time. This could be stated as a fraction, decimal or ratio.

For example the amount of a sample remaining after four half-lives could be expressed as:

  • a fraction - a ½ of a ½ of a ½ of a ½ remains, which is ½ × ½ × ½ × ½ = 1/16 of the original sample
  • a decimal – 1/16 = 0.0625 of the original sample
  • a ratio – given in the form 'activity after n half-lives : initial activity' , in this case 1:16

This could then be incorporated into other data. So if the half-life is two days, four half-lives is 8 days. So suppose a sample has a count rate of 3,200 Becquerel (Bq) at the start, what its count rate would be after 8 days would be 1/16th of 3,200 Bq = 200 Bq.

Example

The half-life of cobalt-60 is 5 years. If there are 100 g of cobalt-60 in a sample, how much will be left after 15 years?

15 years is three half-lives so the fraction remaining will be (\frac{1}{2})^3 = \frac{1}{8} = 12.5g

As a ratio of what was present originally compared to what was left, this would be 100:12.5 or 1:0.125

Question

What is the half-life of a sample where the activity drops from 1,200 Bq down to 300 Bq in 10 days?

Half of 1,200 is 600, half of 600 is 300. So it takes two half-lives to drop from 1,200 Bq to 300 Bq, which is 10 days. So one half-life is five days.