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.

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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 mass to halve.

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 activity to halve from 80 down to 40. It takes another two days for the activity to halve again; this time from 40 to 20.

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

This process continues and, although the activity 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 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 eight days. So suppose a sample has an activity of 3,200 Becquerel (Bq) at the start, after eight days its activity would be 1/16th of 3,200 Bq = 200 Bq.

Example

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

Fifteen years is three half-lives so the fraction remaining will be \left ( \frac{1}{2} \right )^3 = \frac{1}{8} = 0.125

0.125 \times 100~g  =  12.5~g

As a ratio of what was present originally compared to what is 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 ten 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 ten days. So one half-life is five days.