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Radioactive decay is a random process. A block of radioactive material will contain many trillions of nuclei. No 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.

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-Müller tube.

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

From the start of timing it takes 2 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.

A third 2-day period, from 4 to 6 days, sees the count rate halve 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. For example, if a sample of a tree (for example) contains 64 grams (g) of radioactive carbon, it will contain 32 g after 5,730 years and 16 g after another 5,730 years.

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 be after a given length of time. This can be stated as a fraction, decimal or ratio.

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

• A fraction - a ½ of a ½ of a ½ of a ½ remains, which is ½ x ½ x ½ x ½ = 1/16th of the original sample.
• A decimal - 1/16th = 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 is sometimes known as the net decline.

This could then be incorporated into other data. So if the half-life is two days, four half-lives is eight days. For example a sample has an activity of 3,200 becquerel (Bq) at the start, its activity after eight 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?

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