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Why is radioactive decay random, and how do we calculate half-life?

Radioactive decay as a random process, activity and count rate, the meaning of half-life, and calculating half-life or the time for a given decay from a graph or table of count rate.

A focused answer to OCR Gateway GCSE Physics A topic P6 on radioactive decay, covering decay as a random process, activity and count rate measured with a Geiger-Muller tube, the meaning of half-life, and calculating half-life or the time for a given decay.

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  1. What this topic is asking
  2. Radioactive decay is random
  3. Activity and count rate
  4. The meaning of half-life
  5. Calculating half-life and decay
  6. Try this

What this topic is asking

OCR wants you to explain that radioactive decay is a random process, define activity and count rate, explain the meaning of half-life, and calculate the half-life or the time for a given decay from a graph or table. This is part of topic P6.1 of the OCR Gateway Physics A (J249) specification.

Radioactive decay is random

Although a single decay is unpredictable, a sample contains a huge number of nuclei, so the overall behaviour is very predictable: a fixed fraction decays in each fixed time, giving the regular halving pattern of half-life.

Activity and count rate

When measuring a source, you should first record the background radiation count rate (from rocks, cosmic rays and so on) with no source present, and subtract it from later readings to get the count rate due to the source alone.

The meaning of half-life

So after one half-life, half the original nuclei remain; after two half-lives, a quarter remain; after three, an eighth; after nn half-lives, a fraction (12)n\left(\tfrac{1}{2}\right)^n of the original remains. The count rate follows the same pattern, falling by half each half-life.

Calculating half-life and decay

To find the half-life from a graph of count rate against time, read off the time taken for the count rate to fall to half its starting value. To find the time for a given decay, count how many times the count rate must halve to reach the final value, then multiply that number of half-lives by the half-life. Higher candidates may also be asked for the net decline as a ratio, for example stating that the count rate fell to one-eighth of the original after three half-lives.

Try this

Q1. A source has a count rate of 20002000 counts per minute and a half-life of 2 hours2\,\text{hours}. State the count rate after 6 hours6\,\text{hours}. [2 marks]

  • Cue. 6 hours6\,\text{hours} is 33 half-lives, so the count rate halves three times: 2000→1000→500→2502000 \to 1000 \to 500 \to 250 counts per minute.

Q2. State the unit of activity and what one unit means. [1 mark]

  • Cue. The becquerel, equal to one decay per second.

Exam-style practice questions

Practice questions written in the style of OCR exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

OCR 20194 marksThe count rate from a radioactive source falls from 800800 counts per minute to 100100 counts per minute in 2424 hours. Calculate the half-life of the source.
Show worked answer →

A P6 Calculate question on half-life. Each half-life halves the count rate, so work out how many halvings take you from 800800 to 100100: 800→400800 \to 400 (one), 400→200400 \to 200 (two), 200→100200 \to 100 (three). That is 3 half-lives (2 marks for showing the halvings to reach 3). The total time of 24 hours24\,\text{hours} covers 33 half-lives, so one half-life is 24÷3=8 hours24 \div 3 = 8\,\text{hours} (2 marks for dividing and the answer with units). Markers reward the number of half-lives and the division to get 8 hours8\,\text{hours}. A common error is to divide 2424 by 88 (using the wrong number) or to miscount the halvings.

OCR 20213 marksExplain what is meant by the half-life of a radioactive isotope, and explain why radioactive decay is described as a random process.
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A P6 question worth three marks. The half-life is the average time taken for half the unstable (undecayed) nuclei in a sample to decay, or equivalently the time for the count rate (activity) to fall to half its value (2 marks for either correct definition). Decay is random because it is impossible to predict which nucleus will decay next, or when a particular nucleus will decay; we can only state the probability of decay in a given time (1 mark). Markers reward the half-life as the time for half the nuclei (or count rate) to decay, and randomness as the inability to predict the next decay. A common error is to say half-life is the time for the whole sample to decay.

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