Half-lives and the random nature of radioactive decay — AQA GCSE Combined Science
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Half-lives and the random nature of radioactive decay explained
Random decay means you cannot predict when any particular unstable nucleus will decay, nor which nucleus in a sample will decay next.
Read the full explanation
Each nucleus has a fixed probability of decaying in a given time, but the actual moment is a matter of chance. This is why a single count-rate reading is unreliable: the count over the next minute might be higher or lower than the last. However, because a sample contains enormous numbers of nuclei, the overall pattern is predictable. The activity falls by half in each half-life, giving a smooth decay curve. For example, a source with a half-life of 5 days and an initial count rate of 800 counts per minute would be expected to give about 400 counts per minute after 5 days and about 200 counts per minute after 10 days, even though no one can say when an individual nucleus will decay.
The half-life of a radioactive isotope is the time it takes for the number of nuclei of the isotope in a sample to halve, or the time it takes for the count rate (or activity) from a sample containing the isotope to fall to half its initial level.
Half-life is the time for half the unstable nuclei in a sample to decay, so the number of nuclei of that isotope halves. Equivalently, it is the time for the count rate or activity from the sample to fall to half its initial level. For example, if a source gives 800 counts per minute, after one half-life it gives about 400, after two about 200, and after three about 100. Because decay is random, individual nuclei decay unpredictably, but large numbers give a smooth, predictable halving pattern. Half-life is fixed for a given isotope and is unaffected by temperature, pressure or chemical state. It allows dating and medical tracing.
Students should be able to explain the concept of half-life and how it is related to the random nature of radioactive decay.
Radioactive decay is random: you cannot predict which nucleus decays next or exactly when. Yet a large sample shows a steady pattern because many nuclei are present, so the proportion decaying per unit time is stable. Half-life is the time for half the nuclei, or half the count rate, to remain. For example, a source of 600 counts per minute falls to about 300, then 150, then 75 over equal half-lives. Individual decays are chance events, but the overall halving is predictable. This lets scientists date rocks and trace medical isotopes.
Students should be able to determine the half-life of a radioactive isotope from given information.
Half-life is the time taken for the number of unstable nuclei in a radioactive sample to halve, or for the count rate to fall to half its initial value. To determine it from given information, identify the starting count rate, halve it, and read the time at which that value occurs. For example, if a source starts at 800 counts per minute and falls to 400 counts per minute after 6 hours, the half-life is 6 hours. If data are given at intervals, you may need to halve repeatedly: 800 → 400 → 200 → 100 counts per minute, each step taking one half-life. On a graph, find the time for the count rate to halve, or use the time for the activity to fall to half. Always state the unit of time and remember that half-life is constant for a given isotope, regardless of the starting amount.
(HT only) Students should be able to calculate the net decline, expressed as a ratio, in a radioactive emission after a given number of half-lives.
After each half-life, the count rate or activity of a radioactive sample falls to half its previous value. After n half-lives, the remaining fraction is (1/2)ⁿ. The net decline is expressed as a ratio of the final emission to the original emission, which is 1:2ⁿ. For example, after 3 half-lives, the remaining fraction is (1/2)³ = 1/8. The ratio of final to original emission is therefore 1:8. If the starting count rate is 640 counts per minute, after 3 half-lives it is 640 ÷ 8 = 80 counts per minute. The ratio of 80 to 640 simplifies to 1:8. Higher tier students must be able to calculate this net decline without needing a graph, using powers of 2 and clear division, and correctly format the ratio as requested by AQA mark schemes.
Your focus
- Describe radioactive decay as a random process affecting individual nuclei.
- Explain how the random decay of many nuclei produces a predictable half-life pattern.
- Calculate the expected count rate or activity after a given number of half-lives.
Show all 15 objectives
- Define half-life in terms of nuclei number and in terms of count rate or activity.
- Apply repeated halving to find the activity or number of nuclei remaining after a given number of half-lives.
- Explain why random decay still produces a predictable half-life for a large sample.
- Explain why radioactive decay is random for individual nuclei.
- Explain how a large sample produces a predictable half-life despite random decay.
- Relate half-life to a suitable use such as dating or medical tracing.
- Identify the initial count rate or activity from given data, a graph or a table.
- Determine the half-life by halving the count rate and reading the corresponding time, including cases requiring repeated halving.
- State the half-life with the correct unit and explain that it is constant for a given isotope.
- Determine the number of half-lives from a given time and half-life.
- Calculate the remaining fraction or final emission after n half-lives using (1/2)ⁿ.
- Express the net decline as a simplified ratio in the form 1:2ⁿ.
Half-lives and the random nature of radioactive decay exam tips
Marking Points
- Defines random decay as being unable to predict which nucleus decays next or exactly when a given nucleus will decay.
- Explains that each nucleus has a constant probability of decaying per unit time, but the timing of any individual decay is chance.
- Uses the large number of nuclei in a sample to explain why the overall activity follows a predictable half-life pattern.
- Applies half-life reasoning to calculate expected count rate or activity after one or more half-lives, for example halving 800 counts per minute to 400 then 200.
- Distinguishes random decay from a regular or clockwork process, noting that a single short count may not match the expected average.
- State that half-life is the time for the number of nuclei of the isotope in a sample to halve.
- State the equivalent definition: the time for the count rate or activity from the sample to fall to half its initial level.
- Use the halving sequence correctly, for example 800 → 400 → 200 → 100 counts per minute over successive half-lives.
- Explain that random decay of individual nuclei produces a predictable halving pattern only because a sample contains very many nuclei.
- Recognise that half-life is constant for a given isotope and is not changed by temperature, pressure or chemical combination.
- Calculate the number of half-lives from total time divided by half-life, or find activity after a given time by repeated halving.
- Explain that radioactive decay is random, so individual nuclei decay at unpredictable times.
- Explain that a large number of nuclei gives a statistically predictable overall pattern.
- Link the predictable pattern to half-life as the time for half the nuclei or half the count rate to remain.
- Use a numerical example, such as 600 → 300 → 150 → 75 counts per minute, to show repeated halving.
- Distinguish between the random behaviour of one nucleus and the reliable average behaviour of a large sample.
- Apply the concept to a context such as carbon dating or medical tracers, choosing an isotope with a suitable half-life.
- Identify the initial count rate or activity from the data, graph or table before halving it.
- Halve the initial value correctly and read the corresponding time from the graph or table to obtain one half-life.
- For data given at intervals, count the number of halvings needed to reach the stated value and divide the total time by that number.
- State the half-life with the correct unit of time, such as seconds, minutes, hours, days or years.
- Recognise that the half-life is constant for a given isotope and does not depend on the starting count rate or sample size.
- Use a graph correctly by reading across from half the initial count rate to the curve and down to the time axis.
- Identify the number of half-lives, n, from the question or by dividing the total time by the half-life.
- Calculate the remaining fraction after n half-lives using (1/2)ⁿ.
- Calculate the net decline ratio as 1:2ⁿ, representing the final emission compared to the original emission.
- Express the answer as a simplified ratio in the form 1:X (e.g., 1:8 after 3 half-lives).
- Show intermediate values, such as the final count rate, to make the calculation method clear.
Examiner Tips
- 💡Use the phrase 'cannot predict' when defining random decay, and add that the overall behaviour of a large sample is predictable.
- 💡For half-life calculations, count how many half-lives have passed, then halve the starting value that many times, showing each step.
- 💡When interpreting a graph, read the half-life from the curve by finding the time for the count rate to fall to half its initial value, not by joining points with a ruler.
- 💡Write the definition in terms of either nuclei or count rate, and make clear which quantity is halving.
- 💡Show each halving step in a calculation so the examiner can follow the sequence.
- 💡Check whether the question asks for activity remaining or time elapsed, and label units such as counts per minute or seconds.
- 💡Use the phrase random decay and then explain how many nuclei produce a predictable average.
- 💡Support your explanation with a short halving sequence and clear units.
- 💡When asked to explain, link each statement to the next: random decay, large numbers, stable proportion, half-life.
- 💡Underline the initial count rate and the time interval in the question before starting any calculation.
- 💡Show each halving step clearly, for example 800 → 400 → 200 counts per minute, so the examiner can follow your method.
- 💡If the value does not halve exactly, estimate the half-life from the graph and give the unit; do not round the count rate to zero.
- 💡Write down the number of half-lives first, then use powers of 2 to find the ratio quickly.
- 💡Always present the net decline ratio in the format 1:2ⁿ unless specifically instructed otherwise.
Common Mistakes
- Saying that a nucleus 'gets old' and decays after a set time; correct this by stating that decay is not scheduled and the nucleus has the same chance of decaying at any moment.
- Treating a single count-rate measurement as proof that half-life has changed; correct this by explaining that random variation makes individual readings unreliable and repeated readings or longer counts are needed.
- Confusing half-life with the time for all nuclei to decay; correct this by stressing that half-life is the time for half the unstable nuclei to decay, or for the count rate to halve.
- Treating half-life as the time for all the nuclei to decay; correct this by stressing that half remain after each half-life, so the number never reaches zero in a simple halving model.
- Halving the time rather than the count rate or number of nuclei; correct this by applying each half-life to the quantity, not to the time.
- Assuming half-life changes with temperature or chemical state; correct this by stating that nuclear decay is unaffected by external conditions.
- Saying that half-life makes each nucleus decay at a fixed time; correct this by stating that individual decay times remain random.
- Confusing half-life with the time for the count rate to reach zero; correct this by showing the halving sequence continues without reaching zero.
- Believing that a small sample will halve exactly on schedule; correct this by explaining that statistical predictability needs a large number of nuclei.
- Halving the time instead of the count rate: the error is dividing the given time by 2; the correction is to halve the count rate or activity and find the time at which that value occurs.
- Reading the time at which the count rate reaches zero: the error is treating zero as the end point; the correction is to use the point at which the count rate has fallen to half its starting value.
- Forgetting the unit or using the wrong unit: the error is writing a number without a time unit; the correction is to include the unit shown in the data, such as hours or days.
- Using n/2 instead of (1/2)ⁿ. Correction: the process is repeated halving, so use the power (1/2)ⁿ or multiply by 1/2 for each half-life.
- Reversing the ratio to 2ⁿ:1 (e.g., 8:1). Correction: AQA mark schemes require the ratio of final to original emission, so it should be written as 1:2ⁿ (e.g., 1:8).
- Forgetting to simplify the ratio. Correction: if calculating from raw count rates like 80:640, divide both sides by the highest common factor to give 1:8.