Half-lives and the random nature of radioactive decay — AQA GCSE Physics
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Half-lives and the random nature of radioactive decay explained
Radioactive decay is random at the level of a single nucleus.
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You cannot predict which nucleus in a sample will decay next, and you cannot predict when any particular nucleus will decay; an old nucleus is no more likely to go than a fresh one. What is fixed is the chance: every undecayed nucleus of a given isotope has the same constant probability of decaying in the next second. The dice model captures this, because each die has a fixed one in six chance of showing a six on every throw and you cannot say which die will do it. Because a real sample holds an enormous number of nuclei, those individual gambles average out into a smooth, predictable decay curve, and the measured count rate only wobbles a little from reading to reading.
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 has two equivalent definitions and the board accepts either: the time for the number of undecayed nuclei of that isotope in the sample to halve, or the time for the count rate or activity from the sample to fall to half its starting level. The two match because each nucleus has a fixed chance of decaying each second, so the number of decays per second is proportional to the number of nuclei still left. Halve the nuclei and you halve the activity. That is why the decline is repeated halving rather than a steady fall: one half-life leaves a half, two leave a quarter, three leave an eighth. It also means a long half-life gives a low activity. Polonium-209, half-life 125 years, is far less active than polonium-210, half-life 138 days, for the same number of atoms.
Students should be able to explain the concept of half-life and how it is related to the random nature of radioactive decay.
Half-life and randomness fit together through one idea: a constant chance of decay per nucleus per second. No nucleus is scheduled, but each has the same probability of going in the next interval. The number of decays per second is therefore proportional to how many undecayed nuclei are left, so as the sample runs down the decays slow down and the curve gets shallower. Equal time intervals remove equal fractions rather than equal numbers, which is exactly what produces a fixed half-life. So the behaviour of one nucleus is unpredictable while the behaviour of billions is reliable. On a graph of several isotopes, the steepest initial curve belongs to the isotope with the shortest half-life, because it decays quickest and has the smallest percentage left at any later time.
Students should be able to determine the half-life of a radioactive isotope from given information.
There are three routes to a half-life and the data decides which one you use. From a graph: correct for background, halve the initial count rate, read across to the curve and down to the time axis. From a table: find the time interval over which the count rate falls to half, do it again for a second interval and take the mean. From a start and end value: count how many halvings take you from one to the other, then divide the total time by that number. For example, 800 Bq falling to 100 Bq is three halvings, 800 to 400 to 200 to 100, so if this took 24 minutes the half-life is 8 minutes. Always check the unit on the time axis, and quote the answer in the same unit.
(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.
Higher tier only. After n half-lives the fraction of the original emission still present is 1/2ⁿ, so the net decline is written as a ratio such as 1:16 after four half-lives. The method is always the same. Divide the starting value by the final value to get the factor the emission has fallen by, write that factor as a power of two to get the number of half-lives, then multiply by the half-life if the question wants a time. Worked example: a polonium-210 sample falls from 256 000 atoms to 16 000. The factor is 16, which is 2⁴, so four half-lives have passed; at 138 days each that is 4 × 138 = 552 days, and the net decline is 1:16. Ratios have no unit.
Your focus
- State the two things that cannot be predicted about a decaying sample: which nucleus goes next, and when a given nucleus goes.
- Explain why a count-rate graph is a smooth curve even though every individual decay is unpredictable.
- Justify the dice throwing model of decay by matching the fixed one in six chance to the constant chance of decay per nucleus.
Show all 15 objectives
- Give both accepted definitions of half-life, one in terms of undecayed nuclei and one in terms of count rate.
- Read a half-life off a decay curve by halving the corrected count rate and finding the matching time interval.
- Predict which of two isotopes in one sample has the greater activity when their half-lives are given in different units.
- Explain why a fixed chance of decay per nucleus produces a constant half-life rather than a steady decline.
- Identify the shortest half-life from a graph of several isotopes and justify the choice from the gradient or the percentage remaining.
- Account for the flattening of a decay curve using the falling number of undecayed nuclei.
- Determine a half-life from a decay curve by halving the corrected count rate and reading off the interval.
- Calculate a half-life from a start value, an end value and a total time by counting the halvings.
- Decide when a set of readings needs a background correction before a half-life can be found.
- Calculate the number of half-lives that have passed from the starting and final values of an emission.
- Express a net decline as a ratio of the form 1:2ⁿ for a stated number of half-lives.
- Work out the elapsed time from a number of half-lives and a quoted half-life, giving a unit.
Half-lives and the random nature of radioactive decay exam tips
Marking Points
- one mark for saying you cannot predict which nucleus in the sample will decay next
- one mark for saying you cannot predict when a particular nucleus will decay
- one mark for linking the model to the fixed chance of rolling a six on any throw
- one mark for stating that the average behaviour of a very large number of nuclei is still predictable
- one mark for the time taken for the number of undecayed nuclei of the isotope in the sample to halve
- one mark for the alternative definition, the time for the count rate or activity to fall to half its initial level
- one mark for halving the corrected initial count rate and reading across to the time axis on a decay curve
- one mark for stating that the isotope with the longer half-life has the lower activity for the same number of atoms
- one mark for identifying the isotope with the shortest half-life as the one whose curve is steepest at the start
- one mark for the supporting reason, that it decays quickest or has the smallest percentage of nuclei left at any time after the start
- one mark for each undecayed nucleus having the same fixed chance of decaying in a given time, whatever its age
- one mark for linking the falling number of undecayed nuclei to the falling activity, which is why the curve flattens
- one mark for halving the corrected initial count rate and reading the time for that fall
- one mark for counting the number of halvings between the two values given
- one mark for dividing the total time by the number of half-lives
- one mark for the half-life with the correct unit taken from the data
- one mark for the number of half-lives, found by halving repeatedly or by writing the fall as a power of two
- one mark for the net decline expressed as a ratio in the form 1:2ⁿ, such as 1:16 after four half-lives
- one mark for multiplying the number of half-lives by the half-life, such as 4 × 138
- one mark for the final value with the correct unit, such as 552 days
Examiner Tips
- 💡When a question asks for two reasons why the dice model is good, give one about which die and one about when, because they are separate marks.
- 💡Examiners accept answers written in terms of 'rolling a six', so use the model's own language rather than translating it.
- 💡Say randomness applies to individual nuclei before you talk about the sample, so the examiner sees you have separated the two ideas.
- 💡Write 'undecayed nuclei of the isotope', not just 'atoms', so the definition cannot be read as the mass of the sample.
- 💡On a graph question, mark the initial value and its half on the axis in pencil before reading across, so your working is visible.
- 💡If the question compares two isotopes, state the activity comparison first and the half-life reason second, because the reason mark usually depends on it.
- 💡Answer in the order the mark scheme works: name the isotope, then justify it, because the second mark is usually dependent on the first.
- 💡Use the word 'percentage' or 'fraction' when comparing curves, since the graphs are often plotted as percentage of nuclei remaining.
- 💡Link randomness to half-life explicitly with the phrase 'constant chance of decay', which is the bridge the question is testing.
- 💡Write the halving chain out in full, such as 800, 400, 200, 100, so the number of half-lives is on the page for a method mark.
- 💡If a background count rate is quoted anywhere in the question, subtract it from every value before you start.
- 💡Show the reading lines you draw on a printed graph; examiners credit the construction even if the final number is slightly out.
- 💡Write the halving chain, 256 000, 128 000, 64 000, 32 000, 16 000, and count the arrows; that count is the number of half-lives.
- 💡Read whether the question wants a ratio, a fraction or a time, because the three answers look very different from the same working.
- 💡Show the multiplication before you evaluate it, so a substitution mark is available even if the arithmetic slips.
Common Mistakes
- saying that because decay is random the half-life cannot be predicted either
- claiming nuclei decay in turn, so that the oldest or outermost nuclei go first
- writing that decay speeds up or slows down as a nucleus 'gets ready' to decay
- treating the small wobble in successive count-rate readings as a mistake in the experiment
- describing the dice model as random because the thrower cannot aim, rather than because the outcome per die is unpredictable
- defining half-life as the time for the whole sample to decay, or for half the mass to disappear
- halving the time axis instead of halving the count rate when reading a decay curve
- forgetting to subtract the background count rate before halving the reading
- saying the half-life gets longer as the sample runs down, when it is fixed for that isotope
- assuming two half-lives leave nothing, instead of a quarter of the original
- arguing that random decay makes the half-life unpredictable too
- saying equal times remove equal numbers of nuclei instead of equal fractions
- reading the steepest curve as the longest half-life
- giving the reason without first naming the isotope, when the reason mark depends on the identification
- explaining half-life purely as a rule to follow, with no mention of chance per nucleus
- dividing the total time by the ratio of the counts, for example by 8 rather than by the 3 halvings it represents
- using raw counts when a background count rate has been given in the question
- reading the time at which the count rate reaches zero, which it never does
- taking the half-life from the flat tail of the curve where the readings are least reliable
- quoting the answer in seconds when the axis is in minutes or years
- multiplying the half-life by the ratio rather than by the number of half-lives, giving 138 × 16
- giving the ratio the wrong way round when the question asks for the decline in the emission
- treating four half-lives as a fall to one quarter instead of one sixteenth
- attaching a unit such as becquerels to a ratio
- rounding the number of half-lives to a whole number when the data does not actually give a whole number of halvings