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AQA GCSE Physics · Topic 4.4 Atomic structure
Edexcel International GCSE Physics · Section 7 (b) Radioactivity
Half-lives and the Random Nature of Decay
In this lesson you'll learn to: say exactly what random means for a decaying nucleus,
define activity and count rate and give the unit, state the definition of half-life
in the words that earn the mark, read a half-life off a graph, and work out how much of a
source is left after a given number of half-lives.
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Prefer to watch? Scroll to the bottom for the video on this topic
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A video walkthrough is coming soon — it will appear at the bottom of this page
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Start here — watch a sample decay
Don't read anything yet. Every gold circle below is one undecayed nucleus. Pick one at random with your eye and try to guess when it will turn grey — you can't, and neither can anybody else. Now stop watching that one nucleus and watch the whole sample instead: the gold bar, the count rate and the curve all halve in the same, completely predictable time.
Undecayed nucleus
Decayed — it has emitted its radiation
0.00
half-lives have passed
0 seconds
0 seconds
Undecayed
100.0%
0 of 0 nuclei
predicted ½n = 100.0%
Decayed
0.0%
0 of 0 nuclei
Count rate on the Geiger counter
0 becquerel (Bq)
0 becquerel (Bq)
it halves in step with the number of nuclei — which is why you can
measure a half-life without ever counting atoms
White = what actually happened. Dashed gold = the perfect ½n prediction. Faint lines mark ½, ¼ and ⅛.
Try this: set the sample to 200 nuclei and watch the white line go bumpy, then set it to
2400 and watch it settle onto the dashed curve. Nothing about the nuclei changed — only
how many you are averaging over.
▲ One nucleus is unpredictable. A thousand of them are not.
Seen it halve? Now learn how to say it.
The examiner will not ask you to describe the picture. They will ask you to define half-life, read one off a graph, or work out what fraction is left. The notes below give you the exact wording and the method for each one.
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Revision notes
1. Radioactive decay is a random process
An unstable nucleus will decay — but nothing decides when. There is no countdown inside a nucleus and no trigger outside it. That is what "random" means here, and the exam wants it spelled out in two halves:
- you cannot predict which nucleus in a sample will decay next, and
- you cannot predict when any particular nucleus will decay.
Every undecayed nucleus has exactly the same chance of decaying in the next second, however long it has already sat there. A nucleus does not get "old", "due" or "tired".
Memorise this
Random — you cannot predict which nucleus will decay next.
2. Activity and count rate
Two words that sound the same and are not. Learn both, and learn which one you can actually measure.
| Quantity | What it means | Unit |
|---|---|---|
| Activity | The rate at which nuclei in the source decay — how many decays happen per second inside the source itself. | becquerel (Bq) 1 Bq = 1 decay per second |
| Count rate | The number of decays recorded each second by a detector, such as a Geiger–Müller tube. | becquerel (Bq) |
Always write becquerel
Both of these are measured in becquerel (Bq). Some boards will not give you the unit mark for "counts per second" — write Bq every time and you are safe on all of them.
The count rate is always smaller than the activity: the radiation goes off in all directions and the detector only sits in one place, so it catches a fraction of it. That does not matter for half-life, because both halve at the same time.
3. The definition of half-life
There are two accepted versions and they earn the same mark. Learn whichever you find easier to say — but say all of it, and do not drop the word average.
Half-life — learn one of these
Either: the average time taken for the number of nuclei of the isotope to halve.
Or: the average time taken for the activity to halve.
4. The shape of a decay curve
Activity against time is always the same shape, and the shape itself is worth marks:
- it always falls, and never becomes a straight line;
- it gets less steep as time goes on, because fewer nuclei are left to decay;
- it never quite reaches zero — each half-life removes half of what is left, not a fixed amount.
The give-away that it is a half-life curve: pick any starting point on it, and the time to halve from there is the same.
Exam technique
Draw your read-off lines on the graph. Every board expects you to determine a half-life
from a graph, and the construction lines are part of the answer — the examiner needs to
see where your number came from. Start at the initial activity, draw a horizontal line across
at half of it, drop a vertical line down to the time axis, and read the time off there.
Then do it once more from a different starting point: if the two readings do not agree, you
have misread the curve.
5. Counting half-lives
After n half-lives, the fraction of the original still undecayed is ½n. You will almost never need more than five, so it is quicker to learn the table than the formula.
| Half-lives passed | Fraction left | % left | % decayed |
|---|---|---|---|
| 0 | 1 | 100% | 0% |
| 1 | ½ | 50% | 50% |
| 2 | ¼ | 25% | 75% |
| 3 | ⅛ | 12.5% | 87.5% |
| 4 | 1/16 | 6.25% | 93.75% |
| 5 | 1/32 | 3.125% | 96.875% |
In the exam you halve step by step and write every step down. It is faster than it looks and it protects the marks if you slip at the end.
Exam technique
“Decayed by 80%” — turn it round first. If a question asks for the time
for a percentage to decay, start by working out how much is undecayed. Decayed by
80% means 20% is left. Then read that off the graph: draw a horizontal line at 20% of
the original value, drop a vertical line down from where it meets the curve, and read
the time off the time axis. The mistake to avoid is drawing your line at 80% — the graph
shows what is left, not what has gone.
If the question mentions background
If a question gives you a background count, subtract it from every reading before you start halving. Background radiation is still there when the source has gone, so a reading that levels off at 20 Bq has not stopped decaying — you are just seeing the background. There is a whole lesson on background radiation — see that one for the detail.
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How to write each answer
Three questions come up again and again. These are the sentences that get the marks — learn the shape, then swap in the numbers from your question.
“Define half-life.”
The half-life is the average time taken for the
number of nuclei of the isotope to
halve.
“Define random decay.”
It is impossible to predict which nucleus will decay next.
“Find the half-life from this graph.”
The initial count rate is ___ Bq, so I read off the time at
half of that, ___ Bq. This gives a half-life of
___.
what is halving the time / the randomness the answer itself
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Worked examples
Cover the answer, try it, then check. The working matters as much as the number.
Example 1 — reading a half-life off a graph
The count rate from a source starts at 800 Bq and has fallen to 400 Bq after 6 days. What is the half-life?2 marks
Step 1 — halve the starting value. Half of 800 Bq is 400 Bq.
Step 2 — read the time at that value. The count rate reaches 400 Bq at 6 days.
Answer: half-life = 6 days
Step 2 — read the time at that value. The count rate reaches 400 Bq at 6 days.
Answer: half-life = 6 days
Mark scheme: one mark for halving the initial count rate, one for the correct time with the unit. A number with no unit scores zero.
Example 2 — activity after a number of half-lives
A source of sodium-24 has an activity of 2400 Bq. The half-life of sodium-24 is 15 hours. What is the activity after 60 hours?3 marks
Step 1 — how many half-lives? 60 ÷ 15 = 4 half-lives
Step 2 — halve four times, writing every step:
2400 → 1200 → 600 → 300 → 150
Answer: activity = 150 Bq
Step 2 — halve four times, writing every step:
2400 → 1200 → 600 → 300 → 150
Answer: activity = 150 Bq
Mark scheme: one mark for 4 half-lives, one for the halving, one for 150 Bq. Common slip: dividing 2400 by 4 to get 600 — you halve four times, you do not divide by four.
Example 3 — working backwards to a time
The count rate from a sample falls from 6400 Bq to 200 Bq. The half-life of the isotope is 4 days. How long did this take?3 marks
Step 1 — halve until you reach 200, counting as you go:
6400 → 3200 → 1600 → 800 → 400 → 200 that is 5 half-lives
Step 2 — multiply by the half-life. 5 × 4 days = 20 days
or, if you'd rather add than multiply:
4 days + 4 days + 4 days + 4 days + 4 days = 20 days
Answer: 20 days
6400 → 3200 → 1600 → 800 → 400 → 200 that is 5 half-lives
Step 2 — multiply by the half-life. 5 × 4 days = 20 days
or, if you'd rather add than multiply:
4 days + 4 days + 4 days + 4 days + 4 days = 20 days
Answer: 20 days
Tip: counting the arrows, not the numbers, is what stops the off-by-one. There are six numbers here but only five halvings.
Example 4 — fraction remaining
A sample of a radioactive isotope is left for 3 half-lives. What fraction of the isotope remains?2 marks
Step 1 — halve once for each half-life. ½ × ½ × ½
Step 2 — write it as a fraction.
Answer: ⅛ of the isotope remains
Step 2 — write it as a fraction.
Answer: ⅛ of the isotope remains
Mark scheme: one mark for halving three times, one for ⅛. Watch out: read the question — "what fraction remains" wants ⅛, but "what fraction has decayed" wants ⅞.
Example 4 — checking a half-life twice
Use the decay curve above to find the half-life, then check your answer using a different starting point.3 marks
First reading: 800 Bq halves to 400 Bq. That happens between 0 and 6 days, so
half-life = 6 days.
Check from elsewhere: 400 Bq halves to 200 Bq, between 6 and 12 days — again 6 days.
Answer: half-life = 6 days, confirmed from two independent readings.
Check from elsewhere: 400 Bq halves to 200 Bq, between 6 and 12 days — again 6 days.
Answer: half-life = 6 days, confirmed from two independent readings.
Why it earns marks: 7.13 asks you to determine the half-life from a graph. Two consistent readings show the curve really is exponential and protect you from one careless read-off.
📘 Now do it in your workbook
Half-life read-offs, decay curves and step-by-step halving questions — with worked answers at the back so you can mark your own.
AQA GCSE Physics Workbook · ATOMIC STRUCTURE · pages 9 & 10 Edexcel iGCSE Physics Workbook · RADIOACTIVITY · pages 9 & 10
⬇ Free: pages 9 & 10 (PDF)
⬇ Free: pages 9 & 10 (PDF)
Get the full workbook on Amazon
Get the full workbook on Amazon
Free sample = the pages for this lesson. The full workbook covers the whole topic for your board, with exam-style questions and worked answers.
✅ Can you do it? Tick as you go
0% complete🎉 Nice work! You've ticked off every objective for this spec point.
Don't forget to hit “Mark complete” at the bottom of the lesson.
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Prefer to watch? Here's the whole thing
Video walkthrough — coming soon
Everything above, explained out loud — useful for a last-minute recap, or if you'd rather hear it than read it.
Everything above, explained out loud — useful for a last-minute recap, or if you'd rather hear it than read it. The video is on its way.
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Video coming soon
The walkthrough for this lesson is being filmed and will appear right here. Until then, the notes and the free workbook pages above cover every mark.
▲ In GHL you can also use the lesson's built-in video field instead of this embed.
Now put it into practice
Watching is the easy bit — half-life questions are only safe once you have halved your way down a few of them on paper. Pages 9 & 10 of the workbook are free, and the full book covers every spec point in Topic 4 Atomic Structure with worked answers.
Watching is the easy bit — half-life questions are only safe once you have read a few off a graph yourself. Pages 9 & 10 of the workbook are free, and the full book covers the whole of Section 7 Radioactivity and particles with worked answers.
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