Half-Life and Radioactive Decay
Radioactive decay is random, but with huge numbers of nuclei the average behaviour is completely predictable — and that predictability is the half-life.
1. The definition
Half-life is the time taken for HALF the radioactive nuclei in a sample to decay. Equivalently: the time for the activity (count rate) to fall to half its original value.
Measured in seconds, hours, days or years — half-lives range from fractions of a second to billions of years.
Either definition earns the mark, but be precise: it is half the undecayed nuclei, or half the count rate — not half the total mass of the object.
Why it works despite randomness: you cannot predict when any one nucleus will decay, but with billions of nuclei the proportion decaying in a given time is reliable.
2. Calculating with half-lives
Each half-life halves what remains:
| Half-lives elapsed | Fraction remaining |
|---|---|
| 0 | 1 |
| 1 | 1/2 |
| 2 | 1/4 |
| 3 | 1/8 |
| 4 | 1/16 |
| n | (1/2)ⁿ |
Method:
- Find how many half-lives have passed: n = total time ÷ half-life
- Halve the starting value n times
Example. A source has an activity of 800 Bq and a half-life of 5 days. What is its activity after 20 days?
- n = 20 ÷ 5 = 4 half-lives
- 800 → 400 → 200 → 100 → 50 Bq
Example. A 240 g sample decays to 30 g. How many half-lives?
- 240 → 120 → 60 → 30, so 3 half-lives
- If the half-life is 6 hours, the time is 3 × 6 = 18 hours
Halve repeatedly — don’t divide by the number of half-lives. Dividing 800 by 4 to get 200 is the classic error. Four half-lives means halving four times, giving 1/16 of the original.
Count the number of halvings carefully. Miscalculating the number of half-lives was the most-recorded error in this topic. Write out the chain (800 → 400 → 200 → 100 → 50) rather than doing it mentally.
Working backwards: to find the half-life, count how many halvings took you from the start value to the end value, then divide the total time by that number.
Activity never reaches exactly zero — each half-life removes half of what is left, so it approaches zero without arriving.
3. Decay graphs
A graph of activity (or number of nuclei) against time gives a characteristic decreasing curve that gets shallower — an exponential decay.
Reading the half-life from a graph:
- Read the initial activity from the y-axis
- Halve it
- Read across to the curve, then down to the time axis
- That time is the half-life
Check with a second pair of points. Start from any activity, halve it, and the time taken should be the same — that constancy is the defining property of exponential decay, and it is a good check on your reading.
Read the graph carefully and use the axis scales. Misinterpreting graph data for a half-life calculation was recorded.
Remember to subtract background count rate from readings before using them, if the question gives a background level.
4. Activity and units
Activity is the number of decays per second, measured in becquerels (Bq).
Count rate is what a detector measures, usually in counts per second or counts per minute — check which the question uses.
Activity decreases over time, because there are fewer undecayed nuclei left to decay.
5. Uses of half-life
Choosing an isotope is always about matching the half-life to the job:
| Use | Half-life needed |
|---|---|
| Medical tracer | short (hours) — does its job, then decays away quickly |
| Cancer treatment | moderate, with gamma emission |
| Smoke detector | long — so it doesn’t need replacing |
| Carbon dating | comparable to the age being measured |
Carbon dating: living things absorb carbon-14 (half-life ≈ 5730 years). When they die, absorption stops and the C-14 decays. Comparing the remaining C-14 with a living sample gives the age.
Dating rocks uses isotopes with much longer half-lives, such as uranium.
A short half-life means the source is highly active but short-lived; a long half-life means weak but persistent. Match this reasoning to whichever use the question asks about.
6. Scope note
Some lessons used the decay constant and exponential formulae.
The decay constant is not on the 0625 syllabus. At IGCSE you work with whole numbers of half-lives and read values from graphs. The exponential decay equation belongs to A-level — check your own syllabus document.
7. Mistakes that cost marks
Dividing by the number of half-lives instead of halving repeatedly.
Miscounting the number of half-lives.
Saying activity reaches zero.
Defining half-life as “half the time for decay”.
Forgetting to subtract background count rate.
Misreading the decay graph or its scales.
Choosing an isotope with an unsuitable half-life in an application question.
Omitting units — Bq, or counts per second.
Frequently asked questions
What is half-life? The time for half the radioactive nuclei to decay, or for the count rate to halve.
How do I calculate the activity after several half-lives? Halve the starting value once per half-life.
What fraction remains after 3 half-lives? 1/8.
How do I find the number of half-lives? Total time ÷ half-life — or count the halvings from start value to end value.
How do I read half-life from a graph? Halve the initial activity, read across to the curve and down to the time axis.
Does the activity ever reach zero? No — it halves each time, approaching zero.
What is activity measured in? Becquerels (Bq) — decays per second.
Why does activity decrease? Fewer undecayed nuclei remain.
Why is a short half-life used for medical tracers? So the source decays away quickly and doesn’t stay in the body.
Do I need the decay constant? No — not for 0625.
Quick revision checklist
- I can define half-life in both forms
- I know decay is random but predictable in bulk
- I halve repeatedly rather than dividing
- I can find the fraction remaining after n half-lives
- I can work backwards to find the half-life
- I know activity never reaches zero
- I can read half-life from a decay graph
- I check my reading with a second pair of points
- I subtract background count rate
- I know activity is measured in becquerels
- I can match half-life to an application
- I can explain carbon dating
- I know the decay constant is beyond 0625
These notes cover half-life and radioactive decay in the Cambridge IGCSE Physics (0625) syllabus and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Physics lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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