The big idea: Radioactive decay is random — you can't say when one nucleus will decay. But on average a sample gets weaker with time.
The half-life is the time it takes for half the radioactive nuclei to decay.
In that same time the activity and the count rate also halve.
New words, plainly: Activity = how many nuclei decay each second, measured in becquerel (Bq), where 1 Bq = 1 decay per second.
Count rate = how many of those decays a detector actually records each second (clicks per second).
Half-life = the time for the activity (or count rate) to fall to half.
Halving keeps going: Every half-life that passes, you halve again — not subtract a fixed amount.
So after 1, 2, 3 half-lives the count rate is 1/2, 1/4, 1/8 of the start. It flattens but never reaches zero.
At SL you only deal with whole numbers of half-lives, so you never need the exponential equation. You just halve, halve, halve, once for each half-life that has passed.
The rule (learn this — it is NOT in the data booklet): After n whole half-lives, the count rate (or activity) is the start value multiplied by (1/2)ⁿ.
Work out n = total time ÷ half-life, then halve that many times.
| Half-lives passed | Fraction of original left | Count rate from 80 s⁻¹ |
|---|---|---|
| 0 | 1 (all of it) | 80 s⁻¹ |
| 1 | 1/2 | 40 s⁻¹ |
| 2 | 1/4 | 20 s⁻¹ |
| 3 | 1/8 | 10 s⁻¹ |
| n | (1/2)ⁿ | 80 × (1/2)ⁿ |
Subtract the background FIRST: A detector always records some background radiation (from rocks, cosmic rays, etc.) even with no source.
The true count rate from the source = measured count rate − background count rate. Correct for background before you start halving.
Worked example — count rate after two half-lives
A fresh source gives a measured count rate of 204 counts per second. The background count rate is 4 counts per second. The source has a half-life of 6.0 hours. Find the MEASURED count rate 12 hours later.
Solution
- Correct for background first. True count rate from the source now:
- Find how many half-lives have passed: n = total time ÷ half-life.
- Halve the source count rate twice (once per half-life):
- Add the background back on — the detector still records it:
Final answer
The measured count rate after 12 hours is 54 counts per second.
Memorize terms 3x faster
Smart flashcards show you cards right before you forget them. Perfect for definitions and key concepts.
How this is tested: Half-life is a classic Paper 1A (multiple-choice) one-step calculation, and turns up in Paper 2 structured questions too.
- Paper 1A — halve over whole half-lives: given a start value and a half-life, find the count rate / activity after a whole number of half-lives. - Paper 1A — correct for background: subtract the background count before halving (and add it back if asked for the measured value). - Paper 1A — ratios: two samples with the same half-life keep the same ratio of activities as both halve together. - Paper 2: use the activity to work out an effect, such as an ionisation current.
Classic trap: forgetting the background, or subtracting a fixed amount each half-life instead of halving.
Ratios stay put: If two sources have the same half-life, after the same time both have halved the same number of times.
The ratio of their activities is unchanged — e.g. 5 : 1 stays 5 : 1.
IB-style question — (a) activity after three half-lives
A radioactive sample has an initial activity of 6.4 × 10⁶ Bq and a half-life of 8.0 days. Determine its activity after 24 days.
Solution
- Find the number of half-lives: n = total time ÷ half-life.
- Halve the activity three times (once per half-life):
- Tidy the answer with its unit (Bq = decays per second):
Final answer
After 24 days (3 half-lives) the activity is 8.0 × 10⁵ Bq — one-eighth of the start.
IB-style question — (b) ratio of two samples
Two samples X and Y have the SAME half-life. Right now sample X is four times as active as sample Y. State the ratio of their activities after two half-lives, and explain your answer.
Solution
- Both samples have the same half-life, so in the same time both halve the same number of times.
- After two half-lives each activity is multiplied by (1/2)² = 1/4 — the same factor for both.
- Multiplying both by the same factor leaves their ratio unchanged.
Final answer
The ratio stays 4 : 1. Equal half-lives means both fall by the same factor, so the ratio is unchanged.