4.4.2.3a - Half-life and Random Decay
Radioactive decay is random: you cannot predict exactly when one unstable nucleus will decay. In a large sample, though, the overall pattern is predictable enough to measure. Half-life describes that pattern by asking how long it takes for the number of undecayed nuclei, count rate, or activity to halve.
Random Decay
An unstable nucleus may decay at any moment. It is not possible to look at one nucleus and say exactly when it will decay, even if you know the isotope and have measured many similar samples before.
Random radioactive decay
Radioactive decay is random because the exact time at which a particular unstable nucleus decays cannot be predicted.
This does not mean a radioactive sample behaves with no pattern at all. A sample usually contains a very large number of unstable nuclei, so the random decays combine to make a smooth overall trend. The count rate from the sample may wobble slightly from one reading to the next, but the average count rate falls in a predictable way.
Use two ideas together:
- single nucleus: unpredictable
- large sample: statistically predictable
The key exam phrase is not simply "decay is random"; it is "the decay of an individual nucleus is random". Half-life belongs to the behaviour of many nuclei in a sample, or to the count rate/activity measured from that sample.
What Half-life Means
Half-life is a time, not an amount. It tells you how long a radioactive isotope takes to halve according to one of three linked measurements.
Half-life
The half-life of a radioactive isotope is the time taken for the number of radioactive nuclei in a sample to halve, or for the count rate or activity from the sample to fall to half its initial value.
Activity is the number of nuclear decays per second. Its unit is the becquerel, Bq. Count rate is the reading from a detector, such as counts per second or counts per minute. In GCSE questions, if the setup is fixed and the count rate has been corrected where needed, count rate can be used to find half-life.
Suppose a sample has a corrected count rate of 80 counts/s and a half-life of 4 minutes. After 4 minutes, the count rate is 40 counts/s. After another 4 minutes, it is 20 counts/s. The same half-life halves what is left; it does not subtract the same number of counts each time.
Each half-life halves the remaining nuclei, count rate, or activity.
The wording "half its initial value" is often used for the first half-life, such as 80 counts/s falling to 40 counts/s. After that, the new value becomes the starting point for the next halving, such as 40 counts/s falling to 20 counts/s over the next half-life.
Determining Half-life From Tables
When a table gives count rate, activity, or number of nuclei at different times, look for two values where one is half the other. The half-life is the time difference between those two readings.
For example:
| Time in minutes | Corrected count rate in counts/s |
|---|---|
| 0 | 96 |
| 3 | 68 |
| 6 | 48 |
| 9 | 34 |
| 12 | 24 |
The count rate falls from 96 counts/s to 48 counts/s between 0 minutes and 6 minutes.
So the half-life is:
6 min - 0 min = 6 min
There is a second check in the same data: 48 counts/s falls to 24 counts/s between 6 minutes and 12 minutes. That is also 6 minutes, so the table is consistent.
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