Radioactive Half Life
Radioactive decay describes how unstable atomic nuclei lose energy, a random process that can be precisely modelled for large populations using half-life. This topic is essential for interpreting decay graphs and performing multi-step calculations involving rates of change, a core skill for ESAT physics questions.
Part of the ESAT Physics syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The half-life of an isotope is constant and is not affected by physical conditions like temperature or pressure, nor by the initial amount of the substance.
- After 'n' half-lives, the fraction of remaining undecayed nuclei is (1/2)n, and the activity will have also reduced by this same fraction.
- A decay graph of activity vs. time is an exponential curve. The half-life is the time interval on the x-axis that corresponds to the activity on the y-axis falling to half its initial value.
- For a parent isotope X decaying into a stable daughter isotope Y, the percentage of X decreases exponentially while the percentage of Y increases, such that at any time, %X + %Y = 100%.
- Measurements of count rate must be corrected for background radiation before being used in half-life calculations. The source's true count rate is the measured rate minus the background rate.
Diagram
› Why does this happen?
Why is the decay graph a curve, not a straight line?
The rate of decay (the 'activity') depends on how many unstable nuclei are left to decay. At the start, there are lots of unstable nuclei, so the activity is high and many nuclei decay each second. This makes the graph very steep at the beginning. As the nuclei decay, there are fewer unstable ones left, so the rate of decay slows down. This makes the graph get shallower over time, creating the classic decay curve.
Why don't temperature and pressure affect half-life?
Radioactive decay happens right in the centre of an atom, in the nucleus. It's a 'nuclear' process, controlled by very strong forces inside that nucleus. Chemical reactions, on the other hand, only involve the electrons orbiting on the outside of the atom. Changing temperature and pressure affects how fast atoms move and how their outer electrons interact, which can speed up or slow down a chemical reaction. But these changes aren't nearly powerful enough to affect the nucleus at the centre. This means the chance of any single nucleus decaying stays the same, so the half-life does not change.
Formulae
Activityfinal = Activityinitial × (1/2)n To calculate the final activity (or number of nuclei) after 'n' half-lives have passed. Can be rearranged to find initial activity or n.
n = totaltime / Thalf To determine the number of half-lives ('n') that have occurred over a given total time, where Thalf is the half-life.
Definitions
- Half-life (T_half)
- The average time required for half of the unstable nuclei in a radioactive sample to undergo decay, or equivalently, the time it takes for the count rate or activity of the sample to decrease to half of its initial value.
Worked example
A sample of radioactive Iodine-131 has its activity measured. At the start (t=0), the count rate is 480 counts per second (cps). After 24 days, the rate is 50 cps. The experiment is conducted in a location with a stable background count of 20 cps. What is the half-life of Iodine-131 in days?
- 1
First, correct the measured count rates for background radiation to find the source's true activity.
Initial source activity = 480 - 20 = 460 cps.
Final source activity = 50 - 20 = 30 cps.
This step is incorrect, as the final count rate is what remains from the source plus background, so the final source activity is 50-20 = 30cps.
The initial rate is 480-20 = 460cps.
Wait, the prompt implies the total rate.
Let's re-evaluate.
Initial source activity = 480(total) - 20(bg) = 460 cps.
Final source activity = 50(total) - 20(bg) = 30cps.
This seems right.
Let's re-calculate to make the numbers easy for ESAT.
Let's change the numbers.
Start:
660 cps.
After 24 days:
60 cps.
Background:
20 cps.
- 2
Step 1:
Correct the initial and final measurements by subtracting the background count rate.
Initial source activity = 660 - 20 = 640 cps.
Final source activity = 60 - 20 = 40 cps.
- 3
Step 2:
Determine how many times the initial source activity has halved to reach the final activity.
We are going from 640 cps to 40 cps.
- 4
Step 3:
Count the halvings:
640 → 320 (1), 320 → 160 (2), 160 → 80 (3), 80 → 40 (4).
This took 4 half-lives.
- 5
Step 4:
The total time of 24 days is equivalent to 4 half-lives.
Therefore, one half-life is the total time divided by the number of half-lives:
24 days / 4 = 6 days
Answer: 6 days
Common mistakes
- ×Forgetting to subtract background radiation. All half-life calculations must be done using the count rate from the source alone, not the total measured rate.
- ×Confusing the direction of time. When calculating a past activity, you must double the current activity for each half-life you go back, not halve it.
- ×Misinterpreting decay product graphs. A curve rising towards a maximum represents the formation of the stable daughter product, while the decaying curve represents the original radioactive parent isotope.
- ×Making an 'off-by-one' error when counting half-lives. Explicitly write down the sequence of halvings (e.g., 100 → 50 → 25) to ensure you count the number of arrows, not the numbers in the list.
No-calculator tips
- ✓To find the number of half-lives between two values, start with the smaller number and double it repeatedly until you reach the larger one. The number of times you doubled is the number of half-lives.
- ✓Memorise the first few powers of 2 for fractions: 1 half-life = 1/2 left; 2 = 1/4; 3 = 1/8; 4 = 1/16; 5 = 1/32. This allows for rapid calculation of the remaining fraction.
- ✓If you are asked to find the time for a sample to decay to a specific fraction (e.g., 1/8th), you can immediately identify the number of half-lives (in this case, 3, since 23 = 8) and multiply by the half-life period.