🔮 Half-Life 🔮
The time it takes for half the radioactive nuclei to decay — or for the activity to halve. Calculations, graphs, and the randomness of decay.
✦ Definition
Half-Life (t½)
The half-life of a radioactive isotope is the time taken for half the unstable nuclei in a sample to decay, OR the time for the count rate (activity) to halve.
Radioactive decay is a random process — you cannot predict when an individual nucleus will decay, but with large numbers of nuclei the overall pattern is predictable.
Key Points
- Each isotope has its own fixed half-life — it cannot be changed by temperature, pressure, or chemical reactions
- Half-lives range from fractions of a second to billions of years
- After each half-life, half the remaining unstable nuclei decay
- The activity (count rate) and number of undecayed nuclei both halve each half-life
✦ The Decay Curve
✦ Worked Example
How to Calculate
A radioactive isotope has a half-life of 10 minutes and starts with an activity of 800 counts per second. Find the time for activity to drop to 200 counts per second.
✦ Reading a Half-Life Graph
Method
- Pick any starting count rate on the y-axis
- Find half of that value
- Draw a horizontal line across to the curve, then straight down to the x-axis
- The difference in time = one half-life
- Repeat to check — you should get the same time each time
Don't Forget Background Radiation!
In real experiments, always subtract background radiation from your readings first. The corrected count rate = total count − background count.
✦ Summary: What Halves?
Draw a horizontal line from 240 Bq across to the curve, then draw a vertical line down to the x-axis [1].
The time reads 6 hours. Half-life = 6 hours [1].
Check: 480 → 240 at 6h, 240 → 120 at 12h (another 6h) ✓
Time = 2 × 10 = 20 minutes [1].
(a) Find the half-life of Strontium-90 [2]
(b) State the activity after 3 half-lives [1]
(b) After 3 half-lives: 1000 → 500 → 250 → 125 Bq [1].
(a) After how many hours will the activity of isotope X have fallen to 75 Bq? [2]
(b) After 45 years, what will the activity of isotope Y be? [1]
(b) 45 ÷ 15 = 3 half-lives. 600 → 300 → 150 → 75 Bq [1].
At t=0: 340 − 20 = 320 cpm (corrected) Half of 320 = 160 cpm. Total reading at half = 160 + 20 = 180 cpm [1].
Read from graph: 180 cpm occurs at t = 5 minutes [1].
Half-life = 5 minutes [1].
Check: Corrected 320 → 160 → 80 → 40 at t = 0, 5, 10, 15 min ✓
(a) What is the corrected count rate? [1]
(b) What will the corrected count rate be after 90 minutes? [2]
(b) 90 ÷ 30 = 3 half-lives. 240 → 120 → 60 → 30 counts per minute [1]. Total reading = 30 + 20 = 50 cpm [1].
(b) What is the activity after 24 days? [1]
(c) How many half-lives have passed after 24 days? [1]
(b) Read from graph at 24 days ≈ 50 Bq [1].
(c) 24 ÷ 8 = 3 half-lives [1]. Check: 400 → 200 → 100 → 50 ✓
(b) A radioactive source has an activity of 6400 Bq and a half-life of 3 days. Calculate the activity after 15 days. [2]
(b) 15 ÷ 3 = 5 half-lives [1]. 6400 → 3200 → 1600 → 800 → 400 → 200 Bq [1].
Check: 600 → 300 in the next 10 seconds (halved again) ✓ [1].
Half-life = 10 seconds [1].
At 40 seconds = 4 half-lives: 1200 → 600 → 300 → 150 → 75 Bq [1].
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