Half-lives and the random nature of radioactive decay
A radioactive sample contains many unstable nuclei. Each nucleus has a fixed probability of decaying per second, but you can't say which one will decay next. Over many nuclei the behaviour is statistical and predictable.
📖Definition— Definition of half-life
The half-life ($T_{1/2}$) is the time taken for half the nuclei in a sample to decay. Equivalently, the time for the count rate to fall to half.
After 1 half-life: 1/2 remain. After 2 half-lives: 1/4 remain. After 3 half-lives: 1/8 remain. After $n$ half-lives: $(1/2)^n$ remain.
The half-life is fixed for a given isotope and doesn't depend on temperature, chemistry, age or sample size.
Reading a half-life from a graph
Plot count rate (or remaining nuclei) vs time. Find where the curve drops to half its initial value: read off the time. That's $T_{1/2}$. Also halves again after 2$T_{1/2}$, etc.
✦Worked example— Worked example 1 — fraction remaining
A radioactive isotope has a half-life of 5 years. What fraction remains after 20 years?
- 20/5 = 4 half-lives.
- Fraction remaining = (1/2)⁴ = 1/16.
✦Worked example— Worked example 2 — finding count rate
Initial activity 800 counts/min. Half-life 3 hours. What's the activity after 12 hours?
- 12/3 = 4 half-lives.
- Activity = 800 × (1/2)⁴ = 800/16 = 50 counts/min.
"Net decline" calculations
Sometimes asked: by what fraction does activity decline in $n$ half-lives?
- After 3 half-lives: 1/8 remain → 7/8 has decayed → 87.5% decline.
Activity and the count rate
The activity is the number of decays per second (becquerel, Bq). A Geiger-Müller (GM) tube measures count rate (counts per second), which is proportional to activity but reduced by the detector's geometry and efficiency.
We always subtract the background count (measured separately) from the GM reading before calculating fractions.
⚠Common mistakes
- Computing $1/n$ instead of $(1/2)^n$ for $n$ half-lives.
- Forgetting to subtract background count when reading from a graph.
- Saying half-life is the time to fully decay — it's the time to halve.
- Treating half-life as variable with mass — it's an isotope property only.
➜Try this— Quick check
A sample drops from 1024 Bq to 32 Bq. How many half-lives passed?
- 1024/32 = 32 = 2⁵ → 5 half-lives.
AI-generated · claude-opus-4-7 · v3-deep-physics