Radioactive Decay & Half-Life
Unstable atomic nuclei break apart at random, one by one, and there is no way to predict exactly when any single nucleus will decay. But across a large sample, the pattern is completely predictable: after one half-life, exactly half the original nuclei remain, and after each further half-life, half of what is left decays away.
The formula
N = N₀ · (½)^(t/t½)
- N — remaining amount (g): the amount of the substance still remaining at time t
- N₀ — initial amount (g): the amount present at the very start, when t equals zero
- t — elapsed time (s): how much time has passed since the start
- t½ — half-life (s): the time it takes for half of the sample to decay
Worked example
An 80 gram sample of a radioactive isotope has a half-life of 5 days. How much is left after 10 days?
- N₀ = 80 g
- t½ = 5 days
- N = N₀ · (½)^(t/t½)
- N = 80 g · (½)^(10/5) = 80 g · (½)²
N = 20 g
Test yourself
The same 80 gram sample keeps decaying. How much is left after 15 days?
- 40 g
- Correct answer: 10 g
- 26.7 g
Right! Three half-lives: 80 g → 40 g → 20 g → 10 g.
After 3 half-lives have passed, what fraction of the original sample is still left?
- Correct answer: 1/8
- 1/3
- 1/6
Correct! One half times one half times one half equals one eighth.
Where you see this
Archaeologists date ancient wood and bone by carbon-14, whose half-life of 5,730 years steadily drops its concentration after death — measuring what is left reads the calendar. Hospitals use the same decay law with short-lived tracers that glow through the body and fade within a day.
Common mistakes
The arithmetic error is treating each half-life as subtracting a fixed amount — it multiplies by one half, so three half-lives leave one eighth (½ × ½ × ½), never zero and never three-halves gone: an 80 g sample goes 80 → 40 → 20 → 10 g. The conceptual error is expecting single nuclei to obey the law — any one nucleus decays at a random, unknowable moment; only large samples become predictable.
How it connects
Decay is probability done with nuclei — the same law-of-large-numbers logic as thermodynamics' entropy — and it hands the next lesson its question: WHY are some nuclei unstable and others not? The answer, binding energy, also brings back relativity's c² to price the difference.