Exponential Decay
Half-life is the time for a quantity to fall to half its value. The defining property of exponential decay is that this time is constant — it takes as long to go from 100 to 50 as from 50 to 25:
N(t) = N₀ × (½)t / t½
Decay Chart
| Half-lives elapsed | Fraction remaining | Percentage |
|---|---|---|
| 0 | 1 | 100% |
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 4 | 1/16 | 6.25% |
| 5 | 1/32 | 3.125% |
| 7 | 1/128 | 0.78% |
| 10 | 1/1024 | 0.098% |
The rule of thumb in pharmacology and radiation safety: after five half-lives roughly 97% is gone, and after ten, over 99.9%. Nothing ever reaches exactly zero, which is why decay is described by thresholds rather than end points.
Half-Life, Decay Constant and Mean Lifetime
Three ways to express the same rate:
- Half-life t½ — time to fall by half.
- Decay constant λ = ln(2) / t½ — the fractional loss per unit time.
- Mean lifetime τ = 1/λ = t½ / ln(2) — the average survival time of an individual atom, about 1.44 half-lives.
Common Half-Lives
| Substance | Half-life | Used for |
|---|---|---|
| Carbon-14 | 5,730 years | Dating organic material up to ~50,000 years |
| Uranium-238 | 4.5 billion years | Dating rocks and the age of the Earth |
| Caesium-137 | 30 years | Fallout contamination |
| Iodine-131 | 8 days | Thyroid treatment and imaging |
| Technetium-99m | 6 hours | Medical imaging |
| Caffeine | about 5 hours | Why an afternoon coffee affects sleep |
| Ibuprofen | about 2 hours | Dosing intervals |
Radiocarbon Dating
Living things maintain a constant ratio of carbon-14 to carbon-12 by exchange with the atmosphere. When they die, exchange stops and the C-14 decays. Measuring the remaining ratio gives the time since death. Beyond about ten half-lives — 50,000 years — too little remains to measure reliably, which is why older material requires potassium-argon or uranium-series dating instead.
Frequently Asked Questions
Does anything ever fully decay?
Mathematically no — the curve approaches zero without reaching it. Physically yes, once the last atom decays, which is why the model breaks down at very small quantities.
Can half-life be changed?
Not by ordinary means. Temperature, pressure and chemistry leave nuclear decay essentially untouched. This constancy is exactly what makes radiometric dating reliable.
Why is biological half-life different from radioactive half-life?
Because the body also excretes the substance. The effective half-life combines both processes and is always shorter than either alone.