Core answer: After n half-lives, the remaining amount = initial amount × (1/2)ⁿ. Formula: N = N₀ × (1/2)^(t/T), where N₀ is the initial amount, t elapsed time, T the half-life. After 7 half-lives less than 1% remains; after 10, under 0.1% — conventionally "eliminated".
The half-life formula
N = N₀ × (1/2)^(t/T)
Given any three quantities, solve for the fourth:
- Remaining amount: substitute directly
- Time: t = T × log₂(N₀/N)
- Half-life: T = t ÷ log₂(N₀/N)
Decay table (0–10 half-lives)
| Half-lives elapsed | Remaining | Intuition |
|---|---|---|
| 1 | 50% | Halved |
| 2 | 25% | A quarter |
| 3 | 12.5% | An eighth |
| 4 | 6.25% | A sixteenth |
| 5 | 3.125% | ~3% |
| 6 | 1.56% | |
| 7 | 0.78% | <1% |
| 8 | 0.39% | |
| 9 | 0.20% | |
| 10 | 0.098% | <0.1%, treated as eliminated |
Key insight: decay is exponential, not linear — each half-life removes half of *what is currently left*, so the amount approaches zero but never exactly reaches it.
Example 1: drug metabolism
Example: a drug with a 6-hour half-life, one 400 mg dose, minimum effective blood level 50 mg.
- After 6h: 200 mg → 12h: 100 mg → 18h: 50 mg → 24h: 25 mg
- Below the effective threshold after 18 hours — hence prescriptions of 2–3 doses daily
- From 400 mg to "essentially cleared" (<1%, 4 mg) takes ~6.6 half-lives ≈ 40 hours
This also explains "why some drugs must be stopped 5–7 half-lives before surgery".
Example 2: carbon-14 dating
Carbon-14's half-life is 5,730 years. Living organisms keep C-14 in equilibrium with the environment; after death it only decays.
Example: unearthed bone contains 12.5% of the modern C-14 level:
12.5% = (1/2)³ → 3 half-lives elapsed → age ≈ 3 × 5,730 = 17,190 years
The method's ceiling is ~50,000 years (beyond that, <0.2% remains and errors explode); older samples switch to potassium-argon dating (half-life 1.25 billion years).
Half-life quick reference
| Substance / context | Half-life |
|---|---|
| Caffeine (human body) | ~5 hours |
| Ibuprofen (human body) | ~2 hours |
| Iodine-131 (nuclear medicine) | 8 days |
| Cobalt-60 (radiotherapy) | 5.27 years |
| Carbon-14 (dating) | 5,730 years |
| Uranium-238 | 4.47 billion years |
| Polonium-214 | 164 microseconds |
Half-life vs mean lifetime
- Half-life: time for the quantity to halve
- Mean lifetime: average survival time of a particle = half-life ÷ ln2 ≈ half-life × 1.443
Example: iodine-131 with an 8-day half-life has a mean lifetime of ~11.5 days. Medical dose conversions use this relationship.
Common mistakes and myths
- "Two half-lives and it is gone" — no; each half-life removes half of the *current* amount. Mathematically there is always something left, just approaching zero.
- "Half-life means losing half per unit time" — exponential decay is not "minus a fixed X per hour" but "minus a fixed *percentage* of the current value per hour".
- "Radioactive waste just needs time, with no end" — engineering practice uses 10 half-lives (0.1% left) as a disposal benchmark, but long-lived nuclides (plutonium-239, 24,000 years) demand geological-scale containment.
- "An afternoon coffee is gone by night" — with a 5-hour half-life, a 3 PM cup (200 mg) leaves ~75–100 mg in your system at 10 PM; sensitive people will still lose sleep.
Use the [Half-Life Calculator](/c/science/half-life) to get remaining amounts and the decay curve from initial amount, half-life, and time, and the [Exponent Calculator](/c/math/exponent) to check (1/2)ⁿ by hand.