Radioactive Decay and Half-Life
The half-life is the time for half of a radioactive sample to decay. It is constant regardless of sample size — a fundamental property of each isotope.
Decay Formula
N(t) = N₀ × (½)^(t/t½) = N₀ × e^(-λt)
N₀ = initial quantity
t = elapsed time
t½ = half-life
λ = decay constant = ln(2)/t½ ≈ 0.693/t½Worked Examples
Iodine-131 (t½=8.02 days): 1000 mg initial
After 24 days (3 half-lives):
N = 1000 × (½)³ = 125 mg
After 40 days:
N = 1000 × e^(-0.693/8.02 × 40) = 31.3 mgCommon Half-Lives
- Carbon-14: 5730 years (radiocarbon dating)
- Uranium-238: 4.47 billion years
- Iodine-131: 8.02 days (medical imaging)
- Radon-222: 3.82 days
- Strontium-90: 28.8 years
- Potassium-40: 1.25 billion years
Calculate radioactive decay: Free Half-Life Calculator
Radioactive Half-Life Quick-Reference Table
| Isotope | Half-life | Decay type | Application |
|---|---|---|---|
| Carbon-14 (¹⁴C) | 5,730 years | β⁻ | Archaeological dating |
| Uranium-238 (²³⁸U) | 4.47 billion years | α | Geological dating |
| Iodine-131 (¹³¹I) | 8.02 days | β⁻, γ | Thyroid cancer treatment |
| Technetium-99m (⁹⁹ᵐTc) | 6.01 hours | γ | Medical imaging (SPECT) |
| Plutonium-239 (²³⁹Pu) | 24,100 years | α | Nuclear weapons/reactors |
| Radon-222 (²²²Rn) | 3.82 days | α | Indoor air quality concern |
How Half-Life Works
Half-life (t₁/₂) is the time for half of a radioactive sample to decay. After n half-lives, the fraction remaining is (½)ⁿ. The governing equation is N(t) = N₀ × e^(−λt), where λ = ln(2)/t₁/₂ is the decay constant. The relationship between amount remaining and time is exponential — no matter the sample size, always exactly half decays each half-life period.
Radiocarbon dating uses the known half-life of ¹⁴C (5,730 years) and the known initial ¹⁴C/¹²C ratio in living organisms to date organic material up to ~50,000 years old. Medical isotopes like ⁹⁹ᵐTc are chosen for short half-lives (patient exposure is brief) combined with emissions suited to gamma cameras. Nuclear waste management requires safe storage for many half-lives until activity falls to safe levels.
Common Mistakes
- Linear thinking: After 2 half-lives, 25% (not 0%) remains. After 10 half-lives, ~0.1% remains. The decay is exponential — it never reaches exactly zero.
- Confusing activity with amount: Activity (becquerels = decays per second) = λ × N. As N decreases, activity decreases proportionally. Cutting the number of atoms in half also halves the activity.
- Using wrong units for λ and t: If t₁/₂ is in years, λ = ln(2)/t₁/₂ is in yr⁻¹; use t in years in the decay equation. Mixing units (days vs. hours) produces wrong answers.
Frequently Asked Questions
Living organisms continuously exchange carbon with the atmosphere, maintaining a constant ¹⁴C/¹²C ratio of ~1.2×10⁻¹². At death, uptake stops and ¹⁴C decays. Measuring the remaining ratio tells how many half-lives have passed: t = −t₁/₂ × log₂(N/N₀). A sample with half the original ¹⁴C is ~5,730 years old. Calibration curves (tree rings, corals) correct for past variations in atmospheric ¹⁴C.
Fission produces a range of products. Some short-lived isotopes (days to years) decay quickly. But actinides like Pu-239 (t₁/₂ = 24,100 yr) and Np-237 (t₁/₂ = 2.14 million yr) persist for geological timescales. Safe storage requires the waste to remain isolated for roughly 10 half-lives — up to 200,000+ years for some isotopes — which is why deep geological repositories are proposed.
Biological half-life accounts for both radioactive decay and physiological elimination. The effective half-life t_eff = (t_physical × t_biological) / (t_physical + t_biological). ¹³¹I has a physical t₁/₂ of 8 days but is eliminated by the kidneys with a biological half-life of ~80 days, giving t_eff ≈ 7.3 days — the actual time for radiation dose to halve in the body.