Radioactive Decay Formula (Half-Life)
The radioactive decay formula shows how unstable atoms disintegrate over time.
Calculate remaining quantity using half-life or decay constant.
The Formula
This formula describes how a radioactive substance decreases over time. Every half-life period, exactly half of the remaining atoms decay into a different element or isotope.
Equivalent Form (Decay Constant)
Both forms are equivalent. The decay constant λ and the half-life t½ are related by:
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| N(t) | Remaining quantity at time t | atoms, grams, or Bq |
| N₀ | Initial quantity (at t = 0) | same as N(t) |
| t | Elapsed time | seconds, years, etc. |
| t½ | Half-life (time for half to decay) | same unit as t |
| λ | Decay constant (rate of decay) | per unit time |
| e | Euler's number ≈ 2.71828 | dimensionless |
Example 1 — Carbon-14 Dating
Carbon-14 has a half-life of 5,730 years. A sample starts with 200 grams. How much remains after 11,460 years?
Number of half-lives = 11,460 / 5,730 = 2
N(t) = 200 × (½)² = 200 × 0.25
N(t) = 50 grams remaining
Example 2 — Iodine-131 Medical Use
Iodine-131 (used in thyroid treatment) has a half-life of 8 days. A patient receives 400 MBq. What activity remains after 24 days?
Number of half-lives = 24 / 8 = 3
N(t) = 400 × (½)³ = 400 × 0.125
N(t) = 50 MBq remaining
Reference — Common Half-Lives
| Isotope | Half-Life | Use |
|---|---|---|
| Carbon-14 | 5,730 years | Archaeological dating |
| Uranium-238 | 4.5 billion years | Geological dating |
| Iodine-131 | 8 days | Medical thyroid treatment |
| Technetium-99m | 6 hours | Medical imaging |
| Radon-222 | 3.8 days | Indoor air hazard |
| Cobalt-60 | 5.27 years | Cancer radiotherapy |
When to Use It
Use the radioactive decay formula when:
- Calculating the age of archaeological or geological samples (carbon dating, uranium-lead dating)
- Determining safe waiting times before handling radioactive medical waste
- Estimating radiation dose from a known radioactive source over time
- Understanding nuclear reactor fuel consumption and waste
- Planning radiation safety protocols in hospitals and laboratories