The Mathematics of Exponential Radioactive Decay
Radioactive decay is a first-order stochastic process in which unstable atomic nuclei lose energy by emitting ionizing radiation. Because the probability of decay per unit time is constant for each specific nuclide, macroscopic decay follows an exact exponential decay curve: N(t) = N₀ (½)t/t½ = N₀ e-λt.
Radioactive Decay Governing Equations
1. Remaining Quantity N(t)
N(t) = N₀ · (½)t / t½
Exponential half-life power
2. Decay Constant (λ)
λ =
ln(2)t½
λ ≈ 0.69315 ÷ t½
3. Elapsed Time (t)
t = t½ · log₂(
N₀N
)Logarithmic age solver
Half-Life Step Sequence Table
1 t½ = 50.0%|2 t½ = 25.0%|3 t½ = 12.5%|4 t½ = 6.25%|5 t½ = 3.125%
Step-by-Step Calculation Breakdown
Example 1: Carbon-14 Dating of an Ancient Wooden Artifact (2,000 Years Elapsed)
1. Parameters: Initial quantity N₀ = 100.00%, Half-life t½ = 5,730 years, Elapsed t = 2,000 years
2. Calculate half-lives elapsed: n = 2,000 ÷ 5,730 = 0.3490 half-lives
3. Compute remaining: N(2000) = 100 × (0.5)0.3490 = 78.51% (21.49% decayed)
Example 2: Iodine-131 Medical Dose Decay (8.02 Days Half-Life, 24.06 Days Elapsed)
1. Parameters: Initial dose N₀ = 50.00 mCi, Half-life t½ = 8.02 days, Time t = 24.06 days
2. Calculate half-lives: n = 24.06 ÷ 8.02 = 3.00 exact half-lives
3. Compute residual activity: 50.00 × (0.5)³ = 50.00 × 0.125 = 6.25 mCi (87.5% eliminated)
Key Radioactive Nuclides & Clinical/Scientific Roles
| Isotope Name | Half-Life Duration | Scientific / Clinical Application |
|---|---|---|
| Carbon-14 (C-14) | 5,730 years | Archaeological organic dating |
| Uranium-238 (U-238) | 4.468 billion years | Geological Earth age dating & nuclear fuel |
| Plutonium-239 (Pu-239) | 24,110 years | Nuclear energy & space RTG batteries |
| Radium-226 (Ra-226) | 1,600 years | Historical luminous paints & radiochemistry |
| Cesium-137 (Cs-137) | 30.17 years | Industrial radiography & medical calibration |
| Strontium-90 (Sr-90) | 28.8 years | Industrial thickness gauges & RTGs |
| Tritium (Hydrogen-3) | 12.32 years | Self-powered emergency exit lighting & fusion |
| Cobalt-60 (Co-60) | 5.27 years | Medical cancer radiation therapy & sterilization |
| Polonium-210 (Po-210) | 138.4 days | Static eliminators & alpha emitter research |
| Iodine-131 (I-131) | 8.02 days | Thyroid medical diagnostic and ablation therapy |
| Technetium-99m (Tc-99m) | 6.01 hours | Hospital diagnostic SPECT organ imaging |
| Fluorine-18 (F-18) | 109.8 minutes | Oncology PET scan metabolic tracer (FDG) |