Radioactive Decay, Half-Life (t½

Nuclear Physics: Radioactive Decay Kinetics & Half-Life

Radioactive decay is a fundamental quantum mechanical Poisson process where unstable atomic nuclei spontaneously transform by emitting ionizing radiation (alpha particles, beta particles, or gamma rays):

• First-Order Exponential Decay: The decay rate) is strictly proportional to the number of radioactive atoms present.

• Half-Life): The time required for exactly one-half of the parent nuclei to undergo radioactive transmutation into daughter nuclei. Half-life is invariant—completely unaffected by temperature, pressure, gravity, or chemical bonding.

• The 10-Half-Life Decommissioning Rule: In nuclear medicine, hazardous radiation waste is typically safely disposed of after 10 half-lives), when residual activity has dropped below 0.1% of its original baseline.

How to Use This Calculator

Enter your Radionuclide / Isotope Preset — choose from Iodine-131 (t½ = 8.02 Days — Thyroid Radiotherapy), Technetium-99m (t½ = 6.01 Hours — Medical SPECT Imaging), Carbon-14 (t½ = 5,730 Years — Radiocarbon Dating), Initial Amount / Activity (N₀ in grams, MBq, or mCi) (e.g. 100), Elapsed Decay Time (Duration t) (e.g. 16.04), Elapsed Time Unit — choose from Hours, Days, Years, Custom Half-Life (t½ [if custom]) (e.g. 8.02), and Custom Half-Life Unit (if custom) — choose from Hours, Days, Years. Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Radioactive Decay Law (Ernest Rutherford, 1902) equations:

N(t) = N_0 · ((1) / (2))^(t) / (t_1/2) = N_0 · e^-λ t

Decay Constant: λ = ((2)) / (t_1/2) ≈ (0.69315) / (t_1/2)

Elapsed Half-Lives: n = (t) / (t_1/2)

Decayed Fraction: Δ N = N_0 - N(t)

Mean Atomic Lifetime: τ = (1) / (λ) = t_1/2(2) ≈ 1.4427 · t_1/2

Worked Example

For example, evaluating a sample of Iodine-131) with initial radioactivity \ after Exactly \ have passed. Residual activity is N(t) = 100 × (0.5)^2 = 25.00 MBq (25.00\% remaining, 75.00\% decayed). The decay constant is with a mean atomic lifetime .

Understanding Your Result

Evaluate your result in the context of your specific use case and relevant industry standards or reference values. A single calculated result is a starting point — test two to three scenarios to understand how sensitive the outcome is to your key input assumptions. For high-stakes decisions, verify your result against a secondary source or subject-matter expert before acting.