Nuclear Decay & Half-Life Calculator

Radioactive Decay Guide

Radioactive Decay Law

N = N₀ × e^(−λt) = N₀ × (1/2)^(t/t½). Where N = quantity remaining, N₀ = initial quantity, λ = decay constant, t = time elapsed, t½ = half-life. The decay constant λ = ln(2)/t½ = 0.693/t½. After one half-life: 50% remains. After two: 25%. After three: 12.5%. After ten half-lives: 0.098% remains (essentially complete decay for practical purposes). The relationship between half-life and decay constant: t½ = 0.693/λ.

Types of Radioactive Decay

Alpha decay (α): emits ⁴₂He nucleus. Mass number decreases by 4, atomic number by 2. Penetrating power: low (stopped by paper or skin). Example: ²³⁸U → ²³⁴Th + ⁴He. Beta minus decay (β⁻): neutron → proton + electron + antineutrino. Mass number unchanged, atomic number increases by 1. Penetrating power: medium (stopped by aluminium). Example: ¹⁴C → ¹⁴N + e⁻. Gamma decay (γ): electromagnetic radiation released when nucleus transitions to lower energy state. No change in mass or atomic number. Pene

Carbon-14 Dating

Carbon-14 (t½ = 5,730 years) forms in the atmosphere from cosmic ray interactions with nitrogen. Living organisms maintain a constant ¹⁴C/¹²C ratio by exchanging carbon with the atmosphere. At death, ¹⁴C decays without replacement. Measuring the remaining fraction: t = −t½/ln(2) × ln(N/N₀). This method accurately dates organic material up to approximately 50,000 years (beyond that, too little ¹⁴C remains to measure accurately). Important assumption: constant atmospheric ¹⁴C levels historically —

Medical and Industrial Applications

Medical imaging: technetium-99m (t½ = 6 hours) is used in 80% of nuclear medicine procedures — short half-life minimises patient radiation dose while providing clear images. Cancer treatment: iodine-131 (t½ = 8 days) targets thyroid tissue. Cobalt-60 (t½ = 5.27 years) in external beam radiotherapy. Smoke detectors: americium-241 (t½ = 432 years) provides a stable alpha source. Food irradiation: gamma rays from cobalt-60 kill pathogens without radioactive contamination of the food. Nuclear waste: