Nuclear Decay & Half-Life Calculator

Radioactive Decay Guide

What should I know about 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.

What do I need to know about 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. Penetrating power is highest for gamma radiation, which requires thick lead or concrete shielding, compared to alpha (stopped by paper) or beta (stopped by a few millimetres of aluminium).

What's the key thing to understand about 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 C-14 levels historically — an assumption that isn't perfectly true, which is why calibration curves derived from tree rings are used to correct raw radiocarbon dates.

What should I know about 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: some fission products have very long half-lives, which is the central challenge in safely storing spent nuclear fuel for the thousands of years needed for radioactivity to decline to safe levels.