Calculate radioactive decay using the half-life formula — find remaining quantity, percentage decayed, decay constant (λ) and mean lifetime (τ).
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Half-life (t½) is the time required for exactly half of a radioactive sample to decay. After one half-life, 50% remains. After two: 25%. After ten: 0.098%. Half-life is a fundamental constant of each isotope — unchanged by temperature, pressure, chemical environment or the amount present.
N(t) = N₀ × (1/2)^(t/t½). N(t) is the remaining quantity at time t, N₀ is the initial quantity and t½ is the half-life. The decay constant λ = ln(2)/t½ ≈ 0.693/t½. Mean lifetime τ = 1/λ = t½/ln(2) ≈ 1.443 × t½.
Carbon-14 (t½ = 5,730 years) is continuously produced in the atmosphere and absorbed by living organisms. After death, C-14 decays without replacement. By measuring the remaining C-14 ratio compared to modern standards, scientists can calculate how long ago an organism died — accurate up to roughly 50,000 years.
Half-lives range enormously: Polonium-214 has a half-life of 164 microseconds; Iodine-131 (medical imaging) = 8 days; Carbon-14 (archaeology) = 5,730 years; Uranium-235 (nuclear fuel) = 703 million years; Uranium-238 = 4.5 billion years (similar to Earth's age). This range reflects vast differences in nuclear stability.
Half-life (t½) is the time for 50% to decay. Mean lifetime (τ) is the average time a single atom survives before decaying. τ = t½ / ln(2) ≈ 1.443 × t½. After one mean lifetime, 1/e ≈ 36.8% of the material remains (not 50%). Mean lifetime is used in more advanced decay physics calculations.
