Half-Life & Radioactive Decay Calculator
Calculate radioactive decay: remaining quantity after time, elapsed time from remaining amount, or half-life from decay data. Includes carbon dating applications.
Half-Life & Radioactive Decay Guide
What Half-Life Means
Half-life is the time for half of a radioactive substance to decay. After one half-life: 50% remains. After two half-lives: 25% remains. After three: 12.5%. After n half-lives: (1/2)^n remains. The decay is exponential, not linear. Formula: N(t) = N₀ × (1/2)^(t/t½). Where: N(t) = quantity at time t. N₀ = initial quantity. t½ = half-life. Alternative form: N(t) = N₀ × e^(-λt). Where λ (decay constant) = ln(2) / t½ = 0.693 / t½. The two forms are equivalent. Why exponential: each atom has a fixed
Carbon Dating
Carbon-14 (radiocarbon) dating uses the known half-life of C-14 (5,730 years) to date organic materials. How it works: living organisms constantly exchange carbon with the atmosphere, maintaining a constant ratio of C-14 to stable C-12. When an organism dies, it stops taking in carbon. The C-14 decays (half-life 5,730 years). C-12 remains stable. By measuring the remaining C-14 ratio, you calculate how long ago the organism died. Calculation: age = t½ × log₂(N₀/N) = 5,730 × log₂(100/percent_rema
Medical and Industrial Uses
Medical isotopes (chosen for specific half-lives): Technetium-99m: 6 hours. Most-used medical imaging isotope. Short half-life means rapid clearance from body. Iodine-131: 8 days. Thyroid treatment and imaging. Fluorine-18: 110 minutes. PET scans. Made on-site (decays too fast to transport far). Cobalt-60: 5.27 years. Cancer radiotherapy, sterilisation. Why half-life matters medically: short half-life: less radiation dose to patient (decays quickly), but must be used soon after production. Long
Decay Constant and Mean Lifetime
Three related quantities: half-life (t½): time for half to decay. Most intuitive. Decay constant (λ): probability of decay per unit time. λ = ln(2)/t½ = 0.693/t½. Used in the exponential decay equation. Mean lifetime (τ): average lifetime of an atom. τ = 1/λ = t½/ln(2) = 1.443 × t½. Always longer than half-life. Relationships: τ = t½ / 0.693 = 1.443 × t½. λ = 1/τ = 0.693/t½. Activity (decay rate): A = λN. Measured in Becquerels (Bq = 1 decay/second) or Curies (Ci = 3.7×10¹⁰ Bq). Activity also de
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