Radioactive Decay Calculator

At a glance:Radioactive decay is the process by which an unstable atomic nucleus spontaneously emits particles or radiation (α particles, β particles, γ rays) and transforms into another nuclide. It is a 'random' but 'statistically highly regular' process — we cannot predict when a single nucleus will decay, but for a large ensemble the decay follows a definite exponential law. The most common measure of decay speed is the half-life (T½): the time required for half of a given quantity of radioactive nuclei to decay. The half-life is an intrinsic property of each radionuclide, independent of temperature, pressure, and chemical state, and ranges from less than a second to billions of years. From the definition of the half-life we obtain the decay formula: remaining amount N = N₀·(1/2)^(t/T½), where N₀ is the initial amount (atom count, mass, or activity), t is the elapsed time, and T½ is the half-life (t and T½ must share the same time unit). The core logic: after each half-life the remaining amount is multiplied by 1/2; after n half-lives (n=t/T½) the remaining amount is N₀ times (1/2)ⁿ. Using the defaults: N₀=1000, T½=5 (years), t=10 years → n=10/5=2 half-lives → N=1000×(1/2)²=250. The formula can also be written with the natural exponential: N=N₀·e^(−λt), where λ=ln2/T½ is the decay constant; the two forms are exactly equivalent. Applications: (1) radiometric dating — using nuclides of known half-life to date samples, e.g. carbon-14 dating (T½≈5730 yr) for archaeological and biological remains, uranium–lead dating for rocks and Earth's age; (2) nuclear medicine — radioactive isotopes (iodine-131, technetium-99m) for diagnosis and therapy, where the half-life controls dose and timing; (3) nuclear energy and waste management — assessing decay and storage safety periods; (4) tracer technology in industry, agriculture, and biology. Cautions: t and T½ must use the same time unit; the formula describes the statistical behavior of a large ensemble (small amounts show statistical fluctuations); units of N₀ and N can be atom count, mass, or activity as long as they are consistent; decay is irreversible and N monotonically decreases toward (but never reaches) zero; this calculator requires non-negative initial amount and time and a half-life greater than 0.