Radioactive Decay Chain Calculator
Simulate sequential radioactive decay chains.
Calculate activity and remaining quantity of parent and daughter nuclides over time using Bateman equations.
What Is a Radioactive Decay Chain? Most heavy radioactive nuclei do not decay directly to a stable product. Instead they decay through a sequence of daughter nuclides — each radioactive — until a stable element is reached. The uranium-238 decay chain contains 14 steps before reaching stable lead-206. The thorium-232 chain has 10 steps; the uranium-235 chain has 11 steps. These chains produce naturally occurring radioactive materials (NORM) in soil, groundwater, and building materials.
The Bateman Equations For a chain A → B → C (stable): N_A(t) = N_A(0) × e^(−λ_A × t) N_B(t) = N_A(0) × λ_A/(λ_B − λ_A) × (e^(−λ_A×t) − e^(−λ_B×t)) + N_B(0) × e^(−λ_B×t) Where λ = ln(2)/t½ is the decay constant for each nuclide. Activity A = λ × N (disintegrations per second, in Becquerels).
Secular Equilibrium When the parent half-life is much longer than the daughter (t½_parent » t½_daughter), secular equilibrium is reached. At secular equilibrium: A_parent = A_daughter (activities are equal). Example: Ra-226 (1600 yr) → Rn-222 (3.8 days). After ~38 days, Rn reaches secular equilibrium. This is why uranium ore in equilibrium has equal activities of all 14 decay products.
Transient Equilibrium When the parent half-life is only somewhat longer (10–100×), transient equilibrium is reached. The daughter/parent activity ratio becomes constant but not equal. Example: Mo-99 (66 hr) → Tc-99m (6 hr) — the workhorse of nuclear medicine imaging. Tc-99m generators exploit this: Mo-99 “grows in” Tc-99m, which is eluted every 24 hours.
Famous Decay Chains Uranium-238 → (14 steps) → Lead-206. Half-lives range from 4.47 billion years to 164 microseconds. Radium-226 in this chain produces radon-222 — a radioactive gas, the second leading cause of lung cancer after smoking. Thorium-232 → Lead-208 chain produces Ra-228 and Ac-228 found in Brazil nuts (natural thorium uptake). Nuclear power: fission products like I-131 (8 days) → Xe-131 (stable) are key in reactor safety.
Units of Radioactivity Becquerel (Bq): 1 disintegration per second. SI unit since 1975. Curie (Ci): 3.7 × 10¹⁰ Bq. Originally defined as the activity of 1 gram of radium-226. 1 mCi = 37 MBq. 1 µCi = 37 kBq.
Ingrowth: Growing Daughters from Pure Parent Starting with only parent atoms (N_B(0) = 0), the daughter activity grows. At time t = 0 there is no daughter; activity builds up exponentially toward equilibrium. The daughter peaks at t_max = ln(λ_B/λ_A)/(λ_B − λ_A).