Radioactive Decay & Chain

Radioactive decay follows an exponential law: the probability per unit time that a given nucleus decays is constant, which leads to N(t) = N₀ e^{−λt} with decay constant λ related to the half-life by λ = (ln 2)/T₁/₂. This simulator uses arbitrary but consistent time units; you choose T₁/₂ directly and see the corresponding λ in the sidebar. In single-nuclide mode it plots the surviving fraction of parent atoms versus time—identical to integrating dN/dt = −λN with N(0) = N₀. Chain mode adds the textbook serial decay parent → daughter → stable product, modeled as dN₁/dt = −λ₁N₁, dN₂/dt = λ₁N₁ − λ₂N₂, dN₃/dt = λ₂N₂ with independent half-lives for parent and daughter. After each numerical step the three populations are renormalized so N₁ + N₂ + N₃ = 1, preserving the interpretation as fractions of the original atom budget. A fourth-order Runge–Kutta integrator keeps the curves smooth. A yellow marker shows the time of maximum daughter fraction, a classic visual for transient equilibrium and secular equilibrium discussions when λ_parent and λ_daughter differ. Activities reported as proportional to λN use the same normalization (relative units, not becquerels). The model omits branching ratios, competing channels, chemistry in the sample, detector efficiency, and statistical fluctuations—appropriate for a first ODE picture in general chemistry or introductory modern physics.