Radioactive Decay Law
The radioactive decay law is the first-order differential equation dN/dt = −λN, and its exponential solution N(t) = N₀e⁻λt, describing the rate at which a large population of identical radioactive nuclei decays — derived directly from the assumption that each individual nucleus has a constant, fixed probability of decaying per unit time, independent of the nucleus's age.
Curious
The Codex's radioactive decay entry introduces the exponential decay formula N(t) = N₀e⁻λt directly; this entry focuses on where that formula actually comes from — the underlying differential equation and the reasoning that leads to its exponential solution — plus a few additional related concepts, like activity and decay chains, that build on it.
The key starting assumption is remarkably simple: each individual radioactive nucleus, at any given moment, has the same fixed probability λ of decaying within the next small time interval, completely independent of how long it has already existed. This single assumption, worked through mathematically, is enough to derive the entire familiar exponential decay curve.
Exploring
| Symbol | Name | SI Unit |
|---|---|---|
| dN/dt | Rate of change of nuclei count | s⁻¹ |
| A(t) | Activity — decay rate at time t | becquerel (Bq), 1 Bq = 1 decay/s |
Deep Dive
The differential equation dN/dt = −λN follows directly from treating radioactive decay as a first-order random process: in a population of N nuclei, the expected number decaying in a small time interval dt is proportional both to N (more nuclei present means more expected decays) and to dt (a longer interval means more expected decays), with λ as the constant of proportionality representing each nucleus's fixed per-unit-time decay probability. Separating variables and integrating this equation directly yields the exponential solution N(t) = N₀e⁻λt.
Decay chains and the Bateman equations: Many practically important radioactive processes involve chains of sequential decays — a parent nucleus decays into a daughter that is itself radioactive, which decays into a granddaughter, and so on, until a stable nuclide is finally reached. The mathematics describing the time-dependent abundance of each species in such a chain (the Bateman equations, 1910) generalise the simple single-species decay law into a system of coupled linear differential equations, with solutions that can display complex, non-monotonic behaviour — a daughter species' abundance, for instance, can rise before eventually falling, as it is simultaneously being produced by the parent's decay and consumed by its own decay.
The radioactive decay law's statistical, first-order character connects it to a broader class of physical and mathematical phenomena obeying the same underlying mathematics — including certain models of population decline, drug elimination kinetics in pharmacology (first-order pharmacokinetics), and any process where the rate of change is proportional to the current quantity present, making the decay law one of the most widely applicable simple differential equations in all of science.
Concept Relationship Map
Requires: radioactive decay
Enables: half-life, medical dosimetry, decay chain modelling
Commonly confused with: assuming activity decays differently from the underlying nuclei count (they share exactly the same decay constant)
Sources
- Rutherford, E. (1900). Philosophical Magazine, 49(300), 1–14.
- Bateman, H. (1910). "The Solution of a System of Differential Equations Occurring in the Theory of Radioactive Transformations." Proceedings of the Cambridge Philosophical Society, 15, 423–427.