Law of Radioactive Decay

Even though an individual radioactive nucleus's decay is intrinsically unpredictable -- there is no way to say in advance exactly when any ONE particular nucleus will decay -- the behaviour of a large POPULATION of identical radioactive nuclei is remarkably predictable, governed by a precise statistical law. The reasoning is this: in a short time interval dt, the number of decays dN occurring should be proportional both to the LENGTH of that interval (a longer interval gives more decays) and to the NUMBER of parent nuclei still present, N(t) (more remaining nuclei means more decays can happen). This gives the fundamental differential equation of radioactive decay, $\frac{dN}{dt}=-\lambda N(t)$, where the constant of proportionality $\lambda$ is called the decay constant (the negative sign reflects that N is decreasing with time).

Integrating this equation from an initial population $N_0$ (at t = 0) gives the famous exponential decay law, $N(t)=N_0e^{-\lambda t}$ -- the number of surviving parent nuclei falls off exponentially with time, never actually reaching zero in finite time, but getting arbitrarily close to it. …