Radioactive decay

An atomic nucleus consists of protons and neutrons; many such combinations are unstable and spontaneously decay (transmute) into another substance — such nuclei are radioactive.

Model. The rate $\dfrac{dA}{dt}$ at which the nuclei of a radioactive substance decay is proportional to the amount $A(t)$ of substance remaining:

Writing $k=-k'$ with $k'>0$ makes the decreasing nature explicit: $\dfrac{dA}{dt}=-k'A$.

Remark. This is the same differential equation as the growth model of §10.8.1 — the difference lies entirely in the sign of the proportionality constant: $k>0$ for growth, $k<0$ for decay. A single differential equation form can thus serve as the mathematical model for very different phenomena, depending only on the sign (and physical interpretation) of its constant.

Solution and half-life. Separating and integrating gives $A(t)=A_0e^{-k't}$, where $A_0$ is the initial mass. The half-life $t_h$ is the time at which the remaining amount has dropped to half the original, $A(t_h)=\dfrac{A_0}{2}$; solving $\dfrac12=e^{-k't_h}$ gives $t_h=\dfrac{\log 2}{k'}$. …