Radioactive decay

Example — Radioactive decay

A radioactive sample follows $N'(t) = -\lambda N(t)$, with $\lambda>0$ the decay constant.

Solution: $N(t) = N_0\,e^{-\lambda t}$.

The half-life $T_{1/2}$ is the time for which $N(T_{1/2}) = N_0/2$. From $e^{-\lambda T_{1/2}} = 1/2$ we obtain $\lambda T_{1/2} = \ln 2$, that is: $\boxed{T_{1/2} = \frac{\ln 2}{\lambda}}$

The curve $N/N_0 = e^{-\lambda t}$ decreases exponentially; the dashed line marks the level $1/2$ reached at $t = T_{1/2}$.

Exponential decay: the fraction of remaining nuclei $N/N_0$ halves after each time $T_{1/2}$.

Links

Topics: Differential equations
Concepts: Decay · Half-life · Separable variables
Functions: Exponential function
Skills: Integrating · Modelling