Question

State the law of radioactive decay. Hence derive the expression $N=N_0{e}^{-\lambda t}$ where symbols have their usual meanings.

Answer 1

Law of radioactive decay : At any instant, the rate of radioactive disintegration is directly proportional to the number of nuclei of the radioactive element present at that instant.
We know by radioactive decay law
$\frac{dN}{dt}\propto N$
or $\frac{dN}{dt}=-\lambda N$
where $\lambda$ is the constant of proportionality called radioactive decay constant. Let $N_0$ be the number of nuclei present at time $t=0$, and $N$ the number of nuclei present at time $t$.
${\int }_{N_0}^N\frac{dN}{N}=-{\int }_0^t\lambda dt=-\lambda {\int }_0^{t}dt$
${\log }_{e}N-{\log }_e{N}_0=-\lambda t$
${\log }_e\left(\frac{N}{N_0}\right)=-\lambda t$
$\frac{N}{N_0}=e^{-\lambda t}$
$N=N_0{e}^{-\lambda t}$