Question

(a) Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus.
(b) How is the mean life of a given radioactive nucleus related to the decay constant?

Answer 1

a) According to radioactive decay law, the rate of decay is proportional to amount of nuclei ,N. Thus, $-\frac{dN}{dt}\propto N$ (negative sign come because the amount of nuclei decreases with time)
So, $\frac{dN}{dt}=-\lambda N$ or $\frac{dN}{N}=-\lambda t$
Integrating, $lnN=-\lambda t+c$ where C is the integrating constant.
At $t=0$, $N=N_0$ so $ln{N}_0=c$
Now, $lnN=-\lambda t+ln{N}_0$
or $ln\left(N/N_0\right)=-\lambda t$ or $N/N_0=e^{-\lambda t}$
Hence, $N=N_0{e}^{-\lambda t}$, this is the mathematical expression of radioactive decay of nuclei.
b) The relation between mean life $\left(t_{avg}\right)$ and decay constant $\left(\lambda \right)$ : $t_{avg}=\frac{1}{\lambda }$