Plot a graph showing the variation of undecayed nuclei $N$ versus time $t$. From the graph, find out how one can determine the half-life and average life of the radioactive nuclei.

Law of Radioactivity defines that the number of Nuclei undergoing number of Nuclei present in the sample at that Instant.

Since from the graph; we have

     $N=N_{-}{e}^{-\lambda t}$    ....1

where $\lambda$=Disintegration constant

For $\therefore$ For $T_{1/2}$ is the time at $N-\frac{1}{2}{N}_D$

$\Rightarrow$ $t=T_{1/2}$

$\frac{N_0}{2}$=$N_0{e}^{-\lambda \frac{T}{2}}$

$\Rightarrow$ $\boxed{T_{1/2}=\frac{Ln2}{\lambda }=\frac{0.693}{\lambda }}$

And for Mean life -we have to sum it over the whole Range for 

$N(t)=N_0{e}^{-\lambda t}$

for number of nuclei which decay in time t to t  t $△$;

$N(t)△t=\lambda N_0{e}^{-\lambda t}△t$

 For Integration it over the Range $T=0to\infty$

$\tau$=$\frac{\lambda N_0{\iint }_0^{\infty }t{e}^{-\lambda t}dt}{N_0}$

$=\lambda {\int }_0^{\infty }t{e}^{\lambda t}dt$

$\boxed{\tau =\frac{1}{\lambda }}$

Mean-life