Question

Obtain a relation between half life of a radioactive substance and decay constant$\left(\lambda \right)$.

Answer 1

Decay constant is proportionality between the size of a population of radioactive atoms and the rate at which the population decreases because of radioactive decay. Suppose $N$ is the size of a population of radioactive atoms at a given time $t$ and $dN$ is the amount by which the population decreases in time $dT$ ;
then the rate of change is given by the equation:

$\Rightarrow \frac{dN}{dT}=-\lambda N$

$\Rightarrow \frac{dN}{N}=-\lambda .dT$

$\Rightarrow N=N_0{e}^{-\lambda T}$

where $N_0$ is the population of initial radioactive atoms at time $T=0$

the time required for half of the original population of radioactive atoms to decay is called Half-life.

$N=\frac{N_0}{2}$ at $T=T_{\left(\frac{1}{2}\right)}$

$\Rightarrow \frac{N_0}{2}=N_0{e}^{-\lambda T_{\left(\frac{1}{2}\right)}}$

$T_{\left(\frac{1}{2}\right)}=\frac{0.693}{\lambda }$