Define half life period and decay constant for a radioactive substance. Deduce the relation between them.

a) Half life period (T): Time taken for the number of radioactive nuclei to disintegrate to half of its original number of nuclei is called half life period (T).
b) Decay constant ( $\mathrm{\lambda }$ ): The ratio of the rate of radioactive decay to the number of nuclei present at that instant. It is a proportional constant, denoted by ‘ $\mathrm{\lambda }$ ’ with units ‘s^-1’.
$\therefore \mathrm{\lambda }=\frac{-\left(\frac{\mathrm{dN}}{\mathrm{dt}}\right)}{\mathrm{N}}$

Relation between half life period and decay constant :
1) The radioactive decay law states that the number of radioactive nuclei in a radioactive sample decreases exponentially with time.
$\mathrm{N}={\mathrm{N}}_0{\mathrm{e}}^{-\mathrm{\lambda t}}$
here $\mathrm{\lambda }$ is decay constant.
2) If N₀ is the number of nuclei at t = 0 and N is the number of nuclei at any constant of time ‘t’ then by definition
at t = T,  $\mathrm{N}=\frac{{\mathrm{N}}_0}{2}\text{ we get }\frac{{\mathrm{N}}_0}{2}={\mathrm{N}}_0{\mathrm{e}}^{-\mathrm{\lambda t}}$
${\mathrm{e}}^{\mathrm{\lambda T}}$= 2 taking logarithms on both sides, we get
$\begin{array}{l}\mathrm{\lambda T}=\ln 2\\ \mathrm{T}=\frac{\ln 2}{\mathrm{\lambda }}=\frac{2.303{\log }_{10}\left(2\right)}{\mathrm{\lambda }}\therefore \mathrm{T}=\frac{0.693}{\mathrm{\lambda }}\end{array}$