Q.Derive the relationship between half life and decay constant of a radioelement.

The half-life, $t_{1/2}$, of a radioelement is the time needed for a given number of its nuclei to decay to exactly half their initial value. Substituting $N = N_0/2$ at $t=t_{1/2}$ into the decay-constant expression gives the simple, reciprocal relationship $t_{1/2} = \frac{0.693}{\lambda}$ (equivalently $\lambda = \frac{0.693}{t_{1/2}}$), where 0.693 is $\ln 2$. A short half-life means a large decay constant (fast decay); a long half-life means a small decay constant (slow decay). Every radioisotope has its own fixed, characteristic half-life, and together with the exponential decay law this relationship …