• # question_answer The half-life of a radioactive substance is 30s. Calculate (i) the decay constant. (ii) time taken for the sample to decay 3/4th of its initial value.

Given, ${{t}_{1/2}}=30\,s$ (i) Disintegration constant             $\lambda =\frac{0.693}{{{t}_{1/2}}}=\frac{0.693}{30}=0.0231\,{{s}^{-1}}$ (ii) Number of atoms disintegrated $=\frac{3}{4}{{N}_{0}}$ Number of atoms left, $N={{N}_{0}}-\frac{3}{4}{{N}_{0}}=\frac{1}{4}{{N}_{0}}$ Number of half-lives in t seconds, $n=t/30$ $\because$       $N={{N}_{0}}{{\left( \frac{1}{2} \right)}^{n}}$ $\therefore$      $\frac{{{N}_{0}}}{4}={{N}_{0}}{{\left( \frac{1}{2} \right)}^{t/30}}\Rightarrow {{\left( \frac{1}{2} \right)}^{2}}={{\left( \frac{1}{2} \right)}^{t/30}}$ $\Rightarrow$   $2=t/30\Rightarrow t=60\,s$