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.

Answer:

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, \[n=t/30\] \[\because \]       \[N={{N}_{0}}{{\left( \frac{1}{2} \right)}^{n}}\] \[\therefore \]      \[\frac{{{N}_{0}}}{4}={{N}_{0}}{{\left( \frac{1}{2} \right)}^{t/30}}\Rightarrow {{\left( \frac{1}{2} \right)}^{2}}={{\left( \frac{1}{2} \right)}^{t/30}}\] \[\Rightarrow \]   \[2=t/30\Rightarrow t=60\,s\]