Question

At a given instant these are $25\%$ undecayed radioactive nuclei in a sample. After $10$ second, the number of undecayed nuclei reduces to $12.5\%$ calculate:
(a) Mean life of nuclei.
(b) The time in which the number of undecayed nuclei will further reduce to $6.25\%$ of the reduced number.

Answer 1

It is given that half-life of radioactive nuclei is 10 s

(a) Mean life of nuclei is

$\tau =\frac{{\tau }_{1/2}}{0.693}$

$=\frac{10}{0.693}$

$=14.43s$

(b) From 12.5% nuclei has to be reduced to 6.25%.

$N=N_0{\left(\frac{1}{2}\right)}^n$

$6.25=100{\left(\frac{1}{2}\right)}^n$

$2^n=16$

$n=4$

So, in 4 half-lives undecayed nuclei will further reduce to 6.25% of the reduced number.

$t=4\times {\tau }_{1/2}=4\times 10=40s$