91Ó°ÊÓ

The time taken by the sample to decay to half the original number of the nuclei is called the half-life of the sample.

The expression of the half-life of a radioactive isotope is given as follows.

\({t_{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\\}

\!\lower0.7ex\hbox{$2$}}}} = \frac{{0.693}}{\lambda }\)

Here,\(\lambda \)is the activity of the sample.

The half-life of the time is in the order as,

\({T_{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\\}

\!\lower0.7ex\hbox{$2$}}}} = {10^9}{\rm{ }}s\)

The half-life of the a generation is twice the half-life.

\(\begin{aligned}{c}1{\rm{ }}life - time &= 2\left( {{T_{{1 \mathord{\left/ {\vphantom {1 2}} \right.\\} 2}}}} \right)\\ &= 2\left( {{{10}^9}{\rm{ }}s} \right)\end{aligned}\)

The time elapsed since\(0{\rm{ }}AD\) is\(2019\)years. The expression for the number of generations passed is,

\(Number{\rm{ }}of{\rm{ generations}} = \frac{{2019}}{{\frac{1}{3} \times life{\rm{ }}time}}\)

Substitute\(2\left( {{{10}^9}{\rm{ }}s} \right)\)for\(1{\rm{ }}life - time\)in the above equation.

\(\begin{aligned}{c}Number{\rm{ }}of{\rm{ generations}} &= \frac{{3 \times \left( {2019{\rm{ years}}} \right)\left( {\frac{{3.15 \times {{10}^9}{\rm{ }}s}}{{1{\rm{ }}year}}} \right)}}{{2\left( {{{10}^9}{\rm{ }}s} \right)}}\\ &= 95.39\\ \approx 95\end{aligned}\)

Hence, the required number of generations that have passed since the year \(0\) AD is \(95\).