Question: How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of\({\bf{1}}{{\bf{0}}^{ - {\bf{22}}}}{\rm{ }}{\bf{s}}\).

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 average lifetime of an unstable atomic nucleus is in the order is

\({t_{nucleus}} = {10^{ - 22}}{\rm{ }}s\)

The average lifetime of human is being \(80{\rm{ }}years\).

In seconds, a life time is

\(\begin{aligned}{}{t_{human}} = 80{\rm{ }}years \times \frac{{365.2{\rm{ }}days}}{{1{\rm{ }}years}} \times \frac{{24{\rm{ }}hr}}{{1{\rm{ }}day}} \times \frac{{{\rm{3600 sec}}}}{{1{\rm{ hr}}}}\\ \approx 2.5 \times {10^9}{\rm{ }}s\end{aligned}\)

The ratio of the average life of a human to the average life of an extremely unstable atomic nucleus can be computed as

\(\begin{aligned}{}\frac{{{t_{nucleus}}}}{{{t_{human}}}} = \frac{{{{10}^{ - 22}}{\rm{ }}s}}{{2.5 \times {{10}^9}{\rm{ }}s}}\\{t_{nucleus}} = 0.4 \times {10^{ - 31}} \times {t_{human}}\end{aligned}\)

\({t_{nucleus}} = 4.0 \times {10^{ - 32}} \times {t_{human}}\)

Here,\(4.0\)multiple may vary according to the average value of the human life time.

If you take\({\bf{70}} - {\bf{80}}{\rm{ }}{\bf{years}}\), the ratio will be about \(3 \times {10^{ - 32}}\), and if we take the human life time as \(100{\rm{ }}years\), then the ratio will be about \(5 \times {10^{ - 32}}\).

Therefore, the life time of a human being is longer than the mean life time of an unstable nucleus by an order \({10^{31}}\).