Question

The decay constant of a radioactive element is \[1.5 \times {10^{ - 9}}\,{\rm{per}}\,{\mathop{\rm second}\nolimits} \]. Its mean life in seconds will be
A. \[1.5 \times {10^9}\]
B. \[4.62 \times {10^8}\]
C. \[6.67 \times {10^8}\]
D. \[10.35 \times {10^8}\]

Answer 1

Hint: The time for which a radioactive element remains active is defined as Mean life and it can also be defined as the ratio of sum of lives of all atoms in an element and total number of atoms.

Formula used:
The expression of mean life is,
\[\tau = \dfrac{1}{\lambda }\]
where \[\lambda \] is decay constant.

Complete step by step solution:
Decay constant of a radioactive element is defined as its probability of decay per unit time. Decay constant has a unit \[{s^{ - 1}}\]and denoted by the symbol \['\lambda '\]. Mean life time of a radioactive element is defined as time for which it remains active, and is equal to the sum of lifetime of individual nuclei in a sample divided by the number of atoms present in the sample. It is denoted by symbol \[\tau \] and mean life time is inversely proportional to the decay constant.

Given here, \[\lambda = 1.5 \times {10^{ - 9}}\,{\rm{per}}\,{\mathop{\rm second}\nolimits} \]
And, mean life \[\tau = \dfrac{1}{\lambda }\]
Therefore,
\[\tau = \dfrac{1}{{1.5 \times {{10}^{ - 9}}}} \\
\therefore \tau = 6.67 \times {10^8}\,s\]
Hence, the mean life of the given radioactive element is \[6.67 \times {10^8}\,\] second.

Therefore, option C is the correct answer.

Note: If \[{\tau _{1,\,}}{\tau _{2,\,}}{\tau _{3,\,}}........{\tau _{n\,}}\] represents the lifetime for \[{N_{1,\,}}{N_{2,\,}}{N_{3,\,}}........{N_{n\,}}\] atoms respectively for an element with n numbers of atom then mean life is given by, \[\tau = \dfrac{{{\tau _1}{N_1} + {\tau _{2\,}}{N_{2\,}} + {\tau _3}{N_{3\,}}........{\tau _{n\,}}{N_n}}}{n}\]. Mean life is comparatively longer than the half life.