Question

How long will it take for 75% of the atoms of a certain radioactive element, originally present to disintegrate? The half-life of the element is 10 days.
A. 240 days
B. 3.6 days
C. 15.6 days
D. 4.15 days

Answer 1

Hint:The above can be resolved by using the formula for the time taken by the element to possess disintegration while undergoing any radioactive process. In this formula, the half-life is given along with data for the percentage of the disintegration of an atom. The further solving the natural logarithm and by substituting the values, one can get the desired result.

Complete step by step answer:
Given:
The percentage of atoms presents is, \[N = 75\% \].
The half life for the element is, \[{t_1} = 10\;{\rm{days}}\].
The expression for the time taken to disintegrate is given as,
\[{t_2} = - \dfrac{{{t_1}}}{{0.693}}\left( {\ln N} \right)\]
Sole by substituting the values in the above equation as,
\[\begin{array}{l}
{t_2} = - \dfrac{{{t_1}}}{{0.693}}\left( {\ln N} \right)\\
{t_2} = - \dfrac{{10\;{\rm{days}}}}{{0.693}}\left( {\ln \left( {\dfrac{{75}}{{100}}} \right)} \right)\\
{t_2} = 4.15\;{\rm{days}}
\end{array}\]
Therefore, the time required by the element to disintegrate is 4.15 days and option (D) is correct.

Note: Try to remember the formula of the time taken by any element to undergo the nuclear reaction or the radioactive process. Besides, the concept of half-life is used, which is the time required to complete the half process under radioactive disintegration. Moreover, some fundamentals of radioactivity also need to be remembered as it accounts for the process, where the evolution of energy takes place along with atomic disintegration. And this energy can be used further for other applications. In addition the examples of some radioactive elements and time required for their half as well as complete disintegration can also be considered.