Question

The half life of a radioactive nucleus is 50 days. The time interval \[({{t}_{2}}-{{t}_{1}})\] between the time \[{{t}_{2}}\] when \[\dfrac{2}{3}\] of it has decayed and the time \[{{t}_{1}}\] when \[\dfrac{1}{3}\] of it had decayed is:
A. 60 days
B. 15 days
C. 30 days
D. 50 days

Answer 1

Hint: In this question, we have already given the amount of decayed substance at two-intervals of time. We can use the equation of radioactive decay to find the answer. By comparing the equation of decay at these two-time intervals, we can calculate the total time taken within these two time periods.

Formula used:
\[N={{N}_{0}}{{e}^{-\lambda t}}\], where \[N\] is the number of remaining radioactive nuclei at time t,\[{{N}_{0}}\] is the number of radioactive nuclei before the decay, \[\lambda \] is the decay constant, and t is the time required for the decay.
\[{{T}_{\dfrac{1}{2}}}=\dfrac{\ln 2}{\lambda }\]

Complete Step-by-Step solution:
We can solve this problem by using the radioactive decay equation. According to the question, the half-life of the radioactive nucleus is 50 days.
\[{{T}_{\dfrac{1}{2}}}=50\text{ days}\]
According to the equation of radioactive decay,
\[N={{N}_{0}}{{e}^{-\lambda t}}\], where \[N\] is the number of remaining radioactive nuclei at time t, \[{{N}_{0}}\] is the number of radioactive nuclei before the decay, \[\lambda \] is the decay constant, and t is the time required for the decay.
We have already given, at the time \[{{t}_{2}}\], \[\dfrac{2}{3}\] of the radioactive nuclei decayed.
So, we can write the equation as,
\[{{N}_{2}}=\dfrac{1}{3}{{N}_{0}}\]
Therefore, the equation of decay can be written as,
\[\dfrac{1}{3}{{N}_{0}}={{N}_{0}}{{e}^{-\lambda {{t}_{2}}}}\]……………..(1)
At the time \[{{t}_{1}}\], \[\dfrac{1}{3}\] of the radioactive nuclei decayed.
\[{{N}_{1}}=\dfrac{2}{3}{{N}_{0}}\]
Therefore, the equation of decay can be written as,
\[\dfrac{2}{3}{{N}_{0}}=\dfrac{1}{3}{{N}_{0}}{{e}^{-\lambda {{t}_{1}}}}\]…………………(2)
Since these two equations contain several similar quantities, we can divide equation (1) by equation (2).
\[\dfrac{\dfrac{1}{3}{{N}_{0}}}{\dfrac{2}{3}{{N}_{0}}}=\dfrac{{{N}_{0}}{{e}^{-\lambda {{t}_{2}}}}}{{{N}_{0}}{{e}^{-\lambda {{t}_{1}}}}}\]
\[\dfrac{1}{2}=\dfrac{{{e}^{-\lambda {{t}_{2}}}}}{{{e}^{-\lambda {{t}_{1}}}}}\]
We can write this as,
\[\dfrac{1}{2}={{e}^{-\lambda ({{t}_{2}}-{{t}_{1}})}}\]
\[\lambda ({{t}_{2}}-{{t}_{1}})=\ln 2\]
\[{{t}_{2}}-{{t}_{1}}=\dfrac{\ln 2}{\lambda }\]……………………..(3)
The half-life of radioactive nuclei can be written as,
\[{{T}_{\dfrac{1}{2}}}=\dfrac{\ln 2}{\lambda }\]………………………..(4)
We can plug this equation into equation (3)
\[{{t}_{2}}-{{t}_{1}}={{T}_{\dfrac{1}{2}}}\]
Therefore, the time interval will be 50 days. So, the correct answer is option D.

Note: Most of the candidates will take the ratio of the decay the same as in the question. In the question, the ratio indicates the amount of decayed substance. But, in the equation of radioactive decay, we required the amount that is remaining after the decay. So as per the question, we have to find the remaining nuclei after the decay from the ratio of decayed nuclei. Since it is a ratio, we can simply find the difference between the initial amount and decayed amount.
At the time \[{{t}_{2}}\], \[\dfrac{2}{3}\] of the radioactive nuclei decayed. Then, \[\dfrac{1}{3}\] of the radioactive nuclei is remaining for the decay.
At the time \[{{t}_{1}}\], \[\dfrac{1}{3}\] of the radioactive nuclei decayed. \[\dfrac{2}{3}\] of the radioactive nuclei is remaining for the decay.
The equation of half-life of radioactive nuclei is important. It is advised to learn for the competitive exams.