Question

The probability of a nucleus to decay in two mean lives is
(A) $ \dfrac{1}{4} $
(B) $ \dfrac{{{e^2} - 1}}{{{e^2}}} $
(C) $ \dfrac{3}{4} $
(D) $ \dfrac{1}{{{e^2}}} $

Answer 1

Hint
Firstly we will use the formula for the rate of disintegration $ \dfrac{{dN}}{{dt}} = \lambda N $ . Then we will find out what is the value of 1 mean life and 2 mean life respectively. It will be found out that decaying a nucleus in two mean lives is $ \dfrac{2}{\lambda } $ . Then we will use the formula of probability to find out the underlying radioactive nucleus. After calculating this we will subtract it from 1 to find the probability of a nucleus to decay in two mean lives.

Complete step by step answer
Let decay constant of radioactive disintegration constant is $ = \lambda $
The time rate of disintegration will be directly proportional to the number of radioactive particles present in the sample at that time
Let at time t the number of radioactive particles present in the sample be N.
In time dt, dN number of particles disintegrate. So, the rate of disintegration is
 $ \dfrac{{dN}}{{dt}} \propto N \\
   \Rightarrow \dfrac{{dN}}{{dt}} = - \lambda N...(i) \\ $
The negative sign indicates the decrease in the number of radioactive elements with time.
If we take only the magnitude of rate of disintegration then $ \dfrac{{dN}}{{dt}} = \lambda N $
The mean life of a radioactive element is the ratio of the total lifetime of all the radioactive atoms to the total number of such atoms in it.
 $ \therefore $ mean life, $ \tau = $ total life time of all atoms/total number of atoms
  $ \tau = \dfrac{1}{\lambda } $
Thus 1 mean life of a radioactive element is the reciprocal of the radioactive constant.
So, according to the question, decaying a nucleus in two mean lives is $ \dfrac{2}{\lambda } $
Now, probability of non decaying radioactive nucleus(P) = undecayed number of nuclei(N)/ total number of nuclei ( $ {N_0} $ )
 $ P = \dfrac{N}{{{N_0}}} = \dfrac{{{N_0}{e^{ - \lambda t}}}}{{{N_0}}} $ $ = {e^{ - \lambda \times \dfrac{2}{\lambda }}} $ $ = \dfrac{1}{{{e^2}}} $
Now we have found that the probability of non decaying radioactive nucleus so to find out the probability of decaying nucleus $ \left( {P'} \right) $ we have to subtract P from 1
  $ P' = 1 - P $ $ = $ $ 1 - \dfrac{1}{{{e^2}}} = \dfrac{{{e^2} - 1}}{{{e^2}}} $
So, the the probability of a nucleus to decay in two mean lives is $ \dfrac{{{e^2} - 1}}{{{e^2}}} $ (Option-B).

Additional Information
Half-life is the time period after which the number of radioactive atoms present in a radioactive sample becomes half of its initial number due to disintegration is called half-life of that radioactive element.
The decay constant is the reciprocal of time during which the number of atoms of a radioactive substance decreases to $ \dfrac{1}{e} $ (or $ 36.8\% $ ) of the number present initially.

Note
One might choose the option- D because after calculating the probability one might think that $ \dfrac{1}{{{e^2}}} $ is the right answer. But one has to make sure that this value is actually the probability of non decaying radioactive nucleus. So we have to apply the basic rule of probability to find out the probability of the decaying nucleus. One thing one might follow that the value of one mean life is actually the reciprocal of the decay constant.