Question

The half-life of $ {}^{215}At $ is $ 100\mu s $ .The time taken for the radioactivity of a sample of $ {}^{215}At $ to decay $ \dfrac{1}{{16}}th $ of initial value is
 $ \left( A \right)400\mu s \\
  \left( B \right)6.3\mu s \\
  \left( C \right)40\mu s \\
  \left( D \right)300\mu s \\ $

Answer 1

Hint: In order to solve this question, we are going to first determine the decay constant from the half-life of $ {}^{215}At $ as given in the question. After that, the time for the concentration of the $ {}^{215}At $ to decay $ \dfrac{1}{{16}}th $ of initial value is calculated by putting the values in the law of radioactive decay equation.
According to the law of radioactive decay, the concentration of $ {}^{215}At $ at a time $ t $ is given by
 $ N\left( t \right) = {N_0}{e^{ - \lambda t}} $
The half-life of the radioactive element is given by
 $ {t_{\dfrac{1}{2}}} = \dfrac{{\ln 2}}{\lambda } $

Complete step by step solution:
According to the law of radioactive decay, the concentration of $ {}^{215}At $ at a time $ t $ is given by
 $ N\left( t \right) = {N_0}{e^{ - \lambda t}} $
Now as we know that the half-life of the radioactive element, i.e. in which the concentration of $ {}^{215}At $ remains half of its initial value is given by
 $ {t_{\dfrac{1}{2}}} = \dfrac{{\ln 2}}{\lambda } $
Now putting the value of the decay constant, $ \lambda = \dfrac{{\ln 2}}{{{t_{\dfrac{1}{2}}}}} $ in the above equation for the concentration measurement
Putting these values for the decay to the $ \dfrac{1}{{16}}th $ of initial value:
 $ \dfrac{1}{{16}} = {e^{ - \left( {\dfrac{{0.693}}{{100}}} \right)t}} \\
   \Rightarrow \ln \left( {\dfrac{1}{{16}}} \right) = - \dfrac{{0.693}}{{100}}t \\
   \Rightarrow t = 400\mu s \\ $
Thus, the option $ \left( A \right)400\mu s $ is the correct answer.

Note:
The probability per unit time that a nucleus will decay is constant, independent of time. The decay constant is represented by $ \lambda $ . The number of radioactive elements undergoing decay per unit time, is proportional to the total number of nuclei in the sample material. Half – life of the radioactive element is the time in which the concentration of the radioactive element remains half of its initial concentration.