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Question: The half-life of \( {}^{215}At \) is \( 100\mu s \) .The time taken for the radioactivity of a sampl...

The half-life of 215At{}^{215}At is 100μs100\mu s .The time taken for the radioactivity of a sample of 215At{}^{215}At to decay 116th\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 \\\

Explanation

Solution

In order to solve this question, we are going to first determine the decay constant from the half-life of 215At{}^{215}At as given in the question. After that, the time for the concentration of the 215At{}^{215}At to decay 116th\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 215At{}^{215}At at a time tt is given by
N(t)=N0e−λtN\left( t \right) = {N_0}{e^{ - \lambda t}}
The half-life of the radioactive element is given by
t12=ln⁡2λ{t_{\dfrac{1}{2}}} = \dfrac{{\ln 2}}{\lambda }

Complete step by step solution:
According to the law of radioactive decay, the concentration of 215At{}^{215}At at a time tt is given by
N(t)=N0e−λtN\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 215At{}^{215}At remains half of its initial value is given by
t12=ln⁡2λ{t_{\dfrac{1}{2}}} = \dfrac{{\ln 2}}{\lambda }
Now putting the value of the decay constant, λ=ln⁡2t12\lambda = \dfrac{{\ln 2}}{{{t_{\dfrac{1}{2}}}}} in the above equation for the concentration measurement
Putting these values for the decay to the 116th\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 (A)400μs\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 λ\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.