Question

What is the half-life of a radioactive substance of any given amount of the substance 1M disintegrate in 40 minutes?
A.10 min
B.160 min
C.13 min 20 sec
D.20 min

Answer 1

Radioactive decay shows disappearance of a constant fraction of activity per unit time. Half-life is the time required to decay a sample to $50\%$ of its initial activity. In the question,
$87.5\%$ of any given amount of the substance 1M disintegrate in 40 minutes and thus the remaining amount of the substance is $12.5\%$. Substituting the values in the equation, $t = \dfrac{1}{K}In\dfrac{{[{R^\circ }]}}{{[R]}}$, we will calculate the half-life.

Complete step by step answer:
It is given in the question that the $87.5\%$ of any given amount of the substance 1M disintegrate in 40 minutes.
Let the half-life of the substance be taken to be ${t_{\dfrac{1}{2}}}$.
$t = \dfrac{1}{K}In\dfrac{{[{R^\circ }]}}{{[R]}}$
Where, $R^\circ$ is the initial concentration = 1 M
R is the remaining concentration $= 12.5\%$
Substituting the values in the above equation, we get
$40 = \dfrac{1}{K}In\dfrac{{[{R^\circ }]}}{{\dfrac{{12.5}}{{100}}[R]}}$
$\Rightarrow$ $K = \dfrac{1}{{40}}In\left( {\dfrac{{100}}{{12.5}}} \right)$
$K = \dfrac{1}{{40}}In\,8$
${t_{\dfrac{1}{2}}} = ln\dfrac{2}{K}$
$= \dfrac{{In\,2}}{{In\dfrac{5}{{40}}}}$
$= \dfrac{{In\,2}}{{In\,8}} \times 40$
$= \dfrac{{40}}{3}$ minutes
= 13.33 minutes
${t_{\dfrac{1}{2}}} = 13$ minutes 20 sec
Hence, the half-life of the substance is 13 min 20 sec.

Therefore, the correct answer is option (C).

Note: The time in which half of the original number of nuclei decay is called as the half-life that is “t sec” in this case. Half of the remaining nuclei decay in the following half-life. Therefore, the number of radioactive nuclei decreases from N to $\dfrac{N}{2}$ in one half-life and $\dfrac{N}{4}$ in the next and to $\dfrac{N}{8}$ in the next and so on. Radioactive decay is a spontaneous process. It cannot be predicted exactly for any single nucleus and can only be described statistically and probabilistically i.e. can only give averages and probabilities.