Question

A radioactive material of half-life time of 69.3 days kept in a container $\dfrac{2}{3}rd$ of the substance remains undecayed after (given, $\ln \dfrac{3}{2}=0.4$)
(A) 20 days
(B) 25 days
(C) 40 days
(D) 50 days

Answer 1

Radioactivity refers to the phenomenon in which the substance decays by emission of radiation. Half-life is defined as the time taken by the material in which the number of undecayed atoms becomes half. A material containing unstable nuclei is considered radioactive

Complete step by step answer: We know there exists a relationship between the decay constant, $\lambda$and half-life ${{T}_{1/2}}$. It states ${{T}_{1/2}}\lambda =0.693$
Given, ${{T}_{1/2}}$= 69.3 days
Thus $\begin{aligned} & 69.3\times \lambda =0.693 \\\ & \lambda =0.01/days \\\ \end{aligned}$
Now we have calculated the decay constant and it is given that $\dfrac{2}{3}rd$of the substance remains undecayed.
Let the number of initial atoms be ${{N}_{0}}$. Then the number of undecayed atoms be N. then according to the question,
$N=\dfrac{2}{3}{{N}_{0}}$
Now using the law of radioactivity, $N={{N}_{0}}{{e}^{-\lambda t}}$

& \dfrac{2}{3}{{N}_{0}}={{N}_{0}}{{e}^{-\lambda t}} \\\ & \dfrac{2}{3}={{e}^{-0.01t}} \\\ \end{aligned}$$ $$0.66={{e}^{-0.01t}}$$ Using the natural log, we get, $$Ln(0.66)=-0.01t$$ -0.41=-0.01t t=40 days Hence, the correct option is (C). Additional Information: Half-life is the time for half the radioactive nuclei in any sample to undergo radioactive decay. For example, after 2 half-lives, there will be one fourth the original material remains, after three half-lives one eight the original material remains, and so on. Half-life is a convenient way to assess the rapidity of a decay **Note:** While solving such problems we have to keep in mind that while using the formula $$N={{N}_{0}}{{e}^{-\lambda t}}$$, the quantity on LHS is the number of atoms or nuclei which are undecayed after time t and $${{N}_{0}}$$is the original number of atoms or nuclei at a time, t=0.