Question

The half-life of a radioactive substance is 20 min. The approximate time interval (${{t}_{2}}-{{t}_{1}}$) between the time ${{t}_{2}}$when 2/3 of it has decayed and time ${{t}_{1}}$when 1/3 of it had decayed is ;
(A) 14 min
(B) 20 min
(C) 28 min
(D) 7 min

Answer 1

Hint:Radioactivity can be defined as the ability of a substance in which the substance decays by emission of radiation. The materials which show such phenomenon are called radioactive substances. 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 by emission of radiation gained stability. Sometimes they can be harmful.

Complete step by step answer:
Given ${{T}_{1/2}}=20\min $, the relationship between decay constant $\lambda $and half life is ${{T}_{1/2}}=\dfrac{0.693}{\lambda }$
$\Rightarrow {{T}_{1/2}}=\dfrac{0.693}{\lambda }$
$\Rightarrow \lambda =\dfrac{0.693}{{{T}_{1/2}}}$
$\Rightarrow \lambda =\dfrac{0.693}{20\times 60}$
$\Rightarrow \lambda =0.00057{{s}^{-1}}$--(*)
Now for time ${{t}_{2}}$, $\dfrac{2}{3}$of the substance has decayed, then remaining is $\dfrac{1}{3}$. Thus, using the formula,
Now using the law of radioactivity, \[N={{N}_{0}}{{e}^{-\lambda t}}\]where N is the number of undecayed Nuclei at time, t and ${{N}_{0}}$are the nuclei in the starting
\[\Rightarrow \dfrac{1}{3}{{N}_{0}}={{N}_{0}}{{e}^{-\lambda {{t}_{2}}}}\]---(1)
Number of undecayed atoms after time ${{t}_{1}}$, $\dfrac{1}{3}$of the substance has decayed, then remaining is $\dfrac{2}{3}$.
\[\Rightarrow \dfrac{2}{3}{{N}_{0}}={{N}_{0}}{{e}^{-\lambda {{t}_{1}}}}\]---(2)
Dividing equation (2) by (1) we get,
$\Rightarrow 2={{e}^{\lambda ({{t}_{2}}-{{t}_{1}})}}$
$\Rightarrow \ln 2=\ln \{{{e}^{\lambda ({{t}_{2}}-{{t}_{1}})}}\}$
$\Rightarrow \ln 2=\lambda ({{t}_{2}}-{{t}_{1}})$
$\Rightarrow \lambda ({{t}_{2}}-{{t}_{1}})=0.3010$
using equation (*) we get,
$\therefore ({{t}_{2}}-{{t}_{1}})=\dfrac{0.3010}{0.00057}=1200s$
Converting in to minutes,${}^{1200}/{}_{60}=20\min $

So, the correct option is B.

Note: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. While solving such problems we have to keep in mind that while using the formula \[N={{N}_{0}}{{e}^{-\lambda t}}\], we have to take undecayed nuclei at that time.