Question

The half cycle of a radioactive nucleus is 50 days. The time interval $({{t}_{2}}-{{t}_{1}})$ between the time ${{t}_{2}}$ when $\dfrac{2}{3}$of it has decayed and the time ${{t}_{1}}$ when $\dfrac{1}{3}$of it has decayed is
A. 50 days
B. 60 days
C. 15 days
D. 30 days

Answer 1

First let us learn what half-life or half cycle of a radioactive substance means. Half-life in radioactivity is the interval of time required for one half of a $\dfrac{N}{{{N}_{o}}}=\,{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{T}}}$ atomic nuclei of a radioactive sample to decay. To solve the above question, we need to use the formula given, since the time is mentioned. There is another formula for half-life you can use, ${{t}_{\dfrac{1}{2}}}=\,\dfrac{\ln (2)}{k}$ where ${{t}_{\dfrac{1}{2}}}$ is the half life and k is the decay constant.

Complete step by step answer:
Let us perform the solution stepwise now.
When time is t 2
Number of decayed nuclei $=\dfrac{2}{3}$
So, number of undecayed nuclei $(1-\dfrac{2}{3})\,at\,\,{{t}_{2}}$
$=\,\,\dfrac{1}{3}\,\left( \dfrac{N}{{{N}_{o}}} \right)$
Again, when time is t 1
Number of decayed nuclei $=\dfrac{1}{3}$
So, number of undecayed nuclei $(1-\dfrac{1}{3})\,$at t1
$=\,\,\dfrac{2}{3}\,\left( \dfrac{N}{{{N}_{o}}} \right)$
At time t 2 $\dfrac{1}{3}\,=\,{{\left( \dfrac{1}{2} \right)}^{\dfrac{{{t}_{2}}}{T}}}$ eq. (1) [Using the formula $\dfrac{N}{{{N}_{o}}}=\,{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{T}}}$]
At time t 1$\dfrac{2}{3}=\,{{\left( \dfrac{1}{2} \right)}^{\dfrac{{{t}_{1}}}{T}}}$ eq. (2)
Now, we divide equation 1 by equation 2
$\dfrac{\dfrac{1}{3}}{\dfrac{2}{3}}\,=\,{{\dfrac{\left( \dfrac{1}{2} \right)}{{{\left( \dfrac{1}{2} \right)}^{\,\dfrac{{{t}_{t}}}{T}}}}}^{\dfrac{{{t}_{2}}}{T}}}$
$\Rightarrow \dfrac{1}{2}={{\left( \dfrac{1}{2} \right)}^{\dfrac{{{t}_{2}}-{{t}_{1}}}{T}}}$
Since, the boxes are equal, we can compute both the powers,
$1=\dfrac{{{t}_{2}}-{{t}_{1}}}{T}$ (Now as you already know, T is the half-life, which is given as 50 in the question)
${{t}_{2}}-{{t}_{1}}=50$ (putting T=50)
Therefore, the time interval ${{t}_{2}}-{{t}_{1}}$ is equal to 50.

Thus, Option A is correct.

Note:
As you can see the half-life of a process is an indication of how fast that process proceeds- a measure of rate or capacity of the process. Also you can say it is the time taken by a substance to diminish to one half of its initial amount. Students must remember that radioactive decay is a process that occurs in an unstable atomic nucleus. If you solve more such problems you will get a better idea about it.