Question

Half-lives of two radioactive elements A and B are 20minutes and 40minutes respectively. Initially the samples have an equal number of nuclei. After 80minutes the ratio of decayed numbers of A and B nuclei will be:
A. 1:16
B. 4:1
C. 1:4
D. 5:4

Answer 1

As the first step, you could recall the expression for the number of nuclei remaining after a radioactive decay. You could thus find the remaining nuclei in both samples A and B. Then, you could find the number of decayed nuclei from the both samples by subtracting the above from the initial number of nuclei present. Then you could simply take their ratio to get the answer.

Formula used:
Remaining nuclei after decay,
$N={{N}_{0}}{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{T}}}$

Complete Step by step solution:
In the question, we are given the half lives of two radioactive elements A and B as 20minutes and 40minutes respectively. We are told that the initial number of nuclei is the same for both samples. We are asked to find the ratio of decayed number of nuclei of both samples after the interval of 80minutes.
We know that, the remaining number of nuclei after the decay is given by,
$N={{N}_{0}}{{\left( \dfrac{1}{2} \right)}^{\dfrac{t}{T}}}$
Where, t is the time interval, T is the half life of the given sample and ${{N}_{0}}$is the initial number of nuclei in the sample.
Now, for sample A, the remaining number of nuclei will be,
${{N}_{A}}={{N}_{0}}{{\left( \dfrac{1}{2} \right)}^{\dfrac{80}{20}}}$
$\Rightarrow {{N}_{A}}={{N}_{0}}{{\left( \dfrac{1}{2} \right)}^{4}}$
$\therefore {{N}_{A}}=\dfrac{{{N}_{0}}}{16}$
Now the number of decayed nuclei will be,
$\Delta {{N}_{A}}={{N}_{0}}-\dfrac{{{N}_{0}}}{16}=\dfrac{15{{N}_{0}}}{16}$ ……………………………….. (1)
Similarly, for sample B,
${{N}_{B}}={{N}_{0}}{{\left( \dfrac{1}{2} \right)}^{\dfrac{80}{40}}}$
$\Rightarrow {{N}_{B}}={{N}_{0}}{{\left( \dfrac{1}{2} \right)}^{2}}$
$\therefore {{N}_{B}}=\dfrac{{{N}_{0}}}{4}$
Now the number of decayed nuclei will be,
$\Delta {{N}_{B}}={{N}_{0}}-\dfrac{{{N}_{0}}}{4}=\dfrac{3{{N}_{0}}}{4}$ ………………………………….. (2)
So, the ratio of decayed numbers of A and B nuclei after 80minutes will be given by,
$\dfrac{\Delta {{N}_{A}}}{\Delta {{N}_{B}}}=\dfrac{\dfrac{15{{N}_{0}}}{16}}{\dfrac{3{{N}_{0}}}{4}}$
$\Rightarrow \dfrac{\Delta {{N}_{A}}}{\Delta {{N}_{B}}}=\dfrac{5}{4}$
$\therefore {{N}_{A}}:{{N}_{B}}=5:4$
Therefore, we found that the ratio of decayed numbers of A and B nuclei after 80minutes will be 5:4.

Hence, option D is found to be the correct answer.

Note:
In questions that are related to nuclei decay where we are asked to find the number of nuclei, we should be careful about the specification as to which nuclei are to be found. It could be remaining nuclei after the decay or it can be the number of decayed nuclei. Also, shouldn’t get confused with t and T which are total time taken for the decay and half life respectively.