JEE Main 2016 · Physics · Atoms, Nuclei and Dual Nature of Radiation · easy

Two radioactive samples A and B contain the same number of nuclei initially. The half-life of A is 2 h and that of B is 4 h. After 8 h the ratio of…

Single correct option

Two radioactive samples

A

and

B

contain the same number of nuclei initially. The half-life of

A

2\ \mathrm{h}

and that of

B

4\ \mathrm{h}

. After

8\ \mathrm{h}

the ratio of the activity of

A

to that of

B

Show answer and solution

Answer: (D) $\dfrac12$

Activity is

\lambda N

with

N=N_0e^{-\lambda t}=N_0\,2^{-t/T}

. In

8\ \mathrm{h}

, sample

A

undergoes

4

half-lives, so

N_A=\dfrac{N_0}{16}

, and sample

B

undergoes

2

half-lives, so

N_B=\dfrac{N_0}{4}

. The decay constants satisfy

\lambda_A/\lambda_B=T_B/T_A=2

. Therefore

\dfrac{A_A}{A_B}=\dfrac{\lambda_AN_A}{\lambda_BN_B}=2\times\dfrac{1/16}{1/4}=2\times\dfrac14=\dfrac12

. Comparing only the numbers of nuclei would give

\dfrac14

, which forgets that the shorter-lived sample has the larger decay constant.

Original question written in the JEE Main 2016 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake