JEE Advanced 2024 Paper 2 · Physics · Atoms, Nuclei and Dual Nature of Radiation · medium

Two radioactive samples A and B contain initially NA and NB nuclei with NA/NB=4. The half-life of A is 1 h and that of B is 2 h. The two samples have…

Numerical answer

Two radioactive samples

A

and

B

contain initially

N_A

and

N_B

nuclei with

N_A/N_B=4

. The half-life of

A

1\ \mathrm{h}

and that of

B

2\ \mathrm{h}

. The two samples have equal activity after a time

t

hours. The value of

t

Show answer and solution

Answer: 6

Activity is

\lambda N

with

N=N_0e^{-\lambda t}

. Since

\lambda\propto1/T_{1/2}

, the ratio of initial activities is

\dfrac{A_A(0)}{A_B(0)}=\dfrac{N_A}{N_B}\cdot\dfrac{\lambda_A}{\lambda_B}=4\times\dfrac{2}{1}=8

. At time

t

(hours):

A_A=A_A(0)\,2^{-t/1}

and

A_B=A_B(0)\,2^{-t/2}

. Equal activities require

8\cdot2^{-t}=2^{-t/2}

, i.e.

2^{3-t}=2^{-t/2}

, so

3-t=-t/2

and

t=6

. At that time

A

has gone through

6

half-lives (factor

1/64

) and

B

through

3

(factor

1/8

), so the activity ratio

8\times\tfrac{1}{64}\div\tfrac18=1

, confirming the result. Hence

t=6

Original question written in the JEE Advanced 2024 Paper 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake