JEE Main 2018 · Physics · Atoms, Nuclei and Dual Nature of Radiation · easy

Two radioactive samples A and B contain equal numbers of nuclei at t=0. Their half-lives are 20 minutes and 40 minutes respectively. The activities…

Single correct option

Two radioactive samples

A

and

B

contain equal numbers of nuclei at

t=0

. Their half-lives are

20

minutes and

40

minutes respectively. The activities of the two samples become equal after a time of

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Answer: (D) $40$ min

Activity is

\lambda N

with

N=N_0e^{-\lambda t}

. With

\lambda_A=2\lambda_B

and equal initial numbers,

\dfrac{A_A}{A_B}=\dfrac{\lambda_A}{\lambda_B}e^{-(\lambda_A-\lambda_B)t}=2\cdot2^{-t/20}\cdot2^{t/40}=2^{\,1-t/40}

. Setting the ratio to

1

gives

1-\dfrac t{40}=0

, so

t=40

min. Check: at

40

min,

N_A=N_0/4

and

N_B=N_0/2

, so

\lambda_AN_A=2\lambda_B\cdot\dfrac{N_0}{4}=\lambda_B\dfrac{N_0}{2}=\lambda_BN_B

Original question written in the JEE Main 2018 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake