JEE Main 2022 · Physics · Atoms, Nuclei and Dual Nature of Radiation · medium

Two radioactive samples contain equal numbers of nuclei at t=0. Sample P has half-life 10 min and sample Q has half-life 20 min. After how many…

Numerical answer

Two radioactive samples contain equal numbers of nuclei at

t=0

. Sample P has half-life

10

min and sample Q has half-life

20

min. After how many minutes (from

t=0

) will the activities of the two samples be equal?

Show answer and solution

Answer: 20

Activity

=\lambda N

and

\lambda\propto1/T_{1/2}

, so initially

A_P=2A_Q

(equal

N_0

\lambda_P=2\lambda_Q

). At time

t

A_P=2A_0'\,2^{-t/10}

and

A_Q=A_0'\,2^{-t/20}

, where

A_0'=\lambda_QN_0

. Setting them equal:

2\cdot2^{-t/10}=2^{-t/20}

, so

1-\dfrac t{10}=-\dfrac t{20}

, giving

1=\dfrac{t}{20}

and

t=20

min. Check: at 20 min

N_P=N_0/4

N_Q=N_0/2

and

\lambda_PN_P=\lambda_QN_Q

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