JEE Main 2025 · Physics · Atoms, Nuclei and Dual Nature of Radiation · medium

A sample contains two radioactive nuclides P and Q that do not decay into each other. The half-life of P is 20 minutes and that of Q is 40 minutes.…

Numerical answer

A sample contains two radioactive nuclides P and Q that do not decay into each other. The half-life of P is

20

minutes and that of Q is

40

minutes. At

t=0

the number of nuclei of P is

8

times the number of nuclei of Q. The time (in minutes) after which the numbers of nuclei of P and Q become equal is ______.

Show answer and solution

Answer: 120

The number of nuclei remaining is

N = N_0\,2^{-t/T_{1/2}}

. So

N_P = 8N_{Q0}\,2^{-t/20}

and

N_Q = N_{Q0}\,2^{-t/40}

. Setting them equal:

2^{3-t/20} = 2^{-t/40}

, hence

3 - \dfrac{t}{20} = -\dfrac{t}{40}

, i.e.

3 = t\left(\dfrac1{20}-\dfrac1{40}\right) = \dfrac{t}{40}

. Therefore

t = 120

minutes. Check: after

6

half-lives of P,

8\to\tfrac18

of Q's initial count, and after

3

half-lives of Q the count is

\tfrac18

as well.

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