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Physics · Nuclear Physics

JEE Main 2025 — 23 January, Morning Shift — Question 54

A radioactive nucleus n_2 has 3 times the decay constant as compared

to the decay constant of another radioactive nucleus n_1. If initial number

of both nuclei are the same, what is the ratio of number of nuclei .

of n_2 to the number of nuclei of n_1, after one half-life of n_1 ?

  1. Option A:

    1/4

    Correct
  2. Option B:

    1/8

  3. Option C:

    4

  4. Option D:

    8

Answer: A

Step-by-step solution

N2=N0e−3λt\mathrm{N}_{2}=\mathrm{N}_{0} \mathrm{e}^{-3 \lambda \mathrm{t}} N1=N0e−λtN_{1}=N_{0} e^{-\lambda t} N2 N1=e−2λt\frac{\mathrm{N}_{2}}{\mathrm{~N}_{1}}=\mathrm{e}^{-2 \lambda \mathrm{t}}

N02=N0e−λt\frac{\mathrm{N}_{0}}{2}=\mathrm{N}_{0} \mathrm{e}^{-\lambda \mathrm{t}} λt=ln⁡2\lambda t=\ln 2 t=ln⁡2λ\mathrm{t}=\frac{\ln 2}{\lambda} =e−2λln⁡2λ=\mathrm{e}^{-2 \lambda \frac{\ln 2}{\lambda}} N2 N1=14\frac{\mathrm{N}_{2}}{\mathrm{~N}_{1}}=\frac{1}{4}

Answer key and solution verified before publishing.

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Exam
JEE Main 2025
Paper
23 January, Morning Shift
Subject
Physics
Chapter
Nuclear Physics
Topic
Laws of Radioactive Decay