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JEE Main 2023 · Physics · Atoms, Nuclei and Dual Nature of Radiation · medium

A radioactive nucleus can decay by two independent modes: by α-emission with partial half-life 6 h and by β-emission with partial half-life 3 h.…

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A radioactive nucleus can decay by two independent modes: by α\alpha-emission with partial half-life 6 h6\ \mathrm{h} and by β\beta-emission with partial half-life 3 h3\ \mathrm{h}. Starting with N0N_0 nuclei, the number of nuclei that have undergone β\beta-decay after 4 h4\ \mathrm{h} is
  1. A. 3N04\dfrac{3N_0}{4}
  2. B. N02\dfrac{N_0}{2}
  3. C. 2N03\dfrac{2N_0}{3}
  4. D. N03\dfrac{N_0}{3}
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Answer: (B) N02\dfrac{N_0}{2}

Decay constants add: λ=λα+λβ=ln⁡2(16+13)=ln⁡22\lambda=\lambda_\alpha+\lambda_\beta=\ln2\left(\dfrac16+\dfrac13\right)=\dfrac{\ln2}{2}, so the effective half-life is 2 h2\ \mathrm{h}. After 4 h4\ \mathrm{h} two half-lives have passed and N0/4N_0/4 remain; 3N04\dfrac{3N_0}{4} have decayed by either mode. Each decay is a β\beta-decay with probability λβ/λ=1/31/2=23\lambda_\beta/\lambda=\dfrac{1/3}{1/2}=\dfrac23, so the β\beta-decays number 23⋅3N04=N02\dfrac23\cdot\dfrac{3N_0}{4}=\dfrac{N_0}{2}.
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