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Atoms and Nuclei question

2021 · Shift 1 · Q42
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Atoms and Nuclei question

2021 · Shift 1 · Q42

JEE AdvancedPhysicsAtoms and NucleiMCQ+3 / −1
A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is
  1. A
    50
  2. B
    75
  3. C
    350
  4. D
    525
View written solutionFree

Correct answer: D

  1. Let the initial number of nuclei be N0=1000N_0 = 1000N0​=1000 and the half-life of nucleus QQQ be T1/2=20 min.T_{1/2} = 20\text{ min}.T1/2​=20 min.

  2. In one hour, t=60 min=3×20 mint = 60\text{ min} = 3\times 20\text{ min}t=60 min=3×20 min so this is equal to 3 half-lives.

  3. After 3 half-lives, the number of undecayed nuclei is N=N0(12)3=1000⋅18=125.N = N_0\left(\frac12\right)^3 = 1000\cdot \frac18 = 125.N=N0​(21​)3=1000⋅81​=125.

  4. Hence, the total number of nuclei that decay in the first hour is 1000−125=875.1000 - 125 = 875.1000−125=875.

  5. Out of all decays, the probability of alpha-decay is 60%=0.660\% = 0.660%=0.6. Therefore, the expected number of alpha-decays is Nα=0.6×875=525.N_\alpha = 0.6 \times 875 = 525.Nα​=0.6×875=525.

  6. So the number of alpha-decays of QQQ in the first one hour is 525.\boxed{525}.525​.

  7. Checking options:

    • A: 50 ❌
    • B: 75 ❌
    • C: 350 ❌
    • D: 525 ✅

Therefore, the correct option is D.

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