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NEET PHYSICSNUCLEIMedium

Question

The fraction of the original number of radioactive atoms that disintegrates (decays) during the average lifetime of a radioactive substance will be:

A

1/e

B

1/(1+e)

C

(e-1)/(e+1)

D

(e-1)/e

Step-by-Step Solution

  1. Identify the Decay Law: The number of undecayed (remaining) nuclei NN at time tt is given by the radioactive decay law: N=N0eλtN = N_0 e^{-\lambda t}, where N0N_0 is the initial number of nuclei and λ\lambda is the decay constant .
  2. Define Average Life (τ\tau): The average or mean life of a radioactive species is defined as the reciprocal of the decay constant, i.e., τ=1/λ\tau = 1/\lambda .
  3. Calculate Remaining Nuclei: Substitute t=τ=1/λt = \tau = 1/\lambda into the decay equation to find the fraction remaining: N=N0eλ(1/λ)=N0e1=N0eN = N_0 e^{-\lambda(1/\lambda)} = N_0 e^{-1} = \frac{N_0}{e}
  4. Calculate Decayed Nuclei: The question asks for the fraction that disintegrates (decays). The number of decayed nuclei is the initial number minus the remaining number: Ndecayed=N0N=N0N0e=N0(11e)=N0(e1e)N_{\text{decayed}} = N_0 - N = N_0 - \frac{N_0}{e} = N_0 \left(1 - \frac{1}{e}\right) = N_0 \left(\frac{e-1}{e}\right)
  5. Determine Fraction: The fraction of the original number that has decayed is: Fraction=NdecayedN0=e1e\text{Fraction} = \frac{N_{\text{decayed}}}{N_0} = \frac{e-1}{e}

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from NUCLEI. Mastering this topic is crucial for scoring well in the upcoming medical entrance examinations. Solving conceptually related problems will help you understand the nuances of these concepts and improve your problem-solving speed.

PHYSICSNUCLEIfractionoriginalnumberradioactivedisintegrates

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