Solved: PHYSICS Question for NEET

NEET PHYSICSMedium

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

1/e

1/(1+e)

(e-1)/(e+1)

(e-1)/e

Identify the Decay Law: The number of undecayed (remaining) nuclei $N$ at time $t$ is given by the radioactive decay law: $N = N_0 e^{-\lambda t}$ , where $N_0$ is the initial number of nuclei and $\lambda$ is the decay constant .
Define Average Life ( $\tau$ ): The average or mean life of a radioactive species is defined as the reciprocal of the decay constant, i.e., $\tau = 1/\lambda$ .
Calculate Remaining Nuclei: Substitute $t = \tau = 1/\lambda$ into the decay equation to find the fraction remaining: $N = N_0 e^{-\lambda(1/\lambda)} = N_0 e^{-1} = \frac{N_0}{e}$
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: $N_{\text{decayed}} = N_0 - N = N_0 - \frac{N_0}{e} = N_0 \left(1 - \frac{1}{e}\right) = N_0 \left(\frac{e-1}{e}\right)$
Determine Fraction: The fraction of the original number that has decayed is: $\text{Fraction} = \frac{N_{\text{decayed}}}{N_0} = \frac{e-1}{e}$

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