The fraction of the original number of radioactive atoms that disintegrates (decays) during the average lifeti

Question

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

Accepted Answer

1. **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 .
2. **Define Average Life ($	au$):** The average or mean life of a radioactive species is defined as the reciprocal of the decay constant, i.e., $	au = 1/\lambda$ .
3. **Calculate Remaining Nuclei:** Substitute $t = 	au = 1/\lambda$ into the decay equation to find the fraction remaining:
   $$ N = N_0 e^{-\lambda(1/\lambda)} = N_0 e^{-1} = \frac{N_0}{e} $$
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:
   $$ N_{	ext{decayed}} = N_0 - N = N_0 - \frac{N_0}{e} = N_0 \left(1 - \frac{1}{e}ight) = N_0 \left(\frac{e-1}{e}ight) $$
5. **Determine Fraction:** The fraction of the original number that has decayed is:
   $$ 	ext{Fraction} = \frac{N_{	ext{decayed}}}{N_0} = \frac{e-1}{e} $$

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