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(A) When a radioactive nucleus decays by two different processes with decay constants $\lambda_1$ and $\lambda_2$,the total decay constant is $\lambda

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$9$

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(A) When a radioactive nucleus decays by two different processes with decay constants $\lambda_1$ and $\lambda_2$,the total decay constant is $\lambda_{	ext{eff}} = \lambda_1 + \lambda_2$.Since the decay constant $\lambda = \frac{\ln 2}{T}$,we can write the relation for half-lives as:$\frac{\ln 2}{T_{	ext{eff}}} = \frac{\ln 2}{T_1} + \frac{\ln 2}{T_2}$$\frac{1}{T_{	ext{eff}}} = \frac{1}{T_1} + \frac{1}{T_2}$Given $T_1 = 10 \ s$ and $T_2 = 100 \ s$,we have:$\frac{1}{T_{	ext{eff}}} = \frac{1}{10} + \frac{1}{100} = \frac{10 + 1}{100} = \frac{11}{100}$$T_{	ext{eff}} = \frac{100}{11} \approx 9.09 \ s$Thus,the effective half-life is close to $9 \ s$.

$A$ radioactive nucleus decays by two different processes. The half-life for the first process is $10 \ s$ and that for the second is $100 \ s$. The effective half-life of the nucleus is close to $..... \ s$.

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