Question
The decay constant of a radioactive sample is $\lambda.$ The half$-$life and the average$-$life of the sample are respectively$:$
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Answer
The half$-$life of a radioactive sample $\Big(\text{t}_{\frac{1}2{}}\Big)$ is defined as the time elapsed before half the active nuclei decays.
Let the initial number of the active nuclei present in the sample be $N_0.$
$\frac{\text{N}_{0}}{2}=\text{N}_{\text{0}}\text{e}^{-\lambda\text{t}_{\frac{1}2{}}}$
$\Rightarrow\text{t}_{\frac{1}{2}}=\frac{\text{In}2}{\lambda}$
Average life of the nuclei, $\text{t}_{\text{av}}=\frac{\text{S}}{\text{N}_{0}}=\frac{1}{\lambda}$
Here, $S$ is the sum of all the lives of all the $N$ nuclei that were active at $t = 0$ and $\lambda$ is the decay constant of the sample.
Let the initial number of the active nuclei present in the sample be $N_0.$
$\frac{\text{N}_{0}}{2}=\text{N}_{\text{0}}\text{e}^{-\lambda\text{t}_{\frac{1}2{}}}$
$\Rightarrow\text{t}_{\frac{1}{2}}=\frac{\text{In}2}{\lambda}$
Average life of the nuclei, $\text{t}_{\text{av}}=\frac{\text{S}}{\text{N}_{0}}=\frac{1}{\lambda}$
Here, $S$ is the sum of all the lives of all the $N$ nuclei that were active at $t = 0$ and $\lambda$ is the decay constant of the sample.
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