Question

The decay constant of a radioactive sample is λ. The half-life and the average-life of the sample are respectively
(a) 1/λ and (In 2/λ)
(b) (In 2/λ) and 1/λ
(c) λ(In 2) and 1/λ
(d) λ/(In 2) and 1/λ.

Answer 1

(b) (ln 2/λ) and 1/λ

The half-life of a radioactive sample ($t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$) 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 N₀.
$$\begin{gathered} \frac{N_o}{2}=N_o{e}^{-\lambda t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} \\ \Rightarrow t_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\frac{\ln 2}{\lambda } \end{gathered}$$

Average life of the nuclei, $t_{av}=\frac{S}{N_o}=\frac{1}{\lambda }$
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.