Question

The decay constant of a radioactive sample is λ. The half-life and the average-life of the sample are respectively

Options

• 1/λ and (In 2/λ)

• (In 2/λ) and 1/λ

• λ(In 2) and 1/λ

• λ/(In 2) and 1/λ.

Answer 1

(ln 2/λ) and 1/λ

The half-life of a radioactive sample `(t_(1"/"2))`  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` .

`N_0/2 = N_0e^(-lambdat_"1/2")`

⇒ `t_"1/2" = ("In 2")/lambda`

Average life of the nuclei, `t_(av) = S/N_0 = 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.