Question

The decay constant of a radioactive sample is $\lambda$. The respective values of its half life and meanlife are
i) $\frac{1}{\lambda }$ and $\left(lo{g}_{e}2\right)$
ii) $\frac{lo{g}_{e}2}{\lambda }$ and $\frac{1}{\lambda }$

Answer 1

We have decay equation as follows

Remaining nuclei , $N=N_0{e}^{-\lambda t}$ ..................(1)

We know that for $t=half-life$ we get remaining nuclei as half of initial nuclei$\left(N_0\right)$

i.e $N=N_0/2$ so using this with $t=t_{1/2}$ in equation-1 we get

$t_{1/2}=\frac{lo{g}_{e}2}{\lambda }$

Mean life is nothing but time(T) for which $N=N_0/e$

so using it in equation-1 we get $T=\frac{1}{\lambda }$

Option 2 is correct.