Question

The decay constant of a radioactive sample is $\lambda$. The half-life and mean life of the sample are respectively:

• A. $1/\lambda ,\ln 2/\lambda$
• B. $\ln 2/\lambda ,1/\lambda$
• C. $\lambda \ln 2,1/\lambda$
• D. $\lambda /{Pn}_2,1/\lambda$

Answer 1

The correct option is B $\ln 2/\lambda ,1/\lambda$

Decay constant =$\lambda$.

Half life ($t_{half})=\frac{1}{\lambda }ln(\frac{N_o}{N_o-N})$

Substituting$N=N_o$, we get $t_{half}=\frac{1}{\lambda }ln2$

Mean life = $\frac{1}{N_o}{\int }_0^{N_o}tdN$

substitute $N=N_o{e}^{-\lambda t}$

Solving this we get, Mean life=$\frac{1}{\lambda }$