Question

The relation between half-life $T$ of a radioactive sample and its mean life $\tau$ is:

• A. $T=0.693\tau$
• B. $\tau =0.693T$
• C. $\tau =T$
• D. $\tau =2.718T$

Answer 1

Answer: (A)

$T=0.693\tau$

Let $\lambda$ is the decay constant.

$N=N_0{e}^{-\lambda t}$

$\frac{N_0}{2}=N_0{e}^{-\lambda T}$

$\frac{1}{2}=e^{-\lambda T}$

$-ln2=-\lambda T$

Half life $T=\frac{ln}{\lambda }$ -----eq.1

Mean life $\tau =\frac{{\int }_0^{\infty }-\lambda Ntdt}{{\int }_0^{\infty }-\lambda Ndt}$

Mean life $\tau =\frac{1}{\lambda }$-----eq.2

from eq.1 and eq.2

$T=ln2\ast \tau T=0.693\tau$

Hence option 'A' is correct.