Q.Define the mean life of a radioactive sample and derive its relation with the decay constant $\lambda$ and with the half-life $T_{1/2}$.

The mean life $\tau$ is the average time for which a nucleus of a radioactive species survives before decaying. Since the decay law gives an exponentially decreasing population $N(t)=N_0e^{-\lambda t}$, the average lifetime, found by averaging $t$ over the whole decaying population, works out to be exactly the reciprocal of the decay constant:

Combining this with the half-life relation $T_{1/2}=\dfrac{0.693}{\lambda}$ from Section 8.5, and substituting $\lambda=1/\tau$:

$T_{1/2} = 0.693\times\frac{1}{\lambda} = 0.693\,\tau$ …

Define mean life as $\tau=1/\lambda$, then substitute $\lambda=1/\tau$ into $T_{1/2}=0.693/\lambda$ to g …

• Assuming mean life equals half-life -- they are related but not equal, since $0.693\tau=T_{1/2}$. …