Define the average life of a radioactive sample and obtain its relation to decay constant and half-life.

(N/A) The average life of a radioactive sample is defined as the sum of the lifetimes of all individual nuclei divided by the total number of nuclei present initially.
Alternatively,it is the time interval during which the number of nuclei of a radioactive element reduces to $1/e$ times its original number.
Let $\tau$ be the average life. The relation between average life and decay constant $\lambda$ is derived as follows:
From the radioactive decay law,$N = N_0 e^{-\lambda t}$.
The number of nuclei decaying in time $dt$ is $dN = \lambda N_0 e^{-\lambda t} dt$.
The total lifetime of all $N_0$ nuclei is $\int_{0}^{\infty} t dN = \int_{0}^{\infty} t (\lambda N_0 e^{-\lambda t}) dt$.
Thus,$\tau = \frac{1}{N_0} \int_{0}^{\infty} t \lambda N_0 e^{-\lambda t} dt = \lambda \int_{0}^{\infty} t e^{-\lambda t} dt$.
Using integration by parts,$\int_{0}^{\infty} t e^{-\lambda t} dt = \frac{1}{\lambda^2}$.
Therefore,$\tau = \lambda \cdot \frac{1}{\lambda^2} = \frac{1}{\lambda}$.
Since half-life $T_{1/2} = \frac{\ln 2}{\lambda}$,we have $\tau = \frac{T_{1/2}}{\ln 2} \approx 1.44 T_{1/2}$.