Define average life of a radioactive substance. Obtain the relation between decay constant and average life.

Explanation

Average life `(tau)` : It is equal to the total life time of all the `N_(0)` nuclei divided by the total number of original nuclei `N_(0)`. It is denoted by `tau`.
Relation between decay constant and average life :
1. Let `N_(0)` be the radioactive nuclei that are present at t = 0 in the radioactive sample. The no' of nuclei which decay between t and t + dt is dN.
2. The total life time of these dN nuclei is t dN. the total life time of all the nuclei present initially in the sample `= int_(0)" t dN"`
3. Average life time `(tau)` is equal to the total life time of all the `N_(0)` nuclei divided by the total number of original nuclei `N_(0)`.
4. Average life `tau = (int " t dN")/(N_(0))`
But `(dN)/(dt)= - lambda N`
`dN= - lambda N dt = N_(0)e^(-lambda t)dt [because N = N_(0) e^(-lambda t)]`
5. `therefore tau = (int_(0)t lambda N_(0)e^(-lambda t)dt)/(N_(0))`
On integration, we get `tau = (1)/(lambda)`
`tau = (T)/(0.693) " " [because lambda = (0.693)/(T)]`
6. From the above equation the reciprocal of the decay constant gives us the average life of a radioactive sample.