Deduce the relation between half life and average life of a radioactive substance.

Explanation

Relation between half life (T) and average life `(tau)` :
1. We know, the radioactive decay law, `N = N_(0)e^(-lambda t)` _____ (1)
2. Consider, `'N'_(0)` is the number of nuclei present at t = 0 and after time 'T', only `(N_(0))/(2)` are left and after a time '2T', only `(N_(0))/(4)` remain and soon.
3. Substituting `N = (N_(0))/(2)` at t = T in eqn. (1) then
`(N_(0))/(2)=N_(0) e^(-lambda T)rArr (1)/(2)=(1)/(e^(lambda T))rArr e^(lambda T) = 2`
Taking `log_(e )` on both sides, we get
`lambda T = log_(e )^(2)=2.303 log_(e )^(2)=0.693`
`therefore T = (0.693)/(lambda)` _____ (2)
4. Average life `tau = (int tdN)/(No)`
5. But `-(dN)/(dt)=lambda N dN = - lambda N_(0) e^(-lambda t) dt [because " from eqn." (1)]`
6. `tau = ._(0)^(oo)int (t(lambda N_(0)e^(-lambda t)dt))/(N_(0))`
on integrating, we get `tau = (1)/(lambda)` _____ (3)
7. From equs (2) and (3) we get `tau = (T)/(0.693)`
This is the relation between average life and half life of radiactive substance.