(a) Write symbolically the `beta`-decay process of `""_(15)^(32)P`.
(b) Derive an expression for the average life of a radionuclide . Give its relationship with the half-life.

Explanation

(b) Average or Mean Life : Average life is defined as the total life time of all the atoms of the element divided by the total number of atoms present in the sample of the element .
By definition , average life time of the radioactive element is
`tau = (int_(0)^(N_(0)) t dN)/(N_(0))`
Since , `" " dN = - lambda N` and `N = N_(0) e^(-lambda t)`
`therefore " " dN = - lambda N_(0) e^(-lambda t)` dt
When N = 0 then `t = oo` and when N = `N_(0)` then t = 0
`tau= (int_(oo)^(0) t. - lambda N_(0) e^(-lambda t) dt)/( N_(0)) = lambda int_(oo)^(0) t e^(-lambda t) dt`
`tau = lambda [ (t e^(-lambda t))/(lambda) - (e^(-lambda t))/(lambda^(2))]_(0)^(oo) " "` (Using integration by parts )
`tau = lambda [ (0-0) - (0 - (1)/(lambda^(2)))] = lambda xx (1)/(lambda^(2)) = (1)/(lambda)`
Since , `" " lambda = (0.693)/(t_(1//2))`
`therefore " " tau = (1)/((0.693)/(t_(1//2))) = (tau_(1//2))/(0.693) = 1.44 tau_(1//2)`
Hence , the average life period of a radioactive element is 1.44 times the half life period of the element .