(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

(a) `._(15)^(32)Prarr._(16)^(32)s+beta^(-)rArr._(15)^(32)Prarr._(16)^(32)S+._(-1)^(0)e`
(b) Average or Mean life: Average life is defined as the total life time of all the atoms 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 elements is
`tau=(underset(0)overset(N_(0))int t dN)/(N_(0))`
Since `dN= -lambdaN` and `N=N_(0)e^(-lambdat)`
`:. dN = -lambdaN_(0)e^(-lambdat)`
when `N=0` the `t=prop` and when `N=N_(0)`
then `t=0`
`:. tau(underset(prop)overset(0)int t-lambdaN_(0)e^(-lambdat)dt)/(N_(0))=lambdaunderset(-prop)overset(0)int te^(-lambdat)dt`
`tau=lambda[(te^(-lambdat))/(lambda)-(e^(-lambdat))/(lambda^(2))]` (Using intergration by parts)
`tau=lambda[(0-0)-(0-(1)/(lambda^(2)))]=lambdaxx(1)/(lambda^(2))=(1)/(lambda)`
Since `lambda=(0.693)/(t_(1//2))`
`:. tau=((1)/(0.693))/(t_(1//2))=(t_(1//2))/(0.693)=1.44t_(1//2)`
Hence the average life period of a radioactive element is `1.44` times tha half life peroid of the element