(a) Write symbolically the `beta`-decay process of phosphorus.
(b) Derive an expression for the average life of a radionuclide. Give its relationship with the half-life.

Explanation

(a) `beta-` decay process
`""_(15)^(32) P to ""_(16)^(32) S + beta^(-) + bar(v) "or"_(15)^(32) P to ""_(16)^(32) X + ""_(-1)e^(0) + bar(v)`
(b) Derivation of average life :
`tau = (lambda N_(0) int_(0)^(**)te^(-lambda t) dt)/( N_(0) ) = lambda int_(0)^(**) te^(-lambda t) dt`
`rArr tau = 1//lambda`
Relation of average life with half life :
`T_(1//2) = (I_(n) 2) /( lambda) = tau (I_(n) 2)`
`T_(1//2) = tau log_(e)^(2)`