(a) Derive the law of radioactive decay `N=N_(0)e^(-lamdat)`
(b) The half - life of `""_(92)^(238)U` undergoing `alpha` - decay is `4.5 xx10^(9)` years. Find its mean life.
(c) What fraction of the initial mass of a radioactive substance will decay in five half - life periods ?

Explanation

(a) According to the law of radioactive decay , we know that rate of disintegration of a radioactive sample is directly proportional to the actual quantity of that material at that instant. Mathematically ,
`(-dN)/(dt)=lamdaN`
where `lamda` is the disintegration (or decay) constant of given radioactive substance
`:. (dN)/N=lamda dt `
On integration , we have `int_(N_0)^(N_t)(dN)/N=-lamdaint_(0)^(t)dtimplies[log_eN]_(N_0)^(N_t)=-lamda[t]_0^(t)`
`implies log_(e) N_(t)-log_(e)N_(0)=-lamdat" or " log_(e)(N_t/N_0)=-lamdat`
or `N_t/N_0=e^(-lamdat)impliesN_t=N_(0)e^(-lamdat)`
(b) As half life period of `""_(92)^(238)U,T_(1/2)=4.5xx10^9` years
`:.` Mean life `tau=(T_(1/2))/(0.693)=(4.5xx10^9)/(0.693)=6.5xx10^(9)` years
(c) Fraction of radioactive subsance left intact after n = 5 half lives is = `[1/2]^n=[1/2]^(5)=1/32`
`:.` Fraction of the initial mass of a radioactive substance decayed in 5 half lives
`=1-(1)/32 = 31 / (32) ` or 96.875%