For radioactive disintegration of a radioactive substance, show that
`N = N_(0)e^(-lambdat)`
where the terms have their usual meaning,

Explanation

The number of atoms disintegrated per second at any instant is directly proportional to the number of radioactive atoms actually present in the sample at that instant. Mathematically,
`(dN)/(dt) prop N`
where `lambda` is disintegration constant. `(dN)/(dt ) =- lambdaN`
`(dN)/(N) = -lambdadt`
Integrating on both sides we find
`int (dN)/(N) =- lambda int dt`
log N =`-lambdat +C`
At t=0 , N` =N_(0)`
i.e., `C= log_(e) N_(0)`
Thus from above equations
`log_(e) N= -lambdat+ log_(e) N_(0)`
`log(N)/(N_(0))= - lambdat`
As `N= N_(0)e^(-lambdat)`
This is radioactive decay law which shows that decay is exponenetial in nature. Hence proved.