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State the law of radioactivity and hence...

State the law of radioactivity and hence, show that `N=N_(0)e^(-lambda t)`.

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Statement : .The rate of decay or disintegration is directly proportional to the number of radioactive atoms present in the sample at that instant..
Let `N_(0)` be the initial number of radioactive atoms present in the sample at `t=0`, Let N be the number of atoms at any instant of time .t. .
`(d N) /( dt) = - lambda N` where `lambda=` decay constant
`(dN)/( N) = - lambda dt`
Integrating we get `int_(dN)/( N) =int - lambda t`
`log_(e) N= - lambda t + C " "...(i)`
When `t=0, N=N_(0) log_(e) N_(0) = C " "...(ii)`
Substituting (ii) in (i) we get
`log_(e) N = - lambda t + log_(e) N_(0)`
`log_(e) ((N)/( N_0))=- lambda t`
`(N)/( N_0) = e^(- lambda t)`
`N=N_(0) e^(- lambda t)`
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