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Starting with the law of radioactive dis...

Starting with the law of radioactive disintegration, show that : `N=N_(0)e^(-lamdat)`, where the terms have their usual meaning.

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Let N be the number of atoms present in a radioactive substance at any instant. Then,
`-(dN)/(dt)=lamdaN`
Where `lamda` is decay constant.
`(dN)/(N)=-lamdadt`
On integrating, we get
`log_(e)N=-lamdat+C" "......(1)`
where C is the integration constant. Suppose there were `N_(0)` atoms in the beginning.
Then, `N=N_(0)` at t=0`
Putting it in equation (1), we get
`log_(e)N_(0)=C`.
Now, the equation (1) becomes
`log_(e)N=-lamdat+log_(e)N_(0)`
`log_(e)N-log_(e)N_(0)=-lamdat`
or `log_(e)""(N)/(N_(0))=-lamdat`
or `(N)/(N_(0))=e^(-lamdat)`
`N=N_(0)e^(-lamdat)`
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