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

Explanation

Radio active law:- The rate of disintegration of radio active atoms present in the sample of an element is directly proportional to the number of radioactive atoms present at that instant.
i.e., `(dN)/(dt) propN`
or `(dv)/(dt)=-lambdaN` where `'lambda` is known as the disintegration constant . The-ve sign indicates that the number of radio active nuclei/atoms decreases with the passage of time .
Hence, `(dN)/(N)=-lambda dt`
Integrating both sides we get `log_(e)=- lambda t +C`

Applying the initial condition for `t=0,N=N_(0)`
We get, `C=log_(e) N_(0)`
i.e., `log_(e)((N)/(N_(0)))=- lambdat`
or `(N)/(N_(0))=e^(-lambda t)`
Hence, `N=N_(0)e^(-lambdat)`
A graph 'N' v/s time 't' gives the exponential curve as shown in the figure.