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) prop N`
Or `(dv )/(dt) =- lamda N ` where `. lamda .` is known as the disinte gration constant . the =- ve sign indicates that the number of radio active nuclei / atoms decreases with the passage of time .
Hence , ` (dN )/( N ) =-lamda dt `
integrating both sides we get
` log _(e) =- lamda 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))=-lamda t `
or ` (N)/(N_0)=e^(-lamda t) `

hence `N=N _0 e^(-lamda t)`
A graph `.N. v//s ` time .t. gives the exponential curve as shown in the figure .