What is radioactivity ? State the law of radioactive decay. Show that radioactive decay is exponential in nature.

Explanation

1. Radioactivity : The nuclei of certain elements disintegrate spontaneously by emitting alpha `(alpha)`, beta `(beta)` and gamma `(gamma)` rays. This phenomenon is called Radioactivity or Natural radioactivity.
2. Law of radioactivity decay : ''The rate of radioactive decay `((dN)/(dt))` (or) the number of nuclei decaying per unit time at any instant, is directly proportional to the number of nuclei (N) present at that instant is called law of radioactivity decay''.
3. Radioactive decay is exponential in nature : Consider a radioactive substance. Let the number of nuclei present in the sample at t = 0, be `N_(0)` and let N be the radioactive nuclei remain at an instant t.
`(dN)/(dt)prop N rArr (dN)/(dt) = - lambda N`
`dN = - lambda Ndt " "` _________ (1)
The proportionally constant `lambda` is called decay constant or disintegration constant. The negative sign indicates the decrease in the number of nuclei.
4. From eq. `(dN)/(N)= - lambda dt " "` _________ (2)
5. Integrating on both sides
`int (dN)/(N)= - lambda int dt`
`l n N = - lambda t + C " "` _________ (3)
Where C = Integration constant.
6. At `t = 0, N = N_(0)`. Substituting in eq. (3), we get `l n N_(0) = C`
`therefore l n N = - lambda t + l n N_(0)`
`l n N - l n N_(0) =- lambda t`
`l n ((N)/(N_(0)))= - lambda t`
`therefore N = N_(0)e^(-lambda t)`
The above equation represents radioactive decay law.
7. It states that the number of radioactive nuclei in a radioactive sample decreases exponentially with time.