Obtain the law of radioactivity. Law of radioactive decay

Explanation

(i) At any instant t, the number of decays per unit time, called rate of decay `((dN)/(dt))` is proportional to the number of nuclei (N) at the same instant.
`((dN)/(dt))propN`
(ii) By introducing a proportionality constant, the relation can be written as
`(dN)/(dt)=-lamdaN" "...(1)`
(iii) Here proportionality constant `lamda` is called decay constant which is different for different radioactive sample and the negative sign in the equation implies that the N is decreasing with time.
By rewriting the equation (1), we get
`dN=-lamdaNdt" "...(2)`
(iv) Here dN represents the number of nuclei decaying in the time interval dt.
(v) Let us assume that at time t = 0s, the number of nuclei present in the radioactive sample is `N_(0)`. By integrating the equation (2), we can calculate the number of undecayed nuclei N at any time t.
(vi) From equation (2), we get
`(dN)/N=-lamdadt" "...(3)`
`int_(N_(0))^(N)(dN)/N=int_(0)^(1)lamdadt`
`["In"N]_(N_(0))^(N)=-lamdat`
In `[N/N_(0)]=-lamdat`
Taking exponentials on both sides, we get
`N=N_(0)e^(-lamdat)" "...(4)`
[Note : `e^("inx")=e^(y)rArrx=e^(y)`]
(vii) Equation (4) is called the law of radioactive decay. Here N denotes the number of undecayed nuclei present at any time t and `N_(0)` denotes the number of nuclei at initial time t = 0.
(viii) Note that the number of atoms is decreasing exponentially over the time. This implies that the time taken for all the radioactive nuclei to decay will be infinite.