A radioactive substance `"A"` having `N_(0)` active nuclei at `t=0`, decays to another radioactive substance `"B"` with decay constant `lambda_(1)`. `B` further decays to a stable substance `"C"` with decay constant `lambda_(2)`. (a) Find the number of nuclei of `A, B` and `C` time `t`. (b) What should be the answer of part (a) if `lambda_(1) gt gt lambda_(2)` and `lambda_(1) lt lt lambda_(2)`

Explanation

(a) The decay scheme is as shown
`{:(,Aoverset(lambda_(1))rarr,Boverset(lambda_(2))rarr,C,("Stable")),(t=0,N_(0),0,0,),(t,N_(1),N_(2),N_(3),):}`
Here `N_(1), N_(2)` and `N_(3)` represent the nuclei of `A, B` and `C` at any time `t`
For `A`, we can write
`N_(1)=N_(0)e^(-lambda_(1)t)`......(1)
For `B`, we can write
`(dN_(2))/(dt)=lambda_(1)N_(1)-lambda_(2)N_(2)`......(2)
or `(dN_(2))/(dt)+lambda_(2)N_(2)=lambda_(1)N_(1)`
This is a linear differential equation with intergrating factor
`I.F=e^(lambda_(1)t)`
`e^(lambda_(2)t)(dN_(2))/(dt)+e^(lambda_(2)t)lambda_(2)N_(2)=lambda_(1)N_(1)e^(lambda_(2)t)`
`int d (N_(2)e^(lambda_(2)t))=intlambda_(1)N_(1)e^(lambda_(2)t)dt`
`N_(2)e^(lambda_(2)t)=lambda_(1)N_(0)int e^(-lambda_(1)t)e^(lambda_(2)t)dt` using.....(1)
`N_(2)e^(lambda2t)=lambda_(1)N_(0)=(e^((lambda_(2)-lambda_(1))t))/(lambda_(2)-lambda_(1))+C`.......(3)
At `t=0, N_(2)=0" " 0=(lambda_(1)N_(0))/(lambda_(2)-lambda_(1))+C`
Hence `C=(lambda_(1)N_(0))/(lambda_(1)-lambda_(2))`
Using `C` in eqn. (3), we get
`N_(2)=(lambda_(2)N_(0))/(lambda_(2)-lambda_(1))(e^(-lambda_(2_(t)t)-e^(-lambda_(2)t)))` and `N_(1)+N_(2)+N_(3)=N_(0)`
`:. N_(3)=N_(0)-(N_(1)+N_(2))`
(b) For `lambda_(1) gt gt lambda_(2)" " N_(2)=(lambdaN_(0))/(-lambda_(1))(-e^(-lambda_(2)t))" " =N_(0)e^(-lambda_(2)t)`
For `lambda_(1) lt lt lambda_(2) " " N_(2)=(lambda_(1)N_(0))/(lambda_(2))(e^(-lambda_(1)t))=0`
If `lambda_(1) gt lambda_(2)` that means `A` will decay very fast to `'B'` and `B` will then decay slowly. We can say that practically `N_(1)` vanishes in very short time & `B` has initial no. of atoms as `N_(0)`
`:. lambda_(1)N_(1)~~lambda_(2)N_(2) " " rArrN_(2)=(lambda_(1)N_(1))/(lambda_(2))=(lambda_(1)N_(0))/(lambda_(2))(e^(-lambda_(1)t))`