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Consider a radioactive disintegration ac...

Consider a radioactive disintegration according to the equation `ArarrBrarrC`. Decay constant of A and B is same and equal to `lambda`. Number of nuclei of A, B and C are `N_0` , 0, 0 respectively at `t=0`. Find
(a) number of nuclei of B as function of time t.
(b) time t at which the activity of B is maximum and the value of maximum activity of B.

Text Solution

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The correct Answer is:
A, B, D
(a) A B C
At `t=0` `N_0` 0 0
`At t `N_1` `N_2` `N_3`
Here, `N_1=N_0e^(-lambdat)` …(i)
`(dN_2)/(dt)=lambda(N_1-N_2)`
or `(dN_2)/(dt)=lambdaN_0e^(-lambdat)-lambdaN_2`
or `dN_2+lambdaN_2dt=lambdaN_0e^(-lambdat)`
`:.` `e^lambdatdN_2+lambdaN_2e^lambdat=lambdaN_0dt`
or `d(N_2e^lambdat)=lambdaN_0dt`
`:.` `N_2e^lambdat=lambdaN_0t+C`
At `t=0`, `N_2=0`,
`:.` `C=0`
`:.` `N_2=lambdaN_0(te^-lambdat)` (b) Activity of B is
`R_2=lambdaN_2=lambda^2N_0(te^(-lambdat))`
For maximum activity, `(dR_2)/(dt)=0`
`:.` `t=1/lambda`
`:.` `R_(max)=(lambdaN_0)/(e)`
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