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Nuclei of a radioactive element A are be...

Nuclei of a radioactive element A are being produced at a constant rate `alpha`. The element has a decay constant `lambda`. At time `t=0`, there are `N_0` nuclei of the element.
(a) Calculate the number N of nuclei of A at time t.
(b) If `alpha=2N_0lambda`, calculate the number of nuclei of A after one half-life of A, and also the limiting value of N as `trarroo`.

Text Solution

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The correct Answer is:
A, B, C, D
(a) Let at time t, number of radioactive nuclei are N.
Net rate of formation of nuclei of A
`(dN)/(dt)=alpha-lambdaN` or `(dN)/(alpha-lambdaN)=dt`
or `int_(N_0)^N(dN)/(alpha-N)=int_0^tdt`
Solving this equation, we get
`N=1/lambda[alpha-(alpha-lambdaN_0)e^(-lambdat)]` ...(i)
(b) (i) Substituting `alpha=2lambdaN_0` and `t=t_(1//2)=(1n2)/(lambda)` in Eq. (i), we get `N=3/2N_0`
(ii) Substituting `alpha=2lambdaN_0` and `trarroo` in Eq. (i), we get
`N=(alpha)/(lambda)=2N_0`
or `N=2N_0`
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