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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`.

A

Number of nuclei of `A` at time `t` is `(1)/(lambda)[alpha-(alpha-lambdaN_(0))e^(-lambdat)]`

B

Number of nuclei of `A` at time `t` is `(1)/(lambda)[(alpha-lambdaN_(0))e^(-lambdat)]`

C

If `alpha=2N_(0)lambda`, then the limiting value of number of nuclei of `A (rarr prop)` will be `2N_(0)`

D

If `alpha=2N_(0)lambda`, then the number of nuclei of `A` after one half-life of `A` will be `N_(0)//2`

Text Solution

Verified by Experts

The correct Answer is:
A, C
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`
`-(1)/(lambda)[l n(alpha-lambdaN)]_(No)^(N)=t`
`l n(alpha-lambdaN)/(alpha-N_(0))= -lambdat`
`N=(1)/(lambda)[alpha-(a-lambdaN_(0))e^(-lambdat)]`
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