જો \(f\) એ ક્ષય પામેલા ન્યુક્લિયસની સંખ્યા \(\left(N_{d}\right)\) અને \(t=0\) સમયે ન્યુક્લિયસની સંખ્યા \(\left({N}_{0}\right)\) નો ગુણોત્તર દર્શાવે તો રેડિયોએક્ટિવ ન્યુક્લિયસોના ગ્રુપ માટે \(f\) નો સમય સાપેક્ષ ફેરફારનો દર ......... વડે આપી શકાય. \([\lambda\) એ રેડિયોએક્ટિવ ક્ષય નિયાતાંક છે.]

A \(\lambda\left(1-{e}^{-\lambda t}\right)\)
B \(-\lambda e^{-\lambda t}\)
C \(\lambda e^{-\lambda t}\)
D \(-\lambda\left(1-{e}^{-\lambda . t}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\lambda e^{-\lambda t}\)

Step-by-step Solution

Detailed explanation

\(N = N _{0} e ^{-\lambda t}\) \(N _{ d }= N _{0}- N\) \(N _{ d }= N _{0}\left(1- e ^{-\lambda t }\right)\) \(\frac{ N _{ d }}{ N _{0}}= f =1- e ^{-\lambda t }\) \(\frac{ df }{ dt }=\lambda e ^{-\lambda t }\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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