If \('f^{\prime}\) denotes the ratio of the number of nuclei decayed \(\left(N_{d}\right)\) to the number of nuclei at \(t=0\) \(\left({N}_{0}\right)\) then for a collection of radioactive nuclei, the rate of change of \('f'\) with respect to time is given as: \([\lambda\) is the radioactive decay constant]

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Answer & Solution

Correct Answer

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

Step-by-step Solution

Detailed explanation

\(N = N _{0} e ^{-\lambda t}\)

\(N _{ d }= N _{0}- N\)

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

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

\(\frac{ df }{ dt }=\lambda e ^{-\lambda t }\)

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