Nuclei of a radioactive element is produced at constant rate α and they decay with decay constant λ. At t=0, the number of nuclei is 0. The number of active nuclei at time t is
A
αλ(1−e−λt)
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B
α(1−e−λt)
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C
αλ−e−λt
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D
αe−λt
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Solution
The correct option is Aαλ(1−e−λt) According to the information, dNdt=−λN+α
Rearranging and integrating, ∫N0dN−λN+α=∫t0dt
Let −λN+α=p ⇒−λdN=dp ⇒dN=−dpλ
Substituting back in integral, ∫pαdpp=∫t0−λdt
This gives p=αe−λt
Putting the expression for p, we get p=−λN+α=αe−λt
This gives N=αλ(1−e−λt)