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Question

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(1eλt)
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B
α(1eλt)
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C
αλeλt
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D
αeλt
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Solution

The correct option is A αλ(1eλt)
According to the information,
dNdt=λN+α
Rearranging and integrating,
N0dNλN+α=t0dt

Let λN+α=p
λdN=dp
dN=dpλ

Substituting back in integral,
pαdpp=t0λdt
This gives p=αeλt
Putting the expression for p, we get
p=λN+α=αeλt
This gives
N=αλ(1eλt)


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