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Question

Nuclei of a radioactive element are being produced at a constant rate α. The element has a decay constant λ. At time t = 0, there are N0 nuclei of the element.

The number N of nuclei at time t is


A

1λ(αλN0)eλt

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B

1λ(α+λN0)eλt]

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C

1λ[α+(αλN0)eλt]

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D

1λ[α(αλN0)eλt]

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Solution

The correct option is D

1λ[α(αλN0)eλt]


To calculate the number N of nuclei at time t (given that the number at t = 0 is N0), we integrate the above expression from N = N0 to N = N and from t = 0 to t = t, We have

NN0dNαλN=t0dt

or 1λ|loge(αλN)|NN0=|t|t0

or 1λloge(αλNαλN0)=t

or loge(αλNαλN0)=λt or αλNαλN0=eλt

or N=1λ[α(αλN0)eλt]


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