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Question

A radioactive nuclide is produced at the constant rate of n per second. It decays into a stable nucleus with decay constant λ. If N0 is a number of nuclei at t=0, then expected number N of nuclei in existence after t seconds is given by

A
N=nλ+N0eλt
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B
N=N0eλt
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C
N=nλ+(N0+nλ)eλt
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D
N=nλ+(N0nλ)eλt
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Solution

The correct option is D N=nλ+(N0nλ)eλt
The population is increasing by n and simultaneously decreasing by λN.

dNdt=nλN

Integrating the above equation within proper limit,

NN0dNnλN=t0dt

1λ[ln(nλN)ln(nλN0)]=t

ln[nλNnλN0]=λt

nλNnλN0=eλt

nλN=(nλN0)eλt

nneλt+λN0eλt=λN

N=nλ[1eλt]+N0eλt

N=nλ+(Nonλ)eλt

Hence, (D) is the correct answer.

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