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  • A radioactive nuclide is produced at the constant rate of n per second (say, by bombarding a target with neutrons). The expected number N of nuclei in existence t seconds after the number is <img a

A radioactive nuclide is produced at the constant rate of n per second (say, by bombarding a target with neutrons). The expected number N of nuclei in existence t seconds after the number is N_{0} is given by

Where \lambda is the decay constant of the sample.

Option: 1

\mathrm{N}=\mathrm{N}_0 \mathrm{e}^{-\lambda t}


Option: 2

\mathrm{N}=\frac{\mathrm{n}}{\lambda}+\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}}


Option: 3

\mathrm{N}=\frac{\mathrm{n}}{\lambda}+\left(\mathrm{N}_0-\frac{\mathrm{n}}{\lambda}\right) \mathrm{e}^{-\lambda \mathrm{t}}


Option: 4

None of these


Answers (1)

best_answer

\frac{\mathrm{dN}}{\mathrm{dt}}=\mathrm{n}-\lambda \mathrm{N}

\begin{aligned} & \mathrm{dN}=(\mathrm{n}-\lambda \mathrm{N}) \mathrm{dt} \\ & \int_{\mathrm{N}_0}^{\mathrm{N}} \frac{\mathrm{dN}}{\mathrm{n}-\lambda \mathrm{N}}=\int_0^{\mathrm{t}} \mathrm{dt} \Rightarrow-\frac{1}{\lambda} \int_{\mathrm{N}_0}^{\mathrm{N}} \frac{-\lambda \mathrm{dN}}{\mathrm{n}-\lambda \mathrm{N}}=\mathrm{t} \\ & \Rightarrow-\frac{1}{\lambda}\left[\log _{\mathrm{e}}(\mathrm{n}-\lambda \mathrm{N})\right]_{\mathrm{N}_0}^{\mathrm{N}}=\mathrm{t} \\ & \Rightarrow-\frac{1}{\lambda}\left[\log _{\mathrm{e}}\left(\frac{\mathrm{n}-\lambda \mathrm{N}}{\mathrm{n}-\lambda \mathrm{N}_0}\right)\right]=\mathrm{t} \end{aligned}

\begin{aligned} & \Rightarrow \lambda \mathrm{t}=\left[\log _{\mathrm{e}}\left(\frac{\mathrm{n}-\lambda \mathrm{N}_0}{\mathrm{n}-\lambda \mathrm{N}}\right)\right] \\ & \mathrm{e}^{\lambda \mathrm{t}}=\frac{\mathrm{n}-\lambda \mathrm{N}_0}{\mathrm{n}-\lambda \mathrm{N}} \\ & \mathrm{n}-\lambda \mathrm{N}=\left(\mathrm{n}-\lambda \mathrm{N}_0\right) \mathrm{e}^{-\lambda \mathrm{t}} \\ & \frac{\mathrm{n}}{\lambda}-\left(\frac{\mathrm{n}}{\lambda}-\mathrm{N}_0\right) \mathrm{e}^{-\lambda \mathrm{t}}=\mathrm{N} \end{aligned}

 

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mansi

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