## Question

###### The formation of an isotope in a nuclear reactor is given by $$ \frac{d N_{2}}{d t}=n v \sigma_{1} N_{10}-\lambda_{2} N_{2}(t)-n v \sigma_{2} N_{2}(t) $$ Here, the product $n v$ is the neutron flux, neutrons per cubic centimeter, times centimeters per second mean velocity; $\sigma_{1}$ and $\sigma_{2}\left(\mathrm{~cm}^{2}\right)$ are measures of the probability of neutron absorption by the original isotope, concentration $N_{10}$. which is assumed constant and the newly formed isotope, concentr

The formation of an isotope in a nuclear reactor is given by $$ \frac{d N_{2}}{d t}=n v \sigma_{1} N_{10}-\lambda_{2} N_{2}(t)-n v \sigma_{2} N_{2}(t) $$ Here, the product $n v$ is the neutron flux, neutrons per cubic centimeter, times centimeters per second mean velocity; $\sigma_{1}$ and $\sigma_{2}\left(\mathrm{~cm}^{2}\right)$ are measures of the probability of neutron absorption by the original isotope, concentration $N_{10}$. which is assumed constant and the newly formed isotope, concentration $N_{2}$, respectively. The radioactive decay constant for the isotope is $\lambda_{2}$. (a) Find the concentration $N_{2}$ of the new isotope as a function of time. (b) If the original element is $\mathrm{Eu}^{153}, \sigma_{1}=400 \mathrm{barns}=400 \times 10^{-24} \mathrm{~cm}^{2}$, $\sigma_{2}=1000$ barns $=1000 \times 10^{-24} \mathrm{~cm}^{2}$, and $\lambda_{2}=1.4 \times 10^{-9} \mathrm{~s}^{-1}$. If $N_{10}=10^{20}$ and $(n v)=10^{9} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}$, find $N_{2}$, the concentration of Eu $^{154}$ after one year of continuous irradiation. Is the assumption that $N_{1}$ is constant justified?