Nuclei Set-1

Q1. A nucleus at rest emits an $\alpha$‑particle (mass number $4$) with kinetic energy $E_\alpha=5.00\ \mathrm{MeV}$. Treat nuclear masses as proportional to mass numbers and neglect electron masses. Estimate the recoil kinetic energy of the daughter nucleus with mass number $A_d=200$.

Q2. Assertion (A): The binding energy per nucleon $B/A$ attains a maximum near mass number $A\approx 56$.
Reason (R): A nucleus at this maximum cannot release energy by either fusion or fission because any change in mass number moves it to nuclei with lower $B/A$.

Q3. A freshly prepared pure radioactive sample gives counting rates (background subtracted) $1600\ \text{counts/min}$ at $t=0$ and $200\ \text{counts/min}$ at $t=30\ \text{min}$. Assuming simple exponential decay, the half‑life $t_{1/2}$ of the radionuclide is closest to:

Q4. The table gives approximate binding energy per nucleon $B/A$ (in MeV) for selected mass numbers: $A=20:\ 7.5$, $A=40:\ 8.5$, $A=120:\ 8.6$, $A=240:\ 8.4$. Which of the following statements about the reactions is correct?
(i) Fusion: two $A=20$ nuclei $\to$ one $A=40$ nucleus.
(ii) Fission: one $A=240$ nucleus $\to$ two $A=120$ fragments.
(iii) Fusion: two $A=120$ nuclei $\to$ one $A=240$ nucleus.

Q5. Given $m_p=1.00728\ \mathrm{u}$, $m_n=1.00867\ \mathrm{u}$ and nuclear mass $m(^{16}\mathrm{O})=15.99491\ \mathrm{u}$. Using $1\ \mathrm{u}=931.5\ \mathrm{MeV}/c^2$, the binding energy per nucleon of $^{16}\mathrm{O}$ is approximately:

Q6. Assertion (A): If beta decay produced only the daughter nucleus and an electron, the emitted electrons would all have a single (discrete) kinetic energy.
Reason (R): The observed continuous beta‑spectrum led to the postulate of an additional neutral particle (neutrino) which carries a variable share of energy and momentum.

Q7. Assume $1.00\ \mathrm{kg}$ of $^{235}\mathrm{U}$ undergoes complete fission and each fission releases $200\ \mathrm{MeV}$. Take $N_A=6.0\times10^{23}\ \mathrm{mol}^{-1}$, $1\ \mathrm{MeV}=1.6\times10^{-13}\ \mathrm{J}$ and specific heat of water $c=4184\ \mathrm{J\,kg^{-1}K^{-1}}$. Roughly how many kilograms of water can be heated from $20^\circ\mathrm{C}$ to $100^\circ\mathrm{C}$ by this energy (assume all fission energy is absorbed by water)?

Q8. Consider a decay chain $P \xrightarrow{\lambda_P} D \xrightarrow{\lambda_D} \text{stable}$ with half‑lives $T_P=30\ \text{days}$ and $T_D=2\ \text{days}$. A pure sample of $P$ reaches secular equilibrium. The fraction of atoms that are in the daughter state $D$ at equilibrium is approximately:

Q9. Assertion (A): If the mass difference between parent and daughter nuclei is less than $2m_ec^2$, positron emission ($\beta^+$) is energetically forbidden while electron capture (EC) can still occur.
Reason (R): Positron emission requires creation of an $e^+e^-$ pair costing at least $2m_ec^2\ (\approx 1.022\ \mathrm{MeV})$, whereas EC converts an orbital electron into a neutrino and does not require creating a positron.

Q10. A nuclear excited state shows a resonance width $\Gamma=2.0\ \mathrm{eV}$. Using the energy–time uncertainty relation $\Delta E\,\Delta t\approx \hbar$ with $\hbar=6.58\times10^{-16}\ \mathrm{eV\cdot s}$, the lifetime $\tau$ of the state is approximately: