Nuclei Set-2

Q1. A $^4\text{He}$ nucleus has measured nuclear mass $m(^{4}\text{He})=4.001506\ \text{u}$. Given $m_p=1.007276\ \text{u}$, $m_n=1.008665\ \text{u}$ and $1\ \text{u}=931.5\ \text{MeV}/c^2$, the binding energy per nucleon of $^4\text{He}$ is closest to:

Q2. For the fusion reaction $^2_1\text{H}+^3_1\text{H}\to ^4_2\text{He}+n+Q$, given $m(^2\text{H})=2.014102\ \text{u}$, $m(^3\text{H})=3.016049\ \text{u}$, $m(^4\text{He})=4.002603\ \text{u}$ and $m(n)=1.008665\ \text{u}$ with $1\ \text{u}=931.5\ \text{MeV}/c^2$, the released energy $Q$ is approximately:

Q3. Assertion (A): In nuclear $\beta^-$ decay the emitted electron is one of the orbital electrons of the parent atom.
Reason (R): The $\beta^-$ electron is created inside the nucleus during the weak interaction and is therefore not an orbital electron.

Q4. A parent nucleus P (half-life $T_p=20\ \text{days}$) decays to daughter D (half-life $T_d=5\ \text{days}$). Initially $N_P(0)=10^6$ and $N_D(0)=0$. The time $t$ (in days) when the activities of parent and daughter become equal is closest to:

Q5. For three isobars with the same mass number $A$ the binding energies are given as $B(A,Z-1)=500\ \text{MeV}$, $B(A,Z)=498\ \text{MeV}$ and $B(A,Z+1)=503\ \text{MeV}$. Which of the following is the most likely decay tendency of the nucleus $(A,Z)$?

Q6. A $^{235}\text{U}$ nucleus splits into two fragments of mass numbers $95$ and $140$. If the average binding energies per nucleon are $8.5\ \text{MeV}$ for the $A=95$ fragment, $8.6\ \text{MeV}$ for the $A=140$ fragment and $7.6\ \text{MeV}$ for $^{235}\text{U}$, the energy released in this fission (approximately) is:

Q7. A plot of $\ln(\text{Activity})$ vs time for a sample is a straight line of slope $-0.3\ \text{day}^{-1}$ for the first few days, then at late times it becomes a straight line of slope $-0.01\ \text{day}^{-1}$. The best physical explanation is:

Q8. Using a simplified semi-empirical mass formula keeping only Coulomb and asymmetry terms one finds the $Z$ that maximizes binding energy for a given $A$ approximately as

For $A=120$, with $a_c=0.72\ \text{MeV}$ and $a_a=23\ \text{MeV}$, the nearest integer value of the most stable $Z$ is:

Q9. Assertion (A): For $\alpha$-emitters the logarithm of the decay constant $\lambda$ shows an approximately linear dependence on $1/\sqrt{Q_\alpha}$ (Geiger–Nuttall relation).
Reason (R): Quantum tunnelling through the Coulomb barrier leads to a transmission probability (Gamow factor) $\sim\exp\!\big(-\text{const}/\sqrt{E}\,\big)$, which yields the observed dependence of $\lambda$ on $1/\sqrt{Q_\alpha}$.

Q10. In a lab frame a neutron (mass $m_n\approx1\ \text{u}$) strikes a stationary nucleus of mass $m_A=20\ \text{u}$ to induce an endothermic reaction with $Q=-5.0\ \text{MeV}$. Using the threshold relation $E_{\text{th}}=(-Q)\left(1+\dfrac{m_n}{m_A}\right)$, the minimum kinetic energy of the neutron required is approximately: