Practice Questions

Define the atomic mass unit (u).

Define nuclear fission and nuclear fusion.

The radius of a nucleus is found to be $3.6 \text{ fm}$. If $R_0 = 1.2 \text{ fm}$, calculate the mass number of this nucleus.

Propose one key reason why nuclear fusion is often considered a 'cleaner' energy source compared to nuclear fission.

List the three main types of radioactive decay.

Recall the empirical formula that relates the radius of a nucleus to its mass number A.

Describe the composition of a nucleus using the terms atomic number (Z), neutron number (N), and mass number (A).

Identify the number of protons and neutrons in the nucleus of a gold isotope, ${ }_{79}^{197} \text{Au}$.

Calculate the radius of a ${}_{26}^{56}\text{Fe}$ nucleus, given that the constant $R_0 = 1.2 \times 10^{-15} \text{ m}$.

The mass of a proton is $1.67262 \times 10^{-27} \text{ kg}$. Solve for its mass in atomic mass units (u), given that $1 \text{ u} = 1.660539 \times 10^{-27} \text{ kg}$.

Calculate the energy equivalent in MeV for a mass of $1.0 \times 10^{-30} \text{ kg}$. (Given $c = 3 \times 10^8 \text{ m/s}$ and $1 \text{ MeV} = 1.6 \times 10^{-13} \text{ J}$).

Evaluate the key properties of the nuclear force and justify why it must be significantly stronger than the electrostatic force for a nucleus to be stable.

Analyze the statement: 'The mass of a nucleus is always less than the sum of the masses of its constituent free nucleons.' Explain the physical significance of this mass difference.

Explain the concept of 'mass defect' of a nucleus.

Justify why binding energy per nucleon, rather than the total binding energy, is a more appropriate measure for comparing the stability of different nuclei.

Summarize the main features observed in the plot of binding energy per nucleon versus mass number A.

Analyze why a heavy nucleus like Uranium (${}_{92}^{238}\text{U}$) is unstable, while a medium-sized nucleus like Iron (${}_{26}^{56}\text{Fe}$) is highly stable, by referring to the binding energy per nucleon curve.

Critique the feasibility of achieving controlled nuclear fusion at room temperature. Justify your answer by evaluating the role of the Coulomb barrier.

Propose a self-sustaining chain reaction mechanism for the fission of ${}_{92}^{235}U$. Justify why a 'critical mass' is essential to sustain this reaction and evaluate the role of the neutrons produced in the process.

Apply the concept of mass-energy interconversion to explain why energy is released in a nuclear reaction where the total binding energy of the products is greater than the total binding energy of the reactants.

Calculate the approximate radius of the ${ }_{26}^{56} \text{Fe}$ nucleus, given that the constant $R_0 = 1.2 \text{ fm}$.

Explain the terms 'isotopes' and 'isobars' with one example for each.

Propose an experimental method to verify the relationship for nuclear radius, $R = R_0 A^{1/3}$. Justify your choice of projectile particles for this experiment.

Summarize three key properties of the nuclear force.

Recall Einstein's mass-energy equivalence relation and explain its significance in nuclear reactions.

Formulate a hypothesis to explain why the mass of a stable nucleus is consistently less than the sum of the masses of its individual protons and neutrons. Evaluate this hypothesis using Einstein's mass-energy equivalence principle.

Compare the nuclear density of a gold nucleus (${}_{79}^{197}\text{Au}$) with that of an iron nucleus (${}_{26}^{56}\text{Fe}$). Show your reasoning.

The nucleus of gold is denoted by ${}_{79}^{197}\text{Au}$. Solve for the number of protons and neutrons in its nucleus. From the following list of nuclides, identify one isotone and one isobar of a different element: ${}_{80}^{197}\text{Hg}$, ${}_{79}^{198}\text{Au}$, ${}_{78}^{196}\text{Pt}$.

Compare and contrast nuclear fission and nuclear fusion as sources of energy. Refer to the binding energy per nucleon curve to support your analysis.

A student claims that since the binding energy per nucleon for Uranium ($A=238$) is $7.6 \text{ MeV}$ and for Iron ($A=56$) is $8.75 \text{ MeV}$, Iron must be more radioactive. Critique this statement.

Calculate the energy released in the following fusion reaction: ${}_{1}^{2}\text{H} + {}_{1}^{3}\text{H} \rightarrow {}_{2}^{4}\text{He} + {}_{0}^{1}\text{n}$. Atomic masses are: $m({}_{1}^{2}\text{H}) = 2.014102 \text{ u}$, $m({}_{1}^{3}\text{H}) = 3.016049 \text{ u}$, $m({}_{2}^{4}\text{He}) = 4.002603 \text{ u}$, $m({}_{0}^{1}\text{n}) = 1.008665 \text{ u}$.

Calculate the energy equivalent of one atomic mass unit (1 u) in MeV.

Demonstrate that the density of nuclear matter is nearly constant and independent of the mass number A. Start from the empirical formula for the radius of a nucleus, $R = R_0 A^{1/3}$, and clearly state any assumptions made.

Examine the statement that the nuclear force is short-ranged, using the constancy of the binding energy per nucleon for nuclei with $30 < A < 170$ as evidence.

A possible fission reaction for Uranium-235 is ${}_{92}^{235}\text{U} + {}_{0}^{1}\text{n} \rightarrow {}_{54}^{140}\text{Xe} + {}_{38}^{94}\text{Sr} + 2{}_{0}^{1}\text{n}$. Calculate the Q-value for this reaction and analyze whether the reaction is exothermic. Given atomic masses: $m({}_{92}^{235}\text{U}) = 235.04393 \text{ u}$, $m({}_{54}^{140}\text{Xe}) = 139.92164 \text{ u}$, $m({}_{38}^{94}\text{Sr}) = 93.91536 \text{ u}$, $m({}_{0}^{1}\text{n}) = 1.008665 \text{ u}$.

Calculate the mass defect and binding energy per nucleon for a helium nucleus (${}_{2}^{4}\text{He}$). Given: mass of helium nucleus $m({}_{2}^{4}\text{He}) = 4.001506 \text{ u}$, mass of proton $m_p = 1.00727 \text{ u}$, and mass of neutron $m_n = 1.00866 \text{ u}$. Use $1 \text{ u} = 931.5 \text{ MeV/c}^2$.

Critique the statement: 'The density of a Uranium nucleus is significantly greater than that of a Helium nucleus because Uranium is a much heavier element.' Justify your answer by deriving a general expression for nuclear matter density.

Explain why the density of nuclear matter is nearly constant for all nuclei.

Design a conceptual model to explain the saturation property of nuclear forces. Justify how your model explains the near-constancy of binding energy per nucleon for nuclei with mass numbers in the range $30 < A < 170$.

Explain how both nuclear fission and nuclear fusion processes release energy, with reference to the binding energy per nucleon curve.