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title: "A heavy nucleus \\(X\\) undergoes induced nuclear fission after absorbing a slow-moving neutron. The reaction produces two lighter product nuclei, \\(Y\\) and \\(Z\\), along with \\(k\\) free neutrons, where \\(k\\) is an integer greater than 1. The reaction is represented by the following equation:  \\[ _{0}^{1}\\text{n} + \\text{X} \\rightarrow \\text{Y} + \\text{Z} + k_{0}^{1}\\text{n} \\]  The rest masses of the nuclei are \\(m_X\\), \\(m_Y\\), and \\(m_Z\\). The mass of a single neutron is \\(m_n\\). Figure 1 shows a graph of the average binding energy per nucleon as a function of mass number. The positions of nuclei \\(X\\), \\(Y\\), and \\(Z\\) are indicated on the curve."
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url: "https://nerd-notes.com/ubq/117689/"
date_modified: "2026-08-04T07:53:22+00:00"
---

# A heavy nucleus \(X\) undergoes induced nuclear fission after absorbing a slow-moving neutron. The reaction produces two lighter product nuclei, \(Y\) and \(Z\), along with \(k\) free neutrons, where \(k\) is an integer greater than 1. The reaction is represented by the following equation:

\[ _{0}^{1}\text{n} + \text{X} \rightarrow \text{Y} + \text{Z} + k_{0}^{1}\text{n} \]

The rest masses of the nuclei are \(m_X\), \(m_Y\), and \(m_Z\). The mass of a single neutron is \(m_n\). Figure 1 shows a graph of the average binding energy per nucleon as a function of mass number. The positions of nuclei \(X\), \(Y\), and \(Z\) are indicated on the curve.

A heavy nucleus \(X\) undergoes induced nuclear fission after absorbing a slow-moving neutron. The reaction produces two lighter product nuclei, \(Y\) and \(Z\), along with \(k\) free neutrons, where \(k\) is an integer greater than 1. The reaction is represented by the following equation:

\[ _{0}^{1}\text{n} + \text{X} \rightarrow \text{Y} + \text{Z} + k_{0}^{1}\text{n} \]

The rest masses of the nuclei are \(m_X\), \(m_Y\), and \(m_Z\). The mass of a single neutron is \(m_n\). Figure 1 shows a graph of the average binding energy per nucleon as a function of mass number. The positions of nuclei \(X\), \(Y\), and \(Z\) are indicated on the curve.

![A line graph plotting 'Binding Energy per Nucleon (MeV)' on the vertical axis versus 'Mass Number' on the horizontal axis. The vertical axis has no numbers but has a label. The horizontal axis has tick marks labeled 0, 50, 100, 150, 200, 250. A solid curve starts near the origin, rises steeply to a rounded peak around Mass Number 60, and then slowly and steadily decreases as it extends to the right towards 250. Three solid dots are explicitly plotted and labeled on the curve. Point Y is plotted around Mass Number 90. Point Z is plotted around Mass Number 140. Point X is plotted around Mass Number 235. The vertical positions (Binding Energy per Nucleon) for dots Y and Z are noticeably higher than the vertical position for dot X. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830002-cvbMC9.jpg)

**Part a)** Student 1 analyzes Figure 1 and makes the following claim: "Because nuclei \(Y\) and \(Z\) have a higher binding energy per nucleon than nucleus \(X\), their nucleons are more tightly bound together. Therefore, the system must absorb energy from its surroundings during the fission reaction in order to create this stronger binding." **Evaluate** Student 1's claim. **Justify** your answer using physical principles. *(3 points)*

**Part b)** **Derive** an expression for the total energy released, \( E_{rel} \), during this fission reaction. Express your answer in terms of \(m_X\), \(m_Y\), \(m_Z\), \(m_n\), \(k\), and fundamental constants as appropriate. *(3 points)*

**Part c)** Student 2 correctly states that the total rest mass of the products after the fission reaction is less than the total rest mass of the reactants before the reaction. **Explain** how the expression derived in Part (b) demonstrates that energy is released to the surroundings. Then, **justify** how a decrease in total rest mass of the system is physically consistent with the fact that the product nuclei have a higher binding energy per nucleon. *(3 points)*

**Part d)** Assume that in a specific instance of this fission reaction, the kinetic energies of the product nuclei and the free neutrons are negligible, and all of the released energy \( E_{rel} \) is instead carried away by a single gamma-ray photon. **Derive** an expression for the momentum \( p \) of this gamma-ray photon. Express your answer in terms of \( E_{rel} \) and fundamental constants as appropriate. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117689/*