**Correct Option: B** ### Explanation of the Correct Answer The relationship between mass defect and nuclear binding energy is a fundamental concept in nuclear physics. Let's break it down step-by-step: 1. **Understanding Mass Defect**: - The mass defect of a nucleus is the difference between the total mass of the individual nucleons (protons and neutrons) when they are free and the actual mass of the nucleus when these nucleons are bound together. - Mathematically, it can be expressed as: \[ \text{Mass Defect} = (Z \cdot m_p + N \cdot m_n) - m_{\text{nucleus}} \] where \(Z\) is the number of protons, \(N\) is the number of neutrons, \(m_p\) is the mass of a proton, \(m_n\) is the mass of a neutron, and \(m_{\text{nucleus}}\) is the mass of the nucleus. 2. **Nuclear Binding Energy**: - The nuclear binding energy is the energy required to disassemble a nucleus into its individual nucleons. It is a measure of the stability of the nucleus; the higher the binding energy, the more stable the nucleus. - According to Einstein's famous equation \(E=mc^2\), mass can be converted into energy. This means that the mass defect can be converted into binding energy. 3. **Direct Proportionality**: - The binding energy (\(E_b\)) is directly proportional to the mass defect (\(\Delta m\)): \[ E_b = \Delta m \cdot c^2 \] - Here, \(c\) is the speed of light in a vacuum (approximately \(3 \times 10^8 \, \text{m/s}\)). This equation shows that as the mass defect increases, the binding energy also increases, indicating a more stable nucleus. ### Why Other Options Are Incorrect - **Option A: The mass defect is equal to the total mass of the nucleons.** - This statement is incorrect because the mass defect is not equal to the total mass of the nucleons; rather, it is the difference between the total mass of the nucleons and the mass of the nucleus. The mass defect is a measure of how much mass is "lost" when nucleons bind together, which is converted into binding energy. - **Option C: The mass defect indicates the number of nucleons in the nucleus.** - This option is misleading. While the mass defect is related to the number of nucleons (since more nucleons generally lead to a larger mass defect), it does not directly indicate the number of nucleons. The mass defect is a specific calculation based on the binding energy and the actual mass of the nucleus, not a direct count of nucleons. - **Option D: The nuclear binding energy decreases with an increase in mass defect.** - This statement is fundamentally incorrect. As established, the nuclear binding energy increases with an increase in mass defect. A larger mass defect means more energy is released when nucleons bind together, resulting in a higher binding energy. ### Summary of Key Points - The mass defect is the difference between the mass of free nucleons and the mass of the nucleus. - Nuclear binding energy is the energy required to separate a nucleus into its individual nucleons. - The binding energy is directly proportional to the mass defect, as described by \(E=mc^2\). - Understanding the relationship between mass defect and binding energy is crucial for grasping nuclear stability and reactions. This thorough understanding of mass defect and nuclear binding energy is essential for students preparing for professional exams in physics.