### Correct Option: A. The mass defect is directly proportional to the binding energy. #### Detailed Explanation: 1. **Understanding Mass Defect**: - The mass defect of a nucleus is the difference between the mass of the individual nucleons (protons and neutrons) when they are free and the mass of the nucleus when they 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. **Binding Energy**: - The binding energy of a nucleus is the energy required to disassemble the 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 mass-energy equivalence principle, the energy equivalent of mass can be expressed as: \[ E = mc^2 \] where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light in a vacuum (approximately \(3 \times 10^8 \, \text{m/s}\)). 3. **Connecting Mass Defect and Binding Energy**: - The mass defect is directly related to the binding energy through the equation: \[ \text{Binding Energy} = \text{Mass Defect} \times c^2 \] - This means that if the mass defect increases, the binding energy also increases proportionally, as both are linked by the constant \(c^2\). Therefore, the correct answer is that the mass defect is directly proportional to the binding energy. #### Why Other Options Are Incorrect: - **Option B: The mass defect is inversely proportional to the binding energy.** - This option is incorrect because an increase in mass defect leads to an increase in binding energy, not a decrease. The relationship is not inverse; they move in the same direction. - **Option C: The mass defect has no effect on the binding energy.** - This option is also incorrect. The mass defect is fundamentally linked to the binding energy. A nucleus with a larger mass defect will have a higher binding energy, indicating that the nucleons are held together more tightly. - **Option D: The mass defect is equal to the binding energy multiplied by the speed of light squared.** - This option is misleading. While the binding energy is calculated using the mass defect multiplied by \(c^2\), the mass defect itself is not equal to the binding energy multiplied by \(c^2\). Instead, it is the other way around: binding energy equals mass defect times \(c^2\). ### Summary for Revision: - The mass defect of a nucleus is the difference between the mass of its individual nucleons and the mass of the nucleus. - Binding energy is the energy required to separate a nucleus into its individual nucleons. - According to Einstein's mass-energy equivalence, the binding energy is directly proportional to the mass defect: \( \text{Binding Energy} = \text{Mass Defect} \times c^2 \). - An increase in mass defect results in an increase in binding energy, indicating greater nuclear stability.