### Correct Option: A. The mass defect is directly proportional to the nuclear 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 itself. - When nucleons come together to form a nucleus, some mass is converted into energy due to the binding forces that hold the nucleus together. This lost mass is what we refer to as the mass defect. 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 mass-energy equivalence principle, expressed by the equation \(E = mc^2\), the mass defect can be converted into energy. 3. **Relationship Between Mass Defect and Binding Energy**: - The mass defect (\(\Delta m\)) is directly related to the nuclear binding energy (\(E_b\)) through the equation: \[ E_b = \Delta m \cdot c^2 \] - 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 proportionally, since \(c^2\) is a constant. 4. **Why Option A is Correct**: - Since the mass defect is directly proportional to the nuclear binding energy, an increase in mass defect results in an increase in binding energy, confirming that option A is correct. #### Analysis of Other Options: - **Option B: The mass defect is inversely proportional to the nuclear binding energy.** - This option is incorrect because it contradicts the established relationship. If the mass defect were inversely proportional to binding energy, an increase in mass defect would lead to a decrease in binding energy, which is not supported by the mass-energy equivalence principle. - **Option C: The mass defect has no effect on the nuclear binding energy.** - This option is also incorrect. The mass defect is fundamentally linked to the binding energy. If there were no mass defect, there would be no binding energy, as the nucleons would not be held together in a nucleus. - **Option D: The mass defect is equal to the nuclear binding energy multiplied by the speed of light squared.** - This option is misleading. While it correctly states the relationship involving \(c^2\), it incorrectly suggests that the mass defect equals the binding energy multiplied by \(c^2\). In reality, the binding energy is derived from the mass defect multiplied by \(c^2\), not the other way around. ### Summary: - The mass defect is the difference in mass between separate nucleons and the nucleus. - Nuclear binding energy is the energy required to separate a nucleus into its individual nucleons. - The mass defect is directly proportional to the nuclear binding energy, as described by \(E_b = \Delta m \cdot c^2\). - Understanding this relationship is crucial for grasping nuclear stability and energy release in nuclear reactions.