### 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 energy that holds the nucleus together. This lost mass is what we refer to as the mass defect. 2. **Understanding 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 energy associated with the mass defect can be calculated. 3. **Relationship Between Mass Defect and Binding Energy**: - The mass defect (\(\Delta m\)) is directly related to the nuclear binding energy (BE) through the equation: \[ BE = \Delta m \cdot c^2 \] - 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 direct proportionality. 4. **Why Option A is Correct**: - Since the mass defect is directly proportional to the nuclear binding energy, if the mass defect increases, the binding energy increases as well. This means that a nucleus with a larger mass defect will have a greater binding energy, making it more stable. #### Why the Other Options are Incorrect: - **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 the 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 and nuclear binding energy are unrelated.** - This option is also incorrect. The mass defect and binding energy are fundamentally related through the equation \(BE = \Delta m \cdot c^2\). They cannot be considered unrelated as they describe different aspects of the same physical phenomenon. - **Option D: The mass defect equals the nuclear binding energy multiplied by the speed of light squared.** - This option is incorrect because it misrepresents the relationship. The correct relationship is that the binding energy is equal to the mass defect multiplied by the speed of light squared, not the other way around. ### Summary: - The mass defect is the difference in mass between free 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 \(BE = \Delta m \cdot c^2\). - Understanding this relationship is crucial for grasping nuclear stability and energy release in nuclear reactions.