The correct option is **C. Nuclear binding energy is equal to the mass defect multiplied by the speed of light squared.** ### Detailed Explanation 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. - This mass defect arises because some mass is converted into energy when nucleons come together to form a nucleus, according to Einstein's mass-energy equivalence principle, expressed by the famous equation \(E = mc^2\). 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. - The binding energy can be calculated using the mass defect. When nucleons bind together, the energy released (or the energy required to separate them) is equivalent to the mass defect multiplied by the square of the speed of light (\(c^2\)). 3. **The Relationship**: - The relationship can be expressed mathematically as: \[ E_b = \Delta m \cdot c^2 \] where: - \(E_b\) is the binding energy, - \(\Delta m\) is the mass defect, - \(c\) is the speed of light (\(approximately 3 \times 10^8 \, m/s\)). - This equation shows that the binding energy is directly proportional to the mass defect. As the mass defect increases, the binding energy also increases, indicating a more stable nucleus. ### Why Other Options Are Incorrect - **Option A: Nuclear binding energy is directly proportional to mass defect.** - While this statement is partially true, it is incomplete. It does not mention the crucial factor of \(c^2\) in the relationship. The binding energy is not just proportional to the mass defect; it is equal to the mass defect multiplied by \(c^2\). - **Option B: Nuclear binding energy is inversely proportional to mass defect.** - This statement is incorrect. An inverse relationship would imply that as the mass defect increases, the binding energy decreases, which contradicts the established relationship. More mass defect means more energy is released when nucleons bind, leading to higher binding energy. - **Option D: Nuclear binding energy and mass defect are unrelated.** - This option is also incorrect. The binding energy is fundamentally related to the mass defect, as explained above. They are not only related but are quantitatively linked through the equation \(E_b = \Delta m \cdot c^2\). ### Common Pitfalls - **Misunderstanding Mass-Energy Equivalence**: Students often confuse mass defect with binding energy. Remember that mass defect is the mass lost when nucleons bind, and binding energy is the energy equivalent of that mass loss. - **Ignoring Units**: When calculating binding energy, ensure that the mass defect is in kilograms (kg) to use the speed of light in standard units (m/s) for accurate results. - **Forgetting the Factor of \(c^2\)**: Always remember that the binding energy is not just the mass defect; it is the mass defect multiplied by the speed of light squared. ### Revision Summary - 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 relationship is given by \(E_b = \Delta m \cdot c^2\), indicating that binding energy is equal to the mass defect multiplied by the speed of light squared. - Higher mass defect leads to higher binding energy, indicating a more stable nucleus.