The correct option is **A. Mass defect is directly proportional to nuclear binding energy.** ### Detailed Explanation 1. **Understanding Mass Defect**: - The mass defect of a nucleus is defined as 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: - \(Z\) = number of protons - \(N\) = number of neutrons - \(m_p\) = mass of a proton - \(m_n\) = mass of a neutron - \(m_{\text{nucleus}}\) = mass of the nucleus 2. **Understanding Nuclear Binding Energy**: - The nuclear binding energy is the energy required to disassemble a nucleus into its constituent 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 Einstein's mass-energy equivalence principle, \(E = mc^2\), where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light. - The relationship can be expressed as: \[ \text{Binding Energy} = \text{Mass Defect} \times c^2 \] 3. **Direct Proportionality**: - From the equations above, we can see that the mass defect is directly related to the binding energy. If the mass defect increases, the binding energy also increases, and vice versa. This is because a larger mass defect indicates that more energy is released when the nucleons come together to form the nucleus, thus resulting in a higher binding energy. ### Why Other Options Are Incorrect - **Option B: Mass defect is inversely proportional to nuclear binding energy**: - This 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: Mass defect is independent of nuclear binding energy**: - This option is also incorrect. The mass defect and binding energy are fundamentally linked through the mass-energy equivalence principle. They cannot be independent of each other. - **Option D: Mass defect is equal to the total mass of the nucleus**: - This statement is incorrect. The mass defect is the difference between the total mass of the individual nucleons and the mass of the nucleus. It is not equal to the total mass of the nucleus; rather, it is a measure of how much mass is "lost" when the nucleons bind together. ### 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. - Mass defect and nuclear binding energy are directly proportional; as one increases, so does the other. - The relationship is derived from Einstein's mass-energy equivalence principle, \(E = mc^2\). This understanding is crucial for grasping concepts in nuclear physics and the stability of atomic nuclei.