### Correct Option: D. The mass defect is directly proportional to the binding energy of the nucleus. #### 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. - 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 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; a higher binding energy means a more stable nucleus. - The relationship between mass defect and binding energy is given by Einstein's mass-energy equivalence principle, expressed as: \[ E = mc^2 \] where: - \(E\) = energy (binding energy in this context) - \(m\) = mass defect - \(c\) = speed of light in a vacuum (approximately \(3 \times 10^8 \, \text{m/s}\)) 3. **Direct Proportionality**: - From the equation \(E = mc^2\), we can see that the binding energy \(E\) is directly proportional to the mass defect \(m\). This means that as the mass defect increases, the binding energy also increases, and vice versa. - Therefore, if a nucleus has a larger mass defect, it will have a larger binding energy, indicating that the nucleons are held together more tightly. #### Why Other Options Are Incorrect: - **Option A: Mass defect increases as binding energy decreases.** - This statement is incorrect because it contradicts the direct proportionality between mass defect and binding energy. If the binding energy decreases, the mass defect would also decrease, not increase. - **Option B: Mass defect has no effect on binding energy.** - This option is incorrect because it ignores the fundamental relationship established by \(E = mc^2\). The mass defect is crucial in determining the binding energy of the nucleus. - **Option C: The mass defect is equal to the binding energy divided by the speed of light squared.** - This statement is misleading. While it is true that \(E = mc^2\) can be rearranged to express mass defect as \(m = \frac{E}{c^2}\), this does not accurately reflect the relationship in the context of binding energy. The mass defect is not simply equal to the binding energy divided by \(c^2\); rather, it is the mass defect that contributes to the binding energy. ### Summary: - The mass defect is the difference between the mass of free nucleons and the mass of the nucleus. - Binding energy is the energy required to separate the nucleus into its individual nucleons. - There is a direct proportionality between mass defect and binding energy, as described by \(E = mc^2\). - Understanding this relationship is crucial for grasping nuclear stability and reactions. ### Revision Points: - Mass defect = mass of free nucleons - mass of nucleus. - Binding energy = energy needed to separate nucleons. - Direct relationship: More mass defect = more binding energy. - Use \(E = mc^2\) to connect mass defect and binding energy.