Calculate (in MeV) the total binding energy and the binding energy per nucleon for ¹²⁹I and for ¹²⁹Xe.

검증된 단계별 안내

Determine the number of protons (Z) and neutrons (N) for each isotope. For ¹²⁹I (Iodine), Z = 53 (atomic number of Iodine) and N = 129 - 53 = 76. For ¹²⁹Xe (Xenon), Z = 54 (atomic number of Xenon) and N = 129 - 54 = 75.

Use the semi-empirical mass formula or look up the experimental atomic masses of ¹²⁹I and ¹²⁹Xe in atomic mass units (u). Convert these masses into energy using the relation: \( E = m \cdot c^2 \), where \( c^2 = 931.5 \; \text{MeV/u} \).

Calculate the total binding energy for each isotope. The total binding energy is given by: \( B = \left[ Z \cdot m_p + N \cdot m_n - m_{nucleus} \right] \cdot c^2 \), where \( m_p \) is the mass of a proton, \( m_n \) is the mass of a neutron, and \( m_{nucleus} \) is the mass of the nucleus.

Determine the binding energy per nucleon for each isotope. This is calculated by dividing the total binding energy by the total number of nucleons (A): \( B_{per\;nucleon} = \frac{B}{A} \), where \( A = Z + N \).

Compare the binding energy per nucleon for ¹²⁹I and ¹²⁹Xe to understand their relative nuclear stability. A higher binding energy per nucleon indicates a more stable nucleus.

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주요 개념

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Binding Energy

Binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is a measure of the stability of a nucleus; the higher the binding energy, the more stable the nucleus. This energy can be calculated using the mass defect, which is the difference between the mass of the individual nucleons and the mass of the nucleus itself.

Mass Defect

The mass defect refers to the difference in mass between the total mass of individual nucleons and the actual mass of the nucleus. This discrepancy arises because some mass is converted into energy during the formation of the nucleus, according to Einstein's equation E=mc². The mass defect is crucial for calculating the binding energy of a nucleus.

Binding Energy per Nucleon

Binding energy per nucleon is the average energy that binds each nucleon (proton or neutron) in the nucleus. It is calculated by dividing the total binding energy by the number of nucleons. This value is important for comparing the stability of different nuclei; generally, a higher binding energy per nucleon indicates a more stable nucleus.