31.6 Binding Energy

Nuclear stability and binding energy are fundamental concepts in understanding atomic nuclei. They explain why some nuclei are stable while others undergo radioactive decay. Binding energy, the force holding nucleons together, is key to nuclear stability.

Calculations of binding energy per nucleon reveal patterns across the periodic table. This knowledge is crucial for understanding nuclear reactions like fusion and fission, which power stars and nuclear plants. The strong nuclear force plays a central role in these processes.

Nuclear Stability and Binding Energy

Concept of binding energy

• Energy required to disassemble a nucleus into its constituent protons and neutrons
• Measure of the strength of the nuclear force holding the nucleus together
• Nuclear stability determined by the binding energy per nucleon
  • Higher binding energy per nucleon indicates more stable nuclei (iron-56)
  • Lower binding energy per nucleon indicates less stable nuclei more likely to undergo radioactive decay (uranium-235)
• Mass defect is the difference between the mass of a nucleus and the sum of the masses of its individual protons and neutrons
• Mass defect related to binding energy through Einstein's equation $E = mc^2$
• Binding energy calculated from mass defect using equation $E_b = (\Delta m)c^2$
  • $E_b$ is binding energy
  • $\Delta m$ is mass defect
  • $c$ is speed of light
• Strong nuclear force is responsible for holding nucleons together within the nucleus

Calculation of binding energy

• Binding energy per nucleon is total binding energy divided by number of nucleons (protons and neutrons) in nucleus
• Measure of average energy required to remove a single nucleon from nucleus
• Steps to calculate binding energy per nucleon:
  1. Determine mass defect ($\Delta m$) by subtracting actual mass of nucleus from sum of masses of individual protons and neutrons
  2. Calculate total binding energy using equation $E_b = (\Delta m)c^2$
  3. Divide total binding energy by total number of nucleons (A) to obtain binding energy per nucleon $E_b/A$
• Example calculation for carbon-12 ($^{12}C$):
  • Mass of $^{12}C$ nucleus: 12.000000 u
  • Mass of 6 protons and 6 neutrons: 12.098940 u
  • Mass defect: $\Delta m = 12.098940 - 12.000000 = 0.098940$ u
  • Binding energy: $E_b = (0.098940)(931.5 \text{ MeV/u}) = 92.16 \text{ MeV}$
  • Binding energy per nucleon: $E_b/A = 92.16 \text{ MeV} / 12 = 7.68 \text{ MeV/nucleon}$
• Atomic mass is used in these calculations and is expressed in atomic mass units (u)

Binding energy across periodic table

• Binding energy per nucleon varies with number of nucleons (A) in nucleus
  • Generally increases with increasing A, reaching maximum around A = 56 (iron-56)
  • For A > 56, binding energy per nucleon gradually decreases
• Elements with highest binding energy per nucleon (around iron-56) are most stable
• Elements with lower binding energy per nucleon are less stable and more likely to undergo nuclear reactions
• Nuclear fusion reactions combine lighter nuclei to form heavier nuclei
  • Fusion reactions exothermic (release energy) for elements lighter than iron-56
  • Fusion of light elements powers stars and potentially used for energy production on Earth (hydrogen)
• Nuclear fission reactions split heavier nuclei into lighter fragments
  • Fission reactions exothermic for elements heavier than iron-56
  • Fission of heavy elements used in nuclear power plants and atomic weapons (uranium-235)
• Variation in binding energy per nucleon across periodic table drives both fusion and fission reactions
  • Systems tend to move towards state of higher stability (higher binding energy per nucleon)

Nuclear Stability and Models

• Nuclear stability curve illustrates the relationship between the number of protons and neutrons in stable nuclei
• Nuclear shell model explains nuclear stability and magic numbers in terms of energy levels and shell structure of nucleons