Binding Energy (AQA A Level Physics)

• In order to compare nuclear stability, it is useful to look at the binding energy per nucleon
• The binding energy per nucleon is defined as:

The binding energy of a nucleus divided by the number of nucleons in the nucleus

• A higher binding energy per nucleon indicates a higher stability
• In other words, more energy is required to separate the nucleons contained within a nucleus

By plotting a graph of binding energy per nucleon against nucleon number, the stability of elements can be inferred

Key Features of the Graph

• At low values of A:
  • Nuclei have lower binding energies per nucleon than at large values of A, but they tend to be stable when N = Z
  • This means light nuclei have weaker electrostatic forces and will undergo fusion
  • The gradient is much steeper compared to the gradient at large values of A
  • This means that fusion reactions release a greater binding energy than fission reactions

• At high values of A:
  • Nuclei have generally higher binding energies per nucleon, but this gradually decreases with A
  • This means the heaviest elements are the most unstable and will undergo fission
  • The gradient is less steep compared to the gradient at low values of A
  • This means that fission reactions release less binding energy than fission reactions

• Iron (A = 56) has the highest binding energy per nucleon, which makes it the most stable of all the elements

• Helium (⁴He), carbon (¹²C) and oxygen (¹⁶O) do not fit the trend
  • Helium-4 is a particularly stable nucleus hence it has a high binding energy per nucleon
  • Carbon-12 and oxygen-16 can be considered to be three and four helium nuclei, respectively, bound together

Comparing Fusion & Fission

Similarities

• In both fusion and fission, the total mass of the products is slightly less than the total mass of the reactants
• The mass defect is equivalent to the binding energy that is released
• As a result, both fusion and fission reactions release energy

Differences

• In fusion, two smaller nuclei combine into a larger nucleus
• In fission, an unstable nucleus splits into two smaller nuclei

• Fusion occurs between light nuclei (A < 56)
• Fission occurs in heavy nuclei (A > 56)

• In light nuclei, attractive nuclear forces dominate over repulsive electrostatic forces between protons, and this contributes to nuclear stability 
• In heavy nuclei, repulsive electrostatic forces between protons begin to dominate over attractive nuclear forces, and this contributes to nuclear instability

• Fusion releases much more energy per kg than fission
• Fusion requires a greater initial input of energy than fission

Part (a)

Step 1: Use the graph to identify each isotope’s binding energy per nucleon

• • Binding energy per nucleon (U-235) = 7.5 MeV
  • Binding energy per nucleon (Sr-91) = 8.2 MeV
  • Binding energy per nucleon (Xe-142) = 8.7 MeV

Step 2: Determine the binding energy of each isotope

Binding energy = Binding Energy per Nucleon × Mass Number

• • Binding energy of U-235 nucleus = (235 × 7.5) = 1763 MeV
  • Binding energy of Sr-91 = (91 × 8.2) = 746 MeV
  • Binding energy of Xe-142 = (142 × 8.7) = 1235 MeV

Step 3: Calculate the energy released

Energy released = Binding energy after (Sr + Xe) – Binding energy before (U)

Energy released = (1235 + 746) – 1763 = 218 MeV

Part (b)

Step 1: Calculate the energy released by 1 mol of uranium-235

• • There are N_A (Avogadro’s number) atoms in 1 mol of U-235, which is equal to a mass of 235 g
  • Energy released by 235 g of U-235 = (6 × 10²³) × 218 MeV

 

Step 2: Convert the energy released from MeV to J

• • 1 MeV = 1.6 × 10^–13 J
  • Energy released = (6 × 10²³) × 218 × (1.6 × 10^–13) = 2.09 × 10¹³ J

Step 3: Work out the proportion of uranium-235 in the sample

• • 1 kg of uranium which is 3% U-235 contains 0.03 kg or 30 g of U-235

Step 4: Calculate the energy released by the sample

Energy released from 1 kg of Uranium =