6.10.3 Mass Defect & Binding Energy

Mass Defect & Binding Energy

• Experiments into nuclear structure have found that the total mass of a nucleus is less than the sum of the masses of its constituent nucleons
• This difference in mass is known as the mass defect or mass deficit
• Mass defect is defined as:

The difference between the measured mass of a nucleus and the sum total of the masses of its constituents

A system of separated nucleons has a greater mass than a system of bound nucleons

• Due to the equivalence of mass and energy, this decrease in mass implies that energy is released in the process
• Since nuclei are made up of neutrons and protons, there are forces of repulsion between the positive protons
  • Therefore, it takes energy, ie. the binding energy, to hold nucleons together as a nucleus

• Binding energy is defined as:

The energy required to break a nucleus into its constituent protons and neutrons

• Energy and mass are proportional, so, the total energy of a nucleus is less than the sum of the energies of its constituent nucleons
• The formation of a nucleus from a system of isolated protons and neutrons is therefore an exothermic reaction - meaning that it releases energy

• In order to compare nuclear stability, it is more 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, it requires more energy to pull the nucleus apart

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

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 tend to have a lower binding energy per nucleon, hence, they are generally less stable
  • This means the lightest elements have weaker electrostatic forces and are the most likely to undergo fusion

• 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

• At high values of A:
  • The general binding energy per nucleon is high and gradually decreases with A
  • This means the heaviest elements are the most unstable and likely to undergo fission

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