23.1.3 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
• Mass defect is defined as:

The difference between the mass of a nucleus and the sum of the individual masses of its protons and neutrons

• The mass defect Δm of a nucleus can be calculated using:

Δm = Zm_p + (A – Z)m_n – m_total

• Where:
  • Z = proton number
  • A = nucleon number
  • m_p = mass of a proton (kg)
  • m_n = mass of a neutron (kg)
  • m_total = measured mass of the nucleus (kg)

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
• This can be calculated using the equation:

E = Δmc²

• 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:            Calculate the mass defect

Number of protons, Z = 26

Number of neutrons, A – Z = 56 – 26 = 30

Mass defect, Δm = Zm_p + (A – Z)m_n – m_total

Δm = (26 × 1.673 × 10^-27) + (30 × 1.675 × 10^-27) – (9.288 × 10^-26)

Δm = 8.680 × 10^-28 kg

Step 2:            Calculate the binding energy of the nucleus

Binding energy, E = Δmc²

E = (8.680 × 10^-28) × (3.00 × 10⁸)² = 7.812 × 10^-11 J

Step 3:            Calculate the binding energy per nucleon

Step 4:            Convert to MeV

J → eV: divide by 1.6 × 10^-19

eV → MeV: divide by 10⁶