Mass Defect And Nuclear Binding Energy (Copy)

• Energy–Mass Equivalence: E = mc²

• Mass and energy are interchangeable, as stated by Einstein’s equation:
  E = mc²
  where:
  E = energy (Joules)
  m = mass (kg)
  c = speed of light in vacuum = 3.00 × 10⁸ m/s
• A small amount of mass corresponds to a large amount of energy.

• Nuclear Reactions Representation

• Reactions are written in the form:
  ⁷₃Li + ⁴₂He → ⁸₄Be + ¹₁H
• Superscript = nucleon number (mass number)
• Subscript = proton number (atomic number)
• Both nucleon and proton numbers must balance on both sides of the equation.

• Mass Defect (Δm)

• The mass of a nucleus is less than the sum of the masses of its individual nucleons.
• This missing mass is the mass defect:
  Δm = (Z × m_p) + (N × m_n) – m_nucleus
  where:
  Z = number of protons
  N = number of neutrons
  m_p = mass of proton
  m_n = mass of neutron
  m_nucleus = actual mass of the nucleus

• Binding Energy (BE)

• Binding energy is the energy required to separate a nucleus into its individual nucleons.
• It is also the energy released when a nucleus is formed from separate nucleons.
• Binding energy is related to mass defect by:
  E = Δm × c²
  (E in Joules when Δm in kg; use 1 u = 1.66 × 10⁻²⁷ kg if using atomic mass units)

• Binding Energy per Nucleon

• Defined as:
  BE per nucleon = Total binding energy ÷ Number of nucleons
• Useful for comparing nuclear stability.
• Greater BE per nucleon ⇒ more stable nucleus

• Graph: Binding Energy per Nucleon vs Nucleon Number

• Curve shows:
  • Low BE/nucleon for light nuclei (e.g. hydrogen)
  • Rapid rise for elements up to iron (Fe, A ≈ 56)
  • Peak at iron-56 (most stable nucleus)
  • Gradual decrease for heavy elements (e.g. uranium)

• Nuclear Fission

• Definition: A heavy nucleus (e.g. ²³⁵₉₂U) splits into two smaller nuclei, releasing energy.
• Mass of products < Mass of reactants ⇒ mass defect → energy release
• Used in nuclear reactors and atomic bombs.

• Nuclear Fusion

• Definition: Two light nuclei (e.g. ¹₁H + ¹₁H) combine to form a heavier nucleus, releasing energy.
• More energy released than fission (per unit mass).
• Requires very high temperature and pressure due to repulsive electrostatic forces.
• Powers stars like the Sun.

• Relevance of Binding Energy per Nucleon

• In fusion, small nuclei combine to form one with higher BE per nucleon, releasing energy.
• In fission, large nucleus splits into ones with higher BE per nucleon, also releasing energy.
• Energy comes from the increase in total BE.

• Calculating Energy Released in Nuclear Reactions

• Step 1: Find mass defect:
  Δm = Total mass of reactants – Total mass of products
• Step 2: Convert Δm to kg if given in atomic mass units (1 u = 1.66 × 10⁻²⁷ kg)
• Step 3: Use:
  E = Δm × c²
  or, for quick calculation in MeV:
  1 u ≈ 931.5 MeV
  ⇒ E (MeV) = Δm (u) × 931.5

Worked Example
Given: Mass of reactants = 4.032 u, mass of products = 4.0026 u
Δm = 4.032 – 4.0026 = 0.0294 u
Energy released = 0.0294 × 931.5 = 27.4 MeV

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course