IBPhysicsHLRevision NotesNuclear & Quantum PhysicsRadioactive DecayMass Defect & Nuclear Binding Energy

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Mass Defect & Nuclear Binding Energy (HL IB Physics)

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Mass Defect & Nuclear 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
    • In other words, the combined mass of 6 separate protons and 6 separate neutrons is more than the mass of a carbon-12 nucleus 
    • This difference in mass is known as the mass defect 
  • Mass defect is defined as:

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

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

increment m space equals space Z m subscript p space plus space left parenthesis A space – space Z right parenthesis m subscript n space – space m subscript t o t a l end subscript

  • Where:
    • Z = proton number
    • A = nucleon number
    • mp = mass of a proton (kg)
    • mn = mass of a neutron (kg)
    • mtotal = measured mass of the nucleus (kg)

Binding Energy, downloadable AS & A Level Physics revision notes

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

  • Due to mass-energy equivalence, a decrease in mass infers that energy must be released
  • Energy and mass are proportional, so, the total energy of a nucleus is less than the sum of the energies of its constituent nucleons
  • Binding energy is defined as:

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

  • The formation of a nucleus from a system of isolated protons and neutrons releases energy

Worked example

The binding energy per nucleon is 7.98 MeV for an atom of Oxygen-16 (16O).

Determine an approximate value for the energy required, in MeV, to completely separate the nucleons of this atom.

Answer:

Step 1: List the known quantities

  • Binding energy per nucleon, E = 7.98 MeV

Step 2: State the number of nucleons

  • The number of nucleons is 8 protons and 8 neutrons, therefore 16 nucleons in total

Step 3: Find the total binding energy

  • The binding energy for oxygen-16 is:

7.98 × 16 = 127.7 MeV

Step 4: State the final answer

  • The approximate total energy needed to completely separate this nucleus is 127.7 MeV

Exam Tip

Binding energy is named in a confusing way, so be careful!

Avoid describing the binding energy as the energy stored in the nucleus – this is not correct – it is energy that must be put into the nucleus to pull it apart.

Mass-Energy Equivalence

  • Einstein showed in his Theory of Relativity that matter can be considered a form of energy and hence, he proposed:
    • Mass can be converted into energy
    • Energy can be converted into mass
  • This is known as mass-energy equivalence, and can be summarised by the equation:

increment E space equals space increment m c squared

  • Where:
    • E = energy (J)
    • m = mass (kg) 
    • c = the speed of light  (m s-1)
  • Some examples of mass-energy equivalence are:
    • The fusion of hydrogen into helium in the centre of the sun
    • The fission of uranium in nuclear power plants
    • Nuclear weapons
    • High-energy particle collisions in particle accelerators

Worked example

Calculate the binding energy per nucleon, in MeV, for the radioactive isotope potassium-40 open parentheses K presubscript 19 presuperscript 40 close parentheses.

You may use the following data:

  • Nuclear mass of potassium-40 = 39.953 548 u
  • Mass of one neutron = 1.008 665 u
  • Mass of one proton = 1.007 276 u

Answer:

Step 1: Identify the number of protons and neutrons in potassium-40

  • Proton number, Z = 19
  • Neutron number, N = 40 – 19 = 21

Step 2: Calculate the mass defect, Δm

  • Proton mass, mp = 1.007 276 u
  • Neutron mass, mn = 1.008 665 u
  • Mass of potassium-40, mtotal = 39.953 548 u

Δm = Zmp + Nmn – mtotal

Δm = (19 × 1.007276) + (21 × 1.008665) – 39.953 548

Δm = 0.36666 u

Step 3: Convert mass units from u to kg

  • 1 u = 1.661 × 10–27 kg

Δm = 0.36666 × (1.661 × 10–27) = 6.090 × 10–28 kg

Step 4: Write down the equation for mass-energy equivalence

E = Δmc2

  • Where c = 3.0 × 108 m s–1

Step 5: Calculate the binding energy, E

E = 6.090 × 10–28 × (3.0 × 108)2 = 5.5 × 10–11 J

Step 6: Determine the binding energy per nucleon and convert J to MeV

  • Take the binding energy and divide it by the number of nucleons
  • 1 MeV = 1.6 × 10–13 J

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    Katie M

    Author: Katie M

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.