Mass Defect & Nuclear Binding Energy (DP IB Physics: HL): Revision Note

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:

$∆m={Zm}_p+(A–Z)m_n–m_{total}$

• 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 mass-energy equivalence, a decrease in mass implies 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

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:

$∆E=∆m{c}^2$

• 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

Atomic Mass Unit

• The atomic mass unit is commonly used in nuclear physics to express the mass of subatomic particles

• It is defined as: 

Exactly one twelfth $\left(\frac{1}{12}\right)$ the mass of a neutral atom of carbon-12

• Atomic mass unit u is roughly equal to the mass of one proton or neutron:

  • 1 u = 1.661 × 10⁻²⁷ kg

• Using more precise values for well-known constants, a useful conversion factor can be determined

• A particle with a mass of 1 u has an equivalent energy of 

$E=m{c}^2=\left(1.66053907\times {10}^{-27}\right)\times {\left(2.99792458\times {10}^8\right)}^2=1.49241809\times {10}^{-10}\mathrm{J}$

• Converting to eV by using the precise value of the elementary charge gives

$E=\frac{1.49241809\times {10}^{-10}}{1.60217663\times {10}^{-19}}=931.494\mathrm{MeV}$

• Therefore, the unified atomic mass unit can be used to quickly convert between nuclear mass and energy using:

  • 1 u = 1.661 × 10⁻²⁷ kg = 931.5 MeV c⁻²