Mass Defect & Nuclear Binding Energy (DP IB Physics: SL)

Energy & Mass 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 = mc²

• 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

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 an atom's mass and the sum of the 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 released when a nucleus forms from constituent nucleons

OR

The (minimum) energy needed to break a nucleus up into its constituent nucleons (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

• Binding energy can often be described per nucleon
  • This is simply found by dividing the relevant binding energy by the number of nucleons in the nucleus

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

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, m_p = 1.007 276 u
  • Neutron mass, m_n = 1.008 665 u
  • Mass of potassium-40, m_total = 39.953 548 u

Δm = Zm_p + Nm_n – m_total

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