8.4.2 Mass Difference & 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 amount of energy required to separate 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 a typical nucleus, binding energies are usually measured in MeV
  • This is considerably larger than the few eV associated with the binding energy of electrons in the atom

• Nuclear reactions involve changes in the nuclear binding energy whereas chemical reactions involve changes in the electron binding energy
  • This is why nuclear reactions produce much more energy than chemical reactions

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 K-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 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 = speed of light

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

• The unified atomic mass unit (u) is roughly equal to the mass of one proton or neutron:
  • 1 u = 1.66 × 10⁻²⁷ kg

• It is sometimes abbreviated to a.m.u
• This value will be given on your data sheet in the exam
• The a.m.u is commonly used in nuclear physics to express the mass of subatomic particles. It is defined as

The mass of exactly one-twelfth of an atom of carbon-12

• Therefore, one atom of carbon-12 has a mass of exactly 12 u
• Since mass and energy are interchangeable, the a.m.u can also be expressed in MeV
  • 1 u is equivalent to 931.5 MeV

Table of common particles with mass in a.m.u

• The mass of an atom in a.m.u is roughly equal to the sum of its protons and neutrons (nucleon number)
  • For example, the mass of Uranium-235 is roughly equal to 235u

• A.m.u might be quoted in kg or MeV since mass and energy are equivalent via E = mc²
  • MeV is a unit of energy whilst kg is a unit of mass