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 = Zmp + (A – Z)mn – mtotal
- 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)
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
Worked example
Calculate the binding energy per nucleon, in MeV, for the radioactive isotope potassium-40 (19K).
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
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 K-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 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
Exam Tip
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.Make sure to learn the definitions of mass defect and binding energy as these are common exam questions!