Mass defect & binding energy
Mass defect
- 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 the mass of a nucleus and the sum of the individual 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)
Mass defect of carbon-12
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
- Binding energy is defined as:
The energy required to break 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
- This means that it releases energy
- This energy 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
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 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
binding energy per nucleon = (5.5 × 10–11)/40 = 1.375 × 10–12 J
binding energy per nucleon = (1.375 × 10–12)/(1.6 × 10–13) = 8.594 MeV
Examiner Tip
The terms binding energy and mass defect can cause students confusion, so be careful when using them in your explanations.
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 separate all the nucleons.
The same goes for the term mass defect, make sure to only use this when all the nucleons are separated and not to describe the decrease in mass which occurs during radioactive decay.