# 8.4.4 Binding Energy

### Binding Energy per Nucleon Graph

• In order to compare nuclear stability, it is more useful to look at the binding energy per nucleon
• The binding energy per nucleon is defined as:

The binding energy of a nucleus divided by the number of nucleons in the nucleus

• A higher binding energy per nucleon indicates a higher stability
• In other words, it requires more energy to pull the nucleus apart
• Iron (A = 56) has the highest binding energy per nucleon, which makes it the most stable of all the elements

#### Key Features of the Graph

• At low values of A:
• Nuclei tend to have a lower binding energy per nucleon, hence, they are generally less stable
• This means the lightest elements have weaker electrostatic forces and are the most likely to undergo fusion
• Helium (4He), carbon (12C) and oxygen (16O) do not fit the trend
• Helium-4 is a particularly stable nucleus hence it has a high binding energy per nucleon
• Carbon-12 and oxygen-16 can be considered to be three and four helium nuclei, respectively, bound together
• At high values of A:
• The general binding energy per nucleon is high and gradually decreases with A
• This means the heaviest elements are the most unstable and likely to undergo fission

#### Worked Example

Step 1: Calculate the mass defect

Number of protons, Z = 26

Number of neutrons, A – Z = 56 – 26 = 30

Mass defect, Δm = Zmp + (A – Z)mn – mtotal

Δm = (26 × 1.673 × 10-27) + (30 × 1.675 × 10-27) – (9.288 × 10-26)

Δm = 8.680 × 10-28 kg

Step 2: Calculate the binding energy of the nucleus

Binding energy, E = Δmc2

E = (8.680 × 10-28) × (3.00 × 108)2 = 7.812 × 10-11 J

Step 3: Calculate the binding energy per nucleon

Step 4: Convert to MeV

J → eV: divide by 1.6 × 10-19

eV → MeV: divide by 106

#### Exam Tip

Checklist on what to include (and what not to include) in an exam question asking you to draw a graph of binding energy per nucleon against nucleon number:

• You will be expected to draw the best fit curve AND a cross to show the anomaly that is helium
• Do not begin your curve at A = 0, this is not a nucleus!
• Make sure to correctly label both axes AND units for binding energy per nucleon
• You will be expected to include numbers on the axes, mainly at the peak to show the position of iron (56Fe)
• At low values of A:
• Attractive nuclear forces between nucleons dominate over repulsive electrostatic forces between protons
• In the right conditions, nuclei undergo fusion
• In fusion, the mass of the nucleus that is created is slightly less than the total mass of the original nuclei
• The mass defect is equal to the binding energy that is released, since the nucleus that is formed is more stable
• At high values of A:
• Repulsive electrostatic forces between forces begin to dominate, and these forces tend to break apart the nucleus rather than hold it together
• In the right conditions, nuclei undergo fission
• In fission, an unstable nucleus is converted into more stable nuclei with a smaller total mass
• This difference in mass, the mass defect, is equal to the binding energy that is released
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