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Binding Energy per Nucleon Curve (DP IB Physics: SL)

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Binding Energy per Nucleon Curve

  • 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

By plotting a graph of binding energy per nucleon against nucleon number, the stability of elements can be inferred

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

Comparing Fusion & Fission

  • Fusion occurs at low values of A because:
    • Attractive nuclear forces between nucleons dominate over repulsive electrostatic forces between protons

  • 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

  • Fission occurs at high values of A because:
    • Repulsive electrostatic forces between protons begin to dominate, and these forces tend to break apart the nucleus rather than hold it together

  • 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

  • Fusion releases much more energy per kg than fission
  • The energy released is the difference in binding energy caused by the difference in mass between the reactant and products
    • Hence, the greater the increase in binding energy, the greater the energy released

  • At small values of A (fusion region), the gradient is much steeper compared to the gradient at large values of A (fission region)
  • This corresponds to a larger binding energy per nucleon being released

Worked example

Step 1: Calculate the mass defect Δm

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 E 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

Binding energy per nucleon = E over A

E over A equals fraction numerator 7.812 cross times 10 to the power of negative 11 end exponent over denominator 56 end fraction equals 1.395 space cross times space 10 to the power of negative 12 end exponent space straight J

Step 4: Convert to MeV

J → eV: multiply by 6.2 × 10-18

eV → MeV: divide by 106

Binding energy per nucleonequals open parentheses 1.395 space cross times space 10 to the power of negative 12 end exponent close parentheses space cross times space open parentheses 6.2 space cross times space 10 to the power of 18 close parentheses equals space 8 space 649 space 000 space eV space equals space 8.6 space MeV space left parenthesis 2 space straight s. straight f. right parenthesis

Worked example

The equation below represents one possible decay of the induced fission of a nucleus of uranium-235.

U presubscript 92 presuperscript 235 plus n presubscript 0 presuperscript 1 rightwards arrow S presubscript 38 presuperscript 91 r plus scriptbase X e end scriptbase presubscript 54 presuperscript 142 plus 3 n presubscript 0 presuperscript 1

The graph shows the binding energy per nucleon plotted against nucleon number A.Worked Example - Binding Energy Graph, downloadable AS & A Level Physics revision notes

Calculate the energy released:

a) By the fission process represented by the equation

b) When 1.0 kg of uranium, containing 3% by mass of U-235, undergoes fission

Part (a)

Step 1: Use the graph to identify each isotope’s binding energy per nucleon

8-4-4-worked-example---binding-energy-graph-ans-new

    • Binding energy per nucleon (U-235) = 7.5 MeV
    • Binding energy per nucleon (Sr-91) = 8.2 MeV
    • Binding energy per nucleon (Xe-142) = 8.7 MeV

Step 2: Determine the binding energy of each isotope

Binding energy = Binding Energy per Nucleon × Mass Number

    • Binding energy of U-235 nucleus = (235 × 7.5) = 1763 MeV
    • Binding energy of Sr-91 = (91 × 8.2) = 746 MeV
    • Binding energy of Xe-142 = (142 × 8.7) = 1235 MeV

Step 3: Calculate the energy released

Energy released = Binding energy after (Sr + Xe) – Binding energy before (U)

Energy released = (1235 + 746) – 1763 = 218 MeV

Part (b)

Step 1: Calculate the energy released by 1 mol of uranium-235

    • There are NA (Avogadro’s number) atoms in 1 mol of U-235, which is equal to a mass of 235 g
    • Energy released by 235 g of U-235 = (6 × 1023) × 218 MeV

 

Step 2: Convert the energy released from MeV to J

    • 1 MeV = 1.6 × 10–13 J
    • Energy released = (6 × 1023) × 218 × (1.6 × 10–13) = 2.09 × 1013 J

Step 3: Work out the proportion of uranium-235 in the sample

    • 1 kg of uranium which is 3% U-235 contains 0.03 kg or 30 g of U-235

Step 4: Calculate the energy released by the sample

Energy released from 1 kg of Uranium = left parenthesis 2.09 space cross times space 10 to the power of 13 right parenthesis space cross times space 30 over 235 space equals space 2.67 space cross times space 10 to the power of 12 space straight J

Examiner 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:

  • 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)

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Katie M

Author: Katie M

Expertise: Physics

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.