8867 / 2027
Nuclear Physics overview

Lesson 9 of 9

Energy from fission and fusion

  • I can sketch binding energy per nucleon against nucleon number.Syllabus

    Syllabus 8867, 11(p). Sketch binding energy per nucleon against nucleon number.

  • I can explain how binding energy per nucleon relates to fusion and fission.Syllabus

    Syllabus 8867, 11(q). Explain how binding energy per nucleon relates to fusion and fission.

A reaction releases energy when its products have less total rest mass than its reactants. The binding-energy-per-nucleon curve helps explain why both the fusion of light nuclei and the fission of heavy nuclei can do this.

Read the binding-energy curve

Binding energy per nucleon rises rapidly across the light nuclei, reaches a broad maximum near nucleon numbers 50-60 in the iron/nickel region, then decreases gradually for heavier nuclei. It measures an average binding energy, not the total binding energy of a nucleus.

Binding energy per nucleon on the actual nucleon-number scale

Binding energy per nucleon on the actual nucleon-number scaleThe vertical axis is binding energy per nucleon from zero to nine megaelectronvolts per nucleon. The horizontal axis is nucleon number 1 to 240 on a uniform numerical scale. The values at nucleon numbers 2 and 4 correctly remain close to the left boundary rather than being artificially spread across the graph. The approximate anchors pair nucleon numbers 1, 2, 4, 16, 56, 92, 141 and 235 with binding energies per nucleon 0, 1.11, 7.07, 8.0, 8.8, 8.6, 8.3 and 7.6 MeV respectively. The trend rises sharply for light nuclei, has a broad maximum near nucleon number 56 and then falls gradually for heavy nuclei. The smooth connecting curve is an approximate trend through supplied anchors, not a precision measured isotope dataset. Individual isotopes can have additional local structure.Binding energy per nucleon/ MeV per nucleon0369160120180240Peak regionNucleon number AApproximate trend; equal A spacing.

The steep light-nucleus rise is compressed by the true horizontal scale. Read the separate enlarged panel below for A = 2 and A = 4. Energy release for a reaction still requires the complete, balanced initial and final account.

Light nuclei: a separately enlarged nucleon-number range

Light nuclei: a separately enlarged nucleon-number rangeThe vertical axis is binding energy per nucleon from zero to nine megaelectronvolts per nucleon. This separately labelled horizontal enlargement covers nucleon number zero to twenty. The trend begins with a single nucleon; no nucleus with zero nucleons is plotted. The same vertical scale as the main plot is retained. Approximate anchors are 1.11 MeV per nucleon at nucleon number 2, 7.07 at nucleon number 4, and 8.0 at nucleon number 16. The smooth connecting curve is an approximate trend through supplied anchors, not a precision measured isotope dataset. Individual isotopes can have additional local structure.Binding energy per nucleon/ MeV per nucleon03690248121620Nucleon number AHorizontal enlargement: A = 0 to 20

Only the horizontal range is enlarged here; the vertical quantity and scale are unchanged. A larger total binding energy is not automatically a larger binding energy per nucleon. The curve is an approximate trend, not an exact isotope mass table.

The main plot preserves the actual spacing of nucleon number A. The separate A = 0 to 20 view enlarges the light-nucleus region horizontally. These are approximate trend illustrations; the ordinate is binding energy per nucleon, not total binding energy.

Products with greater total binding energy have lower total rest energy for the same complete nucleon account. Multiply each nucleus's binding energy per nucleon by its own A before comparing totals. Local isotope differences exist, and this curve alone does not determine a reaction rate or every kind of nuclear stability.

For a sketch question

Label these features when you sketch the curve

Put nucleon number A on the x-axis and binding energy per nucleon (in MeV) on the y-axis. Then show these features:

  1. A steep rise for light nuclei. Hydrogen-2 is only about 1.1 MeV per nucleon.
  2. Helium-4 sits above its neighbours. At about 7.1 MeV per nucleon it is a spike above hydrogen-3, helium-3 and lithium-6. Helium-4 is unusually tightly bound.
  3. A broad peak of about 8.8 MeV per nucleon near A = 56. Label it iron or nickel. These are the most tightly bound nuclei.
  4. A slow fall for heavy nuclei. Uranium is down to about 7.6 MeV per nucleon at A = 235 to 238.
  5. Mark the two directions. Draw an arrow up the left side labelled "fusion" and an arrow up the right side, towards the peak, labelled "fission".

Explain fission and fusion in mark-scheme order

A common 3-mark question asks why both fission and fusion release energy. Write the links in this order:

  1. Fusion: light nuclei are on the rising part of the curve. When they join, the product has a higher binding energy per nucleon.
  2. Fission: a heavy nucleus is past the peak. When it splits into two medium nuclei, the products are nearer the peak and have a higher binding energy per nucleon.
  3. In both cases the total binding energy of the products is greater than that of the reactants. So the products have less total rest mass. This mass difference Δm is released as energy, E = Δmc2, mostly as kinetic energy of the products.

Iron and nickel sit at the peak. Neither fusing nor splitting them raises the binding energy per nucleon, so neither process releases energy from them.

Fusion combines light nuclei

In the supplied deuterium-tritium reaction, a hydrogen-2 nucleus and a hydrogen-3 nucleus form an alpha particle and a neutron:

+→+

Both sides have five nucleons and charge +2e. Use all products when comparing rest masses. The supplied nuclear masses, in u, are 2.01355321, 3.01550072, 4.00150618 and 1.00866492 in the displayed order.

Worked fusion energy

Include the outgoing neutron

Δm = (2.01355321 + 3.01550072)
- (4.00150618 + 1.00866492)
= 0.01888283 u
Q = 0.01888283(931.5)
= 17.5894 MeV ≈ 17.6 MeV

Here Q is the total energy released per complete reaction, using supplied 1 u c2 = 931.5 MeV. Leaving out the neutron would give a huge false mass deficit. Positive released energy does not remove the initial electric repulsion between the positively charged reactants.

Put these in order

Explain why energy is released when a uranium-235 nucleus undergoes fission. Put the links in order.

Show the answer

1. A slow neutron is absorbed by a uranium-235 nucleus, making it unstable. The unstable nucleus cannot hold together and splits.

2. The nucleus splits into two smaller nuclei and two or three neutrons. The fragments sit nearer the peak of the binding energy per nucleon curve than uranium does.

3. The fragments have a higher binding energy per nucleon than uranium-235, so the total binding energy increases. A larger total binding energy means the products are lighter than the reactants.

4. The total mass of the products is less than the total mass of the reactants. The missing mass is released as energy according to E = (change in m) c2.

5. The mass difference appears as kinetic energy of the fragments and neutrons, E = (change in m) c2.

Fission splits a heavy nucleus

One possible neutron-induced fission channel is:

+→++ 3

The totals are A = 236 and charge +92e on both sides. Other fragment combinations are possible; this is one supplied channel, not a unique outcome of every fission event.

Approximate supplied binding energies for a fission estimate
NucleusABinding energy / MeV per nucleon
Uranium-2352357.60
Barium-1411418.30
Krypton-92928.60
Initial total binding = 235(7.60) = 1786.0 MeV
Final total binding = 141(8.30) + 92(8.60)
= 1170.3 + 791.2 = 1961.5 MeV
Energy released ≈ 1961.5 - 1786.0
= 175.5 MeV

Free neutrons contribute no nuclear binding energy. This is an estimate from the supplied rounded averages, not a precision energy value for the channel. The products are more tightly bound overall, so the decrease in total rest energy is available as kinetic energy and radiation.

For both fusion and fission, compare the complete initial and final accounts. Subtracting two per-nucleon values without weighting by nucleon number does not give the reaction energy.

Check your understanding

For deuteron + triton -> alpha particle + neutron, supplied nuclear masses are 2.01355321, 3.01550072, 4.00150618 and 1.00866492 u respectively. With 1 u c2 = 931.5 MeV, what is the total released energy?

  1. About 3.52 MeV.
  2. About 17.6 MeV.
  3. About 957 MeV, using the alpha particle alone as the final mass.
Show the answer

About 17.6 MeV.

Initial minus final rest mass is 2.01355321 + 3.01550072 - 4.00150618 - 1.00866492 = 0.01888283 u. Multiplying by 931.5 gives 17.5894 MeV.

Practise exam questions

Write your answer first, then tick the mark points it makes.

Exam-style question

Use the variation of binding energy per nucleon with nucleon number to explain why energy is released both when light nuclei fuse and when a heavy nucleus splits. [3]

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Exam-style question

Sketch a graph of binding energy per nucleon against nucleon number A for A from 1 to 240. Label the values and features an examiner would look for. [4]

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