Nuclear fusion and the binding energy per nucleon curve

Key ideas

Nuclear fusion is the joining of two light nuclei into a single heavier nucleus, releasing energy — the opposite process to fission, but releasing energy for a related reason.
As with fission, the Q-value of a fusion reaction is found from the mass difference:

$Q = \Delta m \, c^2 = (m_{before} - m_{after})c^2$

Fusion requires the two light nuclei to get extremely close together (within range of the strong nuclear force) despite both being positively charged and so electrostatically repelling each other.
- This requires very high temperatures and pressures (so nuclei collide with enough kinetic energy to overcome the electrostatic repulsion) — the conditions found in the cores of stars, including the Sun.

The binding energy per nucleon curve

Plotting binding energy per nucleon, $E_B/A$ $E_{B} / A$ , against mass number, $A$ $A$ , for all known nuclei produces a curve with a characteristic shape:
- $E_B/A$ rises steeply for light nuclei (small $A$ ),
- peaks around iron/nickel ( $A \approx 56$ ), where nucleons are most tightly bound,
- then falls slowly for heavy nuclei (large $A$ ).
This single curve explains why both fission and fusion release energy:
- Fusion (small $A$ → larger $A$ , moving up the steep left-hand side of the curve): the product nucleus sits higher on the curve, meaning more tightly bound per nucleon, so mass is lost and energy is released.
- Fission (large $A$ → smaller $A$ s, moving up the gentler right-hand side of the curve): the fragment nuclei sit higher on the curve than the original heavy nucleus, again meaning more tightly bound per nucleon, again releasing energy.
- Both processes move nuclei toward the peak of the curve, toward iron/nickel — the most stable nuclei. Any nuclear reaction that moves a nucleus's mass number toward $A \approx 56$ increases the binding energy per nucleon and releases energy; moving away from the peak would require an input of energy.
This is also why fusion of nuclei heavier than iron, or fission of nuclei lighter than iron, does not release energy — those reactions would move away from the peak.

Worked ExampleQ-value of a fusion reaction

Two deuterium nuclei ( $^2\text{H}$ , mass $2.0141$ u each) fuse to form a helium-3 nucleus (mass $3.0155$ u) and a free neutron ( $1.0087$ u). Find the energy released, in MeV.

Step 1 — Total mass before fusion

$m_{before} = 2(2.0141) = 4.0282 \text{ u}$

Step 2 — Total mass after fusion

$m_{after} = 3.0155 + 1.0087 = 4.0242 \text{ u}$

Step 3 — Mass defect and Q-value

$\Delta m = 4.0282 - 4.0242 = 0.0040 \text{ u}$ $Q = \Delta m \, c^2 = 0.0040 \times 931.5 = 3.73 \text{ MeV}$

Note the Q-value per reaction is much smaller than a typical fission event (compare $\sim 170$ MeV) — but fusion releases far more energy per unit mass of fuel, because light nuclei sit on the much steeper part of the binding energy curve.

Tips

Both fission and fusion are explained by the same curve — say "moves toward the peak," not "fission releases energy because nuclei split" as if that were the whole story. Full-mark explanations reference the binding-energy-per-nucleon curve directly.
If asked why fusion doesn't happen spontaneously at room temperature, the answer is electrostatic repulsion between the positively charged nuclei, not "not enough neutrons" or unrelated ideas — nuclei need extreme kinetic energy (extreme temperature) to get close enough for the strong force to take over.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

In a fusion reaction, the total mass before is $4.0282$ u and the total mass after is $4.0248$ u.

Calculate the energy released, in MeV.

Merit

Using the binding energy per nucleon curve, explain why fusing two light nuclei into a heavier one releases energy, referring to how binding energy per nucleon changes.

Excellence

Using the binding energy per nucleon curve, explain why BOTH fission of a heavy nucleus and fusion of light nuclei release energy, and explain why fusion or fission of nuclei near iron (A ≈ 56) would not release energy.

Nuclear fusion and the binding energy per nucleon curve

Covers AS91525

Key ideas

Nuclear fusion is the joining of two light nuclei into a single heavier nucleus, releasing energy — the opposite process to fission, but releasing energy for a related reason.
As with fission, the Q-value of a fusion reaction is found from the mass difference:

$Q = \Delta m \, c^2 = (m_{before} - m_{after})c^2$

Fusion requires the two light nuclei to get extremely close together (within range of the strong nuclear force) despite both being positively charged and so electrostatically repelling each other.
- This requires very high temperatures and pressures (so nuclei collide with enough kinetic energy to overcome the electrostatic repulsion) — the conditions found in the cores of stars, including the Sun.

The binding energy per nucleon curve

Plotting binding energy per nucleon, $E_B/A$ $E_{B} / A$ , against mass number, $A$ $A$ , for all known nuclei produces a curve with a characteristic shape:
- $E_B/A$ rises steeply for light nuclei (small $A$ ),
- peaks around iron/nickel ( $A \approx 56$ ), where nucleons are most tightly bound,
- then falls slowly for heavy nuclei (large $A$ ).
This single curve explains why both fission and fusion release energy:
- Fusion (small $A$ → larger $A$ , moving up the steep left-hand side of the curve): the product nucleus sits higher on the curve, meaning more tightly bound per nucleon, so mass is lost and energy is released.
- Fission (large $A$ → smaller $A$ s, moving up the gentler right-hand side of the curve): the fragment nuclei sit higher on the curve than the original heavy nucleus, again meaning more tightly bound per nucleon, again releasing energy.
- Both processes move nuclei toward the peak of the curve, toward iron/nickel — the most stable nuclei. Any nuclear reaction that moves a nucleus's mass number toward $A \approx 56$ increases the binding energy per nucleon and releases energy; moving away from the peak would require an input of energy.
This is also why fusion of nuclei heavier than iron, or fission of nuclei lighter than iron, does not release energy — those reactions would move away from the peak.

Worked ExampleQ-value of a fusion reaction

Two deuterium nuclei ( $^2\text{H}$ , mass $2.0141$ u each) fuse to form a helium-3 nucleus (mass $3.0155$ u) and a free neutron ( $1.0087$ u). Find the energy released, in MeV.

Step 1 — Total mass before fusion

$m_{before} = 2(2.0141) = 4.0282 \text{ u}$

Step 2 — Total mass after fusion

$m_{after} = 3.0155 + 1.0087 = 4.0242 \text{ u}$

Step 3 — Mass defect and Q-value

$\Delta m = 4.0282 - 4.0242 = 0.0040 \text{ u}$ $Q = \Delta m \, c^2 = 0.0040 \times 931.5 = 3.73 \text{ MeV}$

Tips

Both fission and fusion are explained by the same curve — say "moves toward the peak," not "fission releases energy because nuclei split" as if that were the whole story. Full-mark explanations reference the binding-energy-per-nucleon curve directly.
If asked why fusion doesn't happen spontaneously at room temperature, the answer is electrostatic repulsion between the positively charged nuclei, not "not enough neutrons" or unrelated ideas — nuclei need extreme kinetic energy (extreme temperature) to get close enough for the strong force to take over.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

In a fusion reaction, the total mass before is $4.0282$ u and the total mass after is $4.0248$ u.

Calculate the energy released, in MeV.

Merit

Using the binding energy per nucleon curve, explain why fusing two light nuclei into a heavier one releases energy, referring to how binding energy per nucleon changes.

Excellence