PhysicsISC Class 12 · Nuclei

Nuclear Fusion and the Binding Energy per Nucleon Curve

Updated 15 October 2026 · Fact-checked

Nuclear fusion is the joining of two light nuclei to form a heavier one, as in the Sun, where four hydrogen nuclei become one helium nucleus releasing about 26.7 MeV. Light nuclei have low binding energy per nucleon, so the product is more tightly bound and energy is released. Fusion needs temperatures of millions of kelvin to overcome electric repulsion.

Understand Nuclear Fusion and the Binding Energy Curve

The binding energy per nucleon curve. Plot BE/A (MeV) against mass number A:

It rises steeply for light nuclei: $^{2}$ H about 1.1 MeV, $^{3}$ He about 2.6 MeV, with a sharp peak at $^{4}$ He (about 7.1 MeV), and smaller peaks at $^{12}$ C and $^{16}$ O.
It reaches a broad maximum of about 8.8 MeV near A = 56 (iron, nickel): the most stable nuclei.
It falls slowly for heavy nuclei, to about 7.6 MeV for $^{238}$ U, because the growing Coulomb repulsion among protons weakens binding.
For A between about 30 and 170 it is nearly flat (about 8 to 8.8 MeV), because the nuclear force is short-range and saturates: a nucleon bonds only with its near neighbours.

What the curve predicts (qualitative, as the syllabus asks). Energy is released whenever nucleons move to a state of higher BE/A:

Fission: a heavy nucleus (low BE/A, right side) splits into two middle nuclei (higher BE/A). For A = 240 at 7.6 MeV splitting into two A = 120 at 8.5 MeV, $Q \approx 240 \times (8.5 - 7.6) \approx 216$ MeV, about 200 MeV.
Fusion: two light nuclei (low BE/A, left side) combine into a heavier one (higher BE/A). Because the curve is so steep on the left, fusion releases more energy per nucleon than fission.

In both cases the products have less mass than the reactants; the mass defect appears as energy.

Fusion in the Sun and stars. In the core of the Sun (temperature about $1.5 \times 1 0^{7}$ K), a chain of reactions (the proton-proton cycle) has the net effect
$4_{1}^{1} H \to_{2}^{4} He + 2_{+ 1}^{0} e + 2 ν + 26.7 MeV .$
About 0.7% of the hydrogen mass is turned into energy. This thermonuclear energy is the source of the light and heat of the Sun and stars.

Why fusion is hard. Nuclei are positively charged and repel each other. To come within the range of the nuclear force (a few fm), they need large kinetic energy, so the gas must be at a very high temperature, of the order of $1 0^{6}$ to $1 0^{7}$ K or more (in a star, gravity holds the hot gas together; quantum tunnelling helps at lower energies). On Earth, it is very difficult to heat a gas so much and confine it long enough; fusion reactors are still experimental.

Hydrogen bomb. An uncontrolled fusion reaction (of hydrogen isotopes) triggered by the very high temperature of a fission bomb explosion.

Fusion versus fission.

Fusion	Fission
Light nuclei join	Heavy nucleus splits
Needs very high temperature	Needs slow neutrons
More energy per nucleon	Less energy per nucleon
Products largely non-radioactive; fuel (deuterium) abundant	Radioactive waste; fuel limited
Not yet controlled for power on Earth	Controlled in reactors

Nuclear Fusion and the Binding Energy Curve: formulas and key facts

Net solar fusion: $4_{1}^{1} H \to_{2}^{4} He + 2 e^{+} + 2 ν + 26.7 MeV$; Proton-proton cycle.
Q from masses: $Q = [m_{r e a c t an t s} - m_{p r o d u c t s}] \times 931 MeV$; Positive Q: energy released.
Q from the curve: $Q = \sum (A \cdot \frac{B E}{A})_{p r o d u c t s} - \sum (A \cdot \frac{B E}{A})_{r e a c t an t s}$; Free nucleons have zero BE.
Temperature estimate: $\frac{3}{2} k T \approx \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r}$; Rough barrier estimate.
Fission estimate: $Q \approx 240 \times (8.5 - 7.6) \approx 200 MeV$; Syllabus example.

How to solve Nuclear Fusion and the Binding Energy Curve questions

Decide whether data are masses or BE/A values; then subtract in the right order.

1Check the reaction balances in A and Z.
2Masses: Q = (reactants − products) × 931 MeV; positive means energy is released.
3BE/A data: total BE = A × BE/A for each nucleus; Q = total BE (products) − total BE (reactants).
4Per-nucleon comparisons: divide Q by the number of nucleons taking part.
5Explain-type: point to the curve: moving towards A ≈ 56 releases energy.

Nuclear Fusion and the Binding Energy Curve in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

How much energy from a kilogram of hydrogen in the Sun

Suppose 1 kg of hydrogen in the Sun's core is fully fused into helium, each set of four protons releasing 26.7 MeV.

Estimate the energy released and the fraction of mass converted.

Try it first, then show the solutionSolution

Number of H atoms $= \frac{1000}{1.007825} \times 6.02 \times 1 0^{23} \approx 5.97 \times 1 0^{26}$ ; reactions $= \frac{5.97 \times 1 0 ^{26}}{4} \approx 1.49 \times 1 0^{26}$ .
$E \approx 1.49 \times 1 0^{26} \times 26.7 \times 1.6 \times 1 0^{- 13} \approx 6.4 \times 1 0^{14}$ J.
Check: $\frac{E}{c ^{2}} \approx 7.1 \times 1 0^{- 3}$ kg, about 0.7% of the mass.

Answer: ≈ $6.4 \times 1 0^{14}$ J, from about 7 g of mass.

Memory hook: Seven grams in every kilogram become sunshine.

Why stars stop at iron

Imagine a massive star that has fused hydrogen, then helium, carbon and heavier elements, until its core is iron.

Why can it get no more energy from fusing iron?

Try it first, then show the solutionSolution

Iron (A ≈ 56) is at the peak of the BE/A curve.
Fusing iron into heavier nuclei would lower BE/A, so it would absorb energy instead of releasing it; the core can no longer support itself.

Answer: Iron sits at the top of the curve; fusion beyond it costs energy.

Memory hook: Up the curve from either side, never past the summit.

Tricks to remember Nuclear Fusion and the Binding Energy Curve

Short memory aids. Say each one aloud once, then use it on a question the same day.

Climb towards iron
Light nuclei fuse up the left slope; heavy nuclei split up the right slope.
Use it when: Why both release energy.
Four H, one He, 26.7
4 ¹H → ⁴He + 26.7 MeV.
Use it when: Solar fusion.
Steep left, gentle right
The curve rises steeply for light nuclei, falls gently for heavy.
Use it when: Sketching and explaining.
Hot to beat the push
High temperature gives nuclei the speed to beat Coulomb repulsion.
Use it when: Why fusion is hard.

Nuclear Fusion and the Binding Energy Curve: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Fuel of the Sun · 1 mark · MCQBoard level

The main energy-producing process in the Sun converts:
(a) helium into hydrogen
(b) hydrogen into helium
(c) uranium into barium and krypton
(d) carbon into iron

Try it first, then show the solutionSolution

Net reaction: 4 ¹H → ⁴He + energy.

Answer: (b).

Example 2: Fusion energy from the curve · 2 marks · Short answerBoard level

Using BE/A of $^{2}$ H = 1.11 MeV and $^{4}$ He = 7.07 MeV, find the energy released when two deuterons fuse to form $^{4}$ He.

Try it first, then show the solutionSolution

BE of two deuterons = 2 × 2 × 1.11 = 4.44 MeV; BE of ⁴He = 4 × 7.07 = 28.28 MeV. (1)
Q = 28.28 − 4.44 = 23.84 MeV. (1)

Answer: ≈ 23.8 MeV (the same reaction, by masses, was asked in ISC 2024 Q19(i)(b)).

Example 3: Features of the curve · 4 marks · Long answerHarder board level

Sketch the variation of binding energy per nucleon with mass number. State three features and use the curve to explain why both fission and fusion release energy.

Try it first, then show the solutionSolution

Sketch: steep rise for small A (peak at ⁴He), broad maximum ≈ 8.8 MeV near A = 56, slow fall to ≈ 7.6 MeV at A ≈ 238. (1)
Features: low BE/A for light and heavy nuclei; maximum for middle nuclei (most stable); nearly constant for A ≈ 30 to 170 (short-range, saturating nuclear force). (1)
Fission: a heavy nucleus splits into middle nuclei with higher BE/A; the extra binding energy is released. (1)
Fusion: light nuclei combine into a nucleus of higher BE/A; again the difference is released. (1)

Answer: Energy is released whenever products have greater BE/A, i.e. move towards A ≈ 56.

Example 4: Deuterium and helium-3 · 3 marks · Short answerHarder board level

Find the energy released in $_{1}^{2}$ H + $_{2}^{3}$ He → $_{2}^{4}$ He + $_{1}^{1}$ H. (Atomic masses: ²H 2.014102 u, ³He 3.016029 u, ⁴He 4.002604 u, ¹H 1.007825 u; 1 u = 931 MeV)

Try it first, then show the solutionSolution

Reactants: 2.014102 + 3.016029 = 5.030131 u. (1)
Products: 4.002604 + 1.007825 = 5.010429 u; Δm = 0.019702 u. (1)
Q = 0.019702 × 931 ≈ 18.3 MeV. (1)

Answer: ≈ 18.3 MeV.

Example 5: Estimating the fusion temperature · 3 marks · Short answerHOTS / competency

Two deuterons must come within about 4 fm for fusion. Estimate the Coulomb potential energy at that separation (use $\frac{e ^{2}}{4 π ε _{0}}$ = 1.44 MeV fm) and the temperature at which the average thermal energy $\frac{3}{2} k T$ equals it ( $k = 1.38 \times 1 0^{- 23}$ J/K). Comment.

Try it first, then show the solutionSolution

$U = \frac{1.44}{4} = 0.36$ MeV $= 5.76 \times 1 0^{- 14}$ J. (1)
$T = \frac{2 U}{3 k} = \frac{2 \times 5.76 \times 1 0 ^{- 14}}{3 \times 1.38 \times 1 0 ^{- 23}} \approx 2.8 \times 1 0^{9}$ K. (1)
Stars fuse at about $1 0^{7}$ K because some nuclei are much faster than average and quantum tunnelling lets them through the barrier; still, millions of kelvin are essential. (1)

Answer: 0.36 MeV; ≈ $2.8 \times 1 0^{9}$ K by this rough estimate.

Example 6: Fusion or fission? · 4 marks · Case-basedHOTS / competency

Suppose a science club compares the solar fusion reaction (26.7 MeV from 4 nucleons) with fission of U-235 (about 200 MeV from 236 nucleons).
(i) Find the energy per nucleon in each. (1 mark)
(ii) Which releases more energy per kilogram of fuel? (1 mark)
(iii) Give two reasons fusion is attractive for future power. (1 mark)
(iv) Why is it not yet used in power stations? (1 mark)

Try it first, then show the solutionSolution

(i) Fusion: $\frac{26.7}{4} \approx 6.7$ MeV per nucleon; fission: $\frac{200}{236} \approx 0.85$ MeV per nucleon.
(ii) Fusion, by a factor of about 8 per nucleon (so per kilogram).
(iii) Fuel (deuterium from sea water) is abundant; products are largely non-radioactive.
(iv) Nuclei must be kept at millions of kelvin and confined long enough; this is not yet achieved economically.

Answer: (i) ≈ 6.7 and ≈ 0.85 MeV per nucleon; (ii) fusion; (iii) abundant fuel, less radioactive waste; (iv) temperature and confinement.

Where students lose marks in Nuclear Fusion and the Binding Energy Curve

The slips examiners see most often, and what to write instead.

Mistake: Vague answers for where fusion happens
The ISC 2025 examiners (Q19(i)(b)(1)) saw answers like "solar system", "earth" or "moon".
Do this: Answer "in the Sun and other stars (their cores)".
Usually costs: 1 mark.
Mistake: Using only one reactant
In 2024 Q19(i)(b) (D + D → ⁴He) some used the mass of only one reactant, or left the answer as a mass difference.
Wrong working
For ²H + ³H → ⁴He + n: Δm = 3.016049 − 4.002604.
Right working
Δm = (2.014102 + 3.016049) − (4.002604 + 1.008665) = 0.018882 u; E = 0.018882 × 931 ≈ 17.58 MeV.
Do this: Add all reactants, subtract all products, then multiply by 931.
Usually costs: 1 to 2 marks.
Mistake: Wrong units for Q
Also 2024: answers left in u or with no unit.
Do this: Convert to MeV with the 931 given.
Usually costs: 1 mark.
Mistake: Saying fusion needs neutrons
Confusion with fission.
Do this: Fusion needs very high temperature (thermonuclear), not slow neutrons.
Usually costs: 1 mark.

Exam tips

Sketch the BE/A curve with the peak near A = 56 and ⁴He marked.
Show the subtraction direction clearly: reactants − products for masses; products − reactants for binding energies.
Write "thermonuclear" when explaining energy in stars.
Compare fusion and fission in a short table if asked for differences.

Practice questions on Nuclear Fusion and the Binding Energy Curve

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Energy is released when:
(a) a nucleus of A = 56 splits
(b) two light nuclei fuse
(c) a nucleus of A = 56 fuses with another
(d) helium splits into deuterons
Hint
Move towards the peak.
Answer
(b).
Q2 · 1 mark · Short answerBoard level
Write the net nuclear reaction for energy production in the Sun.
Hint
Four protons.
Answer
4 $_{1}^{1}$ H → $_{2}^{4}$ He + 2 $_{+ 1}^{0}$ e + 2ν + 26.7 MeV.
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Nuclear fusion requires a very high temperature.
Reason (R): The nuclei must have enough kinetic energy to overcome their electrostatic repulsion.
(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.
Hint
Coulomb barrier.
Answer
(a).
Q4 · 2 marks · Short answerHarder board level
Find the energy released in $^{2}$ H + $^{2}$ H → $^{3}$ H + $^{1}$ H. (²H 2.014102 u, ³H 3.016049 u, ¹H 1.007825 u; 1 u = 931 MeV)
Hint
Δm = 0.00433 u.
Answer
≈ 4.03 MeV.
Q5 · 2 marks · Short answerHarder board level
Find the energy released in $^{2}$ H + $^{2}$ H → $^{3}$ He + n. (³He 3.016029 u, n 1.008665 u)
Hint
Δm = 0.00351 u.
Answer
≈ 3.27 MeV.
Q6 · 2 marks · Short answerHOTS / competency
Using BE/A values ⁶Li 5.33 MeV, ²H 1.11 MeV and ⁴He 7.07 MeV, find the energy released in ⁶Li + ²H → 2 ⁴He.
Hint
Total BE of each side.
Answer
8 × 7.07 − (6 × 5.33 + 2 × 1.11) ≈ 22.4 MeV.
Q7 · 2 marks · Short answerHOTS / competency
Why is the BE/A curve nearly flat for medium nuclei? What does this tell us about the nuclear force?
Hint
Range of the force.
Answer
Each nucleon interacts only with its nearest neighbours, so adding nucleons does not raise the binding per nucleon: the nuclear force is short-range and saturates.
Q8 · 4 marks · Case-basedHOTS / competency
Case study: Suppose an experimental fusion device uses the reaction D + D → ³He + n (3.27 MeV per reaction).
(i) How many deuterium nuclei are there in 1 kg of deuterium (2.014102 u per atom)? (1 mark)
(ii) How many reactions can occur? (1 mark)
(iii) Find the energy released. (1 mark)
(iv) Where does fusion happen naturally and continuously in the Universe? (1 mark)
Hint
$N = \frac{1000}{2.014} \times N_{A}$ .
Answer
(i) ≈ $2.99 \times 1 0^{26}$ . (ii) ≈ $1.49 \times 1 0^{26}$ (two nuclei per reaction). (iii) ≈ $7.8 \times 1 0^{13}$ J. (iv) In the Sun and other stars.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Energy from 4 ¹H → ⁴He?Show answer
About 26.7 MeV.
2.Peak of the BE/A curve?Show answer
Near A = 56, about 8.8 MeV.
3.Why does fusion need high temperature?Show answer
To overcome Coulomb repulsion between nuclei.
4.Where does fusion occur naturally?Show answer
In the Sun and stars.
5.Which gives more energy per nucleon, fusion or fission?Show answer
Fusion.

Nuclear Fusion and the Binding Energy Curve: frequently asked questions

Why does the BE/A curve fall for heavy nuclei?

Proton-proton Coulomb repulsion acts over the whole nucleus and grows with the number of protons, while the nuclear attraction is short-range; so binding per nucleon decreases.

What is thermonuclear fusion?

Fusion made possible by very high temperature, which gives nuclei enough kinetic energy to approach each other closely.

Is a hydrogen bomb fission or fusion?

Fusion; but a fission explosion is used to produce the very high temperature needed to start it.

What does the ISC 2027 syllabus need here?

A qualitative prediction from the BE/A curve that fusing light nuclei and splitting heavy nuclei both release energy, the Q ≈ 200 MeV estimate for A = 240, fusion with the example 4 ¹H → ⁴He, the need for very high temperature, the hydrogen bomb and energy production in the Sun and stars.

How was this asked recently?

ISC 2024 Q19(i)(b) (2 marks) asked the energy released in D + D → ⁴He; ISC 2025 Q19(i)(b)(1) asked where fusion takes place continuously.