Nuclear reactions, mass defect and binding energy

Key idea: Nuclear reactions conserve nucleon number, charge and total mass–energy. Binding energy explains why a bound nucleus has less rest mass than its separated nucleons and why fusion or fission can release energy.

GCE A-Level H1 Physics 2027

H1 Physics 8867 · Lesson 3 of 3

Check your understanding

By the end of this lesson, you should be able to

Balance simple nuclear equations.
Calculate mass defect and binding energy using E = mc².
Interpret the binding-energy-per-nucleon curve to explain fusion and fission.

Learn the idea

Big question: Why can a nucleus weigh less than the separated particles from which it is made?

Balance what nuclear reactions conserve

In a nuclear equation, total nucleon number A and total charge number Z balance independently. This often identifies a missing nuclide or emitted particle.

Rest mass need not balance by itself. Total mass–energy is conserved, so a decrease in rest mass appears as released energy and an energy input can increase rest mass.

Check your understanding: What two numbers should you check first in a nuclear equation?

Total nucleon number and total charge number on both sides.

Interpret mass defect and binding

Mass defect is separated-nucleon mass minus bound-nucleus mass. Binding energy Δmc² is the energy needed to separate the nucleus completely, and the same amount is released when that nucleus forms from separated nucleons.

Binding energy per nucleon rises towards iron/nickel and then falls slowly. Fusion of light nuclei and fission of very heavy nuclei can form products with greater total binding energy; the corresponding rest-mass decrease is released.

Check your understanding: Does greater binding energy mean a bound nucleus has greater rest mass?

No. A more tightly bound system has lower rest mass than its separated nucleons by E/c².

Scroll diagram horizontally to read all labels.

Fusion of light nuclei and fission of very heavy nuclei can move products toward greater binding energy per nucleon. The increase in total binding energy is released.

Key ideas

Balance A and Z independently in every nuclear equation.
Greater binding energy means a more tightly bound, lower-rest-mass system.
Energy release depends on total binding energy, not simply the height of one point on the curve.

Relationships to know

mass defect Δm = mass of separated nucleons − nuclear mass
binding energy = Δmc²
1 u corresponds to 931.5 MeV/c²

Follow the reasoning

Worked example

Find binding energy from a mass defect

Question: A helium-4 nucleus has mass 4.00151 u. Use proton mass 1.00728 u and neutron mass 1.00866 u to find its mass defect, total binding energy and binding energy per nucleon. Use 1 u c² = 931.5 MeV.

Step 1: Add separated nucleon masses
Why: Helium-4 contains two protons and two neutrons.
Working: mseparated = 2(1.00728) + 2(1.00866) = 4.03188 u.
Step 2: Find mass defect
Why: The bound nucleus has the smaller mass.
Working: Δm = 4.03188 − 4.00151 = 0.03037 u.
Step 3: Convert and divide
Why: Total binding energy and per-nucleon value answer different questions.
Working: E = 0.03037(931.5) = 28.29 MeV; E/A = 28.29/4 = 7.07 MeV per nucleon.

Answer: Mass defect = 0.03037 u, binding energy = 28.3 MeV and binding energy per nucleon = 7.07 MeV.

Check: The positive mass defect follows the definition separated minus bound; a negative result would signal reversed subtraction.

Now try it with support

Practise with support

Complete ¹⁴₇N + ⁴₂He → ¹⁷₈O + ? and explain the balancing.

Hints

The missing nucleon number is 18 − 17.
The missing charge is 9 − 8.

View the guided answer

The missing particle is ¹₁H. Nucleon numbers balance: 14 + 4 = 17 + 1; charge numbers balance: 7 + 2 = 8 + 1.

Your turn

Practise independently

Given a deuteron mass defect of 0.00239 u, calculate its binding energy in MeV using 1 u = 931.5 MeV/c² and state the conserved quantities in its formation.

Check your answer

Binding energy = 0.00239 × 931.5 = 2.23 MeV. In forming the deuteron, total nucleon number, charge and total mass–energy are conserved; the lower rest mass of the bound system corresponds to released binding energy.

Common mistakes and exam guidance

Watch out for

Balancing only charge and forgetting nucleon number.
Saying mass disappears instead of converting a rest-mass decrease into released energy.

In an exam

On a binding-energy curve, explain the movement towards greater binding energy per nucleon and the resulting increase in total binding energy.
Keep c² or the 931.5 MeV/u conversion attached to the mass-defect step.

Put the ideas together

Exam-style practice [8 marks]

Complete ²³⁵₉₂U + ¹₀n → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + x ¹₀n. Then explain, using the binding-energy-per-nucleon curve and mass–energy conservation, why this fission can release energy. Avoid saying mass or energy disappears.

Plan before you answer

Balance A and Z separately.
Compare initial and final total binding, not just one curve height.
Link released energy to a rest-mass decrease.

View the marking points and model answer

Marking points

Balances nucleon numbers to obtain x = 3.
Checks charge 92 = 56 + 36.
States very heavy uranium lies below the peak in binding energy per nucleon.
States medium-mass products have greater binding energy per nucleon.
Links this to greater total binding energy of products.
States products are more tightly bound/lower rest mass.
Uses ΔE = Δmc² for released energy.
States total mass–energy remains conserved.

Model answer

Nucleon balance gives 235 + 1 = 141 + 92 + x, so x = 3; charge already balances because 92 = 56 + 36. Uranium lies on the heavy side below the binding-energy-per-nucleon peak. The medium-mass products are more tightly bound and have greater total binding energy. Their total rest mass is therefore smaller; the decrease releases energy according to ΔE = Δmc². Total mass–energy is conserved—neither mass nor energy simply disappears.

Finish from memory

Three-question recap

Define mass defect.
Check
Mass of separated nucleons minus mass of the bound nucleus.
What does binding energy represent?
Check
Energy required to separate a nucleus completely, or released when it forms.
Why can fusion and fission both release energy?
Check
Both can produce nuclei with greater total binding energy and lower total rest mass.

Try this next

Practise one nuclear equation, one mass-defect calculation and one curve explanation as a connected sequence.

Check the whole topic Return to Nuclear Physics Review Radioactive decay, half-life, uses and hazards

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Course and syllabus information

Course: GCE A-Level H1 Physics
Edition: GCE A-Level H1 Physics 2027