Use mass–energy equivalence and balance nuclear equations
Use E = mc² to relate mass and energy, then conserve nucleon number and proton number in simple nuclear reactions.
A Level · Cambridge International Physics 9702, 2025–2027 syllabus. The Topic 23 outcomes were also checked against Cambridge’s 2028–2030 syllabus, which reports no significant teaching changes. Online practice is self-checked and cannot replace supervised practical work, full timed assessment or human subject-teacher review.
Before you begin
- Use scientific notation, SI units, proportional reasoning and the particle numbers in nuclide notation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which two totals must be conserved in a nuclear equation?
What you will learn
- Recall and use E = mc² with consistent SI units
- Read nuclide notation
- Complete simple nuclear equations by conserving nucleon and proton numbers
Mass and energy are equivalent
The relation E = mc² gives the energy equivalent of a mass m. Use mass in kilograms and c in metres per second to obtain energy in joules. The speed of light is 3.00 × 10⁸ m s⁻¹, so even a small mass difference corresponds to a large energy.
In a nuclear reaction, compare the total mass before and after only when the relevant masses are supplied. A decrease in total rest mass can correspond to energy released; E = mc² relates the size of the mass change to the energy change.
Nuclear equations conserve two counts
In notation ᴬZX, the top number A is the nucleon number and the bottom number Z is the proton number. A nucleus contains A − Z neutrons. An element symbol is determined by Z.
For a nuclear equation, the sum of the top numbers must match on both sides, and the sum of the bottom numbers must also match. Use these two conservation checks to find a missing particle or identify an impossible equation.
Pause and explain
A missing product has nucleon number 1 and proton number 0. Which particle is it?
Worked example
Complete ¹⁴₇N + ⁴₂He → ¹⁷₈O + ?
Show the worked solution
- Nucleon numbers on the left total 14 + 4 = 18, so the missing particle has A = 1.
- Proton numbers on the left total 7 + 2 = 9; oxygen contributes 8, so the missing particle has Z = 1.
- A particle with A = 1 and Z = 1 is a proton, ¹₁H.
Answer ¹⁴₇N + ⁴₂He → ¹⁷₈O + ¹₁H
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A reaction has total nucleon number 236 on each side. The known products have nucleon numbers 141, 92 and one neutron. How many further neutrons are required?
Give me a hint
Subtract the known product total from the reactant total.
Compare my reasoning
- Known products total 141 + 92 + 1 = 234.
- The remaining nucleon number is 236 − 234 = 2.
- Each neutron contributes one nucleon.
Two further neutrons.
Look for these in your work
- Adds the known product nucleon numbers
- Subtracts from 236
- Counts one nucleon per neutron
Complete ²³₁₁Na + ¹₀n → ²⁴₁₁Na + ?
Give me a hint
Compare the nucleon-number and proton-number sums.
Compare my reasoning
- The left totals are A = 24 and Z = 11.
- The named sodium product already has A = 24 and Z = 11.
- The missing product therefore contributes A = 0 and Z = 0.
No additional particle is needed; the equation as written is balanced.
Look for these in your work
- Checks both totals
- Does not invent a particle
- Recognises the supplied product already balances the equation
A proposed product in a reaction has A = 4 and Z = 2. Identify it and state its neutron number.
Give me a hint
Use Z to identify the element, then calculate A − Z.
Compare my reasoning
- Z = 2 identifies helium.
- A = 4 gives four nucleons.
- Neutrons = 4 − 2 = 2.
An alpha particle, ⁴₂He, with two neutrons.
Look for these in your work
- Identifies helium from Z
- Uses A − Z
- Names the alpha-particle nucleus
Common mistakes
- Adding nucleon numbers but forgetting to conserve proton number
- Treating the element symbol as an independent quantity instead of checking Z
- Using grams in E = mc² without converting to kilograms
What can you explain now?
Which two totals must be conserved in a nuclear equation?
Compare with the explanation
Nucleon number and proton number
Check the sums of the superscripts and subscripts separately.
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Make it stick
How do the two nuclide numbers help you check a nuclear equation?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.