The big idea: When a nucleus decays, the products are very slightly lighter than the nucleus you started with.
That tiny bit of missing mass does not vanish — it turns into energy, which the products fly off with.
The link between the missing mass and the energy is one famous equation: E = mc².
Before the decay
- One parent nucleus, sitting still
- It has the larger total mass
- No kinetic energy yet (at rest)
After the decay
- Two products fly apart (e.g. a daughter nucleus + an alpha)
- Their total mass is slightly smaller — some mass has vanished
- That missing mass has turned into kinetic energy of the products
New words, plainly: Mass defect (Δm) = how much lighter the products are than the parent.
Released energy (Q) = the energy that the mass defect turns into; also called the disintegration energy.
u = the unified atomic mass unit, the handy unit for nuclear masses (1 u is about the mass of one proton).
| Nucleus / particle | Mass / u |
|---|---|
| Parent (before) | 226.025410 |
| Daughter (after) | 222.017580 |
| Alpha particle (after) | 4.002600 |
| Mass that vanished (Δm) | 0.005230 |
A shortcut for the energy: You could turn Δm into kilograms and multiply by c² — but the data booklet gives a faster route.
1 u is worth 931.5 MeV of energy. So once you have the mass defect in u, just multiply by 931.5 to get the energy released in MeV.
To find the energy released you need just two steps: find the mass defect (parent mass − total product mass), then turn that mass into energy with E = mc².
- energy released, called the disintegration energy Q (J)
- mass defect Δm — the mass that disappears in the decay (kg)
- speed of light, 3.00 × 10⁸ m s⁻¹ (given constant)
Two units, one idea: In joules: put Δm in kilograms (1 u = 1.661 × 10⁻²⁷ kg) and multiply by c².
In MeV (faster): keep Δm in u and multiply by 931.5 (because 1 u = 931.5 MeV c⁻²).
Both give the same energy — pick whichever the question asks for.
IB-style question — show the decay releases about 5 MeV
A nucleus at rest decays by emitting an alpha particle. The masses are: parent = 226.025410 u, daughter = 222.017580 u, alpha = 4.002600 u. Show that the energy released is about 5 MeV. (1 u = 931.5 MeV c⁻².)
Solution
- First find the mass defect Δm = parent mass − total mass of the products:
- Work out the bracket and subtract:
- Now turn that mass into energy with the given equation . Using 1 u = 931.5 MeV c⁻², the c² is already built in, so just multiply Δm by 931.5:
- Work it out — keep the unit:
Final answer
Δm = 0.005230 u, so E = 0.005230 × 931.5 ≈ 4.87 MeV ≈ 5 MeV. The missing mass became the released energy.
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How this is tested: This idea is a classic Paper 2 short-answer pair, usually two ‘show that’ parts.
- Paper 2 — show ≈ 5 MeV: given atomic masses, find the mass defect and use E = mc² (the 931.5 shortcut) to show the released energy. - Paper 2 — show ≈ 98%: use conservation of momentum to show the light product (the alpha) carries almost all of the energy. - Paper 1A: a quick multiple-choice version — which product gets more kinetic energy, or which way Δm and Q go.
Classic trap: thinking the two products share the energy equally, or that the heavier one gets more. The opposite is true — the lighter one gets most of it.
Why the light product gets most of the energy: The parent starts at rest, so its total momentum is zero. After the decay the two products must have equal and opposite momentum (p) so they still add to zero.
Kinetic energy is KE = p² ÷ 2m. They share the same p, so the one with the smaller mass (the alpha) ends up with the bigger kinetic energy.
[Diagram: phys-recoil] - Available in full study mode
Light product (alpha)
- Same size momentum p as the daughter (opposite direction)
- Small mass m
- KE = p² ÷ 2m → small m on the bottom means a BIG kinetic energy
Heavy product (daughter)
- Same size momentum p (recoils the other way)
- Large mass m
- KE = p² ÷ 2m → large m on the bottom means a small kinetic energy
Momentum sharing — what to compare
- Parent at rest → total momentum = 0 before and after
- Two products → equal and opposite momentum (same size p)
- Same p, but KE = p² ÷ 2m → lighter mass = more energy
- Energy split ratio: KE_{alpha} : KE_{daughter} = m_{daughter} : m_{alpha}
IB-style question — show the alpha carries about 98% of the energy
The same decay gives a daughter of mass 222 u and an alpha of mass 4 u, released from a parent at rest. Show that the alpha particle carries about 98% of the released energy.
Solution
- Momentum is conserved and the parent was at rest, so the two products have equal and opposite momentum — call its size p (same for both).
- Kinetic energy of each product is . With the same p, the energy a product gets is proportional to 1 ÷ m — so the share of the total going to the alpha is the daughter's mass over the total mass:
- Put in the masses (222 u and 4 u):
- Work it out as a percentage:
Final answer
KEalpha ÷ KEtotal = mdaughter ÷ (mdaughter + malpha) = 222 ÷ 226 ≈ 0.98, so the alpha carries about 98% of the released energy. The heavy daughter barely recoils.