IB Physics SLTopic 5 — The Atomic & Nuclear WorldPaper 1 & 2E = Δmc2~8 min read
Mass Defect & Binding Energy
Weigh a nucleus and then weigh its protons and neutrons separately, and something strange happens: the separate parts weigh more than the whole. The missing mass didn’t vanish — it was converted into the energy that binds the nucleus together, following Einstein’s E = mc2. That tiny lost mass is the key to nuclear energy.
📘 What you need to know
The mass defect (Δm) is the difference between the total mass of a nucleus’s separate nucleons and the measured mass of the nucleus — the nucleus is always lighter
Calculate it with Δm = Zmp + (A − Z)mn − mtotal
By mass–energy equivalence, E = Δmc2: the missing mass equals the energy released when the nucleus formed
Binding energy is the energy needed to completely separate a nucleus into its individual nucleons
It is not energy stored in the nucleus — it’s energy you must put in to pull it apart
Masses are often given in atomic mass units (u), where 1 u = 1.661 × 10−27 kg
The handy shortcut 1 u = 931.5 MeV c−2 converts a mass defect in u straight into energy
The Missing Mass
Take a carbon-12 nucleus. Now imagine pulling out all 6 protons and 6 neutrons and weighing them on their own. Add those masses up and you’ll find the total is bigger than the mass of the intact nucleus. That difference is the mass defect, Δm — the mass “lost” when free nucleons come together to form a bound nucleus.
The same nucleons weigh more spread apart than they do bound together. That mass difference is the mass defect — released as binding energy when the nucleus forms, and required to pull it apart again.
Because the separate nucleons are heavier, forming the nucleus releases energy (the extra mass is converted away), and pulling the nucleus back apart requires putting that energy back in. You find the mass defect by adding up the nucleon masses and subtracting the nucleus’s measured mass:
Mass defectΔm = Zmp + (A − Z)mn − mtotal
Here Z is the proton number, A the nucleon number (so A − Z is the neutron count), mp and mn are the proton and neutron masses, and mtotal is the measured mass of the whole nucleus.
Turning Mass into Energy
Einstein’s theory of relativity says mass and energy are two faces of the same thing, linked by the most famous equation in physics. A change in mass corresponds to a change in energy:
Mass–energy equivalenceE = Δmc2
So the mass defect isn’t just a curiosity — multiply it by c2 and you get the binding energy, the energy released when the nucleus formed (and the energy you’d need to tear it apart).
separated nucleons
→ bind together, lose Δm →
nucleus + energy released
What Binding Energy Really Means
Be careful with the words here, because examiners test them precisely. Binding energy is defined as the energy needed to completely separate a nucleus into its individual protons and neutrons. It is not energy stored inside the nucleus — it’s energy you have to supply from outside to break the nucleus apart. A bigger binding energy means a more tightly held, more stable nucleus.
The Atomic Mass Unit Shortcut
Nuclear masses are usually quoted in atomic mass units (u), defined as one-twelfth the mass of a carbon-12 atom, where 1 u = 1.661 × 10−27 kg. Rather than converting to kilograms and multiplying by c2 every time, you can use a direct conversion between mass and energy:
Mass–energy conversion
1 u = 931.5 MeV c−2
So a mass defect of, say, 0.03 u is worth about 0.03 × 931.5 ≈ 28 MeV of binding energy — no need to touch kilograms at all.
Quick recap: the nucleus is lighter than its separate parts by the mass defect Δm = Zmp + (A−Z)mn − mtotal; convert it to binding energy with E = Δmc2, or use the shortcut 1 u = 931.5 MeV c−2; binding energy is the energy needed to pull the nucleus fully apart.
🧭 Finding a binding energy
Count the nucleons: protons = Z, neutrons = A − Z
Add up the separate masses: Zmp + (A−Z)mn
Subtract the nucleus’s measured mass to get the mass defect Δm (in u)
Convert to energy: either × 1.661×10−27 then × c2, or simply × 931.5 to get MeV
For binding energy per nucleon, divide the total binding energy by A
WE 1
The binding energy per nucleon of carbon-12 is 7.68 MeV. Calculate the total energy needed to completely separate a carbon-12 nucleus into its individual nucleons.
Step 1 — count the nucleons
carbon-12 has 6 protons + 6 neutrons = 12 nucleons
Step 2 — total binding energy = per-nucleon × A
total = 7.68 × 12
total = 92.16 MeV≈ 92.2 MeVThis is the energy you’d have to put in to pull the nucleus completely apart.
WE 2
A helium-4 nucleus has a measured mass of 4.001506 u. Using mp = 1.007276 u and mn = 1.008665 u, calculate (a) the mass defect, and (b) the binding energy per nucleon in MeV.
Part (a) — mass defect
He-4 has Z = 2, so 2 protons and 2 neutrons
Δm = (2 × 1.007276) + (2 × 1.008665) − 4.001506
Δm = 4.031882 − 4.001506 = 0.030376 uΔm ≈ 0.0304 uPart (b) — binding energy per nucleon
total BE = 0.030376 × 931.5
total BE ≈ 28.3 MeV
per nucleon = 28.3 ÷ 4
≈ 7.1 MeV per nucleonHelium-4 is unusually stable, which is why its binding energy per nucleon is so high.
💡 Top tips
Add nucleon masses first, subtract the nucleus last. The nucleus is always lighter, so Δm comes out positive
Use 1 u = 931.5 MeV c−2 to skip the kilogram conversion — multiply Δm (in u) straight by 931.5 for MeV
Binding energy per nucleon = total binding energy ÷ A — this is what lets you compare stability between nuclei
Say it precisely: binding energy is the energy needed to separate the nucleus, not energy stored inside it
⚠ Common mistakes
Describing binding energy as energy stored in the nucleus — it’s the energy needed to pull it apart
Using A instead of (A − Z) for the number of neutrons — remember to subtract the protons
Calling the mass loss in radioactive decay a “mass defect” — that term is only for fully separating a nucleus into nucleons
Forgetting to divide by A when the question asks for binding energy per nucleon
Up next: Binding Energy per Nucleon Curve. Now you can calculate binding energy per nucleon — next we plot it against nucleon number to see why iron is the most stable element, and where fusion and fission each release energy.
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