Here’s one of the strangest facts in physics: a nucleus weighs less than the protons and neutrons that make it up. Take a nucleus apart and the separate pieces are heavier than the whole. That missing mass, the mass defect, hasn’t vanished — it was converted into the energy that holds the nucleus together, through Einstein’s famous E = mc2. Understanding this is the key to where nuclear energy actually comes from.
📚 What you need to know
Mass defectΔm is the difference between the measured mass of a nucleus and the total mass of its separate nucleons
A nucleus is always lighter than its separated parts
Δm = Zmp + (A−Z)mn − mtotal
Binding energy is the energy needed to break a nucleus into its separate protons and neutrons
Forming a nucleus from separate nucleons releases this energy
Mass–energy equivalence:ΔE = Δmc2
The atomic mass unit u: 1 u = 1.661 × 10−27 kg = 931.5 MeV c−2
The mass defect
Weigh a carbon-12 nucleus, then weigh 6 separate protons plus 6 separate neutrons. The separate parts come out heavier. The missing mass is the mass defect, defined as the difference between the mass of the whole nucleus and the sum of its individual nucleons.
A bound carbon-12 nucleus is lighter than its 6 protons + 6 neutrons taken separately. The missing mass is the mass defect.
Mass defectΔm = Zmp + (A − Z)mn − mtotalZ protons of mass mp • (A−Z) neutrons of mass mn • mtotal = measured nucleus mass
Mass–energy equivalence
Where did the missing mass go? Einstein’s theory of relativity says mass and energy are two forms of the same thing, linked by:
Mass–energy equivalenceΔE = Δmc2E = energy (J) • m = mass (kg) • c = speed of light (3.0 × 108 m s−1)
So the lost mass became energy — specifically the energy released when the nucleons snapped together. Because c2 is enormous, even a tiny mass defect corresponds to a huge amount of energy.
Binding energy
Binding energy is the energy needed to pull a nucleus completely apart into its separate protons and neutrons. It’s the exact same amount that was released when the nucleus formed. Mass defect and binding energy are two sides of the same coin, connected by ΔE = Δmc2.
Binding energythe energy required to break a nucleus into its separate protons and neutrons
Watch your wording here — examiners are strict about it. Binding energy is not “energy stored in the nucleus”. It’s the energy you’d have to put in to rip the nucleus apart. And “mass defect” only refers to fully separating the nucleons — don’t use it to describe the small mass lost in radioactive decay. Precise language earns the marks.
Separate nucleons
combine (release energy)
Bound nucleus
split (needs energy)
Separate nucleons
The atomic mass unit
Nuclear masses are tiny, so we use the atomic mass unit (u), defined as exactly one-twelfth the mass of a carbon-12 atom. It’s roughly the mass of one proton or neutron. A handy conversion links mass in u directly to energy in MeV:
The u ↔ MeV conversion1 u = 1.661 × 10−27 kg = 931.5 MeV c−2
This shortcut means a mass defect of 1 u is equivalent to 931.5 MeV of binding energy — no need to run through Δmc2 in full each time if the mass is already in u.
WE 1
The binding energy per nucleon of oxygen-16 (16O) is 7.98 MeV. Estimate the total energy, in MeV, needed to completely separate the nucleons of this atom.
Step 1 — count the nucleons
Oxygen-16 has 8 protons + 8 neutrons = 16 nucleonsStep 2 — total binding energytotal BE = 7.98 × 16 = 127.7 MeVtotal binding energy = 127.7 MeVTotal binding energy = (binding energy per nucleon) × (number of nucleons). This total is exactly the energy needed to pull all 16 nucleons apart.
WE 2
Calculate the binding energy per nucleon of potassium-40 (4019K), in MeV. Nuclear mass = 39.953548 u, mp = 1.007276 u, mn = 1.008665 u. (1 u = 1.661 × 10−27 kg, c = 3.0 × 108 m s−1, 1 MeV = 1.6 × 10−13 J)
Step 1 — nucleons: Z = 19, N = 40 − 19 = 21Step 2 — mass defect in uΔm = (19 × 1.007276) + (21 × 1.008665) − 39.953548Δm = 0.36666 uStep 3 — convert to kg, then energyΔm = 0.36666 × 1.661×10⁻²⁷ = 6.09×10⁻²⁸ kgE = Δmc² = 6.09×10⁻²⁸ × (3.0×10⁸)² = 5.5×10⁻¹¹ JStep 4 — divide by nucleons, convert to MeVper nucleon = (5.5×10⁻¹¹ ÷ 40) ÷ 1.6×10⁻¹³≈ 8.6 MeV per nucleonBuild up in order: mass defect → total binding energy (via Δmc²) → divide by nucleons → convert to MeV. Keep the mass in u until you’re ready to convert to kg.
⚛ Working a mass defect / binding energy question
Count nucleons:Z protons, (A−Z) neutrons.
Mass defect: total nucleon mass − measured nucleus mass.
Convert u → kg (× 1.661×10−27) if needed.
Binding energy:ΔE = Δmc2 (or use 1 u = 931.5 MeV).
Per nucleon? Divide the total binding energy by A.
💡 Top tips
A nucleus is always lighter than its separated nucleons.
Binding energy = energy to break apart, not “energy stored”.
Use 1 u = 931.5 MeV as a shortcut when mass is in u.
Total binding energy = per-nucleon value × number of nucleons.
Keep mass in u until the step where you convert to kg.
⚠ Common mistakes
Describing binding energy as “energy stored in the nucleus” — it’s energy needed to separate it
Using “mass defect” for the small mass lost in decay — it means fully separated nucleons
Forgetting to square the speed of light in Δmc2
Using nucleon number where you need neutron number (A−Z)
Forgetting to divide by A when asked for binding energy per nucleon
Quick recap: A nucleus is lighter than its separate nucleons; that missing mass defect became the binding energy holding it together, via ΔE = Δmc2. Mass defect = total nucleon mass − nucleus mass. Binding energy is the energy needed to break the nucleus apart. Use 1 u = 931.5 MeV to convert quickly, and divide the total by A for binding energy per nucleon.
Total binding energy tells you how tightly a whole nucleus is held, but to compare stability across different elements we need binding energy per nucleon. Plotting that against nucleon number gives one of the most important graphs in physics — the curve that explains both fission and fusion. Next page: Binding Energy per Nucleon.
Mass defect and binding energy still fuzzy?
Book a free meeting and we’ll drill Δm, ΔE = Δmc², the u-to-MeV shortcut, and the exact wording examiners want.
