Nuclear Binding Energy, Fission and Fusion
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Direct answer
The binding energy per nucleon curve peaks near iron-56 at about 8.8 MeV per nucleon, and the entire logic of nuclear energy hangs on that hump: light nuclei fuse and heavy nuclei split, both climbing toward the valley of stability and releasing the difference as energy. Binding energy is BE = [Z m(p) + (A − Z) m(n) − M(nucleus)] c^2, converted through 1 u c^2 = 931.5 MeV. Fission of uranium-235 by a slow neutron yields two mid-mass fragments plus two to three neutrons and about 200 MeV per event — the chain-reaction economy, moderated and controlled in reactors. Fusion of hydrogen into helium packs about 26.7 MeV per helium-4 but demands 10^7-kelvin temperatures to defeat Coulomb repulsion.
What you must remember
- Mass defect and units: BE = (Z m(p) + N m(n) − M) c^2; 1 u = 931.5 MeV/c^2 — every nuclear energy calculation begins with this conversion.
- BE per nucleon curve: rises steeply from deuterium (about 1.1 MeV per nucleon), peaks near 8.8 MeV around iron-56/nickel-62, falls slowly to about 7.6 MeV at uranium — energy release means moving toward the peak from either side.
- Fission arithmetic: U-235 + n gives fragments near A = 95 and 140, two to three neutrons, roughly 200 MeV per fission; 1 kg of U-235 releases energy of order 10^14 J.
- Chain-reaction engineering: moderators (water, heavy water, graphite) slow neutrons to thermal speeds; control rods (cadmium, boron) absorb them; critical mass is the smallest mass sustaining the chain.
- Fusion facts: the proton-proton chain converts four protons into one helium-4 plus two positrons and two neutrinos, releasing 26.7 MeV; it needs about 10^7 K, hence the name thermonuclear.
- Indian engineering anchor: India's PHWR fleet uses heavy water as moderator, allowing natural (unenriched) uranium fuel — a national-programme detail quoted in comprehension passages.
- Stability extras: even-even nuclei dominate the stable list; magic numbers 2, 8, 20, 28, 50, 82, 126 mark extra-stable nucleon counts.
- Pattern note: Main computes mass-defect energies; Advanced builds Q-value chains and compares fission versus fusion yield per kilogram.
Working a mass table honestly
The nucleus of helium-4: 2 protons (1.007276 u each), 2 neutrons (1.008665 u each), measured mass 4.001506 u. Sum of parts: 2 × 1.007276 + 2 × 1.008665 = 4.031882 u; defect = 0.030376 u; BE = 0.030376 × 931.5 = 28.3 MeV, or 7.07 MeV per nucleon. That single calculation, run honestly, explains why fusion of hydrogen into helium pays and why alpha particles are exceptionally tightly bound — the exact arithmetic of every Main numerical here, with only the isotope changed.
Fission pays by the same ledger from the other end. Uranium's 7.6 MeV per nucleon against the fragments' roughly 8.5 means about 0.9 MeV per nucleon released across 235 nucleons — about 200 MeV per event. Per kilogram, fusion of hydrogen beats fission of uranium several-fold, though sustaining the 10^7 K plasma is the engineering frontier.
Where students slip
Using atomic masses where nuclear masses are needed is the standard slip: atomic masses include electrons, which cancel in balanced nuclear equations but wreck proton-mass bookkeeping when counted twice — electrons balance automatically across a reaction, so atomic masses are safe for Q-values, dangerous for raw BE sums unless handled consistently. Second, binding energy is not energy the nucleus possesses but energy needed to dismantle it; a larger BE means a lighter, more tightly bound nucleus, and mass decreases as binding increases. Third, the fission neutron economy: moderators slow neutrons without absorbing them (heavy water excels because deuterium barely captures), control rods deliberately absorb — confusing the two reverses reactor physics. Fourth, Q-value sign discipline: energy released means total final mass less than total initial, and computing Q = (initial − final) mass × 931.5 in MeV keeps the sign honest.
Frequently asked questions
How is binding energy of a nucleus computed?
From the mass defect: BE = [Z m(p) + (A − Z) m(n) − M] c^2, converted at 1 u c^2 = 931.5 MeV; it is the energy needed to pull the nucleus fully apart.
Why does the BE-per-nucleon curve explain both fission and fusion?
Energy releases whenever nuclei move toward the peak near iron: heavy nuclei split and light nuclei merge, both increasing average binding per nucleon and emitting the difference.
How much energy does one fission event release?
About 200 MeV per uranium-235 nucleus split by a thermal neutron, carried as fragment kinetic energy, neutron kinetic energy and gamma radiation.
Why can't iron be used as a fusion or fission fuel?
Iron sits at the top of the binding-energy curve; splitting or merging it moves nuclei away from maximum binding, absorbing energy rather than releasing it.
Why does fusion require temperatures of the order of 10^7 K?
Bare nuclei must collide with enough kinetic energy to overcome their Coulomb repulsion, and only at such plasma temperatures does the Maxwellian tail penetrate the barrier often enough to sustain burning.