Nuclear Fission and Fusion: E = mc², Binding Energy & How They Work

The Sun has been shining for 4.6 billion years and will continue for another 5 billion. The energy source is not chemical — no amount of burning hydrogen could sustain that output for geological time. The answer is nuclear fusion: hydrogen nuclei merging into helium, converting a tiny fraction of their mass directly into energy via E = mc². The c² factor is enormous — roughly 9 × 10¹⁶ m²/s² — which is why even tiny mass differences translate into extraordinary energy release.

Fission works the other way: heavy nuclei like uranium-235 split into lighter fragments, also releasing energy. Both processes tap the same underlying physics — the binding energy curve — but from opposite ends. Understanding that curve tells you everything about why fission releases energy for heavy elements, why fusion releases energy for light ones, and why iron is the dead end in the middle.

The Mass Defect and Binding Energy

The mass of a nucleus is always less than the sum of the masses of its constituent protons and neutrons (nucleons). This difference is the mass defect Δm:

where Z is the proton number, A is the mass number, m_p and m_n are proton and neutron masses, and M_nucleus is the actual nuclear mass. This missing mass has been converted to binding energy by E = mc²:

Binding energy is the energy needed to completely separate all nucleons — the energy holding the nucleus together. The more binding energy per nucleon, the more stable the nucleus.

The key graph in nuclear physics is binding energy per nucleon vs mass number A:

• Peaks around iron-56 (Fe-56) and nickel-62 — the most stable nuclei.

• Light nuclei (hydrogen, helium) have low binding energy per nucleon — fusion to heavier nuclei releases energy.

• Heavy nuclei (uranium, thorium) have slightly lower binding energy per nucleon than iron — fission to medium-weight products releases energy.

• Energy can only be extracted by moving toward the iron peak — either by fusing light nuclei or fissioning heavy ones.

Nuclear Fission: Splitting Heavy Nuclei

The most important fission reaction in nuclear power uses uranium-235:

A slow (thermal) neutron is absorbed by U-235, making an unstable U-236 nucleus that immediately splits into two fission fragments (krypton-92 and barium-141 in this case — but many fragment pairs are possible) plus 2–3 fast neutrons and approximately 200 MeV of energy.

The energy release per fission: ~200 MeV = 3.2 × 10⁻¹¹ J. That seems small, but per kilogram of uranium-235, the energy release is ~8.2 × 10¹³ J — about 2 million times more energy per kilogram than burning coal (3.3 × 10⁷ J/kg). This extraordinary energy density is why nuclear plants can generate large amounts of power from small amounts of fuel.

Chain Reactions and Critical Mass

Each U-235 fission releases 2–3 neutrons. If each of those neutrons causes another fission, the reaction is self-sustaining — a chain reaction. The multiplication factor k determines the behaviour:

• k < 1: sub-critical — chain reaction dies out.

• k = 1: critical — steady, controlled chain reaction (nuclear reactor).

• k > 1: supercritical — exponentially growing reaction (nuclear weapon or reactor runaway).

The critical mass is the minimum mass of fissile material needed to sustain a chain reaction (k ≥ 1). For pure U-235: ~52 kg as a bare sphere. For Pu-239: ~10 kg. Neutron reflectors (beryllium, graphite) and geometric compression can reduce critical mass significantly — the "gun-type" and implosion designs of nuclear weapons use these principles.

In nuclear reactors, control rods (boron or hafnium) absorb neutrons to keep k = 1 exactly — maintaining a controlled, steady chain reaction. Coolant (water, CO₂, or liquid sodium) removes the heat, which drives steam turbines to generate electricity.

Nuclear Fusion: Powering the Stars

The Sun converts approximately 600 million tonnes of hydrogen to helium every second via the proton-proton chain:

Four hydrogen nuclei (protons) fuse to produce one helium-4 nucleus, two positrons, two neutrinos, and 26.7 MeV of energy. The mass of four protons exceeds the mass of one helium nucleus by 0.7% — this 0.7% converts to energy via E = mc². Over the Sun's lifetime, it converts ~4 million tonnes of mass to energy every second.

The most promising fusion reaction for future power generation uses deuterium and tritium (isotopes of hydrogen):

The products are helium-4 (3.5 MeV) and a fast neutron (14.1 MeV). Deuterium is abundant in seawater; tritium can be bred from lithium. The fuel for fusion power is essentially inexhaustible.

Why Fusion Is Hard on Earth

Fusion requires nuclei to approach within ~10⁻¹⁵ m — nuclear distance — despite the enormous electrostatic repulsion between positively charged protons. Classically, this requires temperatures of ~10¹⁰ K. In practice, quantum tunnelling reduces this to ~10⁸ K (100 million degrees) — still far above any material melting point.

Controlled fusion requires confining plasma at 100–150 million °C — ten times hotter than the Sun's core (the Sun relies on its immense gravitational pressure to compensate for lower temperature). Two main confinement approaches:

Magnetic confinement (tokamak): powerful magnetic fields in a toroidal (doughnut-shaped) chamber confine the plasma. The ITER project in France — the world's largest tokamak under construction — aims to produce 500 MW of fusion power from 50 MW of input heating — Q = 10. JET (UK) held the fusion energy record of 59 MJ (2022).

Inertial confinement: intense laser beams simultaneously compress and heat a tiny pellet of D-T fuel, achieving fusion conditions for nanoseconds. The National Ignition Facility (NIF) at Livermore, USA achieved fusion ignition in December 2022 — producing more fusion energy (3.15 MJ) than laser energy delivered to the target (2.05 MJ) — a historic first.

Mass-Energy Equivalence and Binding Energy

Both fission and fusion release energy through E = mc². The mass of a nucleus is always less than the sum of its constituent protons and neutrons — the "missing" mass is called the mass defect Δm, and it represents the binding energy released when the nucleus formed: E_binding = Δmc².

The binding energy per nucleon (BE/A) peaks at iron-56 (~8.8 MeV/nucleon). Elements lighter than iron release energy by fusion (two small nuclei combine → one larger, closer to iron on the curve). Elements heavier than iron release energy by fission (large nucleus splits → two medium nuclei, closer to iron on the curve). Iron is the most stable nucleus — neither fission nor fusion releases energy from iron-56.

Nuclear Fission

A neutron strikes uranium-235: ⁰₁n + ²³⁵₉₂U → ²³⁶₉₂U* → ¹⁴¹₅₆Ba + ⁹²₃₆Kr + 3⁰₁n. The three neutrons released can trigger three more fissions — a chain reaction. Energy released per fission ≈ 200 MeV = 3.2 × 10⁻¹¹ J. One kilogram of U-235 fully fissioned releases ~82 × 10¹² J = 82 TJ — equivalent to ~20 kilotonnes of TNT. A nuclear power station moderates the chain reaction to controlled rate; a bomb allows supercritical exponential growth.

Nuclear Fusion

The most accessible fusion reaction (used in tokamaks and planned fusion power plants): ²₁H + ³₁H → ⁴₂He + ⁰₁n. Energy released ≈ 17.6 MeV per reaction. The proton-proton chain in the Sun: 4¹₁H → ⁴₂He + 2e⁺ + 2ν_e, releasing 26.7 MeV. The Sun converts 4 million tonnes of mass to energy every second via fusion — yielding 3.85 × 10²⁶ W. Fusion requires temperatures of ~100 million kelvin to overcome Coulomb repulsion between nuclei — achievable in tokamaks using magnetic confinement.

Worked Example: Energy from Fission

Calculate the energy released when 1 g of U-235 undergoes fission (200 MeV/reaction). Molar mass of U-235 = 235 g/mol.

Number of atoms in 1 g: N = (1/235) × 6.022 × 10²³ = 2.56 × 10²¹

Equivalent to burning ~2,400 tonnes of coal. From 1 gram of uranium.

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