Mass–Energy Equivalence E=mc²

The equation and what it actually says

Einstein's 1905 result, E = mc², states that mass and energy are the same physical quantity measured in different units — the conversion factor is the speed of light squared, an enormous number, which is why converting even a small mass releases a huge amount of energy. It is not a statement that mass turns into a mysterious different substance called energy; it says the rest energy locked in any mass m at rest is E = mc², and any process that changes a system's total mass changes its total energy by exactly that factor.

E = m c^2
c = 299,792,458 m/s  →  c^2 ~= 9.0 x 10^16 m^2/s^2

1 gram converted completely  →  ~9.0 x 10^13 joules
                                  (~21.5 kilotons of TNT equivalent)

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Binding energy and the mass defect

Nuclei are lighter than the sum of their free protons and neutrons. That missing mass, the mass defect Δm, is exactly the energy that was released when the nucleons bound together, via E_binding = Δm c². Plotting binding energy per nucleon against mass number produces the famous curve that peaks around iron-56 and nickel-62 — the most tightly bound nuclei in nature — and falls off on both sides. That single curve explains why both fission and fusion release energy: either process moves nuclei toward the peak, converting a small amount of mass into a large amount of energy in the process.

Fission: splitting a heavy nucleus

Uranium-235 sits well below the binding-energy peak. When it absorbs a slow (thermal) neutron it becomes the unstable U-236, which splits into two smaller, more tightly bound fragments — typically nuclei with mass numbers roughly in the 90-140 range — plus two or three free neutrons and about 200 MeV of energy per fission, mostly as kinetic energy of the fragments. Because each fission releases more neutrons than it consumed, and each of those neutrons can trigger another fission, the process can sustain a chain reaction: one fission per generation on average keeps the reaction critical, more than one makes it grow exponentially.

Fusion: combining light nuclei

At the other end of the curve, light nuclei gain binding energy per nucleon by fusing. The easiest reaction to trigger, and the one used in most fusion research and weapons, is deuterium-tritium fusion:

D + T  →  He-4 (3.5 MeV)  +  n (14.1 MeV)     total ~17.6 MeV per reaction

requires ~100 million K to overcome the Coulomb (electrostatic) repulsion
between the two positively charged nuclei before the strong force can bind them

Per unit mass of fuel, fusion releases several times more energy than fission, because hydrogen and helium sit much further from the binding-energy peak than uranium does. The obstacle is the Coulomb barrier: two nuclei must get close enough, against their mutual electrostatic repulsion, for the short-range strong nuclear force to take over and bind them, which is why fusion needs extreme temperature and pressure (as in the Sun's core) or powerful confinement (as in a tokamak or an inertial-confinement laser array) to happen at any useful rate.

Chain reactions and criticality

Whether a mass of fissile material sustains, dies out, or runs away depends on the neutron multiplication factor k: the average number of neutrons from one fission that go on to cause another. k < 1 is subcritical and the reaction dies out; k = 1 is critical and steady; k > 1 is supercritical and the reaction rate grows, doubling on a timescale set by how far above 1 k sits and by the neutron lifetime. Reactors are engineered to hold k at almost exactly 1 during normal operation; weapons are engineered to briefly push k well above 1 before the assembly disassembles itself.

Frequently asked questions

Does E=mc² mean mass literally disappears?

Not literally vanish — the total mass-energy of an isolated system is always conserved. In fission or fusion, a small amount of rest mass converts into kinetic energy and radiation, so if you carefully weighed every fragment and every emitted particle and photon you would find slightly less total rest mass than you started with, with the difference accounted for exactly by E=Δmc².

Why does fusion release more energy per kilogram than fission?

Because hydrogen and helium sit much further from the peak of the binding-energy-per-nucleon curve (around iron-56) than uranium does, so fusing light nuclei together climbs a steeper part of that curve than splitting a heavy nucleus does, converting proportionally more mass to energy per kilogram of fuel.

What is the difference between critical and supercritical?

Critical (k=1) means the fission chain reaction sustains itself at a constant rate, which is how a power reactor runs at steady output. Supercritical (k>1) means each generation produces more fissions than the last, so the power grows exponentially — reactors briefly go supercritical to raise power, then return to k=1 to hold steady.

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E=mc²: Mass Defect, Binding Energy, Fission and Fusion