Mass-Energy Equivalence Calculator — E = mc² in Every Direction
Mass and energy are not two things that convert into one another. They are one thing, and c² is the exchange rate between the units we happened to invent for measuring it. This mass-energy equivalence calculator works that exchange rate in both directions and in every context it shows up: the rest energy of a mass, the mass defect and Q-value of a nuclear reaction, the relativistic energyof a moving particle, particle–antiparticle annihilation, partial conversion at a real process efficiency, and the energy–momentum invariant.
Why the number is so large
The whole subject turns on one multiplication. The speed of light is 299 792 458 m/sexactly — exactly, because since 1983 the metre has been defined from it — so c² = 8.987551787 × 10¹⁶ m²/s². A single kilogram is therefore worth 8.98755 × 10¹⁶ J, which is 21.5 megatons of TNT or 2.4965 × 10¹⁰ kWh. Nothing else in physics has a conversion factor that violent, and it is why the mass change in a chemical reaction — real, but around one part in 10¹⁰ — has never been weighed.
Mass defect, binding energy and the 931.494 shortcut
A nucleus weighs less than the sum of its nucleons. That missing mass is the binding energy, and it is what a nuclear reaction trades in. Because nuclear masses are tabulated in atomic mass units to eight or nine decimals, and a mass defect is a difference between two nearly identical numbers, the sensible route is to subtract in u first and convert once at the end: 1 u ≡ 931.494 MeV. Converting each mass to kilograms first throws away most of your significant figures to floating-point cancellation before you have even subtracted.
Deuterium–tritium fusion is the standard worked example. 2.014102 + 3.016049 = 5.030151 u of reactants against 4.002603 + 1.008665 = 5.011267 u of products leaves Δm = 0.018884 u, so Q = 17.590 MeVper reaction. Scaled by Avogadro’s number that is 1.697 × 10⁹ kJ/mol, and per kilogram of fuel 3.374 × 10¹⁴ J/kg— all from converting just 0.375 % of the reactant mass. A negative defect is not an error; it means the products are heavier and the reaction is endothermic, so the calculator labels it as energy absorbed rather than rejecting it.
c— which is off by a factor of 300 million — and mixing atomic mass units with grams. A periodic table’s “grams per mole” figure is already the mass in u; it needs no conversion at all. This tool normalises every input to SI before any arithmetic runs and converts back only at the display boundary, which removes that entire class of error.Once the particle moves
For a moving particle the rest energy is only part of the story. The Lorentz factor γ = 1/√(1 − β²) scales it up to a total energy E = γm₀c², leaving KE = (γ − 1)m₀c² as the kinetic part and p = γm₀v as the momentum. An electron at 0.99c has γ = 7.0888, E = 3.6224 MeV, KE = 3.1114 MeV and p = 3.58616 MeV/c— a figure some sources misprint as 3.5878, and one the invariant pc = √(E² − (m₀c²)²) settles independently.
At everyday speeds the direct subtraction in γ − 1 is numerically hopeless: for an airliner γ differs from 1 in the thirteenth decimal place, and double-precision arithmetic has already run out of digits. The calculation switches to the series β²/2 + 3β⁴/8 + … below β = 10⁻³, which reproduces ½mv² to full precision and shows why the classical formula was never wrong, merely truncated.
The invariant, and the massless case
E² = (pc)² + (m₀c²)² holds for every particle in every frame. Leave any one of the three blank and it is solved for: a proton with 500 MeV/c of momentum has E = √(500² + 938.272²) = 1063.181 MeV and so KE = 124.909 MeV. Set m₀ = 0 and the relation collapses to E = pc, the photon case — momentum without rest mass, always travelling at exactly c.
γm₀a “mass” suggests an object genuinely gets heavier, and that Newton’s second law still works with it substituted in. Neither is true — the notion breaks down entirely for a force applied along the direction of motion. Modern practice is that mass means rest mass, one invariant number, with all the velocity dependence living in the energy and momentum instead. The figure is shown here only because older textbooks still quote it.Annihilation, efficiency and honest comparisons
An electron and a positron at rest convert entirely: 1.022 MeV total, emitted as two back-to-back 511 keV gamma photons at λ = 2.4263 pm. That is the coincidence signal every PET scanner is built to detect, and read backwards it is the pair-production threshold. Real energy sources are far less thorough: fission converts about 0.09 % of its fuel mass, giving 8.09 × 10¹³ J— 22.5 GWh — from a kilogram of uranium, solar fusion about 0.7 %, and chemical burning around 10⁻¹⁰.
The everyday-equivalence panel exists to make those magnitudes intuitive, and every row carries the figure it divides by. Some are definitions and exact — a tonne of TNT is 4.184 × 10⁹ J, a kilowatt-hour is 3.6 × 10⁶ J. Others are stated assumptions: 34.2 MJ per litre of petrol, a 15 kt Hiroshima yield, and 2 700 kWh of electricity per household per year. That last one is worth reading carefully, because household energy claims are routinely quoted without a basis: 2 700 kWh is typical UK domestic electricity, whereas a combined gas-and-electricity household total is nearer 11 000 kWh, and which you pick moves the answer by a factor of four.