The most famous equation in physics
In 1905, Albert Einstein, then a patent clerk in Bern, published a short paper showing that the mass of a body depends on its energy content. The result is usually written
E = mc²
- E is the energy, in joules.
- m is the mass, in kilograms.
- c is the speed of light, exactly 299,792,458 meters per second.
Because c² is about 9 × 10¹⁶, a small mass corresponds to a colossal energy.
Worked example
How much energy is in 1 gram of matter, about the mass of a paper clip?
E = 0.001 kg × (299,792,458 m/s)² = 8.99 × 10¹³ J
That is equal to:
- 24,965,422 kWh, about what 2,300 average US homes use in a year.
- 21,481 tons of TNT, close to the yield of the bomb dropped on Nagasaki in 1945.
- 25 hours of output from a 1,000 MW nuclear reactor.
- The energy in about 740,800 gallons of gasoline.
Where mass turns into energy
- Nuclear fission: splitting a uranium-235 nucleus releases about 200 MeV, and the fragments weigh about 0.1% less than the original nucleus. A reactor running at full power for a year converts only about a kilogram of mass into energy.
- Nuclear fusion: the Sun converts about 4 million tons of mass into energy every second, by fusing hydrogen into helium with a mass loss of 0.7%.
- Chemical reactions also change mass, but by about a billionth of the total, far too small to weigh.
- Annihilation: when matter meets antimatter, all of the mass becomes energy. Hospital PET scanners detect the gamma rays from positrons annihilating with electrons.
The other direction
Energy also has mass. A charged battery weighs a tiny bit more than a flat one, and most of the mass of a proton comes not from its quarks but from the energy binding them. Choose “Energy to mass” to find the mass equivalent of an energy: a full year of US electricity generation, about 4,200 TWh, corresponds to about 170 kg.
Limits of the comparisons
The household comparison uses about 10,800 kWh per year, the US average, and the gasoline comparison 33.7 kWh per gallon, the EPA’s value. Real power plants convert only a third of their heat into electricity, so producing that much electricity would take three times more fuel energy.