E = mc² Calculator

E = m × c² c = 299792458 m/s (exact, by definition) 1 kilowatt hour = 3.6 × 10⁶ J 1 kiloton of TNT = 4.184 × 10¹² J

m

Mass in grams — the field also takes kilograms, tonnes and pounds, and grams is the unit it starts in because one gram is the case everyone asks about

c

The speed of light in a vacuum, exactly 299792458 m/s. Since 1983 this is a defined value rather than a measurement: the metre is defined as the distance light travels in 1/299792458 of a second, so c has no uncertainty at all and neither does anything computed from it

E

The energy released, reported in terajoules, joules, kilowatt hours and kilotons of TNT. The joules are the raw answer; the other three are the same quantity in units people can picture — a power bill, a power station, a news report

Use this page to get a feel for the scale of mass energy: how much a gram would give if it were annihilated, what fraction of that a nuclear reactor or a chemical reaction actually extracts, why the mass defect in fission is small but not negligible, or how much fuel a given energy demand really implies. It is also the page for settling arguments — the equation is quoted constantly and almost always with the units wrong or with the impression that a kilogram of anything releases 9 × 10¹⁶ joules in a power station. This calculator gives the theoretical ceiling; the interesting question in practice is always what fraction of it a real process reaches, and the answer is anything from a ten-billionth for burning petrol to about a thousandth for fission. Two habits make the output useful. First, read the kilotons row when you are thinking about explosions and the kilowatt hours row when you are thinking about electricity, because the joules are too large to have any feel for. Second, remember that the mass here is the mass that disappears — for a real reaction, that is the difference between the mass of the fuel and the mass of everything left over, not the mass of the fuel.

1. One gram of matter

   1. Mass: 1 g = 0.001 kg (the calculation is done in SI)
   2. Energy: 0.001 × 299792458² = 8.9875517873681776 × 10¹³ J
   3. In the headline unit: 89875517873682 J = 89.876 TJ
   4. As electricity: 89875517873682 ÷ 3.6 × 10⁶ = 24965421.6 kWh — 25 gigawatt hours
   5. As TNT: 89875517873682 ÷ 4.184 × 10¹² = 21.481 kilotons

   This is the number the equation is quoted for, and the TNT row is the one that makes it land: 21.5 kilotons against the 15 kilotons of the Hiroshima bomb, so a gram of matter is a bigger explosion than the first one used on a city. The electricity row says the same thing in a different currency — 25 gigawatt hours is roughly what a large power station produces in a day, from a gram. Read the joules as well: 89875517873682 is fourteen digits, and the page shows all of them on purpose, because writing 9.0 × 10¹³ would quietly hide the thing the equation is famous for.

2. One kilogram, the figure usually quoted for mass energy

   1. The only thing that changed is the mass: 1000 g instead of 1 g, a factor of a thousand
   2. Energy: 89875517873682 × 1000 = 8.987551787368176 × 10¹⁶ J
   3. In the headline unit: 89875.518 TJ
   4. As electricity: 24965421631.6 kWh — 25 terawatt hours
   5. As TNT: 21480.764 kilotons, which is 21.5 megatons

   21.5 megatons is the figure that appears in every textbook and every bad argument about nuclear weapons, and it is worth being precise about what it means. It is not what a kilogram of uranium does in a reactor — that releases about 8 × 10¹³ J, a thousandth of this — and it is certainly not what a kilogram of petrol does, which is about 4.6 × 10⁷ J, a factor of two billion short. It is what you would get if the entire kilogram were converted, which in practice only happens when matter meets antimatter. Compare this row with the one above and you have the whole equation: the mass went up by a thousand and every energy went up by a thousand, because c² is just a constant.

3. One pound, for readers working in imperial units

   1. 1 lb = 453.59237 g (the field takes grams, and the pound is defined exactly in terms of the kilogram)
   2. Energy: 0.45359237 × 299792458² = 4.076684915730068 × 10¹⁶ J
   3. In the headline unit: 40766.849 TJ
   4. As electricity: 11324124765.9 kWh — 11.3 terawatt hours
   5. As TNT: 9743.511 kilotons — 9.7 megatons

   A pound of anything, fully converted, is just under ten megatons — a bit less than half the kilogram row, which is what you would expect from the pound being a bit less than half a kilogram. It is included because the pound is defined exactly in terms of the kilogram, so this is an exact conversion rather than a rounded one, and because it is the fastest way to sanity-check a figure someone has quoted in imperial units. Nothing here depends on which substance the pound is made of; the equation has no chemistry in it at all.

This page computes complete conversion of mass into energy, and nothing in ordinary experience does that. Burning petrol converts about one part in ten billion of the fuel's mass; nuclear fission converts roughly one part in a thousand; nuclear fusion in the Sun manages about seven parts in a thousand. All of those are real processes and all of them fall far short of the number this page prints, which is a ceiling rather than a prediction. The TNT equivalence uses a defined figure, 4.184 × 10⁹ joules per tonne, agreed in 1948 when the thermochemical calorie was fixed — but real explosives release somewhat less than that per tonne, because not all of the material reacts and the blast does work on the air rather than on the target. The kilowatt hour row is a unit conversion and nothing more; it does not mean a power station can extract this much from a gram, only that this many joules equals that many kilowatt hours. There is a display limit worth knowing: below about 5.5 micrograms the terajoule row rounds to 0.000, so for tiny masses read the joules instead — the energy is still there, it has simply fallen below three decimal places in a unit chosen for grams and kilograms. Nothing here is relativistic in the sense of being frame-dependent; c² is a constant and the equation holds in every frame, but the kinetic energy of a moving object is not this formula, and at speeds near c the total energy of a body is larger than mc² by the Lorentz factor.

What does E = mc² actually calculate?

The energy equivalent of a mass: multiply the mass in kilograms by the square of the speed of light, 299792458 m/s. One gram gives 0.001 × 299792458² = 8.99 × 10¹³ joules, which is 89.9 terajoules or 21.5 kilotons of TNT. It is the total energy locked in the mass, not the energy any particular process will release — see the note below on what fraction real reactions extract.

Why is the speed of light squared?

Because that is what the derivation gives, and the reason it is so large is that c is large and the squaring makes it larger still: 299792458² is about 9 × 10¹⁶. In the original 1905 paper the equation appears as m = E/c², which reads more naturally — a body's inertia is its energy divided by c², and since c² is enormous, a small mass corresponds to a vast energy.

How much of this energy does a nuclear reactor actually get?

About a thousandth of it. Fission converts roughly 0.1 percent of the fuel's mass into energy, so a kilogram of uranium releases around 8 × 10¹³ joules rather than the 9 × 10¹⁶ this page prints — which is still millions of times more than burning the same mass of coal. Fusion is better, around 0.7 percent, which is why the Sun's fuel lasts so long. Chemical reactions are far behind both: burning petrol converts about one part in ten billion of its mass.

Why kilotons of TNT rather than joules?

Because nobody has a feel for 9 × 10¹³ joules. A kiloton of TNT is defined as 4.184 × 10¹² joules, so the conversion is exact, and it happens to be the unit news reports use for explosions: a gram of matter comes out at 21.5 kilotons, against roughly 15 kilotons for the Hiroshima bomb. The joule row is still shown, at full length, because the size of that number is the point of the equation.

Is the mass in the equation the mass of the fuel?

For a real reaction, no — it is the mass that disappears, which is the mass of the fuel minus the mass of everything left over. In fission that difference is about 0.1 percent of the fuel, and it is that fraction which becomes energy. This page takes the mass you type as being converted completely, which is the ceiling rather than what any reactor achieves; if you want the energy from a real fuel, take the mass defect rather than the fuel mass.

Why does the terajoule row show 0.000 for very small masses?

Because the terajoule is a very large unit and the row is shown to three decimal places. Below about 5.5 micrograms the answer is under 0.0005 TJ and rounds to nothing. The energy has not vanished — read the joule row instead, which is where a microgram still shows up clearly at 89.9 million joules, roughly the energy in two and a half litres of petrol.

• Mass–energy equivalence — the derivation, the mass defect in fission and fusion, and the fractions of rest mass that real processes actually convert — Wikipedia
• TNT equivalent — the 4.184 × 10⁹ J per tonne convention, its origin in the thermochemical calorie, and why real explosives fall short of it — Wikipedia
• NIST Guide to the SI — the metre as defined by the speed of light, and the exact kilogram, pound and joule factors used above — National Institute of Standards and Technology