Chapter 04 · The Science

E = mc² — Mass & Energy

A short paper that Einstein added almost as an afterthought in September 1905 — a body that gives off energy L loses mass L/c² — became the most recognised equation in human history, E = mc². It says something staggering: mass and energy are two faces of the same thing — and a tiny amount of one hides an enormous amount of the other.

What each symbol means

E = mc2

E

Energy, measured in joules.

m

Mass, measured in kilograms.

c²

The speed of light, squared — a colossal number: about 90,000,000,000,000,000.

The genius is in that c². Because the speed of light is so vast (≈ 300,000 km/s), squaring it gives a multiplier so huge that even a feather's worth of mass, fully converted, would release staggering energy. Mass is, in effect, frozen energy.

Sources:
  1. Wikipedia: Mass–energy equivalence.
  2. A. Einstein, "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?", Annalen der Physik 18 (1905).

Try it yourself

Turn mass into energy

Pick something — a paperclip, yourself, a car — and see how much energy its mass contains if it were completely converted, as Einstein's equation allows.

Your prediction

According to E = mc², how much energy is there in a paperclip with a mass of 1 gram?

Interactive · E = mc²

Energy calculator

—

In practice we can only convert a sliver of mass — under one percent even in nuclear fission and fusion (about 0.1 % in uranium fission, 0.7 % in the Sun's hydrogen fusion), almost nothing in chemical reactions. Only matter meeting antimatter turns all its mass into energy. But the principle is exact, and it powers the stars.

Answer

BAbout 25 million kilowatt-hours – more than the Hiroshima bomb released.

1 gram corresponds to about 9 · 10¹³ joules, that is about 25 million kilowatt-hours – roughly 1.4 times what the Hiroshima bomb released in 1945. You would only get that much out if the whole mass were converted, for example when matter meets antimatter. In nuclear fission only about a thousandth of the mass becomes energy, in chemical reactions almost nothing.

For the curious: the calculation

E = m · c² = 0.001 kg · (299,792,458 m/s)² ≈ 8.99 · 10¹³ J. One kilowatt-hour is 3.6 · 10⁶ J, so ≈ 2.5 · 10⁷ kWh. Hiroshima bomb: about 15 kilotons of TNT ≈ 6.3 · 10¹³ J.

Sources: Wikipedia: Mass–energy equivalence · Wikipedia: Little Boy

Challenges0 of 2 done

  • Find the mass whose energy would supply an average US home with electricity for a whole year.
  • Find the mass whose energy equals that of the 1945 Hiroshima bomb (about 6.3 · 10¹³ J). It is less than one gram.
Embed this calculator on your webpage

Copy the HTML code below to embed the interactive calculator on your own webpage:

Sources:
  1. Energy comparisons are order-of-magnitude illustrations. Household figure: about 10,800 kWh per year for a US home (U.S. EIA, 2022); Hiroshima bomb: about 15 kilotons of TNT, 6.3 × 1013 J (Wikipedia: Little Boy). Wikipedia: Mass–energy equivalence.

Your prediction

On the left lie two protons and two neutrons, separately. On the right you join four identical particles into a helium nucleus. What does the balance show?

Interactive · mass defect

The nuclear scale

A helium nucleus is made of two protons and two neutrons. On the left of the scale these four building blocks lie separately. On the right you build a helium nucleus from another set of the same four. What will the scale show once the nucleus is complete?

Drag the particles onto the right-hand pan or tap them (keyboard: Tab and Enter). Then join them together.

pnpn separate pnpn28.3 MeV escape your nucleus
4.031 883 u
left: separate
0.000 000 u
right: your side
4.031 883 u
difference

Left: two protons and two neutrons. Right: nothing yet.

This is E = mc² at work: when particles bind tightly together, they give off energy, and the energy takes mass with it. A similar loss of mass makes the Sun shine, where hydrogen becomes helium.

The tilt of the scale is not to scale: the difference is only 0.75%. Masses in atomic mass units u (1 u is one twelfth of the mass of a carbon-12 atom), values from CODATA 2022.

And in everyday life? A cup of tea

Heat is energy too, so it has mass. Heat 250 g of tea from 20 °C to the temperature below, and the tea becomes heavier by m = E/c²:

+0.70 nanograms heavier
62 760 joules of heat · 2.8 · 10−12 of the tea’s mass

Nobody can measure that. Even the best mass comparators compare weights only to about one part in ten billion (10⁻¹⁰); even at 95 °C the difference for the tea is about thirty times smaller still. Besides, the rising steam carries off far more mass at every moment.

For the curious: the calculation
  • Proton mp = 1.007 276 466 6 u · neutron mn = 1.008 664 916 1 u · helium nucleus mα = 4.001 506 179 1 u
  • Separately: 2 mp + 2 mn = 4.031 882 765 u
  • Mass defect: Δm = 4.031 882 765 u − 4.001 506 179 u = 0.030 376 586 u, that is 0.75%
  • Energy: 1 u · c² = 931.494 MeV, so E = Δm · c² = 0.030 376 586 · 931.494 MeV ≈ 28.30 MeV (7.07 MeV per nucleon)
  • Tea: E = m · cwater · ΔT = 0.25 kg · 4184 J/(kg · K) · 60 K = 62 760 J; Δm = E/c² = 62 760 J / (299 792 458 m/s)² ≈ 7.0 · 10−13 kg = 0.70 ng
  • Tables often give the mass of the whole helium atom, which also contains two electrons. For the scale only the nucleus counts.
Answer

BThe helium nucleus is a little lighter.

The finished nucleus is about 0.75 % lighter than its four parts on their own. When they join, 28.3 MeV of binding energy escapes, and by E = mc² mass goes with that energy. To take the nucleus apart again, you have to put exactly this energy back in.

For the curious: the calculation

Separately: 2 · 1.007276 u + 2 · 1.008665 u = 4.031882 u. Helium nucleus: 4.001506 u. Difference: 0.030376 u, about 0.75 %. As energy: 0.030376 u · 931.494 MeV per u ≈ 28.3 MeV.

Source: Wikipedia: Nuclear binding energy

Sources:
  1. CODATA 2022 values (NIST): proton mass, neutron mass, alpha particle (helium nucleus) mass, 1 u in MeV.
  2. Wikipedia: Nuclear binding energy.
  3. Wikipedia: Specific heat capacity: water: 4184 J/(kg · K); PTB: mass comparators: standard deviation below 0.1 µg for 1 kg in vacuum, i.e. about 10⁻¹⁰.
  4. Own calculation of the mass defect and of the tea; see “For the curious”.

Where it shows up in the real world

The Sun

Every second, the Sun converts about 4 million tonnes of mass into energy through nuclear fusion. That lost mass, via E = mc², is the sunlight that makes life possible.

Nuclear power

Reactors split heavy nuclei; the products weigh slightly less than the originals. That missing mass becomes heat — and then electricity.

The atomic bomb

The same conversion, released all at once. Einstein's equation explains the energy, though he played no part in building the weapon.

PET scans

In hospitals, antimatter meets matter and both vanish into pure energy as gamma rays — mass-to-energy conversion used to image the living body.

It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing — a somewhat unfamiliar conception for the average mind.Albert Einsteinsoundtrack of the film Atomic Physics (J. Arthur Rank Organization), 1948
Sources:
  1. Wikipedia: Nuclear fusion (the Sun); mass–energy equivalence.