Einstein was Right, Energy Equals Mass

John Fyffe
March 23, 2015

Submitted as coursework for PH241, Stanford University, Winter 2014


Fig. 1: This is an example of a mass defect curve that shows the binding energy per nucleon (nucleons are the proton and neutron subatomic particles) vs mass number. [1] (Source: J. Fyffe)

Most people have heard the quote "E = mc2". Some know that Albert Einstein is the one who developed the mass-energy equivalence, but I wonder how many actually understand how to use it or what it means. I have known about E = mc2 for most of my life, but never really understood it beyond the mind altering thought that matter is energy and energy is matter. It is hard to move much past this simple point, because most scientists, engineers, and philosophers have trouble actually defining mass or energy. So what does E = mc2 mean?

I won't attempt to define mass beyond the common notion, because I don't claim to understand where it comes from other than some vague notion that Higgs and his Boson are somehow involved. Energy I'll define in an unsatisfactory but seemingly the only way possible, "That which is conserved and can be transferred as heat, work, radiation, or matter."

E = mc2

Now, let's move on to E = mc2. It might be thought of as, if we took a piece of matter and made it "disappear", that we would get mc2 amount of energy released, but does that ever really happen? Do we ever just make a chunk of matter disappear? No, not really. So, let us instead think about an atom and what it is made up of: protons, neutrons, and electrons. If we measured each of these things - called subatomic particles - independently, we would find they each have their own mass (mp is the mass of the proton, mn the neutron, and me the electron). Now, what we find is that when we form a hydrogen atom from a proton and an electron is that the mass of the hydrogen is not equal to the mass of the sum of its parts. There is now a mass defect. So, what happened to that mass? It was converted to energy, meaning that forming a hydrogen atom from subatomic particles would release energy! Thus, the mass defect is exactly related to the change in energy by Einstein's equation: E = mc2.

We can also think about this process in reverse. If we imagine a hydrogen atom in a box and we split it into a proton and an electron, we would end up with more mass. Therefore, assuming we didn't add any particles with mass to the box, we know that mass must have come from putting energy into the box. This energy is the amount of energy it would take to split a hydrogen atom, also called the binding energy.

Mass Defect Curve

Looking at Fig. 1, we see that the binding energy per nucleon - i.e. subatomic particle - peaks around a mass number of 56, which is atomic iron. [1] To construct the curve in Fig. 1, physicists had to measure mass and energy extremely precisely on a subatomic particle and atomic scale to pull out the subtle changes in mass from element to element. Starting from smaller atoms, the amount of energy it takes to break an atom into subatomic particles goes up as the mass goes up. That is because each subatomic particle you put in a nucleus is very strongly attracted to its nearest neighbors by the nuclear strong force. So, if we have three subatomic particles in a nucleus (He-3) it takes about 2.6 MeV of energy to separate the two protons and one neutron from each other. However, if we add a neutron to form He-4, each of the three original particles are now being held in place by another attractive force from the new neutron. Therefore, the binding energy increases to 4.7 MeV. This trend continues as we add particles until we reach a tipping point at iron (excepting the quantum shell effects seen near mass number 10, which we won't go into here).

The tipping point where the binding energy starts to decrease as we increase mass number is a result of the very short range effect of the nuclear strong force and the accumulation of repulsive charge effects. The nuclear strong force only acts over a very small distance, so when we add a particle to a large nucleus, it is no longer applying the attractive to all of the other particles in the nucleus, only the ones close by. So, each additional particle's effect diminishes as the nucleus gets larger. However, as we add more and more protons to the nucleus, they repel each other because they are all positively charged. This force acts over a larger distance than the nuclear strong force, and therefore is always repelling all of the other protons in the nucleus. So iron is where the repulsive effect becomes more dominant than the attractive strong force effect of each additional particle, causing the binding energy to start decreasing with additional mass number.

Applying Einstein's Equation

Now that we understand the basics of the mass defect curve we can see why nuclear fission reactors work, how the Sun works, and why iron is the most stable element and if we allowed entropy to do its job, all of the elements in the universe would end up as iron.

A Direct Test of E = mc2

A team of scientists recently (2005) tested the famous equation by measuring the masses of silicon and sulphur before and after a neutron was absorbed (as well as measuring the mass of the free neutron). [2] When silicon or sulphur absorb a neutron, they release a gamma-ray. This gamma-ray's wavelength can be measured to give a very accurate value of its energy. The mass difference Δm was measured by simultaneous comparisons of the cyclotron frequencies (which are inversely proportional to the mass) of ions of the initial and final isotopes of silicon and sulfur. This measurement was done with the isotopes confined over a period of weeks in a Penning trap. They did all of these measurements with the utmost care to try and measure the energy and masses as precisely as possible. In doing so, they found that the equation E = mc2 held to within 0.00004%. Table 1 shows the difference in masses before and after the neutron absorption for silicon and sulphur as well as the difference between E and Δmc2.

Reaction Δm (amu) 1 - (Δmc2)/E
M[32S] + M[H] - M[33S] 0.00843729682(30) 2.1(5.2) x 10-7
M[28Si] + M[H] - M[29Si] 0.00825690198(24) -9.7(8.0) x 10-7
Table 1: Nuclear reaction mass and energy equivalence results. [2] Numbers in parenthesis indicate uncertainty on the last digits.

Another Example: Nuclear Fission

Uranium has a binding energy of around 7.25 MeV. If U-238 alpha decays (releases a Helium atom, ie. 2 protons and 2 neutrons) it will form a Helium atom and a Th-234 atom. Helium atoms we said have a binding energy of 2.5 MeV and Th-234 has a binding energy of 7.6 MeV. So we can find the energy released by comparing the reactant binding energy (U-238) and the product binding energies (Th-234 and He-4). We can think of it as breaking up the U-238 into all of the individual protons and neutrons and then reforming the products from those particles. It takes 7.25 MeV to break the U-238 into its subatomic particles. Then, when we use 2 protons and 2 neutrons to form the Helium atom we get 2.5 MeV released and we get 7.6 MeV released when we form the Th-234 atom. Therefore, we get a net energy release of 2.85 MeV per reaction!

δE = 2.5 MeV + 7.6 MeV - 7.25 MeV = 2.85 MeV = 4.5662 × 10-13 J

We can also determine the amount of matter that was destroyed (but let's be clear, it wasn't destroyed, it was converted to energy!)

δm = E/c2 = 4.5662 × 10-13 J / (299,792,458 m/s)2 = 5.0806 × 10-30 kg

(This is about .003 amu. Remember, a proton and neutron are about 1 amu each. So, we "lost" about 3/1000th of the mass of a proton in the reaction.)

But What Does It Mean?!

Now doesn't E = mc2 seem more practical? More useful? More meaningful? We now know how to use e = mc2 to calculate reaction energies and design and engineer power plants or understand our solar system better. This doesn't answer the question of how or why energy and mass are convertible, so I leave you to think about that, and what that means to you. I know I think it is a fun idea!

© John Fyffe. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


[1] E. Lewis, Fundamentals of Nuclear Reactor Physics (Academic Press, 2008).

[2] S. Rainville et al., "World Year of Physics: A Direct Test of E = mc2," Nature 438, 1096 (2005).