# E=mc2 is more complicated that I originally thought

I've been reading up on radiation and it appears to me that if you add up all of the protons, neutrons, and electrons in the universe, the number would never change (except for temporary positron borrowing). I guess this is no surprise considering conservation of matter. On the other hand, because E=mc2, I thought that matter was converted to energy - I see now that m has nothing to do with matter and everything to do with inertial mass. So I am hoping someone can help me out of my confusion - how is it that subatomic particles (at rest) change their mass? does it simply have to do with the strong nuclear force? Maybe my question should be what is mass?

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Staff Emeritus
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I've been reading up on radiation and it appears to me that if you add up all of the protons, neutrons, and electrons in the universe, the number would never change (except for temporary positron borrowing).
I don't think that's true.

Suppose you have an atom of tritium. N=2, P=1, and E=1, so their sum is 4. It decays to 3He, with N=1, P=2 and E=2, which sums to 5.

Andrew Mason
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I've been reading up on radiation and it appears to me that if you add up all of the protons, neutrons, and electrons in the universe, the number would never change (except for temporary positron borrowing). I guess this is no surprise considering conservation of matter. On the other hand, because E=mc2, I thought that matter was converted to energy - I see now that m has nothing to do with matter and everything to do with inertial mass. So I am hoping someone can help me out of my confusion - how is it that subatomic particles (at rest) change their mass? does it simply have to do with the strong nuclear force? Maybe my question should be what is mass?
Mass or Inertia simply exists. We cannot explain it in terms of something simpler. It is a fundamental property of matter. So if you are looking for an explanation of mass in terms of simpler concepts you will be disappointed.

Mass or inertia certainly has something to do with the strong nuclear force. We can think of matter as being made up of neutrons, protons and electrons. And for most purposes we can think of inertia or mass as being directly proportional to the number of neutrons and proton-electron pairs. But if you want to measure mass precisely, you have to take into account the binding energies of those neutrons and protons in the nucleus. You will see from the periodic table that atomic mass of an element is not the atomic number multiplied by the mass of a single proton/electron pair + the number of neutrons multiplied by the mass of a single neutron. It is a bit more than that. The extra mass is due to the binding energy of the protons and neutrons in the nucleus due to the strong nuclear force. This binding energy, E, adds mass m to the nucleus ie: m = E/c^2.

AM

jbriggs444
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[...]You will see from the periodic table that atomic mass of an element is not the atomic number multiplied by the mass of a single proton/electron pair + the number of neutrons multiplied by the mass of a single neutron. It is a bit more than that. The extra mass is due to the binding energy of the protons and neutrons in the nucleus due to the strong nuclear force. This binding energy, E, adds mass m to the nucleus ie: m = E/c^2.
Binding energy is negative. The mass of an atomic nucleus is less than the sum of the masses of the neutrons and protons in that nucleus.

One proton is about 1.00727 atomic mass units.
One neutron is about 1.00866 atomic mass units.

The nucleus of an atom of Carbon-12 has 6 protons and 6 neutrons for an unbound total of 12.09558 atomic mass units.

In fact, Carbon-12 has, by definition, a mass of 12 atomic mass units exactly. The deficit is the binding energy.

I've been reading up on radiation and it appears to me that if you add up all of the protons, neutrons, and electrons in the universe, the number would never change (except for temporary positron borrowing).
nope. Most have disappeared into black holes where the vast majority of information in our universe is 'hidden' to us. In addition nuclear reactions, fission and fusion, converts a tiney portion of matter to energy... maybe about 2% or so for fission, for example.

I guess this is no surprise considering conservation of matter. On the other hand, because E=mc2, I thought that matter was converted to energy - I see now that m has nothing to do with matter and everything to do with inertial mass.
m c2 = sqrt [E2 −(pc)2]. The values of m and p are different in different frames, but they are always different in ways that make this equality hold in every frame.

So I am hoping someone can help me out of my confusion - how is it that subatomic particles (at rest) change their mass? does it simply have to do with the strong nuclear force? Maybe my question should be what is mass?
The mass of an electron, for example, appears different in different confinements....say in a lattice, for example...One way of describing such changes is as Andrew Mason posted; another view is to say that the degrees of freedom in a structure affect it's energy and hence it's observed 'mass'. For example, a proton-electron system [an atom] as whole can absorb the photon whereas a single, free electron can't due to energy-momentum conservation. In order to absorb a photon there must be additional degrees of freedom that can be excited.....

jbriggs444
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[...]In addition nuclear reactions, fission and fusion, converts a tiny portion of matter to energy... maybe about 2% or so for fission, for example.
More like 0.1%

If you consult the periodic chart, you'll see that when measured in atomic mass units, the actual mass of U-235, Ce-140, Zr-94 and a free neutron are all within about 0.1% of the number of nucleons in each.

Note that nuclear fission conserves baryon number. Even though the sum of the masses of the fission products is less than the mass of the U-235 that was fissioned, the total number of protons plus neutrons is unchanged.

Drakkith
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I don't think that's true.

Suppose you have an atom of tritium. N=2, P=1, and E=1, so their sum is 4. It decays to 3He, with N=1, P=2 and E=2, which sums to 5.
Where did your 2nd electron come from in the He3?

Nugatory
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Where did your 2nd electron come from in the He3?
Presumably from the $n \Rightarrow p^+ + e^- + \overline{\nu}$ process for neutron decay?

Drakkith
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Presumably from the $n \Rightarrow p^+ + e^- + \overline{\nu}$ process for neutron decay?
Ah ok. Thanks Nugatory.

nuclear reactions, fission and fusion, converts a tiney portion of matter to energy... .
Are you saying that matter is converted into energy as well as mass? If you are, is the equation E=mc2 the same for matter?

jtbell
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What is the difference between "matter" and "mass"?

Binding energy is negative. The mass of an atomic nucleus is less than the sum of the masses of the neutrons and protons in that nucleus.
Only true for elements lighter then iron. That is why elements lighter then iron release energy when fused and require energy to split but elements heavier then iron release energy when split but require energy to fuse.

Drakkith
Staff Emeritus
Only true for elements lighter then iron. That is why elements lighter then iron release energy when fused and require energy to split but elements heavier then iron release energy when split but require energy to fuse.
I think nuclear binding energy is always negative. If you take Uranium and remove all of it's protons and neutrons, it will require energy. It's simply that the DIFFERENCE in binding energy between different elements can result in a release when you convert one to another via fusion or fission. I believe this is one of the main reasons that fission and other decays typically results in things like Alpha particles being emitted. Their binding energy per nucleon is much higher than anything else on that end of the periodic table so you don't have to expend as much energy to eject them as you would individual protons.

Nugatory
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Only true for elements lighter then iron. That is why elements lighter then iron release energy when fused and require energy to split but elements heavier then iron release energy when split but require energy to fuse.
No, jbriggs had it right. The binding energy is always negative (otherwise, the nucleus wouldn't hold together at all). What's special about iron is that it sits at the minimum of the curve of binding energy per nucleon; this value is always negative, but it's more negative for iron than anything else.

I've been reading up on radiation and it appears to me that if you add up all of the protons, neutrons, and electrons in the universe, the number would never change (except for temporary positron borrowing). I guess this is no surprise considering conservation of matter. On the other hand, because E=mc2, I thought that matter was converted to energy - I see now that m has nothing to do with matter and everything to do with inertial mass. So I am hoping someone can help me out of my confusion - how is it that subatomic particles (at rest) change their mass? does it simply have to do with the strong nuclear force? Maybe my question should be what is mass?
Energy is conserved! Mass is not converted into energy:

"If a body gives off the energy L in the form of radiation, its mass diminishes by L/c².
[..] The mass of a body is a measure of its energy-content"
- http://www.fourmilab.ch/etexts/einstein/E_mc2/www/
What is the difference between "matter" and "mass"?
In a closed system, matter can be converted to radiation. In contrast, the system's mass is conserved.

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What is the difference between "matter" and "mass"?

Correct me if I am off on this, but I think of matter as the particles that make up "stuff". And I think of Mass a little differently. For example, lets assume that electrons are massless and you had 100 protons and 120 neutrons - they would make up a certain amount of mass, say 1000 units. But if you combine these particles to make certain elements, some of the mass is converted to energy (strong nuclear force)and the mass would be less than 1000 units, even though you still have the same amount of matter (100 protons and 120 neutrons). When I hear someone say that matter is converted to energy, I'm thinking of particles converting into energy - so that was my question to Naty1. Did he mean particles were being converted to energy, and if so, does e=mc2 apply.

"Matter" is ill defined in science. Its best to leave the word out completely if you want to get technical. The words mass and energy should suffice.

BruceW
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When I hear someone say that matter is converted to energy, I'm thinking of particles converting into energy - so that was my question to Naty1. Did he mean particles were being converted to energy, and if so, does e=mc2 apply.
That's right, particles can be converted into energy. For example, an electron and positron can collide, and turn into two photons. The photons are particles as well, but they don't have rest mass, so I think they would be equivalent to your definition of 'energy'.

and e=mc2 does apply. This equation could mean one of two things: 1) the 'm' corresponds to the rest mass of a particle, so e would be the energy of the particle when it is measured at rest relative to the lab. 2) 'm' could instead correspond to the relativistic mass, and so this shows that energy and relativistic mass are the same thing, simply measured in different units.

Back to the example of a positron and electron colliding to make 2 photons, the energy and momentum will be conserved. Also, the invariant mass will be conserved. But if you count up the rest masses of the particles, this is clearly not conserved. Since to begin with, you have the rest masses of electron and positron, then after collision, none of the particles have any rest mass.

Edit for clarity: when I say the electron and positron collide and turn into two photons, I mean that the electron and positron are 'used up' in the collision, so they don't exist after the collision.

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Correct me if I am off on this, but I think of matter as the particles that make up "stuff". And I think of Mass a little differently. For example,[..], some of the mass is converted to energy [...] I'm thinking of particles converting into energy [..]
Not exactly; energy is conserved. You can however convert matter in radiation, see post #15 (just before yours, so you may have missed it). bruceW, are all the particles you mention massless? What about Protons or neutrons? Would they be able to convert into energy (either directly or indirectly)?

harrylin, can you clarify your post #15? I do not understand what you mean when you state that "mass is not converted into energy". I read the link you posted to Einstein's 1905 paper, which states "The mass of a body is a measure of its energy-content". Additionally, isn't the essence of the e=mc2 to state that mass can be converted to energy and visa-verse?

Nugatory
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bruceW, are all the particles you mention massless? What about Protons or neutrons? Would they be able to convert into energy (either directly or indirectly)?
An anti-proton and a proton, if they meet, will convert directly to energy in the form of photons. Likewise for a neutron and an anti-neutron.

A single proton cannot be converted to energy, but that's not because there's anything wrong with mass-energy equivalence, it's because the process would violate various conservation laws.

BruceW
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Yes, what nugatory said.

Also, electrons have 'rest mass', as do protons and neutrons, but photons don't.

[..]
harrylin, can you clarify your post #15? I do not understand what you mean when you state that "mass is not converted into energy". I read the link you posted to Einstein's 1905 paper, which states "The mass of a body is a measure of its energy-content". Additionally, isn't the essence of the e=mc2 to state that mass can be converted to energy and visa-verse?
Sure! The essence of E=mc2 is not how you heard it but what I cited (using your symbols):

If a body gives off the energy ΔE in the form of radiation, its mass diminishes by ΔE/c²
In equation: Δm=ΔE/c²

A lot of people are sloppy and say "mass" when they really mean "matter", and "energy" when they really mean "radiation". That can be very confusing. Even Einstein became sloppy in later years, but he formulated it very precisely in his first paper on that topic. What happens is that matter can carry internal energy away by means of radiation, just as he explained.

His paper is based on conservation of energy; that means that the total amount of energy is constant. Thus an amount of energy equal to the radiation energy is now lacking in the atom. That reduced internal energy can be indirectly measured as a reduction in weight. If you could convert mass into energy, then you would end up with more energy than before. Last edited:
I am happy to have understood the last 3 posts, but I am left with another question regarding the fact that mass is not converted into energy. For example, say U-235 was to take on an extra neutron and split into Lanthanum-146 + Bromine-87 + 3 neutrons. The atomic weight of U is more than the combined atomic weight of La + Br + 3 neutrons. I always thought the lost mass was converted into energy. But I think you are saying the lost matter is converted into radiation (gamma and x-ray?). If that is the case, what makes the boom in the nuclear bomb?

Nugatory
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I always thought the lost mass was converted into energy. But I think you are saying the lost matter is converted into radiation (gamma and x-ray?). If that is the case, what makes the boom in the nuclear bomb?
Radiation is a form of energy, so you don't want to spend a lot of time worrying about the distinction between mass being converted into energy and mass being converted into radiation; the latter is a more specific statement of the former.

As Harrylin says, just about all mass conversion processes release most of their energy in the form of radiation. However, it's not all radiation all the time. Fission (which was unknown when Einstein developed E=mc2) imparts a fair amount of kinetic energy to the daughter nuclei because they're both positively charged so fly away from each other at high speed. Alpha particles carry kinetic energy for the same reason, although much less.

So when a nuclear weapon is detonated, two things are released. One is a flood of fast-moving massive particles (alpha particles and fission daughter nuclei) which transfer their kinetic energy to everything nearby as heat. The other is a flood of high-energy gamma and X radiation, which also heats everything nearby as it is absorbed. Thus, a substantial fraction of the energy released by the weapon ends up as heat in the immediate neighborhood... And that produces the shock wave and blast effects.