Really?

a) No, both chemical and nuclear reactions will convert mass into energy. However, the mass fraction converted in chemical reactions is very small, on the order of a millionth of that converted in nuclear reactions. In practice, it’s pretty much undetectable.

b) Yes, c = 1 is used in physics because c ~ 300,000 km/s is an accidental result of its units (depending on the accident of the earth’s radius, its period of rotation, the Babylonian number system, and the French Revolution).

HTH.

Greene is correct. The measured mass of an object is equal to its total energy, so any way of adding or removing energy to it affects the measured mass. Heating up a box will cause it to become slightly heavier, for example.

A good way to think of it is that measured mass is really just another word for “total internal energy.” They’re not two separate quantities; you can’t change one into the other. You can change one ‘type’ of energy into another, and you can transfer energy around, thereby changing the measured mass.

So, eg, burning a quantity of gasoline in your car would not actually change its mass if you didn’t allow any of the reaction products (including heat!) to escape. It would just change chemical potential energy into thermal energy. If heat or light escape, then the mass of the car will decrease corresponding. And if you transform some of this energy into motion (ie kinetic energy), then the proper mass of your car will again decrease correspondingly.

There’s nothing intrinsically different about nuclear reactions. However, there’s a vast quantitative difference. In a typical chemical reaction, the energy change in an atom is a negligibly tiny fraction of the total energy (same thing as mass, remember) of that atom. A ballpark figure would be around 1 part in a trillion. So for *practical* purposes we can ignore it — this is why you see sometimes see books (erroneously) say that mass doesn’t change in chemical reactions. For nuclear reactions the energies involved aren’t so negligible compared to the total; the change in mass might be of the order of 1%.

—

I would agree that E=m is in some ways a better formulation. c^2 is just the constant that establishes the distance/time unit conversion ratio; the key idea is that mass is just another name for total energy. Indeed we high-energy physicists prefer to use units in which the speed of light is simply 1, and so the equation becomes simply E=m.

Lies! Damned lies!

Live preview told me that all my pretty superscripts and subscripts would show up, and they do not. I can only attribute this to the most far-reaching of conspiracies within the open source community to stifle free speech and the aesthetically agreeable representation of mathematical expressions.

Mass can’t be just another name for total energy because mass with momentum has more energy than mc^2.

Another comment to chime in and agree with your first two commenters. Brian Greene is perfectly right.

For a comparison between chemical binding energy and nuclear binding energy contributing to mass, here’s a remark from a Wikipedia Talk page:

Strictly speaking, you are correct that the chemical binding energy contributes mass. However, this mass is very small compared to the mass due to the nuclear binding energy and so can usually be neglected. For example, the chemical binding energy in a hydrogen molecule contributes a mass of ~2.4E-9 g/mol. By contrast the nuclear binding energy in Deuterium contributes a mass of ~0.0029 g/mol (over six orders of magnitude greater than the chemical binding energy).

It’s of course the same for the mass loss via changes in chemical binding energy. A quick over-the-thumb calculation for the mass equivalent of the energy needed to accelerate a car from zero to about 50 km/h, neglecting friction (bad thing…) gives me a mass loss of the order of 10(-9)g for what also amounts to a mole or a fraction of a mole of molecules (hydrogen, water, other) produced in the combustion process — the mass equivalent to the breaking of chemical bonds.

As for setting c=1, so-called natural units, that’s fine because Einstein’s formula is universal, meaning that mass and energy are fundamentally the same thing and can be measured in the same units. But then you need to measure mass in eV/c². If you prefer giving c in miles/hour, better leave c in the formula.

(As an aside, I’m aghast that someone would teach that E=mc² is only valid for nuclear reactions. This is an obstacle to the students’ understanding of the difference between [what are considered] universal equations, i.e. the principles of the physical world, and ad-hoc, approximate formulae that are only valid under certain conditions.)

Thanks for asking the question, John — the same thing occurred to me while reading the article, I wanted to ask somebody but the number of high-energy physicists who read my blog is far far fewer than the corresponding number for CT — so it is more useful for you to ask the question.

John: all energy counts as mass as you put it. So if for instance you burn two molecules of hydrogen using a molecule of oxygen, the resulting 2 molecules of water will weigh a bit less than the original 3 molecules. The energy released during the reaction in the form of heat & light will be equal to the difference in mass using the E=mc^2 equation.

abb1: Of course it isn’t. The E=mc^2 equation is the energy of mass when the speed is 0. This page has the formula of relativistic kinetic energy. E=mc^2 is the result when you set v = 0.

Phil, if that’s true that 2 molecules of hydrogen and 1 molecule of oxygen have greater mass than 1 molecule water (if that’s what you’re saying), then I learned something new here. Thanks.

Units are important, but there are different systems of units. While freshmen use the metric system of units, working physicists typically use a system more appropriate to their work, such as natural units. Unlike the metric system, where you have different named units, there is only one unit in natural units and it’s nameless.

We begin defining natural units by setting c=1. This means that our unit of length is the distance travelled by light in one second. You can still do dimensional analysis in natural units. Distance quantities have one unit and time quantities also have one unit, but velocity as distance/time has zero units. Acceleration is distance/time^2 and so has -1 units.

We continue defining natural units by setting Planck’s constant, hbar = 1. As h = mass*length*length/time, we find that mass quantities have -1 units. That’s basically all you need to start doing dimensional analysis in natural units. As you can imagine, it’s much easier to perform calculations without the large number of c’s and h’s that you get in typical relativistic quantum mechanics calculations. You can apply a conversion factor to your natural value at the end of your calculation to get a metric value if you need one.

Einstein’s full energy equation in natural units is E^2 = m^2 + p^2.

E=mc^2 is a specific form of the longer equation to which it collapses under certain circumstances far too remote in my memory of Quantum Physics class to recall. However, c is the speed of light. It is always the speed of light. It is a constant. It does not come out of that equation (it was measured by Roemer in 1676, far in advance of Einstein). 3 x 10^8 m/s is a good estimate.

With tongue slightly planted in cheek, someone recently calculated that people who commute at speeds considerably slower than light will tend to live slightly longer than people who don’t. A few seconds over the course of a lifetime, I think it came out to be. Sorry, I can’t seem to find a link (wouldn’t surprise me if it was The Onion, though).

Like the difference between high school chemistry and P-chem.
Like the difference between Newtonian and Einsteinian physics.
Like the difference between an experiment conducted in an open Erhlenmeyer flask or a study performed in a bomb calorimeter.

Er, superscripts, that is. But the formula worked…. The trick is to use the carat character to surround what you want superscripted. Sort of like

E^2
^

but all on one line — I can’t figure out how to make Textile not turn carats into superscripts

Suppose I took elevator to the second floor. Now I have more potential gravitational energy than I had on the first floor, just like 2xH2 + O2 have more potential chemical energy than 2xH2O. Correct? Do I have larger mass on the second floor?

This is what we get when we attempt to debate science in what is intrinsically a liberal arts forum. We all use the same words, but to some of us–the scientists–those words have concrete meaning reflecting underlying mathematical constructs, while to everybody else, the meaning of words is fluid and capricious.

For the record, ALL formulations of Einstein’s famous relationship are equivalent, by definition. Exactly so, a priori, forever amen. Period.

That this relationship takes different forms on paper is entirely a matter of notational convenience, no more significant to the operation of the universe than it matters to the playgoer whether the actors’ copies of Shakespeare’s script are rendered in a serif or a sans-serif font.

Energy IS mass, and vice versa. That they appear distinct to us is casual life reflects an idiosyncratic coincidence driven by our physical context and the arrangement of our sensory apparatus, nothing more.

It probably sounds like I’m advising all the liberal-arts types out there to go talk about something else. I’m not. Instead, I’m suggesting that the concepts discussed in this conversation are ones that only retain their entire content when expressed in the language of mathematics. Learn the math, and the personal rewards will be beyond what you are currently equipped to imagine.

Don’t bother, and your experience in these sorts of discussions will amount to simple mental masturbation.

John:
I can accept that the decrease in mass is too small to be measured, but why isn’t the extra energy detected in experiments testing conservation of energy.

It’s not too small to measure. In my high school chemistry class, lo these many years ago, one of the experiments we did measured the difference in mass before and after an exothermic reaction, and compared it to the mass-energy calculation.

Or is the claim that what I was taught to call chemical energy actually counts as mass?

Yep, it not only counts as mass, it actually is mass. Pretty freaky, but it’s true.

I always regarded the “c^2?” part of the equation to be a placeholder for the phrase “an unbelievably huge number”. Why the heck should the speed of light – squared – have any particular significance? I think that the equation that Einstein used, which solves to e=mc^2, is intended to convey the idea that a given mass would convert to “an unbelieveably huge” amount of energy.

@eudoxis: That’s what I wrote myself, in comment nr. 10, as well.

But we were talking physical principles here, not reasonable approximations or, indeed, counting atoms of all types.

This discussion needs the term “rest mass”. If you consider a chemical reaction as mess_of_quarks_and_electrons -> different_arrangements_of_the_mess you can see that the rest mass stays the same – that is, the individual particles in isolation are unchanged hence the total rest mass is unchanged. What happens is that the binding energy of the mess changes, and the total mass if you exclude the radiated energy.

In nuclear reactions you have mess_0 -> mess_1, with different particles on either side, plus different binding energies.

Considering the difference in mass for one the examples that Greene uses, take the combustion of methane (in the same range as Greene’s gasoline engine example). The enthalpy change = -890 kJ/mol. From there (handwave) we calculate a relative change in mass for that reaction of 9.8 * 10^-12 kg. That’s out of a total mass of 80g.

For a 10kg of methane car tank, the change in relative mass after burning is about 1*10^-8 kg. That’s a trivial change and one that can be ignored for ordinary chemical reactions.

What Greene is getting at, in my view, is that E=mc^2 is true – even for different scales of practical measurability.

For the moving car example and the elevator question, the relative change in mass is even smaller. Something on the order of 10^16 kg.

By the way, Einstein never wrote the famous equation E = mc2. Albert wrote L/V2 as I explain on my blog.

Chris W. “But we were talking physical principles here, not reasonable approximations or, indeed, counting atoms of all types.”

Why is this directed at me? I didn’t say anything about counting atoms, and, yes, indeed, most of the comments are about reasonable approximations, especially ones that exclude miniscule changes in relative mass.

So, eudoxis, you’re saying that when I’m lifted to the second floor my mass does increase, is this correct?

This sounds like quite an extraordinary claim, because while in the chemical reaction example you could confuse me with quarks and electrons – here’s no change in the matter taking place whatsoever, only in the location of the object. If my mass on the second floor of the building is different from my mass on the first floor of the same bulding, doesn’t it make the concept of ‘mass’ completely meaningless?

Thanks.

So, Matt, once again: is my mass on the second floor (from my point of view) the same as my mass on the first floor (from my point of view) or is it different? I am the observer.

Once I’m on the second floor, I can now generate some energy by sliding down to the first floor. And if ‘energy’ and ‘mass’ are the same, then I’m bound to loose some mass in the process, correct?

Since I’m the observer, there shouldn’t be any relativity involved here, correct?

abb1: It is the relativemass that increases and only while you are moving in the elevator. You will have the same rest mass when you come to a stop on the second floor as you did on the first floor. It’s useful to talk about relative mass only as it relates to energy.

What all this misses is that sciences are nests of models. Generally speaking the simplest models are easy to apply but incomplete and the most general are much more difficult to apply.

What a good scientist does is select and use the simplest possible model that describes a situation and seek to extend the most general models. Selection of the simple model requires knowledge of the limits of both models.

On that basis, if you want to understand chemistry, you can ignore mass changes during chemical reactions. You are thus ignoring the equivalence between mass and energy and treating each separately. To the limits of our ability to measure this is the case, so the retreat to a simpler model is fully justified. On the otherhand we recognize that as an absolute matter there will be a change, and that it is calculable, just not measureable.

We also recognize that for nuclear reactions we must fully accout for mass-energy equivalence and again, this is absolutely clear within the range of applicability of both models.

Someone will reply, what are six orders of magnitude among friends.

xmas: You are wrong. When hydrogen reacts with oxygen energy is released. The gravitational equivalent is an object falling: it’s potential energy is converted to kinetic energy.

When you perform electrolysis to separate water into hydrogen and oxygen you supply energy to break the bonds. The equivalent is lifting an object where you have to supply energy, increasing its potential energy. Hydrogen and oxygen molecules have greater potential energy than water.

abb1: However I wouldn’t extend the gravitational analogy to say that two identical objects at different heights have a different mass. As soon as you start talking about acceleration and/or gravity you enter the realm of general relativity and I don’t know anything about that.

Regarding the constant c in the E=mc^2 equation you need to bear in mind that even though it is a constant, it has dimensions (like the gravitational constant G, and unlike Pi which is just a number). In any equation the dimensions (M mass, L length, T time) on one side must be the same as the dimensions on the other. On the left had of the equation you have energy (dimensions ML^2T^-2), and on the right hand of the equation you have mass (M) times a constant (L^2T-2 or (LT^-1)^2). So the square root of the constant has the dimension of length / time, i.e. speed. And regardless of what units of measurement you use, the square root of the constant (by constant I mean the whole c^2 part, and by square root just c) is the speed of light.

People have already covered this, but the difference between chemical and nuclear reactions is one of degree. Exothermic chemical reactions typically convert about one part in billions or so of the mass of the reactants into energy. In particular, exploding TNT converts one part in 20 billion (10*9) of its mass into energy.. A kilogram of TNT exploding releases about 4 x 10*6 Joules. A kilogram of uranium that undergoes fission releases about 20 million times as much energy, which means about one part in one thousand of its mass is converted into energy.
One kg of uranium undergoing fission therefore equals 20 kilotons of TNT.
The complete conversion of 1 kilogram of mass into energy would be 9 x 10*16 Joules, or roughly 20 Megatons of TNT.

OK, forget gravity. Spring. You stressed a metal spring, coil of wire, and you keep it stressed. Has its mass increased? Then you release it, it generates energy – does it mean it lost some of its mass now?

Thanks.

abb1, pretty much yes. Let’s shed a little light on the discussion. Rather, let’s muddle it up by bringing in photons, and quantum mechanics to boot.

First off, mass represents inertial response. It starts off getting interpreted in Newtonian mechanics as the resistance of an object to change in motion as a force is applied. The more mass a particle has, the bigger a force one has to apply to get a certain acceleration. At this point, one is assuming that mass is a constant. But to start to see what is really happening, one must use Newton’s form of the relevant equation,

F = dp/dt.

He actually used rate of change in momentum, not F = m*a. Now we jump to relativity using the spatial part of the momentum 4-vector, which is just plain old Newtonian momentum. One can calculate a mass from relativistic momentum, and this momentum varies depending on velocity. One can also calculate energy, which it related to the time part of the momentum 4-vector and also depends on velocity.

Photons are massless particles (or rather, they have zero rest mass), but one can calculate an inertial response based on a photon’s energy calculated from quantum mechanics and the relationships relating momentum and energy in relativity and photon energy on quantum mechanics (think gravitational lensing). One can see pretty quickly that energy corresponds to inertia, i.e., energy is related to mass. The same sorts of considerations apply to particles with non-zero rest mass.

Special relativity is, among other things, about showing that energy is related to the environment the particle lives in. Generaln relativity is, in this context, about saying that there are more environments that behave this way that we thought special relativity applied to.

All of which is to say, mass is essentially a form of energy (the case of the photon illustrates which is more “fundamental”, thought really, the longer you are in physics, the more you think that momentum is the most fundamental quantity), and that yes, adding or removing energy changes the mass of a system; this includes chemical binding energy. One of the traditional introductory modern physics problems is to calculate how much energy is involved in going from a collection of unbound particles to a collection of bound particles and back (don’t remember if it was an atom or a molecule); go look at the problem set in Alfred Beiser’s Survey of Modern Physics book (chapter 1, I think) for it. It asks you to calculate the sum of the masses of the constituent particles, compare that to the measured mass of the aggregate, and calculate the energy associated with the difference of the two. This energy, stored in the arranging, is precisely the chemical energy that our cells get from turning ATP to ADP or we get from burning gasoline or …

Hope this helps.

Abb1, Okay, I’ve asked around, and I’m assured from physicists that, if you carry a ball up, say, the Empire State Building, then you are, in a tiny way, adding to its mass. This does seem rather counter-intuitive to me, but apparently it’s correct. I’m tempted not to trust my intuition, given how many other strange things happen in G.R. (I mean, time seems for absolute than mass, and yet we know that’s a fiction).

Of course, you yourself will weight a bit less after walking up the Empire State building, as you’ll be using energy. My friend was undecided on the point of a spaceship leaving orbit around the sun and travelling into clear space: he thought that maybe the loss of mass from the ship’s engines would cancel out any possible gain in mass. The problem is that it’s impossible to think of an isolated system in G.R. as you cannot remove everything, or you remove the gravity as well!

One can explain the spring more easily: ultimately, it’s a change in the atomic behaviour of the lattice of atoms which makes up the metal which makes the spring: if molecules can have less mass than their constituate atoms, then it doesn’t seem too surprising that stressing a spring can change its mass a small amount.

I think your collegue is wrong: mass most certainly *is* converted into energy in a chemical reaction. A water molecule literally has (a tiny tiny amount) less mass than 2 hydrogen atoms and an oxygen atom. If you read the Wikipedia on S.R. you’ll find that there is a difference between mass and momentum: a photon is massless but it has momentum; I have mass, and if I speed up, I gain momentum! An electron has mass though, so I’m not quite sure what you collegue meant.

Abb1, Okay, yes, but mass *is* lost: others have said this, and I’m certain it’s true: when the electrons move to lower energy orbits, they also loose a small amount of mass. Read again what Ali Soleimani says in #4.

I refuse to believe this, dammit. And I refuse to believe that a wind-up toy is heavier wound-up than unwound. This is a hoax.

eudoxis, there is no such thing as gravitational mass. There is just mass, and photons don’t have any.

Matt, I have no problem with time; I can explain it to myself as something roughly analogous to doppler effect, which is perfectly obvious. But this mass thing is really really counter-intuitive.

Suppose you’re flying away from earth on a spaceship and looking at a clock left on earth. As you’re moving away and it takes time for the light reflected off the clock on earth to reach your eye, it’ll take you a while to observe the minute hand on the clock jumping from 6:24 to 6:25, it’ll take longer to observe it jumping from 6:25 to 6:26, yet longer to see it jumping from 6:26 to 6:27 and so on. Meanwhile the clock in your spaceship is ticking as usual. This is how I imagine it.

One of the standard derivations (going back to Einstein, I think) of E = mc*2 is the flashlight in a box example. You’ve got a box floating in space with a flashlight buried in the wall on one side of the box. Turn the flashlight on. It’s facing inside, so the light leaves the wall, heading to the other side. The momentum of the light is E/c. The recoil momentum of the box is E/c. The box is moving slowly enough so that one can use the classical momentum for it, so it travels at E/Mc, where capital M is the mass of the box. The light reaches the other side in a time L/c (the box is moving so slowly we can neglect its speed in calculating this). The light is absorbed on the other side and the box stops.

The overall center of mass of the system doesn’t change, because all this happened internally, yet the box shifted ever so slightly to the left. In fact it shifted (E/Mc) times (L/c) or EL /M c*2).
To counter this leftward motion of the box, the energy of the light was moved a distance L, so if the overall center of mass didn’t change, we’d have

mass of box times distance it moved = mass associated with E times the distance it moved or

M times EL/(M c*2) = m L, where m is the mass associated with the energy of the light that moved. Solve for m and you get m = E/c*2.

The energy that went into making the light beam was originally stored as chemical energy in the battery and now it is thermal energy added to the opposite side of the box, but in-between it was the energy of the photons. But anyway, when in its chemical or thermal energy form, it added to the mass of the box by the amount E/c*2. That mass was transferred to the other side of the box.