# Mass, Energy, and the Sun

These days pretty much everyone knows that mass and energy are two sides of the same coin, as discovered by Einstein. In fact this is so well known that the average man on the street – knowing nothing at all of physics – would still recognize the expression even if he didn’t know what it meant or how to use it:

E is energy, m is mass, c is the speed of light. As with all such formulas, you need to have properly matching units. You can’t just use mass in pounds, e in British thermal units, and speed in furlongs per fortnight. But any properly consistent set of units will work; for macroscopic purposes we usually use joules, kilograms, and meters per second.

Since the speed of light c = 299,792,458 m/s, the speed of light squared is about 8.99×1016 m^2/s^2. As such one kilogram of mass is the equivalent of a ridiculously huge amount of energy, as we see in nuclear reactors and weapons. It’s a little less commonly understood that this isn’t just true for nuclear reactions, it’s true for everything. Burn some logs in your fireplace and the total mass of the wood and oxygen will have been just microscopically higher than the final mass of the combustion products. The remainder has gone into the energy released to warm your house. It’s the very same equation on a much smaller scale.

To get a specific number for this, we need to know how much energy is released by wood. Some googling seems to indicate a figure around 15 million joules per kilogram, so if we assume the burning of 10 kilograms of wood, we get 150 million joules total. Divide that by the speed of light squared and you’ll get a mass of about 1.7 micrograms. Very tiny, much smaller than your average sand grain.

In fact the total solar power output absorbed by the earth is about 1.8 17 watts. Over the course of a year, this comes out to a staggering 5.6824 joules. It’s a huge amount of energy, which you might expect since it’s responsible for warming the earth, driving the weather, and powering essentially the entire biosphere. Dividing that by c^2 and you’ll see that a total of about 63 million kilograms of solar mass is converted into sunlight each year in order to illuminate the earth. It sounds like a lot, but the same mass of water would fit in a cube 40 meters on a side.

The sun is quite a bit bigger than that, and though it doesn’t come close to achieving perfect conversion efficiency, it probably won’t be running out of nuclear fuel any time soon.

1. #1 himik
November 2, 2009

Matt–

I enjoy reading your blog, but some of this post is not entirely correct in my opinion. E=mc2 works if you want to calculate the energy released in nuclear reactions, but for chemical reactions (like burning of wood) it does not really work. Mass is conserved in chemical reactions and the release (or uptake) of energy is determined by the difference in enthalpies of formation between the products and reagents. In some cases, it can be simplified even more to differences in bond energies.

One can always calculate the equivalent mass of a certain amount of energy released during the reaction, but there is no change in physical mass in chemical processes.

2. #2 D. C. Sessions
November 2, 2009

Dividing that by c^2 and you’ll see that a total of about 63 million kilograms of solar mass is converted into sunlight each year in order to illuminate the earth.

Consider that the sun converts about 4 million tonnes every second. Puts it all into perspective, eh? One sixth of a second of the sun’s output powers the Earth for a year.

3. #3 CCPhysicist
November 2, 2009

You should really do it for Halloween candy or a Big Mac. The former was the subject of a VERY funny Foxtrot cartoon on Sunday.

@1: Of course it is true. Binding energy is binding energy, whether your balance can measure it (nuclear binding energy) or not. What do you think “bond energy” is, a tiny spring? Coulomb forces? Well, the latter is a big part of why the masses are less after fission than before.

Mass number, the number of baryons, is what is conserved in chemical reactions. You just don’t notice the change in mass because it is so small.

4. #4 CCPhysicist
November 2, 2009

Nitpick: You need to count the mass of oxygen as well when burning wood.

5. #5 Anthony
November 2, 2009

Actually, chemical reactions have a change in mass too, it’s just that the change in mass is generally below the resolution of any practical measurement device. A fairly energetic reaction might be on the order of 9 megajoules per kilogram, resulting in a mass change of one part in 10^-10.

6. #6 Uncle Al
November 2, 2009

Wait! Wait! It’s worse than that! BINDING ENERGY! The numbers of electrons, protons, and neutrons in a fission or fusion warhead immediately before and immediately after detonation remain unchanged. No mass-energy conversion there! It’s only a reshouffling of nuclear binding energies.

And worse still. 1.74 solar-mass pulsar (neutron star) PSR J1903+0327 has 27% of its disassembled baryon mass missing as gravitational binding energy. That’s how its equator spins at 11% of lightspeed without fragmenting the mass. The sun is “missing” 1.4×10^(-4)% of its mass as gravitational binding energy. It is a small fraction but a big number.

The first comment, that E = mc^2 does not describe chemistry is, of course, wrong. Fractional eV or 27% overall, E = mc^2.

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