Science For Everyone - Conservation of Mass

Copyright © Karl Dahlke, 2022

Consciousness is not an all or nothing proposition. My dog is conscious of the world around her, but not as conscious as I am. We both understand that the sun rises and sets, but if it didn't rise one day, she would adapt, whereas I would be flummoxed.

Early humans, a million years ago, were starting to understand their world as no animal had before. They understood without words, without language, but they knew the sun would rise, and if it didn't, panic would ensue.

With keen eyes and a developing brain, early humans witnessed wild fires from time to time, and they developed an intuitive understanding of the process. Wood disappeared, releasing tremendous amounts of heat and light, and leaving only ashes behind. Through curiosity, or practicality, or both, Homo Erectus found a way to produce fire at home, in a controlled setting, marking a turningpoint in human evolution. Fire offered heat, light, protection, and improved nutrition through cooking, which was key to feeding that hungry brain as it grew in size and complexity. Their mastery over fire enhanced their intuitive understanding of combustion, but the basic idea was the same - wood is consumed, releasing light and heat, and leaving only ash.

A million years later, humans would develop language, including a word for "consume". This reflected what humans had understood for a million years thither - the wood goes away. It is consumed by the flames, and is gone. The wisps of smoke and the modest pile of ash can hardly account for the cord of wood that stood next to the hearth just a few days before. This intuition was wrong, but it caused no trouble, thus evolution had no reason to correct it.

If atoms are uncreated and eternal, as Democritus declared, and if all matter is made of atoms, then matter can't disappear. This led to a fundamental principle of physics, the conservation of mass. In any chemical reaction, the products weigh exactly as much as the reactants, since the atoms are merely rearranged. By analogy, you could take a lego house apart, and build a lego train, and the train, along with any unused blocks from the house, weighs exactly as much as the house did before. That's pretty intuitive, isn't it? If the world is made of atoms, and if atoms are eternal, then conservation of mass almost has to be true.

So where did all the wood go? The answer had to wait for the discovery of oxygen in 1774 by Joseph Priestley and others, and the chemical revolution to follow. Oxygen combines with wood, and the resulting products are almost entirely invisible gases: steam, carbon dioxide, carbon monoxide, and sulfur dioxide. The wood literally goes into the air; it does not disappear.

A good scientist wants to confirm this in the lab, but it's not easy to do. If you contain a fire in a box, in order to properly weigh the results, the fire quickly runs out of oxygen and stops, or the box explodes from the pressure of the hot gases inside. You could use a very big box, with a lot of oxygen inside, and room for the gases to expand, but that is logistically difficult. The principle was confirmed for other reactions however, reactions that combined solids and liquids to produce solids and liquids, with no troublesome gases to burst the container at its seams. Whenever a reaction could be completely contained, the weight did not change. Mass is conserved, as the theory of atoms demands. Matter is neither created nor destroyed.

Forms of Energy

The concept of energy didn't gain traction until the industrial revolution, when heat was harnessed to do useful work. Scientists began to understand that energy comes in many forms, and can be converted from one form into another. Each conversion entailed some losses however, and efficiency was important. James Watt repeatedly improved his steam engine, whereupon later models required half as much coal as earlier models to do the same work.

Several forms of energy are listed below. Only the first two were understood by the Greeks, heat can do useful things like smelt metals, and wood has something in it that allows it to burn.

  1. Heat - a hot object has more energy than a cold object.

  2. Chemical energy - some materials store energy in the arrangements and bonds of their atoms. If the atoms are rearranged by a chemical reaction, energy is released, usually in the form of heat. Fire is the most familiar example. Like a bell ringing far far away, Homo Erectus had a vague intuition that some sort of energy was stored in the wood, and could be released as heat, if he used his flint and his dry leaves and his bark just as his father had taught him when he was a boy.

  3. State - a gas has more energy than a liquid, has more energy than a solid. This is latent heat. Water gives up heat when it freezes, even though it does not drop in temperature. Once again, this energy is a result of the arrangement of the molecules, but you don't need a chemical reaction to release it, you just need to change the state of the substance from liquid to solid.

  4. Motion - anything that moves has energy. This could be a car driving down the highway, but it can also be the molecules vibrating in a solid, wherein the energy of motion is realized as heat, or state. Moving energy is sometimes called kinetic energy, especially when it is applied to real world objects like baseballs and planets.

  5. Potential - an object has more energy when it is higher in a potential field. The most common example is gravity. A book has more energy when it is on a high shelf than when it is resting on the floor. To illustrate this, carry a large box of books up the stairs, and down the stairs, and you'll see the difference. If you're lucky enough to have a grandfather clock, wind it up, and the weights will run the clock for a week.

  6. Sound - again, anything that vibrates has energy, but sound is an extremely weak form of energy.

  7. Radiant - stand in the sunshine and you can feel the heat pouring over you, though it is not clear how the heat gets from the sun to you. Light is also radiant energy - it travels through space from point A to point B and carries energy along with it. When a tree produces an apple for you to eat, it converts light energy into chemical energy. It is the sun's light, not the sun's heat, that makes the plants grow.

  8. Electrical - energy travels through the wires of your home and runs an assortment of appliances. A battery also produces electricity on demand, thanks to the chemical energy stored inside.

  9. Radioactivity - certain atoms are unstable, and release energy as they break down into other atoms. This is one of those rare moments when an atom changes into another atom, like a 2 by 4 lego brick switching into a 1 by 8 lego brick. Democritus couldn't see that one coming.

  10. Fusion - in the center of stars, atoms can combine to make other atoms. This is one of those rare moments when atoms change into other atoms, like two lego bricks merging into one.

Energy Transformations

Follow the steps as we wind up a grandfather clock. Energy begins in the sun. Hydrogen atoms combine to make helium, and release heat in the process. The sun heats up, and stays hot as long as there is hydrogen to burn. Sunlight radiates through space and lands on a field in Kansas, where a plant converts it into wheat. The starch in wheat holds chemical energy, energy that originally came from the sun. You eat the wheat in your cereal for breakfast, digest it, and repackage the chemical energy in different compounds that are more appropriate for humans than for plants. Some energy is lost in translation; in fact some energy is lost at every step. Perhaps some of the energy was converted into heat, but that's not all bad, you're a warm blooded animal and you need to keep your body at 37 °C. Next, the chemical energy is released in the muscles of your arm as you turn the key and raise the weights from the bottom of the clock up to the top. Chemical energy has been converted into potential energy, the energy of the weights high up off the floor. Some energy is lost; the sprockets clink as you turn the key, and the muscles of your arm are a bit warmer after the exertion. Over the next week, the weights descend, and potential energy is converted into sound and heat. You know the sounds of the clock, the gentle tick tock tick tock, and the strike and the chime. The heat is so tiny you can't even measure it, but it's there, primarily in the escapement. After a week, all the potential energy has been converted into heat, even the sound dissipates into heat, and the clock is in the same state it was before, waiting for someone to come and wind it up again.

the gasoline in your car's gas tank has chemical energy. It burns in the engine to produce heat, and forward motion. The heat is a waste of energy; in a perfect world you could put your hand on the tailpipe and it would be cool as a cucumber, but there are always losses. The car contains kinetic energy as it travels down the road. It may accumulate potential energy as well, if it is driving uphill. Now there is a red light and the car must stop. Apply the brakes, and all that kinetic energy is converted into heat, the rotors and pads are hot! (Some cars have regenerative braking, which captures the kinetic energy instead of wasting it as heat, then recycles this energy to move the car forward again, but that's another story.)

A 10 year old girl is swinging in the park. At the back of her arc she is as far up as she can go. For a split second there is no motion, no kinetic energy, but there is potential energy, as she is high up off the ground. Over the next second her potential energy is converted into kinetic energy as she falls, in a controlled way, to the bottom of her arc. At the bottom she is moving forward as fast as she will go; this is kinetic energy. Momentum carries her forward and up, until she is motionless at the front of her arc, and all the energy is potential again. Energy trades between kinetic and potential in a repeating cycle as she swings back and forth. There are losses of course, as she sweeps through the air, and as the chains rub and squeak. If she doesn't pump energy back into the system, she will eventually come to rest at the bottom of the arc.

finally let's compare the gas dryer and the electric dryer. Some of the energy turns the drum, and operates the fan that blows the humid air out of your house, but most of the energy is heat, to dry the clothes, so let's focus on that. A gas dryer runs on methane that comes into your house via pipes. The methane contains chemical energy, which is released as heat when it burns. Some of this heat is wasted through the sides of the dryer, but most of it goes into the clothes, which is where it belongs. In contrast, an electric dryer entails several steps, and is far less efficient. A power plant far far away burns a fuel, coal or methane, to produce heat. The heat drives a motor, considerably more advanced than Watt's steam engine, and the motor turns a generator. The generator produces electricity, which travels through kilometers of wire to your home. In your house, electricity turns back into heat, which dries the clothes. Each step incurs losses. Even the electricity running through the wires produces waste heat. If you could touch a high voltage line overhead, and I don't recommend it, it would be warm, even on a cold winter day. The fewer steps the better, thus a gas dryer is a simpler design, burning the fuel right where the heat is needed. Similar reasoning holds for the furnace, the gas stove, and the hot water heater. If an appliance generates heat as an essential part of its operation, it is best to burn the fuel on site where it is needed, rather than 100 kilometers away.

Conservation of Energy

Conservation of energy asserts that energy cannot be created or destroyed. This is analogous to conservation of mass, wherein matter cannot be created or destroyed. Energy changes forms, but if you could account for all the losses, all the waste heat that leaks into the environment, you'd have exactly the same amount of energy from beginning to end. This is difficult to prove in the lab, but not impossible. It's easy to hold matter inside a box during a chemical reaction, to make sure mass is neither created nor destroyed, but energy leaks out of a box easily, in the form of heat, or even sound. Also, measuring energy, at the start and at the end, is not easy either. You can't just put the box on a scale and weigh it. Still, some experiments have been run, and as best we can determine, energy is conserved at every step.

Mass Energy Equivalence

the fundamental equation of Einstein's theory of special relativity, e = mc2, isn't hard to understand at a conceptual level. It describes an equivalence between mass and energy; both are different manifestations of the same thing.

Take a trip from the United States to Japan and stop at a bank to trade dollars for yen. Hand the banker 100 dollars and he gives you an astonishing ten thousand yen. You feel rich! The exchange rate is very high, and has been so for the past century. In the other direction, you might be disappointed wehn you give the banker ten thousand yen and all you get back is 100 dollars.

People do not experience the equivalence between mass and energy in the real world because the exchange rate is humongous, ginormous, redonkulous! You can't perceive or even measure the miniscule changes in mass that produce energy. einstein's equation establishes the exchange rate as the speed of light squared. e stands for energy, m stands for mass, and c is the speed of light, hence e = mc2. Turn mass into energy and you must multiply by c2. Turn energy back into mass and you must divide by c2. The numerical value for the speed of light depends on the units, but in the metric system it is 3e8 m/s, or 300 million meters per second, or 300,000 kiilometers per second. This number is multiplied by itself to give the exchange rate. If mass is measured in kilograms, and energy is measured in joules, the exchange rate is 9e16, or 90 quadrillion. A tiny bit of matter is equivalent to a huge amount of energy. No wonder nobody noticed. This was not detected experimentally, or even imagined - everyone thought matter and energy were separate worlds. Einstein discovered the equivalence just by thinking about it, including a complete derivation of why it is so. The idea is amazing, and the mind that conceived it is even more amazing.

Let's illustrate with an example. On a good day, a double A battery can deliver 18 kilojoules of energy. This is 18,000 kilograms meters2 / seconds2. Notice that Einstein's formula also gives the proper units, mass times velocity squared. It's always good to check your units. Divide 18 kilojoules by c2 to see how much mass is converted into energy as the AA battery drains.

1.8E3 / 3E82 = 2E-14

The battery is 2E-11 grams lighter after it has discharged its energy. A tiny bitt of mass, 20 picograms, is converted into energy. Nobody's going to notice that! But where did the mass come from, and where did it go? Atoms have not disappeared within the battery, everything remains sealed within its metal jacket. Somehow, the resulting compounds inside the battery, after discharge, are just a bit lighter than the reactants. Rearrange the atoms to produce electricity, and they are, collectively, lighter than they were before. The same number and type of atoms weigh less when rearranged. We can't measure the mass of a battery down to the picogram, so this cannot be directly verified in the lab. A single bacterium weighs a picogram, so if a few bacteria float through the air and land on the battery, or if a few bacteria fall off, or if a microscopic fleck of paint peals away from the casing, that overshadows the effect we are trying to measure. Furthermore, if the battery is even a tiny bit warmer, or cooler, than it was at the start, it expands or contracts, and displaces more or less air in our atmosphere, and that changes its in-the-air weight. You would have to isolate it in a perfect vacuum, and find somee way to weigh it as it remained isolated from the rest of the world. But you can't isolate it from the moon. If the battery discharges in 6 hours, the moon moves in the sky, from low tide to high tide or vice versa, and that changes the apparent weight of the battery. Somehow we need to measure the mass, not the weight, while controlling for every other variable. This is a tremendous engineering challenge, if it can be done at all.

If the battery is rechargeable, then replenish its charge, thus pouring energy back into it - and its mass increases by 20 picograms, back to its original mass. The atoms inside have been pushed back into their original configuration, and the arrangement of those same atoms, collectively, has more chemical energy, and more mass.

To illustrate, imagine a pile of red lego blocks and a pile of white lego blocks. When the red blocks are on top the battery is charged, and contains chemical energy. Red blocks fall to the bottom as the battery generates electricity, and eventually all the red blocks are on the bottom and the battery is drained. It has less energy, and the collection of blocks is 20 picograms lighter. It's the same blocks, just rearranged, yet somehow the collection is lighter because its energy is spent. Put the energy back, by recharging the battery, and the red blocks move back to the top, and 20 picograms of mass magically reappear.

Heat an iron bar from room temperature up to red hot, and it has more mass, because it has more energy. There aren't any more atoms in the bar; the iron atoms are simply moving faster, and that causes them to have more mass. However, if you weigh it with a traditional scale in the open air, it weighs less. The bar has expanded with heat, and displaces more air in our atmosphere, thus it weighs less. Again, you want to weigh it, before and after, in a perfect vacuum and in isolation from everything else.

A spring that is compressed weighs more than a spring that is fully extended.

A baseball hurtling toward you at 100 mph, 44 meters per second, from the hand of Nolan Ryan, weighs more than a baseball at rest. How much more? About 0.000000000001% more.

A book on a high shelf weighs more than the same book resting on the floor.

Two magnets held apart weigh more than those two magnets slapped together.

A cup of water at 0 °C weighs more than the equivalent block of ice, even though its temperature has not changed. It has more energy in the form of latent heat, thus more mass. How much more? About 52 picograms.

A log, plus the oxygen needed to burn it, weighs a little more than the resulting ashes and smoke and gases, since energy has left the system in the form of heat and light.

Einstein introduced a new principle of physics that replaces the conservation of mass, and the conservation of energy, with the conservation of mass energy. If you burn a log inside a box, and keep all the matter and all the energy inside, all the heat and all the light, it weighs the same. But if you let the heat escape, which it almost certainly will over time, the mass decreases accordingly, though it only decreases by a tiny, tiny bit. That tiny bit is heat divided by c2.

Stepping up to a higher level, the atomic bomb that fell on Nagasaki converted one gram of mass into energy. Forget the math and try to gain an intuitive understanding for the enormous exchange rate. Find a piece of paper and hold it in your hand. Feel it's weight pressing ever so slightly against your hand. If the mass of that paper was converted entirely into energy it would obliterate a medium sized city and kill 100,000 people.

This is paltry compared to a star. Each second the sun converts 4 million metric tons of matter into energy. In other words, the sun is 4 million tons lighter each second, as it radiates its heat and light out into space. (The sun is 227 tons, so the decrease in mass, even over several billion years, is negligible, about 0.1%.)

The Speed of Light

If light were confined to a tube, it would race around the earth 7 times a second. That is unimaginably fast. Indeed, everyone thought the speed of light was infinite until the early 1600's. When something happens you see it immediately, wherever you are.

Sound is fast, faster than the wind, faster than any animal can run or any bird can fly, but it clearly has a speed. Shout towards a mountain, or into a canyon, and the echo returns a half second later. During a thunderstorm, the lightening is seen immediately, and the thunder is heard later. If the lightning is far away, the thunder is heard softly, after a substantial delay. If the lightning is close at hand, the thunder is loud, and cracks in less than a second. Count off seconds as you wait for the thunder, and measure the distance to the lightning; 5 seconds a mile, or 3 seconds a kilometer. Thus the speed of sound was understood at an intuitive level, but light seemed infinitely fast.

Galileo showed that light has a finite speed, but he needed large distances to prove it, distances from one planet to another. The next chapter introduces astronomy, in case you need a break from atoms and heat and chemical reactions, and culminates in a crude estimate of the speed of light.

If the speed of light were infinite, then the conversion of one fleck of mass into energy would blow up the universe, as per e = mc2. If it were finite, but significantly faster than it is today, the reactions in your body would produce so much heat that you would blow up. If it were slower, the mass of the sun would produce less energy. There wouldn't be enough fuel for the sun to shine for a billion years, and there wouldn't be enough time for us to evolve. The speed of light is one of those constants that has to be what it is for us to exist.