Chapter 5
In This Chapter
Introducing the laws of thermodynamics
Keeping energy equal on both sides of the equation
Setting up the flow of time
Understanding absolute zero
Figuring out why time runs one way
E instein began his scientific career by studying problems related to the existence of atoms and molecules. Because molecules or atoms can’t be seen, Einstein relied on a branch of physics called statistical mechanics, which uses statistical methods to study substances that are made up of a large number of components, like gases.
Statistical mechanics was developed in the 19th century in part to understand the laws ruling the way heat is transferred between objects. These laws came from another branch of physics called thermodynamics, which was also developed in the 19th century (and was fully developed by the time Einstein was in school).
Einstein’s work later helped with the development of statistical mechanics. And his theories clarified the concept of time, which is intimately related to thermodynamics and statistical mechanics.
In this chapter, I introduce you to the laws of thermodynamics and to statistical mechanics, and I explain how each served as a springboard for Einstein’s work.
Identifying the Laws of Thermodynamics
Why does time flow in only one direction? You grow older, never younger (unfortunately). Can this flow be reversed? All the laws of physics remain unchanged if the flow of time is reversed. They work equally well in either case. All the laws except for one, that is: The second law of thermodynamics spoils it for us. It appears that the flow of time originates in the second law. And it points the arrow of time firmly in one direction.
Thermodynamics is the study of heat and thermal effects. In this branch of physics, the world is divided into systems. A system can be a body or several bodies (like a gas). When you chose one system to study, all the others become part of the environment.
Thermodynamics can be summarized in four laws:
The first law: The total sum of all forms of energy of a system, including its thermal energy, must always remain constant. In an isolated system, you cannot create or destroy energy; you can only change it from one kind to another. This law is also called the principle of conservation of energy. Check out the next section, “Conserving energy at all costs: The first law,” for details.
The second law: On its own, heat does not pass from cold to hot. Heat passes from hot to cold, but never the other way around without help in the form of some kind of energy. Take a look at the section “Mess, laws, and videotape: The second law” later in this chapter for more details.
The third law: An absolute zero temperature cannot be reached. Temperature is a measurement of the average speed of the molecules of a substance. Try as we might, we can’t get molecular motion to completely stop. Read the section “That’s cold! The third law” later in this chapter for the story on why that is.
The zeroth law: If two objects are each in thermal equilibrium (meaning they have the same temperature) with a third object (often a thermometer), they are in thermal equilibrium with one another. Because it seems so obvious, some people think this one’s not really a law of thermodynamics. For more details, see the upcoming section “Getting picky: The zeroth law.”
The second law was discovered first. The first was discovered second. The third was actually third, but it might not be a separate law of thermodynamics (because it can be viewed as an extension of the second law). And the zeroth law was described last, as an afterthought.
Conserving energy at all costs: The first law
What is energy anyway? Energy is one of those physics words that gets into the everyday language and loses the rigor of its meaning. In physics, energy means the ability to do work. If you have energy, you can do work. If you do work, you are using some form of energy.
Living force
In the 16th century, energy was called living force, because people thought that only living things could perform work. In the 18th century, the English physicist Thomas Young realized that nonliving things, like the wind, could also do work, such as by moving a windmill or a sail ship. He invented the word energy from Greek words that mean “work within.”
Anything that changes anywhere in the universe requires energy. The explosion of a distant star, the roar of a tornado, the fall of a feather from a flying bird — all require the exchange of some form of energy.
Many kinds of energy exist, including thermal energy, mechanical energy, and nuclear energy. You can change one into another, but you can’t destroy or create any of it. When you are done changing energy in some sort of process, you’d better come out even. What you end up with must equal what you started out with.
Physicists call this rule the principle of conservation of energy. And they defend it with their lives. They are not willing to violate it for any reason.
Einstein’s famous equation, E = mc2, actually extended the principle of conservation of energy to include mass, the m in the equation. (Mass is a measure of the resistance that you feel when you try to change the motion of an object. Unlike weight, it’s not dependent on gravity.) His equation, which he published in the last paper of his miracle year as a follow-up to his special relativity paper (see Chapter 3), says that mass and energy are equivalent. Mass is a form of energy, and energy is a form of mass.
Consider this example: If you combine one kilogram of hydrogen with eight kilograms of oxygen to form water, you generate enough energy to run your hair drier for ten hours. If you had an extremely precise scale, you would discover that after running the hair drier that long, you would be missing five-billionths of a kilogram (out of the nine kilograms you originally put in). The missing mass is the energy that you used to run your hair drier.
This well-known chemical reaction is one example of the conversion of mass into energy. A nuclear reaction provides a more dramatic example. In this case, the same amount of hydrogen could produce 10 million times more energy, all of it coming from the conversion of mass into energy, according to Einstein’s equation. This reaction generates the energy that powers the sun.
Technically, Einstein’s extension of the principle of conservation of energy has universal validity. But in practice, physicists don’t have to include it in their calculations unless they are dealing with nuclear processes, such as when they study the physics of the sun or the energies produced in nuclear reactors. Why don’t they use it in all their calculations? Consider this example: The increase in the mass of the Space Shuttle due to its motion in orbit is so small that you couldn’t measure it with the most accurate instruments. The uncertainties in the measurements would be larger than the energies that you would be trying to measure.
The first law of thermodynamics is actually a restatement of the principle of conservation of energy, with heat or thermal energy explicitly included for clarity.
The one who dares, wins
In November 1929, the great German physicist Wolfgang Pauli wrote to his colleagues who were gathered at a scientific meeting that he wasn’t able to attend, expressing some concern about discussions related to some missing energy that had been observed in certain processes. In his letter, he said that he’d discovered a “desperate way” to solve the problem of the missing energy. He proposed that there must be a new particle — later called the neutrino — that was taking away the missing energy.
He thought the idea was bold and didn’t want to publish anything on it, since these neutrinos, if they existed, should’ve been detected already. “But only he who dares, wins,” he said.
Pauli won. The neutrinos were discovered 25 years later. Today, we know that they play an important role in the mechanism that makes the sun work.
After Pauli’s daring assumption, physicists adopted his approach of predicting the existence of new particles by measuring the missing energy in some process. The method works because of the principle of conservation of energy.
Mess, laws, and videotape: The second law
The second law of thermodynamics is one of the most significant principles in all of physics. It can be stated very simply: On its own, heat does not pass from cold to hot. But the implications of this simple law are immense.
Dictating the flow of time
This law explains why a swimming pool doesn’t suddenly freeze in the middle of a hot summer day, releasing heat to its surroundings and then solidifying into ice. No other law of physics prohibits this event from happening.
The second law is also the only law that forbids a baseball sitting on the ground from suddenly jumping across a field and hitting a baseball bat, forcing it to swing back in the batter’s hands as the ball bounces back to the pitcher’s glove.
You can actually see these and other similar “forbidden” phenomena if you record video of them and then run the video in reverse. In a very direct way, the second law sets a direction for the flow of time.
Moving toward disorder
The flow of time is related to disorder. Even when you’re not bringing in new things, your home gets disorganized every day. You have to spend a good many hours to reverse this unstoppable trend. Apply that idea to a much larger space, the universe. It’s been getting disorganized from the beginning, and there is no end in sight.
The term entropy measures the degree of disorder. All natural changes take place so that the entropy increases. As a result, the entropy of the universe is constantly increasing.
Why is entropy always increasing? The reason is actually simple: If there’s one right way to put something together, there are many wrong ways to do it. For example, if you pick up all the pieces of a jigsaw puzzle that you accidentally dropped on the floor and quickly put them right side up on the table, you don’t expect the puzzle to be put together the right way. There is only one correct way to put the pieces together and very many incorrect ways. Chances are you won’t put them together the right way the first time.
Think about organizing your house. The number of ways you’d like to arrange things is probably fairly small. When things aren’t set up the way you like them, they are in disorder. Nothing drives things into disorder. It’s just that there are many possible ways of putting them together, and you only like a few.
The same idea is true for molecules. If you drink a can of soda that you left on the kitchen counter for a few hours, it tastes flat. That’s because the carbon dioxide molecules that were trapped along with the air molecules above the liquid in the can have left. With time, additional carbon dioxide molecules that were inside the liquid found their way to the surface and also left. Eventually, little carbon dioxide is left in the liquid — the carbonation is gone. You can wait all you want, but the carbon dioxide molecules won’t make it back to the open can to refresh the taste. They are now mixed with the air molecules in one of the many possible ways they can mix.
If you could somehow videotape the carbon dioxide molecules leaving the soda and then run the tape in reverse, you’d see that there is a directionality of time here too. It’s the second law of thermodynamics at work.
This constant increase in entropy sets the direction of the flow of time. In real life, we never see what our videos running backwards show us. Time flows in the direction of increasing entropy.
The entropy of a black hole
In 1970, Jacob Bekenstein, then a graduate student at Princeton University, suggested that black holes have entropy and that their entropy increases as matter falls into them. (Black holes are objects so dense that not even light can escape.) At the time, the idea was a strange one, because entropy is the measure of the degree of disorder, and black holes didn’t seem to have much in them that could be in a state of disorder.
But Bekenstein thought that as stars and other matter fall into a black hole, they bring their entropy with them, leaving the rest of the universe with less entropy. If you happened to be near a black hole, you could dump all your broken, messy things into it. Your spaceship would be very organized after that, and its entropy would decrease. The second law of thermodynamics requires that the black hole have entropy, which would increase by the same amount that yours decreases.
The famed British physicist Stephen Hawking had shown that the surface of no return in a black hole increases when stars fall into it. Bekenstein concluded that the value of the surface of no return in a black hole was a measure of the black hole’s entropy.
That’s cold! The third law
The third law of thermodynamics says that an absolute zero temperature cannot be reached. The temperature of an object is a measure of the average speeds of its molecules. When the object is warm, it has more molecular motion than when it’s cold. When you place a thermometer in contact with a warm body, the molecules of that body collide with the thermometer, transmitting some of their motion. The thermometer then registers the temperature according to the calibration and the scale the thermometer uses.
At lower temperatures, the molecular speeds are slower, and a thermometer registers fewer and less energetic collisions. If you keep cooling the object, the molecules slow down even more. You can see that at some point, molecular motion and collisions shouldstop. If you could measure the temperature at which molecular collisions should stop, your thermometer would register –273˚C or –480˚F. In 1848, the English physicist William Thomson, known as Lord Kelvin, thought it would make more sense to have a scale with 0 for the lowest possible temperature, so he invented it. We call this temperature absolute zero or zero Kelvin (also 0 K or zero K).
How do we know that absolute zero is the correct value for the minimum temperature, since no thermometer can measure it? Measuring lower and lower values of the temperature, physicists can project what this value should be. Accurate and precise instrumentation at places like the National Institute for Standards has given us a very precise value for zero K.
You may notice that I’ve been careful to say that molecular motion should stop at absolute zero. It turns out that, in reality, molecular motion never stops. Molecules are small bodies whose motion can be studied only with quantum mechanics, the branch of physics that started with Einstein’s introduction of the quantum of energy in 1905.
Quantum mechanics tells us that molecules always keep a minimum amount of energy, called the zero point energy, that can’t ever be removed. As I explain in Chapter 16, there is an interplay between the speed of a molecule and its position in space. The more you try to fix one of these related quantities to a definite value, the larger the other becomes.
The more you slow down a molecule, the more room it needs for its position in space. Needing more room, it bumps into nearby molecules, gaining speed back from these collisions. If you try to stop a molecule completely, its position in space becomes infinite, meaning that it can be anywhere in the universe. But in this case, it can’t avoid bumping into other molecules and regaining speed. So you really can’t reach absolute zero.
Another way to look at this subject is from the point of view of entropy. At absolute zero, the molecules of a system would occupy the ultimate ordered state, where nothing moves and nothing gets out of place — a state where the entropy would actually decrease to zero. Seen this way, the third law is actually an extension of the second law.
Getting picky: The zeroth law
After the first three laws of thermodynamics were discovered, some picky physicists started thinking that there ought to be a law that told them when two objects were at the same temperature. The physicists wanted to formalize the process of determining when two objects are in thermal equilibrium with one another. Thermal equilibrium means that no heat is transferred between the objects; it also means they are at the same temperature.
The three other laws had been in use for some time, and people were used to calling them the first, second, and third laws. But this new law was so basic that it couldn’t logically follow the other three; it had to precede them. So the scientists decided to call this new law the zeroth law. It simply states that if two objects are each in thermal equilibrium with a third object (the thermometer), they are in thermal equilibrium with one another.
Following the Arrow of Time
The only thing that stands in the way of reversing the arrow of time is entropy, the second law of thermodynamics. Nothing in electromagnetism or mechanics says that you can’t reverse the flow of time.
Running the movie backwards
If you record a video of the collision of two billiard balls and show the video running backwards to your friends, they won’t be able to tell the difference. If you had a special videotape that could see at the atomic level, you could do the same thing with two molecules bouncing around in a box (see Figure 5-1).
|
Figure 5-1: Which collision really took place? You can’t tell. Both ways are possible. |
|
If you add a third molecule, your movie would also look the same when you run it in reverse. A fourth molecule won’t give it away either. Or a fifth, and so on.
Now, imagine that you have a container with two compartments separated by a wall (see Figure 5-2). If you fill one compartment with a gas and then make a hole in the wall, the gas will begin to expand to the empty container.
Show your movie in reverse, and your friends will immediately tell you that you’re running it backwards. They know that you can’t start with a gas filling the two sides and eventually have the gas, on its own, empty one side and confine itself to the other side.
But your friends wouldn’t be able to tell the difference if you had only two or three molecules. The molecules would bounce around and pass through the hole in either direction. At times, the three molecules would all be on one side. At other times, they’d be on the other side. And at other times, you’d have molecules on both sides. Either way you showed the movie, it’d look the same.
|
Figure 5-2: Gas will move from a full to an empty compart-ment, but the reverse isn’t true. |
|
If you have ten molecules, you’d have to wait a longer time to find them all in one side only. With fifty, you’d better bring popcorn and maybe even a pillow; it’d take many hours for that to happen.
With a million molecules, you’d wait your entire life. And with a real gas . . . hell would freeze over.
You can see that there is a clear contradiction when you add molecules. With a few, you have a movie that you can show in reverse. With many, your movie is irreversible. Physicists actually call these situations reversible and irreversible.
Unpopping the cork: Statistical mechanics
Scientists originally discovered the laws of thermodynamics by studying the behavior of gases under different environmental conditions. These laws explained the observations, but no one knew exactly why they worked. The understanding came after scientists began to study the behavior of the atoms and molecules that make up these gases.
Because the number of molecules in a real gas is extremely large, and because molecules are extremely small, physicists realized that you can’t possibly study the behavior of individual particles. But you can use statistical methods to discover their properties. That’s what the Austrian physicist Ludwig Boltzmann did.
Boltzmann began thinking about the contradiction between the motion of a few molecules and the motion of the millions of molecules in a gas. After obtaining his doctorate in physics in 1866, Boltzmann became interested in statistical mechanics. This branch of physics was being developed by James Maxwell and others to use statistical methods in the study of substances that were made up of a large number of components, like gases, for example.
Boltzmann believed that the contradiction would be resolved if you could actually wait for an unimaginable length of time. Statistically, a huge number of other possible combinations of the many molecules take place before one particular combination — the one with all the molecules moving to one side — happens. When that combination occurs, you end up in the future with something you had in the past.
I realize that you probably don’t care to wait an unimaginable length of time for a gas to move over to one side of a box. But what if you want recover the molecules that have escaped from that nice bottle of champagne that you left open? Or what if you want to relive a specific unforgettable afternoon, and to do so you need all the molecules of air and those that make up the trees and grass and the smell of the ocean breeze to come back to the same positions they had that day? (Of course, all the molecules in the nice configuration that made up your gorgeous date would need to return as well.)
If you could accomplish either scenario, you would be traveling back in time.
So, how long would it take to get back that nice bottle of champagne? A number of years so large that you would need 60 zeros to write it down. (In comparison, the universe has existed for 18 billion years — a number that needs only 9 zeros for you to write it down.)
Face it: Your champagne is gone. And your gorgeous date. They’re things of the past. You are growing older by the minute, and there’s nothing you can do about it. There is no traveling back in time. The second law of thermodynamics and statistical mechanics are the culprits.
But don’t give up on time travel yet. Einstein will give us other opportunities.
Bringing Einstein into the Equation
So now you know something about thermodynamics, statistical mechanics, and the relationship between the two. What does all this have to do with Einstein?
Einstein’s first foray into the scientific world had to do with both thermodynamics and statistical mechanics. In his first two professional papers, published soon after graduating from college, he used thermodynamic arguments to explain several effects observed in liquids. In his second paper, he dealt with the atomic foundations of thermodynamics.
In his Brownian motion paper of 1905 (see Chapter 3), Einstein used Boltzmann’s statistical techniques to explain the zigzag motion of particles of smoke. The methods that he used in this paper paved the way for the further development of statistical mechanics.
In 1924, Einstein became interested in the work of the Indian physicist Saryendra Bose (see Chapter 16). Using a new way of counting particles that he invented, Bose was able to derive Max Planck’s formula for the radiation of bodies (see Chapter 15). Einstein extended Bose’s work and applied it to atoms and molecules. This new method became part of the modern development of quantum statistical mechanics. With their new method, Einstein and Bose predicted the existence of a new state of matter, the Bose–Einstein condensate. This new state of matter was discovered recently, as I explain in Chapter 16.
