About Thermodynamics

I might as well try to explain the laws of thermodynamics to a bunch of labradoodles.

Sheldon Cooper

 

 

In his lecture The Two Cultures, C.P. Snow, an English physicist and novelist, laments:

A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics, the law of entropy. The response was cold; it was also negative. Yet I was asking something which is about the scientific equivalent of: “Have you read a work of Shakespeare’s?”
I now believe that if I had asked an even simpler question – such as, “What do you mean by mass, or acceleration?”, which is the scientific equivalent of saying, “Can you read?” – not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their Neolithic ancestors would have had.

Although the laws of thermodynamics are among the most fundamental known to science, we have now almost reached the state where even a Master of Chemistry can be completely unaware of the reason chemical reactions run in a given direction.

Next to classical mechanics, the development of the understanding of temperature, heat, and work and the relations between them, which took the better part of the 18th and 19th century, can be viewed as an outstanding achievement of science. Since the main laws of thermodynamics were accepted sometime in the middle of the 19th century, they are considered so fundamental that almost no scientist believes they will ever be violated. And in spite of C.P. Snow’s remark, thermodynamics has penetrated into popular culture. This is evident from the quote given at the beginning of this page, but also from the Simpsons episode The PTA Disbands in which the school teachers go on strike, and the children are sent home. After a few days, Marge and Homer are in bed when the following exchange takes place:

Marge: I’m worried about the kids, Homey. Lisa’s becoming very obsessive. This morning I caught her trying to dissect her own raincoat.
Homer: [scoffs] I know. And this perpetual motion machine she made today is a joke! It just keeps going faster and faster.
Marge: And Bart isn’t doing very well either. He needs boundaries and structure. There’s something about flying a kite at night that’s so unwholesome. [looks out window]
Bart: [creepy voice] Hello, Mother dear.
Marge: [closing the curtains] That’s it: we have to get them back to school.
Homer: I’m with you, Marge. Lisa! Get in here. [Lisa walks in, chuckling nervously] In this house, we obey the laws of thermodynamics!

I have no idea why flying a kite at night should be detrimental to physical, mental, or moral well-being, but it must be a minor offense compared to overthrowing the two most fundamental laws of physics, and Lisa’s machine does just that. Not only does it keep running, but it keeps going faster and faster, suggesting that it can not only convert heat into work completely, but also create energy from nothing. Unfortunately the movie does not show us clearly how the machine works. Apparently there is a lamp involved, and something that looks like an anchor escapement. Feynman’s suggestion for the Brownian motor comes to mind, an imaginary device for converting thermal motion into useful work. However, even if such a motor would work, it would slow down rather than speed up in time. Homer does not have to worry though, and he does not have to enforce the laws in his home. The structure of the universe we live in is such that they are obeyed automatically for every real process or machine.

The two most fundamental laws of physics can be expressed simply. The first law states that energy can not be created or destroyed, and the second law is in fact nothing but a restatement of the observation that heat flows spontaneously only from warm to cold. No exceptions to these laws have ever been found. This is not for lack of trying. Since the dawn of mankind the idea of a free lunch has been a powerful one, but so far all the evidence shows there is indeed no such thing. Innumerous devices have been suggested that violate one of the two laws, or both, but no one has been able to build an actual working contraption. The US patent office has made it an official policy since 1895 that patents for perpetual motion machines are not granted without a working model. This has not at all stopped inventors from trying though, so far without success. A few even managed to slip through the system and were able to get patents. Mostly these are only used to get money for further development from gullible investors.

I gave a very simple formulation of the second law, most people will know it as the law that states that entropy can only increase. Expressing it in that way is dangerous and has led to many misconceptions, the most common of which is that entropy is disorder and disorder can only increase in the course of time. I wish to state emphatically that this is untrue. The argument is often used by creationists and intelligent designers who claim that Darwin’s theory of evolution is in contradiction with the second law. This is not the case at all. The second law leaves plenty of room for increasing complexity in a small part of the universe, provided the rest becomes more disordered in the process.

Thermodynamics is the science of heat and work. In the 17th century scientists, in those days often rich or noble people with a lot of time on their hands, started thinking about the behavior of gases, temperature, and the flow of heat. Direct conversion of heat into work became possible with the invention of the steam engine, and the question arose about the amount of useful work that could be obtained from such an engine. Starting in the beginning of the 17th century the thermometer was developed, and people like Boyle, Gay-Lussac, Charles, and Lavoisier found what are now known as the ideal gas laws. At first there was still a lot of confusion about the true nature of heat and work, but in the middle of the 19th century these concepts became fixed, and their properties were studied more systematically. By the middle of the 19th century the two main laws of thermodynamics were firmly established, but it still took until the end of that century, and the work of Boltzmann, Maxwell, and Gibbs, to get sufficient insight into the physical background of entropy.

Entropy remains a somewhat mysterious quantity. Based on the study of cyclic processes in combination with the principle that heat only flows spontaneously from warm to cold, the Frenchman Sadi Carnot was, in the early 19th century, able to derive that every system can be assigned a number that only depends on its state, not its history. Such properties are called state quantities. It is rather a miracle that Carnot could do this, since virtually nothing was known at the time about the microscopic properties of gases, and in addition he had completely wrong ideas of what heat actually was. He was interested in the efficiency of steam engines, and could derive a maximum value for it, but he and his successors also discovered something far more fundamental: in any spontaneous process the total entropy will always increase.

Later in the nineteenth century Boltzmann and Maxwell started developing physics based on the idea that matter consists of atoms and molecules which are in constant, thermal, motion. Motion governed by Newton’s laws, established centuries earlier. For the other important state quantity that was found later, energy, it was easy to give the microscopic equivalent: the energy of a system is just the sum of all the energies of its constituents. But entropy is much harder to understand, since the fact that it can only increase immediately leads to to a contradiction with the other laws of physics as they were known at the time. Newton’s laws of motion are invariant under time reversal, and it is absolutely impossible to find a function of the coordinates and velocities of the molecules that does not have that same property. Entropy, however, is clearly not invariant under time reversal, in the past its value can only have been lower than it is today. The later development of quantum mechanics has not helped resolve this problem. Schrödinger’s equation is equally invariant under time reversal.

The now more or less generally accepted solution is that entropy is a statistical quantity that has an overwhelming probability of increasing in time, but the discussion has by no means ended. Many books have been written about it, and complete scientific journals are devoted to it. Boltzmann’s own ideas led him to a rather depressing view of the origin of the universe we live in. He thought the total universe was infinite in time and space, and given the finite probability that entropy will decrease in a small part of it, it is only a matter of time before in some remote corner of it a relatively small area spontaneously reaches a low entropy state. For us this low entropy state would have started about 15 billion years ago and would end when so-called heat death is reached. Then we just have to wait an incredibly long time before in some other remote corner the process starts again. In an infinite time, this has happened an infinite number of times already, and will happen another infinite number of times, with all possible variations. I find that rather depressing, but nobody ever claimed with any evidence that there is a purpose to it all. Currently other models prevail, where our universe started with a low entropy big bang – why that was such an extreme low entropy state is still a mystery – and may or may not be open ended. Some of these theories are really not that far removed from Boltzmann’s ideas, especially the models where multiple big-bang-universes are embedded in a super-universe are in fact rather close.

Thermodynamics is considered the most fundamental of the sciences. The two laws apply to any system, regardless of its microscopic properties. This already follows from Carnot’s derivation, for which no microscopic model is needed. It makes the theory very powerful, since it should be applicable to anything from living systems to black holes. In fact the origin of a number of fields in physics can be traced back to problems when applying thermodynamics to it. Thus, Planck developed his ideas about the quantum starting from dissatisfaction with the implementation of thermodynamical laws to a box filled with light, and Hawking derived his theory of radiating black holes from the fact that non-radiating black holes would lead to a violation of the second law. Although Einstein followed another route in developing his general theory of relativity – he did use thermodynamics in a lot of his other papers – the fact that clocks run slower in a gravitational field also prevents the existence of an interesting perpetual motion machine.

I would like to end this introduction with a quote from the astrophysicist Sir Arthur Stanley Eddington, which states clearly how the laws of thermodynamics are currently viewed. They have not lost any of their power in the almost two centuries of their existence, and maintained their relevance in spite of all the other changes physics went through in the past century.

The law that entropy always increases – the second law of thermodynamics – holds, I think, the supreme position among the laws of nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.

A.S. Eddington, The Nature of the Physical World 

[Intended as a preface for a book on Thermodynamics I once contemplated writing]

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