States Of Matter David Goodstein Pdf

[Noah Casanova]

There is a good deal of doubt about precisely what applied physics is, but a reasonably clear picture has emerged of how an applied physics student ought to be educated, at least here at Caltech. There should be a rigorous education in basic physics and related sciences, but one centered around the macroscopic world, from the atom up rather than from the nucleus down: that is, physics, with emphasis on those areas where the fruits of research are likely to be applicable elsewhere. The course from which this book arose was designed to be consistent with this concept.

The course level was designed for first-year graduate students in applied physics, but in practice it has turned out to have a much wider appeal. The classroom is shared by undergraduates (seniors and an occasional junior) in physics and applied physics, plus graduate students in applied physics, chemistry, geology, engineering, and applied mathematics. All are assumed to have a reasonable undergraduate background in mathematics, a course including electricity and magnetism, and at least a little quantum mechanics.

All the problems that appear at the ends of the chapters were used as either homework or examination problems during the first three years in which the course was taught. Some are exercises in applying the material covered in the text, but many are designed to uncover or illuminate various points that arise, and are actually an integral part of the course. Such exercises are usually referred to at appropriate places in the text.

There is an annotated bibliography at the end of each chapter. The bibliographies are by no means meant to be comprehensive surveys even of the textbooks, much less of the research literature of each field. Instead they are meant to guide the student a bit deeper if he wishes to go on, and they also serve to list all the material consulted in preparing the lectures and this book. There are no footnotes to references in the text.

Parts of the manuscript in various stages of preparation have been read and criticized by some of my colleagues and students, to whom I am deeply grateful. Among these I would like especially to thank Jeffrey Greif, Professors Т. C. McGill, C. N. Pings and H. E. Stanley, David Palmer, John Dick, Run-Han Wang, and finally Deepak Dhar, a student who contributed the steps from Eq. (4.5.20) to Eq. (4.5.24) in response to a homework assignment. I am indebted also to Professor Donald Langenberg, who helped to teach the course the first time it was offered. The manuscript was typed with great skill and patience, principally by Mae Ramirez and Ann Freeman. Needless to say, all errors are the responsibility of the author alone.

Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics.

Perhaps it will be wise to approach the subject cautiously. We will begin by considering the simplest meaningful example, the perfect gas, in order to get the central concepts sorted out. In Chap. 2 we will return to complete the solution of that problem, and the results will provide the foundation of much of the rest of the book.

The perfect gas is a large number of particles in the same box, each of them independently obeying Eqs. (1.1.1) to (1.1.4). The particles occupy no volume, have no internal motions, such as vibration or rotation, and, for the time being, no spin. What makes the gas perfect is that the states and energies of each particle are unaffected by the presence of the other particles, so that there are no potential energies in Eq. (1.1.1). In other words, the particles are noninteracting. However, the perfect gas, as we shall use it, really requires us to make an additional, contradictory assumption : we shall assume that the particles can exchange energy with one another, even though they do not interact. We can, if we wish, imagine that the walls somehow help to mediate this exchange, but the mechanism actually does not matter much as long as the questions we ask concern the possible states of the many-particle system, not how the system contrives to get from one state to another.

This we shall do and, in the course of so doing, try to develop some understanding of entropy, irreversibility, and equilibrium. Let us outline the general ideas briefly in this section, then return for a more detailed treatment.

1. If we wait long enough, the initial conditions become irrelevant. This means that whatever the mechanism for changing state, however the particles are able to redistribute energy and momentum among themselves, all memory of how the system started out must eventually get washed away by the multiplicity of possible events. When a system reaches this condition, it is said to be in equilibrium.

The fact is that our assumptions do lead to sensible behavior. The reason is that although the individual states are equally likely, the number of states with energy more or less fairly shared out among the particles is enormous compared to the number in which a single particle takes nearly all the energy. The probability of finding approximately a given situation in the box is proportional to the number of states that approximate that situation.

The two assumptions we have made should seem sensible; in fact, we have apparently assumed as little as we possibly can. Yet they will allow us to bridge the gap between the quantum mechanical solutions that give the physically possible microscopic states of the system and the thermodynamic questions we wish to ask about it. We shall have to learn some new language, and especially learn how to distinguish and count quantum states of many-particle systems, but no further fundamental assumptions will be necessary.

Let us defer for the moment the difficult problem of how to count possible states and pretend instead that we have already done so. We have N particles in a box of volume V = L, with total energy E, and find that there are Γ possible states of the system. The entropy of the system, S, is then defined by

Thus, if we know Γ, we know S, and Γ is known in principle if we know N, V, and E, and know in addition that the system is in equilibrium. It follows that, in equilibrium, S may be thought of as a definite function of E, N, and V,

Furthermore, since S is just a way of expressing the number of choices the system has, it should be evident that S will always increase if we increase E, keeping N and V constant; given more energy, the system will always have more ways to divide it. Being thus monotonic, the function can be inverted

How do we expect a temperature to behave? There are two requirements. One is merely a question of units, and we have already taken care of that by giving the constant k a numerical value; T will come out in degrees Kelvin. The other, more fundamental point is its role in determining whether two systems are in equilibrium with each other. In order to predict whether anything will happen if we put two systems in contact (barring deformation, chemical reactions, etc.), we need only know their temperatures. If their temperatures are equal, contact is superfluous; nothing will happen. If we separate them again, we will find that each has the same energy it started with.

During the time that the two systems are in contact, the combined system fluctuates about among all the states that are allowed by the physical circumstances. We might imagine that at the instant in which contact is broken, the combined system is in some particular quantum state that involves some definite energy in box 1 and the rest in box 2; when we investigate later, these are the energies we will find. The job, then, is to predict the quantum state of the combined system at the instant contact is broken, but that, of course, is impossible. Our fundamental postulate is simply that all states are equally likely at any instant, so that we have no basis at all for predicting the state.

Now suppose that, while contact exists, energy flows from box 1 to box 2. This flow has the effect of decreasing Γ1 and increasing Γ2. By our argument, we are likely to find that it has occurred if the net result is to have increased the total number of available states Γ1 Γ2, or, equivalently, the sum S1 + S2. Obviously, the condition that no net energy flow be the most likely circumstance is just that the energy had already been distributed in such a way that Γ1 Γ2, or S1 + S2, was a maximum. In this case, we are more likely to find the energy in each box approximately unchanged than to find that energy flowed in either direction.

It is easy to show by analogous arguments that if the individual volumes are free to change, the pressures must be equal in equilibrium, and that if the boxes can be exchange particles, the chemical potentials must be equal. We shall, however, defer formal proof of these statements to Secs. 1.2f and 1.2g, respectively.

We are now in a position to sketch a possible procedure for answering the prototype question suggested earlier: At a given temperature, what will the pressure be? Given the quantities E, N, V for a box of perfect gas, we count the possible states to compute S. Knowing E(S, V, N), we can then find

Our arguments have told us that not only is this equation valid, and the meanings of the quantities in it, but also that it is integrable; that is, there exists a function E(S, V, N) for a system in equilibrium. All of equilibrium thermodynamics is an elaboration of the consequences of those statements.

In the course of this discussion we have ignored a number of fundamental questions. For example, let us return to the arguments that led to Eq. (1.1.7). As we can see from the argument, even if the temperatures were equal, contact was not at all superfluous. It had an important effect: we lost track of the exact amount of energy in each box. This realization raises two important problems for us. The first is the question: How badly have we lost track of the energy? In other words, how much uncertainty has been introduced? The second is that whatever previous operations put the original amounts of energy into the two boxes, they must have been subject to the same kinds of uncertainties : we never actually knew exactly how much energy was in the boxes to begin with. How does that affect our earlier arguments? Stated differently: Can we reapply our arguments to the box now that we have lost track of its exact energy?

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