"A large number of exercises of a broad range of difficulty make this book even more useful…a good addition to the literature on thermodynamics at the undergraduate level." — Philosophical Magazine
Although written on an introductory level, this wide-ranging text provides extensive coverage of topics of current interest in equilibrium statistical mechanics. Indeed, certain traditional topics are given somewhat condensed treatment to allow room for a survey of more recent advances.
The book is divided into four major sections. Part I deals with the principles of quantum statistical mechanics and includes discussions of energy levels, states and eigenfunctions, degeneracy and other topics. Part II examines systems composed of independent molecules or of other independent subsystems. Topics range from ideal monatomic gas and monatomic crystals to polyatomic gas and configuration of polymer molecules and rubber elasticity. An examination of systems of interacting molecules comprises the nine chapters in Part Ill, reviewing such subjects as lattice statistics, imperfect gases and dilute liquid solutions. Part IV covers quantum statistics and includes sections on Fermi-Dirac and Bose-Einstein statistics, photon gas and free-volume theories of quantum liquids.
Each chapter includes problems varying in difficulty — ranging from simple numerical exercises to small-scale "research" propositions. In addition, supplementary reading lists for each chapter invite students to pursue the subject at a more advanced level. Readers are assumed to have studied thermodynamics, calculus, elementary differential equations and elementary quantum mechanics.
Because of the flexibility of the chapter arrangements, this book especially lends itself to use in a one-or two-semester graduate course in chemistry, a one-semester senior or graduate course in physics or an introductory course in statistical mechanics.
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An Introduction to Statistical Thermodynamics
By Terrell L. Hill
Dover Publications, Inc.Copyright © 1986 Terrell L. Hill
All rights reserved.
STATISTICAL-MECHANICAL ENSEMBLES AND THERMODYNAMICS
The object of thermodynamics is to derive mathematical relations which connect different experimental properties of macroscopic systems in equilibrium—systems containing many molecules, of the order of, say, 1020 or more. However useful, these interconnections of thermodynamics give us no information at all concerning the interpretation or explanation, on a molecular level, of the observed experimental properties. For example, from thermodynamics we know that experimental values of the two heat capacities Cp and Cv for a given system must be interrelated by an exact and well-known equation, but thermodynamics is unable to furnish any explanation of why particular experimental values of either Cp or Cv, taken separately, should be observed. Such an explanation falls rather within the province of statistical mechanics or statistical thermodynamics, terms which we regard in this book as synonymous. That is, the object of statistical mechanics is to provide the molecular theory or interpretation of equilibrium properties of macroscopic systems. Thus the fields covered by statistical mechanics and thermodynamics coincide. Whenever the question "why?" is raised in thermodynamics—why, for example, a given equilibrium constant, Henry's law constant, equation of state, etc., is observed—we are presented with a problem in statistical mechanics.
Although thermodynamics itself does not provide a molecular picture of nature, this is not always a disadvantage. Thus there are many complicated systems for which a molecular theory is not yet possible; but regardless of complications on the molecular level, thermodynamics can still be applied to such systems with confidence and exactness.
In recent years both thermodynamics and statistical mechanics have been extended somewhat into the nonequilibrium domain. However, the subject is new and changing, and the foundations are still a little shaky; hence we omit this area from our consideration. An exception is the theory of absolute reaction rates, which we discuss in Chapter 11. This approximate theory is based on a quasi-equilibrium approach which makes it possible to include the theory within the framework of equilibrium statistical mechanics.
Aside from the postulates of statistical mechanics themselves, to be introduced in the next section, the foundation on which our subject is based is quantum mechanics. If we seek a molecular interpretation of the properties of a system containing many molecules, as a starting point we must certainly be provided with knowledge of the properties of the individual molecules making up the system and of the nature of the interactions between these molecules. This is information which can in principle be furnished by quantum mechanics but which in practice is usually obtained from experiments based on the behavior of individual molecules (e.g., spectroscopy), pairs of molecules (e.g., the second virial coefficient of an imperfect gas), etc.
Although quantum mechanics is prerequisite to statistical mechanics, fortunately a reasonably satisfactory version of statistical mechanics can be presented without using any quantum-mechanical concepts other than those of quantum-mechanical states, energy levels, and intermolecular forces. Only in Part IV of the book is it necessary to go beyond this stage.
Another very helpful simplification is that the classical limit of quantum mechanics can be used, without appreciable error, in most problems involving significant intermolecular interactions. Problems of this type are very difficult without this simplification (Part IV).
Despite our extensive use of classical statistical mechanics in the applications of Parts II and III, we introduce the principles of statistical mechanics, beginning in the next section, in quantum-mechanical language because the argument is not only more general but is actually much simpler this way.
1-2 Ensembles and postulates.
As mentioned above, our problem is to calculate macroscopic properties from molecular properties. Our general approach is to set up postulates which allow us to proceed directly with this task insofar as "mechanical" thermodynamic properties are concerned; the "nonmechanical" properties are then handled indirectly by an appeal to thermodynamics. By "mechanical" properties we mean, for example, pressure, energy, volume, number of molecules, etc., all of which can be defined in purely mechanical terms (quantum or classical) without, for example, introducing the concept of temperature. Examples of "nonmechanical" thermodynamic variables are temperature, entropy, free energy (Gibbs or Helmholtz), chemical potential, etc.
Let us consider the pressure as a typical mechanical variable. In principle, if we wished to calculate the pressure in a thermodynamic system from molecular considerations, we would have to calculate (by quantum or possibly classical mechanics) the force per unit area exerted on a wall of the system, taking into account the change in the state of the whole system with time. The force itself would be a function of time. What we would need, therefore, is a time average of the force over a period of time sufficiently long to smooth out fluctuations, i.e., sufficiently long to give a time average which is independent, say, of the starting time, t = t0, in the averaging. Because of the tremendous number of molecules in a typical system, and the fact that they interact with each other, such a hypothetical calculation is of course completely out of the question in either quantum or classical mechanics.
Therefore we are forced to turn to an alternative procedure, the ensemble method of Gibbs, based on postulates connecting the desired time average of a mechanical variable with the ensemble average (defined below) of the same variable. The validity of these postulates rests on the agreement between experiment and deductions (such as those in this book) made from the postulates. So far, there is no experimental evidence available that casts doubt on the correctness of the postulates of statistical mechanics.
Before stating the postulates, we must introduce the concept of an ensemble of systems. An ensemble is simply a (mental) collection of a very large number [??] of systems, each constructed to be a replica on a thermodynamic (macroscopic) level of the actual thermodynamic system whose properties we are investigating. For example, suppose the system of interest has a volume V, contains N molecules of a single component, and is immersed in a large heat bath at temperature T. The assigned values of N, V, and T are sufficient to determine the thermodynamic state of the system. In this case, the ensemble would consist of [??] systems, all of which are constructed to duplicate the thermodynamic state (N, V, T) and environment (closed system immersed in a heat bath) of the original system. Although all systems in the ensemble are identical from a thermodynamic point of view, they are not all identical on the molecular level. In fact, in general, there is an extremely large number of quantum (or classical) states consistent with a given thermodynamic state. This is to be expected, of course, since three numbers, say the values of N, V, and T, are quite inadequate to specify the detailed molecular (or "microscopic") state of a system containing something in the order of 1020 molecules.
Incidentally, when the term "quantum state" is used here, it will be understood that we refer specifically to energy states (i.e., energy eigenstates, or stationary states).
At any instant of time, in an ensemble constructed by replication of a given thermodynamic system in a given environment, many different quantum states are represented in the various systems of the ensemble. In the example mentioned above, the calculated instantaneous pressure would in general be different in these different quantum states. The "ensemble average" of the pressure is then the average over these instantaneous values of the pressure, giving the same weight to each system in the ensemble in calculating the average. A similar ensemble average can be calculated for any mechanical variable which may have different values (i.e., which is not held constant) in the different systems of the ensemble.
We now state our first postulate: the (long) time average of a mechanical variable M in the thermodynamic system of interest is equal to the ensemble average of M, in the limit as [??] - ∞, provided that the systems of the ensemble replicate the thermodynamic state and environment of the actual system of interest. That is, this postulate tells us that we may replace a time average on the one actual system by an instantaneous average over a large number of systems "representative" of the actual system. The first postulate by itself is not really helpful; we need in addition, in order to actually compute an ensemble average, some information about the relative probability of occurrence of different quantum states in the systems of the ensemble. This information must be provided in a second postulate.
Note that the ensemble average of M in the limit as R - ∞, referred to above, must be independent of time. Otherwise the original system which the ensemble "represents" is not in equilibrium.
We shall work out details in this chapter for the three most important thermodynamic environments: (a) an isolated system (N, V, and E given, where E = energy); (b) a closed, isothermal system (N, V, and T given); and (c) an open, isothermal system (μ, V, and T given, where μ = chemical potential). N and μ stand for the sets N1, N2, ... and μ1, μ2, ... if the system contains more than one component. Also, V might stand for a set of "external variables" if there are more than one. The representative ensembles in the above three cases are usually called microcanonical, canonical, and grand canonical, respectively. The first postulate is applicable to all these cases and to other ensembles which will be introduced in Section 1-7. The second postulate, however, can be limited to a statement concerning only the microcanonical ensemble. The corresponding statement for other ensembles can then be deduced (as in Section 1-3, for example) from this limited second postulate without any further assumptions.
Our second postulate is: in an ensemble ([??] - ∞) representative of an isolated thermodynamic system, the systems of the ensemble are distributed uniformly, that is, with equal probability or frequency, over the possible quantum states consistent with the specified values of N, V, and E. In other words, each quantum state is represented by the same number of systems in the ensemble; or, if a system is selected at random from the ensemble, the probability that it will be found in a particular quantum state is the same for all the possible quantum states. A related implication of this postulate, when combined with the first postulate, is that the single isolated system of actual interest (which serves as the prototype for the systems of the ensemble) spends equal amounts of time, over a long period of time, in each of the available quantum states. This last statement is often referred to as the quantum "ergodic hypothesis," while the second postulate by itself is usually called the "principle of equal a priori probabilities." The ergodic hypothesis in classical statistical mechanics is mentioned at the end of Section 6-3. For a more detailed discussion, see Tolman, pp. 63-70 and 356-361. (For full identification of works referred to by only the author's last name, see Preface.)
The value of E in the second postulate must be one of the energy levels of the quantum-mechanical system defined by N and V. Since N is extremely large, the energy levels for such a system will be so close together as to be practically continuous, and furthermore, each of these levels will have an extremely high degeneracy. We shall in general denote the number of quantum states (i.e., the degeneracy) associated with the energy level E for a quantum-mechanical system with N and V by Ω(N, V, E). Thus the number of "possible quantum states" referred to in the second postulate is Ω.
A complication in the above discussion is the fact that, from an operational point of view, E cannot be known precisely; there will always be a small uncertainty δE in the value of E. For all thermodynamic purposes this complication is completely inconsequential. Hence for the sake of simplicity we ignore it.
It should also be mentioned that the point of view in the above statement of the second postulate is not so general as it might be. If the energy level E for the system N, V has a degeneracy Ω, there are Ω orthogonal (and therefore linearly independent) wave functions Ψ, which satisfy the Schrödinger equation HΨ = EΨ. The particular choice of the Ω Ψ's is somewhat arbitrary, since other possible choices can always be set up by forming suitable linear combinations of the Ψ's in the first choice. In any case, the "Ω quantum states" mentioned in connection with the second postulate refers to some set of orthogonal Ψ's all "belonging" to the same E. But regardless of the set of Ψ's chosen, the wave function representing the actual quantum-mechanical state of any system selected from the ensemble will in general not be one of the chosen set of Ψ's, but will be some linear combination of all of them. The contrary is really implied in the above statement of the second postulate. Fortunately, this simplification in our statement of the postulate makes no difference in any deductions we shall make that can be compared with experiment.
We turn now to a derivation from the above two postulates of the essential properties of the canonical and grand ensembles.
1-3 Canonical ensemble.
The experimental system of interest here has a fixed volume V, fixed numbers of molecules N (which stands for N1, N2, ... in a multicomponent system), and is immersed in a very large heat bath at temperature T. The heat bath is assumed "very large" to be consistent with the use of the limit [??] -∞ below. Our first objective is to set up the machinery necessary for calculating the average value of mechanical variables, such as energy and pressure, in the system. In view of the first postulate, this means that we need to be able to calculate the ensemble average of such variables. This, in turn, can be done if we know the value of the particular variable in question in a given quantum state and the fraction of systems in the ensemble which are in this quantum state. It might be noted that because the thermodynamic system here is not isolated but is in contact with a heat bath, the energy of the system can fluctuate; therefore quantum states belonging to different energy levels E will have to be reckoned with. Since mechanical variables have well-defined values in a given quantum state (in fact we can use this property as the definition of a "mechanical variable"), the task that remains is to determine the fraction of systems in the ensemble in a given quantum state (or the probability that a system selected arbitrarily from the ensemble is in a given quantum state). This is the problem we now consider.
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Table of ContentsPART I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS
CHAPTER 1. STATISTICAL-MECHANICAL ENSEMBLES AND THERMODYNAMICS
1-2 Ensembles and postulates
1-3 Canonical ensemble
1-4 Canonical ensemble and thermodynamics
1-5 Grand canonical ensemble
1-6 Micronomical ensemble
1-7 Other ensembles
CHAPTER 2. FURTHER DISCUSSION OF ENSEMBLES AND THERMODYNAMICS
2-2 Thermodynamic equivalence of ensembles
2-3 Second law of thermodynamics
2-4 Third law of thermodynamics
PART II. SYSTEMS COMPOSED OF INDEPENDENT MOLECULES OR SUBSYSTEMS AND INDISTINGUISHABLE MOLECULES OR SUBSYSTEMS
CHAPTER 3. GENERAL RELATIONS FOR INDEPENDENT DISTINGUISHABLE AND INDISTINGUISHABLE MOLECULES OR SUBSYSTEMS
3-1 Independent and distinguishable molecules or subsystems
3-2 Independent and indistinguishable molecules or subsystems
3-3 Energy distribution among independent molecules
3-4 "Ensembles" of small, independent "systems"
CHAPTER 4. IDEAL MONATOMIC GAS
4-1 Energy levels and canonical ensemble partion function
4-2 Thermodynamic functions
4-3 Grand ensemble and others
4-4 Internal degrees of freedom
CHAPTER 5. MONATOMIC CRYSTALS
5-1 Einstien model of a monatomic crystal
5-2 General treatment of molecular vibrations in a monatomic crystal
5-3 The Debye approximation
5-4 Exact treatments of the frequency distribution problem
CHAPTER 6. CLASSICAL STATISTICAL MECHANICS
6-1 Introductory examples
6-2 More general systems
6-3 Phase space and ensembles in classical statistics
6-4 Maxwell-Boltzmann velocity distribution
"CHAPTER 7. INTRODUCTION TO LATTICE STATISTICS: ADSORPTION, BINDING, AND TITRATION PROBLEMS"
7-1 Ideal lattice gas (Langmiur adsorption theory)
7-2 Grand partition function for a single independent site or subsystem
7-3 Systems composed of independent and indistinguishable subsystems
7-4 Elasticity of and adsorption on a linear polymer chain
CHAPTER 8. IDEAL DIATOMIC GAS
8-1 Independence of degrees of freedom
8-4 Thermodynamic functions
CHAPTER 9. IDEAL POLYATOMIC GAS
9-1 Potential energy surface
9-4 Thermodynamic functions
9-5 Hindred internal rotation in ethane
9-6 Hindred translation on a surface
CHAPTER 10. CHEMICAL EQUILIBRIUM IN IDEAL GAS MIXTURES
10-1 General relations
10-2 Statistical derivation in a special case
10-3 Fluctuations in a simple chemical equilibrium
10-4 Examples of chemical equilibria
CHAPTER 11. THE RATE OF CHEMICAL REACTIONS IN IDEAL GAS MIXTURES
11-1 Potential surfaces
11-2 Absolute rate theory
11-3 A nonchemical application of the Eyring theory
CHAPTER 12. IDEAL GAS IN AN ELECTRIC FIELD
12-1 Thermodynamic background
12-2 Statistical-mechanical background
12-3 Dilute gas in an electric field
12-4 Lattice of noninteracting magnetic dipoles
CHAPTER 13. CONFIGURATION OF POLYMER MOLECULES AND RUBBER ELASTICITY
13-1 Freely jointed chain
13-2 Gaussian probability distribution for free polymer molecules
13-3 Rubber elasticity
PART III. SYSTEMS OF INTERACTING MOLECULES
CHAPTER 14. LATTICE STATISTICS
14-1 One-dimensional lattice gas (adsorption)
14-2 Elasticity of a linear polymer chain
14-3 Two-dimensional square lattice
14-4 Bragg-Williams approximation
14-5 Quasi-chemical approximation
14-6 First-order phase transitions
CHAPTER 15. IMPERFECT GASES
15-1 Virial expansion of a one-component gas
15-2 One-component classical monatomic gas
15-3 Two-component imperfect gas
15-4 Imperfect gas near a surface
15-5 Imperfect gas in an electric field
CHAPTER 16. APPROXIMATE CELL AND HOLE THEORIES OF THE LIQUID STATE
16-1 The van der Waals equation of state
16-2 Cell theories of liquids
16-3 Hole theories of liquids
16-4 Law of corresponding states
CHAPTER 17. DISTRIBUTION FUNCTIONS IN CLASSICAL MONATOMIC FLUIDS
17-1 Radial distribution function
17-2 Relation of thermodynamic functions to g( r )
17-3 Integral equation for g(r;x)
17-4 Formal definition of distribution functions
17-5 Surface tension
CHAPTER 18. DILUTE ELECTROLYTE SOLUTIONS AND PLASMAS
18-1 Debye-Hückel theory
18-2 Kirkwood theory of solutions
18-3 Electrolyte solutions
CHAPTER 19. DILUTE LIQUID SOLUTIONS
19-1 McMillan-Mayer solution theory
19-2 Applications of the McMillan-Mayer theory
19-3 Constant pressure solution theory
CHAPTER 20. THEORY OF CONCENTRATED SOLUTIONS
20-1 Lattice theory of solutions
20-2 Cell theories of binary solutions
20-3 "Random-mixing, corresponding-states theory "
20-4 Conformal solution theory
CHAPTER 21. POLYMER AND POLYELECTROLYTE SOLUTIONS AND GELS
21-1 Wall theory of rubber elasticity
21-2 Flory-Hugging polymer solution theory
21-3 Swelling of polymer gels
21-4 Swelling of polyelectrolyte gels
21-5 Isolated polymer or polyelectrolyte molecules in solution
21-6 Second Virial coefficient in polymer and polyelectrolyte solutions
CHAPTER 22. QUANTUM STATISTICS
22-1 Introduction to Fermi-Dirac and Bose-Einstein statistics
22-2 Ideal Fermi-Dirac gas; electrons in metals
22-3 Ideal Bose-Einstein gas; helium
22-4 Blackbody radiation (photon gas)
22-5 Quantum statistics with intermolecular interactions
22-6 The factors hn and N! in classical statistics
22-7 Free-volume theories of quantum liquids
22-8 Gas of symmetrical diatomic modules at low temperatures
APPENDIX I. Natural Constants
APPENDIX II. Maximum-Term Method
APPENDIX III. Method of Undetermined Multipliers
APPENDIX IV. The Lennard-Jones Potential
APPENDIX V. Normal Coordinate Analysis in a Special Case
APPENDIX VI. Vibrational Frequency Distribution in a Solid Continuum
APPENDIX VII. Generalized Coordinates