In this chapter, we summarize and expand somewhat on those results from quantum mechanics and spectroscopy most germane to our study of statistical thermodynamics. We then prepare for revisiting intensive properties in Chapter 9 by considering the nature of thermodynamic calculations before the advent of quantum mechanics. From a pedagogical point of view, the previous three chapters have focused on the properties of a single atom or molecule. For our purposes, the most important such properties are the allowed energy levels and degeneracies corresponding to the translational, rotational, vibrational, and electronic energy modes of an independent particle. Exploiting this knowledge, we proceed to a macroscopic assembly of atoms or molecules, with a focus on calculations of thermodynamic properties for any pure ideal gas. Assemblies composed of different particle types subsequently permit the evaluation of properties for both nonreacting and reacting gaseous mixtures, including equilibrium constants for various chemical reactions. Finally, re-applying spectroscopy to such mixtures, we examine the utility of statistical thermodynamics for experimentally determining temperature or concentrations in realistic gaseous mixtures at high temperatures and pressures.

Energy and Degeneracy

Our foray into quantum mechanics and spectroscopy has led to relations giving the energy and degeneracy for all four energy modes – translation, rotation, vibration, and electronic. If we insist on mode independence, any consideration of diatomic molecules also mandates the simplex model, which presumes a combined rigid rotor and harmonic oscillator.

The thermochemical data for the ideal gases of this appendix are taken directly from the Third Edition of the JANAF Thermochemical Tables, as tabulated by Chase et al. (1985). The twelve atoms and molecules selected are frequently cited in papers dealing with the chemistry and physics of flames and plasmas. For the sake of brevity, the plethora of possible hydrocarbons are not included in this appendix, but many of them can, of course, be found in the complete JANAF tables, which are available in the reference section of most science and engineering libraries.

The ordered listing of the twelve chosen gases is as follows: H2, H, O2, O, H2O, OH, CO, CO2, N2, N, NO, and NO2. Each table describes the formation of one mole of the subject species from its elements in their natural physical state at 298.15 K and 1 bar. The reader should refer to the original JANAF tables for a listing of the quantum mechanical and spectroscopic data used to effect the statistical mechanical calculations for each species. The reference state for all of these compounds is the hypothetical ideal gas at a temperature of 298.15 K (Tr) and a pressure of 1 bar (0.1 MPa). The logarithm of the equilibrium constant is to the base 10. By definition, both the enthalpy and Gibbs free energy of formation are identically zero for any ideal gas reference compound, such as H2, O2, or N2.

The atomic number (top left) is the number of protons in the nucleus. The atomic mass (bottom) is weighted by mean isotopic abundances in the earth's surface. Atomic masses are relative to the mass of the carbon-12 isotope, defined to be exactly 12 atomic mass units (amu). If sufficiently stable isotopes do not exist for an element, the mass of its longest-lived isotope is indicated in parentheses. For elements 110–112, the mass numbers of the known isotopes are given. Reference: Schroeder (2000) as extracted from The European Physical JournalC3, 73 (1998).

Now that we have reviewed the essentials of probability and statistics, we are mathematically prepared to pursue our primary goal, which is to understand at a basic statistical level the fundamental laws and relations of classical thermodynamics. To avoid unnecessary complications, we will begin by evaluating the macroscopic properties of simple compressible systems composed of independent particles. The most important thermodynamic systems of this type are those describing the behavior of ideal gases. Recall that all gases behave as independent particles at sufficiently low density because of their weak intermolecular interactions combined with their extremely short-range intermolecular potentials. Such gaseous systems constitute a propitious place to begin our study of statistical thermodynamics because by invoking the assumption of independent particles, our upcoming statistical analyses can be based rather straightforwardly on probability theory describing independent events, as summarized in Chapter 2.

While considering assemblies of independent particles, we will pursue new insight with respect to three basic concepts important to classical thermodynamics. First, we will seek a whole new statistical understanding of entropy. Second, we will develop a related statistical definition of thermodynamic equilibrium. Third, in so doing, we will gain new perspective concerning the significance of temperature in properly defining thermal equilibrium. Once we understand these three major concepts, we will be in a position to develop statistical expressions allowing us to evaluate the thermodynamic properties of an assembly from the quantum mechanical properties of its individual particles.

To this point in your career, you have probably dealt almost exclusively with the behavior of macroscopic systems, either from a scientific or engineering viewpoint. Examples of such systems might include a piston–cylinder assembly, a heat exchanger, or a battery. Typically, the analysis of macroscopic systems uses conservation or field equations related to classical mechanics, thermodynamics, or electromagnetics. In this book, our focus is on thermal devices, as usually described by thermodynamics, fluid mechanics, and heat transfer. For such devices, first-order calculations often employ a series of simple thermodynamic analyses. Nevertheless, you should understand that classical thermodynamics is inherently limited in its ability to explain the behavior of even the simplest thermodynamic system. The reason for this deficiency rests with its inadequate treatment of the atomic behavior underlying the gaseous, liquid, or solid states of matter. Without proper consideration of constituent microscopic systems, such as a single atom or molecule, it is impossible for the practitioner to understand fully the evaluation of thermodynamic properties, the meaning of thermodynamic equilibrium, or the influence of temperature on transport properties such as the thermal conductivity or viscosity. Developing this elementary viewpoint is the purpose of a course in statistical thermodynamics. As you will see, such fundamental understanding is also the basis for creative applications of classical thermodynamics to macroscopic devices.

The Statistical Foundation of Classical Thermodynamics

Since a typical thermodynamic system is composed of an assembly of atoms or molecules, we can surely presume that its macroscopic behavior can be expressed in terms of the microscopic properties of its constituent particles.

Our applications of statistical thermodynamics have thus far been mainly concerned with the gaseous state. We now move on to the solid state, with a particular focus in this chapter on metallic crystals, such as copper and silver. While semiconductors are, of course, very significant in an electronic age, their study is beyond the purview of this book, although our introduction to the solid state will surely pave the path for any future work that you might contemplate in this lucrative field.

In general, metallic crystals display a lattice structure, which features lattice sites occupied by metallic ions, but with their accompanying electrons migrating throughout the crystal. Because these ions are immobilized within a crystalline structure, they can vibrate about their equilibrium positions but they clearly cannot rotate or translate. Contrastingly, the electrons can surely translate, as indicated by their movement through any electrically conducting solid. In fact, as for individual metallic ions, electrons can be taken as independent particles because the electrostatic forces created between electrons and ions or among the electrons themselves are so long range that the electrons essentially move within a constant potential throughout the lattice structure. On the other hand, the mass of the electron is so tiny that our criterion for the dilute limit cannot possibly hold in this case, as verified by Eq. (9.5). Therefore, for the electron gas, we have no recourse but to summon basic Fermi–Dirac statistics.

We have come a long way together on this journey through the intricacies of statistical thermodynamics. Did the voyage meet your expectations? Were there any surprises along the way? While such questions are part and parcel of any journey, sober reflection on your itinerary offers convincing evidence that macroscopic thermodynamics can indeed be built on a solid foundation of microscopic thermodynamics. More importantly, you have surely found that new insights, deeper understanding, and novel applications follow from a statistical approach to basic thermodynamics. In this concluding chapter, we review highlights from our journey and muse on future experiences awaiting you in the realm of statistical thermodynamics. A significant outcome of this first excursion is that you are now prepared for many new journeys to come. I encourage you to be open to this further understanding and to explore the multitude of evolving venues which seek to exploit both the fundamentals and applications of statistical thermodynamics.

Reprising the Journey

We began our quest by introducing the two basic postulates of statistical thermodynamics: the ergodic hypothesis and the principle of equal a priori probability. The first postulate asserts that any thermodynamic variable characterizing an isolated system of independent particles can be evaluated by suitably averaging over all possible microstates, that is, over all feasible distributions of particles among energy states. The required statistical perspective is encapsulated by the second postulate, which holds that all microstates are equally likely.