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.

In this chapter, we examine equilibrium radiation, which represents our third and final application of statistical mechanics to independent particles beyond the dilute limit. For simplicity in mathematically modeling the radiant field, we apply the methods of statistical thermodynamics to electromagnetic waves enclosed in a cubical blackbody cavity. The enclosed radiation is at both thermal and radiative equilibrium if the walls of the cavity are at constant temperature with equal rates of emission and absorption, respectively.

Bose–Einstein Statistics for the Photon Gas

From a quantum perspective, the electromagnetic radiation within a blackbody cavity can be modeled as an assembly of independent photons. Given this representation, we recall from Section 5.9 that photons are particles of zero spin; hence, a photon gas must follow Bose–Einstein statistics, whose equilibrium particle distribution, following Eq. (3.31), is normally given by

However, in comparison to the usual assumptions associated with Eq. (14.1), photons do not obey particle conservation as they are constantly being formed and destroyed at the walls of a blackbody cavity. Of course, thermodynamic equilibrium still mandates conservation of energy at these same walls. As a result, nothing prevents, for example, the replacement of one incoming photon with two outgoing photons, each having half its energy.

In preparation for our study of statistical thermodynamics, we first review some fundamental notions of probability theory, with a special focus on those statistical concepts relevant to atomic and molecular systems. Depending on your background, you might be able to scan quickly Sections 2.1–2.3, but you should pay careful attention to Sections 2.4–2.7.

Probability: Definitions and Basic Concepts

Probability theory is concerned with predicting statistical outcomes. Simple examples of such outcomes include observing a head or tail when tossing a coin, or obtaining the numbers 1, 2, 3, 4, 5, or 6 when throwing a die. For a fairly-weighted coin, we would, of course, expect to see a head for 1∕2 of a large number of tosses; similarly, using a fairly-weighted die, we would expect to get a four for 1∕6 of all throws. We can then say that the probability of observing a head on one toss of a fairly-weighted coin is 1∕2 and that for obtaining a four on one throw of a fairly-weighted die is 1∕6. This heuristic notion of probability can be given mathematical formality via the following definition:

Here, sample space designates the available Ns occurrences while random event A denotes the subset of sample space given by Ne ≤ Ns.

We have previously shown that the translational energy mode for an ideal gas, even through a shock wave, invariably displays classical equilibrium behavior. In contrast, the rotational, vibrational, and electronic modes generally require significant time for re-equilibration upon disturbances in their equilibrium particle distributions. On this basis, we may expand our statistical discourse to nonequilibrium topics by grounding any dynamic redistribution on the presumption of translational equilibrium. For this reason, we now shift to elementary kinetic theory, which focuses solely on the translational motion of a gaseous assembly. Specifically, in this chapter, we consider equilibrium kinetic theory and its applications to velocity distributions, surface collisions, and pressure calculations. We then proceed to nonequilibrium kinetic theory with particular emphasis on calculations of transport properties and chemical reaction rates, as pursued in Chapters 16 and 17, respectively.

The Maxwell–Boltzmann Velocity Distribution

In Section 9.1, we showed that the translational energy mode for a dilute assembly displays classical behavior because of the inherently minute spacing between its discrete energy levels (Δε ≪ kT).

Parameters describing internal energy modes for molecular systems are required for statistical calculations of thermodynamic properties. This appendix includes such parameters for both diatomic and polyatomic molecules. Energy-mode parameters are tabulated for selected diatomic molecules in their ground electronic states. Term symbols for these electronic states are included, along with relevant bond lengths and dissociation energies. Similar parameters are also given for diatomic molecules in accessible upper electronic states. Finally, term symbols, rotational constants, and vibrational frequencies (cm−1) are tabulated for selected polyatomic molecules, with a particular focus on triatomic species. All diatomic data have been extracted from Huber and Herzberg (1978), while the polyatomic data have been taken from Herzberg (1991). Additional spectroscopic data are available electronically from NIST(http://webbook.nist.gov/chemistry/).