The chapter begins with a classic topic: the efficiency of heat engines. The topic remains relevant today—for environmental reasons, among others—and it also provides the foundation for William Thomson's definition of absolute temperature, an item discussed later (in chapter 4). Next, the chapter develops a method for computing the entropy change when a process occurs rapidly. An extended example—the Otto cycle—and a discussion of “reversibility” conclude the chapter.

The most important thermodynamic cycle: the Carnot cycle

Engineers and environmentalists are interested in cyclic processes, because one can do them again and again. Figure 3.1 shows the essence of the cycle for water and steam in a typical power plant that generates electricity. A simpler cycle, however, is more instructive theoretically, and so let us consider the cycle shown in figure 3.2, performed slowly throughout. We may take the substance to be some real gas, such as nitrogen (as distinguished from an ideal gas). There are four stages, characterized as follows.

Stage 1 to 2: isothermal expansion. From state 1 to state 2, the gas is allowed to expand while being maintained at constant, high temperature Thot. The pressure drops because the volume increases. [The ideal gas law, P = (N/V)kT, valid for a dilute gas, suggests such a pressure drop—qualitatively—for a real, not-necessarily-dilute gas, too.] The gas does work as it expands and would drop in temperature if energy were not supplied by heating.

The preface noted that “the book's conceptual core consists of four linked elements: entropy and the Second Law of Thermodynamics, the canonical probability distribution, the partition function, and the chemical potential.” By now, three of those items are familiar. The present chapter introduces the last item and, for illustration, works out a typical application. The chemical potential plays a significant role in most of the succeeding chapters.

Discovering the chemical potential

The density of the Earth's atmosphere decreases with height. The concentration gradient—a greater concentration lower down—tends to make molecules diffuse upward. Gravity, however, pulls on the molecules, tending to make them diffuse downward. The two effects are in balance, canceling each other, at least on an average over short times or small volumes. Succinctly stated, the atmosphere is in equilibrium with respect to diffusion.

In general, how does thermal physics describe such a diffusive equilibrium? In this section, we calculate how gas in thermal equilibrium is distributed in height. Certain derivatives emerge and play a decisive role. The underlying purpose of the section is to discover those derivatives and the method that employs them. We will find a quantity that measures the tendency of particles to diffuse.

Figure 7.1 sets the scene. Two volumes, vertically thin in comparison with their horizontal extent, are separated in height by a distance H. A narrow tube connects the upper volume Vu to the lower volume Vl.

This chapter has two goals. The first is to develop the quantum version of multiplicity. That will show us how entropy is expressed in quantum theory. The second goal is to develop a general quantitative definition of temperature. Entropy plays a key role in that definition.

The density of states

When a physical system has reached thermal equilibrium, its macroscopic properties do not change with time. In quantum theory, the energy eigenstates of an isolated system provide predictions and estimates that are constant in time; therefore such states are appropriate for a quantum description of thermal equilibrium.

To be sure, the information at hand will not enable us to select a single state as uniquely the correct state to use. We will be driven to consider many states and to form sums over them. This section develops a mathematical technique for working with such sums.

When the system is both isolated and of finite size (as we shall specify here), the energy eigenstates form a discrete set, whose members we can arrange and label in order of increasing energy. (If any two distinct states happen to have the same energy, we just assign them consecutive labels.) Typically, the states will be densely spaced in energy, and so a sum over a range of states can often be approximated adequately by an integral with respect to energy, provided that we have constructed an appropriate density of states: a function that specifies the number of energy eigenstates per unit energy interval.

This chapter explores in greater depth the connections among entropy, temperature, and energy transfer by heating (or cooling). The sections are diverse but related, albeit loosely. The first section develops a general expression for entropy in terms of probabilities. The next section provides an example and also prepares the ground for section 14.3, which discusses a classic method of reaching low temperatures. The section entitled “The Third Law of Thermodynamics” explores the constraints on methods for approaching absolute zero. In section 14.6, we discover that temperatures numerically below absolute zero have been produced in the lab, and we explore that context. Finally, the last three sections reflect on the notions of temperature, heating, and entropy.

In short, the chapter's title, “Approaching zero,” highlights the central four sections, sections 14.3 to 14.6. The sections that precede and follow those sections are loosely related to the central sections and are also valuable in their own right.

Entropy and probability

Chapter 2 used the idea of multiplicity and its evolution to state the Second Law of Thermodynamics. A quantitative study of how multiplicity changes during a slow isothermal expansion led to entropy, defined as S ≡ k In (multiplicity). In chapter 5, we used the entropy concept to derive the canonical probability distribution; thereby we introduced an explicit expression for probabilities at thermal equilibrium. Now we complete the loop by learning how the probabilities can be used to calculate the entropy.

Several aims guided me while I wrote. My first goal was to build from the familiar to the abstract and still get to entropy, conceived microscopically, in the second chapter. I sought to keep the book crisp and lean: derivations were to be succinct and simple; topics were to be those essential for physics and astronomy. From the professor's perspective, a semester is a short time, and few undergraduate curricula can devote more than a semester to thermal physics.

Modularity was another aim. Instructors' tastes vary greatly, and so I sought maximal flexibility in what to teach and when to cover it. The book's logical structure is displayed in figure P1. Chapters 1 to 3 develop topics that appear in the typical fat textbook for introductory physics but are rarely assimilated by students in that course, if the instructor even gets to the topics. Thus the book presumes only an elementary knowledge of classical mechanics and some rudimentary ideas from quantum theory, primarily the de Broglie relationship p = h/λ and the idea of energy eigenstates.

A benefit of modularity is that one can study chapter 13—the classical theory—any time after chapter 5. I placed the classical theory so far back in the book because I think students should get to use the quantum machinery of chapters 4 and 5 on some important physics before they face the development of more formalism.

If we know the temperature of a system and the values of its external parameters, how can we estimate its physical properties, such as energy, pressure, magnetic moment, and distribution of molecular velocities? The question is answered in this chapter: we derive the canonical probability distribution, learn some techniques for applying it efficiently, and work out two major examples.

Probabilities

Probabilities enter into thermal physics because the available data are insufficient to determine the individual properties of 1020 molecules. Moreover, even if such finely detailed data were available, no person or computer could cope with it. Out of necessity, one turns to a statistical analysis. A handful of data and some plausible reasoning lead to predictions whose success is nothing short of astonishing.

Before we look at what a “probability” means, we should note that probabilities always arise in a context. For example, the probability of a 4 appearing, given that I roll a die once, is 1/6. The probability of a four appearing, if I were to count the number of letters that come in my daily mail, would be quite different. Sometimes the context is made explicit; at other times, it is left implicit; but always there is a context.

In a loose sense, this chapter is about the coexistence of solids, liquids, and gases, usually taken two at a time. What physical condition must be met if coexistence is to occur? What other relations follow? The chapter provides some answers. And in its last section, it develops a classic equation—the van der Waals equation of state—that was the first to describe gas and liquid in coexistence.

Phase diagram

Figure 12.1 displays three phases of water as a function of pressure and temperature. By the word phase, one means here a system or portion of a system that is spatially homogenous and has a definite boundary. [In turn, homogeneity means that the chemical composition (including relative amounts), the crystalline structure (if any), and the mass density are uniform in space. A continuous variation produced by gravity, however, is allowed, as in the case of a finite column of air in the Earth's gravitational field.] The liquid phase is typified by a glass of water; the solid, by an ice cube; and the vapor, by the “dry” steam (that is, steam without water droplets) in the turbine of a power plant, as was illustrated in figure 3.1.

Of more interest, however, are the curves in the drawing, locations where two phases coexist. When a pond freezes over and develops a 20 centimeter layer of ice, the lower surface of the ice and the top of the remaining liquid water coexist at a temperature and pressure that lie along the leftward (and upward) tilted curve that emanates from the triple point.

By now, the classical ideal gas should be a familiar system. We can turn to the full quantum theory of an ideal gas. The applications are numerous, and some will be examined in chapter 9. The present chapter develops an effective technique for working with a quantum ideal gas.

Coping with many particles all at once

Every macroscopic system consists of a great many individual particles. Coping with all of them simultaneously, even if only statistically, can be a daunting task. A flow chart will help us to see some of our accomplishments and to see what steps lie ahead. Figure 8.1 displays such a chart on the supposition that the system consists of one species of particle only, for example, N electrons or N helium atoms (of a specific isotope).

If the forces between the particles must be included, then the analysis is difficult. Moreover, there is no general algorithm for proceeding, not even by computer. To be sure, the canonical probability distribution usually is applicable and provides estimates in principle, but the actual evaluation of the partition function, say, defies a direct approach. We have, however, successfully dealt with one such difficult situation. The Debye model for a solid incorporates the inter-particle forces, for example, those between adjacent copper atoms in a crystal. The forces ensure that the motions of adjacent atoms are correlated, and the correlation leads to sound waves, the theoretical basis of the Debye model.

All too easily, a book on thermal physics leaves the impression of a collection of applications: blackbody radiation, Debye theory, conduction electrons, van der Waals equation, and so on. Those applications are vital; they connect the theoretical and experimental worlds. Yet even more important is to come away from the book with a sense of its underlying theoretical structure. What is its equivalent of the equation F = ma in a mechanics text? Or of Maxwell's equations in a book on electromagnetism?

In this book, the key theoretical ideas are the Second Law of Thermodynamics, the canonical probability distribution, the partition function, and the chemical potential. Of these, the Second Law of Thermodynamics comes first logically and is most nearly the central organizing principle. The majority of the applications, however, found us using the latter three items: P(Ψj) = (1/Z)exp(−Ej/kT), Z, or µ. By comparison with mechanics or electromagnetism, thermal physics suffers in that its most basic principle, the Second Law of Thermodynamics, is separated by layers of secondary theory from the physical applications. Perhaps that reflects the inherent difficulty of coping with 1020 particles all at once, or perhaps the root lies in the diversity of topics that can be addressed.

Here, in brief outline, is the run of the chapter. Section 16.1 presents two experiments to introduce the topic of “critical phenomena,” as that phrase is used in physics. The next section illustrates the mathematical behavior of certain physical quantities near a critical point. Section 16.3 constructs a theoretical model: the Ising model. The following two sections develop methods for extracting predictions from the model. The last three sections draw distinctions, outline some general results, and collect the chapter's essentials.

Achieving a sound theoretical understanding of critical phenomena was the major triumph of thermal physics in the second half of the twentieth century. Thus the topic serves admirably as the culmination of the entire book.

Experiments

Liquid–vapor

We begin with an experiment that you may see as a lecture demonstration, performed either with carbon dioxide or with Freon, a liquid once used in refrigerators. A sealed vertical cylinder contains a carefully measured amount of CO2 (say) under high pressure. To begin with, the system is in thermal equilibrium at room temperature, as sketched in part (a) of figure 16.1. A distinct meniscus separates the clear liquid phase from the vapor phase above it.

Now heat the system with a hair dryer. The meniscus rises (because the liquid expands), and gentle boiling commences. Next the meniscus becomes diffuse, and then it disappears. The CO2 has become spatially homogeneous in density, as illustrated in figure 16.1 (b).