Mere Thermodynamics

Mere Thermodynamics

DON S. LEMONS

© 2009 The Johns Hopkins University Press

All rights reserved. Published 2009

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Lemons, Don S. (Don Stephen), 1949–

Mere thermodynamics / Don S. Lemons.

p. cm.

Includes bibliographical references and index.

ISBN-13: 978-0-8018-9014-7 (hardcover : alk. paper)

ISBN-10: 0-8018-9014-4 (hardcover : alk. paper)

ISBN-13: 978-0-8018-9015-4 (pbk. : alk. paper)

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1. Thermodynamics. I. Title.

QC311.L358 2008

536′7—dc22 2008007841

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Contents

Preface

1. Definitions 1

1.1. Thermodynamics

1.2. System

1.3. Boundary, Environment, and Interactions

1.4. States and State Variables

1.5. Equations of State

1.6. Work

1.7. Heat

Problems

2. Equilibrium

2.1. Equilibrium

2.2. Zeroth Law of Thermodynamics

2.3. Empirical Temperature

2.4. Traditional Temperature Scales

2.5. Equilibrium Processes

Problems

3. Heat

3.1. Quantifying Heat

3.2. Calorimetry

3.3. What is Heat?

Problems

4. The First Law

4.1. Count Rumford

4.2. Joule’s Experiments

4.3. The First Law of Thermodynamics

4.4. Thermodynamic Cycles

4.5. Cycle Adjustment

Problems

5. The Second Law

5.1. Sadi Carnot

5.2. Statements of the Second Law

5.3. Equivalence and Inequivalence

5.4. Reversible Heat Engines

5.5. Refrigerators and Heat Pumps

Problems

6. The First and Second Laws

6.1. Rudolph Clausius

6.2. Thermodynamic Temperature

6.3. Clausius’s Theorem

Problems

7. Entropy

7.1. The Meaning of Reversibility

7.2. Entropy

7.3. Entropy Generation in Irreversible Processes

7.4. The Entropy Generator

7.5. Entropy Corollaries

7.6. Thermodynamic Arrow of Time

Problems

8. Fluid Variables

8.1. What Is a Fluid?

8.2. Reversible Work

8.3. Fundamental Constraint

8.4. Enthalpy

8.5. Helmholtz and Gibbs Free Energies

8.6. Partial Derivative Rules

8.7. Thermodynamic Coefficients

8.8. Heat Capacities

Problems

9. Simple Fluid Systems

9.1. The Ideal Gas

9.2. Room-Temperature Elastic Solid

9.3. Cavity Radiation

Problems

10. Nonfluid Systems

10.1. Nonfluid Variables

10.2. The Theoretician’s Rubber Band

10.3. Paramagnetism

10.4. Surfaces

10.5. Chemical Potential

10.6. Multivariate Systems

Problems

11. Equilibrium and Stability

11.1. Mechanical and Thermal Systems

11.2. Principle of Maximum Entropy

11.3. Other Stability Criteria

11.4. Intrinsic Stability of a Fluid

Problems

12. Two-Phase Systems

12.1. Phase Diagrams

12.2. Van der Waals Equation of State

12.3. Two-Phase Transition

12.4. Maxwell Construction

12.5. Clausius-Clapeyron Equation

12.6. Critical Point

Problems

13. The Third Law

13.1. The Principle of Thomsen and Berthelot

13.2. Entropy Change

13.3. Unattainability

13.4. Absolute Entropy

Problem

Appendixes

A: Physical Constants and Standard Definitions

B: Catalog of 21 Simple Cycles

C: Glossary of Terms

D: Selected Worked Problems

E: Answers to Problems

Annotated Bibliography

Index

Preface

Thermodynamics is a physical theory at once beautiful and useful: beautiful because its laws are simply expressed and deductions from them universally applicable, and useful because in distinguishing the possible from the impossible thermodynamics saves us from much fruitless effort. For these reasons, as well as for its undeniable empirical success, thermodynamics deeply impresses those who grasp its essentials. Albert Einstein once boldly claimed that thermodynamics, alone among physical theories, will never be overthrown.

Thermodynamics is nevertheless notoriously difficult to teach and difficult to learn. While much of classical and quantum physics can be framed in terms of easily pictured concepts or easily remembered equations, classical thermodynamics grows out of the austere logic of possibility and impossibility encapsulated in its first and second laws. That these laws are usually given verbal rather than mathematical expression hinders rather than aids readers unused to such formulations.

Even so, most texts don’t bother much with the laws of thermodynamics. While no one questions the laws per se, everyone hurries to get beyond them. What lies beyond the laws of thermodynamics is, on the one hand, a multitude of applications and, on the other, the special models of classical and quantum statistical mechanics. The diligent student of such applications and such models, while able to reproduce many complicated patterns of thought, may lack a coherent vision of thermodynamics itself. On too many occasions accomplished scientists have told me, I never understood thermodynamics.

Mere Thermodynamics presents a vision of thermodynamics itself, its laws, their essential corollaries, useful methods, and important applications. This ordering—first laws, then corollaries, methods, and applications—informs the whole text. My aim has been to present the subject in its most orderly and most plausible aspect, and to write a concise yet balanced text that allows the structure of classical thermodynamics to stand out—its limitations as well as its achievements.

In highlighting the intellectual structure of thermodynamics I have found it impossible to ignore the historical drama of its unfolding. Certain contours of the subject have been permanently shaped by its history, and for this reason, this volume follows the traditional historically oriented approach with its heat engines, reversible cycles, and laws. The historical approach also naturally introduces, in its course, important phenomena and experimental outcomes.

The text aims to reward the reader’s attention with a maximal understanding of the subject’s most difficult parts: the second law of thermodynamics and the concept of entropy. Formally, the meaning of any statement or law consists of what can be deduced from it. For this reason, and also because the second law was discovered before the first, I develop consequences of each law apart from the other—the first law in Chapter 4 and the second law in Chapter 5 and Appendix B—before developing, in Chapter 6, consequences of the two combined. The first and second laws of thermodynamics together lead to the concept of entropy. There is hardly any development of physical theory more impressive than that taking us from the first law and a simple verbal statement of the second law to the existence of entropy as a state variable. This transition, from words to mathematical expression, in Chapter 7, is crucial to thermodynamics. But it also illustrates the distinction between theoretical physics, which in its fullness continually makes transitions of this sort, and the mathematical manipulation of physically meaningful variables.

Chapters 1–7 present concepts, laws, and important corollaries, Chapters 8–9 useful methods and essential applications. This material and related end-of-chapter problems compose a brief course on classical thermodynamics. The remaining four chapters, 10–13, present more loosely sequenced, if fairly standard, topics: nonfluid variables, equilibrium and stability, two-phase systems, and the third law. In spite of the book’s title, I do, on occasion, invoke the molecular hypothesis—that matter is composed of molecules—in order to motivate unfamiliar equations of state. But statistical methods, quantum concepts, and chemical reactions define, by their exclusion, the boundary of Mere Thermodynamics.

This book is designed for second-, third-, and fourth-year physics, chemistry, and pre-engineering undergraduate students who have studied or are concurrently studying multivariate calculus. Often these students have been inadequately introduced to basic thermodynamics in their first physics course. Sometimes, in my own course, I create time for students to report on topics that address their own interests in thermodynamics.

The Annotated Bibliography describes several books and a few articles that provide foundation for, complement, or build upon the ideas presented here. I am indebted to their authors. But I am especially pleased to acknowledge students, friends, and colleagues who have personally contributed to Mere Thermodynamics. Ralph Baierlein, Galen Gisler, and Joel Krehbiel each read and commented on the whole text; Jeff Buller, Rickey Faehl, Bob Harrington, Blake Johnson, Rhon Keinigs, Dwight Neunschwander, Margaret Penner, Bill Peter, Paul Regier, and Bryce Schmidt each read and commented on part of the text. Chapter 13, on the third law of thermodynamics, could not have been written without the aid of Ralph Baierlein, who gave generously of his time and expertise. Margaret Penner’s senior thesis inspired Appendix B, on the logical consequences of the second law as shown in 21 simple cycles. Willis Overholt skillfully drew or redrew the book’s figures. I wrote much of the text during a sabbatical leave from Bethel College of North Newton, Kansas. Finally, I offer heartfelt thanks to my editor at the Johns Hopkins University Press, Trevor Lipscombe. His encouragement and expertise helped me enjoy the writing process. I dedicate this book to my wife, Allison, and two sons, Nathan and Micah. They stood by me during difficult times.

Mere Thermodynamics

ONE

Definitions

1.1 Thermodynamics

The Greek roots of the word thermodynamics, thermo (heat) and dynamics (power or capacity), neatly compose a definition. Etymologically, thermodynamics means the power created by heat, or as we would now say, the work created by heat. Because engines of many kinds produce work from heat, their study belongs to the science of thermodynamics. The concept of work is familiar from mechanics, but what is heat? Thermodynamics assigns its own special meanings not only to the word heat but also to the terms system, boundary, and state.

1.2 System

A thermodynamic system is simply that part of the universe with which we are concerned. We may, for instance, focus on a bucket of seawater or on a beam of iron. Thermodynamic systems may be composed of several chemically distinct components (like seawater), or exist in several phases (as solid, liquid, or gas), or occupy spatially separate parts. We make progress most rapidly by attending first to simple homogeneous, single-phase systems.

1.3 Boundary, Environment, and Interactions

Each thermodynamic system is surrounded by a boundary separating the system from its environment. Boundaries regulate the interaction between the system and its environnment or between two systems. Boundaries can be divided into several kinds: those that permit or forbid work to be done on or by the system and those that permit or forbid heat to be absorbed or rejected by the system. For instance, a movable boundary allows mechanical work to be done on or by the system (see Fig. 1.1), while a rigid one does not.

1.4 States and State Variables

A set of quantities called state variables define the state of each thermodynamic system. State variables include those appropriate to a simple fluid (pressure and volume), to a surface (surface tension and area), to black-body radiation (energy density and radiation pressure), and to an electrical contact (contact potential and current). Different sets of state variables conveniently describe different thermodynamic systems. When a system interacts with its environment, its state variables change.

FIGURE 1.1 Idealized thermodynamic system with a movable boundary that allows work FΔx to be done on the system.

Appropriate state variables can be identified only after a thorough investigation of the system. Thermodynamic state variables are measured with laboratory instruments—that is, macroscopic devices (pressure gauge, balance, and meter stick)—or are inferred from such measurements.

1.5 Equations of State

State variables enter into relations among themselves called equations of state. These relations can be defined by tables of data, by numerical fits to those data, or by analytic models. An example of an analytic equation of state, which we take up in Chapter 9, is P = E/3V, where P is the pressure, E the internal energy, and V the volume of so-called cavity radiation. The equation of state V = Vo(1 + βoT – κoP) relates the state variables of a solid, where T is the system temperature and Vo, βo, and κo are characterizing constants. Each equation of state reduces the number of independent state variables by one.

Thermodynamics is concerned only with systems described by state variables that are related by equations of state. A thermodynamic description per se makes no direct claim about a system’s ultimate components, about its atoms and molecules, their interactions, and their positions and velocities. Thermodynamics differs from most other sciences in not being reductionist. (See Problem 1.1.)

1.6 Work

Performing work on a system changes its state. Work may be performed in a number of ways and not only, as in Figures 1.1 or 1.2a, by pushing a piston into a cylinder that contains the system. Mechanics and electrodynamics books describe different ways of performing work. For instance, a torque τ applied to a rod that turns a paddle through an angle Δθ does work τΔθ on the liquid in which the paddle is immersed (Fig. 1.2b). One means of doing electrical work is to include within the system a circuit element across which a potential difference ΔΦ is applied. A charge q, in passing through the circuit element, loses energy qΔΦ to the system at a rate IΔΦ, where I = dq/dt (Fig. 1.2c). Another way is to apply an externally generated electric field E to the system. Then, as the system charges move and develop an electric dipole moment ΔP, the environment does a quantity of work E · ΔP on the system (Fig. 1.2d). (See Problem 1.2.)

FIGURE 1.2 Four ways of doing work on a system: (a) compressing the system, (b) rotating a paddle wheel immersed in the system, (c) energizing an electric circuit element that is part of the system, and (d) applying an electric field to charges within the system.

1.7 Heat

A system may interact with its environment in ways other than work. We know this because on occasion the state of a system changes even when no work is done on the system. One has only to think of what happens to a cup of hot coffee sitting on a table; inevitably it becomes colder. Let’s imagine a system with idealized boundary. The boundary is rigid, prohibiting mechanical work interactions; metallic, through which an externally applied electric field cannot penetrate; and impermeable, so that mass and charge cannot enter or leave. Further imagine that this idealized boundary prohibits every other kind of work interaction. Even so, another kind of interaction with the environment can still change the system’s state variables. By definition, that