Ebook Physical chemistry Part 1

Physical Chemistry

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Physical Chemistry
David W. Ball
Cleveland State University

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Contents

Preface xv

1 Gases and the Zeroth Law of Thermodynamics 1
1.1 Synopsis 1
1.2 System, Surroundings, and State 2
1.3 The Zeroth Law of Thermodynamics 3
1.4 Equations of State 5
1.5 Partial Derivatives and Gas Laws 8
1.6 Nonideal Gases 10
1.7 More on Derivatives 18
1.8 A Few Partial Derivatives Defined 20
1.9 Summary 21
Exercises 22

2 The First Law of Thermodynamics 24
2.1 Synopsis 24
2.2 Work and Heat 24
2.3 Internal Energy and the First Law of Thermodynamics 32
2.4 State Functions 33
2.5 Enthalpy 36
2.6 Changes in State Functions 38
2.7 Joule-Thomson Coefficients 42
2.8 More on Heat Capacities 46

2.9 Phase Changes 50
2.10 Chemical Changes 53
2.11 Changing Temperatures 58
2.12 Biochemical Reactions 60
2.13 Summary 62
Exercises 63

vii
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viii

CONTENTS

3 The Second and Third Laws of Thermodynamics 66
3.1 Synopsis 66
3.2 Limits of the First Law 66
3.3 The Carnot Cycle and Efficiency 68
3.4 Entropy and the Second Law of Thermodynamics 72
3.5 More on Entropy 75
3.6 Order and the Third Law of Thermodynamics 79
3.7 Entropies of Chemical Reactions 81
3.8 Summary 85
Exercises 86

4 Free Energy and Chemical Potential 89
4.1
4.2
4.3

4.4
4.5
4.6
4.7
4.8

Synopsis 89
Spontaneity Conditions 89
The Gibbs Free Energy and the Helmholtz Energy 92
Natural Variable Equations and Partial Derivatives 96
The Maxwell Relationships 99
Using Maxwell Relationships 103
Focusing on ⌬G 105
The Chemical Potential and Other Partial Molar
Quantities 108
4.9 Fugacity 110
4.10 Summary 114
Exercises 115

5 Introduction to Chemical Equilibrium 118
5.1 Synopsis 118
5.2 Equilibrium 119
5.3 Chemical Equilibrium 121
5.4 Solutions and Condensed Phases 129
5.5 Changes in Equilibrium Constants 132
5.6 Amino Acid Equilibria 135
5.7 Summary 136
Exercises 138

6 Equilibria in Single-Component Systems 141

6.1 Synopsis 141
6.2 A Single-Component System 145
6.3 Phase Transitions 145
6.4 The Clapeyron Equation 148
6.5 The Clausius-Clapeyron Equation 152
6.6 Phase Diagrams and the Phase Rule 154
6.7 Natural Variables and Chemical Potential 159
6.8 Summary 162
Exercises 163

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CONTENTS

7 Equilibria in Multiple-Component Systems 166
7.1 Synopsis 166
7.2 The Gibbs Phase Rule 167
7.3 Two Components: Liquid/Liquid Systems 169
7.4 Nonideal Two-Component Liquid Solutions 179
7.5 Liquid/Gas Systems and Henry’s Law 183
7.6 Liquid/Solid Solutions 185
7.7 Solid/Solid Solutions 188
7.8 Colligative Properties 193
7.9 Summary 201
Exercises 203

8 Electrochemistry and Ionic Solutions 206
8.1 Synopsis 206
8.2 Charges 207

8.3 Energy and Work 210
8.4 Standard Potentials 215
8.5 Nonstandard Potentials and Equilibrium Constants 218
8.6 Ions in Solution 225
8.7 Debye-Hückel Theory of Ionic Solutions 230
8.8 Ionic Transport and Conductance 234
8.9 Summary 237
Exercises 238

9 Pre-Quantum Mechanics 241
9.1 Synopsis 241
9.2 Laws of Motion 242
9.3 Unexplainable Phenomena 248
9.4 Atomic Spectra 248
9.5 Atomic Structure 251
9.6 The Photoelectric Effect 253
9.7 The Nature of Light 253
9.8 Quantum Theory 257
9.9 Bohr’s Theory of the Hydrogen Atom 262
9.10 The de Broglie Equation 267
9.11 Summary: The End of Classical Mechannics 269
Exercises 271

10 Introduction to Quantum Mechanics 273
10.1
10.2
10.3
10.4
10.5

Synopsis 273
The Wavefunction 274
Observables and Operators 276
The Uncertainty Principle 279
The Born Interpretation of the Wavefunction;
Probabilities 281

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ix


x

CONTENTS

10.6 Normalization 283
10.7 The Schrödinger Equation 285
10.8 An Analytic Solution: The Particle-in-a-Box 288
10.9 Average Values and Other Properties 293
10.10 Tunneling 296
10.11 The Three-Dimensional Particle-in-a-Box 299
10.12 Degeneracy 303
10.13 Orthogonality 306
10.14 The Time-Dependent Schrödinger Equation 308
10.15 Summary 309
Exercises 311

11 Quantum Mechanics: Model Systems and the
Hydrogen Atom 315

11.1 Synopsis 315
11.2 The Classical Harmonic Oscillator 316
11.3 The Quantum-Mechanical Harmonic Oscillator 318
11.4 The Harmonic Oscillator Wavefunctions 324
11.5 The Reduced Mass 330
11.6 Two-Dimensional Rotations 333
11.7 Three-Dimensional Rotations 341
11.8 Other Observables in Rotating Systems 347
11.9 The Hydrogen Atom: A Central Force Problem 352
11.10 The Hydrogen Atom: The Quantum-Mechanical Solution 353
11.11 The Hydrogen Atom Wavefunctions 358
11.12 Summary 365
Exercises 367

12 Atoms and Molecules 370
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10

Synopsis 370
Spin 371
The Helium Atom 374
Spin Orbitals and the Pauli Principle 377

Other Atoms and the Aufbau Principle 382
Perturbation Theory 386
Variation Theory 394
Linear Variation Theory 398
Comparison of Variation and Perturbation Theories 402
Simple Molecules and the Born-Oppenheimer
Approximation 403
12.11 Introduction to LCAO-MO Theory 405
12.12 Properties of Molecular Orbitals 409
12.13 Molecular Orbitals of Other Diatomic Molecules 410
12.14 Summary 413
Exercises 416

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CONTENTS

xi

13 Introduction to Symmetry in Quantum Mechanics 419
13.1 Synopsis 419
13.2 Symmetry Operations and Point Groups 419
13.3 The Mathematical Basis of Groups 423
13.4 Molecules and Symmetry 427
13.5 Character Tables 430
13.6 Wavefunctions and Symmetry 437
13.7 The Great Orthogonality Theorem 438
13.8 Using Symmetry in Integrals 441
13.9 Symmetry-Adapted Linear Combinations 443

13.10 Valence Bond Theory 446
13.11 Hybrid Orbitals 450
13.12 Summary 456
Exercises 457

14 Rotational and Vibrational Spectroscopy 461
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9
14.10
14.11
14.12

Synopsis 461
Selection Rules 462
The Electromagnetic Spectrum 463
Rotations in Molecules 466
Selection Rules for Rotational Spectroscopy 471
Rotational Spectroscopy 473
Centrifugal Distortions 479
Vibrations in Molecules 481
The Normal Modes of Vibration 483
Quantum-Mechanical Treatment of Vibrations 484
Selection Rules for Vibrational Spectroscopy 487

Vibrational Spectroscopy of Diatomic and Linear
Molecules 491
14.13 Symmetry Considerations for Vibrations 496
14.14 Vibrational Spectroscopy of Nonlinear Molecules 498
14.15 Nonallowed and Nonfundamental Vibrational Transitions 503
14.16 Fingerprint Regions 504
14.17 Rotational-Vibrational Spectroscopy 506
14.18 Raman Spectroscopy 511
14.19 Summary 514
Exercises 515

15 Introduction to Electronic Spectroscopy and Structure 519
15.1
15.2
15.3
15.4
15.5

Synopsis 519
Selection Rules 520
The Hydrogen Atom 520
Angular Momenta: Orbital and Spin 522
Multiple Electrons: Term Symbols and Russell-Saunders
Coupling 526

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.


xii

CONTENTS

15.6
15.7
15.8
15.9

Electronic Spectra of Diatomic Molecules 534
Vibrational Structure and the Franck-Condon Principle 539
Electronic Spectra of Polyatomic Molecules 541
Electronic Spectra of ␲ Electron Systems:
Hückel Approximations 543
15.10 Benzene and Aromaticity 546
15.11 Fluorescence and Phosphorescence 548
15.12 Lasers 550
15.13 Summary 556
Exercises 558

16 Introduction to Magnetic Spectroscopy 560
16.1 Synopsis 560
16.2 Magnetic Fields, Magnetic Dipoles, and Electric Charges 561
16.3 Zeeman Spectroscopy 564
16.4 Electron Spin Resonance 567
16.5 Nuclear Magnetic Resonance 571
16.6 Summary 582
Exercises 584

17 Statistical Thermodynamics: Introduction 586
17.1
17.2

17.3
17.4

Synopsis 586
Some Statistics Necessities 587
The Ensemble 590
The Most Probable Distribution: Maxwell-Boltzmann
Distribution 593
17.5 Thermodynamic Properties from Statistical Thermodynamics 600
17.6 The Partition Function: Monatomic Gases 604
17.7 State Functions in Terms of Partition Functions 608
17.8 Summary 613
Exercises 614

18 More Statistical Thermodynamics 616
18.1 Synopsis 617
18.2 Separating q: Nuclear and Electronic Partition Functions 617
18.3 Molecules: Electronic Partition Functions 621
18.4 Molecules: Vibrations 623
18.5 Diatomic Molecules: Rotations 628
18.6 Polyatomic Molecules: Rotations 634
18.7 The Partition Function of a System 636
18.8 Thermodynamic Properties of Molecules from Q 637
18.9 Equilibria 640
18.10 Crystals 644
18.11 Summary 648
Exercises 649

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

CONTENTS

19 The Kinetic Theory of Gases 651
19.1 Synopsis 651
19.2 Postulates and Pressure 652
19.3 Definitions and Distributions of Velocities of Gas
Particles 656
19.4 Collisions of Gas Particles 666
19.5 Effusion and Diffusion 671
19.6 Summary 677
Exercises 678

20 Kinetics 680
20.1 Synopsis 680
20.2 Rates and Rate Laws 681
20.3 Characteristics of Specific Initial Rate Laws 685
20.4 Equilibrium for a Simple Reaction 694
20.5 Parallel and Consecutive Reactions 696
20.6 Temperature Dependence 702
20.7 Mechanisms and Elementary Processes 706
20.8 The Steady-State Approximation 710
20.9 Chain and Oscillating Reactions 714
20.10 Transition-State Theory 719
20.11 Summary 725
Exercises 726

21 The Solid State: Crystals 731
21.1. Synopsis 731
21.2 Types of Solids 732

21.3 Crystals and Unit Cells 733
21.4 Densities 738
21.5 Determination of Crystal Structures 740
21.6 Miller Indices 744
21.7 Rationalizing Unit Cells 752
21.8 Lattice Energies of Ionic Crystals 755
21.9 Crystal Defects and Semiconductors 759
21.10 Summary 760
Exercises 762

22 Surfaces 765
22.1
22.2
22.3
22.4
22.5
22.6

Synopsis 765
Liquids: Surface Tension 766
Interface Effects 771
Surface Films 777
Solid Surfaces 778
Coverage and Catalysis 783

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

xiii

xiv

CONTENTS

22.7 Summary 788
Exercises 790

Appendixes 792
1
2
3
4
5

Useful Integrals 792
Thermodynamic Properties of Various Substances 794
Character Tables 797
Infrared Correlation Tables 802
Nuclear Properties 805

Answers to Selected Exercises 806
Photo Credits 817
Index 819

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.


Preface

Subject: physical chemistry

“Is this subject hard?”
—The entire text of a Usenet posting to sci.chem, September 1, 1994

W

HAT THIS PERSON’S QUESTION LACKED IN LENGTH, it

made up for in angst.
I spent almost an hour composing a response, which I posted. My
response generated about half a dozen direct responses, all supporting my statements. Curiously, only half of the responses were from students; the other half
were from professors.
Generally, I said that physical chemistry isn’t inherently harder than any
other technical subject. It is very mathematical, and students who may have formally satisfied the math requirements (typically calculus) may still find physical chemistry a challenge because it requires them to apply the calculus. Many
instructors and textbooks can be overly presumptuous about the math abilities
of the students, and consequently many students falter—not because they can’t
do the chemistry, but because they can’t follow the math.
Also, in some cases the textbooks themselves are inappropriate for the level
of a junior-year course (in my opinion). Many textbooks contain so much
information that they blow the students away. Many of them are great books—
for reference, on a professor’s bookshelf, or for a graduate student studying for
cumulative exams. But for undergraduate chemistry and chemical engineering
majors taking physical chemistry for the first time? Too much! It’s like using the
Oxford English Dictionary as a text for English 101. Sure, the OED has all the
vocabulary you would ever need, but it’s overkill. Many physical chemistry texts
are great for those who already know physical chemistry, but not for those who
are trying to learn physical chemistry. What is needed is a book that works as a
textbook, not as an encyclopedia, of physical chemistry.
This project is my attempt to address these ideas. Physical Chemistry is meant
to be a textbook for the year-long, calculus-based physical chemistry course for
science and engineering majors. It is meant to be used in its entirety, and it does

not contain a lot of information (found in many other physical chemistry
books) that undergraduate courses do not cover. There is some focus on mathematical manipulations because many students have forgotten how to apply
calculus or could use the review. However, I have tried to keep in mind that this
should be a physical chemistry text, not a math text.
xv

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xvi

PREFACE

Most physical chemistry texts follow a formula for covering the major topics: 1/3 thermodynamics, 1/3 quantum mechanics, and 1/3 statistical thermodynamics, kinetics, and various other topics. This text follows that general formula. The section on thermodynamics starts with gases and ends in electrochemistry, which is a fairly standard range of topics. The eight-chapter section
on quantum mechanics and its applications to atoms and molecules starts on a
more historical note. In my experience, students have little or no idea of why
quantum mechanics was developed, and consequently they never recognize its
importance, conclusions, or even its necessity. Therefore, Chapter 9 focuses on
pre-quantum mechanics so students can develop an understanding of the state
of classical science and how it could not explain the universe. This leads into an
introduction to quantum mechanics and how it provides a useful model.
Several chapters of symmetry and spectroscopy follow. In the last six chapters,
this text covers statistical thermodynamics (intentionally not integrated with
phenomenological thermodynamics), kinetic theory, kinetics, crystals, and surfaces. The text does not have separate chapters on photochemistry, liquids,
molecular beams, thermal physics, polymers, and so on (although these topics
may be mentioned throughout the text). This is not because I find these topics
unimportant; I simply do not think that they must be included in an undergraduate physical chemistry textbook.
Each chapter opens with a synopsis of what the chapter will cover. In other
texts, the student reads along blindly, not knowing where all the derivations and
equations are leading. Indeed, other texts have a summary at the end of the

chapters. In this text, a summary is given at the beginning of the chapter so the
students can see where they are going and why. Numerous examples are
sprinkled throughout all of the chapters, and there is an emphasis on the units
in a problem, which are just as important as the numbers.
Exercises at the end of each chapter are separated by section so the student
can better coordinate the chapter material with the problem. There are over
1000 end-of-chapter exercises to give students an opportunity to practice the
concepts from the text. Although some mathematical derivations are included
in the exercises, the emphasis is on exercises that make the students use the concepts, rather than just derive them. This, too, has been intentional on my part.
Many answers to the exercises are included in an answer section at the back of
the book. There are also end-of-chapter exercises that require symbolic mathematics software like MathCad or Maple (or even a high-level calculator), to
practice some manipulations of the concepts. Only a few per chapter, they
require more advanced skills and can be used as group assignments.
For a school on the quarter system, the material in physical chemistry almost
naturally separates itself into three sections: thermodynamics (Chapters 1–8),
quantum mechanics (Chapters 9–16), and other topics (Chapters 17–22). For
a school on the semester system, instructors might want to consider pairing the
thermodynamics chapters with the later chapters on kinetic theory (Chapter
19) and kinetics (Chapter 20) in the first term, and including Chapters 17 and
18 (statistical thermodynamics) and Chapters 21 and 22 (crystalline solids and
surfaces) with the quantum mechanics chapters in the second term.
Professors: For a year-long sequence, you should be able to cover the entire
book (and feel free to supplement with special topics as you see fit).
Students: For a year-long sequence, you should be able to read the entire
book. You, too, can do it.
If you want an encyclopedia of physical chemistry, this is not the book for
you. Other well-known books will serve that need. My hope is that students and
teachers alike will appreciate this as a textbook of physical chemistry.

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

PREFACE

xvii

Acknowledgments
No project of this magnitude is the effort of one person. Chris Conti, a former
editor for West Publishing, was enthusiastic about my ideas for this project long
before anything was written down. His expressions of enthusiasm and moral
support carried me through long periods of indecision. Lisa Moller and Harvey
Pantzis, with the help of Beth Wilbur, got this project rolling at Brooks/Cole.
They moved on to other things soon after I started, but I was fortunate to get
Keith Dodson to serve as developmental editor. His input, guidance, and suggestions were appreciated. Nancy Conti helped with all the paper-shuffling and
reviewing, and Marcus Boggs and Emily Levitan were there to see this project
to its final production. I am in awe of the talents of Robin Lockwood (production editor), Anita Wagner (copy editor), and Linda Rill (photo editor). They
made me feel as if I were the weakest link on the team (perhaps as it should be).
There are undoubtedly many others at Brooks/Cole who are leaving their
indelible mark on this text. Thanks to everyone for their assistance.
At various stages in its preparation, the entire manuscript was class-tested by
students in several physical chemistry offerings at my university. Their feedback
was crucial to this project, since you don’t know how good a book is until you
actually use it. Use of the manuscript wasn’t entirely voluntary on their part
(although they could have taken the course from some other instructor), but
most of the students took on the task in good spirits and provided some valuable comments. They have my thanks: David Anthony, Larry Brown, Robert
Coffman, Samer Dashi, Ruot Duany, Jim Eaton, Gianina Garcia, Carolyn Hess,
Gretchen Hung, Ed Juristy, Teresa Klun, Dawn Noss, Cengiz Ozkose, Andrea
Paulson, Aniko Prisko, Anjeannet Quint, Doug Ratka, Mark Rowitz, Yolanda
Sabur, Prabhjot Sahota, Brian Schindly, Lynne Shiban, Tony Sinito, Yelena
Vayner, Scott Wisniewski, Noelle Wojciechowicz, Zhiping Wu, and Steve

Zamborsky. I would like to single out the efforts of Linnea Baudhuin, a student who
performed one of the more comprehensive evaluations of the entire manuscript.
I would like to thank my faculty colleagues Tom Flechtner, Earl Mortensen,
Bob Towns, and Yan Xu for their support. One regret is that my late colleague
John Luoma, who read several parts of the manuscript and made some very
helpful suggestions, did not see this project to its end. My appreciation also goes
to the College of Arts and Science, Cleveland State University, for support of a
two-quarter sabbatical during which I was able to make substantial progress on
this project.
External reviewers gave feedback at several stages. I might not have always
followed their suggestions, but their constructive criticism was appreciated.
Thanks to:
Samuel A. Abrash, University of
Richmond
Steven A. Adelman, Purdue
University
Shawn B. Allin, Lamar University
Stephan B. H. Bach, University of
Texas at San Antonio
James Baird, University of
Alabama in Huntsville
Robert K. Bohn, University of
Connecticut
Kevin J. Boyd, University of New
Orleans

Linda C. Brazdil, Illinois
Mathematics and Science
Academy
Thomas R. Burkholder,

Central Connecticut State
University
Paul Davidovits, Boston College
Thomas C. DeVore, James
Madison University
D. James Donaldson, University
of Toronto
Robert A. Donnelly, Auburn
University

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xviii

PREFACE

Robert C. Dunbar, Case Western
Reserve University
Alyx S. Frantzen, Stephen F. Austin
State University
Joseph D. Geiser, University of
New Hampshire
Lisa M. Goss, Idaho State
University
Jan Gryko, Jacksonville State
University
Tracy Hamilton, University of
Alabama at Birmingham
Robert A. Jacobson, Iowa State

University
Michael Kahlow, University of
Wisconsin at River Falls
James S. Keller, Kenyon College
Baldwin King, Drew University
Stephen K. Knudson, College of
William and Mary
Donald J. Kouri, University of
Houston
Darius Kuciauskas, Virginia
Commonwealth University
Patricia L. Lang, Ball State
University
Danny G. Miles, Jr., Mount
St. Mary’s College
Randy Miller, California State
University at Chico

Frank Ohene, Grambling State
University
Robert Pecora, Stanford University
Lee Pedersen, University of North
Carolina at Chapel Hill
Ronald D. Poshusta, Washington
State University
David W. Pratt, University of
Pittsburgh
Robert Quandt, Illinois State
University
Rene Rodriguez, Idaho State

University
G. Alan Schick, Eastern Kentucky
University
Rod Schoonover, California
Polytechnic State University
Donald H. Secrest, University
of Illinois at Urbana at
Champaign
Michael P. Setter, Ball State
University
Russell Tice, California
Polytechnic State University
Edward A. Walters, University of
New Mexico
Scott Whittenburg, University of
New Orleans
Robert D. Williams, Lincoln
University

I am indebted to Tom Burkholder of Central Connecticut State University
and Mark Waner of John Carroll University for their assistance in performing
accuracy reviews.
In a project such as this, it is extremely unlikely that perfection has been
attained, so I would be grateful to anyone who points out any typo or misprint.
Finally, thanks to my wife Gail, who endured many an evening with me
pounding away at the word processor instead of our sharing a few relaxing
hours together. I hope you think it was worth it, after all.
David W. Ball
Cleveland, Ohio
(216) 687-2456

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.


Physical Chemistry

Copyright 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.


1
1.1 Synopsis
1.2 System, Surroundings, and
State
1.3 The Zeroth Law of
Thermodynamics
1.4 Equations of State
1.5 Partial Derivatives and
Gas Laws
1.6 Nonideal Gases
1.7 More on Derivatives
1.8 A Few Partial Derivatives
1.9 Summary

Gases and the Zeroth Law
of Thermodynamics

M

UCH OF PHYSICAL CHEMISTRY CAN BE PRESENTED IN A

DEVELOPMENTAL MANNER: one can grasp the easy ideas first and
then progress to the more challenging ideas, which is similar to how these
ideas were developed in the first place. Two of the major topics of physical
chemistry—thermodynamics and quantum mechanics—lend themselves naturally to this approach.
In this first chapter on physical chemistry, we revisit a simple idea from general chemistry: gas laws. Gas laws—straightforward mathematical expressions
that relate the observable properties of gases—were among the first quantifications of chemistry, dating from the 1600s, a time when the ideas of alchemy
ruled. Gas laws provided the first clue that quantity, how much, is important
in understanding nature. Some gas laws like Boyle’s, Charles’s, Amontons’s, and
Avogadro’s laws are simple mathematically. Others can be very complex.
In chemistry, the study of large, or macroscopic, systems involves thermodynamics; in small, or microscopic, systems, it can involve quantum mechanics. In systems that change their structures over time, the topic is kinetics. But
they all have basic connections with thermodynamics. We will begin the study
of physical chemistry with thermodynamics.

1.1 Synopsis
This chapter starts with some definitions, an important one being the thermodynamic system, and the macroscopic variables that characterize it. If we are
considering a gas in our system, we will find that various mathematical relationships are used to relate the physical variables that characterize this gas.
Some of these relationships—“gas laws”—are simple but inaccurate. Other gas
laws are more complicated but more accurate. Some of these more complicated
gas laws have experimentally determined parameters that are tabulated to be
looked up later, and they may or may not have physical justification. Finally,
we develop some relationships (mathematical ones) using some simple calculus. These mathematical manipulations will be useful in later chapters as we
get deeper into thermodynamics.
1

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2

CHAPTER 1

Gases and the Zeroth Law of Thermodynamics

System:: the part of the
universe of interest to you

Su

e
o u d in s v e r t h i g l s
:

Figure 1.1 The system is the part of the uni-

verse of interest, and its state is described using
macroscopic variables like pressure, volume, temperature, and moles. The surroundings are everything else. As an example, a system could be a refrigerator and the surroundings could be the rest
of the house (and the surrounding space).

1.2 System, Surroundings, and State
Imagine you have a container holding some material of interest to you, as in
Figure 1.1. The container does a good job of separating the material from
everything else. Imagine, too, that you want to make measurements of the
properties of that material, independent from the measurements of everything
else around it. The material of interest is defined as the system. The “everything
else” is defined as the surroundings. These definitions have an important function because they specify what part of the universe we are interested in: the system. Furthermore, using these definitions, we can immediately ask other questions: What interactions are there between the system and the surroundings?
What is exchanged between the system and the surroundings?
For now, we consider the system itself. How do we describe it? That depends
on the system. For example, a glass of milk is described differently from the interior of a star. But for now, let us pick a simple system, chemically speaking.
Consider a system that consists of a pure gas. How can we describe this system? Well, the gas has a certain volume, a certain pressure, a certain temperature, a certain chemical composition, a certain number of atoms or molecules,
a certain chemical reactivity, and so on. If we can measure, or even dictate, the

values of those descriptors, then we know everything we need to know about
the properties of our system. We say that we know the state of our system.
If the state of the system shows no tendency to change, we say that the system is at equilibrium with the surroundings.* The equilibrium condition is a
fundamental consideration of thermodynamics. Although not all systems are
at equilibrium, we almost always use equilibrium as a reference point for understanding the thermodynamics of a system.
There is one other characteristic of our system that we ought to know: its
energy. The energy is related to all of the other measurables of our system (as
the measurables are related to each other, as we will see shortly). The understanding of how the energy of a system relates to its other measurables is called
thermodynamics (literally, “heat movement’’). Although thermodynamics
(“thermo’’) ultimately deals with energy, it deals with other measurables too,
and so the understanding of how those measurables relate to each other is an
aspect of thermodynamics.
How do we define the state of our system? To begin, we focus on its physical description, as opposed to the chemical description. We find that we are
able to describe the macroscopic properties of our gaseous system using only
a few observables: they are the system’s pressure, temperature, volume, and
amount of matter (see Table 1.1). These measurements are easily identifiable
and have well-defined units. Volume has common units of liter, milliliter, or
cubic centimeter. [The cubic meter is the Système International (SI) unit of
volume but these other units are commonly used as a matter of convenience.]
Pressure has common units of atmosphere, torr, pascal (1 pascal ϭ 1 N/m2 and
is the SI unit for pressure), or bar. Volume and pressure also have obvious minimum values against which a scale can be based. Zero volume and zero pressure are both easily definable. Amount of material is similar. It is easy to specify an amount in a system, and having nothing in the system corresponds to
an amount of zero.
*Equilibrium can be a difficult condition to define for a system. For example, a mixture
of H2 and O2 gases may show no noticeable tendency to change, but it is not at equilibrium.
It’s just that the reaction between these two gases is so slow at normal temperatures and in
the absence of a catalyst that there is no perceptible change.

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1.3 The Zeroth Law of Thermodynamics

Table 1.1

Variable

3

Common state variables and their units
Symbol
Common units

Pressure

p

Volume

V

Temperature

T

Amount

n

Atmosphere, atm (ϭ 1.01325 bar)
Torricelli, torr (ϭ ᎏ7160ᎏ atm)

Pascal (SI unit)
1
ᎏ
Pascal, Pa (ϭ ᎏ
100,000 bar)
Millimeters of mercury, mmHg (ϭ 1 torr)
Cubic meter, m3 (SI unit)
3
Liter, L (ϭ ᎏ101ᎏ
00 m )
Milliliter, mL (ϭ ᎏ101ᎏ
00 L)
Cubic centimeter, cm3 (ϭ 1 mL)
Degrees Celsius, °C, or kelvins, K
°C ϭ K Ϫ 273.15
Moles (can be converted to grams using molecular weight)

The temperature of a system has not always been an obvious measurable of
a system, and the concept of a “minimum temperature” is relatively recent. In
1603, Galileo was the first to try to quantify changes in temperature with a water thermometer. Gabriel Daniel Fahrenheit devised the first widely accepted
numerical temperature scale after developing a successful mercury thermometer in 1714, with zero set at the lowest temperature he could generate in his lab.
Anders Celsius developed a different scale in 1742 in which the zero point was
set at the freezing point of water. These are relative, not absolute, temperatures.
Warmer and colder objects have a temperature value in these relative scales
that is decided with respect to these and other defined points in the scale. In
both cases, temperatures lower than zero are possible and so the temperature
of a system can sometimes be reported as a negative value. Volume, pressure,
and amount cannot have a negative value, and later we define a temperature
scale that cannot, either. Temperature is now considered a well-understood
variable of a system.

1.3 The Zeroth Law of Thermodynamics
Thermodynamics is based on a few statements called laws that have broad application to physical and chemical systems. As simple as these laws are, it took
many years of observation and experimentation before they were formulated
and recognized as scientific laws. Three such statements that we will eventually
discuss are the first, second, and third laws of thermodynamics.
However, there is an even more fundamental idea that is usually assumed
but rarely stated because it is so obvious. Occasionally this idea is referred to
as the zeroth law of thermodynamics, since even the first law depends on it. It
has to do with one of the variables that was introduced in the previous section,
temperature.
What is temperature? Temperature is a measure of how much kinetic energy
the particles of a system have. The higher the temperature, the more energy a
system has, all other variables defining the state of the system (volume, pressure, and so on) being the same. Since thermodynamics is in part the study of
energy, temperature is a particularly important variable of a system.
We must be careful when interpreting temperature, however. Temperature
is not a form of energy. Instead, it is a parameter used to compare amounts of
energy of different systems.

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4

CHAPTER 1

Gases and the Zeroth Law of Thermodynamics

System A

System B

TA

TB

System A

System B

Tϭ?
Figure 1.2 What happens to the temperature

when two individual systems are brought together?

Consider two systems, A and B, in which the temperature of A is greater
than the temperature of B (Figure 1.2). Each is a closed system, which means
that matter cannot move in or out of each system but energy can. The state of
each system is defined by quantities like pressure, volume, and temperature.
The two systems are brought together and physically joined but kept separate
from each other, as shown. For example, two pieces of metal can be brought
into contact with each other, or two containers of gas can be connected by a
closed stopcock. Despite the connection, matter will not be exchanged between
the two systems or with the surroundings.
What about their temperatures, TA and TB? What is always observed is that
energy transfers from one system to another. As energy transfers between the
two systems, the two temperatures change until the point where TA ϭ TB. At
that point, the two systems are said to be at thermal equilibrium. Energy may
still transfer between the systems, but the net change in energy will be zero and
the temperature will not change further. The establishment of thermal equilibrium is independent of the system size. It applies to large systems, small systems, and any combination of large and small systems.

The transfer of energy from one system to another due to temperature differences is called heat. We say that heat has flowed from system A to system B.
Further, if a third system C is in thermal equilibrium with system A, then
TC ϭ TA and system C must be in thermal equilibrium with system B also. This
idea can be expanded to include any number of systems, but the basic idea
illustrated by three systems is summed up by a statement called the zeroth law
of thermodynamics:
The zeroth law of thermodynamics: If two systems (of any size) are in
thermal equilibrium with each other and a third system is in thermal
equilibrium with one of them, then it is in thermal equilibrium with
the other also.
This is obvious from personal experience, and fundamental to thermodynamics.
Example 1.1
Consider three systems at 37.0°C: a 1.0-L sample of H2O, 100 L of neon gas
at 1.00 bar pressure, and a small crystal of sodium chloride, NaCl. Comment
on their thermal equilibrium status in terms of the varying sizes of the systems. Will there be any net transfer of energy if they are brought into contact?
Solution
Thermal equilibrium is dictated by the temperature of the systems involved,
not the sizes. Since all systems are at the same temperature [that is, T(H2O) ϭ
T(Ne) ϭ T(NaCl)], they are all in thermal equilibrium with each other. To
invoke the zeroth law, if the water is in thermal equilibrium with the neon
and the neon is in thermal equilibrium with the sodium chloride, then the
water is in thermal equilibrium with the sodium chloride. No matter what
the relative sizes of the systems are, there should be no net transfer of energy
between any of the three systems.
The zeroth law introduces a new idea. One of the variables that defines the
state of our system (the state variables) changes its value. In this case, the temperature has changed. We are ultimately interested in how the state variables
change and how these changes relate to the energy of our system.

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1.4 Equations of State

5

System A

pϭ1
Vϭ1
T ϭ 100

pϭ3
Vϭ1
T ϭ 300

pϭ2
Vϭ1
T ϭ 200

Same state
System B

pϭ1
Vϭ1
T ϭ 100

pϭ2
Vϭ1
T ϭ 200

Figure 1.3 The state of a system is determined by what the state variables are, not how the
system got there. In this example, the initial and final states of the two Systems (A) and (B)
are the same, regardless of the fact that System (A) was higher in temperature and pressure in the
interim.

The final point with respect to the system and its variables is the fact that
the system does not remember its previous state. The state of the system is dictated by the values of the state variables, not their previous values or how they
changed. Consider the two systems in Figure 1.3. System A goes to a higher
temperature before settling on T ϭ 200 temperature units. System B goes directly from the initial conditions to the final conditions. Therefore, the two
states are the same. It does not matter that the first system was at a higher temperature; the state of the system is dictated by what the state variables are, not
what they were, or how they got there.

1.4 Equations of State
Phenomenological thermodynamics is based on experiment, on measurements
that you might make in a lab, garage, or kitchen. For example, for any fixed
amount of a pure gas, two state variables are pressure, p, and volume, V. Each
can be controlled independently of each other. The pressure can be varied while
the volume is kept constant, or vice versa. Temperature, T, is another state variable that can be changed independently from p and V. However, experience has
shown that if a certain pressure, volume, and temperature were specified for a
particular sample of gas at equilibrium, then all measurable, macroscopic properties of that sample have certain specific values. That is, these three state variables determine the complete state of our gas sample. Notice that we are implying the existence of one other state variable: amount. The amount of material
in the system, designated by n, is usually given in units of moles.
Further, arbitrary values for all four variables p, V, n, and T are not possible
simultaneously. Again, experience (that is, experiment) shows this. It turns out
that only two of the three state variables p, V, and T are truly independent for
any given amount of a gas. Once two values are specified, then the third one
must have a certain value. This means that there is a mathematical equation into
which we can substitute for two of the variables and calculate what the remaining variable must be. Say such an equation requires that we know p and V
and lets us calculate T. Mathematically, there exists some function F such that
F(p, V) ϭ T

at fixed n

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(1.1)


6

CHAPTER 1

Gases and the Zeroth Law of Thermodynamics

where the function is written as F(p, V ) to emphasize that the variables are
pressure and volume, and that the outcome yields the value of the temperature
T. Equations like equation 1.1 are called equations of state. One can also define
equations of state that yield p or V instead of T. In fact, many equations of state
can be algebraically rearranged to yield one of several possible state variables.
The earliest equations of state for gases were determined by Boyle, Charles,
Amontons, Avogadro, Gay-Lussac, and others. We know these equations as the
various gas laws. In the case of Boyle’s gas law, the equation of state involves
multiplying the pressure by the volume to get a number whose value depended
on the temperature of the gas:
p и V ϭ F(T)

at fixed n

(1.2)

whereas Charles’s gas law involves volume and temperature:

V
ᎏᎏ ϭ F(p)
T

at fixed n

(1.3)

Avogadro’s law relates volume and amount, but at fixed temperature and
pressure:
V ϭ F(n)

at fixed T, p

(1.4)

In the above three equations, if the temperature, pressure, or amount were kept
constant, then the respective functions F(T ), F(p), and F(n) would be constants. This means that if one of the state variables that can change does, the
other must also change in order for the gas law to yield the same constant. This
leads to the familiar predictive ability of the above gas laws using the forms
p1V1 ϭ F(T) ϭ p2V2

or

p1V1 ϭ p2V2

(1.5)

Similarly, using equations 1.3 and 1.4, we can get
V

V
ᎏᎏ1 ϭ ᎏᎏ2
T1
T2

(1.6)

V
V
ᎏᎏ1 ϭ ᎏᎏ2
n1
n2

(1.7)

All three gas laws involve volume, and they can be rewritten as
1
V ϰ ᎏᎏ
p
VϰT
Vϰn
where the symbol ϰ means “is proportional to.’’ We can combine the three proportionalities above into one:
nT
V ϰ ᎏᎏ
p

(1.8)

Since p, V, T, and n are the only four independent state variables for a gas, the
proportionality form of equation 1.8 can be turned into an equality by using

a proportionality constant:
nT
V ϭ R и ᎏᎏ
p

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(1.9)