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Physical Chemistry

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vii

Contents

• 1 Gases and the Zeroth Law of Thermodynamics Preface xv
  • 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 Defined
  • 1.9 Summary
  • Exercises
• 2 The First Law of Thermodynamics
  • 2.1 Synopsis
  • 2.2 Work and Heat
  • 2.3 Internal Energy and the First Law of Thermodynamics
  • 2.4 State Functions
  • 2.5 Enthalpy
  • 2.6 Changes in State Functions
  • 2.7 Joule-Thomson Coefficients
  • 2.8 More on Heat Capacities
  • 2.9 Phase Changes
  • 2.10 Chemical Changes
  • 2.11 Changing Temperatures
  • 2.12 Biochemical Reactions
  • 2.13 Summary
  • Exercises
• 3 The Second and Third Laws of Thermodynamics
  • 3.1 Synopsis
  • 3.2 Limits of the First Law
  • 3.3 The Carnot Cycle and Efficiency
  • 3.4 Entropy and the Second Law of Thermodynamics
  • 3.5 More on Entropy
  • 3.6 Order and the Third Law of Thermodynamics
  • 3.7 Entropies of Chemical Reactions
  • 3.8 Summary
  • Exercises
• 4 Free Energy and Chemical Potential
  • 4.1 Synopsis
  • 4.2 Spontaneity Conditions
  • 4.3 The Gibbs Free Energy and the Helmholtz Energy
  • 4.4 Natural Variable Equations and Partial Derivatives
  • 4.5 The Maxwell Relationships
  • 4.6 Using Maxwell Relationships
  • 4.7 Focusing on  G
    • Quantities 4.8 The Chemical Potential and Other Partial Molar
  • 4.9 Fugacity
  • 4.10 Summary
  • Exercises
• 5 Introduction to Chemical Equilibrium
  • 5.1 Synopsis
  • 5.2 Equilibrium
  • 5.3 Chemical Equilibrium
  • 5.4 Solutions and Condensed Phases
  • 5.5 Changes in Equilibrium Constants
  • 5.6 Amino Acid Equilibria
  • 5.7 Summary
  • Exercises
• 6 Equilibria in Single-Component Systems
  • 6.1 Synopsis
  • 6.2 A Single-Component System
  • 6.3 Phase Transitions
  • 6.4 The Clapeyron Equation
  • 6.5 The Clausius-Clapeyron Equation
  • 6.6 Phase Diagrams and the Phase Rule
  • 6.7 Natural Variables and Chemical Potential
  • 6.8 Summary
  • Exercises
• 7 Equilibria in Multiple-Component Systems
  • 7.1 Synopsis
  • 7.2 The Gibbs Phase Rule
  • 7.3 Two Components: Liquid/Liquid Systems
  • 7.4 Nonideal Two-Component Liquid Solutions
  • 7.5 Liquid/Gas Systems and Henry’s Law
  • 7.6 Liquid/Solid Solutions
  • 7.7 Solid/Solid Solutions
  • 7.8 Colligative Properties
  • 7.9 Summary
  • Exercises
• 8 Electrochemistry and Ionic Solutions
  • 8.1 Synopsis
  • 8.2 Charges
  • 8.3 Energy and Work
  • 8.4 Standard Potentials
  • 8.5 Nonstandard Potentials and Equilibrium Constants
  • 8.6 Ions in Solution
  • 8.7 Debye-Hückel Theory of Ionic Solutions
  • 8.8 Ionic Transport and Conductance
  • 8.9 Summary
  • Exercises
• 9 Pre-Quantum Mechanics
  • 9.1 Synopsis
  • 9.2 Laws of Motion
  • 9.3 Unexplainable Phenomena
  • 9.4 Atomic Spectra
  • 9.5 Atomic Structure
  • 9.6 The Photoelectric Effect
  • 9.7 The Nature of Light
  • 9.8 Quantum Theory
  • 9.9 Bohr’s Theory of the Hydrogen Atom
  • 9.10 The de Broglie Equation
  • 9.11 Summary: The End of Classical Mechannics
  • Exercises
• 10 Introduction to Quantum Mechanics - 10.1 Synopsis - 10.2 The Wavefunction - 10.3 Observables and Operators - 10.4 The Uncertainty Principle - Probabilities 10.5 The Born Interpretation of the Wavefunction;
  • 10.6 Normalization
  • 10.7 The Schrödinger Equation
  • 10.8 An Analytic Solution: The Particle-in-a-Box
  • 10.9 Average Values and Other Properties
  • 10.10 Tunneling
  • 10.11 The Three-Dimensional Particle-in-a-Box
  • 10.12 Degeneracy
  • 10.13 Orthogonality
  • 10.14 The Time-Dependent Schrödinger Equation
  • 10.15 Summary
  • Exercises
  • Hydrogen Atom 11 Quantum Mechanics: Model Systems and the
  • 11.1 Synopsis
  • 11.2 The Classical Harmonic Oscillator
  • 11.3 The Quantum-Mechanical Harmonic Oscillator
  • 11.4 The Harmonic Oscillator Wavefunctions
  • 11.5 The Reduced Mass
  • 11.6 Two-Dimensional Rotations
  • 11.7 Three-Dimensional Rotations
  • 11.8 Other Observables in Rotating Systems
  • 11.9 The Hydrogen Atom: A Central Force Problem
  • 11.10 The Hydrogen Atom: The Quantum-Mechanical Solution
  • 11.11 The Hydrogen Atom Wavefunctions
  • 11.12 Summary
  • Exercises
• 12 Atoms and Molecules
  • 12.1 Synopsis
  • 12.2 Spin
  • 12.3 The Helium Atom
  • 12.4 Spin Orbitals and the Pauli Principle
  • 12.5 Other Atoms and the Aufbau Principle
  • 12.6 Perturbation Theory
  • 12.7 Variation Theory
  • 12.8 Linear Variation Theory
  • 12.9 Comparison of Variation and Perturbation Theories
    • Approximation 12.10 Simple Molecules and the Born-Oppenheimer
  • 12.11 Introduction to LCAO-MO Theory
  • 12.12 Properties of Molecular Orbitals
  • 12.13 Molecular Orbitals of Other Diatomic Molecules
  • 12.14 Summary
  • Exercises
• 13 Introduction to Symmetry in Quantum Mechanics
  • 13.1 Synopsis
  • 13.2 Symmetry Operations and Point Groups
  • 13.3 The Mathematical Basis of Groups
  • 13.4 Molecules and Symmetry
  • 13.5 Character Tables
  • 13.6 Wavefunctions and Symmetry
  • 13.7 The Great Orthogonality Theorem
  • 13.8 Using Symmetry in Integrals
  • 13.9 Symmetry-Adapted Linear Combinations
  • 13.10 Valence Bond Theory
  • 13.11 Hybrid Orbitals
  • 13.12 Summary
  • Exercises
• 14 Rotational and Vibrational Spectroscopy
  • 14.1 Synopsis
  • 14.2 Selection Rules
  • 14.3 The Electromagnetic Spectrum
  • 14.4 Rotations in Molecules
  • 14.5 Selection Rules for Rotational Spectroscopy
  • 14.6 Rotational Spectroscopy
  • 14.7 Centrifugal Distortions
  • 14.8 Vibrations in Molecules
  • 14.9 The Normal Modes of Vibration
  • 14.10 Quantum-Mechanical Treatment of Vibrations
  • 14.11 Selection Rules for Vibrational Spectroscopy
    • Molecules 14.12 Vibrational Spectroscopy of Diatomic and Linear
  • 14.13 Symmetry Considerations for Vibrations
  • 14.14 Vibrational Spectroscopy of Nonlinear Molecules
  • 14.15 Nonallowed and Nonfundamental Vibrational Transitions
  • 14.16 Fingerprint Regions
  • 14.17 Rotational-Vibrational Spectroscopy
  • 14.18 Raman Spectroscopy
  • 14.19 Summary
  • Exercises
• 15 Introduction to Electronic Spectroscopy and Structure
  • 15.1 Synopsis
  • 15.2 Selection Rules
  • 15.3 The Hydrogen Atom
  • 15.4 Angular Momenta: Orbital and Spin
    • Coupling 15.5 Multiple Electrons: Term Symbols and Russell-Saunders
  • 15.6 Electronic Spectra of Diatomic Molecules
  • 15.7 Vibrational Structure and the Franck-Condon Principle
  • 15.8 Electronic Spectra of Polyatomic Molecules
    • Hückel Approximations 15.9 Electronic Spectra of  Electron Systems:
  • 15.10 Benzene and Aromaticity
  • 15.11 Fluorescence and Phosphorescence
  • 15.12 Lasers
  • 15.13 Summary
  • Exercises
• 16 Introduction to Magnetic Spectroscopy
  • 16.1 Synopsis
  • 16.2 Magnetic Fields, Magnetic Dipoles, and Electric Charges
  • 16.3 Zeeman Spectroscopy
  • 16.4 Electron Spin Resonance
  • 16.5 Nuclear Magnetic Resonance
  • 16.6 Summary
  • Exercises
• 17 Statistical Thermodynamics: Introduction
  • 17.1 Synopsis
  • 17.2 Some Statistics Necessities
  • 17.3 The Ensemble
    • Distribution 17.4 The Most Probable Distribution: Maxwell-Boltzmann
  • 17.5 Thermodynamic Properties from Statistical Thermodynamics
  • 17.6 The Partition Function: Monatomic Gases
  • 17.7 State Functions in Terms of Partition Functions
  • 17.8 Summary
  • Exercises
• 18 More Statistical Thermodynamics
  • 18.1 Synopsis
  • 18.2 Separating q : Nuclear and Electronic Partition Functions
  • 18.3 Molecules: Electronic Partition Functions
  • 18.4 Molecules: Vibrations
  • 18.5 Diatomic Molecules: Rotations
  • 18.6 Polyatomic Molecules: Rotations
  • 18.7 The Partition Function of a System
  • 18.8 Thermodynamic Properties of Molecules from Q
  • 18.9 Equilibria
  • 18.10 Crystals
  • 18.11 Summary
  • Exercises
• 19 The Kinetic Theory of Gases
  • 19.1 Synopsis
  • 19.2 Postulates and Pressure
    • Particles 19.3 Definitions and Distributions of Velocities of Gas
  • 19.4 Collisions of Gas Particles
  • 19.5 Effusion and Diffusion
  • 19.6 Summary
  • Exercises
• 20 Kinetics
  • 20.1 Synopsis
  • 20.2 Rates and Rate Laws
  • 20.3 Characteristics of Specific Initial Rate Laws
  • 20.4 Equilibrium for a Simple Reaction
  • 20.5 Parallel and Consecutive Reactions
  • 20.6 Temperature Dependence
  • 20.7 Mechanisms and Elementary Processes
  • 20.8 The Steady-State Approximation
  • 20.9 Chain and Oscillating Reactions
  • 20.10 Transition-State Theory
  • 20.11 Summary
  • Exercises
• 21 The Solid State: Crystals
  • 21.1. Synopsis
  • 21.2 Types of Solids
  • 21.3 Crystals and Unit Cells
  • 21.4 Densities
  • 21.5 Determination of Crystal Structures
  • 21.6 Miller Indices
  • 21.7 Rationalizing Unit Cells
  • 21.8 Lattice Energies of Ionic Crystals
  • 21.9 Crystal Defects and Semiconductors
  • 21.10 Summary
  • Exercises
• 22 Surfaces
  • 22.1 Synopsis
  • 22.2 Liquids: Surface Tension
  • 22.3 Interface Effects
  • 22.4 Surface Films
  • 22.5 Solid Surfaces
  • 22.6 Coverage and Catalysis
    • 22.7 Summary
    • Exercises
• Appendixes
  • 1 Useful Integrals
  • 2 Thermodynamic Properties of Various Substances
  • 3 Character Tables
  • 4 Infrared Correlation Tables
  • 5 Nuclear Properties
• Answers to Selected Exercises
• Photo Credits
• Index

Most physical chemistry texts follow a formula for covering the major top-

ics: 1/3 thermodynamics, 1/3 quantum mechanics, and 1/3 statistical thermo-

dynamics, kinetics, and various other topics. This text follows that general for-

mula. The section on thermodynamics starts with gases and ends in electro-

chemistry, 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 sur-

faces. 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 under-

graduate 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 con-

cepts, 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 mathe-

matics 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.

xvi P R E F A C E

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 sug-

gestions 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 (produc-

tion 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 valu-

able 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:

P R E F A C E xvii

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

Physical Chemistry

11

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 nat-

urally to this approach.

In this first chapter on physical chemistry, we revisit a simple idea from gen-

eral chemistry: gas laws. Gas laws—straightforward mathematical expressions

that relate the observable properties of gases—were among the first quantifi-

cations 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 thermo-

dynamics; in small, or microscopic, systems, it can involve quantum mechan-

ics. 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 ther-

modynamic system, and the macroscopic variables that characterize it. If we are

considering a gas in our system, we will find that various mathematical rela-

tionships 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 calcu-

lus. These mathematical manipulations will be useful in later chapters as we

get deeper into thermodynamics.

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