Physical chemistry for the chemical sciences

Physical chemistry for the chemical sciences / Raymond Chang, John W.. Physical Chemistry for the Chemical Sciences is intended for use in a one-year intro-ductory course in physical ch

University Science Books

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Production Management: Jennifer Uhlich at Wilsted & Taylor

Manuscript Editing: John Murdzek

Design: Robert Ishi, with Yvonne Tsang at Wilsted & Taylor

Composition & Illustrations: Laurel Muller

Cover Design: Genette Itoko McGrew

Printing & Binding: Marquis Book Printing, Inc.

This book is printed on acid-free paper.

Copyright © 2014 by University Science Books

ISBN 978-1-891389-69-6 (hard cover)

ISBN 978-1-78262-087-7 (soft cover), only for distribution outside of North America and Mexico

by the Royal Society of Chemistry.

Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful Requests for permission or further information should be addressed to the Permissions Department, University Science Books.

Library of Congress Cataloging-in-Publication Data

Chang, Raymond.

Physical chemistry for the chemical sciences / Raymond Chang, John W Thoman, Jr.

pages cm

Includes index.

ISBN 978-1-891389-69-6 (alk paper)

1 Chemistry, Physical and theoretical—Textbooks I Thoman, John W., Jr., 1960– II Title QD453.3.C43 2014

Printed in Canada

10 9 8 7 6 5 4 3 2 1

About the cover art: Tunneling in the quantum harmonic oscillator The red horizontal

line represents the zero-point energy_1–

2h νi and the shaded region is the classically

forbidden region in which K0, 0^see Chapter 11h

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Physical Chemistry for the Chemical Sciences is intended for use in a one-year

intro-ductory course in physical chemistry that is typically offered at the junior level (the

third year in a college or university program) Students in the course will have taken

general chemistry and introductory organic chemistry In writing this book, our aim is

to present the standard topics at the appropriate level with emphasis on readability and

clarity While mathematical treatment of many topics is necessary, we have provided

a physical picture wherever possible for understanding the concepts Only the basic

skills of differential and integral calculus are required for working with the equations

The limited number of integral equations needed to solve the end-of-chapter problems

may be readily accessed from handbooks of chemistry and physics or software such

as Mathematica

The 20 chapters of the text can be divided into three parts Chapters 1–9 cover

thermodynamics and related subjects Quantum mechanics and molecular

spectros-copy are treated in Chapters 10–14 The last part (Chapters 15–20) describes

chemi-cal kinetics, photochemistry, intermolecular forces, solids and liquids, and statistichemi-cal

thermodynamics We have chosen a traditional ordering of topics, starting with

ther-modynamics because of the accessibility of the concrete examples and the closeness

to everyday experience For instructors who prefer the “atoms first” or molecular

approach, the order can be readily switched between the first two parts without loss

of continuity

Within each chapter, we introduce topics, define terms, and provide relevant

worked examples, pertinent applications, and experimental details Many chapters

include end-of-chapter appendices, which cover more detailed derivations,

back-ground, or explanation than the body of the chapter Each chapter concludes with a

summary of the most important equations introduced within the chapter, an extensive

and accessible list of further readings, and many end-of-chapter problems Answers

to the even-numbered numerical problems may be found in the back of the book

The end-of-book appendices provide some review of relevant mathematical concepts,

basic physics definitions relevant to chemistry, and thermodynamic data A glossary

enables the student to quickly check definitions Inside of the front and back covers,

we include tables of information that are generally useful throughout the book The

second color (red) enables the student to more easily interpret plots and elaborate

dia-grams and adds a pleasing look to the book

An accompanying Solutions Manual, written by Helen O Leung and Mark D.

Marshall, provides complete solutions to all of the problems in the text This

supple-ment contains many useful ideas and insights into problem-solving techniques

xv

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xvi Preface

The lines drawn between traditional disciplines are continually being modified asnew fields are being defined This book provides a foundation for further study at themore advanced level in physical chemistry, as well as interdisciplinary subjects thatinclude biophysical chemistry, materials science, and environmental chemistry fieldssuch as atmospheric chemistry and biogeochemistry We hope that you find our bookuseful when teaching or learning physical chemistry

It is a pleasure to thank the following people who provided helpful commentsand suggestions: Dieter Bingemann (Williams College), George Bodner (PurdueUniversity), Taina Chao (SUNY Purchase), Nancy Counts Gerber (San Francisco StateUniversity), Donald Hirsh (The College of New Jersey), Raymond Kapral (University

of Toronto), Sarah Larsen (University of Iowa), David Perry (University of Akron),Christopher Stromberg (Hood College), and Robert Topper (The Cooper Union)

We also thank Bruce Armbruster and Kathy Armbruster of University ScienceBooks for their support and general assistance We are fortunate to have JenniferUhlich of Wilsted & Taylor as our production manager Her high professional stan-dard and attention to detail greatly helped the task of transforming the manuscriptinto an attractive final product We very much appreciate Laurel Muller for her artisticand technical skills in laying out the text and rendering many figures Robert Ishi andYvonne Tsang are responsible for the elegant design of the book John Murdzek did ameticulous job of copyediting Our final thanks go to Jane Ellis, who supervised theproject and took care of all the details, big and small

Raymond ChangJohn W Thoman, Jr

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Preface xv

CHAPTER 1 Introduction and Gas Laws 1

1.1 Nature of Physical Chemistry 1

1.2 Some Basic Definitions 1

1.3 An Operational Definition of Temperature 2

1.4 Units 3

•Force 4 •Pressure 4 • Energy 5

• Atomic Mass, Molecular Mass, and the Chemical Mole 61.5 The Ideal Gas Law 7

• The Kelvin Temperature Scale 8 • The Gas Constant R 9

1.6 Dalton’s Law of Partial Pressures 11

1.7 Real Gases 13

• The van der Waals Equation 14 • The Redlich–Kwong Equation 15

• The Virial Equation of State 161.8 Condensation of Gases and the Critical State 18

1.9 The Law of Corresponding States 22

Problems 27

CHAPTER 2 Kinetic Theory of Gases 35

2.1 The Model 35

2.2 Pressure of a Gas 36

2.3 Kinetic Energy and Temperature 38

2.4 The Maxwell Distribution Laws 39

2.5 Molecular Collisions and the Mean Free Path 45

2.6 The Barometric Formula 48

3.1 Work and Heat 73

•Work 73 • Heat 793.2 The First Law of Thermodynamics 80

3.3 Enthalpy 83

• A Comparison of ΔU and ΔH 84

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3.8 Bond Energies and Bond Enthalpies 110

• Bond Enthalpy and Bond Dissociation Enthalpy 111Appendix 3.1 Exact and Inexact Differentials 116

Problems 120

4.1 Spontaneous Processes 1294.2 Entropy 131

• Statistical Definition of Entropy 132 • Thermodynamic Definition ofEntropy 134

4.3 The Carnot Heat Engine 135

• Thermodynamic Efficiency 138 • The Entropy Function 139

• Refrigerators, Air Conditioners, and Heat Pumps 1394.4 The Second Law of Thermodynamics 142

4.5 Entropy Changes 144

• Entropy Change due to Mixing of Ideal Gases 144 • Entropy Changedue to Phase Transitions 146 • Entropy Change due to Heating 1484.6 The Third Law of Thermodynamics 152

• Third-Law or Absolute Entropies 152 • Entropy of ChemicalReactions 155

4.7 The Meaning of Entropy 157

• Isothermal Gas Expansion 160 • Isothermal Mixing of Gases 160

•Heating 160 • Phase Transitions 161 • Chemical Reactions 1614.8 Residual Entropy 161

Appendix 4.1 Statements of the Second Law of Thermodynamics 165Problems 168

CHAPTER 5 Gibbs and Helmholtz Energies and Their Applications 175

5.1 Gibbs and Helmholtz Energies 1755.2 The Meaning of Helmholtz and Gibbs Energies 178

• Helmholtz Energy 178 • Gibbs Energy 1795.3 Standard Molar Gibbs Energy of Formation (ΔfG–°) 1825.4 Dependence of Gibbs Energy on Temperature and Pressure 185

• Dependence of G on Temperature 185 • Dependence of G on

Pressure 1865.5 Gibbs Energy and Phase Equilibria 188

• The Clapeyron and the Clausius–Clapeyron Equations 190

• Phase Diagrams 192 • The Gibbs Phase Rule 1965.6 Thermodynamics of Rubber Elasticity 196

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Contents

Appendix 5.1 Some Thermodynamic Relationships 200

Appendix 5.2 Derivation of the Gibbs Phase Rule 203

6.2 Partial Molar Quantities 215

• Partial Molar Volume 215 • Partial Molar Gibbs Energy 2166.3 Thermodynamics of Mixing 218

6.4 Binary Mixtures of Volatile Liquids 221

• Raoult’s Law 222 • Henry’s Law 2256.5 Real Solutions 228

• The Solvent Component 228 • The Solute Component 2296.6 Phase Equilibria of Two-Component Systems 231

•Distillation 231 •Solid–Liquid Equilibria 2376.7 Colligative Properties 238

• Vapor-Pressure Lowering 239 • Boiling-Point Elevation 239

• Freezing-Point Depression 243 • Osmotic Pressure 245Problems 255

CHAPTER 7 Electrolyte Solutions 261

7.1 Electrical Conduction in Solution 261

• Some Basic Definitions 261 • Degree of Dissociation 266

• Ionic Mobility 268 • Applications of Conductance Measurements 2697.2 A Molecular View of the Solution Process 271

7.3 Thermodynamics of Ions in Solution 274

• Enthalpy, Entropy, and Gibbs Energy of Formation of Ions in Solution 2757.4 Ionic Activity 278

7.5 Debye–Hückel Theory of Electrolytes 282

• The Salting-In and Salting-Out Effects 2867.6 Colligative Properties of Electrolyte Solutions 288

• The Donnan Effect 291Appendix 7.1 Notes on Electrostatics 295

Appendix 7.2 The Donnan Effect Involving Proteins Bearing Multiple Charges 298

Problems 301

8.1 Chemical Equilibrium in Gaseous Systems 305

• Ideal Gases 305 • A Closer Look at Equation 8.7 310

• A Comparison of ΔrG° with ΔrG 311 • Real Gases 3138.2 Reactions in Solution 315

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viii Contents

8.3 Heterogeneous Equilibria 316

•Solubility Equilibria 3188.4 Multiple Equilibria and Coupled Reactions 319

• Principle of Coupled Reactions 3218.5 The Influence of Temperature, Pressure, and Catalysts on the EquilibriumConstant 322

• The Effect of Temperature 322 • The Effect of Pressure 325

• The Effect of a Catalyst 3278.6 Binding of Ligands and Metal Ions to Macromolecules 328

• One Binding Site per Macromolecule 328 •n Equivalent Binding Sitesper Macromolecule 329 • Equilibrium Dialysis 332

Appendix 8.1 The Relationship Between Fugacity and Pressure 335

Appendix 8.2 The Relationships Between K1 and K2 and the Intrinsic Dissociation

Constant K 338

Problems 342

9.1 Electrochemical Cells 3519.2 Single-Electrode Potential 3539.3 Thermodynamics of Electrochemical Cells 356

• The Nernst Equation 360 • Temperature Dependence of EMF 3629.4 Types of Electrodes 363

• Metal Electrodes 363 • Gas Electrodes 364 • Metal-InsolubleSalt Electrodes 364 • The Glass Electrode 364 •Ion-SelectiveElectrodes 365

9.5 Types of Electrochemical Cells 365

•Concentration Cells 365 • Fuel Cells 3669.6 Applications of EMF Measurements 367

• Determination of Activity Coefficients 367 • Determination of pH 3689.7 Membrane Potential 368

• The Goldman Equation 371 • The Action Potential 372Problems 378

10.1 Wave Properties of Light 38310.2 Blackbody Radiation and Planck’s Quantum Theory 38610.3 The Photoelectric Effect 388

10.4 Bohr’s Theory of the Hydrogen Emission Spectrum 39010.5 de Broglie’s Postulate 397

10.6 The Heisenberg Uncertainty Principle 40110.7 Postulates of Quantum Mechanics 40310.8 The Schrödinger Wave Equation 40910.9 Particle in a One-Dimensional Box 412

• Electronic Spectra of Polyenes 41810.10 Particle in a Two-Dimensional Box 420

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Contents

10.11 Particle on a Ring 425

10.12 Quantum Mechanical Tunneling 428

• Scanning Tunneling Microscopy 431Appendix 10.1 The Bracket Notation in Quantum Mechanics 433

•Intensity 453 • Selection Rules 455 • Signal-to-Noise Ratio 456

• The Beer–Lambert Law 45711.2 Microwave Spectroscopy 458

• The Rigid Rotor Model 458 • Rigid Rotor Energy Levels 463

• Microwave Spectra 46411.3 Infrared Spectroscopy 469

• The Harmonic Oscillator 469 • Quantum Mechanical Solution tothe Harmonic Oscillator 471 • Tunneling and the Harmonic OscillatorWave Functions 474 • IR Spectra 475 • Simultaneous Vibrationaland Rotational Transitions 479

11.4 Symmetry and Group Theory 482

• Symmetry Elements 482 • Molecular Symmetry and DipoleMoment 483 • Point Groups 484 • Character Tables 48411.5 Raman Spectroscopy 486

• Rotational Raman Spectra 489Appendix 11.1 Fourier-Transform Infrared Spectroscopy 491

Problems 496

CHAPTER 12 Electronic Structure of Atoms 503

12.1 The Hydrogen Atom 503

12.2 The Radial Distribution Function 505

12.3 Hydrogen Atomic Orbitals 510

12.4 Hydrogen Atom Energy Levels 514

12.5 Spin Angular Momentum 515

12.6 The Helium Atom 517

12.7 Pauli Exclusion Principle 519

x Contents

CHAPTER 13 Molecular Electronic Structure and the Chemical Bond 557

13.1 The Hydrogen Molecular Cation 55713.2 The Hydrogen Molecule 561

13.3 Valence Bond Approach 56313.4 Molecular Orbital Approach 56713.5 Homonuclear and Heteronuclear Diatomic Molecules 570

• Homonuclear Diatomic Molecules 570 • Heteronuclear DiatomicMolecules 573 • Electronegativity, Polarity, and Dipole Moments 57613.6 Polyatomic Molecules 578

• Molecular Geometry 578 • Hybridization of Atomic Orbitals 57913.7 Resonance and Electron Delocalization 585

13.8 Hückel Molecular Orbital Theory 589

• Ethylene^C2H4h 590 • Butadiene^C4H6h 595

• Cyclobutadiene^C4H4h 59813.9 Computational Chemistry Methods 600

•Molecular Mechanics^Force Fieldh Methods 601 • Empirical andSemi-Empirical Methods 601 • Ab Initio Methods 602

Problems 605

CHAPTER 14 Electronic Spectroscopy and Magnetic Resonance

Spectroscopy 611

14.1 Molecular Electronic Spectroscopy 611

• Organic Molecules 613 •Charge-Transfer Interactions 616

• Application of the Beer–Lambert Law 61714.2 Fluorescence and Phosphorescence 619

•Fluorescence 619 •Phosphorescence 62114.3 Lasers 622

• Properties of Laser Light 62614.4 Applications of Laser Spectroscopy 629

•Laser-Induced Fluorescence 629 • Ultrafast Spectroscopy 630

•Single-Molecule Spectroscopy 63214.5 Photoelectron Spectroscopy 63314.6 Nuclear Magnetic Resonance Spectroscopy 637

• The Boltzmann Distribution 640 • Chemical Shifts 641

• Spin–Spin Coupling 642 • NMR and Rate Processes 644

• NMR of Nuclei Other Than1H 646 • Solid-State NMR 648

• Fourier-Transform NMR 649 • Magnetic Resonance Imaging

^MRIh 65114.7 Electron Spin Resonance Spectroscopy 652Appendix 14.1 The Franck–Condon Principle 657Appendix 14.2 A Comparison of FT-IR and FT-NMR 659Problems 665

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•Zero-Order Reactions 673 • First-Order Reactions 674

• Second-Order Reactions 678 • Determination of Reaction Order 68115.3 Molecularity of a Reaction 683

• Unimolecular Reactions 684 • Bimolecular Reactions 686

• Termolecular Reactions 68615.4 More Complex Reactions 686

• Reversible Reactions 686 • Consecutive Reactions 688

• Chain Reactions 69015.5 The Effect of Temperature on Reaction Rate 691

• The Arrhenius Equation 69215.6 Potential-Energy Surfaces 694

15.7 Theories of Reaction Rates 695

• Collision Theory 696 • Transition-State Theory 698

• Thermodynamic Formulation of Transition-State Theory 69915.8 Isotope Effects in Chemical Reactions 703

15.9 Reactions in Solution 705

15.10 Fast Reactions in Solution 707

• The Flow Method 708 • The Relaxation Method 70915.11 Oscillating Reactions 712

15.12 Enzyme Kinetics 714

• Enzyme Catalysis 715 • The Equations of Enzyme Kinetics 716

• Michaelis–Menten Kinetics 717 • Steady-State Kinetics 718

• The Significance of KM and Vmax 721Appendix 15.1 Derivation of Equation 15.9 724

Appendix 15.2 Derivation of Equation 15.51 726

• Formation of Nitrogen Oxides 755 • Formation of O3 755

• Formation of Hydroxyl Radical 756 • Formation of OtherSecondary Pollutants 757 • Harmful Effects and Prevention ofPhotochemical Smog 757

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xii Contents

16.5 Stratospheric Ozone 759

• Formation of the Ozone Layer 759 • Destruction of Ozone 760

• Polar Ozone Holes 762 • Ways to Curb Ozone Depletion 76316.6 Chemiluminescence and Bioluminescence 764

•Chemiluminescence 764 •Bioluminescence 76516.7 Biological Effects of Radiation 766

• Sunlight and Skin Cancer 766 •Photomedicine 767

• Light-Activated Drugs 768Problems 774

17.1 Intermolecular Interactions 77917.2 The Ionic Bond 780

17.3 Types of Intermolecular Forces 782

• Dipole–Dipole Interaction 782 • Ion–Dipole Interaction 784

• Ion–Induced Dipole and Dipole–Induced Dipole Interactions 785

•Dispersion, or London, Interactions 788 • Repulsive and TotalInteractions 789

17.4 Hydrogen Bonding 79117.5 The Structure and Properties of Water 796

• The Structure of Ice 797 • The Structure of Water 798

• Some Physiochemical Properties of Water 80017.6 Hydrophobic Interaction 801

Problems 806

18.1 Classification of Crystal Systems 80918.2 The Bragg Equation 812

18.3 Structural Determination by X-Ray Diffraction 814

• The Powder Method 816 • Determination of the Crystal Structure ofNaCl 817 • The Structure Factor 820 •Neutron Diffraction 82218.4 Types of Crystals 823

• Metallic Crystals 823 • Ionic Crystals 829 •CovalentCrystals 834 • Molecular Crystals 835

Appendix 18.1 Derivation of Equation 18.3 836Problems 840

19.1 Structure of Liquids 84319.2 Viscosity 845

• Blood Flow in the Human Body 84819.3 Surface Tension 851

• The Capillary-Rise Method 852 • Surface Tension in the Lungs 85419.4 Diffusion 856

• Fick’s Laws of Diffusion 857

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20.1 The Boltzmann Distribution Law 875

20.2 The Partition Function 878

20.3 Molecular Partition Function 881

• Translational Partition Function 881 • Rotational PartitionFunction 883 • Vibrational Partition Function 884 •ElectronicPartition Function 886

20.4 Thermodynamic Quantities from Partition Functions 886

• Internal Energy and Heat Capacity 887 •Entropy 88820.5 Chemical Equilibrium 893

20.6 Transition-State Theory 898

• Comparison Between Collision Theory and Transition-State Theory 900

Appendix 20.1 Justification of Q 5 q N/N! for Indistinguishable Molecules 903

Problems 905

Appendix A Review of Mathematics and Physics 907

Appendix B Thermodynamic Data 917

Glossary 923

Answers to Even–Numbered Computational Problems 937

Index 941

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c h a p t e r

1

1

Introduction and Gas Laws

And it’s hard, and it’s hard, ain’t it hard, good Lord.

— Woody Guthrie*

1.1 Nature of Physical Chemistry

Physical chemistry can be described as a set of characteristically quantitative

approaches to the study of chemical problems A physical chemist seeks to predict

and/or explain chemical events using certain models and postulates

Because problems encountered in physical chemistry are diversified and often

complex, they require a number of different approaches For example, in the study

of thermodynamics and rates of chemical reactions, we employ a

phenomenologi-cal, macroscopic approach But a microscopic, molecular approach based on quantum

mechanics is necessary to understand the kinetic behavior of molecules and reaction

mechanisms Ideally, we study all phenomena at the molecular level, because it is here

that change occurs In fact, our knowledge of atoms and molecules is neither extensive

nor thorough enough to permit this type of investigation in all cases, and we

some-times have to settle for a good, semiquantitative understanding It is useful to keep in

mind the scope and limitations of a given approach

1.2 Some Basic Definitions

Before we discuss the gas laws, it is useful to define a few basic terms that will be used

throughout the book We often speak of the system in reference to a particular part of

the universe in which we are interested Thus, a system could be a collection of helium

molecules in a container, a NaCl solution, a tennis ball, or a Siamese cat Having defined

a system, we call the rest of the universe the surroundings There are three types of

systems An open system is one that can exchange both mass and energy with its

sur-roundings A closed system is one that does not exchange mass with its surroundings

but can exchange energy An isolated system is one that can exchange neither mass nor

energy with its surroundings^Figure 1.1h To completely define a system, we need to

understand certain experimental variables, such as pressure, volume, temperature, and

composition, which collectively describe the state of the system.

* “Hard, Ain’t It Hard.” Words and Music by Woody Guthrie TRO-© Copyright 1952

Ludlow Music, Inc., New York, N.Y Used by permission

A system is separated fromthe surroundings by adefinite boundary, such aswalls or surfaces

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