Physical chemistry for the chemical sciences by raymond chang, jr thoman john pdf

PHYSICAL CHEMISTRY
for the Chemical Sciences

Raymond Chang
WILLIAMS COLLEGE

John W. Thoman, Jr.
WILLIAMS COLLEGE

University Science Books

www.uscibooks.com


University Science Books
www.uscibooks.com
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–
QD453.3.C43 2014
541—dc23
2013038473

II. Title.

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 _1–2 hνi and the shaded region is the classically
forbidden region in which K0 , 0 ^see Chapter 11h.


Preface

Physical Chemistry for the Chemical Sciences is intended for use in a one-year introductory 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 spectroscopy are treated in Chapters 10–14. The last part (Chapters 15–20) describes chemical kinetics, photochemistry, intermolecular forces, solids and liquids, and statistical
thermodynamics. We have chosen a traditional ordering of topics, starting with thermodynamics 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, background, 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 diagrams 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 supplement contains many useful ideas and insights into problem-solving techniques.

xv


xvi

Preface

The lines drawn between traditional disciplines are continually being modified as
new fields are being defined. This book provides a foundation for further study at the
more advanced level in physical chemistry, as well as interdisciplinary subjects that
include biophysical chemistry, materials science, and environmental chemistry fields
such as atmospheric chemistry and biogeochemistry. We hope that you find our book
useful when teaching or learning physical chemistry.
It is a pleasure to thank the following people who provided helpful comments
and suggestions: Dieter Bingemann (Williams College), George Bodner (Purdue
University), Taina Chao (SUNY Purchase), Nancy Counts Gerber (San Francisco State
University), 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 Science
Books for their support and general assistance. We are fortunate to have Jennifer
Uhlich of Wilsted & Taylor as our production manager. Her high professional standard and attention to detail greatly helped the task of transforming the manuscript
into an attractive final product. We very much appreciate Laurel Muller for her artistic
and technical skills in laying out the text and rendering many figures. Robert Ishi and
Yvonne Tsang are responsible for the elegant design of the book. John Murdzek did a
meticulous job of copyediting. Our final thanks go to Jane Ellis, who supervised the
project and took care of all the details, big and small.
Raymond Chang
John W. Thoman, Jr.


Contents

Preface
CHAPTER

1

xv

Introduction and Gas Laws 1

1.1
1.2
1.3
1.4

Nature of Physical Chemistry 1
Some Basic Definitions 1
An Operational Definition of Temperature 2
Units 3
• Force 4
• Pressure 4
• Energy 5
• Atomic Mass, Molecular Mass, and the Chemical Mole 6
1.5 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
• The Virial Equation of State 16
1.8 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
2.7 Gas Viscosity 50
2.8 Graham’s Laws of Diffusion and Effusion 53
2.9 Equipartition of Energy 56
Appendix 2.1 Derivation of Equation 2.29 63
Problems 66
CHAPTER

3
3.1
3.2
3.3

The First Law of Thermodynamics 73
Work and Heat 73
• Work 73
• Heat 79
The First Law of Thermodynamics 80
Enthalpy 83

• A Comparison of ΔU and ΔH 84

15


vi

Contents

3.4
3.5

A Closer Look at Heat Capacities 88
Gas Expansion 91
• Isothermal Expansion 92
• Adiabatic Expansion 92
3.6 The Joule–Thomson Effect 96
3.7 Thermochemistry 100
• Standard Enthalpy of Formation 100
• Dependence of Enthalpy of
Reaction on Temperature 107
3.8 Bond Energies and Bond Enthalpies 110
• Bond Enthalpy and Bond Dissociation Enthalpy 111
Appendix 3.1 Exact and Inexact Differentials 116
Problems 120
CHAPTER

4

The Second Law of Thermodynamics 129

4.1
4.2

Spontaneous Processes 129
Entropy 131
• Statistical Definition of Entropy 132
• Thermodynamic Definition of
Entropy 134
4.3 The Carnot Heat Engine 135
• Thermodynamic Efficiency 138
• The Entropy Function 139
• Refrigerators, Air Conditioners, and Heat Pumps 139
4.4 The Second Law of Thermodynamics 142
4.5 Entropy Changes 144
• Entropy Change due to Mixing of Ideal Gases 144
• Entropy Change
due to Phase Transitions 146
• Entropy Change due to Heating 148
4.6 The Third Law of Thermodynamics 152
• Third-Law or Absolute Entropies 152
• Entropy of Chemical
Reactions 155
4.7 The Meaning of Entropy 157
• Isothermal Gas Expansion 160
• Isothermal Mixing of Gases 160
• Heating 160
• Phase Transitions 161
• Chemical Reactions 161
4.8 Residual Entropy 161

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

5
5.1
5.2
5.3
5.4

5.5

5.6

Gibbs and Helmholtz Energies and Their Applications 175
Gibbs and Helmholtz Energies 175
The Meaning of Helmholtz and Gibbs Energies 178
• Helmholtz Energy 178
• Gibbs Energy 179
–
Standard Molar Gibbs Energy of Formation (ΔfG°) 182
Dependence of Gibbs Energy on Temperature and Pressure 185
• Dependence of G on Temperature 185
• Dependence of G on
Pressure 186
Gibbs Energy and Phase Equilibria 188
• The Clapeyron and the Clausius–Clapeyron Equations 190
• Phase Diagrams 192
• The Gibbs Phase Rule 196
Thermodynamics of Rubber Elasticity 196

Contents

Appendix 5.1 Some Thermodynamic Relationships 200
Appendix 5.2 Derivation of the Gibbs Phase Rule 203
Problems 207

CHAPTER

6

Nonelectrolyte Solutions

213

6.1

Concentration Units 213
• Percent by Weight 213
• Mole Fraction ^xh 214
• Molarity ^Mh 214
• Molality ^mh 214
6.2 Partial Molar Quantities 215
• Partial Molar Volume 215
• Partial Molar Gibbs Energy 216
6.3 Thermodynamics of Mixing 218
6.4 Binary Mixtures of Volatile Liquids 221
• Raoult’s Law 222
• Henry’s Law 225

6.5 Real Solutions 228
• The Solvent Component 228
• The Solute Component 229
6.6 Phase Equilibria of Two-Component Systems 231
• Distillation 231
• Solid–Liquid Equilibria 237
6.7 Colligative Properties 238
• Vapor-Pressure Lowering 239
• Boiling-Point Elevation 239
• Freezing-Point Depression 243
• Osmotic Pressure 245
Problems 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 269
7.2 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 275
7.4 Ionic Activity 278

7.5 Debye–Hückel Theory of Electrolytes 282
• The Salting-In and Salting-Out Effects 286
7.6 Colligative Properties of Electrolyte Solutions 288
• The Donnan Effect 291
Appendix 7.1 Notes on Electrostatics 295
Appendix 7.2 The Donnan Effect Involving Proteins Bearing Multiple Charges 298
Problems 301

CHAPTER

8
8.1

8.2

Chemical Equilibrium 305
Chemical Equilibrium in Gaseous Systems 305
• Ideal Gases 305
• A Closer Look at Equation 8.7 310
• A Comparison of Δ rG° with Δ rG 311
• Real Gases 313
Reactions in Solution 315

vii


viii

Contents

8.3

Heterogeneous Equilibria 316
• Solubility Equilibria 318
8.4 Multiple Equilibria and Coupled Reactions 319
• Principle of Coupled Reactions 321
8.5 The Influence of Temperature, Pressure, and Catalysts on the Equilibrium
Constant 322
• The Effect of Temperature 322
• The Effect of Pressure 325
• The Effect of a Catalyst 327
8.6 Binding of Ligands and Metal Ions to Macromolecules 328
• One Binding Site per Macromolecule 328
• n Equivalent Binding Sites
per Macromolecule 329
• Equilibrium Dialysis 332
Appendix 8.1 The Relationship Between Fugacity and Pressure 335
Appendix 8.2 The Relationships Between K1 and K 2 and the Intrinsic Dissociation
Constant K 338
Problems 342

CHAPTER

9

Electrochemistry 351

9.1
9.2
9.3

Electrochemical Cells 351
Single-Electrode Potential 353
Thermodynamics of Electrochemical Cells 356
• The Nernst Equation 360
• Temperature Dependence of EMF 362
9.4 Types of Electrodes 363
• Metal Electrodes 363
• Gas Electrodes 364
• Metal-Insoluble
Salt Electrodes 364
• The Glass Electrode 364
• Ion-Selective
Electrodes 365
9.5 Types of Electrochemical Cells 365
• Concentration Cells 365
• Fuel Cells 366
9.6 Applications of EMF Measurements 367
• Determination of Activity Coefficients 367
• Determination of pH 368
9.7 Membrane Potential 368
• The Goldman Equation 371
• The Action Potential 372
Problems 378

CHAPTER

10
10.1
10.2

10.3
10.4
10.5
10.6
10.7
10.8
10.9

Quantum Mechanics 383

Wave Properties of Light 383
Blackbody Radiation and Planck’s Quantum Theory 386
The Photoelectric Effect 388
Bohr’s Theory of the Hydrogen Emission Spectrum 390
de Broglie’s Postulate 397
The Heisenberg Uncertainty Principle 401
Postulates of Quantum Mechanics 403
The Schrödinger Wave Equation 409
Particle in a One-Dimensional Box 412
• Electronic Spectra of Polyenes 418
10.10 Particle in a Two-Dimensional Box 420


Contents

10.11 Particle on a Ring 425
10.12 Quantum Mechanical Tunneling 428
• Scanning Tunneling Microscopy 431
Appendix 10.1 The Bracket Notation in Quantum Mechanics
Problems 437

CHAPTER

11

433

Applications of Quantum Mechanics to Spectroscopy 447

11.1

Vocabulary of Spectroscopy 447
• Absorption and Emission 447
• Units 448
• Regions of the
Spectrum 448
• Linewidth 449
• Resolution 452
• Intensity 453
• Selection Rules 455
• Signal-to-Noise Ratio 456
• The Beer–Lambert Law 457
11.2 Microwave Spectroscopy 458
• The Rigid Rotor Model 458
• Rigid Rotor Energy Levels 463
• Microwave Spectra 464
11.3 Infrared Spectroscopy 469
• The Harmonic Oscillator 469
• Quantum Mechanical Solution to
the Harmonic Oscillator 471

• Tunneling and the Harmonic Oscillator
Wave Functions 474
• IR Spectra 475
• Simultaneous Vibrational
and Rotational Transitions 479
11.4 Symmetry and Group Theory 482
• Symmetry Elements 482
• Molecular Symmetry and Dipole
Moment 483
• Point Groups 484
• Character Tables 484
11.5 Raman Spectroscopy 486
• Rotational Raman Spectra 489
Appendix 11.1 Fourier-Transform Infrared Spectroscopy 491
Problems 496

CHAPTER

12

Electronic Structure of Atoms 503

12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8

The Hydrogen Atom 503
The Radial Distribution Function 505
Hydrogen Atomic Orbitals 510
Hydrogen Atom Energy Levels 514
Spin Angular Momentum 515
The Helium Atom 517
Pauli Exclusion Principle 519
Aufbau Principle 523
• Hund’s Rules 524
• Periodic Variations in Atomic Properties
12.9 Variational Principle 530
12.10 Hartree–Fock Self-Consistent-Field Method 536
12.11 Perturbation Theory 540
Appendix 12.1 Proof of the Variational Principle 546
Problems 551

528

ix


x

Contents

CHAPTER

13

Molecular Electronic Structure and the Chemical Bond

557

13.1
13.2
13.3
13.4
13.5

The Hydrogen Molecular Cation 557
The Hydrogen Molecule 561
Valence Bond Approach 563
Molecular Orbital Approach 567
Homonuclear and Heteronuclear Diatomic Molecules 570
• Homonuclear Diatomic Molecules 570
• Heteronuclear Diatomic
Molecules 573
• Electronegativity, Polarity, and Dipole Moments 576
13.6 Polyatomic Molecules 578
• Molecular Geometry 578
• Hybridization of Atomic Orbitals 579
13.7 Resonance and Electron Delocalization 585
13.8 Hückel Molecular Orbital Theory 589
• Ethylene ^C2H4h 590
• Butadiene ^C4H6h 595
• Cyclobutadiene ^C4H4h 598
13.9 Computational Chemistry Methods 600
• Molecular Mechanics ^Force Fieldh Methods 601
• Empirical and

Semi-Empirical Methods 601
• Ab Initio Methods 602
Problems 605

CHAPTER

14
14.1

Electronic Spectroscopy and Magnetic Resonance
Spectroscopy 611

Molecular Electronic Spectroscopy 611
• Organic Molecules 613
• Charge-Transfer Interactions 616
• Application of the Beer–Lambert Law 617
14.2 Fluorescence and Phosphorescence 619
• Fluorescence 619
• Phosphorescence 621
14.3 Lasers 622
• Properties of Laser Light 626
14.4 Applications of Laser Spectroscopy 629
• Laser-Induced Fluorescence 629
• Ultrafast Spectroscopy 630
• Single-Molecule Spectroscopy 632
14.5 Photoelectron Spectroscopy 633
14.6 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 Than 1H 646
• Solid-State NMR 648
• Fourier-Transform NMR 649
• Magnetic Resonance Imaging
^MRIh 651
14.7 Electron Spin Resonance Spectroscopy 652
Appendix 14.1 The Franck–Condon Principle 657
Appendix 14.2 A Comparison of FT-IR and FT-NMR 659
Problems 665


Contents

CHAPTER

15

Chemical Kinetics 671

15.1
15.2

Reaction Rate 671
Reaction Order 672
• Zero-Order Reactions 673
• First-Order Reactions 674
• Second-Order Reactions 678
• Determination of Reaction Order 681
15.3 Molecularity of a Reaction 683

• Unimolecular Reactions 684
• Bimolecular Reactions 686
• Termolecular Reactions 686
15.4 More Complex Reactions 686
• Reversible Reactions 686
• Consecutive Reactions 688
• Chain Reactions 690
15.5 The Effect of Temperature on Reaction Rate 691
• The Arrhenius Equation 692
15.6 Potential-Energy Surfaces 694
15.7 Theories of Reaction Rates 695
• Collision Theory 696
• Transition-State Theory 698
• Thermodynamic Formulation of Transition-State Theory 699
15.8 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 709
15.11 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 K M and Vmax 721
Appendix 15.1 Derivation of Equation 15.9 724
Appendix 15.2 Derivation of Equation 15.51 726
Problems 731
CHAPTER

16
16.1

16.2

16.3
16.4

Photochemistry 743
Introduction 743
• Thermal Versus Photochemical Reactions 743
• Primary Versus
Secondary Processes 744
• Quantum Yields 744
• Measurement of
Light Intensity 746
• Action Spectrum 747
Earth’s Atmosphere 748
• Composition of the Atmosphere 748
• Regions of the
Atmosphere 749
• Residence Time 750
The Greenhouse Effect 751
Photochemical Smog 754
• Formation of Nitrogen Oxides 755
• Formation of O3 755
• Formation of Hydroxyl Radical 756
• Formation of Other
Secondary Pollutants 757

• Harmful Effects and Prevention of
Photochemical Smog 757

xi


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 763
16.6 Chemiluminescence and Bioluminescence 764
• Chemiluminescence 764
• Bioluminescence 765
16.7 Biological Effects of Radiation 766
• Sunlight and Skin Cancer 766
• Photomedicine 767
• Light-Activated Drugs 768
Problems 774
CHAPTER

17

Intermolecular Forces

779

17.1
17.2
17.3

Intermolecular Interactions 779
The Ionic Bond 780
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 Total
Interactions 789
17.4 Hydrogen Bonding 791
17.5 The Structure and Properties of Water 796
• The Structure of Ice 797
• The Structure of Water 798
• Some Physiochemical Properties of Water 800
17.6 Hydrophobic Interaction 801
Problems 806
CHAPTER

18

The Solid State

809

18.1
18.2
18.3

Classification of Crystal Systems 809
The Bragg Equation 812
Structural Determination by X-Ray Diffraction 814
• The Powder Method 816
• Determination of the Crystal Structure of
NaCl 817
• The Structure Factor 820
• Neutron Diffraction 822
18.4 Types of Crystals 823
• Metallic Crystals 823
• Ionic Crystals 829
• Covalent
Crystals 834
• Molecular Crystals 835
Appendix 18.1 Derivation of Equation 18.3 836
Problems 840
CHAPTER

19
19.1
19.2
19.3
19.4

The Liquid State 843

Structure of Liquids 843
Viscosity 845
• Blood Flow in the Human Body 848
Surface Tension 851
• The Capillary-Rise Method 852
• Surface Tension in the Lungs
Diffusion 856
• Fick’s Laws of Diffusion 857

854


Contents

19.5

Liquid Crystals 863
• Thermotropic Liquid Crystals 864
• Lyotropic Liquid Crystals
Appendix 19.1 Derivation of Equation 19.13 869
Problems 872

CHAPTER

20

868

Statistical Thermodynamics 875

20.1
20.2
20.3

The Boltzmann Distribution Law 875
The Partition Function 878
Molecular Partition Function 881
• Translational Partition Function 881
• Rotational Partition
Function 883
• Vibrational Partition Function 884
• Electronic
Partition Function 886
20.4 Thermodynamic Quantities from Partition Functions 886
• Internal Energy and Heat Capacity 887
• Entropy 888
20.5 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 qN/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

xiii

chapter

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 phenomenological, 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 sometimes 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 surroundings. 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 from
the surroundings by a
definite boundary, such as
walls or surfaces.

1


2

Chapter 1: Introduction and Gas Laws
Water vapor

Heat

Heat

(a)

(b)

(c)

Figure 1.1
(a) An open system allows the exchange of both mass and energy; (b) a closed system allows
the exchange of energy but not mass; and (c) an isolated system allows exchange of neither
mass nor energy.

Most of the properties of matter may be divided into two classes: extensive properties and intensive properties. Consider, for example, two beakers containing the same
amounts of water at the same temperature. If we combine these two systems by pouring
the water from one beaker to the other, we find that the volume of the water is doubled
and so is its mass. On the other hand, the temperature and the density of the water do
not change. Properties whose values are directly proportional to the amount of the
material present in the system are called extensive properties; those that do not depend
on the amount are called intensive properties. Extensive properties include mass, area,
volume, energy, and electrical charge. As already mentioned, temperature and density
are both intensive properties, and so are pressure and electrical potential. Note that
intensive properties are normally defined as ratios of two extensive properties, such as
pressure 5
density 5

force
area
mass
volume

1.3 An Operational Definition of Temperature
Temperature is a very important quantity in many branches of science, and not surprisingly, it can be defined in a number of different ways. Daily experience tells us
that temperature is a measure of coldness and hotness, but for our purposes we need
a more precise operational definition of temperature. Consider the following system
of a container of gas A. The walls of the container are flexible so that its volume can
expand and contract. This is a closed system that allows heat, but not mass, to flow into
and out of the container. Initially, the pressure and volume are PA and VA, respectively.

Now we bring the container in contact with a similar container of gas B at P B and VB.
Heat exchange will take place until thermal equilibrium is reached. At equilibrium the
pressure and volume of A and B will be altered to PA′, VA′ and PB′, VB′. It is possible to


3

1.4 Units

remove container A temporarily, readjust its pressure and volume to PA″ and VA″, and
still have A in thermal equilibrium with B at PB′ and VB′. In fact, an infinite set of such
values ^PA′, VA′h, ^PA″, VA″h, ^PA‴, VA‴h, … can be obtained that will satisfy the equilibrium
conditions. Figure 1.2 shows a plot of these points.
For all these states of A to be in thermal equilibrium with B, they must have the
same value of a certain variable, which we call temperature. It follows from the discussion above that if two systems are in thermal equilibrium with a third system, then
they must also be in thermal equilibrium with each other. This statement is generally
known as the zeroth law of thermodynamics. The curve in Figure 1.2 is the locus of all
the points that represent the states that can be in thermal equilibrium with system B.
Such a curve is called an isotherm, or “same temperature.” At another temperature, a
different isotherm is obtained.

1.4 Units
In this section we shall review the units chemists use for quantitative measurements.
For many years scientists recorded measurements in metric units, which are related
decimally, that is, by powers of 10. In 1960, however, the General Conference of Weights
and Measures, the international authority on units, proposed a revised metric system
called the International System of Units ^abbreviated SIh. The advantage of the SI
system is that many of its units are derivable from natural constants. For example, the SI
system defines meter ^mh as the length of the path traveled by light in vacuum during a
time interval of 1/299,792,458 of a second. The unit of time, the second, is equivalent to

9,192,631,770 cycles of the radiation associated with a certain electronic transition of the
cesium atom. In contrast, the fundamental unit of mass, the kilogram ^kgh, is defined in
terms of an artifact, not in terms of a naturally occurring phenomenon. One kilogram
is the mass of a platinum–iridium alloy cylinder kept by the International Bureau of
Weights and Measures in Sevres, France.
Table 1.1 gives the seven SI base units and Table 1.2 shows the prefixes used with
SI units. Note that in SI units, temperature is given as K without the degree sign ° and
the unit is plural—for example, 300 kelvins or 300 K. ^More will be said of the Kelvin
Table 1.1
SI Base Units
Base quantity

Name of unit

Symbol

Length

meter

m

Mass

kilogram

kg

Time

second

s

Electrical current

ampere

A

Temperature

kelvin

K

Amount of substance

mole

mol

Luminous intensity

candela

cd

PAЈ,VAЈ
PAЉ,VAЉ

P
PAٞ,VAٞ
PAЉЉ,VAЉЉ
V

Figure 1.2
Plot of pressure versus
volume at constant
temperature for a given
amount of a gas. Such a
graph is called an isotherm.

In 2007, it was discovered
that the alloy had
mysteriously lost about
50 μg!


4

Chapter 1: Introduction and Gas Laws

Table 1.2
Prefixes Used with SI and Metric Units
Prefix

Symbol

Meaning

Example

Tera-

T

1,000,000,000,000, or 1012

Giga-

G

1,000,000,000, or 109

Mega-

M

1,000,000, or 106

Kilo-

k

1,000, or 103

Deci-

d

1/10, or 10−1

1 terameter ^Tmh 5 1 × 1012 m

1 gigameter ^Gmh 5 1 × 109 m

1 megameter ^Mmh 5 1 × 106 m
1 kilometer ^kmh 5 1 × 103 m
1 decimeter ^dmh 5 0.1 m

1 centimeter ^cmh 5 0.01 m

10−2

Centi-

c

1/100, or

Milli-

m

1/1,000, or 10−3

Micro-

μ

1/1,000,000, or 10−6

Nano-

n

1/1,000,000,000, or 10−9

Pico-

p

1/1,000,000,000,000, or 10−12 1 picometer ^pmh 5 1 × 10−12 m

1 millimeter ^mmh 5 0.001 m

1 micrometer ^μmh 5 1 × 10−6 m

1 nanometer ^nmh 5 1 × 10−9 m

temperature scale in Section 1.4.h A number of physical quantities can be derived from
the list in Table 1.1. We shall discuss only a few of them here.

Force
1 N is roughly equivalent to
the force exerted by Earth’s
gravity on an apple.

The unit of force in the SI system is the newton ^Nh, after the English physicist Sir
Isaac Newton ^1642–1726h, defined as the force required to give a mass of 1 kg an

acceleration of 1 m s−2; that is,
1 N 5 1 kg m s−2

Pressure
Pressure is defined as
pressure 5

force
area

The SI unit of pressure is the pascal ^Pah, after the French mathematician and physicist Blaise Pascal ^1623–1662h, where
1 Pa 5 1 N m−2
The following three relations are exact:
1 bar 5 1 × 105 Pa 5 100 kPa
1 atm 5 1.01325 × 105 Pa 5 101.325 kPa
1 atm 5 760 torr


1.4 Units

The torr is named after the Italian mathematician Evangelista Torricelli ^1608–1674h.
The standard atmosphere ^1 atmh is used to define the normal melting point and boiling point of substances and the bar is used to define standard states in physical chemistry. We shall use all of these units in this text.
Pressure is sometimes expressed in millimeters of mercury ^mmHgh, where 1
mmHg is the pressure exerted by a column of mercury 1 mm high when its density
is 13.5951 g cm−3 and the acceleration due to gravity is 980.67 cm s−2. The relation
between mmHg and torr is
1 mmHg 5 1 torr
One instrument that measures atmospheric pressure is the barometer. A simple
barometer can be constructed by filling a long glass tube, closed at one end, with
mercury, and then carefully inverting the tube in a dish of mercury, making sure that

no air enters the tube. Some mercury will flow down into the dish, creating a vacuum
at the top ^Figure 1.3h. The weight of the mercury column remaining in the tube is
supported by atmospheric pressure acting on the surface of the mercury in the dish.
The device that is used to measure the pressure of gases other than the atmosphere
is called a manometer. The principle of operation of a manometer is similar to that of a
barometer. There are two types of manometers, shown in Figure 1.4. The closed-tube
manometer is normally used to measure pressures lower than atmospheric pressure
^Figure 1.4ah. The open-tube manometer is more suited for measuring pressures equal
to or greater than atmospheric pressure ^Figure 1.4bh.

5

With currently accepted
definitions and accuracy,
1 mmHg is 0.142 parts
per million larger than
1 torr. In metrology, this is
significant.

76 cm
Atmospheric
pressure

Energy

The SI unit of energy is the joule ^Jh 6after the English physicist James Prescott Joule
^1818–1889h@. Because energy is the ability to do work and work is force × distance,
we have
1J51Nm

Vacuum

h

Gas

Pgas
(a)

Ph

h

Gas

Pgas

Figure 1.3
A barometer for measuring
atmospheric pressure.
Above the mercury in the
tube is a vacuum. The
column of mercury is
supported by atmospheric
pressure.

Ph
(b)

Patm

Figure 1.4
Two types of manometers used to measure
gas pressures. (a) Gas pressure is less than
atmospheric pressure. (b) Gas pressure is
greater than atmospheric pressure.


6

Chapter 1: Introduction and Gas Laws

Some chemists have continued to use the non-SI unit of energy, the calorie ^calh,
where
1 cal 5 4.184 J

^exactlyh

Most physical quantities have units and in general we can express such a quantity as
physical quantity 5 numerical value × unit
For example, the speed of light ^ch in vacuum is given by
c 5 3.00 × 108 m s−1
Thus, we can write

c
5 3.00 × 108
m s−1
We shall use this convenient format in tables and figures.

Atomic Mass, Molecular Mass, and the Chemical Mole

By international agreement, an atom of the carbon-12 isotope, which has six protons
and six neutrons, has a mass of exactly 12 atomic mass units ^amuh. One atomic mass
unit is defined as a mass exactly equal to one-twelfth the mass of one carbon-12 atom.
Experiments have shown that a hydrogen atom is only 8.400 percent as massive as the
standard carbon-12 atom. Thus, the atomic mass of hydrogen must be 0.08400 × 12 5
1.008 amu. Similar experiments show that the atomic mass of oxygen is 16.00 amu and
that of iron is 55.85 amu.
When you look up the atomic mass of carbon in a table such as the one on the
inside front cover of this book, you will find it listed as 12.01 amu rather than 12.00
amu. The reason for the difference is that most naturally occurring elements ^including carbonh have more than one isotope. This means that when we measure the atomic
mass of an element, we must generally settle for the average mass of the naturally
occurring mixture of isotopes. For example, the natural abundances of carbon-12
and carbon-13 are 98.90 percent and 1.10 percent, respectively. The atomic mass of
carbon-13 has been determined to be 13.00335 amu. Thus, the average atomic mass of
carbon can be calculated as follows:
average atomic mass of carbon 5 ^0.9890h^12 amuh + ^0.0110h^13.00335 amuh
5 12.01 amu
Because there are many more carbon-12 isotopes than carbon-13 isotopes, the average atomic mass is much closer to 12 amu than 13 amu. Such an average is called a
weighted average.
If we know the atomic masses of the component atoms, then we can calculate the
mass of a molecule. Thus, the molecular mass of H2O is
2^1.008 amuh 1 16.00 amu 5 18.02 amu


1.5 The Ideal Gas Law

A mole ^abbreviated molh of any substance is the mass of that substance which
contains as many atoms, molecules, ions, or any other entities as there are atoms in
exactly 12 g of carbon-12. It has been determined experimentally that the number of
atoms in one mole of carbon-12 is 6.0221415 × 1023. This number is called Avogadro’s

number, after the Italian physicist and mathematician Amedeo Avogadro ^1776–1856h.
Avogadro’s number has no units, but dividing this number by mol gives us Avogadro’s
constant ^NAh, where
NA 5 6.0221415 × 1023 mol−1
For most purposes, NA can be taken as 6.022 × 1023 mol−1. The following examples
indicate the number and kind of particles in one mole of any substance.
1. One mole of helium atoms contains 6.022 × 1023 He atoms.
2. One mole of water molecules contains 6.022 × 1023 H2O molecules, or
2 × ^6.022 × 1023h H atoms and 6.022 × 1023 O atoms.
3. One mole of NaCl contains 6.022 × 1023 NaCl units, or
6.022 × 1023 Na+ ions and 6.022 × 1023 Cl− ions.

The molar mass of a substance is the mass in grams or kilograms of 1 mole of the
substance. Thus, the molar mass of atomic hydrogen is 1.008 g mol−1, of molecular
hydrogen is 2.016 g mol−1, and of hemoglobin is 65,000 g mol−1. In many calculations,
molar masses are more conveniently expressed as kg mol−1.

1.5 The Ideal Gas Law
Studying the behavior of gases has given rise to a number of chemical and physical
theories. In many ways, the gaseous state is the easiest to investigate. We start by
examining the properties of an ideal gas, which has the following characteristics: the
molecules of an ideal gas possess no intrinsic volume and they neither attract nor repel
one another. The equation of state, that is, the equation that relates the state variables
of the ^gaseoush system for an ideal gas is
PV 5 nRT

^1.1h

where n is the number of moles of the gas, T is the temperature in kelvins, and R is the
gas constant, to be defined shortly. No ideal gas exists in nature, but under relatively

high temperatures ^$ 25°Ch and low pressures ^# 10 atmh this equation roughly predicts the behavior of most gases.
The ideal gas equation is the accumulation of the work of the English chemist
Robert Boyle ^1627–1691h and the French physicists Jacques Charles ^1746–1823h and
Joseph Gay-Lussac ^1778–1850h. The gas laws associated with these scientists can be
derived from Equation 1.1 under different conditions. For example, at constant temperature and amount of gas ^nh we write
PV 5 constant

7


8

Chapter 1: Introduction and Gas Laws

which is Boyle’s law. At constant pressure and amount of gas, Equation 1.1 becomes
V
5 constant
T
The relation is called the law of Charles and Gay-Lussac or simply Charles’ law.
Charles’ law takes the following form if the volume and amount of gas are kept
constant:
P
5 constant
T
Another law, attributed to Avogadro, states that at constant pressure and temperature, equal volume of gases contain the same number of molecules. From Equation
1.1, we write
V
5 constant
n
The Kelvin Temperature Scale

As mentioned, the ideal gas equation holds only at low pressures. Therefore, in the
limit of P approaching zero, Equation 1.1 can be rearranged to yield
–
PV
T 5 lim
P→0 R

^1.2h

–
where V is called the molar volume, equal to V/n. Equation 1.2 defines the fundamental temperature scale, called the Kelvin scale, which is based on the ideal gas equation.
–
Because P and V cannot take on negative values, the minimum value of T is zero.
The relationship between kelvins and degrees Celsius is obtained by studying the
variation of the volume of a gas with temperature at constant pressure. At any given
pressure, the plot of volume versus temperature yields a straight line. By extending the
line to zero volume, we find the intercept on the temperature axis to be 2273.15°C. At
another pressure, we obtain a different straight line for the volume–temperature plot,
but we get the same zero-volume temperature intercept at 2273.15°C ^Figure 1.5h. ^In
practice, we can measure the volume of a gas over only a limited temperature range,
because all real gases condense at low temperatures to form liquids.h
In
cr

V

ea

273.15° C
Temperature

e

P4

sur

P3

res

Figure 1.5
Plots of the volume of a given amount of gas
versus temperature (t°C) at different pressures.
All gases ultimately condense if they are cooled
to low enough temperatures. When these lines
are extrapolated, they all converge at the point
representing zero volume and a temperature of
−273.15°C.

gp

P2

sin

P1


1.5 The Ideal Gas Law

9

In 1848 the Scottish mathematician and physicist William Thomson ^Lord Kelvin,
1824–1907h realized the significance of this phenomenon. He identified −273.15°C as
absolute zero, which is theoretically the lowest attainable temperature. Then he set up
an absolute temperature scale, now called the Kelvin temperature scale, with absolute
zero as the starting point. On the Kelvin scale, one kelvin ^Kh is equal in magnitude
to one degree Celsius. The only difference between the absolute temperature scale
and the Celsius scale is that the zero position is shifted. The relation between the two
scales is
^1.3h

T/ K 5 t/ °C 1 273.15

Note that the Kelvin scale does not have the degree sign. Important points on the two
scales match up as follows:

Absolute zero
Freezing point of water
Boiling point of water

Kelvin Scale

Celsius Scale

0K
273.15 K
373.15 K

2273.15°C
0°C
100°C

In most cases we shall use 273 instead of 273.15 as the term relating K and °C. In this text
we shall use T to denote absolute ^Kelvinh temperature and t to indicate temperature on
the Celsius scale. The Kelvin temperature scale has major theoretical significance; absolute temperatures must be used in gas law problems and thermodynamic calculations.
The Gas Constant R
The value of R can be obtained as follows. Experimentally it is found that 1 mole of
an ideal gas occupies 22.414 L at 1 atm and 273.15 K ^a condition known as standard
temperature and pressure, or STPh. Thus,
R5

^1 atmh^22.414 Lh
5 0.08206 L atm K−1 mol−1
^1 molh^273.15 Kh

To express R in units of J K−1 mol−1, we use the conversion factors
1 atm 5 1.01325 × 105 Pa ^1 Pa 5 1 N m−2h
1 L 5 1 × 10−3 m3
and obtain
R5

^1.01325 × 105 N m−2h^22.414 × 10−3 m3h
^1 molh^273.15 Kh

5 8.314 N m K−1 mol−1
5 8.314 J K−1 mol−1

^1 J 5 1 N mh

The normal boiling point
and normal freezing point
are measured at 1 atm.


10

Chapter 1: Introduction and Gas Laws

From the two values of R we can write
0.08206 L atm K−1 mol−1 5 8.314 J K−1 mol−1
or
1 L atm 5 101.3 J
and
1 J 5 9.872 × 10−3 L atm
To express R in units of bar, we use the conversion factor 1 atm 5 1.01325 bar and
write
R 5 ^0.08206 L atm K−1 mol−1h^1.01325 bar atm−1h
5 0.08315 L bar K−1 mol−1

EXAMPLE 1.1
Air entering the lungs ends up in tiny sacs called alveoli from which oxygen diffuses
into the blood. The average radius of the alveoli is 0.0050 cm and the air inside
contains 14 mole percent oxygen. Assuming that the pressure in the alveoli is 1.0
atm and the temperature is 37°C, calculate the number of oxygen molecules in one of
the alveoli.
ANSWER

The volume of one alveoli, assumed spherical, is

4
4
V 5 πr3 5 π^0.0050 cmh3
3
3
5 5.2 × 10−7 cm3 5 5.2 × 10−10 L

^1 L 5 103 cm3h

The number of moles of air in one alveoli is given by
n5

PV
^1.0 atmh^5.2 × 10−10 Lh
5
5 2.0 × 10−11 mol
RT ^0.08206 L atm K−1 mol−1h^310 Kh

Because the air inside the alveoli is 14% oxygen, the number of oxygen molecules is
2.0 × 10−11 mol air ×

14% O2 6.022 × 1023 O2 molecules
×
100% air
1 mol O2

5 1.7 × 1012 O2 molecules


1.6 Dalton’s Law of Partial Pressures

1.6 Dalton’s Law of Partial Pressures
So far we have discussed the pressure–volume–temperature behavior of a pure
gas. Frequently, however, we work with mixtures of gases. For example, a chemist
researching the depletion of ozone in the atmosphere must deal with several gaseous
components. For a system containing two or more different gases, the total pressure
^PTh is the sum of the individual pressures that each gas would exert if it were alone
and occupied the same volume. Thus,
PT 5 P1 + P2 + ? ? ? 5 ∑ Pi
i

^1.4h

where P1, P2, … are the individual or partial pressures of components 1, 2, …, and
Σ is the summation sign. Equation 1.4 is known as Dalton’s law of partial pressures,
after the English chemist and school teacher John Dalton ^1766–1844h.
Consider a system containing two gases ^1 and 2h at temperature T and volume
V. The partial pressures of the gases are P1 and P2, respectively. From Equation 1.4,
P1V 5 n1RT

or

P1 5

n1RT
V

P2V 5 n2RT

or

P2 5

n2RT
V

where n1 and n2 are the numbers of moles of the two gases. According to Dalton’s
law,
PT 5 P1 + P2
5 n1

RT
RT
+ n2
V
V

5 ^n1 + n2 h

RT
V

Dividing the partial pressures by the total pressure and rearranging, we get
P1 5

n1
P 5 x1PT
n1 + n2 T

P2 5

n2
P 5 x2P T
n1 + n2 T

and

where x1 and x2 are the mole fractions of gases 1 and 2, respectively. A mole fraction,
defined as the ratio of the number of moles of one gas to the total number of moles of
all gases present, is a dimensionless quantity. Furthermore, by definition, the sum of
all the mole fractions in a mixture must be unity:

∑ xi 5 1
i

^1.5h

11


12

Chapter 1: Introduction and Gas Laws

In general, the partial pressure of the ith component ^Pi h is related to the total pressure
as follows:
Pi 5 x i P T

^1.6h

How are partial pressures determined? A manometer can measure only the total
pressure of a gaseous mixture. To obtain partial pressures, we need to know the mole
fractions of the components. The most direct method of measuring partial pressures
is using a mass spectrometer. The relative intensities of the peaks in a mass spectrum
are directly proportional to the amounts, and hence to the mole fractions, of the gases
present.
The gas laws played a key role in the development of atomic theory, and there
are many practical illustrations of the gas laws in everyday life. Here we shall briefly
describe two examples that are particularly important to scuba divers. Seawater has a
slightly higher density than fresh water—about 1.03 g mL−1 compared to 1.00 g mL−1.
The pressure exerted by a column of 33 ft ^10 mh of seawater is equivalent to 1 atm
pressure. What would happen if a diver were to rise to the surface rather quickly,
holding his breath? If the ascent started at 40 ft under water, the decrease in pressure
from this depth to the surface would be ^40 ft /33 fth × 1 atm, or 1.2 atm. Assuming
constant temperature, when the diver reached the surface, the volume of air trapped in
his lungs would have increased by a factor of ^1 + 1.2h atm /1 atm, or 2.2 times! This
sudden expansion of air could damage or rupture the membranes of his lungs, seriously injuring or killing the diver.
Dalton’s law has a direct application to scuba diving. The partial pressure of
oxygen in air is about 0.2 atm. Because oxygen is essential for our survival, it is sometimes hard to believe that it could be harmful to breathe more than our normal share.
In fact, the toxicity of oxygen is well documented.* Physiologically, our bodies function best when the partial pressure of oxygen is 0.2 atm. For this reason, the composition of the air in a scuba tank is adjusted when the diver is submerged. For example,
at a depth where the total pressure ^hydrostatic plus atmospherich is 4 atm, the oxygen
content in the air supply should be reduced to 5 percent by volume to maintain the
optimal partial pressure ^0.05 × 4 atm 5 0.2 atmh. At a greater depth, the oxygen content must be even lower. Although nitrogen seems to be the obvious choice for mixing
with oxygen in a scuba tank, because it is the major component of air, it is not the
best choice. When the partial pressure of nitrogen exceeds 1 atm, a sufficient amount
will dissolve in the blood to cause nitrogen narcosis. Symptoms of this condition,
which resembles alcohol intoxication, include light-headedness and impaired judgment. Divers suffering from nitrogen narcosis have been known to do strange things,
such as dancing on the sea floor and chasing sharks. For this reason, helium is usually
employed to dilute oxygen in diving tanks. Helium, an inert gas, is much less soluble
in blood than nitrogen, and it does not produce narcotic effects.

* At partial pressures above 2 atm, oxygen becomes toxic enough to produce convulsions
and coma. Years ago, newborn infants placed in oxygen tents often developed retrolental
fibroplasia, damage of the retinal tissues by excess oxygen. This damage usually resulted in
partial or total blindness.