Main Principles of inorganic chemistry

Principles of inorganic chemistry

,
Aimed at senior undergraduates and first-year graduate students, this book offers a principles-based approach to inorganic chemistry that, unlike other texts, uses chemical applications of group theory and molecular orbital theory throughout as an underlying framework. This highly physical approach allows students to derive the greatest benefit of topics such as molecular orbital acid-base theory, band theory of solids, and inorganic photochemistry, to name a few. Takes a principles-based, group and molecular orbital theory approach to inorganic chemistry The first inorganic chemistry textbook to provide a thorough treatment of group theory, a topic usually relegated to only one or two chapters of texts, giving it only a cursory overview Covers atomic and molecular term symbols, symmetry coordinates in vibrational spectroscopy using the projection operator method, polyatomic MO theory, band theory, and Tanabe-Sugano diagrams Includes a heavy dose of group theory in the primary inorganic textbook, most of the pedagogical benefits of integration and reinforcement of this material in the treatment of other topics, such as frontier MO acid--base theory, band theory of solids, inorganic photochemistry, the Jahn-Teller effect, and Wade's rules are fully realized Very physical in nature compare to other textbooks in the field, taking the time to go through mathematical derivations and to compare and contrast different theories of bonding in order to allow for a more rigorous treatment of their application to molecular structure, bonding, and spectroscopy Informal and engaging writing style; worked examples throughout the text; unanswered problems in every chapter; contains a generous use of informative, colorful illustrations
Year: 2015
Edition: 1
Publisher: Wiley
Language: english
Pages: 760 / 763
ISBN 10: 1118859022
ISBN 13: 9781118859025
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PRINCIPLES OF
INORGANIC
CHEMISTRY

PRINCIPLES OF
INORGANIC
CHEMISTRY
Brian W. Pfennig

Copyright © 2015 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form
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Library of Congress Cataloging-in-Publication Data:
Pfennig, Brian William.
Principles of inorganic chemistry / Brian W. Pfennig.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-85910-0 (cloth)
1. Chemistry, Inorganic–Textbooks. 2. Chemistry, Inorganic–Study and teaching (Higher) 3. Chemistry,
Inorganic–Study and teaching (Graduate) I. Title.
QD151.3.P46 2015
546–dc23
2014043250
Cover image :Courtesy of the author
Typeset in 10/12pt GillSans by Laserwords Private Limited, Chennai, India.
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
1 2015

Contents

Preface xi
Acknowledgements xv
Chapter 1

Chapter 2

| The Composition of Matter
1.1 Early Descriptions of Matter
1.2 Visualizing Atoms 6
1.3 The Periodic Table 8
1.4 The Standard Model 9
Exercises 12
Bibliography 13

1
1

| The Structure of the Nucleus
2.1 The Nucleus 15
2.2 Nuclear Binding Energies 16
2.3 Nuclear Reactions: Fusion and Fission 17
2.4 Radioactive Decay and the Band of Stability
2.5 The Shell Model of the Nucleus 27
2.6 The Origin of the Elements 30
Exercises 38
Bibliography 39

15

22

Chapter 3

| A Brief Review of Quantum Theory
3.1 The Wavelike Properties of Light 41
3.2 Problems with the Classical Model of the Atom 48
3.3 The Bohr Model of the Atom 55
3.4 Implications of Wave-Particle Duality 58
3.5 Postulates of Quantum Mechanics 64
3.6 The Schrödinger Equation 67
3.7 The Particle in a Box Problem 70
3.8 The Harmonic Oscillator Problem 75
Exercises 78
Bibliography 79

41

Chapter 4

| Atomic Structure
4.1 The Hydrogen Atom 81
4.1.1
The Radial Wave Functions 82
4.1.2
The Angular Wave Functions 86
4.2 Polyelectronic Atoms 91
4.3 Electron Spin and the Pauli Principle 93
4.4 Electron Configurations and the Periodic Table
4.5 Atomic Term Symbols 98

81

96

vi

CONTENTS

4.5.1
Extracting Term Symbols Using Russell–Saunders Coupling 100
4.5.2
Extracting Term Symbols Using jj Coupling 102
4.5.3
Correlation Between RS (LS) Coupling and jj Coupling 104
4.6 Shielding and Effective Nuclear Charge 105
Exercises 107
Bibliography 108
Chapter 5

| Periodic Properties of the Elements
5.1 The Modern Periodic Table 109
5.2 Radius 111
5.3 Ionization Energy 118
5.4 Electron Affinity 121
5.5 The Uniqueness Principle 122
5.6 Diagonal Properties 124
5.7 The Metal–Nonmetal Line 125
5.8 Standard Reduction Potentials 126
5.9 The Inert-Pair Effect 129
5.10 Relativistic Effects 130
5.11 Electronegativity 133
Exercises 136
Bibliography 137

109

Chapter 6

| An Introduction to Chemical Bonding
6.1 The Bonding in Molecular Hydrogen 139
6.2 Lewis Structures 140
6.3 Covalent Bond Lengths and Bond Dissociation Energies
6.4 Resonance 146
6.5 Polar Covalent Bonding 149
Exercises 153
Bibliography 154

139

144

Chapter 7

| Molecular Geometry
7.1 The VSEPR Model 155
7.2 The Ligand Close-Packing Model 170
7.3 A Comparison of the VSEPR and LCP Models 175
Exercises 176
Bibliography 177

155

Chapter 8

| Molecular Symmetry
8.1 Symmetry Elements and Symmetry Operations
8.1.1
Identity, E 180
8.1.2
Proper Rotation, Cn 181
8.1.3
Reflection, 𝜎 182
8.1.4
Inversion, i 183
8.1.5
Improper Rotation, Sn 183
8.2 Symmetry Groups 186
8.3 Molecular Point Groups 191
8.4 Representations 195
8.5 Character Tables 202
8.6 Direct Products 209
8.7 Reducible Representations 214
Exercises 222
Bibliography 224

179
179

vii

CONTENTS

Chapter 9

Chapter 10

Chapter 11

| Vibrational Spectroscopy
9.1 Overview of Vibrational Spectroscopy 227
9.2 Selection Rules for IR and Raman-Active Vibrational Modes 231
9.3 Determining the Symmetries of the Normal Modes of Vibration 235
9.4 Generating Symmetry Coordinates Using the Projection Operator Method
9.5 Resonance Raman Spectroscopy 252
Exercises 256
Bibliography 258
| Covalent Bonding
10.1 Valence Bond Theory 259
10.2 Molecular Orbital Theory: Diatomics 278
10.3 Molecular Orbital Theory: Polyatomics 292
10.4 Molecular Orbital Theory: pi Orbitals 305
10.5 Molecular Orbital Theory: More Complex Examples
10.6 Borane and Carborane Cluster Compounds 325
Exercises 334
Bibliography 336
| Metallic Bonding
11.1 Crystalline Lattices 339
11.2 X-Ray Diffraction 345
11.3 Closest-Packed Structures 350
11.4 The Free Electron Model of Metallic Bonding 355
11.5 Band Theory of Solids 360
11.6 Conductivity in Solids 374
11.7 Connections Between Solids and Discrete Molecules
Exercises 384
Bibliography 388

227

243

259

317

339

383

Chapter 12

| Ionic Bonding
12.1 Common Types of Ionic Solids 391
12.2 Lattice Enthalpies and the Born–Haber Cycle 398
12.3 Ionic Radii and Pauling’s Rules 404
12.4 The Silicates 417
12.5 Zeolites 422
12.6 Defects in Crystals 423
Exercises 426
Bibliography 428

391

Chapter 13

| Structure and Bonding
13.1 A Reexamination of Crystalline Solids 431
13.2 Intermediate Types of Bonding in Solids 434
13.3 Quantum Theory of Atoms in Molecules (QTAIM) 443
Exercises 449
Bibliography 452

431

Chapter 14

| Structure and Reactivity
14.1 An Overview of Chemical Reactivity 453
14.2 Acid–Base Reactions 455
14.3 Frontier Molecular Orbital Theory 467

453

viii

CONTENTS

14.4 Oxidation–Reduction Reactions 473
14.5 A Generalized View of Molecular Reactivity
Exercises 480
Bibliography 481

475

Chapter 15

| An Introduction to Coordination Compounds
15.1 A Historical Overview of Coordination Chemistry 483
15.2 Types of Ligands and Nomenclature 487
15.3 Stability Constants 490
15.4 Coordination Numbers and Geometries 492
15.5 Isomerism 498
15.6 The Magnetic Properties of Coordination Compounds 501
Exercises 506
Bibliography 508

Chapter 16

| Structure, Bonding, and Spectroscopy of Coordination Compounds
509
16.1 Valence Bond Model 509
16.2 Crystal Field Theory 512
16.3 Ligand Field Theory 525
16.4 The Angular Overlap Method 534
16.5 Molecular Term Symbols 541
16.5.1 Scenario 1—All the Orbitals are Completely Occupied 546
16.5.2 Scenario 2—There is a Single Unpaired Electron in One of the Orbitals 546
16.5.3 Scenario 3—There are Two Unpaired Electrons in Two Different Orbitals 546
16.5.4 Scenario 4—A Degenerate Orbital is Lacking a Single Electron 547
16.5.5 Scenario 5—There are Two Electrons in a Degenerate Orbital 547
16.5.6 Scenario 6—There are Three Electrons in a Triply Degenerate Orbital 547
16.6 Tanabe–Sugano Diagrams 549
16.7 Electronic Spectroscopy of Coordination Compounds 554
16.8 The Jahn–Teller Effect 564
Exercises 566
Bibliography 570

Chapter 17

| Reactions of Coordination Compounds
17.1 Kinetics Overview 573
17.2 Octahedral Substitution Reactions 577
17.2.1 Associative (A) Mechanism 578
17.2.2 Interchange (I) Mechanism 579
17.2.3 Dissociative (D) Mechanism 580
17.3 Square Planar Substitution Reactions 585
17.4 Electron Transfer Reactions 593
17.5 Inorganic Photochemistry 606
17.5.1 Photochemistry of Chromium(III) Ammine Compounds 607
17.5.2 Light-Induced Excited State Spin Trapping in Iron(II) Compounds 611
17.5.3 MLCT Photochemistry in Pentaammineruthenium(II) Compounds 615
17.5.4 Photochemistry and Photophysics of Ruthenium(II) Polypyridyl Compounds
Exercises 622
Bibliography 624

Chapter 18

| Structure and Bonding in Organometallic Compounds
18.1 Introduction to Organometallic Chemistry 627
18.2 Electron Counting and the 18-Electron Rule 628

483

573

617

627

ix

CONTENTS

18.3 Carbonyl Ligands 631
18.4 Nitrosyl Ligands 635
18.5 Hydride and Dihydrogen Ligands 638
18.6 Phosphine Ligands 640
18.7 Ethylene and Related Ligands 641
18.8 Cyclopentadiene and Related Ligands 645
18.9 Carbenes, Carbynes, and Carbidos 648
Exercises 651
Bibliography 654
Chapter 19

| Reactions of Organometallic Compounds
19.1 Some General Principles 655
19.2 Organometallic Reactions Involving Changes at the Metal 656
19.2.1 Ligand Substitution Reactions 656
19.2.2 Oxidative Addition and Reductive Elimination 658
19.3 Organometallic Reactions Involving Changes at the Ligand 664
19.3.1 Insertion and Elimination Reactions 664
19.3.2 Nucleophilic Attack on the Ligands 667
19.3.3 Electrophilic Attack on the Ligands 669
19.4 Metathesis Reactions 670
19.4.1 𝜋-Bond Metathesis 670
19.4.2 Ziegler–Natta Polymerization of Alkenes 671
19.4.3 𝜎-Bond Metathesis 671
19.5 Commercial Catalytic Processes 674
19.5.1 Catalytic Hydrogenation 674
19.5.2 Hydroformylation 674
19.5.3 Wacker–Smidt Process 676
19.5.4 Monsanto Acetic Acid Process 677
19.6 Organometallic Photochemistry 678
19.6.1 Photosubstitution of CO 678
19.6.2 Photoinduced Cleavage of Metal–Metal Bonds 680
19.6.3 Photochemistry of Metallocenes 682
19.7 The Isolobal Analogy and Metal–Metal Bonding in Organometallic Clusters
Exercises 689
Bibliography 691

655

683

Appendix: A Derivation of the Classical Wave Equation
Bibliography 694

693

Appendix: B Character Tables
Bibliography 708

695

Appendix: C Direct Product Tables
Bibliography 713

709

Appendix: D Correlation Tables
Bibliography 721

715

Appendix: E

723

Index

The 230 Space Groups
Bibliography 728

729

Preface

This book was written as a result of the perceived need of mine and several other
colleagues for a more advanced physical inorganic text with a strong emphasis on
group theory and its applications. Many of the inorganic textbooks on the market are
either disjointed—with one chapter completely unrelated to the next—or encyclopedic, so that the student of inorganic chemistry is left to wonder if the only way to
master the field is to memorize a large body of facts. While there is certainly some
merit to a descriptive approach, this text will focus on a more principles-based pedagogy, teaching students how to rationalize the structure and reactivity of inorganic
compounds—rather than relying on rote memorization.
After many years of teaching the inorganic course without a suitable text, I
decided to write my own. Beginning in the summer of 2006, I drew on a variety of
different sources and tried to pull together bits and pieces from different texts and
reference books, finishing a first draft (containing 10 chapters) in August, 2007. I used
this version of the text as supplementary reading for a few years before taking up the
task of writing again in earnest in 2012, subdividing and expanding the upon original
10 chapters to the current 19, adding references and more colorful illustrations, and
including problems at the ends of each chapter.
The book was written with my students in mind. I am a teacher first and a scientist second. I make no claims about my limited knowledge of this incredibly expansive
field. My main contribution has been to collect material from various sources and to
organize and present it in a pedagogically coherent manner so that my students can
understand and appreciate the principles underlying such a diverse and interesting
subject as inorganic chemistry.
The book is organized in a logical progression. Chapter 1 provides a basic
introduction to the composition of matter and the experiments that led to the
development of the periodic table. Chapter 2 then examines the structure and
reactivity of the nucleus. Chapter 3 follows with a basic primer on wave-particle
duality and some of the fundamentals of quantum mechanics. Chapter 4 discusses
the solutions to the Schrödinger equation for the hydrogen atom, the Pauli principle,
the shapes of the orbitals, polyelectronic wave functions, shielding, and the quantum mechanical basis for the underlying structure of the periodic table. Chapter 5
concludes this section of the text by examining the various periodic trends that
influence the physical and chemical properties of the elements. Chapter 6 then
begins a series of chapters relating to chemical bonding by reviewing the basics
of Lewis structures, resonance, and formal charge. Chapter 7 is devoted to the
molecular geometries of molecules and includes not only a more extensive treatment of the VSEPR model than most other textbooks but it also presents the ligand
close-packing model as a complementary model for the prediction of molecular
geometries. Symmetry and group theory are introduced in detail in Chapter 8 and
will reappear as a recurring theme throughout the remainder of the text. Unlike
most inorganic textbooks on the market, ample coverage is given to representations of groups, reducing representations, direct products, the projection operator,
and applications of group theory. Chapter 9 focuses on one of the applications of
group theory to the vibrational spectroscopy of molecules, showing how symmetry
coordinates can be used to approximate the normal modes of vibration of small
molecules. The selection rules for IR and Raman spectroscopy are discussed and

xii

PREFACE

the chapter closes with a brief introduction to resonance Raman spectroscopy. The
next three chapters focus on the three different types of chemical bonding: covalent,
metallic, and ionic bonding. Chapter 10 examines the valence bond and molecular
orbital models, which expands upon the application of group theory to chemical
problems. Chapter 11 then delves into metallic bonding, beginning with a primer on
crystallography before exploring the free electron model and band theory of solids.
Chapter 12 is focused on ionic bonding—lattice enthalpies, the Born–Haber cycle,
and Pauling’s rules for the rationalization of ionic solids. It also has extensive coverage of the silicates and zeolites. The structure of solids is reviewed in greater detail
in Chapter 13, which explores the interface between the different types of chemical bonding in both solids and discrete molecules. Switching gears for a while from
structure and bonding to chemical reactivity, Chapter 14 introduces the two major
types of chemical reactions: acid–base reactions and oxidation–reduction reactions.
In addition to the usual coverage of hard–soft acid–base theory, this chapter also
examines a more general overview of chemical reactivity that is based on the different topologies of the MOs involved in chemical transformations. This chapter
also serves as a bridge to the transition metals. Chapter 15 presents an introduction to coordination compounds and their thermodynamic and magnetic properties.
Chapter 16 examines the structure, bonding, and electronic spectroscopy of coordination compounds, making extensive use of group theory. Chapter 17 investigates
the reactions of coordination compounds in detail, including a section on inorganic
photochemistry. Finally, the text closes with two chapters on organometallic chemistry: Chapter 18 looks at the different types of bonding in organometallics from an
MO point of view, while Chapter 19 presents of a survey of organometallic reaction
mechanisms, catalysis, and organometallic photochemistry and then concludes with
connections to main group chemistry using the isolobal analogy. Throughout the
textbook, there is a continual building on earlier material, especially as it relates to
group theory and MOT, which serve as the underlying themes for the majority of
the book.
This text was originally written for undergraduate students taking an advanced
inorganic chemistry course at the undergraduate level, although it is equally suitable as a graduate-level text. I have written the book with the more capable and
intellectually curious students in my undergraduate courses in mind. The prose is
rather informal and directly challenges the student to examine each new experimental observation in the context of previously introduced principles of inorganic
chemistry. Students should appreciate the ample number of solved sample problems
interwoven throughout the body of the text and the clear, annotated figures and
illustrations. The end-of-chapter problems are designed to invoke an active wrangling
with the material and to force students to examine the data from several different
points of view. While the text is very physical in emphasis, it is not overly mathematical and thorough derivations are provided for the more important physical
relationships. It is my hope that students will not only enjoy using this textbook in
their classes but will read and reread it again as a valuable reference book throughout
the remainder of their chemical careers.
While this book provides a thorough introduction to physical inorganic chemistry, the field is too vast to include every possible topic; and it is therefore somewhat
limited in its scope. The usual group by group descriptive chemistry of the elements,
for example, is completely lacking, as are chapters on bioinorganic chemistry or
inorganic materials chemistry. However, it is my belief that what it lacks in breadth
is more than compensated for by its depth and pedagogical organization. Nonetheless, I eagerly welcome any comments, criticisms, and corrections and have opened a

xiii

PREFACE

dedicated e-mail account for just such a purpose at pfennigtext@hotmail.com. I look
forward to hearing your suggestions.
BRIAN W. PFENNIG
Lancaster, PA
June, 2014

Acknowledgments

This book would not have been possible without the generous contributions of
others. I am especially indebted to my teachers and mentors over the years who
always inspired in me a curiosity for the wonders of science, including Al Bieber,
Dave Smith, Bill Birdsall, Jim Scheirer, Andy Bocarsly, Mark Thompson, Jeff Schwartz,
Tom Spiro, Don McClure, Bob Cava, and Tom Meyer. In addition, I thank some of
the many colleagues who have contributed to my knowledge of inorganic chemistry,
including Ranjit Kumble, Jim McCusker, Dave Thompson, Claude Yoder, Jim Spencer,
Rick Schaeffer, John Chesick, Marianne Begemann, Andrew Price, and Amanda Reig.
I also thank Reid Wickham at Pearson (Prentice-Hall) for her encouragement and
advice with respect to getting published and to Anita Lekwhani at John Wiley &
Sons, Inc. for giving me that chance. Thank you all for believing in me and for your
encouragement.
There is little original content in this inorganic text that cannot be found elsewhere. My only real contribution has been to crystallize the content of many other
authors and to organize it in a way that hopefully makes sense to the student. I
have therefore drawn heavily on the following inorganic texts: Inorganic Chemistry
(Miessler and Tarr), Inorganic Chemistry (Huheey, Keiter, and Keiter), Chemical Applications of Group Theory (Cotton), Molecular Symmetry and Group Theory (Carter),
Symmetry and Spectroscopy (Harris and Bertulucci), Problems in Molecular Orbital Theory (Albright and Burdett), Chemical Bonding and Molecular Geometry (Gillespie and
Hargittai), Ligand Field Theory (Figgis and Hitchman), Physical Chemistry (McQuarrie
and Simon), Elements of Quantum Theory (Bockhoff), Introduction to Crystallography
(Sands), and Organometallic Chemistry (Spessard and Miessler).
In addition, I am grateful to a number of people who have assisted me in the
preparation of my manuscript, especially to the many people who have reviewed
sample chapters of the textbook or who have generously provided permission to
use their figures. I am especially indebted to Lori Blatt at Blatt Communications for
producing many of the amazing illustrations in the text and to Aubrey Paris for her
invaluable assistance with proofing the final manuscript.
I would be remiss if I failed to acknowledge the contributions of my students,
both past and present, in giving me the inspiration and perseverance necessary to
write a volume of this magnitude. I am especially indebted to the intellectual interactions I have had with Dave Watson, Jamie Cohen, Jenny Lockard, Mike Norris,
Aaron Peters, and Aubrey Paris over the years. Lastly, I would like to acknowledge
the most important people in my life, without whose undying support and tolerance
I would never have been able to complete this work—my family. I am especially
grateful to my wonderful parents who instilled in me the values of a good education,
hard work, and integrity; to my wife Jessica for her unwavering faith in me; and to
my incredibly talented daughter Rachel, who more than anyone has suffered from
lack of my attention as I struggled to complete this work.

The Composition
of Matter

1

“Everything existing in the universe is the fruit of chance and necessity.”
—Democritus

1.1

EARLY DESCRIPTIONS OF MATTER

Chemistry has been defined as the study of matter and its interconversions. Thus, in a
sense, chemistry is a study of the physical world in which we live. But how much do
we really know about the fundamental structure of matter and its relationship to the
larger macroscopic world? I have in my rock collection, which I have had since I was a
boy, a sample of the mineral cinnabar, which is several centimeters across and weighs
about 10 g. Cinnabar is a reddish granular solid with a density about eight times that
of water and the chemical composition mercuric sulfide. Now suppose that some
primal instinct suddenly overcame me and I were inclined to demolish this precious
talisman from my childhood. I could take a hammer to it and smash it into a billion
little pieces. Choosing the smallest of these chunks, I could further disintegrate the
material in a mortar and pestle, grinding it into ever finer and finer grains until I was
left with nothing but a red powder (in fact, this powder is known as vermilion and
has been used as a red pigment in artwork dating back to the fourteenth century).
Having satisfied my destructive tendencies, I would nonetheless still have exactly the
same material that I started with—that is, it would have precisely the same chemical
and physical properties as the original. I might therefore wonder to myself if there is
some inherent limitation as to how finely I can divide the substance or if this is simply
limited by the tools at my disposal. With the proper equipment, would I be able to
continue dividing the compound into smaller and smaller pieces until ultimately I
obtained the unit cell, or smallest basic building block of the crystalline structure of
HgS, as shown in Figure 1.1? For that matter (no pun intended), is there a way for
me to separate out the two different types of atoms in the substance?
If matter is defined as anything that has mass and is perceptible to the senses, at
what point does it become impossible (or at the very least impractical) for me to
continue to measure the mass of the individual grains or for them to no longer be
perceptible to my senses (even if placed under an optical microscope)? The ancient
philosopher Democritus (ca 460–370 BC) was one of the first to propose that matter
is constructed of tiny indivisible particles known as atomos (or atoms), the different
varieties (sizes, shapes, masses, etc.) of which form the fundamental building blocks
of the natural world. In other words, there should be some lower limit as to how
Principles of Inorganic Chemistry, First Edition. Brian W. Pfennig.
© 2015 John Wiley & Sons, Inc. Published 2015 by John Wiley & Sons, Inc.

2

1 THE COMPOSITION OF MATTER

(a)

(b)

FIGURE 1.1
Three examples of the same
chemical material ranging from
the macroscopic to the atomic
scale: (a) the mineral cinnabar,
(b) vermilion powder, and (c) the
unit cell of mercuric sulfide.
[Vermilion pigment photo
courtesy of Kremer
Pigments, Inc.]

(c)

finely I can continue to carve up my little chunk of cinnabar. As far back as the Middle
Ages, the alchemists learned that one could decompose a sample of HgS by heating
it up in a crucible. At temperatures above 580 ∘ C, the heat drives off the sulfur and
leaves behind a pool of silvery liquid mercury. Eventually, I could break the molecule
itself apart into its individual atoms, but then I could go no further.

3

1.1 EARLY DESCRIPTIONS OF MATTER

Or could I? In the late 1800s, scientists discovered that if they constructed a hollow glass tube with an anode in one end and a cathode in the other and pumped out
as much of the air as they could, an electrical discharge between the two electrodes
could produce a faint glow within the tube. Later, cathode ray tubes, as they became
to be known, were more sophisticated and contained a phosphorescent coating in
one end of the tube. William Crookes demonstrated that the rays were emitted
from the cathode and that they traveled in straight lines and could not bend around
objects in their path. A while later, Julius Plücker was able to show that a magnet
applied to the exterior of the cathode ray tube could change the position of the
phosphorescence. Physicists knew that the cathode ray carried a negative charge (in
physics, the cathode is the negatively charged electrode and because the beam originated from the cathode, it must therefore be negatively charged). However, they did
not know whether the charge and the ray could be separated from one another. In
1897, Joseph J. Thomson finally resolved the issue by demonstrating that both the
beam and the charged particles could be bent by an electrical field that was applied
perpendicular to the path of the beam, as shown in Figure 1.2. By systematically varying the electric field strength and measuring the angle of deflection, Thomson was
able to determine the charge-to-mass (e/m) ratio of the particles, which he called
corpuscles and which are now known as electrons. Thomson measured the e/m ratio
as −1.76 × 108 C/g, a value that was at least a thousand times larger than the one
expected on the basis of the known atomic weights of even the lightest of atoms,
indicating that the negatively charged electrons must be much smaller in size than a
typical atom. In other words, the atom was not indivisible, and could itself be broken down into smaller components, with the electron being one of these subatomic
particles. As a result of his discovery, Thomson proposed the so-called plum pudding model of the atom, where the atom consisted of one or more of these tiny
electrons distributed in a sea of positive charge, like raisins randomly dispersed in a
gelatinous pudding. Thomson was later awarded the 1906 Nobel Prize in physics for
his discovery of the electron and his work on the electrical conductivity of gases.
In 1909, Robert Millikan and his graduate student Harvey Fletcher determined
the charge on the electron using the apparatus shown in Figure 1.3. An atomizer
from a perfume bottle was used to spray a special kind of oil droplet having a low
vapor pressure into a sealed chamber. At the bottom of the chamber were two
parallel circular plates. The upper one of these plates was the anode and it had a
hole drilled into the center of it through which the oil droplets could fall under the
influence of gravity. The apparatus was equipped with a microscope so that Millikan
could observe the rate of fall of the individual droplets. Some of the droplets became
charged as a result of friction with the tip of the nozzle, having lost one or more
of their electrons to become positively charged cations. When Millikan applied a
potential difference between the two plates at the bottom of the apparatus, the
positively charged droplets were repelled by the anode and reached an equilibrium

Displacement
Negative plate

Positive plate
Cathode

Cathode ray

Anode

FIGURE 1.2
Schematic diagram of a cathode ray tube similar to the one used in J. J. Thomson’s discovery of the electron.
[Blatt Communications.]

4

1 THE COMPOSITION OF MATTER

Atomizer

+
Positively charged plate

FIGURE 1.3
Schematic diagram of the
Millikan oil drop experiment to
determine the charge of the
electron. [Blatt
Communications.]

Source of
ionized
radiation

Telescope

–

Negatively charged plate

state where the Coulombic repulsion of like charges and the effect of gravity were
exactly balanced, so that appropriately charged particles essentially floated there in
space inside the container. By systematically varying the potential difference applied
between the two metal plates and counting the number of particles that fell through
the opening in a given period of time, Millikan was able to determine that each of
the charged particles was some integral multiple of the electronic charge, which
he determined to be −1.592 × 10−19 C, a measurement that is fairly close to the
modern value for the charge on an electron (−1.60217733 × 10−19 C). Using this
new value of e along with Thomson’s e/m ratio, Millikan was able to determine the
mass of a single electron as 9.11 × 10−28 g. The remarkable thing about the mass of
the electron was that it was 1837 times smaller than the mass of a single hydrogen
atom. Another notable feature of Millikan’s work is that it very clearly demonstrated
that the electronic charge was quantized as opposed to a continuous value. The
differences in the charges on the oil droplets were always some integral multiple of
the value of the electronic charge e. Millikan’s work was not without controversy,
however, as it was later discovered that some of his initial data (and Fletcher’s name)
were excluded from his 1913 publication. Some modern physicists have viewed this
as a potential example of pathological science. Nevertheless, Millikan won the 1923
Nobel Prize in physics for this work.
Also in 1909, one of J. J. Thomson’s students, Ernest Rutherford, working with
Hans Geiger and a young graduate student by the name of Ernest Marsden, performed his famous “gold foil experiment” in order to test the validity of the plum
pudding model of the atom. Rutherford was already quite famous by this time, having
won the 1908 Nobel Prize in chemistry for his studies on radioactivity. The fact that
certain compounds (particularly those of uranium) underwent spontaneous radioactive decay was discovered by Antoine Henri Becquerel in 1896. Rutherford was the
first to show that one of the three known types of radioactive decay involved the
transmutation of an unstable radioactive element into a lighter element and a positively charged isotope of helium known as an alpha particle. Alpha particles were
many thousands of times more massive than an electron. Thus, if the plum pudding
model of the atom were correct, where the electrons were evenly dispersed in a
sphere of positive charge, the heavier alpha particles should be able to blow right
through the atom. Geiger and Marsden assembled the apparatus shown in Figure 1.4.
A beam of alpha particles was focused through a slit in a circular screen that had
a phosphorescent coating of ZnS on its interior surface. When an energetic alpha
particle struck the phosphorescent screen, it would be observed as a flash of light.
In the center of the apparatus was mounted a very thin piece of metal foil (although
it is often referred to as the gold foil experiment, it was in fact a piece of platinum foil,
not gold, which was used). While the majority of alpha particles struck the screen

5

1.1 EARLY DESCRIPTIONS OF MATTER

Path of deflected
particles

Source
of alpha
particles
Thin sheet
of Pt foil

Fluorescent screen

Path of undeflected
particles

Beam of
alpha
particles

(a)

Beam of
alpha
particles

(b)

immediately behind the piece of metal foil as expected, much to the amazement of
the researchers, a number of alpha particles were also deflected and scattered at
other angles. In fact, some of the particles even deflected backward from the target.
In his own words, Rutherford was said to have exclaimed: “It was quite the most
incredible event that has ever happened to me in my life. It was almost as incredible
as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit
you. On consideration, I realized that this scattering backwards must be the result
of a single collision, and when I made calculations I saw that it was impossible to get
anything of that order of magnitude unless you took a system in which the greater
part of the mass of an atom was concentrated in a minute nucleus.” Further calculations showed that the diameter of the nucleus was about five orders of magnitude
smaller than that of the atom. This led to the rather remarkable conclusion that matter is mostly empty space—with the very lightweight electrons orbiting around an
incredibly dense and positively charged nucleus, as shown in Figure 1.5. As a matter
of fact, 99.99999999% of the atom is devoid of all matter entirely! On the atomic
scale, solidity has no meaning. The reason that a macroscopic object “feels” at all
hard to us is because the atom contains a huge amount of repulsive energy, so that
whenever we try to “push” on it, there is a whole lot of energy pushing right back.
It wasn’t until 1932 that the final piece of the atomic puzzle was put into place.
After 4 years as a POW in Germany during World War I, James Chadwick returned
to England to work with his former mentor Ernest Rutherford, who had taken over
J. J. Thomson’s position as Cavendish Professor at Cambridge University. It was
not long before Rutherford appointed Chadwick as the assistant director of the
nuclear physics lab. In the years immediately following Rutherford’s discovery that
the nucleus contained protons, which existed in the nucleus and whose charges were

FIGURE 1.4
Schematic diagram of the
Geiger–Marsden experiment,
also known as Rutherford’s gold
foil experiment. [Blatt
Communications.]
FIGURE 1.5
Atomic view of the gold foil
experiment. If the plum pudding
model of the atom were
accurate, a beam of massive
alpha particles would penetrate
right through the atom with
little or no deflections (a). The
observation that some of the
alpha particles were deflected
backward implied that the
positive charge in the atom must
be confined to a highly dense
region inside the atom known as
the nucleus (b). [Blatt
Communications.]

6

1 THE COMPOSITION OF MATTER

equal in magnitude to the electronic charge but with the opposite sign, it was widely
known that the nuclei of most atoms weighed more than could be explained on
the basis of their atomic numbers (the atomic number is the same as the number of
protons in the nucleus). Some scientists even hypothesized that maybe the nucleus
contained an additional number of protons and electrons, whose equal but opposite
charges cancelled each other out but which together contributed to the increased
mass of the nucleus. Others, such as Rutherford himself, postulated the existence
of an entirely new particle having roughly the same mass as a proton but no charge
at all, a particle that he called the neutron. However, there was no direct evidence
supporting this hypothesis.
Around 1930, Bothe and Becker observed that a Be atom bombarded with alpha
particles produced a ray of neutral radiation, while Curie and Joliot showed that this
new form of radiation had enough energy to eject protons from a piece of paraffin
wax. By bombarding heavier nuclei (such as N, O, and Ar) with this radiation and
calculating the resulting cross-sections, Chadwick was able to prove that the rays
could not be attributed to electromagnetic radiation. His results were, however,
consistent with a neutral particle having roughly the same mass as the proton. In
his next experiment, Chadwick bombarded a boron atom with alpha particles and
allowed the resulting neutral particles to interact with nitrogen. He also measured
the velocity of the neutrons by allowing them to interact with hydrogen atoms and
measuring the speed of the protons after the collision. Coupling the results of each
of his experiments, Chadwick was able to prove the existence of the neutron and
to determine its mass to be 1.67 × 10−27 kg. The modern-day values for the charges
and masses of the electron, proton, and neutron are listed in Table 1.1. Chadwick
won the Nobel Prize in physics in 1935 for his discovery of the neutron.

1.2

VISUALIZING ATOMS

At the beginning of this chapter, I asked the question at what point can we divide
matter into such small pieces that it is no longer perceptible to the senses. In a
sense, this is a philosophical question and the answer depends on what we mean as
being perceptible to the senses. Does it literally mean that we can see the individual
components with our naked eye, and for that matter, what are the molecular characteristics of vision that cause an object to be seen or not seen? How many photons
of light does it take to excite the rod and cone cells in our eyes and cause them
to fire neurons down the optic nerve to the brain? The concept of perceptibility
is somewhat vague. Is it fair to say that we still see the object when it is multiplied
under an optical microscope? What if an electron microscope is used instead? Today,
we have “pictures” of individual atoms, such as those shown in Figure 1.6, made by
a scanning tunneling microscope (STM) and we can manipulate individual atoms on

TABLE 1.1 Summary of the properties of subatomic particles.
Particle

Mass (kg)

Mass (amu)

Charge (C)

Electron 9.10938291 × 10−31
0.00054857990946 −1.602176565 × 10−19
Proton
1.672621777 × 10−27 1.007276466812
1.602176565 × 10−19
−27
Neutron 1.674927351 × 10
1.00866491600
0
Source: The NIST Reference on Constants,
(http://physics.nist.gov, accessed Nov 3, 2013).

Units,

and

Uncertainty

7

1.2 VISUALIZING ATOMS

FIGURE 1.6
Scanning tunneling microscopy
of the surface of the (110) face
of a nickel crystal. [Image
originally created by IBM
Corporation.]

a surface in order to create new chemical bonds at the molecular level using atomic
force microscopy (AFM).
But are we really capable of actually seeing an individual atom? Technically speaking, we cannot see anything smaller than the shortest wavelength of light with which
we irradiate it. The shortest wavelength that a human eye can observe is about
400 nm, or 4 × 10−7 m. As the diameter of an atom is on the scale of 10−11 m and
the diameter of a typical nucleus is even smaller at 10−15 m, it is therefore impossible
for us to actually see an atom. However, we do have ways of visualizing atoms. A
scanning tunneling microscope, like the one shown in Figure 1.7, works by moving an
exceptionally sharp piezoelectric tip (often only one atom thick at its point) across
the surface of a conductive solid, such as a piece of crystalline nickel in an evacuated

Piezotube tube
with electrodes

Control voltages for piezotube

Tunneling
current amplifier

Distance control
and scanning unit

Tip

Sample
Tunneling
voltage

Data processing
and display

FIGURE 1.7
Schematic diagram of a scanning
tunneling microscope (STM).
[Blatt Communications.]

8

1 THE COMPOSITION OF MATTER

chamber. When a small voltage is applied to the tip of the STM, a tunneling current
develops whenever the tip is close to the surface of a Ni atom. This tunneling current is proportional to the distance between the tip of the probe and the atoms on
the surface of the crystal. By adjusting the STM so that the tunneling current is a
constant, the tip will move up and down as it crosses the surface of the crystal and
encounters electron density around the nuclei of the nickel atoms. A computer is
then used to map out the three-dimensional contour of the nickel surface and to
color it different shades of blue in this case, depending on the distance that the tip
has moved. The STM can also be used to pick up atoms and to move them around
on a surface. In fact, the scientists who invented the STM (Gerd Binnig and Heinrich
Rohrer, both of whom shared the 1986 Nobel Prize in physics) used an STM to spell
out the name of their sponsoring company IBM by moving around 35 individual Xe
atoms affixed to a Ni surface.
The AFM, which has a smaller resolution than the STM, has the advantage of
being able to visualize nonconductive surfaces. It functions using a cantilever with
a very narrow tip on the end. Instead of interacting directly with the electrons, it
vibrates at a specific frequency and when it encounters an atom, the frequency of
the vibration changes, allowing one to map out the contour of the surface.

1.3

THE PERIODIC TABLE

While chemistry is the study of matter and its interconversions, inorganic chemistry is that subdiscipline of chemistry which deals with the physical properties and
chemistry of all the elements, with the singular exclusion of carbon. An element is
defined by the number of protons in its nucleus. There are 90 naturally occurring
elements (all of the elements up to and including atomic number 92, with the exception of Tc (atomic number 43) and Pm (atomic number 61)). However, if all of the
man-made elements are included, a total of 118 elements are currently known to
exist. It has long been known that many of the elements had similar valences and
chemical reactivity. In the late 1860s and early 1870s, Dmitri Mendeleev and Julius
Lothar Meyer independently discovered that the elements could be arranged into a
table in an orderly manner such that their properties would follow a periodic law. In
his book Principles of Chemistry, Mendeleev wrote: “I began to look about and write
down the elements with their atomic weights and typical properties, analogous elements and like atomic weights on separate cards, and this soon convinced me that
the properties of elements are in periodic dependence upon their atomic weights.”
His resulting periodic table organized the elements into eight broad categories (or
Gruppe) according to increasing atomic mass, as shown in Figure 1.8.
At the time of publication in 1871, only about half of the elements known today
had yet to be discovered. One of the reasons that Mendeleev’s version of the periodic table became so popular was that he left gaps in his table for as yet undiscovered
elements. When the next element on his pile of cards did not fit the periodic trend,
he placed the element in the next group that bore resemblance to it, figuring that a
new element would someday be discovered with properties appropriate to fill in the
gap. Furthermore, by interpolation from the properties of those elements on either
side of the gaps, Mendeleev could use his table to make predictions about the reactivity of the unknown elements. In particular, Mendeleev predicted the properties of
gallium, scandium, and germanium, which were discovered in 1875, 1879, and 1886,
respectively, and he did so with incredible accuracy. For example, Table 1.2 lists the
properties of germanium that Mendeleev predicted 15 years before its discovery
and compares them with the modern-day values. It is this predictive capacity that
makes the periodic table one of the most powerful tools in chemistry. Mendeleev’s
periodic table was organized according to increasing mass. With the discovery of

9

Reibco

1.4 THE STANDARD MODEL

1
2
3
4
5
6
7
8
9
10
11
12

Gruppo I.
—
R1O

Gruppo II.
—
RO

Gruppo III. Gruppo IV.
—
RH4
R1O3
RO1

Gruppo V.
RH2
R1O5

Gruppo VI. Gruppo VII.
RH2
RH
RO3
R2O7

II=1
Li=7

Bo=9.4
B=11
C=12
N=14
O=16
F=19
Na=23
Mg=24
Al=27,8
Si=28
P=31
S=32
Cl=35,5
K=39
Ca=40
—=44
Ti=48
V=51
Cr=52
Mn=55
Fo=56, Co=59,
Ni=59, Cu=63.
(Cu=63)
Zn=65
—=68
—=72
As=75
So=78
Br=80
Rb=85
Sr=87
?Yt=88
Zr=90
Nb=94
Mo=96
—=100
Ru=104, Rh=104,
Pd=106, Ag=108.
(Ag=108)
Cd=112
In=113
Sn=118
Sb=122
To=125
J=127
Cs=183
Ba=187
?Di=188
?Co=140
—
—
—
— — — —
(—)
—
—
—
—
—
—
—
—
?Er=178
?La=180
Ta=182
W=184
—
Os=195, Ir=197,
Pt=198, Au=199.
(Ag=199)
Hg=200
Tl=204
Pb=207
Bi=208
—
—
—
—
—
Th=231
—
U=240
—
— — — —

FIGURE 1.8
Dmitri Mendeleev’s periodic table (1871).

TABLE 1.2 Properties of the element germanium (eka-silicon) as
predicted by Mendeleev in 1871 and the experimental values measured
after its discovery in 1886.
Physical and Chemical Properties
Atomic mass (amu)
Density (g/cm3 )
Specific heat (J/g ∘ C)
Atomic volume (cm3 /mol)
Formula of oxide
Oxide density (g/cm3 )
Formula of chloride
Boiling point of chloride (∘ C)
Density of chloride (g/cm3 )

Predicted

Actual

72
5.5
0.31
13
RO2
4.7
RCl4
<100
1.9

72.3
5.47
0.32
13.5
GeO2
4.70
GeCl4
86
1.84

the nucleus in the early 1900s, the modern form of the periodic table is instead
organized according to increasing atomic number. Furthermore, as we shall see in
a later chapter, the different blocks of groups in the periodic table quite naturally
reflect the quantum nature of atomic structure.

1.4

Gruppo VIII.
—
RO4

THE STANDARD MODEL

As an atom is the smallest particle of an element that retains the essential chemical properties of that substance, one might argue that atoms are the fundamental
building blocks of matter. However, as we have already seen, the atom itself is not
indivisible, as Democritus believed. As early as the 1930s, it was recognized that
there were other fundamental particles of matter besides the proton, the neutron,
and the electron. The muon was discovered by Carl Anderson and Seth Nedermeyer
in 1936. Anderson was studying some of the properties of cosmic radiation when he
noticed a new type of negatively charged particle that was deflected by a magnetic

10

1 THE COMPOSITION OF MATTER

field to a lesser extent than was the electron. The muon has the same charge as the
electron, but it has a mass that is about 200 times larger, which explains why it was
not deflected as much as an electron. Muons are not very stable particles, however;
they have a mean lifetime of only 2.197 × 10−6 s. Muons occur when cosmic radiation interacts with matter and are also generated in large quantities in modern-day
particle accelerators. As it turns out, however, the muon represents just one strange
beast in a whole zoo of subatomic particles that include hadrons, baryons, neutrinos,
mesons, pions, quarks, and gluons—to name just a few, begging the question of just
how divisible is matter and what (if anything) is fundamental?
The standard model of particle physics was developed in the 1970s following
experimental verification of quarks. The standard model incorporates the theory
of general relativity and quantum mechanics in its formulation. According to the
standard model, there are a total of 61 elementary particles, but ordinary matter is composed of only six types (or flavors) of leptons and six types of quarks.
Leptons and quarks are themselves examples of fermions, or particles that have a
spin quantum number of 1/2 and obey the Pauli exclusion principle. It is the various
combinations of these fundamental particles that make up all of the larger particles,
such as protons and neutrons. Thus, for example, a proton is composed of two
up quarks and one down quark (pronounced in such a way that it rhymes with the
word “cork”). Electrons, muons, and neutrinos are all examples of leptons. Both leptons and quarks can be further categorized into one of three different generations,
as shown in Figure 1.9. First-generation particles, such as the electron and the up
and down quarks that make up protons and neutrons, are stable, whereas secondand third-generation particles exist for only brief periods of time following their
generation. Furthermore, each of the 12 fundamental particles has a corresponding
antiparticle. An antiparticle has the same mass as a fundamental particle, but exactly
the opposite electrical charge. The antiparticle of the electron, for instance, is the
positron, which has a mass of roughly 9.109 × 10−31 kg like the electron, but an electrical charge of +1.602 × 10−19 C or +1e. Whenever a particle and its antiparticle
collide, they annihilate each other and create energy. In addition to the 12 fundamental particles and their antiparticles, there are also force-carrying particles, such as

≈2.3 MeV/c2

≈1.275 GeV/c2

≈173.07 GeV/c2

0

≈126 GeV/c2

Charge

2/3

2/3

2/3

0

0

Spin

1/2

1/2

1/2

1

0

Charm

Top

≈95 MeV/c2

≈4.18 GeV/c2

0

–1/3

–1/3

–1/3

0

1/2

1/2

1/2

1

Photon

Down

Strange

Bottom

0.511 MeV/c2

105.7 MeV/c2

1.777 GeV/c2

91.2 GeV/c2

–1

–1

–1

0

1/2

1/2

1/2

1

Tau

Muon

Z boson

<2.2 eV/c2

<0.17 MeV/c2

<15.5 MeV/c2

80.4 GeV/c2

0

0

0

±1

1/2

1/2

Electron
neutrino

1/2

Muon
neutrino

1

Tau
neutrino

Higgs
boson

Gluon

≈4.8 MeV/c2

Electron

Leptons

FIGURE 1.9
The 12 fundamental particles
(leptons in green and quarks in
purple) and the force-carrying
particles (in red) that comprise
the standard model of particle
physics. The newly discovered
Higgs boson, which explains why
some particles have mass, is
shown at the upper right.
[Attributed to MissMJ under the
Creative Commons Attribution
3.0 Unported license (accessed
October 17, 2013).]

Quarks

Up

W boson

Gauge bosons

Mass

11

1.4 THE STANDARD MODEL

Up

Charm

Top

Down

Strange

Bottom

the photon, which carries the electromagnetic force. Collectively, the 12 fundamental particles of matter are known as fermions because they all have a spin of 1/2, while
the force-carrying particles are called bosons and have integral spin. The different
types of particles in the standard model are illustrated in Figure 1.9.
There are four types of fundamental forces in the universe, arranged here in
order of increasing relative strength: (i) gravity, which affects anything with mass;
(ii) the weak force, which affects all particles; (iii) electromagnetism, which affects
anything with charge; and (iv) the strong force, which only affects quarks. There are
six quarks, as shown in Figure 1.10, and they are arranged as pairs of particles into
three generations. The first quark in each pair has a spin of +2/3, while the second
one has a spin of −1/3.
Quarks also carry what is known as color charge, which is what causes them
to interact with the strong force. Color charges can be represented as red, blue,
or green, by analogy with the RGB additive color model, although this is really just
a nonmathematical way of representing their quantum states. Like colors tend to
repel one another and opposite colors attract. Because of a phenomenon known
as color confinement, an individual quark has never been directly observed because
quarks are always bound together by gluons to form hadrons, or combinations of
quarks. Baryons consist of a triplet of quarks, as shown in Figure 1.11. Protons and
neutrons are examples of baryons that form the basic building blocks of the nucleus.
Mesons, such as the kaon and pion, are composed of a pair of particles: a quark and
an antiquark.
Unlike quarks, which always appear together in composite particles, the leptons are solitary creatures and prefer to exist on their own. Furthermore, the
leptons do not carry color charge and they are not influenced by the strong force.
The electron, muon, and tau are all negatively charged particles (with a charge of
−1.602 × 10−19 C), differing only in their masses. Neutrinos, on the other hand,
have no charge and are particularly difficult to detect. The electron neutrino has
an extremely small mass and can pass through ordinary matter. The heavier leptons
(the muon and the tau) are not found in ordinary matter because they decay very
quickly into lighter leptons, whereas electrons and the three kinds of neutrinos are
stable.
Proton

Neutron

1.6 fm

FIGURE 1.10
Cartoon representation of the
six different flavors of quarks
(arranged into pairs by their
generations). The numbers
inside each quark represent their
respective charges. [Blatt
Communications.]

FIGURE 1.11
Representation of a proton,
which is made from two up
and one down quarks, and a
neutron, which is made from
one up and two down quarks.
The diameter of the proton
and neutron are roughly
drawn to scale; however, the
quarks are about 1000 times
smaller than a proton or a
neutron. [Blatt
Communications.]

12

1 THE COMPOSITION OF MATTER

Well, now that we know what matter is made of, we might ask ourselves the
question of what it is that holds it together. Each of the four fundamental forces
(with the exception of gravity, which has not yet fully been explained by the standard
model) has one or more force-carrying particles that are passed between particles of
matter. The photon is the force-carrying particle of electromagnetic radiation. The
photon has zero mass and only interacts with charged particles, such as protons,
electrons, and muons. It is the electromagnetic force that holds atoms together in
molecules—the electrons orbiting one nucleus can also be attracted to the protons in a neighboring nucleus. The electromagnetic force is also responsible for why
particles having the same charge repel one another. Because they are all positively
charged, one might wonder how it is that more than one proton can exist within
the very small confines of the nucleus. The explanation for this conundrum is that
protons are made up of quarks. The quarks are held together in triplets in the proton by the strong force because they have color charge. Likewise, it is the residual
strong force, where a quark on one proton is attracted to a quark on another proton or neutron, which holds the protons and neutrons together inside the nucleus.
The force-carrying particle for the strong force is the gluon. Quarks absorb and emit
gluons very rapidly within a hadron, and so it is impossible to isolate an individual
quark. The weak force is responsible for an unstable heavier quark or lepton disintegrating into two or more lighter quarks or leptons. The weak force is carried by
three different force-carrying particles: the W+ , W− , and Z bosons. The W+ and
W− particles are charged, whereas the Z particle is neutral. The standard model
also predicts the presence of the Higgs boson, popularly known as the god particle, which is responsible for explaining why the fundamental particles have mass.
Recently, scientists working at the LHC (Large Hadron Collider) particle accelerator have finally discovered evidence suggesting the existence of the elusive Higgs
boson. In fact, Peter Higgs, after which the Higgs boson was named, shared the
2013 Nobel Prize in physics for his contributions in the area of theoretical particle physics. The particles that comprise the standard model of particle physics are
to date the most fundamental building blocks of matter. Despite its incredible successes, the standard model has yet to accurately describe the behavior of gravity or
why there are more particles in the universe than antiparticles and why the universe
contains so much dark matter and dark energy. Physicists continue to search for
a grand unified theory of everything, and one is therefore left to wonder whether
anything at all is truly fundamental. In the following chapter, we examine some further properties of the nucleus and show how matter and energy themselves can be
interconverted.

EXERCISES
1.1. In Thomson’s cathode ray tube experiment, the electron beam will not be deflected
unless an external electric or magnetic field has been applied. What does this result
imply about the force of gravity on the electrons (and hence about the mass of an
electron)?
1.2. If a beam of protons were somehow substituted in Thomson’s cathode ray tube experiment instead of a beam of electrons, would their deflection by an electrical field be
larger or smaller than that for an electron? Explain your answer. What would happen if
a beam of neutrons were used?
1.3. The following data were obtained for the charges on oil droplets in a replication of the
Millikan oil drop experiment: 1.5547 × 10−19 , 4.6192 × 10−19 , 3.1417 × 10−19 , 3.0817 ×
10−19 , 1.5723 × 10−19 , 1.5646 × 10−19 , 1.5420 × 10−19 , and 1.5547 × 10−19 C. Use these
data to calculate the average charge on a single electron. Explain how you arrived at
your result.

BIBLIOGRAPHY

1.4. An alpha particle is the same as a helium-4 nucleus: it contains two protons and two
neutrons in the nucleus. Given that the radius of an alpha particle is approximately
2.6 fm, calculate the density of an alpha particle in units of grams per cubic centimeter.
1.5. Given that the mass of an average linebacker at Ursinus College is 250 lbs and the radius
of a pea is 0.50 cm, calculate the number of linebackers that would be required to be
stuffed into the volume of a pea in order to obtain the same density as an alpha particle.
1.6. Given that the radius of the helium-4 nucleus is approximately 2.6 fm, the classical electron radius is 2.8 fm, and the calculated atomic radius of 4 He is 31 pm, calculate the
percentage of the space in a helium-4 atom that is actually occupied by the particles.
1.7. Explain the similarities and differences between scanning tunneling microscopy and
atomic force microscopy.
1.8. At the time when Mendeleev formulated the periodic table in 1871, the element gallium
had yet to be discovered, and Mendeleev simply left a gap in his periodic table for it. By
interpolating data from the elements that surround gallium in the periodic table, predict
the following information about gallium and then compare your predictions to the actual
values: its atomic mass, its density, its specific heat, its atomic volume, its melting point,
the molecular formula for its oxide, the density of its oxide, the molecular formula for
its chloride, and the density of its chloride.
1.9. Which of the following particles will interact with an electromagnetic field? (a) An electron, (b) an up quark, (c) an electron neutrino, (d) a proton, (e) a positron, (f) a muon,
(g) a pion.
1.10. Explain why it is that electrons traveling in the same region of space will always repel
one another, but protons can exist in close proximity with each other in the interior of
the nucleus.

BIBLIOGRAPHY
1. Atkins P, Jones L, Laverman L. Chemical Principles: The Quest For Insight. 6th ed. New
York: W. H. Freeman and Company; 2013.
2. McMurry J, Fay RC. Chemistry. 4th ed. Upper Saddle River, NJ: Pearson Education, Inc;
2004.
3. Nave, C. R. HyperPhysics. http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
(accessed Oct 10, 2013).
4. Schaffner, P. and the Particle Data Group at Lawrence Berkeley National Laboratory,
The Particle Adventure: The Fundamentals of Matter and Force.
http://www.particleadventure.org/index.html (accessed July 2, 2012).
5. Segrè E. From X-Rays to Quarks: Modern Physicists and Their Discoveries. New York: W. H.
Freeman and Company; 1980.

13

The Structure
of the Nucleus

2

“If, as I have reason to believe, I have disintegrated the nucleus of the atom, this
is of greater significance than the war.”
—Ernest Rutherford

2.1

THE NUCLEUS

The defining characteristic of any element is given by the composition of its nucleus.
The nucleus of an atom is composed of the nucleons (protons and neutrons), such
that an element is given the symbol AZ X, where Z is the atomic number (or number
of protons), A is the mass number (also known as the nucleon number), and X is the
one- or two-letter abbreviation for the element. A nuclide is defined as a nucleus
having a specific mass number A. Most elements exist as multiple isotopes, which
differ only in the number of neutrons present in the nucleus. It is important to
recognize that while the different isotopes of an element have many of the same
chemical properties (e.g., react with other elements to form the same stoichiometry
of compounds), they often have very different physical properties. Thus, for example,
while cobalt-59 (59 Co) is a stable isotope and is considered one of the elements
essential to human life, its slightly heavier isotope cobalt-60 (60 Co) is highly unstable
and releases the destructive gamma rays used in cancer radiation therapy. Further,
while “heavy water” or deuterium oxide (D2 O or 2 H2 O) is not radioactive, the
larger atomic mass of the deuterium isotope significantly increases the strength of
a hydrogen bond to oxygen, which slows the rates of many important biochemical
reactions and can (in sufficient quantities) lead to death.
The nucleus of an atom is restricted to a very small radius (typically on the order
of 10−14 –10−15 m). As the majority of an atom’s mass is located in a highly confined
space, the density of a nucleus is exceptionally large (approximately 1014 g/cm3 ). In
fact, it was the presence of a very dense nucleus in the Geiger–Marsden experiment that led to the unexpected observation that some of the alpha particles were
deflected backward toward the source instead of passing directly through the thin
foil. At first glance, this result should be surprising to you, given that the protons in

The nucleus. [Attributed to Marekich, reproduced from http://en.wikipedia.org/wiki/Atomic_nucleus
(accessed October 17, 2013).]

Principles of Inorganic Chemistry, First Edition. Brian W. Pfennig.
© 2015 John Wiley & Sons, Inc. Published 2015 by John Wiley & Sons, Inc.

16

2 THE STRUCTURE OF THE NUCLEUS

a nucleus are positively charged and should therefore repel one another—especially
at short distances. It was not until the 1970s when the strong interaction, one of
the four fundamental forces of nature that comprise the standard model of particle physics, was discovered. The strong force is, as its name implies, the strongest of
these fundamental forces. It is approximately 102 times stronger than the electromagnetic force, which is what causes the protons to repel one another, 106 times
stronger than the weak force, and 1039 times more powerful than the gravitational
force. However, the strong force acts only over very short distances, typically on the
order of 10−15 m. The strong interaction is the force that is carried by the gluons
and holds quarks having unlike color charges together to form hadrons. Over larger
distances, it is the residual strong force that is responsible for holding the protons
and neutrons together in the nucleus of an atom.

2.2

NUCLEAR BINDING ENERGIES

The nuclear binding energy is a measure of how strongly the nucleons are held
together in the nucleus by the strong force. In one sense, it is analogous to
the bond dissociation energy, which measures how strongly atoms are held
together in a molecule. The nuclear binding energy (ΔE) can be calculated from
Equation (2.1), where Δm is the mass defect and c is the speed of light in vacuum
(2.99792458 × 108 m/s):
(2.1)
ΔE = (Δm)c2
According to Einstein’s theory of relativity, matter and energy are interchangeable. It is for this reason that the masses of subatomic particles are often listed
with energy units of MeV/c2 , as shown in Table 2.1. Solving Equation (2.1) for c2
using appropriate units, one obtains the useful equality that c2 = 931.494 MeV/amu.
Because it always takes energy to split a nucleus apart into its isolated nucleon components, the mass of an atom or a nuclide is always less than the sum of its parts.
The mass defect of the particle is therefore defined as the difference in mass between
all the subatomic particles that comprise the atom or nuclide and the mass of the
isotope itself.
Example 2-1. Calculate the nuclear binding energy of an alpha particle if its mass
is 4.00151 amu.
Solution. An alpha particle is a helium-4 nucleus. The sum of the masses
of two neutrons (2 * 1.008665 amu) and two protons (2 * 1.007276 amu) is
4.03188 amu. The mass defect is therefore 4.03188 − 4.00151 = 0.03037 amu.
Given that 1 amu = 1.6605 × 10−27 kg and the speed of light in a vacuum is
2.9979 × 108 m/s:
(
)
1.6605 × 10−27 kg
ΔE = (Δm)c2 = (0.03037 amu)
(2.9979 × 108 m∕s)2
1 amu
ΔE = 4.532 × 10−12 J (6.022 × 1023 mol−1 ) = 2.729 × 1012 J∕mol
As 1 eV = 96485 J/mol, E = 2.829 × 107 eV or 28.29 MeV. It is more useful,
however, to compare the binding energy of one nucleus with that of another
in terms of MeV/nucleon. Therefore, the binding energy of an alpha particle is
28.29 MeV/4 nucleons or 7.072 MeV/nucleon. Alternatively, the nuclear binding
energy can be directly calculated in MeV using the values in the right-most column

17

2.3 NUCLEAR REACTIONS: FUSION AND FISSION

of Table 2.1 as follows:
ΔE = (2 ∗ 938.272 + 2 ∗ 939.565) − (4.00151 ∗ 931.494) = 28.29 MeV
ΔE = 28.29 MeV∕4 = 7.073 MeV∕nucleon

TABLE 2.1 The masses of subatomic particles in different units.
Particle
Proton, mp
Neutron, mn
Electron, me
Atomic mass unit, u

Mass (kg)

Mass (amu)

Mass (MeV/c2 )

1.67262 × 10−27
1.67493 × 10−27
9.10938 × 10−31
1.66054 × 10−27

1.00728
1.00867
5.48580 × 10−4
1

938.272
939.565
0.510999
931.494

Example 2-2. Calculate the nuclear binding energy of a carbon-12 atom.
Solution. Carbon-12 consists of six protons, six neutrons, and six electrons
and weighs exactly 12.0000 amu.
ΔE = (6 ∗ 938.272 + 6 ∗ 939.565 + 6 ∗ 0.510999) − (12.0000 ∗ 931.494)
= 92.16 MeV
ΔE = 92.16 MeV∕12 = 7.680 MeV∕nucleon

In nuclear chemistry, the entropy is usually zero (except in the interiors of
stars) and therefore the nuclear binding energy can be used as a measure of the
stability of a particular nucleus. Because each isotope of an atom has a different
nuclear binding energy, some isotopes will be more stable than others. Figure 2.1
shows the nuclear binding energy curve (per nucleon) as a function of the mass
number. In general, elements having mass numbers around 60 have the largest binding energies per nucleon. Isotopes having these mass numbers belong to Fe and Ni,
which explains the prevalence of these elements in planetary cores. The maximum
in the nuclear binding energy curve occurs for 56 Fe, which helps to justify its overall cosmic abundance. Iron is believed to be the 10th most prevalent element in
the universe, as shown in Figure 2.2, and it is the 4th most abundant in the earth’s
crust.

2.3

NUCLEAR REACTIONS: FUSION AND FISSION

The transmutation of the elements has long been the goal of the alchemists. In
1917, Ernest Rutherford was the first person to realize that dream. Rutherford
converted nitrogen-14 into oxygen-17 and a proton by bombarding a sample of
14 N with a stream of alpha particles, according to the nuclear reaction shown in
Equation (2.2):
14
N
7

+ 42 He → 178 O + 11 H

(2.2)

18

2 THE STRUCTURE OF THE NUCLEUS

FIGURE 2.1
Nuclear binding energy curve
plotting the average binding
energy per nucleon as a function
of the mass number A. The
maximum binding energy occurs
for the “iron group” of isotopes
having mass numbers between
56 and 60. [© Keith Gibbs,
www.schoolphysics.co.uk.]

–2
Energy released by fusion

Binding energy per nucleon (MeV)

–1

–3
–4
–5
–6

238U

Energy released by fission
–7
–8
–9

0

20

40

60

80

100

120

140

160

180

220

240

Mass number (A)
4He 16

56Fe

O

85Br

148

La

12C

Hydrogen

1

Helium

Relative abundance

10–2

Carbon
Oxygen
Neon
Magnesium

10–4

Silicon
Sulfur

Iron

10–6

10–8

FIGURE 2.2
Relative cosmic abundances of
the elements, as compared to
that of hydrogen. [Reproduced
by permission from Astronomy
Today, Chaisson and McMillan,
8th ed., Pearson, 2014.]

200

Boron
Lithium
Beryllium

10–10

10–12
1

10

20
30
Atomic number

40

50

19

2.3 NUCLEAR REACTIONS: FUSION AND FISSION

Whenever writing a nuclear equation, the sum of the atomic numbers of the
reactants must equal the sum of the atomic numbers of the products and the sums of
the mass numbers on each side of the nuclear equation must also be equal. This does
not, however, imply that conservation of mass must apply. Because each isotope has
a unique nuclear binding energy, some mass may be lost or gained in the form of
energy during a nuclear reaction. The energetics of nuclear reactions are measured
in terms of Q, which can be calculated from Equation (2.3), where the masses of
the individual nuclides are recorded in MeV/c2 , as in Table 2.1. If the sign of Q for a
nuclear equation is positive, the reaction is said to be exothermic. By contrast, if the
sign of Q is negative, the nuclear reaction is endothermic and it will require kinetic
energy in order to proceed:
∑
∑
masses −
masses
(2.3)
Q = −ΔH =
reactants

products

Example 2-3. Given that the masses of the isotopes in Equation (2.2) are
14.00307, 4.00151, 16.99913, and 1.00728 amu for 14 N, an alpha particle,
17 O, and a proton, respectively, calculate Q for the nuclear reaction given by
Equation (2.2). Is the reaction endothermic or exothermic?
Solution
Q = [(14.00307 + 4.00151) − (16.99913 + 1.00728) amu] ∗ 931.494 MeV∕amu
= −1.7046 MeV
Because Q is negative, the reaction is endothermic.
Nuclear fusion occurs when two or more small nuclei are joined together to
form a larger nucleus. Typically, when two smaller nuclei fuse together, a tremendous
amount of energy is released—many orders of magnitude larger than the energy
released in an ordinary chemical reaction. Thus, for example, because the average
nuclear binding energy per nucleon (Figure 2.1) is much larger for He than it is for
H, a self-sustaining nuclear fusion reactor would be a fantastic source of energy. One
such example of a typical nuclear reaction occurring in a fusion reactor is given by
Equation (2.4) and is illustrated by the diagram shown in Figure 2.3:
2
1H

+ 31 H → 42 He + 10 n

2H

3H

4He

n + 14.1 MeV

(2.4)

+ 3.5 MeV

FIGURE 2.3
Illustration of the nuclear fusion
reaction given by Equation (2.4).
[Reproduced from http://en.
wikipedia.org/wiki/Nuclear_
fusion (accessed October 17,
2013).]

20

2 THE STRUCTURE OF THE NUCLEUS

Example 2-4. Given that the masses of deuterium, tritium, helium-4, and a neutron are 2.01410, 3.01605, 4.00260, and 1.00867 amu, respectively, prove that
the total energy released by the fusion reaction illustrated in Figure 2.3 and given
by Equation (2.4) is 17.6 MeV.
Solution
Q = [(2.01410 + 3.01605) − (4.00260 + 1.00867) amu] ∗ 931.494 MeV∕amu
= 17.587 MeV

Nuclear fission occurs when a heavier nucleus splits apart to form lighter (or
daughter) nuclei. Fission processes can also release tremendous amounts of energy,
as illustrated by the use of atomic weapons. In his famous letter to President Franklin
D. Roosevelt in August 1939, Albert Einstein, acting on the request of Leo Szilard,
informed the president of the possibility that scientists in Nazi Germany were working on a powerful new weapon based on nuclear fission reactions. Shortly thereafter,
incredible financial and R&D resources were poured into the super-secret Manhattan Project in an effort to produce a viable nuclear weapon. As a result of these
efforts, the first atomic bomb, known simply as The Gadget, was detonated near the
desert town of Alamogordo, NM, on July 16, 1945 (Figure 2.4). Only several weeks
later, the first atomic bombs used in combat were dropped on the Japanese cities of
Hiroshima and Nagasaki on August 6 and 9, respectively. These weapons were credited with ending World War II and saving the lives of the many American soldiers,
which would have been required for a ground invasion. The basic fission reaction
used in the first nuclear weapons is shown by Equation (2.5):
235
U
92

FIGURE 2.4
The Trinity test of the first
atomic bomb in the desert near
Alamogordo, NM. [Photo credit:
US Department of Energy.]

+ 10 n → 140
Ba + 93
Kr + 310 n
56
36

(2.5)

21

2.3 NUCLEAR REACTIONS: FUSION AND FISSION

Conventional
chemical explosive

FIGURE 2.5
Illustration of the way two
subcritical pieces of 235 U are
combined in a nuclear
weapon to initiate the
self-sustaining fission reaction
shown by Equation (2.5).
[Reproduced from
http://en.wikipedia.org/wiki/
Fission_bomb#Fission_weapons
(accessed November 30,
2013).]

Subcritical pieces of
uranium-235 combined

Gun-type assembly method

Fission fragment
140

Fission fragment
140Ba

1n

235

1n

Ba

235U
1n

1n
ENERGY
1n

U

1n
93Kr

ENERGY

140Ba

1n
93

Kr

235U

1n

1n
ENERGY
1n
93Kr

Fission fragment

In order for the reaction to be self-sustaining, a supercritical mass of at least
3% (enriched) 235 U must be assembled using a conventional explosive, as shown in
Figure 2.5. At this high of a concentration, the neutrons produced by the fission
of the uranium-235 isotope have a large enough cross section and sufficient kinetic
energy to initiate the fission of a neighboring 235 U nucleus, leading to a chain reaction,
as shown in Figure 2.6. The earliest atomic bombs had a total energy equivalent to
18 kton of TNT. Modern hydrogen bombs typically have a plutonium core and use
the energy generated from the initial fission reaction to initiate a fusion reaction
of hydrogen nuclei, further enhancing the destructive output. As a result, modern
atomic weapons have a frighteningly large destructive capacity of approximately 1.2
Mton of energy.
Example 2-5. Given the following masses, calculate the energy released by
the fission reaction illustrated by Equation (3.5): 235 U (235.0439 amu), 140 Ba
(139.9106 amu), 93 Kr (92.9313 amu), and a neutron (1.00867 amu).
Solution
Q = [(235.0439 + 1.00867) − (139.9106 + 92.9313 + 3 ∗ 1.00867) amu]
∗ 931.494 MeV∕amu = 172.0 MeV

FIGURE 2.6
Self-sustaining chain reaction
from the fission of 235 U once a
supercritical mass of uranium
has been assembled. [Blatt
Communications.]

22

2 THE STRUCTURE OF THE NUCLEUS

2.4

RADIOACTIVE DECAY AND THE BAND OF STABILITY

Most fission reactions have high activation barriers and are usually very slow. Using
today’s technology, the upper limit for the measurement of nuclear lifetimes is about
1020 years, so that any nuclide with a longer lifetime than this is considered as stable.
There are 266 naturally occurring stable isotopes of the elements. Every element up
to and including atomic number 83 (bismuth) has at least one stable isotope, with
the exceptions of Tc and Pm. In fact, many elements have more than one stable isotope. The masses of stable isotopes can be measured by several different techniques,
but they are most commonly measured using mass spectrometry. A mass spectrometer, such as the one shown in Figure 2.7, can be used to determine the mass of
an isotope relative to the standard value of exactly 12 amu for the 12 C isotope
and can also record its relative abundance. A volatilized sample of the substance is
bombarded by an electron beam to form a stream of positively charged ions. These
ions are then accelerated through an electromagnetic field and separated from one
another by their mass-to-charge (m/Q) ratios. Those ions having a larger positive
charge or a lighter mass will be deflected more strongly by the magnetic field. The
separated ions having different m/Q ratios are then collected and counted by the
detector.
The mass spectrum for a sample of atomic chlorine is shown in Figure 2.8,
indicating that the two most abundant stable isotopes of chlorine are 35 Cl and 37 Cl.

Magnet
Accelerating
grid

Beam of
positive ions

Detector

Heating
filament

Inject
sample

Separation of ions
based on mass
differences

Ionising
electron beam

Recorder

FIGURE 2.7
Schematic diagram of a mass spectrometer. [Blatt Communications.]

Relative abundance

FIGURE 2.8
Mass spectrum for the two
isotopes of chlorine. [Blatt
Communications.]

35

Cl

37

33

34

Cl

35 36 37 38
Mass/charge ratio

39

23

2.4 RADIOACTIVE DECAY AND THE BAND OF STABILITY

The atomic mass of an element as listed in the periodic table is a weighted
average of all the known isotopes having the same atomic number, as shown by
Equation (2.6), where fi is the decimal percentage of the naturally occurring abundance of an isotope and mi is its atomic mass and the sum is for all the known
isotopes having atomic number Z:
[
]
∑
atomic mass on
fi mi
=
periodic table
i

(2.6)

The masses and natural abundances of every known isotope have been tabulated by the National Nuclear Data Center in a booklet known as the Nuclear
Wallet Cards. An electronic version of the nuclear wallet cards can be found at
http://www.nndc.bnl.gov/wallet/wccurrent.html. The mass excess (Δ = M − A) in this
table is given in units of megaelectronvolts and needs to be converted into amu for
all practical calculations (recall that 1 amu = 931.494 MeV).

Example 2-6. Given the mass spectrum for the sample of chlorine shown in
Figure 2.8 and the data in the Nuclear Wallet Cards, calculate the weighted average
atomic mass of a chlorine atom.
Solution. The two stable isotopes of chlorine have the following natural abundances and masses:

35 Cl
37 Cl

75.76%
24.24%

−29.013 MeV
−31.761 MeV

*Calculated as follows: M = Δ + A = −29.013 MeV
34.969 amu

(

34.969 amu*
36.966 amu
1 amu
931.494 MeV

)

+ 35 amu =

Therefore, the weighted average calculated using Equation (2.6) is
atomic mass = 0.7576(34.969 amu) + 0.2424(36.966 amu) = 35.453 amu

Example 2-7. Silicon exists as three stable isotopes (28 Si = 27.977 amu,
29 Si = 28.976 amu, and 30 Si = 29.974 amu). Given that the atomic mass of Si on
the periodic table is 28.086 amu and the natural abundance of 28 Si is 92.23%,
calculate the natural abundances of 29 Si and 30 Si, respectively.
Solution. The total decimal percentage of each isotope must equal unity. Letting
x be the decimal percentage of 29 Si and y be the decimal percentage of 30 Si, then
0.9223 + x + y = 1 or y = 1 − 0.9223 − x = 0.0777 − x.
Substituting these values into Equation (2.6), we obtain the following
equation:
28.086 amu = 0.9223 (27.977 amu) + x(28.976 amu) + (0.0777 − x) (29.974 amu)
Solving for x, we get x = 0.0463 and y = 0.0314. Thus, the percentages of 29 Si
are 4.63 and 3.14%, respectively.
and
30 Si

24

2 THE STRUCTURE OF THE NUCLEUS

Stable
1014 yr
160

1012 yr
1010 yr

140

108 yr
106 yr

120
104 yr
100 yr

100

1 yr
Z=N
80

FIGURE 2.9
Plot of the different isotopes of
the elements as the N/Z ratio.
The solid line is for N/Z = 1 and
the dark band of stability rises
away from this line as the
number of neutrons in the
nucleus increases. Elements on
either side of the band of
stability will undergo
spontaneous radioactive decay.
[Reproduced from http://en.
wikipedia.org/wiki/Table_of_
nuclides_(complete) (accessed
November 30, 2013).]

106 s
104 s
100 s

60

1s
40

10–2 s
10–4 s

20

10–6 s
10–8 s
Unstable

N
Z

20

40

60

80

100

When the 266 known stable isotopes are plotted as N (number of neutrons)
versus Z (number of protons), they form a band of stable nuclides on the positive side
of the line A = 2Z, as shown in Figure 2.9, because the increasing percent composition
of neutrons helps dilute the Coulombic repulsion of the positively charged protons in
the nucleus. The largest stable isotope on this graph occurs for 208 Pb. A maximum
occurs because the strong force can act only over very short ranges to hold the
nucleons together and the radii of the heavier elements eventually becomes larger
than this threshold. The band of stability can be compared to an island rising up out of
an ocean ridge. Those isotopes having the largest nuclear binding energies will have
the highest elevations and those beneath a certain threshold nuclear binding energy
will occur below sea level and will therefore be unstable. Thus, the isotopes on either
side of the dark band of stability in Figure 2.9 are underwater by this analogy and
will always undergo spontaneous radioactive decay to form a more stable nuclide.
The concept of radioactivity was discovered in 1896 by Henri Becquerel, although
the term itself was actually first coined by Marie Curie. This pair (along with Madam
Curie’s husband Pierre) shared the 1903 Nobel Prize in physics for their work on
radioactivity.
The different types of radioactive decays are listed in Table 2.2, according to
the type of particle or radiation that is emitted. Although it is not strictly a form of

25

2.4 RADIOACTIVE DECAY AND THE BAND OF STABILITY

TABLE 2.2 Types of radioactive decay.
Type

Penetration

Speed

Particle

Example

α-Decay

Not very far,
but severe
Moderately
far
Very far

∼0.10c

−42 He2+

226
88 Ra

<0.90c

−−10 e

231
53 I

c

−Photon

60
∗
27 Co

Moderately
far
Low to
moderate
Very far

<0.90c

−01 e

11
6C

∼0.10c

−11 H+

53
27 Co

<0.10c

−10 n

137
53 I

1
→ 136
53 I + 0 n

NA

NA

+−10 e

44
22 Ti

+ −10 e → 44
21 Sc

β-Decay
𝛾-Emission
Positron
emission
Proton
emission
Neutron
emission
Electron
capture∗

2−
→ 222
+ 42 He2+
86 Rn

0
→ 231
52 Xe + −1 e

→ 60
27 Co + γ

→ 115 B + 01 e
1
→ 52
26 Fe + 1 H

∗ Electron capture is actually a nuclear reaction and not a genuine type of radioactive
decay. Therefore, it does not follow first-order kinetics.

radioactive decay, electron capture has also been included in the table. The only way
to increase the atomic number Z is by beta decay, or the emission of an electron.
Thus, any nuclide having a higher mass number than the band of stability can decrease
its N/Z ratio by an increase in Z. On the other hand, isotopes below the band
of stability can decrease Z by one of several different methods: positron emission
(which is most common for low atomic numbers), alpha decay (which is more typical for larger atomic numbers), and neutron emission (which is rare). Extremely
large unstable isotopes can also undergo fission in order to split apart into smaller,
more stable daughter nuclides. The specific type of radioactive decay that an unstable
nucleus will undergo is listed in the Nuclear Wallet Cards.
All radioactive decays occur with first-order kinetics, with the exception of electron capture, which is a two-particle collision. The differential rate law for radioactive
decay is given by Equation (2.7). After integration, an alternative and more useful
form of the rate law is shown by Equation (2.8). The half-life of radioactive decay is
defined as the length of time it takes for the number of unstable nuclides to decrease
to exactly one-half of their original value. The half-life, 𝜏 can be calculated using
Equation (2.9), where k is the first-order rate constant:
rate = −

dN
= kN
dt

N
= e−kt
N0
𝜏=

ln(2)
k

(2.7)
(2.8)
(2.9)

Example 2-8. Radioactive iodine is used to image the thyroid gland. Typically,
a saline solution of Na131 I is administered to the patient by an IV drip. Predict
the most likely type of radioactive decay for this nuclide and calculate Q for the
reaction. Given that the half-life of 131 I is 8.025 days, what percentage of the
isotope will have decayed during the 2.0-h procedure?

26

2 THE STRUCTURE OF THE NUCLEUS

Solution. According to the Nuclear Wallet Cards, the only stable isotope of
iodine is 127 I. Therefore, 131 I lies to the higher side of the band of stability and
will need to increase Z in order to become a stable isotope. The only form of
radioactive decay that increases Z is the emission of a beta particle. Using the
principles of conservation of mass number and conservation of atomic number
during a nuclear reaction, the nuclear equation for beta decay is
131
I
53

→ −10 e + 131
Xe
54

The mass excesses of these particles given by the Nuclear Wallet Cards are
as follows: Δ = −87.442 MeV for iodine-131 and −88.413 MeV for xenon-131.
Using the formula that M = Δ + A and the conversion that 1 amu = 931.494 MeV,
the masses of each isotope can be calculated as follows:
(
)
931.494 MeV
131
I 131 amu
− 87.442 MeV
53
1 amu
= 121, 938.272 MeV (or 130.906 amu)
(
)
931.494 MeV
131
Xe
131
amu
− 88.413 MeV
54
1 amu
= 121, 937.301 MeV (or 130.905 amu)
Q = 121, 938.272 − (0.510999 + 121, 937.301) = 0.460 MeV
The reaction is exothermic, as might be expected for a spontaneous process
(given that the entropy is zero). The kinetics of the process are first-order, so
that the rate constant k can be calculated from Equation (2.9):
k=

ln(2)
0.6931
=
= 0.08637 days−1
𝜏
8.025 days

Next, Equation (2.8) can be used to calculate the ratio of
compared with the initial amount after 2 h (or 0.083 days):

131 I

remaining

−1
N
= e−(0.08637 days )(0.083 days) = 0.9928
N0

Thus, 99.28% of the original 131 I is remaining and only 0.72% has decayed.

Example 2-9. The half-life of lead-214 is 26.8 min. Assuming that the sample
is initially 100% 214 Pb, use a spreadsheet to calculate the percentage of 214 Pb
remaining as a function of time every 10 min for a total of 100 min. Then graph
these data as % 214 Pb versus time.
Solution. Using Equation (2.9), the rate constant k was calculated as
0.0259 min−1 . Using Equation (2.8), the percentage of 214 Pb remaining was
calculated every 10 min for a total period of 100 min using the spreadsheet. The
results are plotted in the graph below the table.

27

2.5 THE SHELL MODEL OF THE NUCLEUS

Time (min)

% Pb-214

0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
100.0

100.0
77.2
59.6
46.0
35.5
27.4
21.1
16.3
12.6
9.7
7.5

Percent lead-214

100.0
80.0
60.0
40.0
20.0
0.0
0.0

20.0

40.0
60.0
Time (min)

80.0

100.0

As the largest stable isotope is 208 Pb, any nuclei heavier than this will undergo
spontaneous radioactive decay to form a smaller, more stable nucleus. However,
this process does not always occur in a single step. More commonly, a number of
nuclear reactions occur until a stable isotope is formed through what is known as
a radioactive decay series. Thus, for example, 238 U emits an alpha particle to form
234 Th, which in turn undergoes two successive beta decays to form the unstable
214 U nucleus. Following a series of five successive alpha decays, the isotope 214 Pb is
formed. Finally, the stable 208 Pb nucleus is reached following two more beta decays,
an alpha decay, two further beta decays, and then one final alpha decay. The entire
sequence of steps is shown in Figure 2.10. All of the steps in the series take place
at different rates until the stable nucleus is achieved in the end. Similar decay series
are known for many other heavier elements.

2.5

THE SHELL MODEL OF THE NUCLEUS

Table 2.3 shows the number of stable isotopes as a function of the even or odd nature
of the number of nucleons. Of the 266 naturally occurring stable isotopes, more
than half have an even number of both protons and neutrons. By comparison, there
are only four stable isotopes having both an odd number of protons and neutrons.
Furthermore, even a cursory inspection of the cosmic abundances of the elements
shown in Figure 2.2 makes it clear that there are more elements having an even
atomic number than there are those with an odd atomic number. In fact, there even
seem to be certain “magic numbers” of nucleons that are consistent with the most
stable nuclei. The nuclear magic numbers are analogous to the common observation

28

2 THE STRUCTURE OF THE NUCLEUS

238

238

U

α

234

234

Th

β 234

Pa

β 234

U

α

230

230

Th

α

Mass number

226

226

222

222

Rn

α

218

218

Po

α

214

214

Pb

β 214

α

210

210

Ti

β 210

Pb
α

FIGURE 2.10
The radioactive decay series for
238
U to 208 Pb. [Blatt
Communications.]

Ra

α

206

206

To

81

β

206

Bi

β 214

Po

α
β 210

Bi

β 210

Po

α

Pb

82

83

84

85 86 87
Atomic number

88

89

90

91

92

TABLE 2.3 Number of the 266 naturally occurring stable nuclides relative
to the numbers of protons and neutrons they contain.
Number of
Protons

Number of
Neutrons

Number of
Stable Nuclides

Examples

Even

Even

157

Even

Odd

55

Odd

Even

50

Odd

Odd

4

16
40
4
208
2 He, 8 O, 20 Ca, 82 Pb
13
29
9
47
4 Be, 6 C, 14 Si, 22 Ti
19
23
89
127
9 F, 11 Na, 39 Y, 53 I
2
6
10
14
1 H, 3 Li, 5 B, 7 N

that certain elements on the periodic table (the noble gases) are especially stable
(atomic numbers 2, 10, 18, 36 54, and 86), except that the magic numbers of nucleons
are different: 2, 8, 20, 28, 50, 82, and 126. Those nuclei that have double magic
numbers are especially stable. Examples of double magic number nuclei include 4 He
(2p, 2n), 16 O (8p, 8n), 40 Ca (20p, 20n), and 208 Pb (82p, 126n). Clearly, there must be
some underlying phenomenon responsible for this rather unusual observation.
Just as it is now understood that the underlying reason for the stability of the
noble gases has to do with the quantum nature of the electrons in atoms (e.g., Bohr’s
shell model of the atom in which the energy levels are quantized), the shell model of
the nucleus states that the different energy levels of the nucleus are also quantized
and that it is this quantization that leads to the enhanced stability of those nuclei
having the nuclear magic numbers listed earlier. The main difference between the
two models has to do with the fact that the electrons in an atom repel each other
through the electromagnetic force (Coulomb’s law), while the nucleons are attracted
to one another through the strong force. As a first approximation, the structure of
the nucleus can be approximated using the harmonic oscillator model (see Chapter 3
for more details), where the attractive forces of the nucleons in Figure 2.11 are given
by the potential energy described in Equation (2.10), where r is the distance between
two nucleons and R is the size of the corresponding square well potential. The solutions to the harmonic oscillator problem are governed by four quantum numbers:
𝜐, the principal quantum number (which takes values of 1, 2, 3, … ), l, the angular

29

2.5 THE SHELL MODEL OF THE NUCLEUS

Harmonic oscillator
Square well

0
R

FIGURE 2.11
Harmonic oscillator versus
square well potential energy
model for the nucleus. [Blatt
Communications.]

R = r0 A1/3

momentum quantum number (which takes values of 0, 1, 2, 3, … and have the corresponding designations s (l = 0), p (l = 1), d (l = 2), f (l = 3), g (l = 4), etc.), ml , the
magnetic substate (which can take 2l + 1 possible values of ±l), and s, the spin state
(which has values of ±1/2). The energies E𝜐l corresponding to the allowed quantum
states are given by Equation (2.11), where ℏ = h/2π and 𝜔 is the angular frequency.
[
]
r2
V(r) = −V0 1 − 2
R
E𝜈l = [2(𝜈 − 1) + l]ℏ𝜔

(2.10)
(2.11)

Because the nucleons are fermions, the Pauli principle must also apply; that is to
say that no two nucleons of the same type (proton or neutron) can have the same
identical set of all four quantum numbers. Because protons have an electric charge
quantum number of +1 while neutrons have a charge quantum number of 0, it is,
however, possible for a proton and a neutron to have the same set of the quantum
numbers 𝜐, l, ml , and s. Even so, the resulting set of nuclear orbitals still could not
explain the observed set of nuclear magic numbers. This problem was later solved
independently in 1949 by Maria Göppert Mayer and Hans Jensen, who shared the
1963 Nobel Prize in physics for their work on the nuclear shell model. It was noted
that because the nucleons were in such close proximity with each other, the orbital
angular momentum and spin angular momentum would couple with one another.
The closest macroscopic analogy would be the way that the moon influences the
tides on earth and how earth’s gravity influences the moon’s rotation. Therefore,
a new quantum number j, which represents the total angular momentum, must be
introduced, as shown in Equation (2.12):
j = l + s = l ± 1∕2

(2.12)

The new quantum mechanical notation for the different energy levels that
includes the spin–orbit coupling term is given by 𝜐lj and has a degeneracy of 2j + 1
nucleons. Assuming that all the nucleons are arranged pairwise into each shell
beginning with the lowest energy level, the rearranged energy level diagram,
shown in Figure 2.12, can be used to explain the experimentally observed magic
numbers.
Evidence supporting the shell model of the nucleus includes the enhanced cosmic abundance of nuclei containing the magic numbers of protons or neutrons (as
shown in Figure 2.13), the stability of isotopes at the end of a radioactive decay series
having magic numbers of protons or neutrons, the nuclear binding energy for a neutron being a maximum for one of the magic numbers and dropping off significantly
with the addition of the next neutron, and the electric quadruple moments being
approximately zero for those isotopes having a magic number. All of these results
are indicative of the added stability of closed nuclear shells.

30

2 THE STRUCTURE OF THE NUCLEUS

Quantum
states angular
momentum

Splitting due
to spin–orbit
coupling

Multiplicity
1g7/2

Total no.
of
nucleons

8

58

1g9/2 10
2p1/2 2
1f5/2 6
2p3/2 4

50
40
38
32

1f7/2

8

28

1d3/2
2s

4
2

20
16

1d5/2

6

14

1p1/2

2

8

1p3/2

4

6

1s

2

2

1g

2s
1d

FIGURE 2.12
Shell model of the nucleus
including strong spin–orbit
coupling. [Blatt
Communications.]

1p
1s

Magic numbers

2p
1f

56Fe
6

Relative abundance

10

FIGURE 2.13
Cosmic abundances of the
elements, showing how many of
the more prevalent elements
have magic numbers of protons
or neutrons. [Blatt
Communications.]

104
Doubly
magic
Z = 82
N = 126

N = 50

102

Z = 50
N = 82

208Pb

0

10

50

100

150

200

Mass number A

Example 2-10. Show all of the different energy levels for a 2d nuclear shell. Also
indicate the degeneracy of each state.
Solution. 𝜐 = 2, l = 2 for 2d. The possible values of j are 5/2 and 3/2. There are
2j + 1 degenerate states associated with each energy level. Thus, the degeneracy
of the 2d5/2 energy level is (6) and the degeneracy of the 2d3/2 energy level is (4).
The 2d shell as an aggregate can therefore hold a total of 10 nucleons.

2.6

THE ORIGIN OF THE ELEMENTS

The famous physicist Carl Sagan once said that “we are star stuff” and he was speaking literally. All of the naturally occurring elements have their origins in the stars.

31

2.6 THE ORIGIN OF THE ELEMENTS

In the beginning, approximately 13.7 billion years ago, all of the known matter and
energy in the universe was concentrated into a single dense region of space known
as the ylem and having a temperature of about 1013 K. As there were no walls to
contain it, the ylem exploded in a cosmic genesis event known as the Big Bang, and
the universe has been expanding ever since. There is a variety of evidence supporting
the theory of a Big Bang. First of all, the electronic spectra from distant galaxies has
been shown to be Doppler-shifted to the red (shifted to longer wavelengths), which
indicates that they are moving away from our solar system. By correlating the magnitude of the red shift, the age of the universe can be calculated to be approximately
13.5–13.9 billion years. Furthermore, there exists a universal cosmic microwave
background radiation from every corner of the universe that is believed to be the
remnant of the initial Big Bang event and that has a black-body radiation temperature
of exactly 2.725 K. Finally, the ratio of He : H in the universe is essentially constant at
0.23 and there is little variation in the isotopic ratios of these two elements across
the entire universe, indicating that the majority of hydrogen and helium was created
as a result of a single cosmic event.
The evolution of the universe has taken place in stages. The first stage following
the Big Bang was the creation of the elementary particles. Initially, matter and energy
existed in a sort of cosmic soup where they could easily interconvert with one
another. As the universe expanded, however, it began to cool to a temperature
between 1010 and 1012 K and the first protons and neutrons were formed. This stage
in the evolution of the universe, which occurred between 10−6 and 1 s after the Big
Bang, is known as baryogenesis. At this point in time, protons could be converted into
neutrons and positrons by bombardment with antineutrinos and neutrons could be
converted back into protons and electrons by the interaction with a neutrino, as
shown in Equations (2.13) and (2.14). A gamma ray could also spontaneously split
apart to form both a positron and an electron in a process known as pair production,
as shown in Equation (2.15):
1
1H
1
n
0

+ 𝜈 → 10 n + 01 e

(2.13)

+ 𝜈 → 11 H + −10 e

(2.14)

𝛾 → 01 e + −10 e

(2.15)

In the first few moments after the Big Bang, the energy density was sufficiently
large that Equations (2.13) and (2.14) occurred at roughly the same rate, so that
the numbers of protons and neutrons were initially equal. However, as the universe
cooled as a result of its expansion, the rate of proton formation began to dominate.
This occurred because the reaction given by Equation (2.13) requires slightly more
energy than the one in Equation (2.14) due to the larger mass of the neutron. As
a result, by the time that the neutrinos and anti-neutrinos stopped interacting with
the other elementary particles (about 1 s after the Big Bang), the composition of the
universe consisted of about 87% protons and 13% neutrons.
At first, the energy density of the universe was still sufficiently strong that the
proton and neutron collisions would immediately break apart again. The next phase
in the evolution of the universe, known as the Big Bang nucleosynthesis (BBN), began
after the universe had cooled even further to a temperature around 109 K, roughly
3 min after the initial expansion so that the density of the universe was approximately
0.1 g/cm3 . In this nucleosynthesis phase, the first nucleons were combined together
to form heavier nuclei. Before this point in time, the next larger nucleus (2 H) was
unable to form because of its low nuclear binding energy. The BBN, which occurred
in the series of fusion reactions shown by Equations (2.16)–(2.19), is therefore

32

2 THE STRUCTURE OF THE NUCLEUS

responsible for the creation of most of the hydrogen and helium in the universe.
1
1H

+ 10 n → 21 H + 𝛾

(2.16)

2
1H

+

2
1H

→ 32 He + 10 n

(2.17)

2
1H

+

2
1H

→ 31 H + 11 H

(2.18)

1
n
0

→ 42 He + 𝛾

(2.19)

3
He
2

+

Because there were initially less neutrons than protons, the former acted as a
limiting reagent, so that by the time the BBN had concluded, the universe consisted
of about 25% He, 74% H, and 1% D, by mass. Trace amounts of Li and Be were also
produced by the reactions given in Equations (2.20) and (2.21). The synthesis of the
heavier elements was significantly reduced by the stability of the double magic number 4 He nucleus. The BBN continued from about 3–20 min following the Big Bang.
The brevity of the BBN also helped prevent the formation of the heavier elements.
As a result, no nuclei heavier than 8 Be were formed by the BBN:
3
He
2

+

4
He
2

→ 74 Be

(2.20)

3
1H

+

4
He
2

→ 73 Li

(2.21)

The BBN was followed by a cooling phase, which allowed the neutrons to decay
into protons. The temperature of the universe at this point in time was no longer
high enough for fusion reactions to occur and it largely consisted of matter. Approximately 379,000 years after the Big Bang, the universe had gradually cooled to a
temperature of about 3000 K. This was finally cool enough for electrons to cling
to the nuclides to form the first neutral atoms. This is sometimes known as the
chemistry phase in the evolution of the universe.
Not much of anything happened again until approximately 109 years after the
Big Bang when the temperature of the universe had cooled even further to about
20 K. At this point in time, clouds of interstellar matter began to coalesce under
the influence of gravity. Gradually, the density of these clouds increased until the
first protostars began to form. Approximately 90% of the stars in the universe at
this point in time were stars having sizes and compositions similar to our sun; stars
that are classified as Main Sequence stars. This type of star was the first kind to form
following the Big Bang. For this reason, the earliest stars are sometimes referred to
as first-generation stars. As the gravitational forces began to contract matter together
in the first stars, the increase in density in their cores caused the temperatures to
rise. At a certain critical point, the increasing temperatures ionized the hydrogen
and helium nuclei. Furthermore, the presence of neutrons was no longer observed,
as they were during the BBN. When the pressure inside a star was sufficiently large,
the interior temperature became hot enough for nuclear fusion to occur. Once the
density increased to ∼100 g/cm3 and the internal temperature of the star reached
at least 4 × 106 K, the gases were ignited and stellar nucleosynthesis was initiated. At
temperatures greater than 107 K, helium-4 can be produced by a process known
as hydrogen burning, given b