Preview Electrons, Atoms, and Molecules in Inorganic Chemistry A Worked Examples Approach by Joseph J. Stephanos Anthony W. Addison (2017)

Preview Electrons, Atoms, and Molecules in Inorganic Chemistry A Worked Examples Approach by Joseph J. Stephanos Anthony W. Addison (2017) Preview Electrons, Atoms, and Molecules in Inorganic Chemistry A Worked Examples Approach by Joseph J. Stephanos Anthony W. Addison (2017) Preview Electrons, Atoms, and Molecules in Inorganic Chemistry A Worked Examples Approach by Joseph J. Stephanos Anthony W. Addison (2017) Preview Electrons, Atoms, and Molecules in Inorganic Chemistry A Worked Examples Approach by Joseph J. Stephanos Anthony W. Addison (2017)

Electrons, Atoms, and Molecules in Inorganic Chemistry

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Contents

2.6 The <!>-Equation 48

2.8 The Radial Equation 57 1.1 Cathode and Anode Rays 2.9 The Final Solution for the Full Wave

1.2 Charge of the Electron 4 Function, 111 ntm(r, 0, (/J) 66

1.3 Mass of Electron and Proton 6 2.10 The Orthonormal Properties of the

1.4 Rutherford's Atomic Model 9 Real Wave Functions 71

1.5 Quantum of Energy 10 2.11 The Quantum Numbers: n I , and m1 76 1.6 Hydrogen Atom Line-Emission Spectra; The Principle Quantum Number, n 76

Electrons in Atoms Exist Only in Very The Qu ntum Numbers I and Angular

Specific Energy States 13 Momentum 76 1.7 Bohr's Quantum Theory of the The Ang lar Momentum Quantum

Hydrogen Atom 15 Numbers, I and m 78 1.8 The Bohr-Sommerfeld Model 18 Picture and Represent Precisely m1

1.9 The Corpuscular Nature of Electrons, Vectors of p- and d-Orbitals 78

Photons, and Particles of Very Small Mass 18 2.12 The Spin Quantum Number, s 84 1.10 Relativity Theory: Mass and Energy, 2.13 The Boundary Surface of s-Orbital 85

Momentum, and Wavelength 2.14 The Boundary Surface of p-Orbitals 87 Interdependence 21 2.15 The Boundary Surface of d-Orbitals 91 1.11 The Corpuscular Nature of 2.16 Calculating the Most Probable Radius 99

Electromagnetic Waves 21 2.17 Calculating the Mean Radius

The Photoele tric Effect 22 of an Orbital 1 0

The Compton Effect 25 2.18 The Structure of Many-Electron Atoms 108 1.12 de Broglie's Considerations 26 2.19 The Pauli Exclusion Principle 109

1.13 Werner Heisenberg's Uncertainty 2.20 Slater Determinant 111

Principle or the Principle of 2.21 Penetration and Shielding 113

Indeterminacy 27 2.22 The Building-Up Principle 11 7 1.14 The Probability of Finding an Electron 2.23 Term Structure for Polyelectron

1.15 Atomic and Subatomic Particles 29 2.24 Term Wave Functions and Single

Suggestions for Further Reading 32 2.25 Spin-Orbital Coupling 131

2.26 Spin-Orbital Coupling in External

Suggestions for Further Reading 144

2.1 The Wave Function (the Schri:idinger

3 Chemical Bonding

2.2 Properties of the Wave Function 39

2.3 Schri:idinger Equation of the 3.1 Ele tronegativity and Electropositivity 149

Hydrogen Atom 40 3.2 Electrone ativity and Electropositivity

Equation From Cartesians to Spherical 3.3 Molecular and Nonmolecular

vii

viii Contents

3.4 Types of Bonds 152 Macrocyclic Effect 193

3.5 Metallic Bonding and General Solvation Enthalpy Differences 196

Conduc vity a d Mobility of Electrons 153 Cavity Size 197

Luster and Free Electron Irradia on 154 Solvent Competition 197

of Valence Electrons 154 Stability and Metal Oxidation State 198 Theories of Bonding in Metals 155 Stability and Metal Ionization Potential 201

Free Electron Theory 155 3.9 Intermolecular Interactions 204

Crystal Structures of Metals (Metallic ion-Induced Dipole Forces,

Structures) 156 ion-Dipole Forces, and Hydrogen

3.6 Ionic Bonding 159 3.10 Covalent Networks and

of Atomic Lattice 159 Graphite, Fullere es, Graphe e, Carbon Born-Haber Cycle and Heat of Formation 163 Nanotubes, and Asbestos 214

Ionic Crystal Structures and the Radius Suggestions for Further Reading 225

Stoichiometric and No stoichiometric

4 Molecular Symmetry

Ionic Ch racter and Covale cy 4.1 Molecular Symmetry 228

Solubility of the Ionic Salts 1 1 Proper Rotation Axis, Cn 230

The Lewis Structures and Octet Rule 173 (Pntpr o Symmetry i 2.15 Exceptions to the Octet Rule 173 Sn: Improper Rotation Axis 236

Bonding and Polarity 174 4.3 The Symmetry and Point Group 238 3.8 Coordinate Covalent Bond 4.4 Some Immediate Applications 239 (Dative Bonding) 175 Dipole Moments and Polarity 239

"18-Eiectron Rule" 176 Equivalent Atoms: (Or Group of Atoms) 244

Nomenclature o Complexes 176 4.5 Group Theory Properties of the

Coordinative Comproportionation 4.6 Similarity Transforms, Conjugation,

Complexation Equilibrium 182 4.7 Matrix Representation 254

Multilgand Complexation 183 Matrices and Vectors 254

Stepwise Formation Con tants and the Matrix Representation of Symmetry

Complex Stability 186 Matrix Representa on of Point Group 258

Hard and Soft Interacti ns, HSAB 186 Irreducible Representa o s 260

Ch mical Features of Hard and Soft Io s, Irreducible and Degen rate

Rule of Interacti ns 190 4.8 Motion Representations of the

Hard-Hard and Soft-Soft Interactions 190 Groups 262

Hard-Soft Interaction and Anion Tra slation Motion 262 Polarizabi I ity 190 Rota onal Motion 264 Chelate Effect 191 4.9 Symmetry Properties of Atomic

Stability and the Geometry of the Mullikan Notation 266

Ch late Ring 193 Atomic Orbital Representation 267

Irreducible Representatio s 272 Wave Function 366

The Direc Product 274 6.9 Band Theory: Molecule Orbital Theory

4.12 Group Theory and Qua tum and Metallic Bonding Orbit 394

Mech nics: Irreducible Representations 6.10 Conductors, Insulators, and

and Wave Function 275 Semiconductors 397

Suggestio s for Further Reading 280 Suggestio s for Furth r Reading 4 1

5.1 Vale ce Bond Theory 282 Vale ce Bond Theory 405

5.2 V EPR Theory and Molecular 7.2 Bases of Crystal Field Theory 405

Geometry 283 d-Orbitals in Cubic Crystal Field 405

5.3 lsoelectronic Species 284 f-Orbitals in Cubic Crystal Field 406

5.4 Procedures to Diagram Molecular 7.3 The Crystal Field Pote tial 407

Structure 284 Octahe ra Crystal field Potential, Voct 407

5.5 Vale ce Bond Theory and Metallic Square Planar Crysta Field Potential, Vsq.PI 4 2

Bonds 288 Tetragon l Crysta Field Potential, VTetrag 4 5

5.6 Orbital Hybridization 290 Teta edral Cry tal Field Potential, V rd 4 7 5.7 Rehybridiza on and Complex Formation 291 7.4 Zero-Order Perturbation Theory 4 9

5.8 Hybridization and o /rr-Bonding 2 4 The Linear Combin ti n of Atomic

5.9 Orbital Hybridization and Molecular Orbitals, LCAO-MO, and Energy

Trigon l Planar Hybri ization 296 The Perturbation Theory for Degenerate

The Extend of d-Orbita Participation in Systems 421

Molecular Bonding 3 1 The Splitting of d-Orbitals in Octahedral

Trigon l Bipyramidal Hybridization 301 Crystal Field Voct 423

Tetragonal Pyramidal Hybridization 303 The Splitting of d-Orbitals in Teta e ral Square Planar Hybridization 304 Crystal Field, Vr d 430

Tetra edra Hybridization 306 The Splitting of d-Orbitals in Tetragonal Octahedral Hybridization 308 Crystal Field, Vo.h 435

5.10 Hybrid Orbitals as Symmetry 7.5 Ty es of Interactions That Affect the

Adapted Lin ar Combination of Atomic Crystal Field Treatment 442

Orbitals (SALC) 311 7.6 Free Jn in Weak Crysta Fields 442

5.11 Molecular Wave Function as Symmetry Problems and the Required

Adapted Linear Combination o Atomic Approximatio s 442

Orbitals (SALC) 322 The Effect of the Crystal Field on S Term 442

Suggestio s for Further Reading 330 The Effect of the Cubic Crystal Field on

6.3 The Linear Combination o Atomic 7.7 Stro g Field Approach 457

Orbitals-Molecular Orbita (LCAO-MO) Determinanta Wave Functions 457

and Hi.ic el Approximatio s 333 The Determinantal Wave Functio s of d2

6.4 Atomic Orbitals Combinations for the in Strong Field of Tetragonal Structure, Second Row Diatomic Molecules 338 Tra s-MLZ 457

X Contents

The Symmetry and the Energy of 9.8 Vibrations of Polyatomic Molecules 532

Determinant Wave Functions of D2 in 9.9 Polyatomic Molecular Motions and

a Strong Field of Trans-ML4Z2 458 Degrees of Freedom 532 The Appropriate Hamiltonian in Strong 9.10 Normal Modes of Vibration, Normal

Field 459 Coordinates, and Polyatomic

The Diagon l Interelectronic Repulsion 460 Molecules 533

The Nondiagonal Interelectronic Repulsion 9.11 Vibrational Energy of Polyatomic

and the Energy of Each Level of the d2 Molecules 536

Configuration in Strong Field of Tra s- 9.12 Vibrational Displacements 536

ML4Z2 463 9.13 Vibrational Energy and Normal

Suggestions for Further Reading 470 Coordinates 537

9.14 Stretching Vibrations of Linear

8.1 The Advantages and Disadvantages of 9.15 Symmetry and Normal Modes of

8.2 Symmetry and Orbital Splitting by 9.16 Assigning the Normal Modes of

8.3 Correlation Table 478 Normal Modes of Vibration for

Orbit l Correlation Table 478 Linear Triatomic Molecule 552 Term Correlation Tables 480 9.17 Force Constants and the GF-Matrix

8.4 Correlation Diagrams of Strong and Method 555 Weak Fields 481 Lagra ge's Equation in Terms of

Correlation Diagram of Strong and Symmetry Coordinates 565

Weak Field States of Oh 481 9.18 Selection Rules 568

Method of Descending Symmetry IR-Selection Rules 568

Raman Selecti n Rules 571

(Descending Multiplicities of the

9.19 Center of Symmetry and the Mutual

Correlation Diagram of Weak and Strong Exclusion Rule 575

Field States of T d 486 9.20 Isolation of a Particular Type of

Orgel Diagram of D Term Configuration 488 9.21 Detecting the Changes of Symmetry

Configuration and Term Interactions 493 Suggestions for Further Reading 582

8.6 Tanabe-Sugano Diagrams 498

dn and d1 - n Diagrams 499 10 Electronic Spectroscopy

Suggestions for Further Reading 503 10.1 Beer-Lambert Law 587

Molar Extinction Coefficient,

9.1 Infrared and Raman Spectroscopy 507 10.2 Allowed Electronic Transition 589 9.2 Permanent Dipole and Polarizability 509 The Transition Moment a d Electronic 9.3 The Classical Explanation of Infrared Transitions 589

and RAMAN Spectroscopy 509 The Born-Oppenheimer Approximation 590 9.4 Rotation of Diatomic Molecules 511 Even and Odd Functions and the

Rigid and Nonrigid Models 514 Symmetry Considerations 590 9.5 Vibration of Diatomic Molecules 515 Symmetry Representations a d the

Vibrational Energy Levels 5 6 Allowed Tra sitions 5 1

Anharmonic Oscillation 521 10.3 Basis of the Selection Rules 592 9.6 The Quantum Mechanics of the 10.4 Selection Rules 593

Translation, Vibration, and Spin, Orbital, and Vibratio a

9.7 Vibration-Rotation Energies of Diatomic 10.5 Unexpected Weak Absorbance 595 Molecules (Vibrational-Rotational State) 529 10.6 Spectroscopy of Electronic Excitations 597

Contents xi

10.7 Electronic Spectra of Selected Examples 601 11 Magnetism

Jahn-Te er Theorem and Vibronic

Cr(H20)63

Simulta eous Pair Exc a o s Bridged 11.8 Spin-Orbital Coupling: In A and E

Dinuclear Metal Centers 612 Ground Terms 6 1

The Low-Spin Vers s High-Spin: d6 Terms 672

Configuration 621 Compounds Contain a Unique

10.8 Spectroscopy of Porphyrins 632 Heterobinuclear Interacti n 696

10.9 The Magnetic Dipole Moment and Gouy's Method 702

the Absorbance Intensity 635 Faraday's Method 705

Kuhn anisotropy Factor and Deducing

This book presents the chemical concepts that govern the chemistry of molecular construction The emphasis is on thebuilding up of an understanding of essential principles and on familiarization with basic inorganic concepts Necessarybackground information is introduced to comprehend the field from both chemical and practical areas The book explainsand details the fundamentals that serve as a source of numerous basic concepts of methods and applications The combi-nation of the basic concepts, methods and applications with example exercises yields a more positive outcome for studentsand teachers

Many inorganic textbooks that are available cover too much material and do not go into the depth needed for mental principles Most of these are seem to be either fairly elementary or very advanced Students might be displeased bythe current selection available, and there is a great need for an inorganic principles of chemical bonding text In our book,

funda-we bridge and integrate both elementary and more advanced principles A student should be able to become familiar withthe topics presented with this one book, rather than learn the basics in one and use another for the more advanced aspects ofthe material

Given the complex and abstract nature of the subject, the book is easy to follow The text is carefully thought throughand laid out The approach of developing the material by answering questions and problems relevant to inorganic chemistry

in extensive mathematical detail is unique and makes the book attractive especially for university students, as a studymaterial source for examinations Every mathematical step in the book is elaborated with close attention to every detail.That necessary mathematical foundation is found in a separate supplement, as an entirely non-mathematical approach will

be of little value for the purpose The bullet point approach of answering common questions rather writing a narrative andstarting each chapter with the circle scheme describing the sections to be discussed are intended to be attention-catching andcould attract certain students and aid them as they study

The content and the style of the book should capture readers from both traditional and modern schools It is useful as areference and text for specialized and graduate courses in physical inorganic or advanced organic chemistry The book isintended for advanced undergraduates and for postgraduates taking courses in chemistry, students studying atomic structureand molecule formation in chemical engineering and material science It should also be of value to research workers in otherfields, who might need an introduction to essential inorganic principles This book is very suitable for self-study; the rangecovered is so extensive that this book can be student’s companion throughout his or her university career At the same time,teachers can turn to it for ideas and inspiration

This book is divided into 11 chapters, and covers a full range of topics in inorganic chemistry: wave-particle duality,electrons in atoms, chemical bonding, molecular symmetry, theories of bonding, valence bond theory, VSEPR theory,orbital hybridization, molecular orbital theory, crystal field theory, ligand field theory, vibrational, rotational, and elec-tronic spectroscopy, magnetism and finally, a mathematics supplement outlining the necessary methods

In the beginning of this book we develop and provide an understanding of the dual wave-particle nature of electrons,photons, and other particles of small mass

Schr€odinger’s method is linked for exploring the modern theory of atomic structure, and to establish the formal ematical framework This framework is employed to set up the final real solution for the full orbital wave function andidentify the four quantum numbersn, l, m, and s, also to compute the most probable radius, mean radius of an orbital,and the boundary surfaces ofs, p and d orbitals The orbital wave equations are used as the key feature in order to explainthe orbital and spin angular momenta, as well the electronic configurations of many-electron atoms, and spin-orbital cou-pling We examine how to identify the term symbols of the ground state and the different terms of the excited microstates ofpolyelectronic atoms, and how many subterms arise when spin-orbit coupling is taken into consideration The splitting ofRussell–Saunders terms into microstates in an external magnetic field and their energies are identified The term wavefunctions and the corresponding single electron wave functions are described in order to understand the effect of theligand field

math-xiii

Then we begin with a review of basic electron accounting procedures for different types of bond formation and proceed

to a model for predicting three-dimensional molecular structure The basic concepts of metallic structure that describe thebonding, define the role of the free valence electrons, and relate the physical properties and theories of metallic bonding areexplored Emphasis is also placed on ionic bonding and the relationships among the lattice energy, thermodynamic param-eters and covalency We review the grounds of the ionic crystal structure, in which radius ratios govern the geometricalarrangements, and review the factors that influence the solubilities The foundation and basics of coordination chemistry arelaid out: firstly, characterization, formulation, formation and stability including hard and soft acid/base interactions, thechelate and macrocycle effects, cavity size, solvation enthalpy, donor atom basicity, solvent competition, steric effects,metal oxidation state, and metal ionization potential An extensive discussion is given of intermolecular forces, exploringtheir r^oles, consequences and significances Considerations are given to the structural and chemical nature of the covalentnetworks in giant molecules such diamond, graphite, fullerenes, graphene, nanotubes and asbestos

In order to deal with molecular structures, where many energy levels of atoms are involved, symmetry concepts areextensively invoked Thus, it is appropriate to explain how to establish a proper system for sorting molecules according

to their structures One purpose of this sorting is to introduce some ideas and mathematical techniques that are essential forunderstanding the structure and properties of molecules and crystals Matrix representations of symmetry operations, pointgroup, translation, rotation motions, and atomic orbital are thoroughly examined This leads to presentation of the charactertables, and finally shows how that the symmetry representations for the atomic orbitals form bases for the molecular wavefunctions

The valence bond theory concept is explained, then we investigate how to predict the shapes and geometries of simplemolecules using the valence shell electron-pair repulsion method The process of predicting the molecule’s structure isreviewed As well, the relationships between the chemical bonds in molecules and its geometry using orbital hybridizationtheory are addressed Special attention is devoted to the angles between the bonds formed by a given atom, also to multiplebonding and σ/π hybridization of atomic orbitals Symmetry-adapted linear combination of atomic wave functions(SALC’s) are composed and detailed, then used to compute the contribution of each atomic orbital to the hybrid orbitals

A brief representation of molecular orbital theory is elaborated Understanding the electronic distribution of someelected small molecules, and approaches to the relative energies of the molecular orbitals are reviewed We then explainhow the electron distribution changes upon going to some low-lying excited electronic states The theory is employed toestimate energy changes in chemical reactions, to study stability & reactivity, to find the delocalization energy, electrondensity, formal charge, bond order, ionization energy, equilibrium constant, and configuration interaction The orbitalcombination introduces the band theory concept that makes it possible to rationalize conductivity, insulation andsemiconductivity

Having now available the valence orbital’s wave functions of the central ion in their real forms, it is possible to explorethe impact of various distributions of ligand atoms around the central ion upon its valence orbitals A quantitative basis ofthis effect in the case of a purely ionic model of coordination, and other degrees of mixing are addressed In this part, weindicate why there is a need for both crystal field theory and ligand field theory The effect of a cubic crystal field ond- andf-electrons is introduced, then the expressions of the Hamiltonian to find the crystal field potential experienced by electrons

in octahedral, square planar, tetragonally distorted octahedral, and tetrahedral ligand arrangements are computed The turbation theory for degenerate systems is used to explain how the crystal field potential of the surrounding ligands perturbsthe degeneracy ofd orbitals of the central ion The energies of the perturbed d-orbitals are calculated by solving the seculardeterminant The obtained energies are fed back into secular equations that are derived from the secular determinant to yieldwave functions appropriate for the presence of the potential The variation in the potential energy of eachd electron due tothe crystal field is determined and the splitting ofd-orbitals so deduced in octahedral, tetrahedral, tetragonally distorted

per-D4hgeometries in terms ofDq,Dt, andDs Problems and the required approximations are discussed for the free ion in weakcrystal field Then, we study the influence of weak field on polyelectronic configuration of free ion terms, and find thesplitting in each term and the wave function for each state In the strong field situation, the first concern was how thestrong field differs from the weak field approach; define the determinant, symmetry and the energy of each state Wecompute the appropriate Hamiltonian and the diagonal and off-diagonal interelectronic repulsion in terms of the Racahparameters A, B, and C

We firstly examine how it is possible to use symmetry and group theory to find what states will be obtained when an ion

is placed into a crystalline environment of definite symmetry Secondly, the relative energies of these states will be tigated Thirdly, we show how the energies of the various states into which the free ion term are split depend on the strength

inves-of the interaction inves-of the ion with its environment The relationship between the energy inves-of the excited states andDqarediscussed using correlation, Orgel, and Tanabe-Sugano diagrams

The vibrational spectra of diatomic molecules establish most of the essential principles that are used for complicatedpolyatomic molecules Since infrared radiation will excite not only molecular vibration but also rotation, there is a need tocomprehend both rotation and vibration of diatomic molecules in order to analyze their spectra Molecular vibrations areexplained by classical mechanics using a simple ball and spring model, whereas vibrational energy levels and transitionsbetween them are concepts taken from quantum mechanics The quantum mechanics of the translation, vibration androtation motions are explored in detail As well, the expression for the vibration-rotation energies of diatomic moleculefor the harmonic and anharmonic oscillator models is introduced Then, we identify the Schr€odinger equation for thevibration system ofn-atom molecules We elucidate how to obtain, monitor, and explain the vibrational–rotational exci-tations, find the quantum mechanical expression for the vibrational and rotational energy levels, predict the frequency of thebands, and compare harmonicvs anharmonic oscillators and between rigid and nonrigid rotor models for the possible exci-tations Lagrange’s equation is used to show the change in the amplitude of displacement with time We examine also how

to calculate the relative amplitudes of motion and the kinetic and potential energies for the vibrational motions of ann-atommolecule Calculation of the force constants using the GF-matrix method are discussed The general steps to determine thenormal modes of vibration, and the symmetry representation of these modes are outlined We examine the relationship andthe differences among the cartesian, internal, and normal coordinates used to characterize the stretching vibrations Then

we show why the normal coordinates are used to calculate the vibrational energy of polyatomic molecules In this part wealso focus on how the molecules interact with the radiation and the chemical information obtainable by measurement of theinfrared and Raman spectra Only the radiation electric field interacts significantly with molecules and is important inexplaining infrared absorption and Raman scattering The requirements and the selection rules for the allowed vibrationaland rotational excitations are explored We investigate the relationship between the center of symmetry and the mutualexclusion rule, how to distinguish among isomers and ligand binding modes, and define the forms of the normal modes

of vibration and which of these modes are infrared and/or Raman active

We then focus on the chemical information obtained from electronic spectra in the visible and ultraviolet regions Weexamine the relationship between the solute concentration and light absorbance, as well the correlation among the molarextinction coefficient, integrated intensity and dipole strength The significance of the Born-Oppenheimer approximation isconsidered, and symmetry considerations are elaborated with respect to allowedness of electronic transitions The conse-quences of spin, orbital and vibrational constraints are investigated to explore the basis of the electronic absorptionselection rules The electronic excitation of functional groups, donor–acceptor complexes, and porphyrins are spectroscop-ically characterized The study outlines the roles of vibronic coupling, configuration interaction, andπ-bonding Factorsthat affect the bandwidth, band intensity, and intense colors of certain metal complexes are also identified The textaddresses the effects of Jahn–Teller distortion, temperature, and reduced symmetry, and elaborates the spectrochemicalseries Comparative studies of octahedral versus tetrahedral, low-spin versus high-spin, anddnversusd10-nconfigurationare conducted We illustrate how to evaluateDqandβ from the positions of the absorption peaks, and discuss the unex-pectedly weak absorbances, simultaneous pair excitations, and the absorption of unpolarized light in general The role of themagnetic dipole moment on the absorbance intensity is investigated, using circular dichroism spectroscopy and the Kuhnanisotropy factor to examine the effects of lower symmetry, absolute configuration, and the energy levels within themolecule

Finally we investigate the types of magnetic behaviors, giving key definitions and concepts leading to the relationshipsrelevant to magnetochemistry These provide a bridge to understand spin and orbital contributions to magnetic moments.Then, thermal spreading using the Boltzmann distribution is employed to investigate and estimate the magnetic momentsand susceptibilities The subsequent section deals with the van Vleck treatment and the second order Zeeman Effect to linkthe spin and orbital contributions to the magnetic susceptibility Requirements and conditions for nonzero orbital contri-bution are discussed The following section is devoted to the effect of imposition of a ligand field on spin-orbital coupling in

A, E, and T ground terms The Curie law, deviations, and data presentation modes are shown Spin-crossover and the effects

of thermal distortion are discussed, followed by the behavior of dinuclear systems with exchange coupling Finally, Gouy’s,Faraday’s, Quincke’s, and NMR methods for susceptibility are described

Joseph J StephanosAnthony W Addison

Chapter 1

Particle Wave Duality

In this chapter we shall develop an understanding of the dual wave-particle nature of electrons, photons, and other particles

of small mass (Scheme 1.1)

The particle nature of the electrons had been confirmed by cathode rays, Millikan’s capacitor, and Thomson’s imentation Models for atomic structures were proposed by Thomson and Rutherford

exper-Studies of black-body radiation shows that energy emits in a small, specific quantity called quanta Also, hydrogenemission spectrum indicates that electrons in atoms exist only in very specific energy states Quantum has been concluded

as the smallest amount of energy that can be lost or gained by an atom Bohr’s and Bohr-Sommerfeld’s atomic models arepresented and discussed Bohr in his atomic models used quantum theory not quantum mechanics, he did not recognize thewave nature of electrons

Wave interference and diffraction are used as evidences to confirm the wave properties of the electrons Einstein’s tionships are explored to describe and clarify the interdependence of mass and energy

rela-Furthermore, the corpuscular nature of light is revealed in the photoelectric effect and the Compton effect de Broglieshows that the dual wave-particle nature is true not only for photon, but for any other material particle as well

Heisenberg, in his uncertainty principle, points out that only the probability of finding an electron in a particular volume

of space can be determined This probability of finding an electron is proportional to the square of the absolute value of thewave function

Subatomic particles are examined and classified as fundamental particles (fermions) and force particle (bosons) thatmediate interactions among fermions

In the following outline, we shall try to understand:

l 1.1: Cathode and anode rays

l 1.2: Charge of the electron

l 1.3: Mass of the electron and proton

l 1.4: Rutherford’s atomic model

l 1.5: Quantum of energy

l 1.6: The hydrogen-atom line-emission spectra

l 1.7: Bohr’s quantum theory of the hydrogen atom

l 1.8: The Bohr-Sommerfeld model

l 1.9: The corpuscular nature of electrons, photons, and particles of very small mass

l 1.10: Relativity theory: mass and energy

l 1.11: The corpuscular nature of electromagnetic waves: the photoelectric effect and the Compton effect

l 1.12: de Broglie’s considerations

l 1.13: Werner Heisenberg’s uncertainty principle, or the principle of indeterminacy

l 1.14: The probability of finding an electron and the wave function

l 1.15: Atomic and subatomic particles

l Suggestions for further reading

1.1 CATHODE AND ANODE RAYS

How do the cathode and anode rays expose and characterize the subatomic particles?

l The discovery of subatomic particles resulted from investigations into the relationship between electricity and matter

l When electric current was passed through various gases at low pressures (cathode-tube), the surface of the tube directlyopposite the cathode glows (Fig 1.1)

Electrons, Atoms, and Molecules in Inorganic Chemistry http://dx.doi.org/10.1016/B978-0-12-811048-5.00001-8

15 22Elementary

particles

Fermions:

matter particles

14 Probability and wave function

2

''Milikan''

Electron's charge

e = 4.77 × 10–10 C 1

Cathode rays and electrons

4 Ratherford's atomic model

3

''Thomson''

Electron's mass

m = 9.109 × 10–31 kg

Energy

Energy particle

10 Relativity theory ''Einstein''

5 Black-Body radiation ''Blank''

Quantum radiation

11, 12 Photoelectic Compton effect

de Broglie's equation

9 Interference and diffraction

21

Bosons:

force particles

13 Heisenberg uncertainty principle

l It has been hypothesized that the glow was caused by a stream of particles, called a cathode ray.

l The ray travels from the cathode to the anode when current is applied

l The following observations are revealed:

If an object placed between the cathode and the anode, it will cause a shadow on the glass This supports the existence

of a cathode ray

If a paddle wheel placed on rails between the electrodes, it will roll along the rails from the cathode toward the anode(Fig 1.2A) This shows that a cathode rayhad sufficient mass to set the wheel in motion

The rays were deflected away from a negatively charged object

Cathode rays were deflected by a magnetic field in the same manner as a wire carrying electric current, which wasknown to have negative charge

l Because cathode rays have the same properties regardless of the element used to produce them, it was concluded that:

The cathode rays are composed of previously unknown negatively charged particles, which were later calledelectrons

Electrons are present in atoms of all elements

l Because atoms are electrically neutral, they must have a positive charge to balance each negative electron

l If one electron is removed from a neutral atom or molecule, the resulting residue has a positive charge equal to the sum

of the negative charges of the electron removed

l Positive ions are formed in the gas discharged tube when electrons from the cathode collide with gaseous atoms(Fig 1.2B)

l The positive ions move toward the cathode, while the negatively charged electrons of the cathode rays move in theopposite direction

Anode Cathode

Cathode

Vacuum pump

Positive ray (Anode ray)

Fluorescence –

Anode

FIG 1.2 (A) Cathode ray had sufficient mass to set the wheel in motion (B) Anode ray produced in gas discharged.

Particle Wave Duality Chapter 1 3

l If canals have been bored in this electrode, the positive ions pass through them and cause fluorescence when they strikethe end of the tube.

l When different gases are used in the discharged tube, different types of positive ions are produced

l The deflections of positive rays in the electrical and magnetic fields were studied

1.2 CHARGE OF THE ELECTRON

Describe Millikan’s oil drop apparatus How could he calculate the charge of the electron?

l The precise determination of the charge of the electron was first computed by Millikan

l The most significant part of Millikan’s apparatus was an electric capacitor inside a thermostated metal chamber(Fig 1.3)

l A fog of small oil droplets was formed in the chamber by an atomizer

l The droplets flow through an aperture in the upper plate of the capacitor

l The movement of the droplets between the plates of the capacitor could be scanned with an eyepiece

l The droplets were ionized by exposure to X-rays emitted from a radiation tube

l By altering the voltage,V, across the plates of the capacitor, it was possible to reach a specific voltage at which theelectric field strength was balanced by the force of the gravity of the charge droplet,ed

As a result:

where

m is the mass of the droplet

g is the acceleration due gravity

E is the strength of the electrical field is the electric force per unit charge at a particular location

However, the electric force between the plates of the capacitor:

Oil atomizer

X-ray tube

FIG 1.3 Millikan apparatus that used to measure the charge on the electron.

4 Electrons, Atoms, and Molecules in Inorganic Chemistry

V is the voltage applied to the plates

d is the distance between plates

The value ofedcan be found if the mass of the droplet,m, is known

l The charged droplets could be forced to go up or down by varying the voltage across the plates

l The mass of the droplet,m, can be evaluated from its falling velocity in absence of the electrical field

l Initially the oil drops are allowed to fall between the plates while the electric field is turned off The droplets reach aterminal velocity due to the air friction

l The field is then turned on; if the voltage is large enough, some of the charged drops will start to go up (This is becausethe upwards electric forceFelis greater than the downwards gravitation force,W¼mg.)

l A perfectly spherical droplet is selected and held in the middle of the field of view by varying the voltage while all theother drops have fallen The experiment is then continued with this one drop

l The drop is allowed to fall in the absence of an electric field and its terminal velocityu1is calculated The force of airfriction for falling drop can then be estimated using Stokes’ Law:

ρairis the density of the air

l For a perfectly spherical droplet, the apparent weight can be written as:

l Oncer is calculated, W, m, and edcan easily be computed (Eqs.1.2.3, 1.2.6)

l In practice this is very hard to accomplish precisely EstimatingFdis complicated because the mass of the oil drop isdifficult to determine without the use of Stokes’ Law

l A more practical approach is to turnV up slightly, so that the oil drop goes up with a new terminal velocity u2Then

if: 6πrη ¼W

u1, then:

edE W ¼ 6πrηu2¼Wu2

u1

l If the mass of the droplet is known,e can be found

Particle Wave Duality Chapter 1 5

l Millikan noticed that the charge on the droplets were always multiples of a certain value of e, the smallest chargeexperimentally found This could be explained by the fact that a droplet can possess whole numbers of electrons,but never a fraction, because an electron is undividable.

l Millikan obtained the following value:e¼4.8031010 esu¼1.602061019coulomb.

1.3 MASS OF ELECTRON AND PROTON

How could Thomson determine the mass of the electron and proton, and what is his atomic model?

l Thomson concluded that all cathode rays are composed of indistinguishable negatively charged particles, which werenamed electrons

l The ratio between the charge of the electron to its mass,e/me, was first established by Thomson

l This ratio is based on the deflection of a beam of electrons in electric and magnetic fields (Fig 1.4)

l Consider a beam of electrons passing between the plates of capacitor (Fig 1.5); the force,fel, that acts on the electron inthe electric field is equal to

Magnetic field perpendicular to the electric field

Electrodes for creating

an electric field

S

+

+ –

N

FIG 1.4 Cathode ray tube for determining the

value of e/m for electron.

Electron beam

Magnetic field

A

B Capacitor plate

d

y

+

– l

FIG 1.5 Lay out for determining e/m

6 Electrons, Atoms, and Molecules in Inorganic Chemistry

fel¼ eE ¼ eV

where

e is the charge of electron

E is the field strength in the capacitor

V is the voltage across the plates

d is the distance between the plates

l This force accelerates the electrons in the direction perpendicular to the original direction of the electron beam

where

a is the acceleration of electron

meis the mass of electron

Substituting fora, using Eq.(1.3.4), therefore,

y¼ 12

The distancey can be found from the distance AB on the screen (Fig 1.5)

The velocity of the electronu can be evaluated from the deviation of the electron in the magnetic field

l When the magnetic field compensates for the deviation of the electron in the electric field, the direction of the electronbeam remains unchanged, and the magnetic and electric fields are equal:

wherefmag.is the force of the magnetic field acting on the electron current andi¼ eu

l According to electrodynamics, the magnetic field will act on the electron moving perpendicular to the field with aforce:

whereH is the intensity of the magnetic field

Therefore, from Eq.(1.3.6),

The only unknown is the value of e

me, which now can be determined:

l The mass of the electron can be calculated ife/meande are known (e¼4.8031010esu).

l This shows that the mass of the electron is 9.1091031kg (about 1/1837 the mass of the hydrogen atom).

l Thomson showed that electron is a particle with

mass

energy

momentum

l Because electrons have so much less mass than atoms, atoms must contain other particles that account for most of their mass

l The values ofe/m were estimated for the positive ions by using basically the same technique employed in the study

are known as protons

have equal charge to that of the electron, but are opposite in sign:

e¼ + 1:6022  1019C

the mass of the proton:

m¼ ee=m ¼

+1:6022  1019C+9:5791  104C=g¼ 1:6726  1024gwhich is about 1837 times heavier than the mass of electron

It is assumed to be a component of all atoms

+

- -

-+

+ +

Thomson atom

8 Electrons, Atoms, and Molecules in Inorganic Chemistry