Electrons, Atoms, and Molecules in Inorganic Chemistry: A Worked Examples Approach

By Joseph J. Stephanos and Anthony W. Addison

Electrons, Atoms, and Molecules in Inorganic Chemistry: A Worked Examples Approach builds from fundamental units into molecules, to provide the reader with a full understanding of inorganic chemistry concepts through worked examples and full color illustrations. The book uniquely discusses failures as well as research success stories. Worked problems include a variety of types of chemical and physical data, illustrating the interdependence of issues.

This text contains a bibliography providing access to important review articles and papers of relevance, as well as summaries of leading articles and reviews at the end of each chapter so interested readers can readily consult the original literature. Suitable as a professional reference for researchers in a variety of fields, as well as course use and self-study. The book offers valuable information to fill an important gap in the field.

• Incorporates questions and answers to assist readers in understanding a variety of problem types
• Includes detailed explanations and developed practical approaches for solving real chemical problems
• Includes a range of example levels, from classic and simple for basic concepts to complex questions for more sophisticated topics
• Covers the full range of topics in inorganic chemistry: electrons and wave-particle duality, electrons in atoms, chemical binding, molecular symmetry, theories of bonding, valence bond theory, VSEPR theory, orbital hybridization, molecular orbital theory, crystal field theory, ligand field theory, electronic spectroscopy, vibrational and rotational spectroscopy

• Language: English
• Publisher: Academic Press
• Release date: Jun 1, 2017
• ISBN: 9780128110492

Joseph J. Stephanos

Joseph J. Stephanos is associate professor of inorganic, bioinorganic, biophysics chemistry at Menoufia University. He has been postdoctoral research associate at Pennsylvania University, postdoctoral instructor and adjunct associate professor at Drexel University and has held various teaching positions leading to the current one. Dr. Stephanos’ research interests concern inorganic studies of biologically active molecules, studying model compounds, their structure and bonding, and the chemistry of metalloproteins and ligands binding, with special reference to mechanistic aspects and structure/function relation. He is the author and coauthor of several articles and one previous book, Chemistry of Metalloproteins: Problems and Solutions in Bioinorganic Chemistry.

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9780128110492_FC

Electrons, Atoms, and Molecules in Inorganic Chemistry

A Worked Examples Approach

First Edition

Joseph J. Stephanos

Anthony W. Addison

Table of Contents

Cover image

Title page

Copyright

Dedication

Preface

Chapter 1: Particle Wave Duality

Abstract

1.1 Cathode and Anode Rays

1.2 Charge of the Electron

1.3 Mass of Electron and Proton

1.4 Rutherford's Atomic Model

1.5 Quantum of Energy

1.6 Hydrogen Atom Line-Emission Spectra; Electrons in Atoms Exist Only in Very Specific Energy States

1.7 Bohr's Quantum Theory of the Hydrogen Atom

1.8 The Bohr-Sommerfeld Model

1.9 The Corpuscular Nature of Electrons, Photons, and Particles of Very Small Mass

1.10 Relativity Theory: Mass and Energy, Momentum, and Wavelength Interdependence

1.11 The Corpuscular Nature of Electromagnetic Waves

1.12 de Broglie's Considerations

1.13 Werner Heisenberg's Uncertainty Principle, or the Principle of Indeterminacy

1.14 The Probability of Finding an Electron and the Wave Function

1.15 Atomic and Subatomic Particles

Chapter 2: Electrons in Atoms

Abstract

2.1 The Wave Function (the Schrödinger Equation)

2.2 Properties of the wave function

2.3 Schrödinger Equation of the Hydrogen Atom

2.4 Transformation of the Schrödinger Equation From Cartesians to Spherical Polar Coordinates

2.5 The Angular Equation

2.6 The Φ-Equation

2.7 The Θ-Equation

2.8 The Radial Equation

2.9 The Final Solution for the Full Wave Function, ψnlm(r,θ,ϕ)

2.10 The Orthonormal Properties of the Real Wave Functions

2.11 The Quantum Numbers: n, l, and ml

2.12 The Spin Quantum Number, s

2.13 The Boundary Surface of s-Orbital

2.14 The Boundary Surface of p-Orbitals

2.15 The Boundary Surface of d-Orbitals

2.16 Calculating the Most Probable Radius

2.17 Calculating the Mean Radius of an Orbital

2.18 The Structure of Many-Electron Atoms

2.19 The Pauli Exclusion Principle

2.20 Slater Determinant

2.21 Penetration and Shielding

2.22 The Building-Up Principle

2.23 Term Structure for Polyelectron Atoms

2.24 Term Wave Functions and Single Electron Wave Functions

2.25 Spin-Orbital Coupling

2.26 Spin-Orbital Coupling in External Magnetic Field

Chapter 3: Chemical Bonding

Abstract

3.1 Electronegativity and Electropositivity

3.2 Electronegativity and Electropositivity Trends

3.3 Molecular and Nonmolecular Compounds

3.4 Types of Bonds

3.5 Metallic Bonding and General Properties of Metals

3.6 Ionic Bonding

3.7 Covalent Bonding

3.8 Coordinate Covalent Bond (Dative Bonding)

3.9 Intermolecular Interactions

3.10 Covalent Networks and Giant Molecules

Chapter 4: Molecular Symmetry

Abstract

4.1 Molecular Symmetry

4.2 The Symmetry Elements

4.3 The Symmetry and Point Group

4.4 Some Immediate Applications

4.5 Group Theory: Properties of the Groups and Their Elements

4.6 Similarity Transforms, Conjugation, and Classes

4.7 Matrix Representation

4.8 Motion Representations of the Groups

4.9 Symmetry Properties of Atomic Orbitals

4.10 Character Tables

4.11 Relation Between any Reducible and Irreducible Representations

4.12 Group Theory and Quantum Mechanics: Irreducible Representations and Wave Function

Chapter 5: Valence Bond Theory and Orbital Hybridization

Abstract

5.1 Valence Bond Theory

5.2 VSEPR Theory and Molecular Geometry

5.3 Isoelectronic Species

5.4 Procedures to Diagram Molecular Structure

5.5 Valence Bond Theory and Metallic Bonds

5.6 Orbital Hybridization

5.7 Rehybridization and Complex Formation

5.8 Hybridization and σ-/π-Bonding

5.9 Orbital Hybridization and Molecular Symmetry

5.10 Hybrid Orbitals as Symmetry Adapted Linear Combination of Atomic Orbitals (SALC)

5.11 Molecular Wave Function as Symmetry Adapted Linear Combination of Atomic Orbitals (SALC)

Chapter 6: Molecular Orbital Theory

Abstract

6.1 Molecular Orbital Theory Versus Valence Bond Theory

6.2 Molecular Orbital Wave Function and Symmetry

6.3 The Linear Combination of Atomic Orbitals-Molecular Orbital (LCAO-MO) and Hückel Approximations

6.4 Atomic Orbitals Combinations for the Second Row Diatomic Molecules

6.5 Heterodiatomic Molecules

6.6 Polyatomic Molecules

6.7 Molecular Orbitals for a Centric Molecule

6.8 Properties Derived From Molecular Wave Function

6.9 Band Theory: Molecule Orbital Theory and Metallic Bonding Orbit

6.10 Conductors, Insulators, and Semiconductors

Chapter 7: Crystal Field Theory

Abstract

7.1 The Advantages and Disadvantages of Valence Bond Theory

7.2 Bases of Crystal Field Theory

7.3 The Crystal Field Potential

7.4 Zero-Order Perturbation Theory (the Effect of Crystal Field on the Orbital Wave Functions of Degenerate Orbitals)

7.5 Types of Interactions That Affect the Crystal Field Treatment

7.6 Free Ion in Weak Crystal Fields

7.7 Strong Field Approach

Chapter 8: Ligand Field Theory

Abstract

8.1 The Advantages and Disadvantages of Crystal Field Theory

8.2 Symmetry and Orbital Splitting by Ligand Field

8.3 Correlation Table

8.4 Correlation Diagrams of Strong and Weak Fields

8.5 Orgel Diagram

8.6 Tanabe-Sugano Diagrams

Chapter 9: Vibrational Rotational Spectroscopy

Abstract

9.1 Infrared and Raman Spectroscopy

9.2 Permanent Dipole and Polarizability

9.3 The Classical Explanation of Infrared and Raman Spectroscopy

9.4 Rotation of Diatomic Molecules

9.5 Vibration of Diatomic Molecules

9.6 The Quantum Mechanics of the Translation, Vibration, and Rotation Motions

9.7 Vibration-Rotation Energies of Diatomic Molecules (Vibrational-Rotational State)

9.8 Vibrations of Polyatomic Molecules

9.9 Polyatomic Molecular Motions and Degrees of Freedom

9.10 Normal Modes of Vibration, Normal Coordinates, and Polyatomic Molecules

9.11 Vibrational Energy of Polyatomic Molecules

9.12 Vibrational Displacements

9.13 Vibrational Energy and Normal Coordinates

9.14 Stretching Vibrations of Linear Molecules

9.15 Symmetry and Normal Modes of Vibration

9.16 Assigning the Normal Modes of Vibration

9.17 Force Constants and the GF-Matrix Method

9.18 Selection Rules

9.19 Center of Symmetry and the Mutual Exclusion Rule

9.20 Isolation of a Particular Type of Motion

9.21 Detecting the Changes of Symmetry Through Reaction

Chapter 10: Electronic Spectroscopy

Abstract

10.1 Beer-Lambert Law

10.2 Allowed Electronic Transition

10.3 Basis of the Selection Rules

10.4 Selection Rules

10.5 Unexpected Weak Absorbance

10.6 Spectroscopy of Electronic Excitations

10.7 Electronic Spectra of Selected Examples

10.8 Spectroscopy of Porphyrins

10.9 The Magnetic Dipole Moment and the Absorbance Intensity

Chapter 11: Magnetism

Abstract

11.1 Magnetic Susceptibility

11.2 Types of Magnetic Behaviors

11.3 Diamagnetic Behavior

11.4 Spin-Only Magnetic Susceptibility, Magnetic Moment, and Thermal Spreading

11.5 Orbital Magnetic Moment

11.6 Second-Order Zeeman Effect and Van Vleck Equation

11.7 Spin-Orbital Coupling and Magnetic Susceptibility

11.8 Spin-Orbital Coupling: In A and E Ground Terms

11.9 Spin-Orbital Coupling: In T Ground Terms

11.10 Curie Law, Deviation, and Data Representations

11.11 The Magnetic Behaviors of Compounds Contain a Unique Magnetic Center

11.12 Structure-Linked Crossover, Thermal Isomerization

11.13 Interactions Between Magnetic Centers

11.14 Measurement of the Magnetic Susceptibility

Mathematics Supplement

Summation Formulas

Quadratic Series

Roots of the Quadratic Equation

Binomial Expansion

Trigonometric Formula

Logarithms

Derivative of a Function

Antiderivatives or Indefinite Integrals

Important Mathematical Functions

Kronecker Delta Function

Hermite Polynomials

Legendre Polynomial

Laguerre Polynomials

Taylor Series

Matrices

Determinants

Character Tables

Nonaxial Groups

Cn Groups

Cnv Groups

Cnh Groups

Dn Groups

Dnd Groups

Dnh Groups

S2n Groups

Cubic Groups

Linear Groups

Index

Copyright

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Dedication

To our Students in Chemistry

Preface

This book presents the chemical concepts that govern the chemistry of molecular construction. The emphasis is on the building up of an understanding of essential principles and on familiarization with basic inorganic concepts. Necessary background information is introduced to comprehend the field from both chemical and practical areas. The book explains and details the fundamentals that serve as a source of numerous basic concepts of methods and applications. The combination of the basic concepts, methods and applications with example exercises yields a more positive outcome for students and teachers.

Many inorganic textbooks that are available cover too much material and do not go into the depth needed for fundamental principles. Most of these are seem to be either fairly elementary or very advanced. Students might be displeased by the current selection available, and there is a great need for an inorganic principles of chemical bonding text. In our book, we bridge and integrate both elementary and more advanced principles. A student should be able to become familiar with the topics presented with this one book, rather than learn the basics in one and use another for the more advanced aspects of the material.

Given the complex and abstract nature of the subject, the book is easy to follow. The text is carefully thought through and 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 study material 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 and starting each chapter with the circle scheme describing the sections to be discussed are intended to be attention-catching and could 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 a reference and text for specialized and graduate courses in physical inorganic or advanced organic chemistry. The book is intended for advanced undergraduates and for postgraduates taking courses in chemistry, students studying atomic structure and molecule formation in chemical engineering and material science. It should also be of value to research workers in other fields, who might need an introduction to essential inorganic principles. This book is very suitable for self-study; the range covered 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 electronic 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.

Schrödinger's method is linked for exploring the modern theory of atomic structure, and to establish the formal mathematical framework. This framework is employed to set up the final real solution for the full orbital wave function and identify the four quantum numbers n, l, m, and s, also to compute the most probable radius, mean radius of an orbital, and the boundary surfaces of s, p and d orbitals. The orbital wave equations are used as the key feature in order to explain the orbital and spin angular momenta, as well the electronic configurations of many-electron atoms, and spin-orbital coupling. We examine how to identify the term symbols of the ground state and the different terms of the excited microstates of polyelectronic atoms, and how many subterms arise when spin-orbit coupling is taken into consideration. The splitting of Russell–Saunders terms into microstates in an external magnetic field and their energies are identified. The term wave functions and the corresponding single electron wave functions are described in order to understand the effect of the ligand field.

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 the bonding, define the role of the free valence electrons, and relate the physical properties and theories of metallic bonding are explored. Emphasis is also placed on ionic bonding and the relationships among the lattice energy, thermodynamic parameters and covalency. We review the grounds of the ionic crystal structure, in which radius ratios govern the geometrical arrangements, and review the factors that influence the solubilities. The foundation and basics of coordination chemistry are laid out: firstly, characterization, formulation, formation and stability including hard and soft acid/base interactions, the chelate 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, exploring their rôles, consequences and significances. Considerations are given to the structural and chemical nature of the covalent networks 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 are extensively 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 for understanding the structure and properties of molecules and crystals. Matrix representations of symmetry operations, point group, translation, rotation motions, and atomic orbital are thoroughly examined. This leads to presentation of the character tables, and finally shows how that the symmetry representations for the atomic orbitals form bases for the molecular wave functions.

The valence bond theory concept is explained, then we investigate how to predict the shapes and geometries of simple molecules using the valence shell electron-pair repulsion method. The process of predicting the molecule's structure is reviewed. As well, the relationships between the chemical bonds in molecules and its geometry using orbital hybridization theory are addressed. Special attention is devoted to the angles between the bonds formed by a given atom, also to multiple bonding 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 some elected small molecules, and approaches to the relative energies of the molecular orbitals are reviewed. We then explain how the electron distribution changes upon going to some low-lying excited electronic states. The theory is employed to estimate energy changes in chemical reactions, to study stability & reactivity, to find the delocalization energy, electron density, formal charge, bond order, ionization energy, equilibrium constant, and configuration interaction. The orbital combination introduces the band theory concept that makes it possible to rationalize conductivity, insulation and semiconductivity.

Having now available the valence orbital's wave functions of the central ion in their real forms, it is possible to explore the impact of various distributions of ligand atoms around the central ion upon its valence orbitals. A quantitative basis of this effect in the case of a purely ionic model of coordination, and other degrees of mixing are addressed. In this part, we indicate why there is a need for both crystal field theory and ligand field theory. The effect of a cubic crystal field on d- and f-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 perturbation theory for degenerate systems is used to explain how the crystal field potential of the surrounding ligands perturbs the degeneracy of d orbitals of the central ion. The energies of the perturbed d-orbitals are calculated by solving the secular determinant. The obtained energies are fed back into secular equations that are derived from the secular determinant to yield wave functions appropriate for the presence of the potential. The variation in the potential energy of each d electron due to the crystal field is determined and the splitting of d-orbitals so deduced in octahedral, tetrahedral, tetragonally distorted D4h geometries in terms of Dq, Dt, and Ds. Problems and the required approximations are discussed for the free ion in weak crystal field. Then, we study the influence of weak field on polyelectronic configuration of free ion terms, and find the splitting in each term and the wave function for each state. In the strong field situation, the first concern was how the strong field differs from the weak field approach; define the determinant, symmetry and the energy of each state. We compute the appropriate Hamiltonian and the diagonal and off-diagonal interelectronic repulsion in terms of the Racah parameters 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 investigated. Thirdly, we show how the energies of the various states into which the free ion term are split depend on the strength of the interaction of the ion with its environment. The relationship between the energy of the excited states and Dq are discussed using correlation, Orgel, and Tanabe-Sugano diagrams.

The vibrational spectra of diatomic molecules establish most of the essential principles that are used for complicated polyatomic molecules. Since infrared radiation will excite not only molecular vibration but also rotation, there is a need to comprehend both rotation and vibration of diatomic molecules in order to analyze their spectra. Molecular vibrations are explained by classical mechanics using a simple ball and spring model, whereas vibrational energy levels and transitions between them are concepts taken from quantum mechanics. The quantum mechanics of the translation, vibration and rotation motions are explored in detail. As well, the expression for the vibration-rotation energies of diatomic molecule for the harmonic and anharmonic oscillator models is introduced. Then, we identify the Schrödinger equation for the vibration system of n-atom molecules. We elucidate how to obtain, monitor, and explain the vibrational–rotational excitations, find the quantum mechanical expression for the vibrational and rotational energy levels, predict the frequency of the bands, and compare harmonic vs. anharmonic oscillators and between rigid and nonrigid rotor models for the possible excitations. 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 an n-atom molecule. Calculation of the force constants using the GF-matrix method are discussed. The general steps to determine the normal modes of vibration, and the symmetry representation of these modes are outlined. We examine the relationship and the 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 we also focus on how the molecules interact with the radiation and the chemical information obtainable by measurement of the infrared and Raman spectra. Only the radiation electric field interacts significantly with molecules and is important in explaining infrared absorption and Raman scattering. The requirements and the selection rules for the allowed vibrational and rotational excitations are explored. We investigate the relationship between the center of symmetry and the mutual exclusion 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. We examine the relationship between the solute concentration and light absorbance, as well the correlation among the molar extinction coefficient, integrated intensity and dipole strength. The significance of the Born-Oppenheimer approximation is considered, and symmetry considerations are elaborated with respect to allowedness of electronic transitions. The consequences of spin, orbital and vibrational constraints are investigated to explore the basis of the electronic absorption selection rules. The electronic excitation of functional groups, donor–acceptor complexes, and porphyrins are spectroscopically characterized. The study outlines the roles of vibronic coupling, configuration interaction, and π-bonding. Factors that affect the bandwidth, band intensity, and intense colors of certain metal complexes are also identified. The text addresses the effects of Jahn–Teller distortion, temperature, and reduced symmetry, and elaborates the spectrochemical series. Comparative studies of octahedral versus tetrahedral, low-spin versus high-spin, and dn versus d¹⁰-n configuration are conducted. We illustrate how to evaluate Dq and β from the positions of the absorption peaks, and discuss the unexpectedly weak absorbances, simultaneous pair excitations, and the absorption of unpolarized light in general. The role of the magnetic dipole moment on the absorbance intensity is investigated, using circular dichroism spectroscopy and the Kuhn anisotropy factor to examine the effects of lower symmetry, absolute configuration, and the energy levels within the molecule

Finally we investigate the types of magnetic behaviors, giving key definitions and concepts leading to the relationships relevant 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 moments and susceptibilities. The subsequent section deals with the van Vleck treatment and the second order Zeeman Effect to link the spin and orbital contributions to the magnetic susceptibility. Requirements and conditions for nonzero orbital contribution 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. Stephanos

Anthony W. Addison

Chapter 1

Particle Wave Duality

Abstract

This chapter is intended to develop an understanding of the dual wave-particle nature of electrons, photons, and other particles of small mass. The particle nature of the electrons had been confirmed by cathode rays, Millikan's capacitor, and Thomson's experimentation. Models for atomic structures were proposed by Thomson and Rutherford. Studies of black-body radiation shows that energy emits in a small, specific quantity called quanta. Also, hydrogen emission 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 are presented and discussed. Bohr in his atomic models used quantum theory, not quantum. Wave interference and diffraction are used as evidence to confirm the wave properties of the electrons. Einstein's relationships are explored to describe and clarify the interdependence of mass and energy. Furthermore, the corpuscular nature of light is revealed in the photoelectric effect and the Compton Effect. de Broglie shows that the dual wave-particle nature is true not only for photons but for any other material particles 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 the wave function. Subatomic particles are examined and classified as fundamental particles (fermions) and force particle (bosons) that mediate interactions among fermions.

Keywords

Electron; Photon; Particles; Wave; Duality; Quantum; Bohr; Light; Photoelectric effect; Compton effect; de Broglie; Heisenberg; Wave function; Subatomic particles; Fermions; Bosons; Quarks; Leptons; Hadrons; Baryons; Mesons; Gluon; Photon;      si1_e ; Z bosons; Graviton; Higgs boson

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).

sch01-01-9780128110485

Scheme 1.1 Schematic chart presents the development of the dual wave-particle nature. Note that mass and energy are entirely different properties of matter. The only conclusion shows that mass of material bodies depends on their motion.

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

Studies of black-body radiation shows that energy emits in a small, specific quantity called quanta. Also, hydrogen emission 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 are presented and discussed. Bohr in his atomic models used quantum theory not quantum mechanics, he did not recognize the wave nature of electrons.

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

Furthermore, the corpuscular nature of light is revealed in the photoelectric effect and the Compton effect. de Broglie shows 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 the wave function.

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

In the following outline, we shall try to understand:

• 1.1: Cathode and anode rays

• 1.2: Charge of the electron

• 1.3: Mass of the electron and proton

• 1.4: Rutherford's atomic model

• 1.5: Quantum of energy

• 1.6: The hydrogen-atom line-emission spectra

• 1.7: Bohr's quantum theory of the hydrogen atom

• 1.8: The Bohr-Sommerfeld model

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

• 1.10: Relativity theory: mass and energy

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

• 1.12: de Broglie's considerations

• 1.13: Werner Heisenberg's uncertainty principle, or the principle of indeterminacy

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

• 1.15: Atomic and subatomic particles

• Suggestions for further reading

1.1 Cathode and Anode Rays

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

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

• When electric current was passed through various gases at low pressures (cathode-tube), the surface of the tube directly opposite the cathode glows (Fig. 1.1).

f01-01-9780128110485

Fig. 1.1 Cathode-tube.

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

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

• 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 ray had sufficient mass to set the wheel in motion.

f01-02-9780128110485

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

○ 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 was known to have negative charge.

• 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 called electrons.

○ Electrons are present in atoms of all elements.

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

• 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.

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

• The positive ions move toward the cathode, while the negatively charged electrons of the cathode rays move in the opposite direction.

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

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

• 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?

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

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

f01-03-9780128110485

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

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

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

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

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

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

○ As a result:

si2_e    (1.2.1)

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:

si3_e    (1.2.2)

where

V is the voltage applied to the plates

d is the distance between plates

By substituting in Eq. (1.2.1):

si4_e

si5_e    (1.2.3)

○ The value of ed can be found if the mass of the droplet, m, is known.

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

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

• Initially the oil drops are allowed to fall between the plates while the electric field is turned off. The droplets reach a terminal velocity due to the air friction.

• The field is then turned on; if the voltage is large enough, some of the charged drops will start to go up. (This is because the upwards electric force Fel is greater than the downwards gravitation force, W=mg.)

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

• The drop is allowed to fall in the absence of an electric field and its terminal velocity u1 is calculated. The force of air friction for falling drop can then be estimated using Stokes' Law:

si6_e    (1.2.4)

where u1 is the terminal velocity of the falling drop (i.e., velocity in the absence of an electric field), η is the viscosity of the air, and r is the radius of the drop.

• The weight W′ of the drop is the volume Vd multiplied by the density ρ and the acceleration due to gravity g.

si7_e    (1.2.5)

• However, the apparent weight in air is the true weight minus the weight of air displaced by the oil drop.

si8_e

ρair is the density of the air

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

si9_e

si10_e    (1.2.6)

• The oil drop is not accelerating at terminal velocity. Therefore, the total force acting on it must be zero and the two forces F and W must cancel one another out (i.e.,      si11_e , Eqs 1.2.4, 1.2.6). This implies

si12_e

si13_e    (1.2.7)

• Once r is calculated, W, m, and ed can easily be computed (Eqs. 1.2.3, 1.2.6).

• In practice this is very hard to accomplish precisely. Estimating Fd is complicated because the mass of the oil drop is difficult to determine without the use of Stokes' Law.

• A more practical approach is to turn V up slightly, so that the oil drop goes up with a new terminal velocity u2 Then

si14_e

• If the mass of the droplet is known, ed can be found.

• Millikan noticed that the charge on the droplets were always multiples of a certain value of e, the smallest charge experimentally 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.

• Millikan obtained the following value: e=4.803×10−10 esu=1.60206×10−19 coulomb.

1.3 Mass of Electron and Proton

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

• Thomson concluded that all cathode rays are composed of indistinguishable negatively charged particles, which were named electrons.

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

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

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Fig. 1.4 Cathode ray tube for determining the value of e / m for electron.

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

f01-05-9780128110485

Fig. 1.5 Lay out for determining e / m e .

si15_e    (1.3.1)

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

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

si16_e    (1.3.2)

where

a is the acceleration of electron

me is the mass of electron

Then

si17_e    (1.3.3)

si18_e    (1.3.4)

• For a period of time t during which the electron is between the plates, the beam is displaced by a distance y. The value of y is determined by

si19_e

where l is the length of the plates and u is the velocity of the electron.

Substituting for a, using Eq. (1.3.4), therefore,

si20_e    (1.3.5)

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

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

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

si21_e    (1.3.6)

where fmag. is the force of the magnetic field acting on the electron current and si22_e .

• According to electrodynamics, the magnetic field will act on the electron moving perpendicular to the field with a force:

si23_e    (1.3.7)

where H is the intensity of the magnetic field.

Therefore, from Eq. (1.3.6),

si24_e

si25_e    (1.3.8)

• By substituting in Eq. (1.3.5):

si20_e    (1.3.5)

The only unknown is the value of si27_e , which now can be determined:

si28_e    (1.3.9)

• It was found that

si29_e

• The mass of the electron can be calculated if e/me and e are known (e=4.803×10−10 esu).

• This shows that the mass of the electron is 9.109×10−31 kg (about 1/1837 the mass of the hydrogen atom).

• Thomson showed that electron is a particle with

○ mass

○ energy

○ momentum

u01-01-9780128110485

Thomson atom

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

• The values of e/m were estimated for the positive ions by using basically the same technique employed in the study of cathode ray.

• When hydrogen gas is used, a positive particle results that has the smallest mass (the largest e/m value) of any positive ion observed:

si30_e

• These particles:

○ are known as protons

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

si31_e

○ the mass of the proton:

si32_e

which is about 1837 times heavier than the mass of electron.

○ It is assumed to be a component of all atoms.

1.4 Rutherford's Atomic Model

What are the experimental observations and the atomic model that concluded by Rutherford, and what are the weaknesses of this model?

• Rutherford, Geiger, and Marsden firstly assumed that mass and charge are uniformly distributed throughout the atoms, in order to experiment that, they bombarded a thin gold foil with fast moving alpha particles (Fig. 1.6).

f01-06-9780128110485

Fig. 1.6 Rutherford bombarded a thin gold foil with fast-moving alpha particles.

• They predicted that alpha particles will pass through with only a slight deflection. However, only 1 in 8000 of the alpha particles had been redirected back toward the source (Fig. 1.7).

f01-07-9780128110485

Fig. 1.7 Few fractions of the alpha particles had been redirected back toward the source.

• Rutherford had proposed the following:

○ The volume of the nucleus is very small compared with the volume of an atom.

○ The electrons encircle the positively charged nucleus like planets around the sun.

○ Because atoms are electrically neutral, a given atom must have as many electrons as protons.

○ To account for the total masses of atoms, he proposed the existence of an uncharged particle, which now called neutron.

u01-02-9780128110485

Rutherford atom

• The weaknesses of this model: according to electromagnetic theory, the atom of Rutherford cannot be real. The electrons circling around the nucleus are accelerating charged particles; therefore, they should emit radiation, spend energy, and execute descending spirals until they collapse into the positive nucleus.

1.5 Quantum of Energy

Describe the ideal black-body radiator. What is the expected relationship between the energy and the frequency of the emitted radiation?

How did Max Planck explain that the intensity of radiation rises with increasing frequency to a maximum and then falls steeply (the ultraviolet catastrophe)?

• For any object to be in equilibrium with its surroundings; the absorbed radiation must be equivalent in wave length and energy to the emitted radiation.

• Consider a red hot block of a metal with a spherical cavity of which the interior is blackened by iron oxide or a mixture of chromium, nickel and cobalt oxides, black-body. The density of the emitted radiant light can be observed and monitored through a small hole (Fig. 1.8).

f01-08-9780128110485

Fig. 1.8 Black-body cavity and emitted radiation.

• An ideal Black-body is a physical body that absorbs all incident electromagnetic radiation (no reflected energy), regardless of frequency or angle of incident.

• As a consequence of the equipartition principle, all frequencies of this emitted radiation should have the same average energy.

• The energy density of the radiated light, si33_e , is simply the number of the oscillators, si34_e n that can occur per unit volume between ν and si35_e times the average energy of an oscillator, si36_e . For electromagnetic radiation, there is an extra factor of two, because both magnetic and electric fields are oscillating.

If:

si37_e

where V is the volume, u is the phase velocity, and ν is the frequency of the electromagnetic wave:

si38_esi39_e

   (1.5.1)

Classic theory predicts that

si40_e    (1.5.2)

where k is the Boltzmann constant and T is the temperature.

If V=1, then

si41_e

   (1.5.3)

• Accordingly, the intensity of black-body radiation (or cavity radiation) should increase endlessly with rising frequency.

• But the experimental data reveal that the intensity of radiation rises to a maximum, then falls steeply with increasing frequency (the ultraviolet catastrophe) (Fig. 1.9).

f01-09-9780128110485

Fig. 1.9 The energy density per unit wavelength in black-body cavity recorded at different temperatures. The energy density increases and the peak shifts to shorter wavelength as the temperature is increased. The total energy density (the area under the curve) increases as temperature is raised.

• When Max Planck was studying the emission of light by a hot object, he proposed that a hot object does not emit electromagnetic energy continuously.

• Planck proposed that the object emits energy in a small, specific quantity called quanta. A quantum is the smallest amount of energy that can be lost or gained by an atom.

• Planck suggested the following relationship between a quantum of energy and the frequency of radiation:

si42_e    (1.5.4)

where

E is the energy in joules, of a quantum radiation

ν is the frequency of radiation emitted

h is the Planck constant; si43_e

• Consider a set of N oscillators having a fundamental vibration frequency, ν.

• If these can take up energy only in multiples of hν, then the allowed energies are: 0hν, 2hν, 3hν, etc.

• According to the Boltzmann formula, if No is the number oscillators in the lowest energy state:

si44_e    (1.5.5)

where Ni is the number of oscillators having energy ɛi, and

si45_e

• Then the total number of the oscillators is

si46_e

   (1.5.6)

and the total energy E is

si47_e

   (1.5.7)

• The average energy of an oscillator, si36_e , is

si49_e    (1.5.8)

si50_e

Then

si51_esi52_esi53_e

Accordingly,

si54_e    (1.5.9)

si55_e

• By substituting si56_e in the energy density,

si57_e

   (1.5.1)

si58_e

Then

si59_e    (1.5.10)

• This expression exactly matches the experimental curve at all wavelengths, and quantum theory had achieved its first great success.

1.6 Hydrogen Atom Line-Emission Spectra; Electrons in Atoms Exist Only in Very Specific Energy States

Why do excited hydrogen atoms emit only specific frequencies of electromagnetic radiation and not a continuous range of frequencies?

What are the proposed mathematic relationships between wave-numbers of the emitted frequencies and the energy state of the excited electrons?

• Classical theory expected that the hydrogen atoms would be excited by any quantity of supplied energy.

• Consequently, scientists expected to detect a continuous spectrum (Fig. 1.10), which is the emission of a continuous range of frequencies of electromagnetic radiation.

f01-10-9780128110485

Fig. 1.10 A continuous visible spectrum is produced when a narrow beam of white light is passed through a prism.

• When researchers passed electric current through a vacuum tube containing hydrogen gas at low pressure, they observed the emission of a characteristic pinkish glow (Fig. 1.11).

f01-11-9780128110485

Fig. 1.11 A series of specific wavelengths of emitted light makes up hydrogen's line-emission spectra.

• When a narrow beam of the emitted pinkish light was shone through a prism, it was separated into a series of specific frequencies, known as the line-emission spectrum.

• Additional series of lines were discovered in the ultraviolet and infrared regions of hydrogen's line-emission spectrum (Fig. 1.12); they are known as the Lyman, Balmer, and Paschen series.

f01-12-9780128110485

Fig. 1.12 Electron energy-level diagram for hydrogen and the energy transitions for the Lyman, Balmer, and Paschen spectral series.

• The fact that hydrogen atoms emit only specific frequencies of light indicated that the energy differences between the atom' energy states were fixed.

• This suggested that the electron of hydrogen atom exist only in very specific energy states.

• Balmer discovered a relationship between the frequencies of the atomic hydrogen line in the visible region of the spectrum; the wave numbers, si60_e , are given by

si61_e    (1.6.1)

with n=3, 4, 5, etc.

The bright red line at λ=656.28 nm corresponds to n=3, the blue line at λ=486.13 nm to n=4, etc. (Figs. 1.11 and 1.12). The constant R is called the Rydberg constant and has the value 109 677.581 cm−1 (1.10×10⁵ cm−1).

• Other hydrogen series were discovered later, which obeyed the more general formula:

si62_e    (1.6.2)

• Lyman found the series with n1=1 in the far ultraviolet and other were found in the infrared.

si63_esi64_esi65_e

In what zone of the electromagnetic spectrum would you expect a spectral line resulting from the electronic transition from the fifth to the tenth level of the hydrogen atom?

si66_e    (1.6.2)

si67_esi68_e

This line would be seen in the infrared zone of the electromagnetic spectrum; it is a member of the Pfund series.

What would be the maximum number of emission lines that you would expect to see in a spectroscope for atomic hydrogen if only seven electronic energy levels were involved (Fig. 1.12)?

• Emission spectrum lines result by electronic transitions from high electronic energy levels to a lower one.

• The number of the emission line for hydrogen = 6 + 5 + 4 + 3 + 2 + 1 = 21 emission lines.

• In general, the maximum number of emission lines is given by=½ (n)(n−1).

si69_e

1.7 Bohr's Quantum Theory of the Hydrogen Atom

What did Bohr conclude in his atomic model, and what are the weaknesses of this model?

How could Bohr drive the radius of the electronic orbits?

What are the radius and the ionization potential of the hydrogen atom?

How could Bohr explain hydrogen's line-emission spectrum?

• The modern theories of atomic structure began with the ideas of Niels Bohr, thus these are essential to be considered and explored.

• Bohr combined Planck's and Rydberg's equations, and suggested a mechanism for discreet emission from excited atomic vapor. He concluded with the following in his atomic model:

○ The electron moves about the nucleus in a circular orbit.

○ Only orbits in which the electron's angular momentum, L, is an integral multiple of ℏ are allowed.

○ The electron does not radiate energy when it is in an allowed orbit.

○ The electron can absorb energy and move to a higher orbit (or it can loss energy and move to a lower orbit) (Fig. 1.13).

f01-13-9780128110485

Fig. 1.13 Absorption and emission of photons by hydrogen atom according to Bohr model.

• Bohr used quantum theory not quantum mechanics; he did not recognize the wave nature of electrons.

• He superimposed the