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 Electrons, Atoms, and Molecules in Inorganic Chemistry A Worked Examples Approach Joseph J Stephanos Anthony W Addison Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1800, San Diego, CA 92101-4495, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom © 2017 Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-811048-5 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: John Fedor Acquisition Editor: Emily McCloskey Editorial Project Manager: Katerina Zaliva Production Project Manager: Paul Prasad Chandramohan Cover Designer: Mathew Limbert Typeset by SPi Global, India To our Students in Chemistry Contents Preface XIII Particle Wave Duality 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; 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 Electrons in Atoms Exist Only in Very Specific Energy States Bohr's Quantum Theory of the Hydrogen Atom The Bohr-Sommerfeld Model The Corpuscular Nature of Electrons, Photons, and Particles of Very Small Mass Relativity Theory: Mass and Energy, Momentum, and Wavelength Interdependence The Corpuscular Nature of Electromagnetic Waves The Photoelectri c Effect The Compton Effect de Broglie's Considerations Werner Heisenberg's Uncertainty Principle, or the Principle of Indeterminacy The Probability of Finding an Electron and the Wave Function Atomic and Subatomic Particles Elementary Particles Suggestions for Further Reading 10 2.5 2.6 2.7 2.8 2.9 2.10 2.11 13 15 18 18 21 21 22 25 26 27 28 29 29 32 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 Electrons in Atoms 2.1 2.2 2.3 2.4 The Wave Function (the Schri:idinger Equation) Properties of the Wave Function Schri:idinger Equation of the Hydrogen Atom Transformation of the Schri:idinger Equation From Cartesians to Spherical Polar Coordinates 37 39 40 47 48 51 57 66 71 76 76 76 78 78 84 85 87 91 99 100 108 109 111 11 11 121 129 131 133 144 Chemical Bonding 3.1 3.2 3.3 41 The Angular Equation The -Equation The 8-Equation The Radial Equation The Final Solution for the Full Wave Function, 111ntm(r, 0, (/J) The Orthonormal Properties of the Real Wave Functions The Quantum Numbers: n, I, and m The Principle Quantum Number, n The Q uantum Numbers I and Angu lar M omentum The Angular M omentum Quantum Numbers, I and m Picture and Represent Precisely m Vectors of p- and d-Orbitals The Spin Quantum Number, s The Boundary Surface of s-Orbital The Boundary Surface of p-Orbitals The Boundary Surface of d-Orbitals Calculating the Most Probable Radius Calculating the Mean Radius of an Orbital The Structure of Many-Electron Atoms The Pauli Exclusion Principle Slater Determinant Penetration and Shielding The Building-Up Principle Term Structure for Polyelectron Atoms Term Wave Functions and Single Electron Wave Functions Spin-Orbital Coupling Spin-Orbital Coupling in External Magnetic Field Suggestions for Further Reading Electronegativity and Electropositivity Electronegativity and Electropositivity Trends Molecular and Nonmolecular Compounds 149 150 151 vii viii Contents 3.4 3.5 3.6 3.7 3.8 Types of Bonds Metallic Bonding and General Properties of Metals Conductiv ity and Mobility of Electrons Luster and Free Electron Irradiation Malleability, Cohesive Force, Number of Valence Electrons Theories of Bonding in M etals Free Electron Theory Bond Lengths Crystal Structures of M etals (Metallic Structures) Alloy and Metallic Compounds Ionic Bonding Lattice Energy and Cohesion of Atom ic Lattice Born-Haber Cycle and Heat of Formation Ion ic Crystal Structures and the Radius Ratio Stoichiometric and Nonstoic hiometri c Defects Ionic Character and Covalency Interference Ionic Character and M elting Point Solubil ity of the Ionic Salts Covalent Bonding The Lewis Structures and Octet Rule Exceptions to the Octet Rule Bonding and Polarity Coordinate Covalent Bond (Dative Bonding) Coordination Number and the " 18-E iectron Ru le" Ligand Denticity Nomenclature of Complexes Complex Formation Coordinative Comproportionation Reaction Complexation Equilibrium Multi ligand Complexation Stepwise Formation Constants and the Sequential Analysis Complex Stabi lity Hard and Soft Interactions, HSAB Chemical Features of Hard and Soft Ions, and Classification Rule of Interactions Hard-Hard and Soft-Soft Interactions Hard-Soft Interaction and Anion Polarizabi Iity Chelate Effect Entropy and Chelate Formation Stability and the Geometry of the Chelate Ring 152 152 153 154 154 155 155 156 156 158 159 159 163 M acrocyclic Effect Solvation Enthalpy Differences Donor Atom Basicity Cavity Size Solvent Competition Steric Effect Stability and Metal Oxidation State Stabi lity and M etal Ionization Potential 3.9 Intermolecular Interactions van der W aals Forces ion-Induced Dipole Forces, ion-Dipole Forces, and Hydrogen Bonding 3.10 Covalent Networks and Giant Molecules Graphite, Ful lerenes, Graphene, Carbon Nanotubes, and Asbestos Suggestions for Further Reading 193 196 196 197 197 198 198 201 204 204 206 212 214 225 164 169 170 71 171 173 173 173 174 Molecular Symmetry 4.1 4.2 4.3 4.4 175 176 176 176 178 4.5 4.6 181 182 183 4.7 184 186 186 189 190 190 4.8 190 191 192 4.9 193 Molecular Symmetry The Symmetry Elements Identity, E Proper Rotation Axis, Cn Plane of Symmetry, a (Pntpr of Symmetry, i Sn: Improper Rotation Axis The Symmetry and Point Group Some Immediate Applications Dipole Moments and Polarity Chirality Equivalent Atoms: (Or Group of Atoms) Crystal Symmetry Group Theory: Properties of the Groups and Their Elements Similarity Transforms, Conjugation, and Classes Matrix Representation M atrices and Vectors M atrix Representation of Symmetry Operation Matrix Representation of Point Group Irreducible Representations Irreducible and Degenerate Representations Motion Representations of the Groups Translation Motion Rotationa l Motion Symmetry Properties of Atomic O rbitals Mullikan Notation Atomic Orbital Representation 228 230 230 230 233 2.15 236 238 239 239 244 244 245 250 252 254 254 255 258 260 261 262 262 264 266 266 267 Contents ix 4.10 Character Tables Properties of the Characters of Representations 4.11 Relation Between any Reducible and Irreducible Representations The Di rect Product 4.12 Group Theory and Quantum M echanics: Irreducible Representations and Wave Function Suggestions for Further Reading 269 270 272 274 275 280 Valence Bond Theory VSEPR Theory and M olecular Geom etry 5.3 lsoelectronic Species 5.4 Procedures to Diagram M olecular Structure 5.5 Valence Bond Theory and M etallic Bonds 5.6 O rbital Hybridization 5.7 Rehybridization and Com plex Formation 5.8 Hybridization and o / rr-Bonding 5.9 Orbital Hybridization and M olecular Symmetry Trigonal Planar Hybridization The Extend of d-Orbital Partic ipation in Molecular Bonding Trigonal Bipyramidal Hybridization Tetragonal Pyramidal Hybridization Square Planar Hybridization Tetrahedral Hybridization Octahedral Hybridization 5.10 Hybrid O rb itals as Symmetry Adapted Linear Combination of Atom ic Orb itals (SALC) 5.11 M olecular Wave Funct ion as Sy mmetry Adapted Linear Combination of Atomic O rbitals (SALC) Suggestions for Further Reading Heterodiatomic M olecules Polyato mic M olecules M olecular O rbitals for a Centric Molecule Properties D erived From M olecular Wave Function 6.9 Band Theory: M olecule O rbital Theory and Metallic Bonding Orbit 6.10 Conductors, Insulators, and Sem iconductors Suggestions for Furt her Reading 7.1 282 283 284 7.2 7.3 284 288 290 29 294 7.4 296 296 301 301 303 304 306 308 311 7.5 7.6 322 330 Molecular Orbital Theory 6.1 6.2 6.3 6.4 M olecular Orbital Theory Versus Valence Bond Theory M olecular O rbital Wave Function and Symmetry The Linear Combination of Ato mic O rbitals-M olecular O rbital (LCAO -MO) and Hi.ickel Approximations At om ic O rbitals Com binations for the Second Row Diatom ic Molecules 347 349 35 366 394 397 40 Crystal Field Theory Valence Bond Theory and Orbital Hybridizatio n 5.1 5.2 6.5 6.6 6.7 6.8 332 333 7.7 333 338 The Advantages and Disadvantages of Valence Bond Theory Bases of Crystal Field Theory d-Orbitals in Cubic Crystal Field f-Orbitals in Cubic Crysta l Field The Crystal Field Potential Octahedral Crystal field Potential, Voct Square Planar Crystal Field Potential, Vsq.PI Tetragonal Crystal Field Potential, VTetrag Tetrahedral Crystal Field Potential, Vrd Zero -O rder Perturbation Theory The Li near Combi nation of Atomic Orbitals, LCAO-MO, and Energy Cil lc:u liltion The Perturbation Theory for Degenerate Systems The Splitting of d-Orbitals in Octahedral Crystal Field, Voct The Splitting of d-Orbitals in Tetrahedral Crystal Field, Vrd The Splitting of d-Orbitals in Tetragonal Crystal Field, Vo.h Types of Interactions That Affect the Crystal Field Treatment Free Jon in Weak Crystal Fields Problems and the Requ ired Approximations The Effect of the Crysta l Field on S Term The Effect of the Cubic Crystal Field on P Term The Effect of a Cubic Crystal Field on D Term The Effect of a Cubic Crystal Field on F Term The Effect of a Cubic Crystal Field on G, H, and I Strong Field Approach Determinantal Wave Functions The Determ inantal Wave Functions of d in Strong Field of Tetragona l Structure, Trans-M L4 Z 405 405 405 406 407 407 12 15 17 19 19 421 423 430 435 442 442 442 442 442 44f) 447 455 457 45 457 20 Electrons, Atoms, and Molecules in Inorganic Chemistry FIG 1.15 A pattern of light and dark is produced by interference of light waves that have passed through two or more opening close in size to the wavelength of light Similar diffraction pattern is produced when a beam of electrons passed through a crystal FIG 1.16 Formation of diffraction pattern when radiation passes through a diffraction grating l l l Diffraction patterns are also obtained by passing beams of electrons through thin gold foil or a thin film of chromium, confirming the wave properties of the electrons Additionally, diffraction patterns have been emitted from crystals placed in beams of neutrons or hydrogen atoms, although these more massive particles also exhibit wave properties When interference and diffraction are observed, they are interpreted as evidence that the passage of waves has taken place Particle Wave Duality Chapter 21 1.10 RELATIVITY THEORY: MASS AND ENERGY, MOMENTUM, AND WAVELENGTH INTERDEPENDENCE What are Einstein’s relationships that describe the interdependence of mass and energy? What are the accepted mass of a photon moving with velocity of light and interdependence of the momentum and the wavelength? l l Einstein established that the mass of a body in motion exceeds its mass at rest, according to the equation: mo m ¼ rffiffiffiffiffiffiffiffiffiffiffiffi u2 1À c (1.10.1) where m is the mass of the body in motion mo is the mass of the body at rest u is the velocity of the body c is the velocity of light in vacuum An increase in the velocity of the body causes a consequent increase in its energy, results in an increase in its mass Einstein also showed that the mass of a body is related to its energy according to E ¼ mc2 (1.10.2) This equation shows the interdependence of the changes in Δm and ΔE in any process: ΔE ¼ Δmc2 l l l l (1.10.3) It cannot be concluded that mass can be converted to energy or vice versa Mass and energy are entirely different properties of matter Eq (1.10.3) only shows that mass of material bodies depends on their motion Albert Einstein established that light energy is not distributed continuously in space, but is quantized in small bundles called photon When a photon is at rest (does not exist), the mass, m, and energy, E, of a photon would be infinitely large A photon has no rest mass because it moves with velocity of light; thus a motionless photon does not exist Therefore, all mass of a photon is dynamic, and according to Planck’s formula: c E ¼ hν ¼ h λ ∵E ¼ mc2 c ; mc2 ¼ h λ h h ¼ λ¼ mc p (1.10.2) (1.10.4) where p is the momentum (impulse) of a photon The momentum is a vector quantity; its direction coincides with that of the velocity Eq (1.10.4) expresses the interdependence of the momentum of the photon mc and the wavelength of light 1.11 THE CORPUSCULAR NATURE OF ELECTROMAGNETIC WAVES How could the dual wave-particle nature of electromagnetic waves or photons be confirmed? What is the most precise technique for determining Planck’s constant? What are the difficulties in explaining the photoelectric effect and the Compton effect, and how could these problems be solved? How could the corpuscular nature of light be revealed in the photoelectric effect and the Compton effect? l l The interference and diffraction experimentally confirm that light consists of transverse electromagnetic vibrations The occurrence of interference and diffraction is a characteristic of any wave process The corpuscular properties of light (that light made up of small particles, which move in straight lines with finite velocity and have kinetic energy) are obviously revealed in two phenomena: the photoelectric effect and the Compton effect 22 Electrons, Atoms, and Molecules in Inorganic Chemistry The Photoelectric Effect l l l l l l l Several materials, such as potassium (and semiconductors), emit electrons when irradiated with visible light, or in certain cases with ultraviolet light The emission of electrons from metal when exposed to light is known as the photoelectric effect Fig 1.18 shows the circuit to monitor and study the photoelectric effect When light is focused on the clean metal surface of the cathode, electrons are emitted If some of these electrons hit the anode, there is a current in the external circuit The number of the emitted electrons reaching the cathode can be increased or decreased by making the anode positive or negative with respect to the cathode At a specific large enough potential, V, nearly all the emitted electrons get to the anode and current attains its maximum (saturation) value (Fig 1.19) Any increase in V does not affect the current, the maximum current is directly proportional to light intensity, and doubling the energy per unit time incident upon the cathode should double the number of electrons emitted Those electrons that are emitted without losing any of their energy to the atoms of the metal surface will have the maximum energy, (Ee)max The maximum energy of the emitted electrons was computed by applying an external electrical field at which the photoelectric current is stopped θ Δy θ Δy λ = Δy sin θ λ = Δy sin θ Intensification of waves Extiction of waves FIG 1.17 Interference of waves on passing through a diffraction grating FIG 1.18 Schematic drawing of photoelectric effect circuit, the anode can be made positive or negative with respect to the cathode, to attract or repel the emitted electrons hn + e V Anode A Cathode Particle Wave Duality Chapter 23 FIG 1.19 Plots of photoelectric current versus voltage V for two values of light intensity I −VO l l l l V In Fig 1.19, if V is less than Vo, no electrons reach the anode The potential Vo is called the stopping potential The experimental results indicate that Vo is independent of the intensity of the incident light The stopping potential, Vo, is related to the maximum kinetic energy of the emitted electrons by   me u2 Ee ịmax ẳ ẳ eVo (1.11a.1) max where me, e, and u are the mass, charge, and velocity of the electron, respectively In fact, increasing the rate of the energy falling on the cathode does not increase the maximum kinetic energy of the electrons emitted, (Ee)max According to wave theory, the energy Ee of the electrons emitted by the metal should be proportional to the intensity of the incident light It was, however, established that:  Ee does not depend on the intensity of the light The increase in the intensity of light only causes a greater number of electrons to be emitted from the metal  Ee depends on its frequency of the light, ν Ee increases with ν Einstein showed that the photoelectric effect could be explained very simply if:  light were regarded as a stream of particles, called photons  a photon is a particle of electromagnetic radiation having zero mass and carries a quantum of energy  the energy of particular photon depends on the frequency of the radiation (ν) Ephoton ¼ hν l l l where h is Planck’s constant In order for an electron to be forced out from a metal surface, the electron must be bumped by a single photon carrying at least the minimum energy required to expel the electron Einstein rationalized the photoelectric effect by suggesting that  electromagnetic radiation is absorbed by matter only in whole numbers of photons  when the intensity of the light of a given frequency is increased, more photons fall on the surface per unit time, but the energy absorbed by each electron is unchanged  for a given metal, no electrons were emitted if the light frequency was below a certain minimum  clearly, the energy of such electrons is equal to the difference between the energy of the photon hν and the work required to overcome the force holding the electron in the metal  if a photon’s frequency is below the minimum, then the electron remains bound to the metal surface  electrons in different metals are bound more or less tightly, so different metals require different minimum frequencies to exhibit the photoelectric effect If Φ is the energy necessary to remove an electron from the surface of a metal, the maximum kinetic energy of the electrons emitted will be 24 Electrons, Atoms, and Molecules in Inorganic Chemistry   ¼ e Vo ¼ h ν À Φ me u max l (1.11a.2) where me, e, u are the mass, charge, and velocity of the electron, respectively Φ is called the work function, the equation is known as Einstein’s photoelectric equation Millikan showed that the Einstein equation is correct, and measurements of h agreed with the value found by Planck Fig 1.20 represents the dependence of the voltage at which the photoelectric current is discontinued on the frequency of the incident light The data fall on a straight line which has a slope h/e This method is one of the most precise techniques for determining Planck’s constant By placing Vo equal to zero in   m e u2 ¼ e Vo ¼ h ν À ϕ max when ν ! νi , then Φ ¼ hνi ¼ hu λi (1.11a.3) Photons of frequency less than νi not have enough energy to eject electrons from the metal Calculate the kinetic energy and the velocity of ejected electrons, when a light with wavelength of 300 nm is incident on a potassium surface for which the work function Φ is 2.26 eV l If the kinetic energy, Ee: Ee ¼   me u2 ¼ eVo ¼ hν À Φ max (1.11a.2) hc λ hc ; Ee ¼ À Φ λ À ÁÀ Á À34 À Á 6:626  10 Js 2:998  10 m=s À ð2:26eVÞ 1:602  10À19 J=ev Ee ¼ À9 300  10 m ∵ hν ¼ Ee ¼ 6:62  10À19 J À 3:62  10À19 J ¼  10À19 J l The velocity, u, of the ejected electron is as follows:      Á Ee 2   10À19 J ¼ ¼ 8:12  105 m=s u¼ m 9:109  10À31 kg Stopping potential (V) FIG 1.20 Dependence of the voltage at which the photoelectric current is discontinued on the frequency of the incident light 30 ni 50 90 70 Frequency (Hz) 110 × 1013 Particle Wave Duality Chapter 25 The Compton Effect l l l l l l l When a photon collides with an electron, it discharges part of its energy to the electron As a result, the radiation is scattered and its wavelength increased (Fig 1.21) Compton noticed that when a variety of substances are exposed to X-rays, the wavelength of the scattered radiation is greater than the initial wavelength The variation in the wavelength, Δλ, does not depend on the nature of the substance or the wavelength of the original radiation It always has a definite value that is determined by the scattering angle ϕ (the angle between the directions of the scattered and original radiation) The accurate expression of the Compton effect could be determined if the hitting of photon and an electron could be considered as an elastic collision of two particles, in which the laws of conservation of energy and impulse are observed Let us assume that a photon of energy hν collides with an electron (Fig 1.21) The energy and the momentum of the electron are taken to be zero After collision:  the energy of the photon become hν0 and scattered by an angle ϕ to the original direction  the kinetic energy of the electron changes and moves in the direction of an angle θ to the direction of the original photon According to the law of conservation of energy, the kinetic energy of the electron, T, is given by T ¼ hν À hν0 ¼ Àhðν0 À ị ẳ h Tẳ me u2 and pe ẳ me u where me, u, and pe are the mass, velocity, and momentum of the electron, therefore, T¼ p2e ¼ ÀhΔν 2me p2e ¼ À2me hΔν l (1.11b.1) According to the law of conservation of momentum, the sum of the vectors of the scattered photon and recoiled electron is equal the momentum of the original photon: pe ¼ pph1 À pph2 (1.11b.2) p2e ¼ p2ph1 + p2ph2 À 2pph1 pph2 cos ϕ where pph1 and pph2 are the magnitudes of the impulses of the original and scattered photons The difference between these magnitudes is negligible, therefore, ;p2ph1 % p2ph2 hν ′ Pph(2) φ hν − φ Pph(1) θ θ − Pe T (A) (B) FIG 1.21 (A) Schematic representation of the collision of the photon and electron (B) Vector addition of momenta of the recoil electron and scattered photon 26 Electrons, Atoms, and Molecules in Inorganic Chemistry ;p2e ¼ 2p2ph1 À 2p2ph1 cos ϕ p2e ¼ 2p2ph1 ð1 cos ị If: cos ị ẳ sin 2 ;p2e ¼ 4p2ph1 sin ϕ h hν ∵p ¼ ¼ λ u (1.11b.3) (1.11b.4) (1.10.4) h2 ν2 ϕ sin u2 (1.11b.5) ∵pe ¼ À2me hΔν (1.11b.1) p2e ¼ hν2 ϕ sin 2 u2 u ∵ν¼ λ u ; @ν ¼ À @λ λ u ; Δν ¼ À Δλ λ   u hν2 ϕ ν2 ; À 2me h À Δλ ¼ sin , and ∵ ¼ u u λ λ À2me hΔν ¼ ;Δλ ¼ l l h ϕ sin me u (1.11b.6) (1.11b.7) ˚ , and is called the Compton wavelength of the The value h/meu has the dimension of length and is equal to 0.0242 A electron Investigation has shown that this equation is in good agreement with the experimental data 1.12 DE BROGLIE’S CONSIDERATIONS How did de Broglie’s show the following? (a) The dual wave-particle nature is true not only for photon but for any other material particle as well (b) nh mur ¼ , Bohr assumption for a stable orbital (1.7.4, this chapter page 16) 2π l l Max Planck proposed that light wave have particle properties Since the particle has wave properties, two fundamental equations need to be obeyed: E ¼ hν, where ν is the frequency E ¼ mu2 , where u is the velocity ; hν ¼ mu2 h u ¼ mu ν u ∵λ ¼ ν ; The de Broglie equation is as follows: λ¼ h h ¼ mu p (1.12.1) Particle Wave Duality Chapter l 27 This is a fundamental relation between the momentum of the electron (as a particle) and the wavelength (de Broglie wave) Every Bohr orbit is associated with a number of waves: Circumference ¼ 2πr ¼ nλ λ¼ 2πr n When n ¼ 1, the number of waves equals 1, etc h 2πr ¼ mu n nh mur ¼ 2π λ¼ (1.7.4) which is simply the original Bohr condition for a stable orbital If an electron in a field of volt/cm has a velocity of × 107 cm/s, find λ λ¼ h 6:63  10À27 ergs ˚ Á À Á ¼ 12:0  10À8 cm ¼ 12A ¼À mu 2:1  10À20 g   107 cm=s Such an electron has a wavelength equivalent to an X-ray l l An X-ray undergoes diffraction, as all waves Electron beams undergo diffraction Therefore electrons have wave properties Previously, Thomson showed that electron is a particle with mass, energy, and momentum 1.13 WERNER HEISENBERG’S UNCERTAINTY PRINCIPLE, OR THE PRINCIPLE OF INDETERMINACY If a particle has the wave nature, is it possible to determine its position and its velocity at the same time? How can you verify the uncertainty principle by diffraction of a beam of particles at slit? Why can we not precisely define the position of the electron without sacrifice information about its momentum (or velocity), and what is the significance of this finding? l Assume a particle of mass m moves along x direction with a speed u (Fig 1.22).The momentum of this particle is in x-direction, px Consequently, the y-component of momentum py is zero At a point xo, the particle is allowed to be diffracted by passing through a slit of width Δy The diffracted beam has momentum p with components in both x and y directions Y p ν Δy θ 2p sin θ θ = Δ py Δy XO X θ Δy sin θ sin θ = ± λ Δy FIG 1.22 Illustration of the path difference and uncertainty principle by diffraction of a beam of particles at the slit 28 l Electrons, Atoms, and Molecules in Inorganic Chemistry The pattern of the diffracted intensity is exposed on the screen placed beyond xo The necessity for reinforcement is sin θ ¼ nλ Δy (1.9.1) where, n ¼ 1, 2, 3… The difference in the path for the two waves must be an integral number, n, of the wavelength, λ The path difference becomes (Figs 1.17 and 1.22) Δy sin θ ¼ nλ l For the first two minima in the diffracted pattern, n ¼ 1: Δy sin θ ¼ λ, then : sin θ ¼ l λ Δy The new direction of the momentum can be defined only within an angular-spread of Ỉθ (Fig 1.22), so that 2pλ Δy (1.13.1) Δpy Á Δy ’ 2pλ ¼ 2h (1.13.2) Δpy ¼ 2p sin θ ¼ ∵λ¼ h h ¼ mu p Therefore, the product of uncertainty in momentum Δpy times the uncertainty in its conjugated coordinate Δy is of the order of h A more accurate derivation shows the following: Δy Á Δpy ! l l h ¼ ℏ 4π If we attempt to locate exactly the position of a particle, we must ignore the information about its momentum (or velocity) Heisenberg suggested that we cannot simultaneously know the exact position, x, and the exact momentum, p, of a particle: ðdxÞðdpÞ ! l l (1.13.3) h 4π The uncertainty principle suggests that we cannot assume that the electron is moving around from point to point, with exact momentum at each point Rather, we have to assume that the electron has only a specific probability of being found at each fixed point in space The concept of an electron following a specific orbit, where its position and velocity are identified exactly, must therefore be replaced by the probability of finding an electron in a certain position or in a particular volume of space 1.14 THE PROBABILITY OF FINDING AN ELECTRON AND THE WAVE FUNCTION How can the electron’s location be determined? l l In quantum mechanics we not ask: what is the location of a particle? However, we may ask: what is the probability of having a particle in an extremely small volume? The probability of finding an electron in any volume element in space is proportional to the square of the absolute value of the wave function, jψj2, integrated over that volume of space This is the physical significance of the wave function Probabilityðx, y, zÞ∝ jψ ðx, y, zÞj2 jψ(x, y, z)j2 is equal to the probability that a particle, whose wave function is ψ, is located in a volume element @V at x, y, z, provided that Z All space ψ @τ ¼ N2 where the normalized condition is @τ and the volume element ¼ @x@y@z Particle Wave Duality Chapter l 29 This interpretation is suggested by the theory of electromagnetic radiation, where intensity  ψ Acceptable solutions to the wave equation must be physically possible, and have certain properties:  ψ must be continuous  ψ must be finite  ψ must be single valued  The probability of finding the electron over all the space from plus infinity to minus infinity must be equal to one 1.15 ATOMIC AND SUBATOMIC PARTICLES Elementary Particles What we mean by elementary particles? l Elementary particles:  are the fundamental units that compose matter  exhibit wave-particle duality, displaying particle-like behavior under certain experimental conditions and wave like behavior in others  have neither known structure nor size  are often formed in high-energy collisions between known particles at particle accelerators  when formed, tend to be unstable and have short half-lives (t1/2: 10À6 À 10À23 s)  have their dynamics governed by quantum mechanics What are the differences between fermions and bosons? Name 12 leptons and 12 quarks What are the chemical characteristic features of leptons, hadrons, baryons, and mesons? l l l Elementary particles are classified as fundamental particles (fermions) and mediating field particles (bosons) Pairs from each classification are grouped together to form a generation, with corresponding particles exhibiting similar physical behavior:  Charged particles of the first generation not decay such electrons  Charged particles of the second and third generations decay with very short half-lives, and are observed in very high-energy environments Elementary fermions: include leptons and hadrons  Leptons: There are six known leptons: electron, muon, tau, and their three associated neutrinos Lepton (negatively charged) Neutrino Generation     eÀ νe 1st μÀ νμ 2nd τÀ ντ 3rd Each of these six leptons has an antiparticles: e+ (positron), νe , μ + , νμ , τ + , and ντ These leptons not break down into small units, and have no measurable size or internal structure Hadrons include baryons and mesons (a) Baryons: protons and neutrons are common examples Baryons have masses equal to or greater than proton (b) Mesons are unstable Baryons are heavier than mesons, and both are heavier than leptons All hadrons are composed of two or three fundamental particles, which are known as quarks There are six quarks that fit together in pairs: up and down, charm and strange, and top and bottom Charge À1/3 quark Charge +2/3 quark Generation d: dawn quark u: up quark 1st s: strange quark c: charm quark 2nd Associated with each quark is an antiquark of oppositecharge: d, u, s, c, b, and t  All quarks have an associated fractional electrical charge À e or + e 3 b: bottom quark t: top quark 3rd 30 Electrons, Atoms, and Molecules in Inorganic Chemistry Baryons Mesons P+ + u e π+ n + u e + u e d – e d – e d – e Proton Neutron u e + K– d + e Pion s – e u – e Kaon FIG 1.23 Examples of baryons and mesons  Baryons and mesons are distinguished by the composition of their internal structure (Fig 1.23): meson: one quark + one antiquark, such as pion and kaon baryon: three quarks, like proton and neutron antibaryon: three antiquarks  The charge of the hadrons is either a multiple of e or zero  The total charge of a hadron is equal to the sum of the charges of its constituent’s quarks (Fig 1.23)  Quarks come in six categories (names): d, u, s, c, b, and t Quarks have an additionally character, which is whimsically called color; however, they are of course not reallyÀ colored Quark “colors” are red (r), blue (b), Á and green (g) Antiquarks have “anticolors”: antired ðrÞ, antiblue b , and antigreen ðgÞ  It is believed that six leptons and six quarks (and their antiparticles) are the fundamental particles Fermions are the building blocks of the matter we see around us What are the main particles that intermediate fundamental interaction? l l Elementary particles that intermediate fundamental interaction are known as bosons Types of bosons include: gluon, photon, W Ỉ , Z boson, graviton, Gauge boson, and Higgs boson  These force carrier particles to intermediate the fundamental interaction among fermions  Such forces result from matter particles exchanging other particles, and are generally referred to as force-intermediating particles  The exchange bosons can mediate an interaction only over a range that is inversely proportional to its mass  Gluon ▪ Strong interaction is mediated by exchange particles called gluons ▪ Gluon is responsible for the binding of protons and neutrons into nuclei, and holds the nucleons together ▪ Force mediated by gluons is the strongest of all the fundamental interactions; it has a very short range (the range of force is about 10À15 m) ▪ A gluon carries one color quark and one anticolor quark ▪ Totally color-neutral gluon such as rr, bb, and gg are not possible This is similar to electric charges: like colors repel, and opposite colors attract  Photon ▪ A photon is the exchange boson that mediates the electromagnetic interaction between electrically charged particles ▪ It is massless and has charge of zero ▪ The electromagnetic interaction is responsible for (1) the attraction of unlike charges and the repulsion of the like charges (2) binding of atoms and molecules ▪ Fig 1.24 represent two electrons repelling each other through the exchange of a photon Because the momentum is conserved, the two electrons change directions ▪ The electromagnetic interaction is a long-range interaction and proportional to 1/r2 Particle Wave Duality Chapter 31 Photon Time e– e– g e– e– FIG 1.24 One electron emits a photon and the other electron absorbs a photon ▪ It is about 10À2 times the strength of the strong interaction at the nuclei  Weak interaction, W Ỉ ▪ Quarks and leptons can interact by a so-called weak interaction ▪ This weak interaction is mediated by particles called W+ or WÀ ▪ W Ỉ carries an electric charge of +1 and À1 and couples to electromagnetic interaction ▪ In this interaction, one quark type is changed into another by the emission of a lepton and neutrino: d ! u + eÀ + v |{z} |{z} |fflfflffl{zfflfflffl}e WÀ À e + e 3 n ! p + eÀ + v |{z} |{z} |fflfflffl{zfflfflffl}e 0e +1e WÀ In the above reactions, d and n emit WÀ boson, which decays to an electron, eÀ, and an antielectronneutrino, ve ▪ It is a short-range nuclear interaction (with a range of force of about 10À18m) ▪ It is involved in β-decay  Z boson ▪ The Z boson permits uncharged elementary particles to participate in weak interactions ▪ The mediator of this interaction needs to be a zero-charged boson, the Z boson ▪ The Z boson decays into an electron, eÀ, and a positron, e+ This provides a clear signature of the Z boson  Higgs boson ▪ The Higgs boson couples to all elementary particles to give them mass This includes the masses of the W Ỉ and Z boson, and the masses of fermions, i.e., the quark and leptons  Graviton ▪ The gravitational interaction is mediated by gravitons ▪ The gravitational force has infinite range ▪ The exchange bosons have a mass of zero What are the main laws that must hold for any reaction or decay? l All physical process have to obey the conservation laws of:  charge  energy  momentum  angular momentum 32 l Electrons, Atoms, and Molecules in Inorganic Chemistry In addition to all these, two fundamental conservation laws must hold:  The total lepton number, lnet: Number of lepton minus the number of antilepton is constant in time lnet ¼ nl À n1 ¼ constant For example, n ! p + |{z} e + ve |{z} lepton antilepton |fflffl{zfflffl} ;lnet ¼ leptonðeÞ À antileptonðνe Þ ¼  The net quark number, Qnet: The difference between the total number of quarks and the total number of antiquarks is constant in time Qnet ¼ nq À nq ¼ constant The conservation law of a net quark number is more often expressed as the law of baryon number conservation  The law of baryon number conservation: The difference between the number of baryons and the number of antibaryons is constant in time SUGGESTIONS FOR FURTHER READING General R.S Berry, V—Atomic orbitals, J Chem Educ 43 (1966) 283 Cathode Rays and Atomic Constituents E Goldstein, Preliminary communications on electric discharges in rarefied gases, Monthly Reports of the Royal Prussian Academy of Science in Berlin, 1876, p 279 J.F Keithley, The Story of Electrical and Magnetic Measurements: From 500 B.C to the 1940s, John Wiley and Sons, New York, NY, 1999 ISBN: 0-7803-1193-0, p 205 M Faraday, VIII Experimental researches in electricity Thirteenth series, Philos Trans R Soc Lond 128 (1838) 125 J.J Thomson, On bodies smaller than atoms, PSM (August) (1901) 323 Charge and Mass of the Electron R.A Millikan, F Harvey, Elektrizit€atsmengen, Phys Zeit 10 (1910) 308 R.A Millikan, On the elementary electric charge and the Avogadro constant, Phys Rev (2) (1913) 109 F.P Michael, Remembering the oil drop experiment, Phys Today 60 (5) (2007) 56 F Harvey, My work with millikan on the oil-drop experiment, Phys Today 43 (June) (1982) A Franklin, Millikan’s oil-drop experiments, Chem Educ (1) (1997) D Goodstein, In defense of Robert Andrews Millikan, Eng Sci 63 (4) (2000) 30 J.J Thomson, Cathode rays, Philos Mag 44 (1897) 293 J.J Thomson, Rays of Positive Electricity, Proc R Soc A 89 (1913) H.A Boorse, L Motz, The World of the Atom, vol 1, Basic Books, New York, NY, 1966 Rutherford’s Atomic Model E.E Salpeter, Models and modelers of hydrogen, in: L Akhlesh (Ed.), Am J Phys., 65, 9, 1996, ISBN: 981-02-2302-1, p 933 E Rutherford, The scattering of α and β particles by matter and the structure of the atom, Philos Mag 21 (1911) Series Quantum Radiation M Planck, On the Theory of the Energy Distribution Law of the Normal Spectrum, Verhandl Deut Ges Phys (1900) 237 A Einstein, Does the Inertia of a Body Depend on its Energy Content? 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Werner Heisenberg’s Uncertainty Principle, the Uncertainty Principle or the Principle of Indeterminacy W Heisenberg, Uber den anschauclichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z Phys 43 (1927) 172 N Bohr, Atomic Physics and Human Knowledge, John Wiley and Sons, New York, NY, 1958 p 38ff W Kauzmann, Quantum Chemistry, Academic Press, New York, NY, 1957 (Chapter 7) W Sherwin, Introduction to Quantum Mechanics, Holt, Rinehart & Winston, New York, NY, 1959 p 130 D.C Cassidy, Uncertainty: The Life and Science of Werner Heisenberg, W H Freeman & Co, New York, NY, 1992 Subatomic Particles S Braibant, G Giacomelli, M Spurio, Particles and Fundamental Interaction: An Introduction to Particle Physics, Springer, New York, NY, 2009 ISBN: 978-94-007-2463-1 K Nakamura, Review of Particle Physics, J Phys G: Nucl Phys 37 (7A) (2010) 075021 (1422pp) W Bauer, G.D Westfall, University Physics With Modern Physics, McGraw-Hill Companies, New York, NY, 2011 ISBN: 978-007-131366-7 R.A Serway, J.S Faughn, Holt Physics, Holt, Rinehart and Winston, Austin, 2002 ISBN: 0-03-056544-8 F Close, Particle Physics: A Very Short Introduction, Oxford University Press, Oxford, 2004 ISBN: 0-19-280434-0 D.J Griffiths, Introduction to Elementary Particles, John Wiley & Sons, Weinheim, 1987 ISBN: 0-471-60386-4 G.L Kane, Modern Elementary Particle Physics, Perseus Books, Reading, MA, United States, 1987 ISBN: 0-201-11749-5 O Boyarkin, Advanced Particle Physics Two-Volume Set, CRC Press, Taylor & Francis Group, Boca, Raton, London, New York, 2011 ISBN: 978-14398-0412-4 .. .Electrons, Atoms, and Molecules in Inorganic Chemistry Electrons, Atoms, and Molecules in Inorganic Chemistry A Worked Examples Approach Joseph J Stephanos Anthony W Addison Academic Press... of Hard and Soft Ions, and Classification Rule of Interactions Hard-Hard and Soft-Soft Interactions Hard-Soft Interaction and Anion Polarizabi Iity Chelate Effect Entropy and Chelate Formation... 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