CHEMICAL BONDING IN SOLIDS AND FLUIDS

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X- ''"V 7

Dr MARK LADD Department of Chemistry, University of Surrey

m ELLIS HORWOOD NEW YORK LONDON TORONTO SYDNEY TOKYO SINGAPORE

Table of contents

Preface xiii

Physical constants and other numerical data xv

Symbol notation xvii

1 Preamble 1

1.1 Atomic nature of matter 1 1.2 Types of interatomic bonding 3

1.2.1 Covalent bonding 3 1.2.2 Ionic bonding 3 1.2.3 Van der Waals' bonding 4

1.2.3.1 Hydrogen-bonding 4 1.2.4 Metallic bonding 5

1.3 States of matter 6 1.3.1 Equipartition and energy states 7 1.3.2 Boltzmann distribution 8 1.3.3 Solid State 8 1.3.4 Liquid State 9 1.3.5 Gaseous State 10

1.3.5.1 Molar arithmetic 10 1.3.6 Other forms of matter 12

1.3.6.1 Glasses 12 1.3.6.2 Liquid crystals 14 1.3.6.3 Solutions 16 1.3.6.4 Colloids 16

Problems for Chapter 1 17

2 Bonding between atoms 18 2.1 Introduction 18 2.2 Classical mechanics—success and failure 19

2.2.1 Black-body radiation 20 2.2.2 Photoelectric effect 24 2.2.3 Compton effect 25 2.2.4 Diffraction of electrons 25

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2.2.5 Wave-particle duality 26 2.2.6 Atomic spectra 27

2.3 Wave equation 29 2.3.1 Born interpretation of the wave equation 30 2.3.2 Uncertainty principle 31 2.3.3 Normalization and quantization 32

2.4 Particle in a box—quantization of translational energy 33 2.4.1 Tunnelling 35 2.4.2 Boxes of higher dimensions 36

2.5 Vibrational and rotational motion 38 2.5.1 Vibrational motion 38 2.5.2 Rotational motion in two dimensions 40 2.5.3 Rotational motion in three dimensions 40 2.5.4 Space and spin quantization 41 2.5.5 Quantum numbers 42

2.6 Structure of the hydrogen atom 43 2.6.1 Atomic orbitals 44 2.6.2 Orbital terminology 48 2.6.3 Selection rules for atoms 50 2.6.4 Atoms with more than one electron 51 2.6.5 Effective atomic number—Screening . . . : 51

2.6.5.1 Slater's rules 51 2.6.6 Aufbau principle 53 2.6.7 Ionization energy 53

2.7 Simple molecules— theoretical principles 59 2.7.1 Variation method 60 2.7.2 Linear combination of atomic orbitals 62

2.7.2.1 Overlap integrals 63 2.7.2.2 Coulomb integrals 64

2.7.3 Orthogonality 64 2.8 Simple molecules - structures. . 65

2.8.1 Molecular orbital method 65 2.8.1.1 Bonding and antibonding orbitals 68

2.8.2 Diatomic molecules 69 2.8.2.1 Homonuclear diatomics 70 2.8.2.2 Symmetry of orbitals 71 2.8.2.3 Heteronuclear diatomics 74 2.8.2.4 Hybridization 74

2.9 Polyatomic molecules 75 2.9.1 H 2 0 molecule—directed bonds 77 2.9.2 Methane 79 2.9.3 Delocalized Systems 80

2.9.3.1 Ethene and ethyne 80 2.10 Hückel molecular orbital theory 82

2.10.1 Delocalization energy 84 2.10.2 Butadiene wave functions 84 2.10.3 % Bond order 85 2.10.4 Free valence 86

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2.10.5 Aromatic Systems 88 2.10.6 Charge distributions 88

2.10.6.1 Aniline and pyridine 90 2.11 Valence-bond approximation 91

2.11.1 Valence-bond model 91 2.11.2 Hydrogen molecule 91

2.11.2.1 Spinfactor 92 2.11.3 Hydrogen fluoride 93 2.11.4 Electronegativity 94

2.12 MO and VB modeis compared 95 2.12.1 VB and MO modeis with homonuclear diatomics 96 2.12.2 Electron density 97

2.13 Ligand-field theory 97 2.14 Apparently abnormal valence 101 2.15 Valence-shell electron-pair repulsion theory 101 2.16 Covalent states of matter 103

2.16.1 Covalent solids 104 2.16.2 Structural and physical properties of the covalent bond . . . 104

2.17 Semi-empirical and ab initio molecular orbital theories 105 Problems for Chapter 2 106

3 Bonding between molecules 109 3.1 Gases 109

3.1.1 Kinetic theory of gases 109 3.1.1.1 Pressure exerted by a gas 109 3.1.1.2 Equipartition 111 3.1.1.3 Mixtures of ideal gases 111 3.1.1.4 Maxwell-Boltzmann distribution of velocities . . . . 112 3.1.1.5 Maxwell-Boltzmann distribution of speeds 114

3.1.2 Imperfect gases 116 3.1.2.1. Van der Waals'equation of State 118 3.1.2.2 Comparing gases 118

3.1.3 Intermolecular attraction 120 3.1.3.1 Molecule volume effect 120 3.1.3.2 Collision frequency 121 3.1.3.3 Mean free path 122

3.1.4 Origin of intermolecular forces 123 3.1.4.1 Electric moments 123 3.1.4.2 Polarizability 123 3.1.4.3 Ion-dipole attraction 124 3.1.4.4 Dipole-dipole interaction 124 3.1.4.5 Dipole-induced dipole interaction 128 3.1.4.6 Induced dipole-induced dipole interaction 129 3.1.4.7 Pairwise additivity 130

3.1.5 Intermolecular potentials 130 3.1.5.1 Lennard-Jones 12-6 potential function 133

3.1.6 Gas viscosity 134

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3.2 Liquids 137 3.2.1 Liquid-gas equilibrium 137 3.2.2 Some thermodynamics of equilibrium 139 3.2.3 Radial distribution function 140

3.2.3.1 Diffraction studies 140 3.2.3.2 Solids and fluids 142

3.2.4 Equation of state for a fluid 144 3.2.5 Monte Carlo method 146 3.2.6 Molecular dynamics 147 3.2.7 Computer Simulation of liquid water 149 3.2.8 Useof modeis 150 3.2.9 Viscosity 152

3.2.9.1 Liquid viscosity by molecular dynamics 152 3.2.10 Quantum liquids 153 3.2.11 Amorphous solids 154

3.3 Molecular solids 156 3.3.1 Packing of molecules 156 3.3.2 Classification of molecular solids 157

3.3.2.1 Noble gases 157 3.3.2.2 Elements 158 3.3.2.3 Small inorganic molecules 160 3.3.2.4 Organic Compounds 161 3.3.2.5 Classification of organic Compounds 165

3.3.3 Structural and physical characteristics of molecular Compounds 165 3.3.3.1 Solubility of molecular Compounds 170

3.3.4 Molecular mechanics 170 3.3.4.1 Some types of procedure 171 3.3.4.2 Importance of molecular mechanics 173

Problems for Chapter 3 174

4 Bonding between ions: 1 177 4.1 Introduction 177 4.2 Attractive energy 177

4.2.1 Madelung constant 178 4.3 Lattice energy—electrostatic model 180 4.4 Lattice energy—thermodynamic model 183

4.4.1 Precision of the thermodynamic lattice energy 186 4.4.1.1 Enthalpy of Sublimation 186 4.4.1.2 Ionization energy 187 4.4.1.3 Dissociation enthalpy 188 4.4.1.4 Electron affinity 188 4.4.1.5 Enthalpy of formation 191

4.5 Polarization in ionic Compounds—precision of the electrostatic model for lattice energy 192 4.5.1 Approximate calculation of lattice energy 193

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4.6 Uses of lattice energies 194 4.6.1 Electron affinities and thermodynamic parameters 194 4.6.2 Compound stability 194 4.6.3 Charge distribution on polyatomic ions 195

4.7 Aspects of crystal chemistry 198 4.7.1 Ionic radii 200 4.7.2 Radius ratio and AX structure types 203 4.7.3 Radius ratio and AX2 structure types 206 4.7.4 Polarization in AX and AX2 structures 207

4.8 Structural and physical characteristics of ionic Compounds 209 4.9 Solubility of ionic Compounds 211

4.9.1 Reference states for solubility 211 4.9.2 Solubility relationships 212 4.9.3 Two example calculations 212

4.9.3.1 Silveriodide 212 4.9.3.2 Lithium fluoride 212

4.9.4 Solubility and energy 213 4.10 Vibrations and defects in ionic Compounds 216

4.10.1 Absorption spectra 216 4.10.2 Heat capacity 217 4.10.3 Defects in crystals 218

4.10.3.1 Schottky defect 218 4.10.3.2 Frenkel defect 220 4.10.3.3 Doping and colour centres 221

4.10.4 Defects and ion mobility 221 4.11 Moltensalts 222 Problems for Chapter 4 223

5 Bonding between ions: II 227 5.1 Introduction 227 5.2 Classical free-electron theory 227

5.2.1 Electrical conductivity 228 5.2.2 Heat capacity 229

5.2.2.1 Classical particles 230 5.2.2.2 Classical solids 230 5.2.2.3 Einstein and Debye solids 232 5.2.2.4 Heat capacity paradox 233

5.3 Wave-mechanical free-electron theory 233 5.3.1 Density of states 236 5.3.2 Fermi-Dirac distribution 238

5.3.2.1 Mean energy of electrons 240 5.3.2.2 Electronic heat capacity 240

5.4 Band theory 241 5.4.1 Energy bands and Brillouin zones 242 5.4.2 Energy bands and molecular orbital theory 245

5.4.2.1 Occupation of Orbitals 246 5.4.2.2 Semiconductors and insulators 247

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5.5 Structures of metals 249 5.5.1 Metallic radii 251 5.5.2 Interstitial sites 252

5.6 Structural and physical characteristics of metallic Compounds . . . . 252 5.7 Superconductivity 255

5.7.1 Simple theory of superconductivity 255 5.7.2 Thermal properties below the critical temperature 257 5.7.3 Superconduction in organic Compounds 257

5.8 Alloys 259 5.8.1 Copper-gold System 259 5.8.2 Silver-cadmium System 261

5.8.2.1 Hume-Rothery rules 262 Problems for Chapter 5 264

Appendix 1 Problem solving and Computers 266 ALI Introduction 266 A1.2 Solving numerical problems 266

Al.2.1 Approach to problems 267 Al.2.2 Suggested procedure 267 Al.2.3 Solving an example problem 268 Al.2.4 Solution 268

A1.3 Computer methods 270 AI.3.1 Program design and availability 270 Al.3.2 Linear least Squares 270 Al.3.3 Gaussian quadrature 270 Al.3.4 Madelung constant 270 Al.3.5 Radial wave functions 271 Al.3.6 Maxwell-Boltzmann distribution 271 Al.3.7 Electron-in-a-box 271 Al.3.8 Monte Carlo calculations 271 Al.3.9 Angular wave functions 272 Al.3.10 Eigenvalue problems 273

Al.3.10.1 Eigenvalues and eigenvectors 273 Al.3.10.2 Diagonal matrices 274 AI.3.10.3 Jacobi diagonalization 275 Al.3.10.4 General matrix diagonalization 277

A 1.3.11 Hückel molecular orbital calculations 277 AI.3.12 Point-group recognition 278 A 1.3.13 Curve fitting and interpolation 281

Problems for Appendix 1 281

Appendix 2 Stereoviewing 283

Appendix 3 The hypsometric formula—an example of the Boltzmann distribution 284

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Appendix 4 Gamma function 286

Appendix 5 Spherical polar coordinates 287 A5.1 Coordinates 287 A5.2 Volume dement, dr 287 A5.3 Laplacian Operator V2 287

Appendix 6 Reduced mass 290

Appendix 7 Solution of a second-order dilFerential equation 292

Appendix 8 Separation of variables 294

Appendix 9 Least-squares line and propagation of errors 295

A9.1 Propagation of errors 296

Appendix 10 Overlap integrals 299

Appendix 11 Equation of State for a solid 300

Appendix 12 Numerical integration 302

A12.1 Numerical and other methods 302 A12.1.1 Simpson's rule 302 A12.1.2 Direct weighing method 303

Appendix 13 Calculation of the Madelung constant 305

Appendix 14 Some thermodynamics of Solutions 307 A14.1 Partial molar volume 307 A14.2 Partial molar entropy 308 A14.3 Measurement of the partial molar entropy of an ion 308 AI4.4 Generalized description of partial molar quantities 309 A14.5 Partial molar enthalpy in Solutions 310 A14.6 Standard State for Solutions 311

Appendix 15 Debye-Hückel limiting law 314

Appendix 16 Average classical thermal energies 316 A16.1 Average kinetic energy 316 A16.2 Average vibrational energy 316

Appendix 17 Fermi-Dirac statistics 318

Solutions to problems 321

Index 340