Principles of Physical Chemistry (eBook)

Cover 1
Front Cover 2
Title Page 2
Copyright 7
Contents 8
Preface Third Edition 28
Acknowledgement 30
List of Symbols 32
About the Companion Website 34
Introduction 36
Chapter 1 Wave–Particle Duality 38
1.1 Overview of Quantum Mechanics 38
1.1.1 Historical Highlights 38
1.1.2 An Approach to Quantum Mechanics 40
1.2 Light 40
1.2.1 Particle Nature of Light: Photoelectric Effect 40
1.2.2 Wave Nature of Light: Diffraction 41
1.2.2.1 Water Surface Waves 41
1.2.2.2 Light Waves 42
1.2.2.3 Wavelength of Light from Interference Pattern 43
1.2.2.4 Relation Between Photon Energy E and Wavelength ? 43
1.2.2.5 Evaluation of Work Function Ework 44
1.2.3 Interpretation of the Experiments 44
1.3 Electrons 45
1.3.1 Particle Nature of Electrons 45
1.3.2 Wave Nature of Electrons 45
1.3.2.1 Diffraction Experiments 45
1.3.2.2 Relation Between the Electron's Speed v and Wavelength ? 46
1.3.3 Interpretation of the Experiments 46
1.3.4 Formal Similarity Between Electron and Photon 47
1.4 Questions Arising About Wave–Particle Duality 47
1.4.1 Single Event: Probability Statement Collective Behavior: Definite Statement
1.4.2 Wave–Particle Duality and the Need to Abandon Familiar Ways of Thinking 48
1.4.2.1 You Cannot Trace the Path of a Photon or an Electron 48
1.4.2.2 Intuitive Aids Assist You to Grasp the Mathematical Formalism Describing Experiments with Photons or Electrons 48
1.4.2.3 Reflect on the Development of our Everyday View to Understand the Strangeness of the Behavior of Photons or Electrons 48
1.4.2.4 Evolutionary and Revolutionary Approach to Quantum Mechanics 49
1.5 Conclusion 49
1.6 Exercises 50
1.7 Problems 51
Chapter 2 Essential Aspects of Structure and Bonding 54
2.1 Introduction 54
2.2 Distinct Energy States 55
2.2.1 Atomic Spectra 55
2.2.2 Franck–Hertz Experiment 56
2.3 Standing Waves 57
2.3.1 Particle Between Parallel Walls 57
2.3.1.1 Vibrating String 57
2.3.1.2 Wavefunction ?n(x) and Kinetic Energy Tn (Potential Energy V& equals
2.3.1.3 Electron Cannot Be at Rest: Possible Energies and Probability Distributions 59
2.3.2 Ill?Posed Questions 59
2.3.3 Electron in a Cubic Box 60
2.4 Ground?State H Atom 60
2.4.1 Result of Rigorous Treatment (See Chapter ) 61
2.4.2 Box Model for H Atom and the Variational Principle 62
2.4.2.1 Average Potential Energy 62
2.4.2.2 Average Kinetic Energy 62
2.4.2.3 Minimization of Total Energy 63
2.4.2.4 Electron Speed and Electron Described as a Cloud 64
2.5 Ground?State H2+ 64
2.5.1 Forming H2+ from an H Atom and a Proton 64
2.5.2 Box Trial Functions for H2+ 65
2.6 Conclusion 66
2.7 Box 2.1. H Atom: Energy for Particle in a Box Trial Function 66
2.8 Box 2.2. H2+ Ground?State Energy from Particle in a Box Trial Functions 67
2.9 Justifications 68
2.9.1 Justification 2.1: Particle in a Box Trial Function for the H Atom 68
2.10 Exercises 69
2.11 Problems 70
References 72
Chapter 3 Schrödinger Equation 73
3.1 Introduction 73
3.2 Wave Equation and Schrödinger Equation 73
3.2.1 Wave Equation 73
3.2.1.1 Vibrating String 74
3.2.1.2 Electron Between Two Walls 75
3.2.2 One?Dimensional Schrödinger Equation 76
3.2.2.1 Potential Energy Well with Walls of Finite Height 76
3.2.2.2 Curvature of ? 77
3.2.2.3 Tunneling 77
3.2.2.4 Tunneling Probability 77
3.2.2.5 Scanning Tunneling Microscope (STM) 79
3.2.3 Three?Dimensional Schrödinger Equation 80
3.2.3.1 Particle in a Three?Dimensional Box 81
3.3 Normalized Wavefunction: ?& equals
3.3.1 One?Dimensional Box Function 83
3.3.2 Three?Dimensional Box Function 84
3.4 Orthogonality of the Wavefunctions 84
3.4.1 Non?Degenerate Wavefunctions 84
3.4.2 Degenerate Wavefunctions 86
3.4.3 Degeneracy Removed in an Electric Field 87
3.5 Bohr Correspondence Principle and Generalized Form of Quantum Mechanics 87
3.5.1 Bohr Correspondence Principle 88
3.5.1.1 Hamilton's Function 88
3.5.1.2 Operators pop and xop 88
3.5.1.3 Hamiltonian Operator and Schrödinger Equation 88
3.5.2 Electron Moving on a Circle 88
3.5.2.1 Representation by Real Wavefunctions 89
3.5.2.2 Representation by Complex Wavefunction: Clockwise and Counterclockwise Rotation 89
3.5.2.3 General Solution and Connection to Angular Momentum 90
3.5.3 Operator i& plankv
3.5.3.1 Relation Between Time?Independent and Time?Dependent Schrödinger Equation 92
3.5.4 Average Value of an Observable (also Called Expectation Value) 93
3.5.5 Heisenberg Uncertainty Principle 93
3.5.5.1 Superposition and the Uncertainty Principle 94
3.6 Summary of Postulates of Quantum Mechanics 95
3.7 Conclusion 96
3.8 Box 3.1. Electron in a Potential Energy Well of Finite Depth 96
3.8.1 Symmetric Solutions 97
3.8.2 Antisymmetric Solutions 97
3.8.3 Numerical Evaluation, Fig. 98
3.9 Foundations 98
3.9.1 Foundation 3.1: Electron in a Potential Energy Well of Finite Depth 98
3.9.2 Foundation 3.2: Hamiltonian Operator and Orthogonality 98
3.9.2.1 Hamiltonian Operator 98
3.9.2.2 Orthogonality 99
3.9.3 Foundation 3.3: Uncertainty Principle for ?x·?px 99
3.9.4 Foundation 3.4: Uncertainty Principle for Two Conjugate Variables 99
3.9.5 Foundation 3.5: Superposition of Wavefunctions: Calculation of Uncertainty Product 99
3.9.6 Foundation 3.6: Tunneling Through a Barrier 99
3.10 Exercises 100
3.11 Problems 101
References 103
Chapter 4 Hydrogen Atom 104
4.1 Introduction 104
4.2 H Atom in the Ground State 104
4.2.1 Wavefunction 105
4.2.2 Energy 105
4.2.3 Radial Probability Distribution of Electron 106
4.2.4 Most Probable Distance and Average Distance 107
4.2.5 Average Potential Energy and Virial Theorem 108
4.3 H Atom in Excited States 109
4.3.1 Energies 109
4.3.2 Wavefunctions 109
4.3.2.1 Nodal Surfaces 110
4.3.3 Radial Probability Distribution and Average Distance 112
4.3.4 Emission Spectra 113
4.3.5 Degeneracy of H?atom Orbitals Can Be Removed by a Magnetic Field 114
4.3.5.1 Representation by Real and Complex Wavefunctions 114
4.3.5.2 Angular Momentum and Energy Splitting in a Magnetic Field 115
4.4 Conclusion 116
4.5 Foundations 116
4.5.1 Foundation 4.1: An Overview of Solution to the Schrödinger Equation for the H?atom 116
4.5.2 Foundation 4.2: Solutions to the Angular and Radial Parts of the H?atom Schrödinger Equation 118
4.5.3 Foundation 4.3: Virial Theorem 118
4.6 Exercises 119
4.7 Problems 120
References 122
Chapter 5 Atoms and Variational Principle 123
5.1 Introduction 123
5.2 Variational Principle 123
5.2.1 Introducing the Variational Principle 123
5.2.2 Justification of the Variational Principle 124
5.2.2.1 Trial Function for the Ground State of an Atom 124
5.2.2.2 Trial Functions for Excited States 126
5.3 He?Atom 127
5.3.1 Ground State of He 127
5.3.1.1 Product of Atomic Wavefunctions Used as Trial Functions 127
5.3.1.2 Self?Consistent Field Method 128
5.3.2 Indistinguishability of Electrons 128
5.3.3 Excited States of He Described by the Product of One?Electron Trial Functions 129
5.4 Electron Spin and Antisymmetry Postulate 130
5.4.1 Energy Splitting in a Magnetic Field, Electron Paramagnetic Resonance 130
5.4.2 Spin Variables 131
5.4.3 Antisymmetry Postulate and Pauli Exclusion Principle 132
5.4.3.1 Electrons are Described by Antisymmetric Wavefunctions 132
5.4.3.2 Antisymmetric Wavefunction for the Ground State of Helium 132
5.4.3.3 Antisymmetric Wavefunctions for the Excited States of Helium 133
5.4.4 Single?Triplet Splitting—Neglecting Magnetic Forces 134
5.5 Many?Electron Atoms 134
5.5.1 Atomic Number Z and Ionization Energy 134
5.5.2 Electronic Structure of the First Three Elements 135
5.5.2.1 Hydrogen Atom 135
5.5.2.2 Helium Atom and He?Like Systems: H?, Li+, Be 2+ 135
5.5.2.3 Lithium Atom 136
5.5.3 Aufbau Principle and Periodic Table 137
5.5.3.1 Orbital Energies 137
5.5.3.2 Self?Consistent Orbital's Energy 139
5.5.3.3 Aufbau Principle 139
5.5.3.4 Periodic Table of the Elements 140
5.5.4 Periodic Properties of the Elements 141
5.5.4.1 Ionization Energy 141
5.5.4.2 Atomic Radii 141
5.6 An Electron Configuration Can Have Multiple Energy States 142
5.6.1 Angular Momentum and Vector Model of Atoms 142
5.6.1.1 Orbital Angular Momentum 142
5.6.1.2 Spin Angular Momentum 143
5.6.1.3 Total Electron Angular Momentum: Spin–Orbit Coupling 143
5.6.2 Atomic Term Symbols 144
5.7 Conclusion 145
5.8 Box 5.1. He Atom: Calculation of Energy 145
5.8.1 Energy in the Ground State 145
5.8.2 Energy in the Excited State 147
5.9 Box 5.2. Current Loop in a Magnetic Field (Bohr Magneton) 147
5.10 Box 5.3. The Electron Spin Postulate and the Antisymmetry Postulate 148
5.10.1 The Electron Spin Postulate 148
5.10.2 The Antisymmetry Postulate 149
5.11 Box 5.4. Slater Wavefunctions 149
5.12 Justifications 150
5.12.1 Justification 5.1: He?Atom: Repulsion Energy in the Ground State 150
5.12.2 Justification 5.2: He?Atom: Repulsion Energy in the Excited State 150
5.12.3 Justification 5.3: Analytical Evaluation of Coulomb Integral 150
5.13 Foundations 151
5.13.1 Foundation 5.1: Proof of Variational Principle 151
5.13.2 Foundation 5.2: First?Order Perturbation Theory 152
5.13.2.1 Example: Electron in a Potential Energy Well 152
5.13.3 Foundation 5.3: Perturbation Theory (Rigorous Treatment) 155
5.13.4 Foundation 5.4: The Hartree–Fock Self?Consistent Field Approximation 155
5.13.5 Foundation 5.5: Atomic Term Symbols 155
5.14 Exercises 155
5.15 Problems 156
References 159
Chapter 6 A Quantitative View of Chemical Bonding 160
6.1 Introduction 160
6.2 H2+ Molecule Ion 161
6.2.1 Electron Described by Exact Wavefunction 161
6.2.1.1 Electronic Energy as a Function of Distance d 162
6.2.1.2 Equilibrium Bond Length 162
6.2.2 Electron Density and the Chemical Bond 162
6.2.2.1 Compression of Electron Cloud 162
6.2.2.2 Coulomb Attraction and Virial Theorem 163
6.2.3 Electron Described by LCAO Wavefunction 163
6.2.3.1 Procedure 164
6.2.3.2 Results 165
6.2.3.3 Failures 165
6.2.3.4 Modifications 165
6.2.4 Comparison of Box Model (Chapter ) and LCAO Method 166
6.2.5 Excited State of H2+: Bonding and Antibonding Orbitals 167
6.3 H2: A Two?Electron System 168
6.3.1 Rigorous Treatment 168
6.3.1.1 Procedure 168
6.3.1.2 Results 169
6.3.2 Product of One?Electron Wavefunctions 169
6.3.2.1 Procedure 169
6.3.2.2 Results 170
6.3.3 Indistinguishability of Electrons 171
6.3.4 Pauli Exclusion Principle: Electronic Wavefunctions Must be Antisymmetric 172
6.4 Tunneling 173
6.4.1 Electron Oscillating Between Protons a and b 173
6.4.2 Tunneling Barrier and Frequency of Oscillation 173
6.4.3 Importance of Tunneling in Redox Chemistry 174
6.5 Conclusion 174
6.6 Box 6.1. Estimation of Evasion Energy (H2 Ground State) 174
6.7 Foundations 175
6.7.1 Foundation 6.1: H2+ Ion Exact Wavefunction and Energy 175
6.7.2 Foundation 6.2: Evaluation of LCAO Integrals in H2+ 175
6.7.3 Foundation 6.3: Oscillation of Electron Between Protons at Distance d (Tunneling) 175
6.8 Exercises 175
6.9 Problems 176
References 178
Chapter 7 Bonding Described by Electron Pairs and Molecular Orbitals 179
7.1 Introduction 179
7.2 Electron Pair Bonds (A Review) 179
7.2.1 Connectivity of Atoms in Three?Atom Molecules 180
7.2.1.1 Double and Triple Bonds 180
7.2.1.2 Connectivity 180
7.2.1.3 Formal Charge 181
7.2.1.4 Dative Bond 181
7.2.1.5 Effective Charge 181
7.2.2 Geometries for Electron Pair Bonds 182
7.2.2.1 VSEPR?Model 182
7.2.2.2 Hybrid Atomic Orbitals 183
7.2.3 Limitations of Electron Pair Model 186
7.3 Molecular Orbitals 187
7.3.1 Homonuclear Diatomics 187
7.3.1.1 Linear Combination of Atomic Orbitals (LCAO) 187
7.3.1.2 Particle?in?Box Wavefunction Description for O2 189
7.3.1.3 The Second Row Diatomics 190
7.3.2 Heteronuclear Diatomics 192
7.3.2.1 Bonding in the HF Molecule 192
7.3.3 Triatomics (H2O) 193
7.3.3.1 Structure and Bonding in H2O 193
7.3.3.2 Geometry 195
7.3.4 Polyatomics 196
7.3.4.1 Divide and Conquer Strategy 196
7.3.4.2 Functional groups 197
7.4 Quantum Chemistry Extension: Orbital?Based Numerical Methods 198
7.5 Polarity, Bond Length, Elasticity 198
7.5.1 Polarity of Bonds and Electronegativity 198
7.5.2 Covalent Radii and Van der Waals Radii 200
7.5.3 Stretching, Bending, and Torsion of Bonds 203
7.5.3.1 Stretching (Hooke's Law) 203
7.5.3.2 Stretching (Morse Potential) 204
7.5.3.3 Bending and Torsion 205
7.6 Conclusion 206
7.7 Justifications 206
7.7.1 Justification 7.1: Tetrahedral Hybrid Function 206
7.8 Foundations 206
7.8.1 Foundation 7.1: LCAO?Treatment of Heteronuclear Diatomic Molecules 206
7.8.2 Foundation 7.2: Hartree–Fock SCF Method for Molecules 209
7.9 Exercises 209
7.10 Problems 210
References 212
Chapter 8 Molecules with ??Electron Systems 214
8.1 Introduction 214
8.2 Bonding Properties of ? Electrons 215
8.2.1 Sigma?Bonded Molecular Skeleton and ??Electrons 215
8.2.2 Overview of Models for ??Systems 216
8.2.2.1 HMO Model 216
8.2.2.2 BCD Model 216
8.2.2.3 Step Potential Model 216
8.2.2.4 FEMO Model 217
8.3 Free Electron Molecular Orbital (FEMO) Model 218
8.3.1 Ethene, Butadiene, Amidinium. ??Electron Chains 218
8.3.2 Benzene. A ??Electron Ring 219
8.3.3 Charge Density dQ/ds 220
8.3.3.1 Total Probability Density ?total(s) and the Bond Length 222
8.3.4 Branched Molecules 223
8.3.4.1 Energies 224
8.3.4.2 Wavefunctions 224
8.3.4.3 ? Electron Density 225
8.4 BCD (Step Potential) Model Applied to Polyenes and Annulenes 225
8.4.1 Total Probability Density and Bond Length 225
8.4.2 Energy Gap in a Long Chain Polyene 226
8.4.3 Annulenes 226
8.4.4 Neutral Soliton 227
8.5 Principles of Density Functional Theory (DFT) 227
8.6 HMO Model 228
8.6.1 Wavefunctions and Energies 228
8.6.2 Application of the HMO Method to ??Electron Chains 230
8.6.3 Bond Length and HMO Bond Order 231
8.6.4 Extensions of HMO Approach 232
8.7 Resonance 232
8.8 Conclusion 234
8.9 Box 8.1. Self?Consistency in BCD (Step Potential) Method 234
8.9.1 Bond Lengths in Hexatriene 234
8.9.2 Bond Lengths in Benzene 235
8.10 Box 8.2. Neglect of the Overlap Integral Sab in HMO 236
8.11 Foundations 236
8.11.1 Foundation 8.1: Construction of V(s) in Fig. 8.3 236
8.11.2 Foundation 8.2:b Free?Electron Model, Branched Systems 236
8.11.3 Foundation 8.3: Free?Electron Model, Representation by Determinants 237
8.11.4 Foundation 8.4: HMO Model 237
8.11.4.1 Finding the Minimum of & epsiv
8.11.4.2 Solving a System of N Linear Equations with Constant Coefficients 238
8.11.5 Example: Butadiene 239
8.11.6 Example: Benzene 240
8.11.7 Foundation 8.5: Fullerene 240
8.12 Exercises 241
8.13 Problems 243
References 244
Chapter 9 Absorption of Light 245
9.1 Introduction 245
9.2 Excitation of ? Electron Systems 246
9.2.1 Basic Experimental Facts 246
9.2.1.1 Transmittance and Absorbance 246
9.2.1.2 Dye Molecules 247
9.2.2 Absorption Maxima of Dye Molecules 249
9.2.2.1 Band Broadening 249
9.2.2.2 Single?Molecule Absorption 250
9.2.2.3 ?max for Cyanine Dyes 250
9.2.3 Strength and Polarization of Absorption Bands 253
9.2.3.1 Interaction of Light With a Molecule 253
9.2.3.2 Polarization of Absorption Bands 256
9.2.4 Hetero?Atoms as Probes for Electron Distribution 256
9.2.5 HOMO–LUMO Gap by Bond Alternation 259
9.2.6 Cyclic ??Systems 261
9.2.6.1 Transition Energies 261
9.2.6.2 Transition Polarization 263
9.2.7 Coupling of ? Electrons 265
9.3 Optical Activity 267
9.3.1 Rotatory Dispersion 267
9.3.2 Ellipticity and Circular Dichroism 269
9.3.3 Anisotropy Factor g for Model in Fig. 9.25 270
9.3.4 Absolute Configuration of Chiral Molecules 270
9.3.4.1 Circular Dichroism of Spirobisanthracene 271
9.3.4.2 Circular Dichroism of Chiral Cyanine Dye 272
9.3.5 Summary 273
9.4 Conclusion 273
9.5 Box 9.1. Lambert–Beer Law 274
9.6 Box 9.2. Relation Between Ellipticity ? and ?& epsiv
9.7 Justifications 275
9.7.1 Justification 9.1: Derivation of Equation (9.63) in Section 9.3.3 275
9.8 Foundations 276
9.8.1 Foundation 9.1: Integrated Absorption: Classical Oscillator 276
9.8.1.1 Absorbed Power in a Radiation Beam and Absorption Coefficient & epsiv
9.8.1.2 Calculating the Absorption Coefficient & epsiv
9.8.1.3 Resonance Curve for & epsiv
9.8.2 Integrated Absorption Power 279
9.8.3 Derivation of Equation (9.77) for the Intensity I 280
9.8.4 Foundation 9.2: Oscillator Strength in the Quantum Mechanical Treatment 281
9.8.4.1 Quantum Mechanical Expression for f 281
9.8.5 Foundation 9.3: Classical and Quantum Mechanical Description of Light Absorption 283
9.8.5.1 Light Absorption Calculated from the Time?Dependent Schrödinger Equation 283
9.8.5.2 Power Absorbed by the Molecule 285
9.8.5.3 Classical Oscillator 286
9.8.5.4 Equivalence of Classical and Quantum Mechanical Treatment 286
9.8.6 Foundation 9.4: Coupling Transitions with Parallel Transition Moments 287
9.8.6.1 Replacing Quantum Mechanical System by Coupled Classical Oscillators 287
9.8.6.2 Doubly Occupied Orbitals 288
9.8.7 Foundation 9.5: Normal Modes of Coupled Oscillators 289
9.8.7.1 Resonance Frequencies of Two Coupled Oscillators 289
9.8.7.2 Conversion of Resonance Frequencies into Excitation Energies 289
9.8.7.3 Oscillator Strength of Two Coupled Oscillators 290
9.8.8 Foundation 9.6: Cyanines in the FEMO and Step Model 291
9.9 Exercises 291
9.10 Problems 294
References 298
Chapter 10 Emission of Light 299
10.1 Introduction 299
10.2 Spontaneous Emission 299
10.2.1 Fluorescence 300
10.2.1.1 Einstein Theory of Emission and Absorption of Radiation 301
10.2.2 Single Molecule Emission 302
10.2.3 Fluorescence Optical Microscopy 302
10.2.4 Phosphorescence and Triplet States 303
10.2.4.1 Energy Shift 303
10.2.4.2 Lifetime 305
10.2.5 Relative Energetics Fluorescence, Phosphorescence, and Absorption 305
10.2.5.1 Shift of Wavelengths 305
10.2.5.2 Jablonski?Diagram 306
10.2.6 Quenching of Fluorescence 306
10.2.7 Absorption from Excited States 307
10.2.7.1 Anthracene: T1?T2 Absorption 308
10.2.7.2 Anthracene: S1?S2 Absorption 309
10.3 Stimulated Emission and Laser Action 309
10.3.1 Inversion of Population 309
10.3.2 Dye Laser Operation 310
10.3.2.1 Pumping 310
10.3.2.2 Laser Action 311
10.3.2.3 Tuning 311
10.3.3 Excimer and Exciplex Laser 312
10.4 Conclusion 312
10.5 Box 10.1. Einstein Coefficients 312
10.6 Foundations 313
10.6.1 Foundation 10.1: Fluorescence Lifetime 313
10.6.1.1 Antenna Equation of Hertz 313
10.6.1.2 Natural Lifetime 314
10.6.1.3 Lifetime and Line Width 314
10.6.2 Foundation 10.2: Calculation of Repulsion Integrals 315
10.6.3 Foundation 10.3: Fluorescence Microscopy: STED, MINSTED 315
10.7 Exercises 315
10.8 Problems 317
References 317
Chapter 11 Nuclei: Particle and Wave Properties 319
11.1 Introduction 319
11.2 Rotational Motion of Molecules 320
11.2.1 Diatomic Quantum?Mechanical Rotator 320
11.2.1.1 Simplified Model 320
11.2.1.2 Rigorous Solution 321
11.2.2 Polyatomic Molecules 322
11.2.2.1 Linear Polyatomic Molecules 322
11.2.2.2 Nonlinear Polyatomic Molecules 323
11.2.3 Rotational Spectra 324
11.2.3.1 Spectra of Diatomic Molecules 324
11.2.3.2 Spectra of Polyatomic Molecules 326
11.3 Vibrational Motion of Molecules 327
11.3.1 Classical Oscillator 327
11.3.2 Quantum?Mechanical Harmonic Oscillator 328
11.3.2.1 Simplified Model 328
11.3.2.2 Rigorous Solution 328
11.3.2.3 Comparison of Quantum?Mechanical Oscillator with Classical Oscillator 329
11.3.3 Vibrational–Rotational Spectra 330
11.3.3.1 Spectra of Diatomic Molecules 331
11.3.3.2 Spectra of Polyatomic Molecules 333
11.3.4 Quantum?Mechanical Anharmonic Oscillator 336
11.3.4.1 Schrödinger Equation for the Anharmonic Oscillator 336
11.3.4.2 Evaluation of kf from Spectroscopic Data 337
11.3.5 Multidimensional Infrared Spectroscopy 339
11.4 Raman Spectra 339
11.4.1 Rayleigh Scattering 339
11.4.2 Raman Spectroscopy 340
11.4.3 Rotational Raman Spectra of Heteronuclear Diatomic Molecules 341
11.4.3.1 Vibrational–Rotational Raman Spectra of Diatomic Molecules 342
11.4.4 Raman Spectra of Polyatomics 344
11.4.4.1 Raman Spectrum of CO2 344
11.4.4.2 Complementarity of IR and Raman Spectra 345
11.4.4.3 IR and Raman Spectra of CH4 345
11.5 Vibrational Structure of Electronic Spectra 346
11.5.1 Franck–Condon Principle 346
11.5.1.1 Types of Vibronic Spectra for Diatomics 347
11.5.2 Photoelectron Spectroscopy 348
11.5.2.1 Evaluation of Bond Energy 349
11.5.3 Polyatomic Molecules 350
11.5.3.1 Vibrational Structure in Naphthalene Spectrum 350
11.5.3.2 Vibrational Structure in Cyanine Dye and Polyene Spectrum 351
11.5.3.3 Jablonski Diagram 351
11.5.3.4 Time Scales of Processes 352
11.5.3.5 Vibrational Dynamics of Anthracene 352
11.6 Conclusion 354
11.7 Box 11.1. Reduced Mass of Diatomic Molecules and Frequency of Vibration 354
11.7.1 Reduced Mass 354
11.7.1.1 Vibration 354
11.7.1.2 Rotation 354
11.7.2 Frequency of Vibration 355
11.8 Box 11.2. Oscillator: Normal Modes of a Triatomic Linear Molecule 355
11.8.1 Reduced Mass ? 355
11.8.1.1 Symmetric Stretching Mode (Fig. 11B.2b) 355
11.8.2 Asymmetric Stretching Mode (Fig. 11B.2c) 356
11.8.2.1 Bending Mode (Fig. 11B.2d) 356
11.8.2.2 Force Constants 356
11.9 Foundations 357
11.9.1 Foundation 11.1: Rigid Rotator Solution of the Schrödinger Equation 357
11.9.2 Foundation 11.2: Harmonic Oscillator: Quantum?Mechanical Treatment 357
11.9.3 Foundation 11.3: Hermitian Polynomials 357
11.9.4 Foundation 11.4: Absorption Spectra, Selection Rules for Rotation of Linear Molecules 357
11.9.4.1 Transition Moment 357
11.9.4.2 Intensity Distribution of Absorption Lines (Rotator in Space) 358
11.9.5 Foundation 11.5: Nonrigid Rotator, Centrifugal Effect on Energy 359
11.9.6 Foundation 11.6: Absorption Spectra, Selection Rules for Vibration 359
11.9.6.1 Diatomic Molecules 359
11.9.6.2 Polyatomic Molecules 360
11.9.7 Foundation 11.7: Raman Spectra, Selection Rules 360
11.9.8 Foundation 11.8: Multidimensional Infrared Spectroscopy 360
11.10 Exercises 361
11.11 Problems 363
References 366
Chapter 12 Nuclear Spin 367
12.1 Introduction 367
12.2 Nuclear Spin: Fundamentals 367
12.2.1 Spin of Protons in H2 367
12.2.2 Antisymmetry of Total Wavefunction of a Molecule Including Electrons and Nuclei 368
12.2.2.1 Ground State of H2: Para?H2 368
12.2.2.2 First Excited Rotational State of H2: Ortho?H2 368
12.2.2.3 No Distinction Between Identical Particles 369
12.2.2.4 Experimental Proof of Indistinguishability by Rotational Raman Spectrum of H2 369
12.2.3 Nuclei with Half?Integer Spin (Antisymmetric Total Wavefunction), Nuclei with Integral Spin (Symmetric Total Wavefunction) 370
12.2.3.1 Fermions and Bosons 370
12.2.3.2 Ortho?D2 and Para?D2 370
12.2.3.3 Raman Spectrum of N2: Alternating Intensities 371
12.2.3.4 Raman Spectrum of O2: Every Second Line Is Missing 371
12.3 Nuclear Magnetic Resonance (NMR) 372
12.3.1 Fundamentals 372
12.3.2 Chemical Shift 374
12.3.3 Fine Structure of NMR Spectra 375
12.3.3.1 Coupling of Two Protons 375
12.3.3.2 Coupling of Three Nonequivalent Protons 376
12.3.3.3 Exchange and Relaxation Effects 377
12.3.4 Nuclear Overhauser Effect (NOE) and 2D Spectroscopy 377
12.3.4.1 Nuclear Overhauser Effect 377
12.3.4.2 NMR Spectroscopy for Determination of Protein Structures in Solution 379
12.3.5 Magnetic Resonance Imaging (MRI) 379
12.4 Conclusion 379
12.5 Justifications 380
12.5.1 Justification 12.1: Raman Spectrum of N2 380
12.5.2 Justification 12.2: Spacing of Raman Lines in O2 and CO2 380
12.5.3 Justification 12.3: Coupling of Two Equivalent Protons 380
12.6 Exercises 380
12.7 Problems 381
References 381
Chapter 13 Solids and Intermolecular Forces 382
13.1 Introduction 382
13.2 Ionic Crystals 383
13.2.1 Bond Energy of an Ion Pair 383
13.2.2 Lattice Types 384
13.3 Metals 386
13.3.1 Free Electron Model for Conduction Electrons 386
13.3.1.1 Particle?in?a?Box Model 387
13.3.1.2 Fermi Energy 387
13.3.1.3 Li Metal: Number g of Electronic States Between E and E+?E 387
13.3.2 Cohesion Energy of Metals 388
13.3.3 Quantum Wires and Nanostructure 390
13.4 Semiconductors 392
13.4.1 Organic Semiconductors: Polyacetylene 392
13.4.2 Silicon 392
13.4.3 Semiconductor Bandgap 393
13.4.3.1 Doping by a Donor 393
13.4.3.2 Doping by Acceptor 393
13.4.3.3 Bandgap Emission 394
13.4.4 Semiconductor Quantum Dots 394
13.5 Molecular Crystals and Intermolecular Forces 395
13.5.1 Electrostatic Forces, the Dipole 396
13.5.2 Hydrogen Bonds 398
13.5.3 Induction Forces 400
13.5.4 Dispersion Forces 401
13.6 Conclusion 403
13.7 Box 13.1. Madelung Constant 403
13.8 Box 13.2. Number and Density of Energy States of a Metal's Electron Gas 404
13.9 Box 13.3. Fermi Energy and Mean Energy of Electrons in a Metal 405
13.9.1 Fermi Energy 405
13.9.2 Calculation of the Mean Energy E? of the Electron Gas 405
13.10 Box 13.4. Dipole and Induction Energies 405
13.10.1 Dipole Energy (Dipoles at an Angle ?) 406
13.10.2 Dipole in the Field of a Point Charge 406
13.10.3 Electric Field Strength of Dipole 406
13.10.3.1 Case 1: Field Strength at Point P in Distance r Along the Direction of the Dipole (Fig. 13B.6) 406
13.10.3.2 Case 2: Field Strength in Distance r Perpendicular to the Direction of the Dipole (Fig. 13B.7) 406
13.11 Box 13.5. Polarizability of a Conducting Plate and of a Conducting Sphere 407
13.12 Foundations 407
13.12.1 Foundation 13.1: Some Features of Crystal Structures and Lattices 407
13.12.2 Foundation 13.2: Calculation of Dipole and Induction Energies 407
13.13 Exercises 407
13.14 Problems 409
References 412
Chapter 14 Thermal Motion of Molecules 413
14.1 Introduction 413
14.2 Kinetic Gas Theory and Temperature 414
14.2.1 Thermal Motion and Pressure 414
14.2.1.1 Calculation of Pressure 414
14.2.1.2 Comparison with Experiment 416
14.2.2 Avogadro's Law 417
14.2.3 Thermal Equilibration and Heat 418
14.2.4 Ideal Gas Law and the Definition of Absolute Temperature 419
14.2.4.1 Temperature Measures the Average Translational Energy 419
14.2.4.2 Ideal Gas Law 420
14.2.4.3 Extensive and Intensive Variables, Molar Quantities 420
14.2.4.4 Measuring Temperature 420
14.2.5 Law of Partial Pressures 421
14.3 Speed and Collisions of Gas Molecules 421
14.3.1 Average Speed of Molecules in a Gas 421
14.3.1.1 Effusion 423
14.3.2 Mean Free Path and Number of Collisions 423
14.3.2.1 Mean Free Path 423
14.3.2.2 Number of Collisions 425
14.3.3 Diffusion 425
14.3.3.1 Mean Displacement in the x?Direction (Random Walk Model) 425
14.3.3.2 Mean Square Displacement in Space 427
14.3.3.3 Mean Square Displacement as a Function of Time 428
14.3.3.4 Equation of Einstein and Smoluchowski 429
14.3.3.5 Fick's First Law 431
14.3.3.6 The Influence of Gravity 432
14.3.4 Viscosity Arising from Collisions of Molecules 433
14.3.4.1 Viscous Flow 433
14.3.4.2 Calculating the Viscosity Coefficient 433
14.3.4.3 Dependence of ? on Temperature 434
14.3.4.4 Dependence of ? on Pressure 435
14.3.4.5 Calculating Collision Diameters 435
14.4 Thermal Motion in Liquids 435
14.4.1 Collisions in Liquids 435
14.4.2 Diffusion Coefficient D of a Liquid 436
14.4.3 Viscosity of a Liquid 437
14.4.4 Stokes–Einstein Equation 437
14.5 Conclusion 438
14.6 Box 14.1. Molar Quantities 439
14.7 Box 14.2. Pressure on the Wall of a Container and Number of Collisions with a Wall 439
14.7.1 Pressure on the Wall 439
14.7.2 Number of Collisions with a Wall 440
14.8 Box 14.3. Averaging the Displacement x2 440
14.9 Box 14.4. Averaging the Free Path ? 441
14.10 Box 14.5. Atmospheric Distribution 441
14.11 Box 14.6. Derivation of the Stokes–Einstein Equation 442
14.12 Foundations 443
14.12.1 Foundation 14.1: The Random Walk in Three Dimensions 443
14.12.2 Foundation 14.2: Intermolecular Forces Affecting the Mean Free Path 443
14.12.3 Foundation 14.3: Law of Hagen–Poiseuille 443
14.12.4 Foundation 14.4: Central Collision of Particles 443
14.13 Exercises 444
14.14 Problems 445
References 447
Chapter 15 Energy Distribution in Molecular Assemblies 448
15.1 Introduction 448
15.2 The Boltzmann Distribution Law 449
15.2.1 System Consisting of Two Quantum States 449
15.2.2 Systems Consisting of Many Quantum States 450
15.2.3 Internal Energy U 451
15.3 Electronic Energy 452
15.4 Vibrational Energy 453
15.4.1 Population Number Nn of Harmonic Oscillator 454
15.4.2 Internal Energy Uvib of Diatomic Molecules 455
15.4.3 Internal Energy Uvib of Polyatomic Molecules 456
15.5 Rotational Energy 457
15.5.1 Population Number NJ of Rigid Rotator (Heteronuclear Molecules) 457
15.5.1.1 High Temperature Limit 458
15.5.2 Internal Energy Urot of Rotation (Heteronuclear Molecules) 459
15.5.2.1 Diatomic and Linear Polyatomic Molecules 459
15.5.2.2 Nonlinear Polyatomic Molecules 460
15.5.3 Homonuclear Diatomic Molecules: Ortho? and Para?Hydrogen 460
15.6 Translational Energy 461
15.6.1 Internal Energy Utrans of Translation According to Quantum Mechanics 461
15.6.2 Maxwell–Boltzmann Distribution of Speeds 462
15.6.2.1 Classical Treatment 462
15.6.2.2 Quantum Mechanical Treatment 463
15.6.2.3 Most Probable Speed, Mean Speed, Root Mean Square Speed 465
15.6.2.4 Supersonic Expansion 466
15.7 Characteristic Temperature 466
15.8 Proving the Boltzmann Distribution for Distinguishable Particles 467
15.8.1 Distribution of N Particles Among 3 Levels with Given Internal Energy 467
15.8.1.1 Number of Representations of a System with Given Internal Energy 468
15.8.1.2 Calculating Population Numbers Ni 470
15.8.2 Rigorous Treatment 470
15.8.2.1 How to Calculate the Population Numbers Ni 471
15.9 Proving the Boltzmann Distribution for Indistinguishable Particles (High Temperature) 471
15.9.1 Number of Available Quantum States 471
15.9.2 Number of Representations and Boltzmann Law 472
15.9.3 Calculating the Internal Energy U of an Ideal Atomic Gas 473
15.9.4 Relation Between ? and T 473
15.10 Energy Distribution of Fermions and Bosons Among Quantum States 474
15.10.1 Photons Obey Boson Statistics: Planck's Radiation Law 475
15.11 Conclusion 476
15.12 Box 15.1. Equipartition Principle in the Classical Limit 477
15.12.1 Translational Degrees of Freedom 477
15.12.2 Rotational Degrees of Freedom 477
15.12.3 Vibrational Degrees of Freedom 477
15.12.4 Summary 478
15.13 Box 15.2. Canonical and Microcanonical Ensembles 478
15.14 Justifications 478
15.14.1 Justification 15.1: Molecular Partition Function Ztrans of Translation 478
15.14.2 Justification 15.2: Distinguishable Particles Occupying Degenerate Energy Levels 479
15.15 Foundations 479
15.15.1 Foundation 15.1: Classical Derivation of the One?Dimensional Maxwell–Boltzmann Distribution of Speeds 479
15.15.1.1 Calculation of Constant A 480
15.15.2 Foundation 15.2: How to Find the Maximum of ln? 480
15.15.3 Foundation 15.3: Energy Distribution of Fermions and Bosons 481
15.15.4 Foundation 15.4: Canonical and Microcanonical Ensembles 481
15.15.5 Foundation 15.5: Internal Energy U as Sum of Contributions Uel, Uvib, Urot, and Utrans 483
15.15.6 Foundation 15.6: Planck's Radiation Law 484
15.16 Exercises 484
15.17 Problems 486
References 489
Chapter 16 Work w, Heat q, and Internal Energy U 490
16.1 Introduction 490
16.2 Thermodynamic Systems 491
16.2.1 Defining System and Surroundings 491
16.2.2 Defining Thermodynamic States 491
16.2.3 Change of State 492
16.3 Change of State at Constant Volume (Isochoric Process) 493
16.3.1 Change of Internal Energy ?U and the Heat q 493
16.3.1.1 Atomic Gas 494
16.3.1.2 Diatomic Molecules at Moderate Temperature 494
16.3.1.3 Conservation of Energy 494
16.3.2 Heat Capacity CV: Definition 494
16.3.2.1 Atomic Gas 495
16.3.2.2 Diatomic Molecules at Moderate Temperature 495
16.3.2.3 Generalization (Full Range of Temperature) 495
16.3.3 Translational Contribution to CV of a Gas 495
16.3.4 Rotational and Vibrational Contributions to CV of a Gas 496
16.3.4.1 Diatomic Molecule: HD 496
16.3.4.2 Polyatomic Molecules: CO2 497
16.3.5 Ortho? and Para?H2: Fascinating Quantum Effects on CV 498
16.3.6 Electronic Contribution to CV (CV,el) 498
16.3.7 Heat Capacity and Characteristic Temperature 499
16.3.8 CV of Solids 500
16.3.8.1 Einstein Model 500
16.3.8.2 Debye Model 500
16.4 Change of State at Constant Pressure (Isobaric Process) 501
16.4.1 Change of Internal Energy ?U Heat q and Work w
16.4.2 Enthalpy H and Heat Capacity CP: Definition 502
16.4.2.1 Defining the Enthalpy H 503
16.4.2.2 Defining the Heat Capacity CP 503
16.4.2.3 Relation Between CV and CP 503
16.4.2.4 General Relation Between CV and CP 503
16.5 Heat Exchange and Chemical Reactions 505
16.5.1 Reaction at Constant Volume: ?U 505
16.5.2 Reaction at Constant Pressure: ?H 506
16.5.3 Temperature Dependence of ?U and ?H 508
16.5.4 Molar Enthalpies of Formation from Elements ?fH? 509
16.5.5 Molar Enthalpy of Reaction ?rH? 509
16.5.6 Bond Enthalpies and Bond Energies 510
16.5.7 Enthalpies and Reaction Cycles 511
16.6 Conclusions 512
16.7 Box 16.1. ?U,q, and w in a Change of State 512
16.8 Box 16.2. Temperature of a Flame 513
16.9 Foundations 514
16.9.1 Foundation 16.1: Deriving the Debye Function (CV of Solids) 514
16.9.2 Foundation 16.2: How to Calculate ?T1T2?CP,m?·dT 514
16.10 Exercises 514
16.11 Problems 516
References 518
Chapter 17 Reversible Work wrev, Reversible Heat qrev, and Entropy S 519
17.1 Introduction 519
17.2 Irreversible and Reversible Changes of State 520
17.2.1 Irreversible Changes 520
17.2.2 Reversible Changes 520
17.2.2.1 Reversible Isothermal Expansion of an Ideal Gas 521
17.2.2.2 Reversible Adiabatic Expansion of an Ideal Gas 523
17.2.2.3 Carnot Cycle 524
17.2.2.4 Efficiency of a Carnot Engine 526
17.3 Counting the Number of Representations of a Thermodynamic State, ? 527
17.3.1 Distribution Possibilities 527
17.3.2 Number of Representations ? of an Atomic Gas in a Given Thermodynamic State 529
17.3.2.1 Approximate Treatment 529
17.3.2.2 Rigorous Treatment 530
17.4 The Entropy: S& equals
17.4.1 Entropy of Subsystems 532
17.4.2 Entropy of Atomic Gas: Sackur–Tetrode Equation 532
17.5 Entropy Change ?S 533
17.5.1 Temperature Equilibration: Entropy Increase 533
17.5.1.1 Calculating the Entropy Change 533
17.5.1.2 Calculating the Change of the Number of Representations 534
17.5.2 Mixing: Entropy Increase 535
17.5.2.1 Mixing of Two Different Gases 535
17.5.2.2 Mixing of Identical Gases 535
17.5.3 Entropy Cannot Decrease for an Isolated System 535
17.5.4 Entropy Can Decrease in Closed Systems 536
17.6 Heat and Entropy Change 536
17.6.1 Heat and Entropy Change for an Ideal Gas 536
17.6.1.1 Expansion at Constant Temperature 536
17.6.1.2 Heating at Constant Volume 537
17.6.1.3 Generalization to an Arbitrary Process 538
17.6.2 Cyclic Processes (?S& equals
17.6.3 Heat and Entropy Change in Arbitrary Processes 539
17.7 Thermodynamic Temperature Scale and Cooling 540
17.7.1 Thermodynamic Temperature Scale 540
17.7.2 How Low Can Temperature Go? 540
17.8 Entropies of Substances 541
17.8.1 Entropy of an Atomic Gas 541
17.8.2 Entropy of Diatomic Gases 543
17.8.2.1 Rotation of Diatomic Molecules 543
17.8.2.2 Vibration of Diatomic Molecules 544
17.8.3 Entropy of Polyatomic Gases 545
17.8.3.1 Rotation of Polyatomic Molecules 545
17.8.3.2 Vibration of Polyatomic Molecules 545
17.8.4 Entropy of Different Substances 546
17.9 Laws of Thermodynamics 547
17.10 Conclusion 549
17.11 Box 17.1. Derivation of Equation (17.35) 549
17.12 Box 17.2. Indistinguishable Particles: Change of ?S When Dividing a System into Subsystems 549
17.12.1 2 Particles Occupying g Quantum States 549
17.12.2 N Particles Occupying g Quantum States (g& gg
17.13 Foundations 550
17.13.1 Foundation 17.1: The Relationship Between Entropy and Heat 550
17.13.1.1 Proof of Proposal 1: ?iqrev,iTi& equals
17.13.1.2 Clausius Inequality 552
17.13.1.3 Proof of Proposal 2: ?iqrev,iTi?0 (Isolated System) 552
17.13.2 Foundation 17.2: Number of Representations ? from Molecular Partition Function Z 553
17.13.2.1 Distinguishable Particles 553
17.13.2.2 Indistinguishable Particles 555
17.13.3 Foundation 17.3: Entropy of Homonuclear Diatomic Gases 556
17.14 Exercises 556
17.15 Problems 558
References 561
Chapter 18 General Conditions for Spontaneity and its Application to Equilibria of Ideal Gases and Dilute Solutions 562
18.1 Introduction 562
18.2 General Conditions for Spontaneity 563
18.2.1 Helmholtz Energy A 564
18.2.2 Gibbs Energy G 565
18.3 ?G and its Dependence on Temperature 566
18.3.1 Molar Gibbs Energy of Formation from Elements ?fG? 566
18.3.2 Molar Gibbs Energy of Reaction ?rG? 566
18.3.3 Temperature Dependence of ?G? 567
18.4 Pressure Dependence of ?G In Ideal Gases 568
18.4.1 Vapor Pressure (Clausius–Clapeyron Equation) 569
18.4.1.1 Evaporation of a Liquid 569
18.4.1.2 Evaporation of a Solid 570
18.4.2 Chemical Evolution of a Gas 570
18.5 ?G and Chemical Equilibrium in Ideal Gases 571
18.5.1 Formal Derivation of Mass Action Law 571
18.5.2 Mass Action Law – A Direct Consequence of the Relation Between Work and Heat 573
18.5.2.1 Reaction in a Closed Container at Constant T and V 573
18.5.2.2 Reaction in a Closed Container at Constant T and P 575
18.5.3 Applying the Mass Action Law 575
18.5.3.1 Dimerization Equilibrium For NO2 575
18.5.3.2 Dissociation of Water at Different Temperatures 577
18.5.4 Temperature Dependence of Equilibrium Constant K 578
18.5.4.1 Estimation of K& equals
18.5.4.2 ?H and ?S from Measured K 579
18.5.5 Pressure Changes and Equilibrium 580
18.5.6 Reactions Involving Gases and Immiscible Condensed Species 581
18.5.6.1 Equilibrium Constant 581
18.5.7 Equilibrium Constant from Molecular Properties 582
18.5.7.1 Isomerization Reaction i?Butane ? n?Butane 582
18.6 ?G and Equilibrium in Dilute Solution 584
18.6.1 Osmotic Pressure and Concentration 585
18.6.1.1 Reversible Change of Concentration 586
18.6.2 Depression of Vapor Pressure (Raoult's Law) 587
18.6.2.1 Nonvolatile Solute and Volatile Solvent 587
18.6.2.2 Two Volatile Components 587
18.6.3 Elevation of Boiling Point and Depression of Melting Point 588
18.6.3.1 Elevation of Boiling Point 588
18.6.3.2 Depression of Melting point 589
18.6.4 Mass Action Law (Solutions of Neutral Particles) 590
18.6.5 Mass Action Law (Solutions of Charged Particles) 591
18.6.6 Gibbs Energy of Formation in Aqueous Solution 591
18.6.6.1 Solutions of Neutral Particles 591
18.6.6.2 Solutions of Charged Particles 592
18.6.7 Part of Reactants or Products in Condensed or Gaseous State 593
18.7 Conclusion 593
18.8 Box 18.1. Temperature Dependence of ?G 594
18.9 Box 18.2. Raoult's Law 594
18.10 Box 18.3. How to Obtain ?fGaq? from ?fG? 595
18.11 Foundations 596
18.11.1 Foundation 18.1: To Calculate ?GT2 from ?GT1 596
18.11.2 Foundation 18.2: Relationship between the Molecular Partition Function and the Equilibrium Constant 597
18.11.3 Foundation 18.3: Isotope Exchange Equilibrium 597
18.11.4 Foundation 18.4: How to Determine ?G? for Ions 597
18.12 Exercises 598
18.13 Problems 600
References 603
Chapter 19 Formal Thermodynamics and Its Application to Phase Equilibria 604
19.1 Introduction 604
19.2 Internal Energy, Enthalpy, Work, and Heat 604
19.2.1 Work 605
19.2.2 Heat 605
19.2.3 Combined Form of the First and Second Laws 606
19.2.4 Relating U and H to Measurable Quantities 607
19.2.4.1 dU in Terms of Measurable Quantities 607
19.2.4.2 dH in Terms of Measurable Quantities 608
19.2.5 An Important Application: Calculating CP?CV 609
19.2.6 Maxwell Relations 610
19.3 Spontaneity and Free Energy 612
19.3.1 Helmholtz Energy A 612
19.3.2 Gibbs Energy G 613
19.4 Phase Equilibria and Phase Transitions 614
19.4.1 Solid–Liquid Equilibria 615
19.4.1.1 Solid–Liquid Phase Coexistence 616
19.4.2 Liquid–Gas and Solid–Gas Equilibria 617
19.4.2.1 Liquid–Gas and Solid–Gas Phase Coexistence 617
19.4.3 Phase Diagrams and Phase Rule 618
19.4.3.1 P–T Phase Diagrams 618
19.4.3.2 P–V Phase Diagram 619
19.4.3.3 Critical Point 619
19.4.3.4 Vapor Pressure 620
19.5 Conclusion 621
19.6 Box 19.1. Clapeyron Equation 621
19.7 Exercises 622
19.8 Problems 624
References 626
Chapter 20 Real Gases 627
20.1 Introduction 627
20.2 ?G for Real Gases and the Fugacity 627
20.3 Equations of State for Real Gases 629
20.3.1 Hard Sphere Gas 629
20.3.2 Van der Waals Equation 630
20.3.3 Critical Point and Van der Waals Constants 631
20.3.4 The Vapor–Liquid Phase Change? 632
20.3.5 A Molecular Model for the Thermodynamics of Gas Condensation 633
20.3.6 Virial Equation of State 635
20.4 Change of State for Real Gases 637
20.4.1 Adiabatic Expansion into Vacuum 637
20.4.2 Joule–Thomson Effect 639
20.4.2.1 Joule–Thomson Coefficient 639
20.4.2.2 Inversion Temperature Ti 640
20.5 Chemical Equilibria Involving Real Gases 641
20.6 Conclusion 642
20.7 Box 20.1. Estimation of V1 in the Hard Sphere Model 643
20.8 Foundations 643
20.8.1 Foundation 20.1: Molecular Perspective on Solid–Gas Equilibria 643
20.8.2 Foundation 20.2: Virial Equation of State 643
20.9 Exercises 644
20.10 Problems 645
References 648
Chapter 21 Real Solutions 649
21.1 Introduction 649
21.2 Partial Molar Quantities and Thermodynamics of Multicomponent Systems 649
21.2.1 Ideal and Non?Ideal Solution 649
21.2.2 Partial Molar Volume 650
21.2.3 Chemical Potential 651
21.2.4 Thermodynamic Relations 651
21.3 Activities and Activity Coefficient for Real Solutions 653
21.3.1 Activity from Vapor Pressure Above a Solution 653
21.3.2 Activity from Osmotic Pressure 655
21.4 Phase Transitions of Solutions 656
21.4.1 Elevation of Boiling Point and Depression of Melting Point 656
21.4.2 Solutions with Two Volatile Components 658
21.5 Mass Action Law For Reactions in Solution 659
21.6 Electrolyte Solutions and the Debye–Hückel Theory 661
21.6.1 Debye–Hückel Theory for Electrolytes 661
21.6.1.1 The Activity Coefficient Depends on rD 661
21.6.1.2 Calculating the Average Distance rD 662
21.6.1.3 Mean Activity Coefficient 663
21.7 Conclusion 663
21.8 Box 21.1. Electrical Work to Transfer Ions into Concentrated Solution 663
21.9 Foundations 664
21.9.1 Foundation 21.1: Activity Coefficient of Solute From Activity Coefficient of Solvent 664
21.9.2 Foundation 21.2: Distribution of Ions in Solution 665
21.10 Exercises 665
21.11 Problems 667
References 669
Chapter 22 Reaction Equilibria in Aqueous Solutions and Biosystems 670
22.1 Introduction 670
22.2 Proton Transfer Reactions: Dissociation of Weak Acids 670
22.2.1 Henderson–Hasselbalch Equation 671
22.2.2 Degree of Dissociation in Aqueous Solution 672
22.2.3 Degree of Dissociation in a Buffer Solution 673
22.2.4 Titration Curve of a Weak Acid 674
22.3 Stepwise Proton Transfer 675
22.3.1 Diprotic Acid 675
22.3.2 Amino Acids 676
22.4 Electron Transfer Reactions 677
22.4.1 Electron Transfer from Metal to Proton: Dissolution of Metals in Acid 678
22.4.2 Electron Transfer from Metal 1 to Metal 2 Ion: Coupled Redox Reactions 678
22.4.3 Electron Transfer to Proton at pH 7: ?G?? 679
22.4.4 Photoinduced Electron Transfer 680
22.5 Electron Transfer Coupled with Proton Transfer 682
22.6 Group Transfer Reactions in Biochemistry 684
22.6.1 Group Transfer Potential 685
22.6.2 Coupled Reactions in Biology 685
22.7 Bioenergetics 686
22.7.1 Synthesis of Glucose 686
22.7.1.1 Energy Supplied by NADPH and ATP 686
22.7.1.2 Sunlight?Induced NADPH and ATP Production 687
22.7.2 Combustion of Glucose 687
22.7.3 Energy Balance of Formation and Degradation (Combustion) of Glucose 687
22.8 Conclusion 687
22.9 Foundations 688
22.9.1 Foundation 22.1: Titration of Acetic Acid by NaOH 688
22.9.1.1 Effect of Increasing Volume 690
22.9.2 Foundation 22.2: Two Coupled Chemical Equilibria 690
22.10 Exercises 691
22.11 Problems 693
References 694
Chapter 23 Chemical Reactions in Electrochemical Cells 695
23.1 Introduction 695
23.2 ?G and Potential E of an Electrochemical Cell 695
23.3 Simple Cells and Nernst Equation 698
23.3.1 Metal/Metal Ions 698
23.3.1.1 Salt Bridge 700
23.3.2 Gas Electrodes 700
23.3.3 Nernst Equation and Standard Potential 701
23.4 Standard Potential E? and Reference Electrodes 702
23.4.1 Practical Determination of E? 703
23.4.2 Absolute Electrode Potential 704
23.4.2.1 Absolute Potential of the Hydrogen Electrode 704
23.4.2.2 Solvation Energy of the Proton 704
23.4.2.3 Absolute Potential of Arbitrary Electrodes Eabs?(M+/M) 705
23.4.2.4 Solvation Energies of Metal Ions 705
23.4.2.5 Born Model 706
23.5 Use of Electrochemical Cells for Thermodynamic Measurements 707
23.5.1 pH Electrodes 707
23.5.2 Measurement of Mean Activity Coefficient ?± 708
23.5.3 Measurement of Equilibrium Constant 710
23.5.4 Liquid–Liquid Junctions 711
23.5.4.1 Ion Selective Electrodes 711
23.6 Applications of Electrochemical Cells 712
23.6.1 Primary Batteries and Rechargeable Batteries 712
23.6.1.1 Leclanché Element (Zn/Graphite) 712
23.6.1.2 Alkaline Manganese Battery 713
23.6.1.3 Lithium Battery 713
23.6.1.4 Rechargeable Lead Battery 714
23.6.2 Fuel Cells 715
23.6.3 Electrolysis and Electrosynthesis 716
23.6.4 Overvoltage 717
23.7 Conductivity of Electrolyte Solutions 718
23.7.1 Mobility of Ions 718
23.7.2 Generalization 719
23.7.3 Mobility of H+ 720
23.7.4 Ion Transport Through Membranes 720
23.8 Conclusion 721
23.9 Exercises 721
23.10 Problems 723
References 725
Chapter 24 Chemical Kinetics 726
24.1 Introduction 726
24.2 Collision Theory for Gas Reactions 727
24.2.1 Counting the Number of Collisions 727
24.2.2 Activation 728
24.2.2.1 Accumulation of Energy 729
24.2.3 Steric Factor 730
24.3 Rate Equation for Elementary Bimolecular Reactions 730
24.3.1 Rate Constant and Frequency Factor for a Gas?Phase Reaction 730
24.3.1.1 Reactants A and B are Identical Molecules 732
24.3.2 Rate Constant and Frequency Factor for a Reaction in Solution 732
24.3.2.1 Reaction Through Activated Complex 732
24.3.2.2 Diffusion Controlled Reaction in Solution 733
24.4 Rate Laws 734
24.4.1 What Is a Rate Law? 735
24.4.2 Zero?Order Rate Law 735
24.4.3 First?Order Rate Law 736
24.4.4 Second?Order Rate Law 737
24.5 Activation Energy and Frequency Factor 739
24.5.1 Arrhenius Plot 739
24.5.2 The Chemist's Rule of Thumb 740
24.6 Combinations of Elementary Reactions 740
24.6.1 Reactions Leading to Equilibrium 740
24.6.2 Caveat on the Interpretation of Kinetic Data 742
24.6.3 Parallel Reactions 742
24.6.3.1 Same Reaction Order in Both Reactions 742
24.6.3.2 Different Reaction Orders in Both Reactions 743
24.6.4 Consecutive Reactions 744
24.7 Complex Reactions 745
24.7.1 Approximation Methods 745
24.7.1.1 Approximation by Rate?Determining Step 745
24.7.1.2 Steady?State Approximation 746
24.7.2 Lindemann–Hinshelwood Mechanism 747
24.7.3 Chain Reactions 748
24.7.3.1 Branching Chain Reactions 749
24.7.4 Enzyme Reactions (Michaelis–Menten Mechanism) 750
24.7.4.1 Hydrolysis of Urea by the Enzyme Urease 750
24.7.4.2 Inhibition of Enzyme Activity 751
24.7.5 Autocatalytic Reactions 752
24.7.5.1 Autocatalytic Oxidation of Ru2+ 753
24.7.5.2 Inhibition of Autocatalysis 755
24.7.6 Bistability 756
24.7.7 Oscillating Reactions 757
24.7.7.1 Lotka–Volterra Mechanism 757
24.7.7.2 Belousov–Zhabotinsky Reaction 757
24.7.8 Chemical Waves 759
24.7.8.1 Circular Waves 759
24.7.8.2 Chemical Pinwheel 761
24.8 Experimental Methods 762
24.8.1 Monitor Reaction Progress and Sampling 762
24.8.2 Flow Methods 762
24.8.3 Quenching Methods 763
24.8.4 Flash Photolysis 764
24.8.5 Relaxation Method 765
24.9 Conclusion 767
24.10 Box 24.1. Activation Factor 767
24.11 Box 24.2. Rate Law for a Chain Reaction 767
24.12 Exercises 768
24.13 Problems 771
References 775
Chapter 25 Transition States and Chemical Reactions 777
25.1 Introduction 777
25.2 Transition State in a Statistical View 777
25.2.1 Transition State Theory for Bimolecular Reactions 778
25.2.1.1 Rough Estimation of Rate Constant kr 779
25.2.1.2 Rigorous Calculation of Rate Constant kr 780
25.2.1.3 Activation Enthalpy and Activation Entropy 782
25.2.2 Transition State Theory for Unimolecular Reactions 783
25.2.3 More on Activation Energy 784
25.2.4 Applications of Transition State Theory 786
25.2.4.1 Isotope Effects 786
25.2.4.2 Linear Gibbs Energy Relationship (Hammett Equation) 788
25.2.4.3 Limits of Transition State Theory 789
25.3 Transition State in a Dynamical View 790
25.3.1 State?to?State Reaction Rates 790
25.3.2 Transition State Spectroscopy 792
25.3.3 Reaction Constant kr(E) from Reaction Cross?Sections 793
25.3.4 Relation Between kr(E) and the Rate Constant kr(T) 795
25.4 Transition State Theory and Reactions in Solution 795
25.4.1 Unimolecular Reactions and Frictional Coupling 795
25.4.1.1 Solvation and Reaction Rates 796
25.4.1.2 Friction and Reaction Rates 796
25.4.2 Dissociation Reactions 799
25.4.3 Proton Transfer Reactions 800
25.5 Conclusion 801
25.6 Box 25.1. Rigorous Derivation of Rate Equation (25.16) 801
25.7 Box 25.2. Rate Constant from Transition State Theory 802
25.8 Foundations 803
25.8.1 Foundation 25.1: Thermal Rate Constants 803
25.8.2 Foundation 25.2: Reaction Rate and Solvent Friction 803
25.9 Exercises 803
25.10 Problems 806
References 810
Chapter 26 Macromolecules 812
26.1 Introduction 812
26.2 Random Coil 812
26.2.1 A Chain of Statistical Chain Elements 814
26.2.1.1 Contour Length 814
26.2.1.2 Length of Statistical Chain Elements 814
26.2.1.3 Probability Distribution of Coil Diameter 815
26.3 Measuring the Length of Statistical Chain Elements 815
26.3.1 Light Scattering 815
26.3.2 Hydrodynamics: Coil Approximated as a Sphere 817
26.3.2.1 Diffusion 818
26.3.2.2 Sedimentation 818
26.3.2.3 Viscosity 819
26.3.3 Hydrodynamics: Macroscopic Modeling 820
26.3.3.1 Diffusion 821
26.3.3.2 Sedimentation 822
26.3.3.3 Viscosity 823
26.4 Uncoiling a Coil and Its Recoiling 824
26.4.1 Unraveling a Coil by Applying a Force at the Chain Ends 824
26.4.1.1 Small Forces: Length of a Statistical Chain Element 825
26.4.1.2 Large Forces: Elasticity of Stretched Chain 825
26.4.2 Fully Unraveling a Coil in a Flowing Medium 826
26.4.2.1 Minimum Speed vc to Unravel a Coil 826
26.4.2.2 Force Acting on Fixed End 827
26.4.3 Partially Unraveling a Coil in a Flowing Medium 827
26.4.4 Restoring a Coil 829
26.5 Proteins as Biopolymers 829
26.6 Motion Through Entangled Polymer Chains 831
26.6.1 Moving Random Coil by Winding Through Meshwork: Gel Electrophoresis of DNA Fragments 832
26.7 Rubber Elasticity 833
26.8 Conclusion 834
26.9 Box 26.1. Statistical Chain Elements 834
26.10 Box 26.2. Bending of a DNA Strand 835
26.11 Box 26.3. Force Contracting Chain Held at Chain Ends 836
26.12 Foundations 837
26.12.1 Foundation 26.1: Light Scattering of Macromolecules 837
26.12.1.1 Polarizability ? of Macromolecules in the Field of an Incident Light Wave 837
26.12.1.2 Forward and Backward Light Scattering 837
26.12.2 Foundation 26.2: Viscosity of Dilute Solutions of Polymers and Macroscopic Models 838
26.12.3 Foundation 26.3: Diffusion of a Random Coil in a Gel in the Absence of an Electric Field and Force?Induced Motion 838
26.13 Exercises 838
26.14 Problems 841
References 844
Chapter 27 Organized Molecular Assemblies 846
27.1 Introduction 846
27.2 Liquid Surfaces and Liquid/Liquid Interfaces 846
27.2.1 Surface Tension and Interfacial Tension 847
27.2.1.1 Capillarity 848
27.2.2 Surface Active Molecules (Surfactants) 849
27.2.2.1 Films of Soluble Surfactants 850
27.2.2.2 Films of Insoluble Surfactants 850
27.3 Films on Solid Surfaces 852
27.3.1 Langmuir–Blodgett Films (LB Films) 852
27.3.2 Self?Assembled Monolayers (SAM) 853
27.3.3 Contact Angle 853
27.4 Micelles 854
27.4.1 Spherical Micelles. Critical Micelle Concentration 855
27.4.1.1 Aggregation of Amphiphilic Molecules 855
27.4.1.2 Critical Micelle Concentration 855
27.4.1.3 Chemical Potential Change for Micelle Formation 856
27.4.2 Geometry of Packing 859
27.5 Membranes 859
27.5.1 Liposomes 859
27.5.2 Soap Lamella 860
27.5.3 Black Lipid Membranes 861
27.6 Biomembranes 862
27.6.1 Lateral Diffusion 862
27.6.2 Ion Transport Through a Membrane 862
27.6.3 Transport of Small Protein Through a Membrane 864
27.7 Liquid Crystals 865
27.8 Conclusion 865
27.9 Box 27.1. Surface Film Gibbs Adsorption Equation
27.10 Foundations 866
27.10.1 Foundation 27.1: Liquid Crystals 866
27.10.2 Foundation 27.2: Clausius–Mosotti Equation 866
27.11 Exercises 866
27.12 Problems 868
References 870
Chapter 28 Supramolecular Machines 872
28.1 Introduction 872
28.2 Idea of a Supramolecular Machine 872
28.2.1 A Simple Energy Transduction Device 873
28.2.2 Programmed Interlocking Molecules 873
28.3 Manipulating Photon Motion 874
28.3.1 Energy Transfer Between Dye Molecules. FRET: Ruler in Nanometer Range. SNOM 874
28.3.1.1 Monolayer Assembly 874
28.3.1.2 FRET: Near?Field Absorption 876
28.3.1.3 Motion of ATP Synthase 877
28.3.1.4 Scanning Near?Field Optical Microscope (SNOM) 877
28.3.1.5 Molecular Switch. Fluorescence of Dye A Quenched by Photoinduced Electron Transfer from B to C 878
28.3.2 Coupled Chromophores and Excitons 879
28.3.2.1 Functional Unit by Coupling Chromophores 879
28.3.2.2 Dye Aggregate as Energy Harvesting Device 880
28.3.2.3 Formation of Excited Domain (Exciton) 881
28.3.2.4 Lifetime of Fluorescence of Exciton 882
28.3.2.5 Speed of Exciton 883
28.3.2.6 Efficiency of Energy Transfer 883
28.3.3 Solar Energy Harvesting in Biosystems 883
28.3.3.1 Purple Bacteria: Three Groups of Collector Molecules 883
28.3.3.2 Green Bacteria: Antenna Molecules in Self?Organized Brickstone Work Arrangement 884
28.3.4 Manipulating Luminescence Lifetime by Programming Echo Radiation Field 884
28.3.4.1 Energy Transfer Quenching by a Mirror 886
28.4 Manipulating Electron Motion 886
28.4.1 Photoinduced Electron Transfer in Designed Monolayer Assemblies 886
28.4.1.1 Tunneling from Donor D to Acceptor A 886
28.4.2 Tunneling Current Through Monolayers 888
28.4.2.1 Tunneling Pathway in MIM Structure 888
28.4.2.2 Tunneling from Electrode Through Chains of Different Compositions and Lengths to Ferrocene 889
28.4.3 Conduction Through Single Molecules 890
28.4.3.1 Tunneling Through Octanedithiol Molecules 890
28.4.3.2 Single Molecule Between Gold Electrodes 892
28.4.4 Electron Transfer in Proteins 894
28.4.4.1 Electron Tunneling in Proteins 894
28.4.4.2 Wiring Proteins to Electrodes 895
28.4.5 Solar Energy Conversion: The Electron Pump of Plants and Bacteria 896
28.4.5.1 Basic Mechanism of Electron Pumping 896
28.4.5.2 Bacterial Electron Pump 896
28.4.5.3 Optimum Yield. Compromise Between Minimizing Energy Loss and Minimizing Recombination Rate 897
28.4.5.4 Tunneling Processes in the Bacterial Reaction Center 897
28.4.6 Electron Transfer in Soft Medium 899
28.4.6.1 Marcus Equation 899
28.4.6.2 Electron Tunneling from D? to D+ 899
28.4.7 Inverted Region of Electron Transfer Reactions 900
28.4.8 Artificial Photoinduced Electron Pumping 901
28.5 Manipulating Nuclear Motion 902
28.5.1 Light?Induced Change of Monolayer Properties 902
28.5.1.1 Periodic Change of Monolayer Area 902
28.5.1.2 Monolayer Commanding Liquid Crystal Orientation 902
28.5.1.3 The Visual System and Cis–Trans Isomerization 903
28.5.2 Mechanical Switching Devices 903
28.5.3 Solar Energy Conversion in Halobacteria 904
28.5.3.1 Photoinduced Proton Pump 904
28.5.4 Translocation of Proton from Cytoplasmic to Extracellular Channel 906
28.5.5 Biomotors 907
28.5.5.1 Motor Portal to Bacteriophage Capsid Packaging Newly Synthesized DNA 907
28.5.5.2 Molecular Motor Kinesin Walking Along a Microtubule 908
28.5.5.3 ATP Synthase 909
28.5.5.4 Mechanism of F0 Motor 910
28.5.6 Conclusion 912
28.6 Box 28.1. Polarization of Retinal in the Coulomb Field of Amino Acids D85 and D212 912
28.7 Box 28.2. Modeling Brownian Motion 913
28.7.1 Calculation of Probabilities ?+ and ?? 913
28.8 Foundations 914
28.8.1 Foundation 28.1: Manipulation of Monolayer Assemblies 914
28.8.1.1 Cleavage at Hydrophobic Interface (Fig. 28F.1a) 914
28.8.1.2 Cleavage at Hydrophilic Interface (Fig. 28F.1b) 915
28.8.1.3 Dry Transfer from Glass to PVA Film (Fig. 28F.1c) 915
28.8.1.4 Transfer of Organized Structure (Fig. 28F.1d) 915
28.8.2 Foundation 28.2: Energy Transfer 915
28.8.2.1 One Donor and One Acceptor in Distance r 915
28.8.2.2 Molecules in a Planar Layer 916
28.8.2.3 Relation Between r0 and d0 917
28.8.2.4 Comparison of Absorbed Power in Far Field and Near Field 918
28.8.3 Foundation 28.3: Energy Transfer from Exciton to Acceptor 918
28.8.4 Foundation 28.4: Radiation Echo Field 918
28.8.5 Foundation 28.5: Electron Transfer Between ??Electron Systems 918
28.8.5.1 General Case 919
28.8.5.2 Rate of Electron Transfer 919
28.8.5.3 Evaluation of & epsiv
28.8.5.4 Evaluation of & epsiv
28.8.5.5 Evaluation of Rate Constant kr 923
28.8.6 Foundation 28.6: Marcus Equation 924
28.8.7 Foundation 28.7: Chloride Ion Pump and Sensory Receptor of Halobacteria 927
28.9 Exercises 927
28.10 Problems 929
References 931
Chapter 29 Origin of Life. Matter Carrying Information 936
29.1 Introduction 936
29.1.1 Definition of Life in the Present Context 936
29.2 Investigation of Complex Systems 937
29.2.1 Increasing Simplification with Increasing Stages of Complexity 937
29.3 Can Life Emerge by Physicochemical Processes? 937
29.3.1 Bioevolution as a Process of Learning How to Survive as a Species 937
29.3.1.1 Multiplication, Variation, Selection 937
29.3.2 Model Case for the Learning Mechanism 938
29.3.2.1 Drawing Structures Adapting to a Tub 938
29.3.3 Optimum Error Frequency 939
29.4 Modeling the Emergence of the Genetic Apparatus 939
29.4.1 Basic Questions and their Refinement 939
29.4.1.1 Evolution of the Universe and Evolution of Life: The Big Bang and the Tiny Bang 940
29.4.1.2 Paradigm of Present Attempt to Understand the Origin of Life 941
29.4.2 General Conditions for Life to Come into Being: Periodicity in Time, Compartmentalization, and Structural Diversity 941
29.4.3 Modeling a Continuous Sequence of Physicochemical Processes Leading to a Genetic Apparatus 942
29.4.3.1 Strategy: Search for Sequences of Small Steps 942
29.4.3.2 Beginning: Replication of short Strands that are Homochiral. Symmetry Breaking 943
29.4.3.3 Sketching the Path 943
29.4.3.4 Basic Evolutionary Principle and Paradigm for Increasing Complexity 944
29.4.3.5 Chemical Features 944
29.4.3.6 Later Evolutionary Steps: Emergence of an Eye with Lens 945
29.5 General Aspects of Life's Emergence and Evolution 945
29.5.1 Information and Knowledge 945
29.5.2 Processing Information, Genesis of Information and Knowledge, and the Maxwell Demon 945
29.5.2.1 Computer and Biosystem. Genetic Information I and Knowledge K 945
29.5.2.2 Entropy Production in Information Processing and in Information and Knowledge Creation 946
29.5.2.3 Minimum Energy Dissipation in a Measurement and the Maxwell Demon 946
29.5.3 Limits of Physicochemical Ways of Thinking 948
29.5.3.1 Device That Develops an Internal Model of the World 948
29.5.3.2 Development of Human Internal Model of an External Physical World 949
29.5.3.3 Limitations in Physical Theory 950
29.6 Conclusion 951
29.7 Box 29.1. Genetic Apparatus 951
29.8 Foundations 952
29.8.1 Foundation 29.1: Search for Logical Conditions Driving the Emergence of a Genetic Translation Device 952
29.8.1.1 A Thought Experiment 952
29.8.1.2 Breakthrough of an Integrative Translation Device 957
29.8.1.3 Computer Simulation of Thought Experiment 958
29.8.2 Foundation 29.2: Attempts to Model the Origin of Life 959
29.8.2.1 The Earliest Phase 961
29.8.2.2 Breaking Symmetry: Chirality of Nucleotides and Its Evolutionary Benefit 961
29.8.2.3 The Earliest HA?Device 962
29.8.2.4 Stage?by?Stage Evolution of the Code 964
29.9 Exercises 965
29.10 Problems 965
References 966
Appendices 967
Index 1016
Back Cover 1042
EULA 1044