Electrochemical Methods

References 57 1.11 Problems 57 2 Potentials and Thermodynamics of Cells 61 2.1 Basic Electrochemical Thermodynamics 61 2.1.1 Reversibility 61 2.1.2 Reversibility and Gibbs Free Energy 64 2.1.3 Free Energy and Cell emf 64 2.1.4 Half-Reactions and Standard Electrode Potentials 66 2.1.5 Standard States and Activity 67 2.1.6 emf and Concentration 69 2.1.7 Formal Potentials 71 2.1.8 Reference Electrodes 72 2.1.9 Potential-pH Diagrams and Thermodynamic Predictions 76 2.2 A More Detailed View of Interfacial Potential Differences 80 2.2.1 The Physics of Phase Potentials 80 2.2.2 Interactions Between Conducting Phases 82 2.2.3 Measurement of Potential Differences 84 2.2.4 Electrochemical Potentials 85 2.2.5 Fermi Energy and Absolute Potential 88 2.3 Liquid Junction Potentials 91 2.3.1 Potential Differences at an Electrolyte-Electrolyte Boundary 91 2.3.2 Types of Liquid Junctions 91 2.3.3 Conductance, Transference Numbers, and Mobility 92 2.3.4 Calculation of Liquid Junction Potentials 96 2.3.5 Minimizing Liquid Junction Potentials 100 2.3.6 Junctions of Two Immiscible Liquids 101 2.4 Ion-Selective Electrodes 101 2.4.1 Selective Interfaces 101 2.4.2 Glass Electrodes 102 2.4.3 Other Ion-Selective Electrodes 106 2.4.4 Gas-Sensing ISEs 111 2.5 Lab Note: Practical Use of Reference Electrodes 112 2.5.1 Leakage at the Reference Tip 112 2.5.2 Quasireference Electrodes 112 2.6 References 113 2.7 Problems 116 3 Basic Kinetics of Electrode Reactions 121 3.1 Review of Homogeneous Kinetics 121 3.1.1 Dynamic Equilibrium 121 3.1.2 The Arrhenius Equation and Potential Energy Surfaces 122 3.1.3 Transition State Theory 123 3.2 Essentials of Electrode Reactions 125 3.3 Butler-Volmer Model of Electrode Kinetics 126 3.3.1 Effects of Potential on Energy Barriers 127 3.3.2 One-Step, One-Electron Process 127 3.3.3 The Standard Rate Constant 130 3.3.4 The Transfer Coefficient 131 3.4 Implications of the Butler-Volmer Model for the One-Step, One-Electron Process 132 3.4.1 Equilibrium Conditions and the Exchange Current 133 3.4.2 The Current-Overpotential Equation 133 3.4.3 Approximate Forms of the i-eta Equation 135 3.4.4 Exchange Current Plots 139 3.4.5 Very Facile Kinetics and Reversible Behavior 139 3.4.6 Effects of Mass Transfer 140 3.4.7 Limits of Basic Butler-Volmer Equations 141 3.5 Microscopic Theories of Charge Transfer 142 3.5.1 Inner-Sphere and Outer-Sphere Electrode Reactions 142 3.5.2 Extended Charge Transfer and Adiabaticity 143 3.5.3 The Marcus Microscopic Model 146 3.5.4 Implications of the Marcus Theory 152 3.5.5 A Model Based on Distributions of Energy States 162 3.6 Open-Circuit Potential and Multiple Half-Reactions at an Electrode 168 3.6.1 Open-Circuit Potential in Multicomponent Systems 169 3.6.2 Establishment or Loss of Nernstian Behavior at an Electrode 170 3.6.3 Multiple Half-Reaction Currents in i-E Curves 171 3.7 Multistep Mechanisms 171 3.7.1 The Primacy of One-Electron Transfers 172 3.7.2 Rate-Determining, Outer-Sphere Electron Transfer 173 3.7.3 Multistep Processes at Equilibrium 173 3.7.4 Nernstian Multistep Processes 174 3.7.5 Quasireversible and Irreversible Multistep Processes 174 3.8 References 177 3.9 Problems 180 4 Mass Transfer by Migration and Diffusion 183 4.1 General Mass-Transfer Equations 183 4.2 Migration in Bulk Solution 186 4.3 Mixed Migration and Diffusion Near an Active Electrode 187 4.3.1 Balance Sheets for Mass Transfer During Electrolysis 188 4.3.2 Utility of a Supporting Electrolyte 192 4.4 Diffusion 193 4.4.1 A Microscopic View 193 4.4.2 Fick's Laws of Diffusion 196 4.4.3 Flux of an Electroreactant at an Electrode Surface 199 4.5 Formulation and Solution of Mass-Transfer Problems 199 4.5.1 Initial and Boundary Conditions in Electrochemical Problems 200 4.5.2 General Formulation of a Linear Diffusion Problem 201 4.5.3 Systems Involving Migration or Convection 202 4.5.4 Practical Means for Reaching Solutions 202 4.6 References 204 4.7 Problems 205 5 Steady-State Voltammetry at Ultramicroelectrodes 207 5.1 Steady-State Voltammetry at a Spherical UME 207 5.1.1 Steady-State Diffusion 208 5.1.2 Steady-State Current 211 5.1.3 Convergence on the Steady State 211 5.1.4 Steady-State Voltammetry 212 5.2 Shapes and Properties of Ultramicroelectrodes 214 5.2.1 Spherical or Hemispherical UME 215 5.2.2 Disk UME 215 5.2.3 Cylindrical UME 221 5.2.4 Band UME 221 5.2.5 Summary of Steady-State Behavior at UMEs 222 5.3 Reversible Electrode Reactions 224 5.3.1 Shape of the Wave 224 5.3.2 Applications of Reversible i-E Curves 226 5.4 Quasireversible and Irreversible Electrode Reactions 230 5.4.1 Effect of Electrode Kinetics on Steady-State Responses 230 5.4.2 Total Irreversibility 232 5.4.3 Kinetic Regimes 234 5.4.4 Influence of Electrode Shape 234 5.4.5 Applications of Irreversible i-E Curves 235 5.4.6 Evaluation of Kinetic Parameters by Varying Mass-Transfer Rates 237 5.5 Multicomponent Systems and Multistep Charge Transfers 239 5.6 Additional Attributes of Ultramicroelectrodes 241 5.6.1 Uncompensated Resistance at a UME 241 5.6.2 Effects of Conductivity on Voltammetry at a UME 242 5.6.3 Applications Based on Spatial Resolution 243 5.7 Migration in Steady-State Voltammetry 245 5.7.1 Mathematical Approach to Problems Involving Migration 245 5.7.2 Concentration Profiles in the Diffusion-Migration Layer 246 5.7.3 Wave Shape at Low Electrolyte Concentration 248 5.7.4 Effects of Migration on Wave Height in SSV 248 5.8 Analysis at High Analyte Concentrations 251 5.9 Lab Note: Preparation of Ultramicroelectrodes 253 5.9.1 Preparation and Characterization of UMEs 254 5.9.2 Testing the Integrity of a UME 254 5.9.3 Estimating the Size of a UME 256 5.10 References 257 5.11 Problems 258 6 Transient Methods Based on Potential Steps 261 6.1 Chronoamperometry Under Diffusion Control 261 6.1.1 Linear Diffusion at a Plane 262 6.1.2 Response at a Spherical Electrode 265 6.1.3 Transients at Other Ultramicroelectrodes 267 6.1.4 Information from Chronoamperometric Results 270 6.1.5 Microscopic and Geometric Areas 271 6.2 Sampled-Transient Voltammetry for Reversible Electrode Reactions 275 6.2.1 A Step to an Arbitrary Potential 276 6.2.2 Shape of the Voltammogram 277 6.2.3 Concentration Profiles When R Is Initially Absent 278 6.2.4 Simplified Current-Concentration Relationships 279 6.2.5 Applications of Reversible i-E Curves 279 6.3 Sampled-Transient Voltammetry for Quasireversible and Irreversible Electrode Reactions 279 6.3.1 Effect of Electrode Kinetics on Transient Behavior 280 6.3.2 Sampled-Transient Voltammetry for Reduction of O 282 6.3.3 Sampled Transient Voltammetry for Oxidation of R 284 6.3.4 Totally Irreversible Reactions 285 6.3.5 Kinetic Regimes 287 6.3.6 Applications of Irreversible i-E Curves 287 6.4 Multicomponent Systems and Multistep Charge Transfers 289 6.5 Chronoamperometric Reversal Techniques 290 6.5.1 Approaches to the Problem 292 6.5.2 Current-Time Responses 293 6.6 Chronocoulometry 294 6.6.1 Large-Amplitude Potential Step 295 6.6.2 Reversal Experiments Under Diffusion Control 296 6.6.3 Effects of Heterogeneous Kinetics 299 6.7 Cell Time Constants at Microelectrodes 300 6.8 Lab Note: Practical Concerns with Potential Step Methods 303 6.8.1 Preparation of the Electrode Surface at a Microelectrode 303 6.8.2 Interference from Charging Current 305 6.9 References 306 6.10 Problems 307 7 Linear Sweep and Cyclic Voltammetry 311 7.1 Transient Responses to a Potential Sweep 311 7.2 Nernstian (Reversible) Systems 313 7.2.1 Linear Sweep Voltammetry 313 7.2.2 Cyclic Voltammetry 321 7.3 Quasireversible Systems 325 7.3.1 Linear Sweep Voltammetry 326 7.3.2 Cyclic Voltammetry 326 7.4 Totally Irreversible Systems 329 7.4.1 Linear Sweep Voltammetry 329 7.4.2 Cyclic Voltammetry 332 7.5 Multicomponent Systems and Multistep Charge Transfers 332 7.5.1 Multicomponent Systems 332 7.5.2 Multistep Charge Transfers 333 7.6 Fast Cyclic Voltammetry 334 7.7 Convolutive Transformation 336 7.8 Voltammetry at Liquid-Liquid Interfaces 339 7.8.1 Experimental Approach to Voltammetry 340 7.8.2 Effect of Interfacial Potential on Composition 341 7.8.3 Voltammetric Behavior 341 7.9 Lab Note: Practical Aspects of Cyclic Voltammetry 344 7.9.1 Basic Experimental Conditions 344 7.9.2 Choice of Initial and Final Potentials 345 7.9.3 Deaeration 347 7.10 References 347 7.11 Problems 349 8 Polarography, Pulse Voltammetry, and Square-Wave Voltammetry 355 8.1 Polarography 355 8.1.1 The Dropping Mercury Electrode 355 8.1.2 The Ilkovi Equation 356 8.1.3 Polarographic Waves 357 8.1.4 Practical Advantages of the DME 358 8.1.5 Polarographic Analysis 358 8.1.6 Residual Current and Detection Limits 359 8.2 Normal Pulse Voltammetry 361 8.2.1 Implementation 362 8.2.2 Renewal at Stationary Electrodes 363 8.2.3 Normal Pulse Polarography 364 8.2.4 Practical Application 366 8.3 Reverse Pulse Voltammetry 367 8.4 Differential Pulse Voltammetry 369 8.4.1 Concept of the Method 370 8.4.2 Theory 371 8.4.3 Renewal vs. Pre-Electrolysis 374 8.4.4 Residual Currents 375 8.4.5 Differential Pulse Polarography 375 8.5 Square-Wave Voltammetry 376 8.5.1 Experimental Concept and Practice 376 8.5.2 Theoretical Prediction of Response 377 8.5.3 Background Currents 380 8.5.4 Applications 381 8.6 Analysis by Pulse Voltammetry 383 8.7 References 385 8.8 Problems 386 9 Controlled-Current Techniques 389 9.1 Introduction to Chronopotentiometry 389 9.2 Theory of Controlled-Current Methods 391 9.2.1 General Treatment for Linear Diffusion 391 9.2.2 Constant-Current Electrolysis--The Sand Equation 392 9.2.3 Programmed Current Chronopotentiometry 394 9.3 Potential-Time Curves in Constant-Current Electrolysis 394 9.3.1 Reversible (Nernstian) Waves 394 9.3.2 Totally Irreversible Waves 394 9.3.3 Quasireversible Waves 395 9.3.4 Practical Issues in the Measurement of Transition Time 396 9.4 Reversal Techniques 398 9.4.1 Response Function Principle 398 9.4.2 Current Reversal 398 9.5 Multicomponent Systems and Multistep Reactions 400 9.6 The Galvanostatic Double Pulse Method 401 9.7 Charge Step (Coulostatic) Methods 403 9.7.1 Small Excursions 404 9.7.2 Large Excursions 405 9.7.3 Coulostatic Perturbation by Temperature Jump 405 9.8 References 406 9.9 Problems 407 10 Methods Involving Forced Convection--Hydrodynamic Methods 411 10.1 Theory of Convective Systems 411 10.1.1 The Convective-Diffusion Equation 412 10.1.2 Determination of the Velocity Profile 412 10.2 Rotating Disk Electrode 414 10.2.1 The Velocity Profile at a Rotating Disk 414 10.2.2 Solution of the Convective-Diffusion Equation 416 10.2.3 Concentration Profile 418 10.2.4 General i-E Curves at the RDE 419 10.2.5 The Kouteck?-Levich Method 420 10.2.6 Current Distribution at the RDE 423 10.2.7 Practical Considerations for Application of the RDE 426 10.3 Rotating Ring and Ring-Disk Electrodes 426 10.3.1 Rotating Ring Electrode 427 10.3.2 The Rotating Ring-Disk Electrode 428 10.4 Transient Currents 432 10.4.1 Transients at the RDE 432 10.4.2 Transients at the RRDE 433 10.5 Modulation of the RDE 435 10.6 Electrohydrodynamic Phenomena 436 10.7 References 439 10.8 Problems 440 11 Electrochemical Impedance Spectroscopy and ac Voltammetry 443 11.1 A Simple Measurement of Cell Impedance 444 11.2 Brief Review of ac Circuits 446 11.3 Equivalent Circuits of a Cell 450 11.3.1 The Randles Equivalent Circuit 451 11.3.2 Interpretation of the Faradaic Impedance 452 11.3.3 Behavior and Uses of the Faradaic Impedance 455 11.4 Electrochemical Impedance Spectroscopy 458 11.4.1 Conditions of Measurement 458 11.4.2 A System with Simple Faradaic Kinetics 460 11.4.3 Measurement of Resistance and Capacitance 465 11.4.4 A Confined Electroactive Domain 466 11.4.5 Other Applications 470 11.5 ac Voltammetry 470 11.5.1 Reversible Systems 470 11.5.2 Quasireversible and Irreversible Systems 473 11.5.3 Cyclic ac Voltammetry 477 11.6 Nonlinear Responses 477 11.6.1 Second Harmonic ac Voltammetry 478 11.6.2 Large Amplitude ac Voltammetry 479 11.7 Chemical Analysis by ac Voltammetry 481 11.8 Instrumentation for Electrochemical Impedance Methods 482 11.8.1 Frequency-Domain Instruments 482 11.8.2 Time-Domain Instruments 483 11.9 Analysis of Data in the Laplace Plane 485 11.10 References 485 11.11 Problems 487 12 Bulk Electrolysis 489 12.1 General Considerations 490 12.1.1 Completeness of an Electrode Process 490 12.1.2 Current Efficiency 491 12.1.3 Experimental Concerns 491 12.2 Controlled-Potential Methods 495 12.2.1 Current-Time Behavior 495 12.2.2 Practical Aspects 497 12.2.3 Coulometry 498 12.2.4 Electrogravimetry 500 12.2.5 Electroseparations 501 12.3 Controlled-Current Methods 501 12.3.1 Characteristics of Controlled-Current Electrolysis 501 12.3.2 Coulometric Titrations 503 12.3.3 Practical Aspects of Constant-Current Electrolysis 506 12.4 Electrometric End-Point Detection 507 12.4.1 Current-Potential Curves During Titration 507 12.4.2 Potentiometric Methods 508 12.4.3 Amperometric Methods 509 12.5 Flow Electrolysis 510 12.5.1 Mathematical Treatment 510 12.5.2 Dual-Electrode Flow Cells 515 12.5.3 Microfluidic Flow Cells 516 12.6 Thin-Layer Electrochemistry 521 12.6.1 Chronoamperometry and Coulometry 521 12.6.2 Potential Sweep in a Nernstian System 524 12.6.3 Dual-Electrode Thin-Layer Cells 526 12.6.4 Applications of the Thin-Layer Concept 526 12.7 Stripping Analysis 527 12.7.1 Introduction 527 12.7.2 Principles and Theory 528 12.7.3 Applications and Variations 529 12.8 References 531 12.9 Problems 534 13 Electrode Reactions with Coupled Homogeneous Chemical Reactions 539 13.1 Classification of Reactions 539 13.1.1 Reactions with One E-Step 541 13.1.2 Reactions with Two or More E-Steps 542 13.2 Impact of Coupled Reactions on Cyclic Voltammetry 545 13.2.1 Diagnostic Criteria 545 13.2.2 Characteristic Times 547 13.2.3 An Example 547 13.2.4 Including Kinetics in Theory 548 13.2.5 Comparative Simulation 551 13.3 Survey of Behavior 552 13.3.1 Following Reaction--case E R c I 552 13.3.2 Effect of Electrode Kinetics in Ec I Systems 556 13.3.3 Bidirectional Following Reaction 558 13.3.4 catalytic Reaction--case E r c ' I 561 13.3.5 Preceding Reaction--Case C r E r 564 13.3.6 Multistep Electron Transfers 569 13.3.7 ECE/DISP Reactions 576 13.3.8 Concerted vs.StepwiseReaction 584 13.3.9 Elaboration of Reaction Schemes 590 13.4 Behavior with Other Electrochemical Methods 591 13.5 References 593 13.6 Problems 595 14 Double-Layer Structure and Adsorption 599 14.1 Thermodynamics of the Double Layer 599 14.1.1 The Gibbs Adsorption Isotherm 599 14.1.2 The Electrocapillary Equation 601 14.1.3 Relative Surface Excesses 601 14.2 Experimental Evaluations 602 14.2.1 Electrocapillarity 602 14.2.2 Excess Charge and Capacitance 603 14.2.3 Relative Surface Excesses 606 14.3 Models for Double-Layer Structure 606 14.3.1 The Helmholtz Model 607 14.3.2 The Gouy-Chapman Theory 609 14.3.3 Stern's Modification 614 14.3.4 Specific Adsorption 617 14.4 Studies at Solid Electrodes 619 14.4.1 Well-Defined Single-Crystal Electrode Surfaces 620 14.4.2 The Double Layer at Solids 623 14.5 Extent and Rate of Specific Adsorption 627 14.5.1 Nature and Extent of Specific Adsorption 628 14.5.2 Electrosorption Valency 629 14.5.3 Adsorption Isotherms 630 14.5.4 Rate of Adsorption 633 14.6 Practical Aspects of Adsorption 634 14.7 Double-Layer Effects on Electrode Reaction Rates 636 14.7.1 Introduction and Principles 636 14.7.2 Double-Layer Effects Without Specific Adsorption of Electrolyte 638 14.7.3 Double-Layer Effects with Specific Adsorption 639 14.7.4 Diffuse Double-Layer Effects on Mass Transport 640 14.8 References 645 14.9 Problems 648 15 Inner-Sphere Electrode Reactions and Electrocatalysis 653 15.1 Inner-Sphere Heterogenous Electron-Transfer Reactions 653 15.1.1 TheRoleoftheElectrodeSurface 653 15.1.2 Energetics of 1e Electron-Transfer Reactions 654 15.1.3 Adsorption Energies 657 15.2 Electrocatalytic Reaction Mechanisms 657 15.2.1 Hydrogen Evolution Reaction 657 15.2.2 Tafel Plot Analysis of HER Kinetics 660 15.3 Additional Examples of Inner-Sphere Reactions 667 15.3.1 Oxygen Reduction Reaction 667 15.3.2 Chlorine Evolution 670 15.3.3 Methanol Oxidation 670 15.3.4 CO 2 Reduction 673 15.3.5 Oxidation of NH 3 to N 2 674 15.3.6 Organic Halide Reduction 676 15.3.7 Hydrogen Peroxide Oxidation and Reduction 677 15.4 Computational Analyses of Inner-Sphere Electron-Transfer Reactions 678 15.4.1 Density Functional Theory Analysis of Electrocatalytic Reactions 679 15.4.2 Hydrogen Evolution Reaction 679 15.4.3 Oxygen Reduction Reaction 681 15.5 Electrocatalytic Correlations 684 15.6 Electrochemical Phase Transformations 688 15.6.1 Nucleation and Growth of a New Phase 688 15.6.2 Classical Nucleation Theory 689 15.6.3 Electrodeposition 699 15.6.4 Gas Evolution 707 15.7 References 713 15.8 Problems 718 16 Electrochemical Instrumentation 721 16.1 Operational Amplifiers 721 16.1.1 Ideal Properties 721 16.1.2 Nonidealities 723 16.2 Current Feedback 725 16.2.1 Current Follower 725 16.2.2 Scaler/Inverter 726 16.2.3 Adders 726 16.2.4 Integrators 727 16.3 Voltage Feedback 728 16.3.1 Voltage Follower 728 16.3.2 Control Functions 729 16.4 Potentiostats 730 16.4.1 Basic Considerations 730 16.4.2 The Adder Potentiostat 731 16.4.3 Refinements to the Adder Potentiostat 732 16.4.4 Bipotentiostats 733 16.4.5 Four-Electrode Potentiostats 734 16.5 Galvanostats 734 16.6 Integrated Electrochemical Instrumentation 736 16.7 Difficulties with Potential Control 737 16.7.1 Types of Control Problems 737 16.7.2 Cell Properties and Electrode Placement 740 16.7.3 Electronic Compensation of Resistance 740 16.8 Measurement of Low Currents 744 16.8.1 Fundamental Limits 744 16.8.2 Practical Considerations 746 16.8.3 Current Amplifier 746 16.8.4 Simplified Instruments and Cells 746 16.9 Instruments for Short Time Scales 748 16.10 Lab Note: Practical Use of Electrochemical Instruments 749 16.10.1 Caution Regarding Electrochemical Workstations 749 16.10.2 Troubleshooting Electrochemical Systems 749 16.11 References 751 16.12 Problems 752 17 Electroactive Layers and Modified Electrodes 755 17.1 Monolayers and Submonolayers on Electrodes 756 17.2 Cyclic Voltammetry of Adsorbed Layers 757 17.2.1 Fundamentals 757 17.2.2 Reversible Adsorbate Couples 758 17.2.3 Irreversible Adsorbate Couples 763 17.2.4 Nernstian Processes Involving Adsorbates and Solutes 766 17.2.5 More Complex Systems 770 17.2.6 Electric-Field-Driven Acid-Base Chemistry in Adsorbate Layers 771 17.3 Other Useful Methods for Adsorbed Monolayers 775 17.3.1 Chronocoulometry 775 17.3.2 Coulometry in Thin-Layer Cells 777 17.3.3 Impedance Measurements 778 17.3.4 Chronopotentiometry 779 17.4 Thick Modification Layers on Electrodes 780 17.5 Dynamics in Modification Layers 782 17.5.1 Steady State at a Rotating Disk 783 17.5.2 Principal Dynamic Processes in Modifying Films 784 17.5.3 Interplay of Dynamical Elements 789 17.6 Blocking Layers 791 17.6.1 Permeation Through Pores and Pinholes 792 17.6.2 Tunneling Through Blocking Films 796 17.7 Other Methods for Characterizing Layers on Electrodes 798 17.8 Electrochemical Methods Based on Electroactive Layers or Electrode Modification 798 17.8.1 Electrocatalysis 799 17.8.2 Bioelectrocatalysis Based on Enzyme-Modified Electrodes 799 17.8.3 Electrochemical Sensors 803 17.8.4 Faradaic Electrochemical Measurements in vivo 809 17.9 References 812 17.10 Problems 817 18 Scanning Electrochemical Microscopy 819 18.1 Principles 819 18.2 Approach Curves 821 18.3 Imaging Surface Topography and Reactivity 825 18.3.1 Imaging Based on Conductivity of the Substrate 825 18.3.2 Imaging Based on Heterogeneous Electron-Transfer Reactivity 826 18.3.3 Simultaneous Imaging of Topography and Reactivity 827 18.4 Measurements of Kinetics 828 18.4.1 Heterogeneous Electron-Transfer Reactions 828 18.4.2 Homogeneous Reactions 831 18.5 Surface Interrogation 835 18.6 Potentiometric Tips 839 18.7 Other Applications 839 18.7.1 Detection of Species Released from Surfaces, Films, or Pores 839 18.7.2 Biological Systems 840 18.7.3 Probing the Interior of a Layer on a Substrate 841 18.8 Scanning Electrochemical Cell Microscopy 841 18.9 References 846 18.10 Problems 849 19 Single-Particle Electrochemistry 851 19.1 General Considerations in Single-Particle Electrochemistry 851 19.2 Particle Collision Experiments 852 19.3 Particle Collision Rate at a Disk-Shaped UME 854 19.3.1 Collision Frequency 854 19.3.2 Variance in the Number of Particle Collisions 855 19.3.3 Time of First Arrival 856 19.4 Nanoparticle Collision Behavior 857 19.4.1 Blocking Collisions 857 19.4.2 Electrocatalytic Amplification Collisions 861 19.4.3 Electrolysis Collisions 864 19.5 Electrochemistry at Single Atoms and Atomic Clusters 870 19.6 Single-Molecule Electrochemistry 875 19.7 References 879 19.8 Problems 881 20 Photoelectrochemistry and Electrogenerated Chemiluminescence 885 20.1 Solid Materials 885 20.1.1 The Band Model 885 20.1.2 Categories of Pure Crystalline Solids 886 20.1.3 Doped Semiconductors 889 20.1.4 Fermi Energy 890 20.1.5 Highly Conducting Oxides 891 20.2 Semiconductor Electrodes 892 20.2.1 Interface at a Semiconducting Electrode in the Dark 892 20.2.2 Current-Potential Curves at Semiconductor Electrodes 896 20.2.3 Conducting Polymer Electrodes 899 20.3 Photoelectrochemistry at Semiconductors 901 20.3.1 Photoeffects at Semiconductor Electrodes 901 20.3.2 Photoelectrochemical Systems 903 20.3.3 Dye Sensitization 905 20.3.4 Surface Photocatalytic Processes at Semiconductor Particles 906 20.4 Radiolytic Products in Solution 908 20.4.1 Photoemission of Electrons from an Electrode 908 20.4.2 Detection and Use of Radiolytic Products in Solution 909 20.4.3 Photogalvanic Cells 909 20.5 Electrogenerated Chemiluminescence 910 20.5.1 Chemical Fundamentals 910 20.5.2 Fundamental Studies of Radical-Ion Annihilation 912 20.5.3 Single-Potential Generation Based on a Coreactant 916 20.5.4 ECL Based on Quantum Dots 917 20.5.5 Analytical Applications of ECL 918 20.5.6 ECL Beyond the Solution Phase 922 20.6 References 922 20.7 Problems 927 21 In situ Characterization of Electrochemical Systems 931 21.1 Microscopy 931 21.1.1 Scanning Tunneling Microscopy 932 21.1.2 Atomic Force Microscopy 934 21.1.3 Optical Microscopy 937 21.1.4 Transmission Electron Microscopy 938 21.2 Quartz Crystal Microbalance 940 21.2.1 Basic Method 940 21.2.2 QCM with Dissipation Monitoring 942 21.3 UV-Visible Spectrometry 942 21.3.1 Absorption Spectroscopy with Thin-Layer Cells 942 21.3.2 Ellipsometry 945 21.3.3 Surface Plasmon Resonance 946 21.4 Vibrational Spectroscopy 947 21.4.1 Infrared Spectroscopy 947 21.4.2 Raman Spectroscopy 950 21.5 X-Ray Methods 953 21.6 Mass Spectrometry 954 21.7 Magnetic Resonance Spectroscopy 955 21.7.1 Esr 955 21.7.2 Nmr 956 21.8 Ex-situ Techniques 957 21.8.1 Electron Microscopy 957 21.8.2 Electron and Ion Spectrometry 958 21.9 References 960 Appendix A Mathematical Methods 967 A.1 Solving Differential Equations by the Laplace Transform Technique 967 A.1.1 Partial Differential Equations 967 A.1.2 Introduction to the Laplace Transformation 968 A.1.3 Fundamental Properties of the Transform 969 A.1.4 Solving Ordinary Differential Equations by Laplace Transformation 970 A.1.5 Simultaneous Linear Ordinary Differential Equations 972 A.1.6 Mass-Transfer Problems Based on Partial Differential Equations 973 A.1.7 The Zero-Shift Theorem 975 A.2 Taylor Expansions 976 A.2.1 Expansion of a Function of Several Variables 976 A.2.2 Expansion of a Function of a Single Variable 977 A.2.3 Maclaurin Series 977 A.3 The Error Function and the Gaussian Distribution 977 A.4 Leibnitz Rule 979 A.5 Complex Notation 979 A.6 Fourier Series and Fourier Transformation 981 A.7 References 982 A.8 Problems 983 Appendix B Basic Concepts of Simulation 985 B.1 Setting Up the Model 985 B.1.1 A Discrete System 985 B.1.2 Diffusion 986 B.1.3 Dimensionless Parameters 987 B.1.4 Time 990 B.1.5 Distance 990 B.1.6 Current 991 B.1.7 Thickness of the Diffusion Layer 992 B.1.8 Diffusion Coefficients 993 B.2 An Example 993 B.2.1 Organization of the Spreadsheet 993 B.2.2 Concentration Arrays 996 B.2.3 Results and Error Detection 996 B.2.4 Performance 997 B.3 Incorporating Homogeneous Kinetics 999 B.3.1 Unimolecular Reactions 999 B.3.2 Bimolecular Reactions 1000 B.4 Boundary Conditions for Various Techniques 1001 B.4.1 Potential Steps in Nernstian Systems 1001 B.4.2 Heterogeneous Kinetics 1002 B.4.3 Potential Sweeps 1003 B.4.4 Controlled Current 1003 B.5 More Complex Systems 1004 B.6 References 1005 B.7 Problems 1005 Appendix C Reference Tables 1007 References 1013 Index 1015