Part I Principles of Biophysical Inquiry
1 Introduction: To the Student - First Edition ................. 3
2 Philosophy and Practice of Biophysical Study ................. 5
2.1 What Is Biophysical Chemistry and Why Study It? ......... 5
2.2 Science Is Not Content but a Unique Method of
Discovery ............................................... 6
2.3 The Progression of Inquiry Guides the Scientific
Modeling Process ........................................ 8
2.4 A Brief History of Human Methods of Inquiry Reveals
Important Aspects of the Scientific Method .............. 9
2.5 The Gedanken Experiment Is a Thought Experiment ........ 12
2.6 The Beginnings of Modern Science- Kepler and Galileo ... 14
2.7 Modern Biophysical Studies Still Follow the Paradigm
of Kepler and Galileo .................................. 16
Further Reading ............................................. 20
Problem Sets ................................................ 22
3 Overview of the Biological System Under Study ............... 23
3.1 Hierarchies of Abstraction Are Essential in the Study
of Biophysical Chemistry ............................... 24
3.2 An Overview of the Cell: The Essential Building
Block of Life .......................................... 25
3.3 Control Mechanisms Are Essential Process Elements
of the Biological State Space .......................... 33
3.4 Biological Energy Transduction Is an Essential
Process that Provides Energy to Ensure the High
Degree of Organization Necessary for Life .............. 34
3.5 The Cell Is a Building Block of Chemical and
Biological Organization and Also a Key to the Study
of Biological Complexity ............................... 44
3.6 A Brief History of Life ................................ 45
3.7 Evolution Can Be Modeled as a Dynamic Process with
Many Bifurcations in the State Space of Life ........... 46
Further Reading ............................................. 54
Problem Sets ................................................ 55
4 Physical Thoughts, Biological Systems - The Application
of Modeling Principles to Understanding Biological
Systems ..................................................... 57
4.1 The Interaction Between Formal Models and Natural
Systems Is the Essence of Physical and Biophysical
Science ................................................ 57
4.2 Observables Are the Link Between Observer and
Reality ................................................ 58
4.3 Systems Science Guides the Linkage of Natural and
Formal Models .......................................... 60
4.4 Abstraction and Approximation May Be Useful but Are
Not Always Correct ..................................... 61
4.5 The Choices Made in Observables and Measurement
Influence What Can Be Known About a System ............. 62
4.6 The Simplifying Concept of Abstraction Is Central to
Both Scientific Understanding and Misconception ........ 64
4.7 Equations of State Capture the System Behavior or
"Systemness" ........................................... 65
4.8 Equivalent Descriptions Contain the Same Information ... 67
4.9 Symmetry and Symmetry Operations Allow Molecules to
Be Placed in Groups .................................... 69
4.10 The Goodness of the Model Depends on Where You Look
with Bifurcation Leading to New Discovery .............. 71
4.11 Bifurcations in State Space Characterize Complex
Systems ................................................ 72
4.12 Catastrophes and Chaos Are Examples of Formal
Mathematical Systems That May Capture Important
Behaviors of Natural Systems ........................... 74
Further Reading ............................................. 78
Problem Sets ................................................ 80
5 Probability and Statistics .................................. 81
5.1 An Overview of Probability and Statistics .............. 82
5.2 Discrete Probability Counts the Number of Ways Things
Can Happen ............................................. 82
5.3 Specific Techniques Are Needed for Discrete Counting ... 84
5.4 Counting Conditional and Independent Events That
Occur in Multistage Experiments Require Special
Considerations ......................................... 87
5.5 Discrete Distributions Come from Counting Up the
Outcomes of Repeated Experiments ....................... 89
5.6 Continuous Probability Is Represented as a Density
of Likelihood Rather Than by Counting Events ........... 94
Further Reading ............................................ 105
Problem Sets ............................................... 105
Part II Foundations
6 Energy and Force - The Prime Observables ................... 109
6.1 Experimental Models Are a Careful Abstraction of
Either Descriptive or Explanatory Models .............. 109
6.2 Potential Energy Surfaces Are Tools that Help Find
Structure Through the Measurement of Energy ........... 110
6.3 Conservative Systems Find Maximal Choice by
Balancing Kinetic and Potential Energies Over Time .... 113
6.4 Forces in Biological Systems Do the Work That
Influences Structure and Function ..................... 115
Further Reading ............................................ 122
Problem Sets ............................................... 123
7 Biophysical Forces in Molecular Systems .................... 125
7.1 Form and Function in Biomolecular Systems Are
Governed by a Limited Number of Forces ................ 126
7.2 Mechanical Motions Can Describe the Behavior of
Gases and the Migration of Cells ...................... 127
7.3 The Kinetic Theory of Gases Explains the Properties
of Gases Based on Mechanical Interactions
of Molecules .......................................... 129
7.4 The Electric Force Is the Essential Interaction that
Leads to the Chemical Nature of the Universe .......... 144
7.5 Wave Motion Is Important in Electromagnetic and
Mechanical Interactions in Biological Systems ......... 162
7.6 Harmonic Waves Are the Result of a Sinusoidal
Oscillation ........................................... 167
7.7 Energy and Intensity of Waves ......................... 174
7.8 Standing Waves ........................................ 176
7.9 Superposition and Interference - Waves of Different
Frequencies ........................................... 179
7.10 Complex Waveforms ..................................... 181
7.11 Wave Packets .......................................... 182
7.12 Dispersion ............................................ 184
7.13 The Wave Equation ..................................... 185
7.14 Waves in Two and Three Dimensions ..................... 186
Further Reading ............................................ 189
Problem Sets ............................................... 189
8 Physical Principles: Quantum Mechanics ..................... 191
8.1 The Story of the Discovery of Quantum Mechanics
Is an Instructive History of How Scientific Ideas
Are Modified .......................................... 192
8.2 From the Standpoint of the Philosophy of
Epistemological Science, the Quantum Revolution
Ended an Age of Certainty ............................. 192
8.3 The Ultraviolet Catastrophe Is a Term That Refers
to a Historical Failure of Classical Theory ........... 194
8.4 The Concept of Heat Capacity Was Modified by Quantum
Mechanical Considerations ............................. 202
8.5 The Photoelectric Effect and the Photon-Particle
Properties of Radiation Could Be Understood Using
Planck's Quanta ....................................... 203
8.6 Electromagnetic Radiation Has a Dual Nature ........... 205
8.7 de Broglie's Postulate Defines the Wavelike
Properties of Particles ............................... 206
8.8 The Electron Microscope Employs Particles as Waves
to Form Images ........................................ 207
8.9 The Uncertainty Principle Is an Essential Conclusion
of the Quantum Viewpoint .............................. 209
8.10 An Historical Approach to Understanding Atomic
Structure and the Atom ................................ 210
8.11 Quantum Mechanics Requires the Classical Trajectory
Across a Potential Energy Surface to be Replaced by
the Wavefunction ...................................... 218
8.12 Solutions to the Time-Independent Schrцdinger
Theory ................................................ 224
8.13 Building the Atomic Model - One-Electron Atoms ........ 235
8.14 Building the Atomic Model - Multi-electron Atoms ...... 238
Further Reading ............................................ 240
Problem Sets ............................................... 241
9 Chemical Principles ........................................ 243
9.1 Knowing the Distribution of Electrons in Molecules
Is Essential for Understanding Chemical Structure
and Behavior .......................................... 243
9.2 The Nature of Chemical Interactions ................... 244
9.3 Electrostatic Forces Describe the Interactions from
Salt Crystals to van der Waals Attraction ............. 244
9.4 Covalent Bonds Involve a True Sharing of Electrons
Between Atoms ......................................... 249
9.5 Hydrogen Bonds Are a Unique Hybrid of Interactions
and Play a Fundamental Role in the Behavior of
Biological Systems .................................... 262
9.6 Biological Systems Are Made from a Limited Number of
Elements .............................................. 264
Further Reading ............................................ 266
Problem Sets ............................................... 267
10 Measuring the Energy of a System: Energetics and the
First Law of Thermodynamics ................................ 269
10.1 Historically Heat Was Thought to Be a Fluid or "the
Caloric" .............................................. 269
10.2 The Thermodynamic Modeling Space Is a Systemic
Approach to Describing the World ...................... 271
10.3 The First Law States that "The Energy of the
Universe Is Conserved" ................................ 276
10.4 The Heat Capacity Is a Property that Can Reflect the
Internal Energy of a System ........................... 283
10.5 Enthalpy Is Defined When a System Is Held at Constant
Pressure .............................................. 285
Thought Questions .......................................... 289
Further Reading ............................................ 289
Problem Sets ............................................... 290
11 Entropy and the Second Law of Thermodynamics ............... 293
11.1 The Arrow of Time and Impossible Existence of
Perpetual Motion Machines Are Both Manifestations of
the Second Law of Thermodynamics ...................... 294
11.1.1 The Movement of a System Toward Equilibrium
Is the Natural Direction ....................... 295
11.2 The Design of a Perfect Heat Engine Is an Important
Thought Experiment .................................... 295
11.3 A Mechanical/Kinetic Approach to Entropy .............. 307
11.4 Statistical Thermodynamics Yields the Same
Conclusions as Classical Treatment of
Thermodynamics ........................................ 310
11.5 Entropy Can Be Described and Understood on
a Statistical Basis ................................... 321
11.6 The Third Law of Thermodynamics Defines an Absolute
Measure of Entropy .................................... 324
Further Reading ............................................ 324
Problem Sets ............................................... 325
12 Which Way Is That System Going? The Gibbs Free Energy ...... 327
12.1 The Gibbs Free Energy Is a State Function that
Indicates the Direction and Position of a System's
Equilibrium ........................................... 327
12.2 The Gibbs Free Energy Has Specific Properties ......... 329
12.3 The Free Energy Per Mole, μ, Is an Important
Thermodynamic Quantity ................................ 333
12.4 The Concept of Activity Relates an Ideal System to
a Real System ......................................... 333
12.5 The Application of Free Energy Considerations to
Multiple-Component Systems ............................ 334
12.6 The Chemical Potential Is an Important Driving
Force in Biochemical Systems .......................... 335
12.7 Entropy and Enthalpy Contribute to Calculations
of the Free Energy of Mixing .......................... 337
12.8 Finding the Chemical Equilibrium of a System Is
Possible by Making Free Energy Calculations ........... 340
12.9 The Thermodynamics of Galvanic Cells Is Directly
Related to the Gibbs Free Energy ...................... 345
12.10 Free Energy Changes Relate the Equilibrium Position
of Biochemical Reactions .............................. 348
Further Reading ............................................ 348
Problem Sets ............................................... 349
13 The Thermodynamics of Phase Equilibria ..................... 351
13.1 The Concept of Phase Equilibrium Is Important
in Biochemical Systems ................................ 351
13.2 Thermodynamics of Transfer Between Phases ............. 353
13.3 The Phase Rule Relates the Number of Variables of
State to the Number of Components and Phases at
Equilibrium ........................................... 353
13.4 The Equilibrium Between Different Phases Is Given
by the Clapeyron Equation ............................. 355
13.5 Surface Phenomena Are an Important Example of Phase
Interaction ........................................... 362
13.6 Binding Equilibria Relate Small Molecule Binding
to Larger Arrays ...................................... 364
Further Reading ............................................ 383
Problem Sets ............................................... 383
Part III Building a Model of Biomolecular Structure
14 Water: A Unique Solvent and Vital Component of Life ........ 389
14.1 An Introduction to the Most Familiar of All Liquids ... 389
14.2 The Physical Properties of Water Are Consistent with
a High Degree of Intermolecular Interaction ........... 391
14.3 Considering the Properties of Water as a Liquid ....... 392
14.4 The Structure of Monomolecular Water Can Be
Described Using a Variety of Models ................... 394
14.5 The Capacity of Water to Form Hydrogen Bonds
Underlies Its Unusual Properties ...................... 398
14.6 The Structure and Dynamics of Liquid Water Results
in "Ordered Diversity" That Is Probably Distinct
from Ice .............................................. 402
14.7 Hydrophobic Forces Reference Interactions Between
Water and Other Molecules ............................. 405
Further Reading ............................................ 407
Problem Sets ............................................... 408
15 Ion-Solvent Interactions ................................... 409
15.1 The Nature of Ion-Solvent Interactions Can Be
Discovered Through the Progression of Inquiry ......... 409
15.2 The Born Model Is a Thermodynamic Cycle That Treats
the Interaction Energy Between a Simplified Ion and
a Structureless Solvent ............................... 410
15.3 Adding Water Structure to the Solvent Continuum ....... 418
15.4 Extending the Ion-Solvent Model Beyond the Born
Model ................................................. 429
15.5 Solutions of Inorganic Ions ........................... 435
15.6 Ion-Solvent Interactions in Biological Systems ........ 437
Further Reading ............................................ 438
Problem Sets ............................................... 438
16 Ion-Ion Interactions ....................................... 441
16.1 Ion-Ion Interactions Can Be Modeled and These Models
Can Be Experimentally Validated and Refined ........... 441
16.2 The Debye-Hückel Model Is a Continuum Model That
Relates a Distribution of Nearby Ions to a Central
Reference Ion ......................................... 444
16.3 The Predictions Generated by the Debye-Hückel Model
Can Be Experimentally Evaluated ....................... 453
16.4 More Rigorous Treatment of Assumptions Leads to an
Improved Performance of the Debye-Hückel Model ....... 455
16.5 Consideration of Other Interactions Is Necessary to
Account for the Limits of the Debye-Hückel Model ..... 457
16.5.1 Bjerrum Suggested That Ion Pairing Could
Affect the Calculation of Ion-Ion
Interactions ................................... 457
Further Reading ............................................ 458
Problem Sets ............................................... 459
17 Lipids in Aqueous Solution ................................. 461
17.1 Biological Membranes Form at the Interface Between
Aqueous and Lipid Phases .............................. 461
17.2 Aqueous Solutions Can Be Formed with Small Nonpolar
Molecules ............................................. 462
17.3 Aqueous Solutions of Organic Ions Are an Amalgam of
Ion-Solvent and Nonpolar Solute Interaction ........... 465
17.4 Lipids Can Be Placed into Several Major Classes ....... 468
17.5 The Organization of Lipids into Membranes Occurs
When Aqueous and Lipid Phases Come in Contact ......... 474
17.6 The Physical Properties of Lipid Membranes ............ 478
17.7 Biological Membranes: A More Complete Picture ......... 482
Further Reading ............................................ 483
18 Macromolecules in Solution ................................. 485
18.1 The Physical Interactions of Polymers in Solution
Are Not Unique but Modeling the Interactions Will
Require Different Considerations Than Those of
Smaller Molecules ..................................... 486
18.2 Thermodynamics of Solutions of Polymers ............... 487
18.3 The Conformation of Simple Polymers Can Be Modeled
by a Random Walk and a Markov Process ................. 505
18.4 The Major Classes of Biochemical Species Form
Macromolecular Structures ............................. 506
18.5 Nonpolar Polypeptides in Solution ..................... 523
18.6 Polar Polypeptides in Solution ........................ 527
18.7 Transitions of State .................................. 531
18.8 The Protein Folding Problem ........................... 538
18.9 Pathological Protein Folding .......................... 542
Further Reading ............................................ 549
Problem Sets ............................................... 551
19 Molecular Modeling - Mapping Biochemical State Space ....... 553
19.1 The Prediction of Macromolecular Structure and
Function Is a Goal of Molecular Modeling .............. 553
19.2 Molecular Modeling Is Built on Familiar Principles .... 554
19.3 Empirical Methods Use Carefully Constructed Physical
Models ................................................ 555
19.4 Computational Methods Are the Ultimate Gedanken
Experiments ........................................... 569
19.5 Molecular Mechanics Is a Newtonian or Classical
Mechanical Modeling Approach .......................... 571
19.6 Quantum Mechanical Methods Are Computational
Difficult but Theoretically "Pure" .................... 579
Further Reading ............................................ 581
Problem Sets ............................................... 582
20 The Electrified Interphase ................................. 583
20.1 The Interphase Is Formed When Phases Meet ............. 583
20.2 A Detailed Structural Description of the Interphase
Is a Task for Physical Study .......................... 587
20.3 The Simplest Picture of the Interphase Is the
Helmholtz-Perrin Model ................................ 589
20.4 The Balance Between Thermal and Electrical Forces Is
Seen as Competition Between Diffuse-Layer Versus
Double-Layer Interphase Structures .................... 590
20.5 The Stern Model Is a Combination of the Capacitor
and Diffuse Layer Models .............................. 591
20.6 A More Complete Picture of the Double-Layer Forms
with Added Detail ..................................... 593
20.7 Colloidal Systems and the Electrified Interface Give
Rise to the Lyophilic Series .......................... 595
20.8 Salting Out Can Be Understood in Terms of
Electrified Interphase Behavior ....................... 599
Further Reading ............................................ 600
Problem Sets ............................................... 600
Part IV Function and Action Biological State Space
21 Transport - A Non-equilibrium Process ...................... 605
21.1 Transport Is an Irreversible Process and Does Not
Occur at Equilibrium .................................. 605
21.2 The Principles of Non-equilibrium Thermodynamics Can
Be Related to the More Familiar Equilibrium Treatment
with the Idea of Local Equilibrium .................... 606
Further Reading ............................................ 610
22 Flow in a Chemical Potential Field: Diffusion .............. 611
22.1 Transport in Chemical, Electrical, Pressure, and
Thermal Gradients Are All Treated with the Same
Mathematics ........................................... 611
22.2 Diffusion or the Flow of Particles Down
a Concentration Gradient Can Be Described
Phenomenologically .................................... 612
22.3 The Random Walk Forms the Basis for a Molecular
Picture of Flux ....................................... 616
Further Reading ............................................ 622
Problem Sets ............................................... 622
23 Flow in an Electric Field: Conduction ...................... 625
23.1 Transport of Charge Occurs in an Electric Field ....... 625
23.2 Describing a System of Ionic Conduction Includes
Electronic, Electrodic, and Ionic Elements ................. 627
23.3 The Flow of Ions Down a Electrical Gradient Can Be
Described Phenomenologically .......................... 631
23.4 A Molecular View of Ionic Conduction .................. 637
23.5 Interionic Forces Affect Conductivity ................. 640
23.6 Proton Conduction Is a Special Case that Has a Mixed
Mechanism ............................................. 643
Further Reading ............................................ 646
Problem Sets ............................................... 647
24 Forces Across Membranes .................................... 649
24.1 Energetics and Force in Membranes ..................... 649
24.2 The Donnan Equilibrium Is Determined by a Balance
Between Chemical and Electrical Potential in a Two-
Phase System .......................................... 650
24.3 Electric Fields Across Membranes Are of Substantial
Magnitude ............................................. 653
24.4 Electrostatic Profiles of the Membrane Are Potential
Energy Surfaces Describing Forces in the Vicinity of
Membranes ............................................. 657
24.5 The Electrochemical Potential Is a Thermodynamic
Treatment of the Gradients Across a Cellular
Membrane .............................................. 661
24.6 Transport Through the Lipid Bilayer of Different
Molecules Requires Various Mechanisms ................. 661
Further Reading ............................................ 666
Problem Sets ............................................... 667
25 Kinetics - Chemical Kinetics ............................... 669
25.1 The Equilibrium State Is Found by Chemical
Thermodynamics but Chemical Kinetics Tells the Story
of Getting There ...................................... 670
25.2 A Historical Perspective on the Development of
Chemical Kinetics ..................................... 671
25.3 Kinetics Has a Specific and Systemic Language ......... 675
25.4 Order of a Reaction Relates the Concentration of
Reactants to the Reaction Velocity .................... 676
25.5 Expressions of the Rate Laws Are Important
Properties of a Reaction .............................. 677
25.6 Elementary Reactions Are the Elements of the
System That Defines a Chemical Mechanism .............. 681
25.7 Reaction Mechanisms Are a System of Interacting
Elements (Molecules) in the Context of a Potential
Energy Surface ........................................ 682
25.8 Solution Kinetics Are More Complicated Than the
Simple Kinetic Behavior of Gases ...................... 699
25.9 Enzymes Are Macromolecular Catalysts with Enormous
Efficiency ............................................ 699
Further Reading ............................................ 710
Problem Sets ............................................... 710
26 Dynamic Bioelectrochemistry - Charge Transfer in
Biological Systems ......................................... 713
26.1 Electrokinetics and Electron Charge Transfer Depend
on Electrical Current Flow in Biochemical Systems ..... 713
26.2 Electrokinetic Phenomena Occur When the Elements of
the Biological Electrical Double Layer Experience
Either Mechanical or Electrical Transport ............. 714
26.3 Electron Transfer Is an Essential Form of Biological
Charge Transfer ....................................... 722
Further Reading ............................................ 736
Problem Sets ............................................... 737
Part V Methods for the Measuring Structure and Function
27 Separation and Characterization of Biomolecules Based
on Macroscopic Properties .................................. 741
27.1 Introduction: Mechanical Motion Interacts with Mass,
Shape, Charge, and Phase to Allow Analysis of
Macromolecular Structure .............................. 742
27.2 Buoyant Forces Are the Result of Displacement of the
Medium by an Object ................................... 742
27.3 Systems Study in the Biological Science Requires
Methods of Separation and Identification to Describe
the "State of a Biological System" .................... 760
27.4 Electrophoresis Is a Practical Application of
Molecular Motion in an Electrical Field Based on
Charge and Modified by Conformation and Size .......... 760
27.5 Chromatographic Techniques Are Based on the
Differential Partitioning of Molecules Between Two
Phases in Relative Motion ............................. 763
27.6 The Motion Induced by a Magnetic Interaction Is
Essential for Determination of Molecular Mass in
Modern Biological Investigations ...................... 767
Further Reading ............................................ 775
Problem Sets ............................................... 775
28 Analysis of Molecular Structure with Electronic
Spectroscopy ............................................... 779
28.1 The Interaction of Light with Matter Allows
Investigation of Biochemical Properties ............... 780
28.2 The Motion of a Dipole Radiator Generates
Electromagnetic Radiation ............................. 780
28.3 Optical Interactions Can Be Treated at Varying
Levels of Abstraction ................................. 780
28.4 Atomic and Molecular Energy levels Are a Quantum
Phenomenon That Provide a Window on Molecular
Structure ............................................. 782
28.5 Absorption Spectroscopy Has Important Applications
to Biochemical Analysis ............................... 789
28.6 Fluorescence and Phosphorescence Occur When Trapped
Photon Energy Is Re-radiated After a Finite
Lifetime .............................................. 794
28.7 Electron Paramagnetic Resonance (EPR) and Nuclear
Magnetic Resonance (NMR) Depend on Interactions
Between Photons and Molecules in a Magnetic Field ..... 797
Further Reading ............................................ 812
Problem Sets ............................................... 813
29 Molecular Structure from Scattering Phenomena .............. 815
29.1 The Interference Patterns Generated by the
Interaction of Waves with Point Sources Is
a Valuable Tool in the Analysis of Structure .......... 815
29.2 Diffraction Is the Result of the Repropagation of
a Wave ................................................ 819
29.3 X-Ray Diffraction Is a Powerful Fool for Structure
Determination ......................................... 822
29.4 Scattering of Light Rather Than Its Absorption Can
Be Used to Probe Molecular Structure and
Interaction ........................................... 831
Further Reading ............................................ 835
30 Analysis of Structure - Microscopy ......................... 837
30.1 Seeing Is Believing ................................... 837
30.2 The Light Microscope Allows Visualization of
Structures on the Dimensional Scale of the
Wavelength of a Photon ................................ 839
30.3 Visualization Requires Solving the Problem of
Contrast .............................................. 843
30.4 Scanning Probe Microscopy Creates an Image of a
Structures by Interactions on a Molecular Scale ....... 849
Further Reading ............................................ 854
Problem Sets ............................................... 855
31 Epilogue ................................................... 857
Now Try .................................................... 857
32 Physical Constants ......................................... 859
Conversions ................................................ 859
Appendix A: Mathematical Methods .............................. 861
A.l Units and Measurement ................................. 861
A.2 Exponents and Logarithms .............................. 862
A.3 Trigonometric Functions ............................... 865
A.4 Expansion Series ...................................... 869
A.5 Differential and Integral Calculus .................... 870
A.6 Partial Differentiation ............................... 870
A.7 Vectors ............................................... 872
Appendix B: Quantum Electrodynamics ........................... 875
Appendix C: The Pre-Socratic Roots of Modern Science .......... 877
Appendix D: The Poisson Function .............................. 879
Appendix E: Assumptions of a Theoretical Treatment of the
Ideal Gas Law .............................................. 881
Appendix F: The Determination of the Field from the
Potential in Cartesian Coordinates ......................... 883
Appendix G: Geometrical Optics ................................ 885
G.l Reflection and Refraction of Light .................... 885
G.2 Mirrors ............................................... 886
G.3 Image Formation by Refraction ......................... 889
G.4 Prisms and Total Internal Reflection .................. 891
Appendix H: The Compton Effect ................................ 893
Appendix I: Hamilton's Principle of Least Action/Fermat's
Principle of Least Time .................................... 895
Appendix J: Derivation of the Energy of Interaction Between
Two Ions ................................................... 897
Appendix K: Derivation of the Statement, qrev > qirrev ......... 899
Appendix L: Derivation of the Clausius-Clapeyron Equation ..... 901
Appendix M: Derivation of the van't Hoff Equation for
Osmotic Pressure ........................................... 903
Appendix N: Fictitious and Pseudoforces - The Centrifugal
Force ...................................................... 905
Appendix O: Derivation of the Work to Charge and Discharge
a Rigid Sphere ............................................. 907
Appendix P: Review of Circuits and Electric Current ........... 909
P.l Current Density and Flux .............................. 909
P.2 Circuits .............................................. 911
P.3 Measuring Instruments ................................. 916
Appendix Q: Fermi's Golden Rule ............................... 919
Appendix R: The Transition from Reactant to Product:
Adiabatic and Non-adiabatic Transitions .................... 921
Index ......................................................... 923
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