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Physical Chemistry : Principles and Applications in Biological Sciences

by ; ; ;
Edition:
4th
ISBN13:

9780130959430

ISBN10:
013095943X
Format:
Paperback
Pub. Date:
8/6/2001
Publisher(s):
Prentice Hall
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Summary

This best-selling volume presents the principles and applications of physical chemistry as they are used to solve problems in biology and medicine.The First Law; the Second Law; free energy and chemical equilibria; free energy and physical Equilibria; molecular motion and transport properties; kinetics: rates of chemical reactions; enzyme kinetics; the theory and spectroscopy of molecular structures and interactions: molecular distributions and statistical thermodynamics; and macromolecular structure and X-ray diffraction.For anyone interested in physical chemistry as it relates to problems in biology and medicine.

Author Biography

Ignacio Tinoco was an undergraduate at the University of New Mexico, a graduate student at the University of Wisconsin, and a postdoctoral fellow at Yale. He then went to the University of California, Berkeley, where he has remained. His research interest has been on the structures of nucleic acids, particularly RNA. He was chairman of the Department of Energy committee that recommended in 1987 a major initiative to sequence the human genome. His present research is on unfolding single RNA molecules by force.

Kenneth Sauer grew up in Cleveland, Ohio, and received his A.B. in chemistry from Oberlin College. Following his Ph.D. studies in gas-phase physical chemistry at Harvard, he spent three years teaching at the American University of Beirut, Lebanon. A postdoctoral opportunity to learn from Melvin Calvin about photosynthesis in plants led him to the University of California, Berkeley, where he has been since 1960. Teaching general chemistry and biophysical chemistry in the Chemistry Department has complemented research in the Physical Biosciences Division of the Lawrence Berkeley National Lab involving spectroscopic studies of photosynthetic light reactions and their role in water oxidation. His other activities include reading, renaissance and baroque choral music, canoeing, and exploring the Sierra Nevada with his family and friends.

James C. Wang was on the faculty of the University of California, Berkeley, from 1966 to 1977. He then joined the faculty of Harvard University, where he is presently Mallinckrodt Professor of Biochemistry and Molecular Biology. His research focuses on DNA and enzymes that act on DNA, especially a class of enzymes known as DNA topoisomerases. He has taught courses in biophysical chemistry and molecular biology and has published over 200 research articles. He is a member of Academia Sinica, the American Academy of Arts and Sciences, and the U.S. National Academy of Sciences.

Joseph Puglisi was born and raised in New Jersey. He received his B.A. in chemistry from The Johns Hopkins University in 1984 and his Ph.D. from the University of California, Berkeley, in 1989. He has studied and taught in Strasbourg, Boston, and Santa Cruz, and is currently professor of structural biology at Stanford University. His research interests are in the structure and mechanism of the ribosome and the use of NMR spectroscopy to study RNA structure. He has been a Dreyfus Scholar, Sloan Scholar, and Packard Fellow.

Table of Contents

Preface xvii
About the Authors xix
Introduction
The Human Genome and Beyond
4(2)
Transcription and Translation
6(4)
Ion Channels
10(1)
References
11(1)
Suggested Reading
12(1)
Problem
12(3)
The First Law: Energy Is Conserved
Concepts
15(1)
Applications
16(1)
Energy Conversion and Conservation
16(13)
Systems and Surroundings
17(1)
Energy Exchanges
18(10)
First Law of Thermodynamics
28(1)
Describing the State of a System
29(16)
Variables of State
29(2)
Equations of State
31(2)
Paths Connecting Different States
33(3)
Dependence of the Energy and Enthalpy of a Pure Substance on P, V, and T
36(8)
Relations Between Heat Exchanges and DE and DH
44(1)
Phase Changes
45(2)
Chemical Reactions
47(9)
Heat Effects of Chemical Reactions
47(3)
Temperature Dependence of ΔH
50(1)
The Energy Change ΔE for a Reaction
51(1)
Standard Enthalpies (or Heats) of Formation
51(2)
Bond Energies
53(3)
Molecular Interpretations of Energy and Enthalpy
56(1)
Summary
57(3)
State Variables
57(1)
Unit Conversions
57(1)
General Equations
57(1)
Pressure-Volume Work Only
58(1)
Solids and Liquids
58(1)
Gases
59(1)
Phase Changes
59(1)
Chemical Reactions
60(1)
Mathematics Needed for Chapter 2
60(1)
References
61(1)
Suggested Reading
61(1)
Problems
61(8)
The Second Law: The Entropy of the Universe Increases
Concepts
69(1)
Applications
69(1)
Historical Development of the Second Law: The Carnot Cycle
69(4)
A New State Function, Entropy
73(2)
The Second Law of Thermodynamics: Entropy Is Not Conserved
75(2)
Molecular Interpretation of Entropy
77(4)
Fluctuations
79(2)
Measurement of Entropy
81(1)
Chemical Reactions
81(1)
Third Law of Thermodynamics
82(5)
Temperature Dependence of Entropy
82(1)
Temperature Dependence of the Entropy Change for a Chemical Reaction
83(1)
Entropy Change for a Phase Transition
84(1)
Pressure Dependence of Entropy
85(2)
Spontaneous Chemical Reactions
87(1)
Gibbs Free Energy
87(10)
ΔG and a System's Capacity to Do Nonexpansion Work
87(1)
Spontaneous Reactions at Constant T and P
88(1)
Calculation of Gibbs Free Energy
89(2)
Temperature Dependence of Gibbs Free Energy
91(3)
Pressure Dependence of Gibbs Free Energy
94(3)
Phase Changes
97(1)
Helmholtz Free Energy
97(1)
Noncovalent Reactions
97(9)
Hydrophobic Interactions
100(1)
Proteins and Nucleic Acids
101(5)
Use of Partial Derivatives in Thermodynamics
106(5)
Relations Among Partial Derivatives
107(4)
Summary
111(2)
State Variables
111(1)
Unit Conversions
111(1)
General Equations
111(1)
ΔG and a System's Capacity to Do Nonexpansion Work
111(1)
Spontaneous Reactions at Constant T and P
111(1)
Changes in Entropy and Gibbs Free Energy
112(1)
References
113(1)
Suggested Reading
113(1)
Problems
113(8)
Free Energy and Chemical Equilibria
Concepts
121(1)
Applications
122(1)
Chemical Potential (Partial Molar Gibbs Free Energy)
122(3)
Gibbs Free Energy and the Chemical Potential
122(1)
The Sum Rule for Partial Molar Quantities
123(1)
Chemical Potential and Directionality of Chemical Reaction
123(2)
Reactions of Gases: The Ideal Gas Approximation
125(5)
Dependence of Chemical Potential on Partial Pressures
125(2)
Equilibrium Constant
127(3)
Nonideal Systems
130(11)
Activity
130(1)
Standard States
131(8)
Activity Coefficients of Ions
139(2)
The Equilibrium Constant and the Standard Gibbs Free Energies of the Reactants and Products
141(12)
Calculation of Equilibrium Concentrations: Ideal Solutions
144(6)
Temperature Dependence of the Equilibrium Constant
150(3)
Galvanic Cells
153(6)
Standard Electrode Potentials
156(2)
Concentration Dependence of
158(1)
Biochemical Applications of Thermodynamics
159(11)
Thermodynamics of Metabolism
165(5)
Biological Redox Reactions
170(6)
NADH-Q Reductase
171(1)
Cytochrome Reductase
172(1)
Cytochrome c Oxidase
172(1)
Double Strand Formation in Nucleic Acids
172(3)
Ionic Effect on Protein-Nucleic Acid Interactions
175(1)
Summary
176(3)
Chemical Potential (Partial Molar Gibbs Free Energy)
176(1)
Standard States and Activities
177(1)
Gibbs Free-Energy Change and Equilibrium Constant for a Chemical Reaction
178(1)
Galvanic Cells
178(1)
Mathematics Needed for Chapter 4
179(1)
References
179(1)
Suggested Reading
179(1)
Problems
179(8)
Free Energy and Physical Equilibria
Concepts
187(1)
Applications
187(1)
Membranes and Transport
187(1)
Ligand Binding
188(1)
Colligative Properties
188(1)
Phase Equilibria
188(25)
One-Component Systems
189(1)
Boiling Point and Freezing Point
189(4)
Solutions of Two or More Components
193(4)
Equilibrium Dialysis
197(1)
The Scatchard Equation
198(4)
Cooperative Binding and Anticooperative Binding
202(4)
Free Energy of Transfer Between Phases
206(4)
Donnan Effect and Donnan Potential
210(3)
Membranes
213(14)
Lipid Molecules
213(1)
Lipid Bilayers
214(2)
Phase Transitions in Lipids, Bilayers, and Membranes
216(2)
Surface Tension
218(4)
Surface Free Energy
222(2)
Vapor Pressure and Surface Tension
224(1)
Biological Membranes
225(2)
Active and Passive Transport
227(4)
Colligative Properties
231(8)
Molecular-Weight Determination
239(2)
Vapor-Pressure Lowering
240(1)
Summary
241(3)
Phase Equilibrium
241(1)
Solutions
242(2)
References
244(1)
Suggested Reading
245(1)
Internet
245(1)
Problems
245(8)
Molecular Motion and Transport Properties
Concepts
253(1)
Applications
254(1)
Kinetic Theory
255(10)
Brownian Motion and Random Molecular Motion
255(1)
Velocities of Molecules, Translational Kinetic Energy, and Temperature
256(5)
Maxwell-Boltzmann Distribution of Velocities
261(4)
Molecular Collisions
265(1)
Mean Free Path
266(1)
Diffusion
267(12)
The Random Walk and Diffusion in a Gas
267(2)
Diffusion Coefficient and Fick's First Law
269(2)
Fick's Second Law
271(1)
Determination of the Diffusion Coefficient
272(1)
Relationship Between the Diffusion Coefficient and the Mean-Square Displacement
273(1)
Determination of the Diffusion Coefficient by Laser Light Scattering
274(1)
Diffusion Coefficient and Molecular Parameters
275(1)
Solvation
276(1)
Shape Factor
277(2)
Diffusion Coefficients of Random Coils
279(1)
Sedimentation
279(6)
Determination of the Sedimentations Coefficient
281(2)
Standard Sedimentation Coefficient
283(2)
Determination of Molecular Weights from Sedimentation and Diffusion
285(4)
Determination of Molecular Weights from Sedimentation Equilibrium
285(3)
Density-Gradient Centrifugation
288(1)
Viscosity
289(2)
Measurement of Viscosity
290(1)
Viscosities of Solutions
291(1)
Electrophoresis
291(10)
Gel Electrophoresis
292(1)
DNA Sequencing
293(1)
Double-Stranded DNA
294(1)
DNA Fingerprinting
294(2)
Conformations of Nucleic Acids
296(1)
Pulsed-Field Gel Electrophoresis
297(2)
Protein Molecular Weights
299(1)
Protein Charge
300(1)
Macromolecular Interactions
301(1)
Size and Shape of Macromolecules
301(1)
Summary
302(5)
Kinetic Theory
302(2)
Diffusion
304(1)
Sedimentation
304(1)
Frictional Coefficient and Molecular Parameters
305(1)
Combination of Diffusion and Sedimentation
305(1)
Viscosity
305(1)
Electrophoresis
306(1)
Gel Electrophoresis
306(1)
References
307(1)
Suggested Reading
307(1)
Problems
307(8)
Kinetics: Rates of Chemical Reactions
Concepts
315(1)
Applications
316(1)
Kinetics
316(25)
Rate Law
318(1)
Order of a Reaction
318(2)
Experimental Rate Data
320(1)
Zero-Order Reactions
321(1)
First-Order Reactions
322(7)
Second-Order Reactions
329(5)
Renaturation of DNA as an Example of a Second-Order Reaction
334(4)
Reactions of Other Orders
338(1)
Determining the Order and Rate Constant of a Reaction
338(3)
Reaction Mechanisms and Rate Laws
341(13)
Parallel Reactions
343(2)
Series Reactions (First Order)
345(4)
Equilibrium and Kinetics
349(2)
Complex Reactions
351(1)
Deducing a Mechanism from Kinetic Data
352(2)
Temperature Dependence
354(3)
Transition-State Theory
357(3)
Electron Transfer Reactions: Marcus Theory
360(2)
Ionic Reactions and Salt Effects
362(1)
Isotopes and Stereochemical Properties
363(2)
Very Fast Reactions
365(7)
Relaxation Methods
365(1)
Relaxation Kinetics
366(6)
Diffusion-Controlled Reactions
372(2)
Photochemistry and Photobiology
374(4)
Vision
377(1)
Photosynthesis
378(3)
Summary
381(6)
Zero-Order Reactions
381(1)
First-Order Reactions
381(1)
Second-Order Reactions
382(1)
Temperature Dependence
383(2)
Electron Transfer Reactions: Marcus Theory
385(1)
Relaxation Kinetics
385(1)
Diffusion-Controlled Reactions
386(1)
Absorption of Light
386(1)
Photochemistry
386(1)
Mathematics Needed for Chapter 7
387(1)
References
387(1)
Suggested Reading
388(1)
Problems
388(13)
Enzyme Kinetics
Concepts
401(1)
Applications
401(2)
Catalytic Antibodies and RNA Enzymes-Ribozymes
401(2)
Enzyme Kinetics
403(3)
Michaelis-Menten Kinetics
406(9)
Kinetic Data Analysis
409(4)
Two Intermediate Complexes
413(2)
Competition and Inhibition
415(8)
Competion
415(1)
Competitive Inhibition
416(2)
Noncompetitive Inhibition
418(1)
Allosteric Effects
419(3)
Single-Molecule Kinetics
422(1)
Summary
423(2)
Typical Enzyme Kinetics
423(1)
Michaelis--Menten Mechanism
424(1)
Monod-Wyman-Changeux Mechanism
425(1)
Mathematics Needed for Chapter 8
425(1)
References
426(1)
Suggested Reading
426(1)
Problems
427(10)
Molecular Structures and Interactions: Theory
Concepts
437(1)
Applications
437(1)
The Process of Vision
438(3)
Origins of Quantum Theory
441(5)
Blackbody Radiation
442(2)
Photoelectric Effect
444(1)
Electrons as Waves
444(1)
Heisenberg Uncertainty Principle
445(1)
Quantum Mechanical Calculations
446(3)
Wave Mechanics and Wavefunctions
446(3)
Schrodinger's Equation
449(6)
Solving Wave Mechanical Problems
451(1)
Outline of wave Mechanical Procedures
452(3)
Particle in a Box
455(8)
Tunneling
463(2)
Simple Harmonic Oscillator
465(3)
Rigid Rotator
468(1)
Hydrogen Atom
469(1)
Electron Distribution
470(19)
Electron Distribution in a Hydrogen Atom
471(5)
Many-Electron Atoms
476(3)
Molecular Orbitals
479(5)
Hybridization
484(2)
Delocalized Orbitals
486(3)
Molecular Structure and Molecular Orbitals
489(4)
Geometry and Stereochemistry
489(2)
Transition Metal Ligation
491(2)
Charge Distributions and Dipole Moments
493(1)
Intermolecular and Intramolecular Forces
493(4)
Bond Stretching and Bond Angle Bending
494(1)
Rotation Around Bonds
495(2)
Noncovalent Interactions
497(17)
Electrostatic Energy and Coulomb's Law
497(3)
Net Atomic Charges and Dipole Moments
500(3)
Dipole-Dipole Interactions
503(2)
London Attraction
505(1)
van der Waals Repulsion
506(1)
London-van der Waals Interaction
507(1)
The Lowest-Energy Conformation
508(2)
Hydrogen Bonds
510(2)
Hydrophobic and Hydrophilic Environments
512(2)
Molecular Dynamics Simulation
514(2)
Monte Carlo Method
514(1)
Molecular Dynamics Method
515(1)
Outlook
516(1)
Summary
517(1)
Photoelectric Effect
517(1)
Wave-Particle Duality
517(1)
Heisenberg Uncertainty Principle
517(1)
Schrodinger's Equation
518(1)
Some Useful Operators
518(5)
Systems Whose Schrodinger Equation Can Be Solved Exactly
519(2)
Coulomb's Law
521(1)
Dipoles and Their Interaction Energy
521(1)
Intramolecular (Within) and Intermolecular (Between) Interactions
521(2)
Mathematics Needed for Chapter 9
References
523(1)
Suggested Reading
523(1)
Problems
524(7)
Molecular Structures and Interactions: Spectroscopy
Concepts
531(1)
Applications
532(1)
Electromagnetic Spectrum
532(1)
Color and Refractive Index
533(2)
Absorption and Emission of Radiation
535(13)
Radiation-Induced Transitions
536(2)
Classical Oscillators
538(1)
Quantum Mechanical Description
538(2)
Lifetimes and Line Width
540(1)
Role of Environment in Electronic Absorption Spectra
541(2)
Beer-Lambert Law
543(5)
Proteins and Nucleic Acids: Ultraviolet Absorption Spectra
548(6)
Amino Acid Spectra
549(1)
Polypeptide Spectra
549(2)
Secondary Structure
551(1)
Origin of Spectroscopic Changes
551(1)
Nucleic Acids
552(1)
Rhodopsin: A Chromorphic Protein
553(1)
Fluorescence
554(13)
Simple Theory
555(1)
Excited-State Properties
556(4)
Fluorescence Quenching
560(1)
Excitation Transfer
561(2)
Molecular Rulers
563(1)
Fluorescence Polarization
564(1)
Phosphorescence
565(1)
Single-Molecule Fluorescence Spectroscopy
565(2)
Optical Rotatory Dispersion and Circular Dichroism
567(6)
Polarized Light
568(3)
Optical Rotation
571(2)
Circular Dichroism
573(1)
Circular Dichroism of Nucleic Acids and Proteins
573(3)
Vibrational Spectra, Infrared Absorption, and Raman Scattering
576(3)
Infrared Absorption
576(1)
Raman Scattering
577(2)
Nuclear Magnetic Resonance
579(4)
The Spectrum
581(2)
Interactions in Nuclear Magnetic Resonance
583(15)
Chemical Shifts
583(2)
Spin-Spin Coupling, Scalar Coupling, or J-Coupling
585(3)
Relaxation Mechanisms
588(2)
Nuclear Overhauser Effect
590(1)
Multidimensional NMR Spectroscopy
590(4)
Determination of Macromolecular Structure by Nuclear Magnetic Resonance
594(2)
Electron Paramagnetic Resonance
596(1)
Magnetic Resonance Imaging
597(1)
Summary
598(3)
Absorption and Emission
598(2)
Excitation Transfer
600(1)
Optical Rotatory Dispersion and Circular Dichroism
600(1)
Nuclear Magnetic Resonance
600(1)
References
601(1)
Suggested Reading
601(2)
Problems
603(12)
Molecular Distributions and Statistical Thermodynamics
Concepts
615(1)
Applications
615(1)
Binding of Small Molecules by a Polymer
616(12)
Identical-and-Independent-Sites Model
617(2)
Langmuir Adsorption Isotherm
619(1)
Nearest-Neighbor Interactions and Statistical Weights
620(2)
Cooperative Binding, Anticooperative Binding, and Excluded-Site Binding
622(3)
N Identical Sites in a Linear Array with Nearest-Neighbor Interactions
625(1)
Identical Sites in Nonlinear Arrays with Nearest-Neighbor Interactions
626(2)
The Random Walk
628(8)
Calculation of Some Mean Values for the Random-Walk Problem
630(4)
Diffusion
634(1)
Average Dimension of a Linear Polymer
634(2)
Helix-Coil Transitions
636(9)
Helix-Coil Transition in a Polypeptide
636(5)
Helix-Coil Transition in a Double-Stranded Nucleic Acid
641(4)
Statistical Thermodynamics
645(14)
Statistical Mechanic Internal Energy
646(1)
Work
647(1)
Heat
648(1)
Most Probable (Boltzmann) Distribution
649(4)
Quantum Mechanical Distributions
653(1)
Statistical Mechanical Entropy
653(1)
Examples of Entropy and Probability
654(4)
Partition Function: Applications
658(1)
Summary
659(3)
Binding of Small Molecules by a Polymer
659(1)
Random-Walk and Related Topics
660(1)
Helix-Coil Transitions
660(1)
Statistical Thermodynamics
661(1)
Mathematics Needed for Chapter 11
662(1)
References
662(1)
Suggested Reading
663(1)
Problems
663(4)
Macromolecular Structure and X-Ray Diffraction
Concepts
667(1)
Applications
667(1)
Visible Images
668(1)
X Rays
668(16)
Emission of X Rays
669(1)
Image Formation
669(1)
Scattering of X Rays
670(5)
Diffraction of X Rays by a Crystal
675(3)
Measuring the Diffraction Pattern
678(1)
Bragg Reflection of X Rays
679(2)
Intensity of Diffraction
681(2)
Unit Cell
683(1)
Determination of Molecular Structure
684(14)
Calculation of Diffracted Intensities from Atomic Coordinates: The Structure Factor
684(2)
Calculation of Atomic Coordinates from Diffracted Intensities
686(2)
The Phase Problem
688(1)
Direct Methods
688(1)
Isomorphous Replacement
688(2)
Multiwavelength Anomalous Diffraction
690(1)
Determination of a Crystal Structure
691(3)
Scattering of X Rays by Noncrystalline Materials
694(1)
Absorption of X Rays
695(1)
Extended Fine Structure of Edge Absorption
696(1)
X Rays from Synchrotron Radiation
697(1)
Electron Diffraction
698(1)
Neutron Diffraction
699(1)
Electron Microscopy
700(4)
Resolution, Contrast, and Radiation Damage
700(1)
Transmission and Scanning Electron Microscopes
701(1)
Image Enhancement and Reconstruction
701(1)
Scanning Tunneling and Atomic Force Microscopy
702(2)
Summary
704(3)
X-ray Diffraction
704(3)
Neutron Diffraction
707(1)
Electron Microscopy
707(1)
Mathematics Needed for Chapter 12
707(1)
References
708(1)
Suggested Reading
708(1)
Problems
709(3)
Appendix 712(13)
Answers 725(3)
Index 728

Excerpts

PREFACE There is a deep sense of pleasure to be experienced when the patterns and symmetry of nature are revealed. Physical chemistry provides the methods to discover and understand these patterns. We think that not only is it important to learn and apply physical chemistry to biological problems, it may even be fun. In this book, we have tried to capture some of the excitement of making new discoveries and finding answers to fundamental questions. This is not an encyclopedia of physical chemistry. Rather, we have written this text specifically with the life-science student in mind. We present a streamlined treatment that covers the core aspects of biophysical chemistry (thermodynamics and kinetics as well as quantum mechanics, spectroscopy, and X-ray diffraction), which are of great importance to students of biology and biochemistry. Essentially all applications of the concepts are to systems of interest to life-science students; nearly all the problems apply to life-science examples. For this fourth edition we are joined by Joseph Puglisi, a new, young author who strengthens the structural biology content of the book. We have also tried to make the book more reader-friendly. In particular, we omit fewer steps in the explanations to make the material more understandable, and we have followed the many helpful and specific recommendations of our reviewers to improve the writing throughout. Important new topics, such as single-molecule thermodynamics, kinetics, and spectroscopy, are introduced. Subjects that have become less pertinent to current biophysical chemistry have been deleted or de-emphasized. Reference lists for each chapter have been updated. However, the format and organization of the book is essentially unchanged. Chapter 1 introduces representative areas of active current research in biophysical chemistry and molecular biology: the human genome, the transfer of genetic information from DNA to RNA to protein, ion channels, and cell-to-cell communication. We encourage students to read the current literature to see how the vocabulary and concepts of physical chemistry are used in solving biological problems. Chapters 2 through 5 cover the laws of thermodynamics and their applications to chemical reactions and physical processes. Essentially all of the examples and problems deal with biochemical and biological systems. For example, after defining work as a force multiplied by the distance moved (the displacement), we discuss the experimental measurement of the work necessary to stretch a single DNA molecule from its random-coiled form to an extended rod. Molecular interpretations of energies and entropies are emphasized in each of the chapters. Chapter 4, "Free Energy and Chemical Equilibria," now starts with the application of the chemical potential td chemical reactions. We think that this will make it easier to understand the logic relating activities and equilibrium constants to free energy. Binding of ligands and equilibria between phases are described in chapter 5, "Free Energy and Physical Equilibria." We discuss in detail the allosteric effect and the cooperative binding of oxygen by hemoglobin. We also describe the formation of lipid monolayers, lipid bilayers, and micelles, and their structures are compared to biological membranes. Chapters 6 through 8 cover molecular motion and chemical kinetics. Chapter 6, "Molecular Motion and Transport Properties," starts with the Brownian motion on an aqueous surface of a single lipid molecule labeled with a fluorescent dye. The random motion of the molecule can be followed to test Einstein's equation relating average distance traveled by a single molecule to a bulk diffusion coefficient. Following this direct experimental demonstration of thermal motion of a molecule, we introduce the kinetic theory of gases and discuss transport properties (diffusion, sedimentation, and electrophoresis) of macromolecules. The next two chap


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