A reissue of a classic book, corrected, edited, and typeset, to be published in the Biological Physics Series. Intended for undergraduate courses in biophysics, biological physics, physiology, medical physics, and biomedical engineering, this book is an introduction to statistical physics with examples and problems from the medical and biological sciences. Topics include the elements of the theory of probability, Poisson statistics, thermal equilibrium, entropy and free energy, and the second law of thermodynamics. Chapters include problems and references. The book can be used as a supplement to standard introductory physics courses, and as a text for medical schools, medical physics courses, and biology departments. The three volumes combined present all the major topics in physics. Originally published in 1974 from the authors's typescript, this reissue will be edited, corrected, typeset, the art redrawn, and an index added. These books are being reissued by Springer in the Biological Physics Series in response to frequent requests to provide these texts to satisfy the growing need among students and practitioners in the medical and biological sciences with a working knowledge of the physical sciences. The books are also in demand in physics departments either as supplements to traditional intro texts or as main text for those departments offering courses with biological or medical physics orientation. A solutions manual will be available. The authors are recognized experts in the field, and are under contract for an upperlevel/grad text in biological physics. Benedeck was the recepient of the 1995 Irving Lanmuir Prize from the American Physical Society for Chemical Physics, and the 1994 Biological Physics Prize by the American Physical Society.
1 Elements of the Theory of Probability: The Binomial Distribution: Applications.- 1.1 Definition of Probability. The Two Rules. Illustrative Examples.- 1.2 Bernoulli Trials. The Binomial Distribution.- 1.3 Mean Values and Variance.- 1.4 Illustrative Applications.- 1.4.A. The Sex Distribution of Children.- 1.4.B. Random Coils: The Conformation of Chain Polymers.- 1.4.C. The Distribution of Electric Charges on the Hemoglobin Molecule.- Appendix to Section 1.4.C: Probabilities for the State of Ionization of a Polar Residue.- 1.5 References and Supplementary Reading.- 1.6 Problems.- 2 Diffusion and Transport Processes.- 2.1 Molecular Movement and the Physical Properties of Gases: A Short Survey.- 2.1.A. Ideal Gas Law. Kelvin Temperature. Avogadro’s Number.- 2.1.B. Mean Kinetic Energy of a Molecule. The Boltzmann Constant.- 2.l.C. The Equipartition Law. Specific Heats.- 2.1.D. Random Motion of a Gas Molecule, Root Mean Square Velocity, Mean Free Path, and Collision Frequency.- 2.2 Random Walk in One and Three Dimensions.- 2.2.A. The Bernoulli Distribution for the Probability PN (X) of a Displacement? in N Steps.- 2.2.B. The Gaussian Form of the Bernoulli Distribution.- 2.2.C. Space-Time Evolution of the Probability Distribution. The Diffusion Constant. The Mean Square Displacement as a Function of Time.- 2.2.D. Probability of Displacements for the Three-Dimensional Random Walk. Numerical Values for Diffusion Constants. Some Elementary Applications.- 2.2.E. Elementary Application: The Transfer of Oxygen and Carbon Dioxide in the Human Lung.- 2.3 The Diffusion Equation.- 2.3.A. The Space-Time Evolution of Particle Distribution. Integral Representations for Concentration C (x, t).- 2.3.B. Application of the Integral Representation for C(x, t). The Experiment of Lam and Poison. Determination of Diffusion Constant D.- 2.3.C. The Diffusion Equation for C (x, t).- 2.3.D. An Application: Smoothing Out of Sinusoidal Variations in Concentration.- 2.4 Particle Conservation, Particle Current, and Fick’s Law.- 2.4.A. Particle Conservation, Current, and the Continuity Equation.- 2.4.B. The Relation Between Current and a Concentration Gradient. Fick’s Law.- 2.4.C. Flow and Diffusion Across Porous Membranes in the Presence of Either a Concentration Difference ?C or a Pressure Difference ?P.- (i) Volume Flow Across a Porous Membrane Under the Influence of a Pressure Gradient. The Hydraulic Permeability Lp.- (ii) Solute Flow Across a Porous Membrane Due to a Concentration Gradient. The Membrane Permeability p.- (iii) Numerical Values for the Filtration Coefficient Lp and Permeability p. Theory and Experiment Compared. The Hindrance Factor.- (iv) Molecular Sieving by Membranes. The Reflection Coefficient a. Introduction to the Relation Between Solute Flow JS, Volume Flow JV, and the Concentration and Pressure Differences ?C and ?p Across the Membrane.- (v) Equalization Time for the Concentration Difference Across a Membrane, a Two-Compartment Problem.- 2.4.D. Hemodialysis. The Artificial Kidney.- (i) Physiological Role of the Kidney.- (ii) Description and Function of the Artificial Kidney.- 2.5 Flow and Diffusion of Particles Under the Action of External Forces and Collisions with Solvent Molecules.- 2.5.A. Flow, Collisions, and Momentum Transfer in a Concentration Gradient.- 2.5.B. Particle Current and the Diffusion Equation in the Presence of a Concentration Gradient and Externally Applied Forces. Drift Velocity.- 2.5.C. Mobility and the Stokes—Einstein Relation.- 2.5.D. Sedimentation Equilibrium: Scale Heights and the Molecular Weights of Macromolecules. Perrin’s Experimental Measurement of Avogadro’s Number.- 2.5.E. Ultracentrifugation.- (i) Design and Performance of the Ultracentrifuge.- (ii) The Sedimentation Coefficient s. Determination of Molecular Weights.- (iii) Determination of Molecular Weights from Sedimentation Equilibrium: Some Data.- 2.6 Flow of Solute and Solvent Across a Membrane in the Presence of Both Pressure and Concentration Gradients.- 2.6.A. Hydrostatic Pressure. Semipermeable Membrane. Osmotic Pressure. Van t’Hoff s Law. Volume Flow (Jv) Across a Semipermeable Membrane in the Presence of Both ?p and ?C.- (i) Hydrostatic Pressure.- (ii) Phenomenological Description of Osmotic Pressure and Volume Flow Across a Semipermeable Membrane. Van t’Hoff’s Law.- (iii) Physical Origin and the Theory for the Osmotic Pressure. Derivation of Van t’Hoff’s Law. Poisseuille Flow and the Flow of Solvent Through a Semipermeable Membrane Under the Influence of Both Pressure and Concentration Differences.- 2.6.B. Coupled Flow of Solute and Solvent Across a Membrane Subject to Both a Hydrostatic and an Osmotic Pressure Difference. The Three Membrane Parameters.- (i) Volume Flow Through a Permeable, Porous Membrane due to a Pressure and Concentration Gradient.- (ii) Solute Flow Through a Permeable Membrane due to Solvent Drag and Diffusion.- (iii) The Symmetrical Form of the Coupled Flow Relations.- (iv) Measurements of Membrane Parameters on Synthetic Membranes. Data on Lp, ?, p.- 2.6.C. Transport of Water and Solute Across Biological Membranes.- (i) The Permeability and Filtration Coefficient of Red Blood Cells.- (ii) The Permeability and Filtration Coefficient of Capillary Walls.- 2.Al Derivation of the Relation (2-6): Total Kinetic Energy = 3/2p V.- 2.A2 Proof of the Equipartition Law for a “Test Particle” of Mass M in a Gas at Temperature T.- 2.A3 Gaussian Integrals.- 2.7 References and Supplementary Reading.- 2.8 Problems.- 3 Poisson Statistics.- 3.1 Introduction.- 3.2 Derivation of the Poisson Probability Distribution.- 3.2.A. The Poisson Distribution and the Sampling of Particles from a Solution.- 3.2.B. The Poisson Distribution and Radioactive Decay.- 3.2.C. The Poisson Distribution and the Photoelectric Effect.- 3.3 Properties of the Poisson Distribution.- 3.3.A. Normalization and Average Value of the Poisson Distribution.- 3.3.B. Fluctuations of n Around the Average: Accurate Measurement of the Average Number of Events.- 3.3.C. Graphs of P(n, n).- 3.3.D. The Form of the Poisson Distribution for Large nx: The Normal, or Gaussian Distribution.- 3.4 Poisson Statistics and the Detection of Light by the Eye.- 3.4.A. The Detection of Light at the Threshold of Vision.- (i) Anatomical and Physiological Conditions for Maximum Sensitivity.- (ii) The Frequency-of-Seeing Curve.- (iii) Theory for the Shape of the Frequency-of-Seeing Curve.- 3.4.B. “Seeing” in the Presence of Background Light. Visual Contrast Thresholds and the Detection of Signals in the Presence of Noise.- (i) Experimental Measurement of the Visual Contrast Threshold.- (ii) The Noise Theory of the Visual Contrast Threshold Curve: For Short-Time, Small-Area Test Sources.- (iii) Visual Contrast Thresholds for Long-Time, Large-Area Test Sources.- 3.4.C. Phototransduction.- 3.5 The Luria-Delbrück Experiment: Mutation as the Source of Bacterial Immunity to Virus Attack.- 3.5.A. Introduction.- 3.5.B. Theory of the Probability Distribution for Phage Resistant Bacteria Under the Hypothesis of Mutation.- (i) Growth of Bacterial Population. Division Time.- (ii) Probability Distribution for Clones of Resistant Bacteria.- (iii) The Mean Value and Variance of the Probability Distribution for the Number of Resistant Bacteria.- 3.5.C. The Experimental Data of Luria and Delbrück. Comparison Between Theory and Experiment. Determination of the Bacterial Mutation Rate.- 3.6 References and Supplementary Reading.- 3.7 Problems.- 4 Thermal Equilibrium. The Boltzmann Factor. Entropy and Free Energy. The Second Law of Thermodynamics. Application to Physics, Chemistry, and Biology.- 4.1 The Statistical Nature of Thermal Equilibrium.- 4.1.A. Introduction: Thermal Equilibrium in Gases, Solids and Fluids. Equilibrium Between Phases. Chemical Reaction Equilibrium. Statistical Physics Versus Thermodynamics.- 4.1.B. Elements of Quantum Physics.- (i) Energy States in Atoms, Molecules, Macromolecules, and Solids.- (ii) Quantum States. Stability of Atoms and Molecules.- (iii) Free Particles. De Broglie Wavelength and Uncertainty Relations.- (iv) The Importance of Quantization for Statistical Physics. The Principle of Detailed Balance.- 4.2 The Probability Distribution of Energy. The Boltzmann Factor.- 4.2.A. Probability Distribution for the Energy of Vibrating Atoms in a Crystalline Solid.- (i) The Einstein Crystal as a Model.- (ii) Definition of the Probability Distribution P(n) for the Energy ?n of an Atom in the Crystal.- (iii) Microstates and Macrostates of the Einstein Crystal. Weight of a Macrostate.- (iv) Numerical Example for a Very Small Crystal.- (v) Finding the Most Probable Macrostate.- (vi) The Probability Distribution P(n) for the Energies of Atoms in the Einstein Crystal. The Boltzmann Factor.- (vii) Physical Interpretation of the Boltzmann Factor.- 4.2.B. Energy Distribution for the Atoms of an Ideal Monoatomic Gas.- (i) Phase Space and Phase-Space Trajectories of Particles.- (ii) Thermal Equilibrium as a Stationary Population in Phase Space.- (iii) The Counting of Population Patterns. The Role of Planck’s Constant h. All Microstates Are Equally Probable.- (iv) The Equilibrium Population Density in Phase Space. Macrocells and Macrostates.- (v) The Weight W of a Macrostate.- (vi) Finding the Most Probable Macrostate.- (vii) The Boltzmann Factor and Temperature.- (viii) The Barometric Formula.- (ix) The Maxwell—Boltzmann Distribution Function of Velocities.- 4.2.C. Thermal Equilibrium Between Solid and Gas. Vapor Pressure of a Solid.- (i) Gas and Solid in Thermal Contact. Demonstration that ? = 1/kT at All Temperatures.- (ii) The Vapor Pressure of a Crystalline Solid.- 4.3 Macroscopic Statement of Equilibrium Conditions. Entropy and the Second Law of Thermodynamics, Minimum Principle for Free Energy. Chemical Potentials.- 4.3.A. Introduction. Simple and Composite Systems. Equations of State. Equilibrium in Composite Systems.- 4.3.B. Entropy of a Simple System and its Properties. The Second Law of Thermodynamics.- (i) The Entropy of the Einstein Crystal.- (ii) The Entropy of the Ideal Gas.- (iii) Physical Interpretation of the Entropy. Ideal Gas Case.- (iv) Illustrative Calculation of the Entropy Difference for Two States of the Ideal Monoatomic Gas.- (v) A Note on the Chemical Potential ?.- (vi) General Statement of the Equilibrium Conditions. The Second Law of Thermodynamics.- (vii) Simple Illustrative Applications of the Entropy Maximum Principle.- 4.3.C. The Free Energy Minimum Principles.- (i) A System at Constant Temperature and Volume. The Helmholtz Free Energy.- (ii) Mathematical Interlude: Maxwell Relations and Their Use.- (iii) Equilibrium at Constant Pressure and Temperature. The Gibbs Free Energy.- 4.4 Applications of the Equilibrium Conditions to Problems in Physics, Chemistry, and Biology.- 4.4.A. Equilibrium Between Phases. The Clausius-Clapeyron Equation.- 4.4.B. Dilute Solutions of Nonelectrolytes.- (i) The Gibbs Free Energy of a Dilute Solution. The Concept of the Ideal Solution.- (ii) Connection Between the Gibbs Free Energy and the Chemical Potentials of Each Species in a Multicomponent Solution.- (iii) Expressions for the Chemical Potentials of Solvent, and Solutes in a Dilute, Ideal Solution.- (iv) Raoult’s Law; the Lowering of the Solvent Vapor Pressure by the Presence of Solute. Elevation of Boiling Point.- (v) Osmotic Pressure Revisited. Van t’ Hoff’s Law.- (vi) The Solubility of Gases. Henry’s Law.- 4.4.C. The Binding of Ligands to Multi-subunit Proteins.- (i) Thermodynamic Equilibrium and the Binding of Ligands to Distinct Sites on Multi-subunit Proteins.- (ii) The Structure and Function of the Oxygen Binding Proteins: Hemoglobin and Myoglobin.- (iii) The Theory of Oxygen Binding to Myoglobin.- (iv) The Theory of Oxygen Binding to Hemoglobin.- (v) Further Models and Applications of the Theory of Ligand Binding.- 4.Al Multinomial Coefficients: Weight of a Macrostate for the Einstein Crystal.- 4.A2 Occupancy of Microcells by Atoms of an Ideal Gas.- 4.A3 The Equipartition Theorem of Classical Statistical Mechanics.- 4.5 References and Supplementary Reading.- 4.6 Problems.- Table of Important Constants.- Table of Units and Conversion Factors.- 4.6.A. Units of Length.- 4.6.B. Units of Area and Volume.- 4.6.C. Units of Force.- 4.6.D. Units of Pressure.- 4.6.E. Units of Energy.