THERMODYNAMICS AND CHEMISTRY: A NON-MATHEMATICAL TREATISE FOR CHEMISTS AND STUDENTS OF CHEMISTRY
1 Principles of Probability 1 The Principles of Probability Are the Foundations of Entropy 1 What Is Probability? 2 The Rules of Probability Are Recipes for Drawing Consistent Inferences 3 Correlated Events Are Described by Conditional Probabilities 7 Combinatorics Describes How to Count Events 10 Collections of Probabilities Are Described by Distribution Functions 14 Distribution Functions Have Average Values and Standard Deviations 18 Summary 22 Problems 25 References 27 Suggested Reading 28 2 Extremum Principles Predict Equilibria 29 What Are Extremum Principles? 29 What Is a State of Equilibrium? 30 An Extremum Principle: Maximizing Multiplicity Predicts the Most Probable Outcomes 32 Simple Models Show How Systems Tend Toward Disorder 33 Summary 37 Problems 38 Suggested Reading 38 3 Heat, Work, & Energy 39 Energy Is a Capacity to Store and Move Work and Heat 39 Energy Is Ubiquitous and Important 39 Some Physical Quantities Obey Conservation Laws 45 Heat Was Thought to Be a Fluid 47 The First Law of Thermodynamics: Heat Plus Work Is Conserved 49 Atoms and Molecules Have Energies 50 Why Does Heat Flow? 52 The Second Law of Thermodynamics Is an Extremum Principle 55 Summary 55 Problems 56 References 56 Suggested Reading 56 4 Math Tools: Multivariate Calculus 59 Multivariate Calculus Applies to Functions of Multiple Variables 59 Partial Derivatives Are the Slopes of the Tangents to Multivariate Functions 60 The Extrema of Multivariate Functions Occur Where the Partial Derivatives Are Zero 63 How Can You Find the Extrema of Multivariate Functions That Are Subject to Constraints? 66 Integrating Multivariate Functions Sometimes Depends on the Pathway of Integration, and Sometimes It Doesn’t 72 The Chain Rule Applies to Functions of Several Variables 75 Summary 77 Problems 78 References 79 Suggested Reading 79 5 Entropy & the Boltzmann Law 81 What Is Entropy? 81 Maximum Entropy Predicts Flat Distributions When There Are No Constraints 85 Maximum Entropy Predicts Exponential Distributions When There Are Constraints 86 What Is the Reason for Maximizing the Entropy? 90 Summary 90 Problems 91 References 92 Suggested Reading 92 6 Thermodynamic Driving Forces 93 Thermodynamics Is Two Laws and a Little Calculus 93 The Fundamental Thermodynamic Equations, S =S(U,V,N) or U =U(S,V,N), Predict Equilibria 95 The Fundamental Equations Define the Thermodynamic Driving Forces 96 The Second Law Predicts Thermal, Mechanical, and Chemical Equilibria 99 Summary 106 Problems 107 Suggested Reading 107 7 The Logic of Thermodynamics 109 Thermodynamics Helps You Reason About Heat, Work, Processes, and Cycles 109 We Start with a Few Basic Premises 110 Four Different Processes Can Change an Ideal Gas 114 Pathways Are Combinations of Processes 117 A Cycle is a Set of Outbound and Returning Pathways 118 The Maximum Possible Work Is Achieved via Reversible Processes 122 Why Is There an Absolute Temperature Scale? 128 Summary 128 Problems 129 Suggested Reading 130 8 Laboratory Conditions & Free Energies 131 Switch from Maximum Entropy to Minimum Free Energy 131 Free Energy Defines Another Extremum Principle 132 Thermochemistry: The Enthalpy of a Molecule Is the Sum of the Enthalpies of Its Covalent Bonds 144 Summary 145 Problems 146 Suggested Reading 147 9 Maxwell’s Relations & Mixtures 149 Thermodynamics Applies to Various Types of Forces 149 Maxwell’s Relations Interrelate Partial Derivatives 152 Susceptibilities Are Measurable and Useful 154 Multicomponent Systems Have Partial Molar Properties 160 Summary 165 Problems 166 Suggested Reading 168 10 The Boltzmann Distribution Law 169 Statistical Mechanics Gives Probability Distributions for Atoms and Molecules 169 The Boltzmann Distribution Law Describes the Equilibria Among Atoms and Molecules 171 What Does a Partition Function Tell You? 176 Thermodynamic Properties Can Be Predicted from Partition Functions 182 Summary 189 Problems 190 References 192 Suggested Reading 192 11 The Statistical Mechanics of Simple Gases & Solids 193 Statistical Mechanics Predicts Macroscopic Properties from Atomic Structures 193 The Particle in a Box Is the Quantum Mechanical Model for Translational Motion 195 Vibrations Can Be Treated by the Harmonic Oscillator Model 201 Rotations Can Be Treated by the Rigid Rotor Model 203 The Electronic Partition Function 205 The Total Partition Function Predicts the Properties of Molecules 207 Ideal Gas Properties Are Predicted by Quantum Mechanics 207 The Equipartition Theorem Says That Energies Are Uniformly Distributed to Each Degree of Freedom 212 Summary 216 Problems 218 References 220 Suggested Reading 220 12 What Is Temperature? What Is Heat Capacity? 221 A Microscopic Perspective on Temperature and Heat Capacity 221 A Graphical Procedure Shows the Steps from Fundamental Functions to Experimental Measurables 225 What Drives Heat Exchange? 227 The Heat Capacity Is Proportional to the Energy Fluctuations in a System 228 Summary 232 Problems 233 References 234 Suggested Reading 234 13 Chemical Equilibria 235 Simple Chemical Equilibria Can Be Predicted from Atomic Structures 235 Le Chatelier’s Principle Describes the Response to Perturbation from Equilibrium 243 The Temperature Dependence of Equilibrium Is Described by the van ’t Hoff Equation 244 Summary 249 Problems 250 Suggested Reading 251 14 Equilibria Between Liquids, Solids, & Gases 253 Phase Equilibria Are Described by the Chemical Potential 253 The Clapeyron Equation Describes p(T) at Phase Equilibrium 257 How Do Refrigerators and Heat Pumps Work? 262 Surface Tension Describes the Equilibrium Between Molecules at the Surface and in the Bulk 265 Summary 267 Problems 268 Suggested Reading 268 15 Solutions & Mixtures 269 A Lattice Model Describes Mixtures 269 Interfacial Tension Describes the Free Energy of Creating Surface Area 276 What Have We Left Out? 278 Summary 279 Problems 280 References 280 Suggested Reading 280 16 Solvation & the Transfer of Molecules Between Phases 283 The Chemical Potential Describes the Tendency of Molecules to Exchange and Partition 283 Solvation Is the Transfer of Molecules Between Vapor and Liquid Phases 284 A Thermodynamic Model Defines the Activity and Activity Coefficient 287 Adding Solute Can Raise the Boiling Temperature of the Solvent 289 Adding Solute Can Lower the Freezing Temperature of a Solvent 291 Adding Solute on One Side of a Semipermeable Membrane Causes an Osmotic Pressure 293 Solutes Can Transfer and Partition from One Medium to Another 295 Dimerization in Solution Involves Desolvation 299 Summary 303 Problems 304 References 306 Suggested Reading 306 17 Physical Kinetics: Diffusion, Permeation, & Flow 307 Applied Forces Cause Liquids to Flow 307 Fick’s Law and Fourier’s Law: Gradients of Particles or Heat Cause Flows 308 The Diffusion Equation Describes How Concentration Gradients Change Over Time and Space 310 The Smoluchowski Equation Describes Particles Driven by Both Applied Forces and Diffusion 317 The Einstein–Smoluchowski Equation Relates Diffusion and Friction 318 Physical and Chemical States Evolve Toward Equilibrium 320 Sources and Sinks Also Contribute to Flows: Examples from Population Biology 321 Onsager Reciprocal Relations Describe Coupled Flows 324 Summary 325 Problems 326 References 327 Suggested Reading 327 18 Microscopic Dynamics 329 The Dynamical Laws Arise from Stochastic Processes 329 Diffusing Particles Can be Modeled as Random Flights 329 Inter-Arrival Rates Are Described by Poisson and Waiting-Time Distributions 340 Single-Particle Kinetics is Described by Trajectories 343 Master Equations Are Differential Equations for Markov Dynamics 345 Brownian Ratchets Convert Binding and Release Events into Directed Motion 348 The Fluctuation–Dissipation Theorem Relates Equilibrium Fluctuations to the Rate of Approach to Equilibrium 351 Summary 353 Problems 354 References 354 Suggested Reading 354 19 Chemical Kinetics & Transition States 357 Chemical Reaction Rates Depend on Temperature 357 The Mass Action Laws Describe Mechanisms in Chemical Kinetics 357 At Equilibrium, Rates Obey Detailed Balance 358 Chemical Reactions Depend Strongly on Temperature 360 Activated Processes Can Be Modeled by Transition-State Theory 363 Catalysts Speed Up Chemical Reactions 372 The Brønsted Law of Acid and Base Catalysis: The Stronger the Acid, the Faster the Reaction it Catalyzes 375 Funnel Landscapes Describe Diffusion and Polymer Folding Processes 379 Summary 380 Problems 381 References 383 Suggested Reading 384 20 Coulomb’s Law of Electrostatic Forces 385 Charge Interactions Are Described by Coulomb’s Law 385 Charge Interactions Are Long-Ranged 387 Charge Interactions Are Weaker in Media 389 Electrostatic Forces Add Like Vectors 392 What Is an Electrostatic Field? 393 Electric Fields Have Fluxes 395 Summary 401 Problems 402 References 402 Suggested Reading 402 21 The Electrostatic Potential 403 We Switch from Fields (Forces) to Electrostatic Potentials (Energies) 403 Dipoles Are Equal and Opposite Charges Separated by a Distance 409 Voltage-Gated Ion Channels May Be Modeled as Orienting Dipoles 411 Poisson’s Equation Is Used to Compute the Potential for Any Constellation of Charges 414 You Can Model Ions Near Interfaces Using Image Charges 418 Summary 421 Problems 422 References 423 Suggested Reading 423 22 Electrochemical Equilibria 425 Electrochemical Potentials Describe Equilibria in Solutions of Charged Species 425 The Nernst Equation: Electrostatic Fields Drive Ion Distributions 426 In Redox Reactions, Electrons Hop from One Molecule to Another 429 Selective Ion Flow Through Membranes Can Create a Voltage 437 Acid–Base Equilibria: Chemical Reactions Are Affected by Electrostatic Interactions 438 Acid–Base Equilibria Are Shifted by the Presence of Charged Surfaces 439 Gradients of Electrostatic Potential Cause Ion Flows 441 Creating a Charge Distribution Costs Free Energy 444 Summary 451 Problems 452 References 453 Suggested Reading 453 23 Salt Ions Shield Charged Objects in Solution 455 Salts Dissociate into Mobile Ions and Shield Charged Objects in Water 455 Electrolytes Are Strong or Weak, Depending on Whether They Fully Dissociate in Water 462 Summary 467 Problems 468 References 469 Suggested Reading 469 24 Intermolecular Interactions 471 Molecules Repel Each Other at Very Short Range and Attract at Longer Distances 471 Short-Ranged Attractions Can Be Explained as Electrostatic Interactions 472 The van der Waals Gas Model Accounts for Intermolecular Interactions 479 Radial Distribution Functions Describe Structural Correlations in Liquids 483 The Lattice Model Contact Energy w Approximates Intermolecular Interactions 485 Summary 486 Problems 487 References 488 Suggested Reading 488 25 Phase Transitions 489 Two States Can Be Stable at the Same Time 489 Phase Diagrams Describe What Phases Are Stable Under Which Conditions 492 Peaks on Energy Landscapes Are Points of Instability 497 Thermal Phase Equilibria Arise from a Balance between Energies and Entropies 500 The Spinodal Curve Describes the Limit of Metastability 502 The Critical Point Is Where Coexisting Phases Merge 504 The Principles of Boiling Are Related to the Principles of Immiscibility 505 Boiling a Liquid Mixture Involves Two Types of Phase Transition 510 Summary 512 Problems 514 Suggested Reading 517 26 Cooperativity: the Helix–Coil, Ising, & Landau Models 519 Abrupt Transitions Are Common in Nature 519 Transitions and Critical Points Are Universal 519 The Landau Model Is a General Description of Phase Transitions and Critical Exponents 522 The Ising Model Describes Magnetization 525 Polymers Undergo Helix–Coil Transitions 527 The Kinetics of Phase Transitions Can Be Controlled by Nucleation 535 Summary 536 Problems 537 References 538 Suggested Reading 539 27 Adsorption, Binding, & Catalysis 541 Binding and Adsorption Processes Are Saturable 541 The Langmuir Model Describes Adsorption of Gas Molecules on a Surface 541 The Langmuir Model Also Treats Binding and Saturation in Solution 546 The Independent-Site Model Describes the Principle of Adsorption Chromatography 548 The Michaelis–Menten Model Describes Saturation in Rate Processes 549 Sabatier’s Principle: Catalysts Should Bind Neither Too Tightly Nor Too Weakly 552 Summary 554 Problems 555 References 556 Suggested Reading 556 28 Multi-site & Cooperative Ligand Binding 559 Binding Polynomials Are Used to Compute Binding Curves 560 The Simplest Model of Binding Cooperativity Involves Two Binding Sites 562 ‘Cooperativity’ Describes the Depletion of Intermediate States 565 Binding Polynomials Can Be Constructed Using the Addition and Multiplication Rules of Probability 567 Aggregation and Micellization Are Cooperative Processes 570 The Model of McGhee and von Hippel Treats Ligands that Crowd Out Other Ligands 575 The Grand Canonical Ensemble Gives the Basis for Binding Polynomials 578 Summary 581 Problems 582 References 583 Suggested Reading 584 29 Bio & Nano Machines 585 Biochemical Machines Are Powered by Coupled Binding Processes 585 Oxygen Binding to Hemoglobin Is a Cooperative Process 586 Binding Polynomials Can Treat Multiple Types of Ligand 590 Rates Can Often Be Treated by Using Binding Polynomials 594 The Molecular Logic of Biology Is Encoded in Coupled-Binding Actions 595 Biochemical Engines Harness Downhill Processes to Drive Uphill Processes 598 Summary 608 Problems 609 References 612 Suggested Reading 612 30 Water 615 Water Has Anomalous Properties 615 Hydrogen Bonds Are a Defining Feature of Water Molecules 615 Pure Water Has Anomalous Properties 620 Summary 627 Problems 628 References 628 Suggested Reading 628 31 Water as a Solvent 629 Oil and Water Don’t Mix: the Hydrophobic Effect 629 The Signature of Hydrophobicity Is Its Temperature Dependence 630 Alcohols Constrict the Volumes in Mixtures with Water 636 Ions Can Make or Break Water Structure 636 Ion Pairing Preferences in Water Depend on Charge Densities 639 Summary 640 References 641 Suggested Reading 642 32 Polymer Solutions 643 Polymer Properties Are Governed by Distribution Functions 643 Polymers Have Distributions of Conformations 643 Polymer Solutions Differ from Small-Molecule Solutions 644 The Flory–Huggins Model Describes Polymer Solution Thermodynamics 646 Flory–Huggins Theory Predicts Nonideal Colligative Properties for Polymer Solutions 651 The Phase Behavior of Polymers Differs from that of Small Molecules 651 The Flory Theorem Says that the Intrinsic Conformations of a Chain Molecule Are Not Perturbed by Surrounding Chains 655 Summary 656 Problems 657 Suggested Reading 657 33 Polymer Elasticity & Collapse 659 Polymeric Materials Are Often Elastic 659 Real Chain Molecules Have Bending Correlations and Stiffness 662 Random-Flight Chain Conformations Are Described by the Gaussian Distribution Function 665 Polymer Elasticity Follows Hooke’s Law 666 The Elasticity of Rubbery Materials Results from the Sum of Chain Entropies 670 Polymers Expand in Good Solvents, Are Random Flights in ? Solvents, and Collapse In Poor Solvents 673 Polymer Solutions Exert Osmotic Pressures 678 Polymer Radius Depends on Polymer Concentration: the Semi-Dilute Cross-Over Regime 681 Summary 683 Problems 684 References 684 Suggested Reading 684 34 Polymers Resist Confinement & Deformation 685 ‘Excluded Volume’ Describes the Large Volume Inside a Polymer Conformation that Is Inaccessible to Other Chains 685 Chain Conformations Are Perturbed Near Surfaces 686 Polymer Conformations Can Be Described by the Diffusion Equation 689 Polymers Tend to Avoid Confined Spaces 691 The Rouse–Zimm Model Describes the Dynamics of Viscoelastic Fluids 693 Reptation Describes the Snakelike Motions of Chains in Melts and Gels 696 Summary 697 Problems 698 References 698 Suggested Reading 698