Physical Chemistry PDF Textbook

 University College Cardiff Science Library Author M ooRE Accessions No. SO 5 2278 Class No. Physical Chemistry Walter J*. Moore Professor of Chemistry University of Sydney LONGMAN  LONGMAN GROUP LIMITED...

 University College Cardiff Science Library Author M ooRE Accessions No. SO 5 2278 Class No. Physical Chemistry Walter J*. Moore Professor of Chemistry University of Sydney LONGMAN  LONGMAN GROUP LIMITED London Associated companies, branches and representatives throughout the world © 1972 by PRENTICE-HALL, INC., Englewood Cliffs, New Jersey. All rights reserved. No part of this publication may be re­ produced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the Copyright owner. First Edition published in the U.S.A., 1950 Second U.S. Edition published in Great Britain, 1956 Second U.S. Edition (with answers to problems) published in Great Britain as Third Edition, 1957 New impressions 1958, 1959, 1960, 1961 and 1962 Third U.S. Edition published in Great Britain as Fourth Edition, 1963 New impressions 1963, 1964, 1965, 1966, 1968 and 1970 Fourth U.S. Edition published in Great Britain by Longman Group Limited, London as Fifth Edition, 1972 New impressions 1974, 1976, 1978 ISBN 0 582 44234 6 Printed in Great Britain by William Clowes and Sons Ltd., London, Colchester and Beccles  Preface Oor universe is like an e 'e Turned in. man's benmaist hert to see. And swamped in subjectivity. But whether it can use its sicht To bring what lies withoot to licht To answer’s still ayont my micht. Hugh MacDiarmid * 1926 Like the last edition, this new edition of Physical Chemistry is the result of a substantial rewriting of the entire book. About 22 years ago, in the preface to the first edition, I said that this book was not designed to be a collection of facts, but rather an introduction to ways of thinking about the world. Actually this edition was written under the title Foundations of Physical Chemistry, and such a title expresses quite well the basic intention of the book. I have tried to emphasize critical discussions of definitions, postulates, and logical operations. The concepts of physical chemistry today are transient states in the progress of the science. The historical background in the book is intended to help the student reach this under­ standing, without which science becomes static and comparatively uninteresting. For some students of physical chemistry the use of mathematics remains a major difficulty. We try to convince students that the scientist must learn mathe­ matics while he studies science. It is neither necessary nor desirable to learn “pure mathematics” first and then to apply it to scientific problems. The level of mathe­ matical difficulty in this edition is somewhat higher than previously, but as a compensation, more careful discussions of mathematical details have been given. Nevertheless, many students would find it worthwhile to acquire one of several excellent books on mathematics for the physical sciences, references to which are made in the text. In this edition, the order of the subject matter has been changed in order to bring statistical mechanics into the text as early as possible, and then to use its methods in subsequent discussions. Examination of current textbooks of general chemistry and physics (universal prerequisites for the study of physical chemistry) *From “The Great Wheel” by Hugh MacDiarmid (C. M. Grieve) in A Drunk Man Looks at the Thistle (Edinburgh: Wm. Blackwood & Sons, 1926). vi Preface indicates that almost all contain sufficient atomic physics and elementary quantum theory to serve as an adequate foundation for the principles of statistical mechanics as given in my Chapter 5. I have tried to follow the recommendations on nomenclature and units of the International Union of Pure and Applied Chemistry, except for the retention of the atmosphere as a unit of pressure, a relic of nonsystematic units, which also should disappear in due course. Probably within a decade, the SI system of units * will be in general use by all scientists. There are not many worked out numerical problems in the text, but Professors William Bunger of Indiana State University and Theodore Sakano of Rose-Hulman Polytechnic Institute have prepared a manual of solutions to all the problems at the ends of chapters. In my experience students will learn most quickly if they obtain this manual as a companion to the text. It is always a pleasant duty to thank my confreres who have contributed so generously with illustrations, corrections, and suggestions to improve the book. So many people have helped that I am sure to forget to mention some, but these also have my thanks. The publishers wisely enlisted Thomas Dunn to provide a general analysis of the book and Jeff Steinfeld to make a critical reading of the manuscript. Walter Kauzmann was a continual source of help, both in the exten­ sive comments he sent me and in the excellent material I found in his clearly written books. By devious pathways, an exegesis of the third edition by George Kistiakowsky fell into my hands, which provided many valuable clarifications. Apart from these major efforts on the total book, much work on individual chapters was done by Peter Langhoff, Edward Bair, Donald McQuarrie, Robert Mortimer, John Bockris, Donald Sands, Edward Hughes, John Ricci, John Griffith, Dennis Peters, Ludvik Bass, Albert Zettlemoyer and Dieter Hummel (who has made a German translation). Lucky is the author who has such good neighbors as these. Acknowledgments to scientists who sent illustrations are included in the text. At Prentice-Hall, Albert Belskie, Editor for Chemistry, was a solid source of support and good counsel at all times. With all this help one may wonder why the book is still so far from an ideal state. The answer must have something to do with the fact that we are not work­ ing closer to absolute zero.t As always, I shall welcome comments from readers and try to correct all the mistakes that they will find. W.J.M. *M. A. Paul, “The International System of Units (SI)—Development and Progress,” J. Chem. Doc. 11, 3 (1971). fA concise summary of thermodynamics has been given: (1) The First Law says you can’t win; the best you can do is break even. (2) The Second Law says you can break even only at absolute zero. (3) The Third Law says you can never reach absolute zero.  Contents 1 Physicochemical Systems 1 1. What Is Science? 2. Physical Chemistry 3. Mechanics: Force 4. Mechanical Work 5. Mechanical Energy 6. Equilibrium 7. The Thermal Properties of Matter 8. Temperature as a Mechanical Property 9. The Spring of the Air and Boyle’s Law 10. The Law of Gay-Lussac 11. Definition of the Mole 12. Equation of State of an Ideal Gas 13. The Equation of State and PVT Relationships 14. PVT Behavior of Real Gases 15. Law of Corresponding States 16. Equations of State for Gases 17. The Critical Region 18. The van der Waals Equation and Liquefaction of Gases 19. Other Equations of State 20. Mixtures of Ideal Gases 21. Mixtures of Nonideal Gases 22. The Concepts of Heat and Heat Capacity 23. Work in Changes of Volume 24. General Concept of Work 25. Reversible Processes Problems 2 Energetics 38 1. History of the First Law of Thermodynamics 2. The Work of Joule 3. Formulation of the First Law 4. The Nature of Internal Energy 5. A Me­ chanical Definition of Heat 6. Properties of Exact Differentials 7. Adiabatic and Isothermal Processes 8. Enthalpy 9. Heat Capacities 10. The Joule Experiment 11. The Joule-Thomson Experiment 12. Application of the First Law to Ideal Gases 13. Examples of Ideal-Gas Calculations 14. Thermo­ chemistry—Heat of Reaction 15. Enthalpies of Formation 16. Experi­ mental Thermochemistry 17. Heat Conduction Calorimeters 18. Heats of Solution 19. Temperature Dependence of Enthalpy of Reaction 20. Bond Enthalpies 21. Chemical Affinity Problems 3 Entropy and Free Energy 77 1. The Carnot Cycle 2. The Second Law of Thermodynamics 3. The Thermodynamic Temperature Scale 4. Relation of Thermodynamic and Ideal-Gas Temperature Scales 5. Entropy 6. First and Second Laws Combined 7. The Inequality of Clausius 8. Entropy Changes in an Ideal Gas 9. Change of Entropy in Changes of State of Aggregation 10. Entropy Changes in Isolated Systems 11. Entropy and Equilibrium 12. Thermo­ v/7 viii Contents dynamics and Life 13. Equilibrium Conditions for Closed Systems 14. The Gibbs Function—Equilibrium at Constant T and P 15. Isothermal Changes in A and G 16. Thermodynamic Potentials 17. Legendre Trans­ formations 18. Maxwell’s Relations 19. Pressure and Temperature Dependence of Gibbs Function 20. Pressure and Temperature Variation of Entropy 21. Applications of Thermodynamic Equations of State 22. The Approach to Absolute Zero 23. The Third Law of Thermodynamics 24. An Illustration of the Third Law 25. Third-Law Entropies Problems 4 Kinetic Theory 116 1. Atomic Theory 2. Molecules 3. The Kinetic Theory of Heat 4. The Pressure of Gas 5. Gas Mixtures and Partial Pressures 6. Kinetic Energy and Temperature 7. Molecular Speeds 8. Molecular Effusion 9. Imperfect Gases—The van der Waals Equation 10. Intermolecular Forces and the Equation of State 11. Molecular Velocities—Directions 12. Collisions of Molecules with a Wall 13. Distribution of Molecular Velocities 14. Velocity in One Dimension 15. Velocity in Two Dimensions 16. Velocity in Three Dimensions 17. Experimental Velocity Analysis 18. The Equipartition of Energy 19. Rotation and Vibration of Diatomic Mole­ cules 20. Motions of Polyatomic Molecules 21. The Equipartition Principle and Heat Capacities 22. Collisions Between Molecules 23. Derivation of Collision Frequency 24. The Viscosity of a Gas 25. Kinetic Theory of Gas Viscosity 26. Molecular Diameters and Intermolecular Force Constants 27. Thermal Conductivity 28. Diffusion 29. Solutions of Diffusion Equa­ tion Problems 5 Statistical Mechanics 167 1. The Statistical Method 2. Entropy and Disorder 3. Entropy and Information 4. Stirling Formula for Nl 5. Boltzmann 6. How the State of a System Is Defined 7. Ensembles 8. Lagrange Method for Constrained Maximum 9. Boltzmann Distribution Law 10. Statistical Thermodynamics 11. Entropy and the Third Law 12. Evaluation of Z for Noninteracting Particles 13. Translational Partition Function 14. Partition Functions for Internal Molecular Motions 15. Classical Partition Function Problems 6 Changes of State 202 1. Phases 2. Components 3. Degrees of Freedom 4. General Equi­ librium Theory: The Chemical Potential 5. Conditions for Equilibrium Between Phases 6. The Phase Rule 7. Phase Diagram for One Compo­ nent 8. Thermodynamic Analysis of PT Diagram 9. The Helium System 10. Vapor Pressure and External Pressure 11. Statistical Theory of Phase Changes 12. Solid-Solid Transformations—The Sulfur System 13. Mea­ surements at High Pressures Problems  Contents ix 7 Solutions 229 1. Measures of Composition 2. Partial Molar Quantities: Partial Molar Volume 3. Activities and Activity Coefficients 4. Determination of Partial Molar Quantities 5. The Ideal Solution—Raoult’s Law , 6.Thermo- dynamics of Ideal Solutions 7. Solubility of Gases in Liquids—Henry’s Law 8. Mechanism of Anesthesia 9. Two-Component Systems 10. Pressure- Composition Diagrams 11. Temperature-Composition Diagrams 12. Fractional Distillation 13. Solutions of Solids in Liquids 14. Osmotic Pressure 15. Osmotic Pressure and Vapor Pressure 16. Deviations from Ideality 17. Boiling Point Diagrams 18. Solubility of Liquids in Liquids 19. Thermodynamic Condition for Phase Separation 20. Thermodynamics of Nonideal Solutions 21. Solid-Liquid Equilibria: Simple Eutectic Diagrams 22. Formation of Compounds 23. Solid Solutions 24. The Iron-Carbon Diagram 25. Statistical Mechanics of Solutions 26. The Bragg-Williams Model Problems 8 Chemical Affinity 279 1. Dynamic Equilibrium 2. Free Enthalpy and Chemical Affinity 3. Condition for Chemical Equilibrium 4. Standard Free Enthalpies 5. Free Enthalpy and Equilibrium in Reactions of Ideal Gases 6. Equilibrium Constant in Concentrations 7. Measurement of Homogeneous Gas Equi­ libria 8. Principle of Le Chatelier and Braun 9. Pressure Dependence of Equilibrium Constant 10. Temperature Dependence of Equilibrium Constant 11. Equilibrium Constants from Heat Capacities and the Third Law 12. Statistical Thermodynamics of Equilibrium Constants 13. Example of a Sta­ tistical Calculation of KP 14. Equilibria in Nonideal Systems—Fugacity and Activity 15. Nonideal Gases—Fugacity and Standard State 16. Use of Fugacity in Equilibrium Calculations 17. Standard States for Components in Solution 18. Activities of Solvent and Nonvolatile Solute from Vapor Pressure of Solution 19. Equilibrium Constants in Solution 20. Thermodynamics of Biochemical Reactions 21. AG^ of Biochemicals in Aqueous Solution 22. Pressure Effects on Equilibrium Constants 23. Effect of Pressure on Activity 24. Chemical Equilibria Involving Condensed Phases Problems 9 Chemical Reaction Rates 324 1. The Rate of Chemical Change 2. Experimental Methods in Kinetics 3. Order of Reaction 4. Molecularity of a Reaction 5. Reaction Mecha­ nisms 6. First-Order Rate Equations 7. Second-Order Rate Equations 8. Third-Order Rate Equations 9. Determination of the Reaction Order 10. Opposing Reactions 11. Principle of Detailed Balancing 12. Rate Constants and Equilibrium Constants 13. Consecutive Reactions 14. Parallel Reactions 15. Chemical Relaxation 16. Reactions in Flow Systems 17. Steady States in Flow Systems 18. Nonequilibrium Thermo­ dynamics 19. The Onsager Method 20. Entropy Production 21. Stationary States 22. Effect of Temperature on Reaction Rate 23. Colli­ x Contents sion Theory of Gas Reactions 24. Reaction Rates and Cross Sections 25. Calculation of Rate Constants from Collision Theory 26. Tests of Simple Hard-Sphere Collision Theory 27. Reactions of Hydrogen Atoms and Molecules 28. Potential Energy Surface for H + H2 29. Transition State Theory 30. Transition-State Theory in Thermodynamic Terms 31. Chemical Dynamics—Monte Carlo Methods 32. Reactions in Molecular Beams 33. Theory of Unimolecular Reactions 34. Chain Reactions: Formation of Hydrogen Bromide 35. Free-Radical Chains 36. Branching Chains—Explosive Reactions 37. Trimolecular Reactions 38. Reactions in Solution 39. Catalysis 40. Homogeneous Catalysis 41. Enzyme Reactions 42. Kinetics of Enzyme Reactions 43. Enzyme Inhibition 44. An Exemplary Enzyme, Acetylcholinesterase Problems 10 Electrochemistry: Ionics 420 1. Electricity 2. Faraday’s Laws and Electrochemical Equivalents 3. Coulometers 4. Conductivity Measurements 5. Molar Conductances 6. The Arrhenius Ionization Theory 7. Solvation of Ions 8. Transport Numbers and Mobilities 9. Measurement of Transport Numbers—Hittorf Method 10. Transport Numbers—Moving Boundary Method 11. Results of Transference Experiments 12. Mobilities of Hydrogen and Hydroxyl Ions 13. Diffusion and Ionic Mobility 14. Defects of the Arrhenius Theory 15. Activities and Standard States 16. Ion Activities 17. Activity Coeffi­ cients from Freezing Points 18. The Ionic Strength 19. Results of Activity Coefficient Measurements 20. A Review of Electrostatics 21. The Debye- Hiickel Theory 22. The Poisson-Boltzmann Equation 23. The Debye- Huckel Limiting Law 24. Theory of Conductivity 25. Ionic Association 26. Effects of High Fields 27. Kinetics of Ionic Reactions 28. Salt Effects on Kinetics of Ionic Reactions 29. Acid-Base Catalysis 30. General Acid-Base Catalysis Problems 11 Interfaces 475 1. Surface Tension 2. Equation of Young and LaPlace 3. Mechanical Work on Capillary System 4. Capillarity 5. Enhanced Vapor Pressure of Small Droplets—Kelvin Equation 6. Surface Tensions of Solutions 7. Gibbs Formulation of Surface Thermodynamics 8. Relative Adsorptions 9. Insoluble Surface Films 10. Structure of Surface Films 11. Dynamic Properties of Surfaces 12. Adsorption of Gases on Solids 13. The Lang­ muir Adsorption Isotherm 14. Adsorption on Nonuniform Sites 15. Sur­ face Catalysis 16. Activated Adsorption 17. Statistical Mechanics of Adsorption 18. Electrocapillarity 19. Structure of the Double Layer 20. Colloidal Sols 21. Electrokinetic Effects Problems 12 Electrochemistry—Electrodics 520 1. Definitions of Potentials 2. Electric Potential Difference for a Galvanic Cell 3. Electromotive Force (EMF) of a Cell 4. The Polarity of an  Contents xi Electrode 5. Reversible Cells 6. Free Energy and Reversible EMF 7. Entropy and Enthalpy of Cell Reactions 8. Types of Half-Cells (Electrodes) 9. Classification of Cells 10. The Standard EMF of Cells 11. Standard Electrode Potentials 12. Calculation of the EMF of a Cell 13. Calcula­ tion of Solubility Products 14. Standard Free Energies and Entropies of Aqueous Ions 15. Electrode-Concentration Cells 16. Electrolyte-Con­ centration Cells 17. Nonosmotic Membrane Equilibrium 18. Osmotic Membrane Equilibrium 19. Steady State Membrane Potentials 20. Nerve Conduction 21. Electrode Kinetics 22. Polarization 23. Diffusion Overpotential 24. Diffusion in Absence of a Steady State—Polarography 25. Activation Overpotential 26. Kinetics of Discharge of Hydrogen Ions Problems 13 Particles and Waves 570 1. Simple Harmonic Motion 2. Wave Motion 3. Standing Waves 4. Interference and Diffraction 5. Black-Body Radiation 6. The Quantum of Energy 7. The Planck Distribution Law 8. Photoelectric Effect 9. Spectroscopy 10. The Interpretation of Spectra 11. The Work of Bohr on Atomic Spectra 12. Bohr Orbits and Ionization Potentials 13. Particles and Waves 14. Electron Diffraction 15. Waves and the Uncertainty Principle 16. Zero-Point Energy 17. Wave Mechanics—The Schrodinger Equation 18. Interpretation of the Functions 19. Solution of the Schrodinger Equation—The Free Particle 20. Solution of Wave Equation —Particle in Box 21. Penetration of a Potential Barrier Problems 14 Quantum Mechanics and Atomic Structure 614 1. Postulates of Quantum Mechanics 2. Discussion of Operators 3. Generalization to Three Dimensions 4. Harmonic Oscillator 5. Harmonic Oscillator Wave Functions 6. Partition Function and Thermodynamics of Harmonic Oscillator 7. Rigid Diatomic Rotor 8. Partition Function and Thermodynamics of Diatomic Rigid Rotor 9. The Hydrogen Atom 10. Angular Momentum 11. Angular Momentum and Magnetic Moment 12. The Quantum Numbers 13. The Radial Wave Function 14. Angular Dependence of Hydrogen Orbitals 15. The Spinning Electron 16. Spin Postulates 17. The Pauli Exclusion Principle 18. Spin-Orbit Interaction 19. Spectrum of Helium 20. Vector Model of the Atom 21. Atomic Orbitals and Energies—The Variation Method 22. The Helium Atom 23. Heavier Atoms—The Self-Consistent Field 24. Atomic Energy Levels— Periodic Table 25. Perturbation Method 26. Perturbation of a Degenerate State Problems 15 The Chemical Bond 670 1. Valence Theory 2. The Ionic Bond 3. The Hydrogen Molecule Ion 4. Simple Variation Theory of HJ 5. The Covalent Bond 6. The Valence- xii Contents Bond Method 7. The Effect of Electron Spins 8. Results of the Heitler- London Method 9. Comparison of M.O. and V.B. Methods 10. Chem­ istry and Mechanics 11. Molecular Orbitals for Homonuclear Diatomic Molecules 12. Correlation Diagram 13. Heteronuclear Diatomic Mole­ cules 14. Electronegativity 15. Dipole Moments 16. Polarization of Dielectrics 17. Induced Polarization 18. Determination of the Dipole Moment 19. Dipole Moments and Molecular Structure 20. Polyatomic Molecules 21. Bond Distances, Bond Angles, Electron Densities 22. Electron Diffraction of Gases 23. Interpretation of Electron Diffraction Pictures 24. Nonlocalized Molecular Orbitals—Benzene 25. Ligand Field Theory 26. Other Symmetries 27. Electron-Excess Compounds 28. Hydrogen Bonds Problems 16 Symmetry and Group Theory 732 1. Symmetry Operations 2. Definition of a Group 3. Further Symmetry Operations 4. Molecular Point Groups 5. Transformations of Vectors by Symmetry Operations 6. Irreducible Representations Problems 17 Spectroscopy and Photochemistry 747 1. Molecular Spectra 2. Light Absorption 3. Quantum Mechanics of Light Absorption 4. The Einstein Coefficients 5. Rotational Levels— Far-Infrared Spectra 6. Intemuclear Distances from Rotation Spectra 7. Rotational Spectra of Polyatomic Molecules 8. Microwave Spectroscopy 9. Internal Rotations 10. Vibrational Energy Levels and Spectra 11. Vibration-Rotation Spectra of Diatomic Molecules 12. Infrared Spectra of Carbon Dioxide 13. Lasers 14. Normal Modes of Vibration 15. Symmetry and Normal Vibrations 16. Raman Spectra 17. Selection Rules for Raman Spectra 18. Molecular Data from Spectroscopy 19. Electronic Band Spectra 20. Reaction Paths of Electronically Excited Mole­ cules 21. Some Photochemical Principles 22. Bipartition of Molecular Excitation 23. Secondary Photochemical Processes: Fluorescence 24. Secondary Photochemical Processes: Chain Reactions 25. Flash Photolysis 26. Photolysis in Liquids 27. Energy Transfer in Condensed Systems 28. Photosynthesis in Plants 29. Magnetic Properties of Molecules 30. Para­ magnetism 31. Nuclear Properties and Molecular Structure 32. Nuclear Paramagnetism 33. Nuclear Magnetic Resonance 34. Chemical Shifts and Spin-Spin Splitting 35. Chemical Exchange in NMR 36. Electron Paramagnetic Resonance Problems 18 The Solid State 828 1. The Growth and Form of Crystals 2. Crystal Planes and Directions 3. Crystal Systems 4. Lattices and Crystal Structures 5. Symmetry Properties 6. Space Groups 7. X-Ray Crystallography 8. The Bragg Treatment  Contents xiii 9. Proof of Bragg Reflection 10. Fourier Transforms and Reciprocal Lattices 11. Structures of Sodium and Potassium Chlorides 12. The Powder Method 13. Rotating Crystal Method 14. Crystal-Structure Determinations 15. Fourier Synthesis of a Crystal Structure 16. Neutron Diffraction 17. Closest Packing of Spheres 18. Binding in Crystals 19. The Bond Model 20. Electron-Gas Theory of Metals 21. Quantum Statistics 22. Cohesive Energy of Metals 23. Wave Functions for Electrons in Solids 24. Semi­ conductors 25. Doping of Semiconductors 26. Nonstoichiometric Com­ pounds 27. Point Defects 28. Linear Defects: Dislocations 29. Effects Due to Dislocations 30. Ionic Crystals 31. Cohesive Energy of Ionic Crystals 32. The Born-Haber Cycle 33. Statistical Thermodynamics of Crystals: Einstein Model 34. The Debye Model Problems 19 Intermolecular Forces and the Liquid State 902 1. Disorder in the Liquid State 2. X-Ray Diffraction of Liquid Structures 3. Liquid Crystals 4. Glasses 5. Melting 6. Cohesion of Liquids— The Internal Pressure 7. Intermolecular Forces 8. Equation of State and Intermolecular Forces 9. Theory of Liquids 10. Flow Properties of Liquids 11. Viscosity Problems 20 Macromolecules 928 1. Types of Polyreactions 2. Distribution of Molar Masses 4. Light Scattering—The Rayleigh Law 5. Light Scattering by Macromolecules 6. Sedimentation Methods: The Ultracentrifuge 7. Viscosity 8. Stereo­ chemistry of Polymers 9. Elasticity of Rubber 10. Crystallinity of Poly­ mers Problems Appendix A 960 Appendix B 961 Name Index 963 Subject Index 969  Physicochemical Systems Nosotros (la indivisa divinidad que opera en nosotros) hemos sonado el mundo. Lo hemos sonado resistente, misterioso, visible, ubicuo en el espacio y firme en el tiempo; pero hemos consentido en su arquitectura tenues y eternos intersticios de sinrazbn. para saber que es fatso. Jorge Luis Borges * 1932 On the planet Earth the processes of evolution created neural networks called brains. Reaching a certain degree of complexity, these networks generated electrical phenomena in space and time called consciousness, volition, and memory. The brains in some of the higher primates, genus Homo, devised a medium called language to communicate with one another and to store information. Some of the human brains persistently sought to analyze the input signals received from the world in which they had their existence. One form of analysis, called science, proved to be especially effective in correlating, modifying, and controlling the sensory input data. Most of the structure of brains was laid down in conformity with information coded into the base sequence of the DNA molecules of the genetic material. Addi­ tional structuring was caused by a relatively uniform experience during their periods of growth and maturation. Thus, heredity and early environment combined to produce adult brains with rather stereotyped capabilities for analysis and communication. Language was effective in communications that dealt with the content of sensory input data, but it did not allow the brains to talk about themselves or their relation to the world without breakdowns into paradox or contradiction. In particular, although it was possible to find thousands of books filled with results of science, to observe thousands of men at work in the fields of science, and to experience the earthshaking effects of science, it was not possible to explain in words what science was or even the mechanism by which it operated. Different views on these questions were eloquently put forth from time to time. *From “La Perpetua Carrera de Achilles e la Tortuga” in Discusion (Buenos Aires: M. Gleizer, 1932). “We (the undivided divinity that operates within us) have dreamed the world. We have dreamed it resistant, mysterious, visible, ubiquitous in space and firm in time; but we have allowed into its architecture tenuous and eternal interstices of unreason to let us understand that it is false.” 1 2 Physicochemical systems Chap. 1 1. What Is Science? According to one view, called conventionalism, the human brains created or invented certain beautiful logical structures called laws of nature and then devised special ways, called experiments, of selecting sensory input data so that they would fit into the patterns ordained by the laws. In the conventionalist view, the scientist was like a creative artist, working not with paint or marble but with the unorganized sensations from a chaotic world. Scientific philosophers supporting this position included Poincare, * Duhem,f and Eddington.J A second view of science, called inductivism, considered that the basic pro­ cedure of science was to collect and classify sensory input data into a form called observable facts. From these facts, by a method called inductive logic, the scientist then drew general conclusions which were the laws of nature. Francis Bacon, in his Novum Organum of 1620, argued that this was the only proper scientific method, and at that time his emphasis on observable facts Was an important antidote to medieval reliance on a formal logic of limited capabilities. Bacon’s definition accords most closely with the layman’s idea of what scientists do, but many com­ petent philosophers have also continued to support the essentials of inductivism, including Russell § and Reichenbach. || A third view of science, called deductivism, emphasized the primary importance of theories. According to Popper,# “Theories are nets cast to catch what we call ‘the world’: to rationalize, to explain, and to master it. We endeavor to make the mesh ever finer and finer.” According to the deductivists, there is no valid inductive logic, since general statements can never be proved from particular instances. On the other hand, a general statement can be disproved by one contrary particular instance. Hence, a scientific theory can never be proved, but it can be disproved. The role of an experiment is therefore to subject a scientific theory to a critical test. The three philosophies outlined by no means exhaust the variety of efforts made to capture science in the web of language. As we are studying the part of science called physical chemistry, we should pause sometimes (but not too often) to ask ourselves which philosophic school we are attending. *Henri Poincare, Science and Hypothesis (New York: Dover Publications, Inc., 1952). tPierre Duhem, The System of the World, 6 Vols. (Paris: Librarie Scientifique Hermann et Cie., 1954). J Arthur Stanley Eddington, The Philosophy of Physical Science (Ann Arbor, Mich.: Univer­ sity of Michigan Press, 1958). §Bertrand Russell, Human Knowledge, Its Scope and Limits (New York: Simon and Schuster, Inc., 1948). 11 Hans Reichenbach, The Rise of Scientific Philosophy (Berkeley: University of California Press, 1963). it Karl R. Popper, The Logic of Scientific Discovery (New York: Harper Torchbooks, 1965). Sec. 3 Mechanics; force 3 2. Physical Chemistry There appear to be two reasonable approaches to the study of physical chemistry. We may adopt a synthetic approach and, beginning with the structure and behavior of matter in its finest known state of subdivision, gradually progress from electrons to atoms to molecules to states of aggregation and chemical reactions. Alterna­ tively, we may adopt an analytical treatment and, starting with matter or chemicals as we find them in the laboratory, gradually work our way back to finer states of subdivision as we require them to explain experimental results. This latter method follows more closely the historical development, although a strict adherence to history is impossible in a broad subject whose different branches have progressed at different rates. Two main problems have been primary concerns of physical chemistry: the question of the position of chemical equilibrium, which is the principal problem of chemical thermodynamics; and the question of the rate of chemical reactions, which is the field of chemical kinetics. Since these problems are ultimately con­ cerned with the interactions of molecules, their complete solutions should be implicit in the mechanics of molecules and molecular aggregates. Therefore, molecular structure is an important part of physical chemistry. The discipline that allows us to bring our knowledge of molecular structure to bear on the problems of equilibrium and kinetics is found in the study of statistical mechanics. We shall begin our study of physical chemistry with thermodynamics, which is based on concepts common to the everyday world. We shall follow quite closely the historical development of the subject, since usually more knowledge can be gained by watching the construction of something than by inspecting the polished final product. 3. Mechanics: Force The first thing that may be said of thermodynamics is that the word itself is evi­ dently derived from dynamics, which is a branch of mechanics dealing with matter in motion. Mechanics is founded on the work of Isaac Newton (1642-1727), and usually begins with a statement of the well-known equation F = ma with _ d\ _ J2r (1-1) dt~ dt2 The equation states the proportionality between a vector quantity F, called the force applied to a particle of matter, and the acceleration a of the par­ ticle, a vector in the same direction, with a proportionality factor m, called the mass. Equation (1.1) may also be written 4 Physicochemical systems Chap. 1 F=^ (1.2) dt where the product of mass and velocity is called the momentum. In the International System of Units (SI), the unit of mass is the kilogram* (kg), the unit of time is the secondf (s), and the unit of length is the metref (m). The SI unit of force is the newton (N). Mass might be introduced in Newton’s Law of Gravitation, F_ Gm1m2 ^12 which states that there is an attractive force between two masses proportional to their product and inversely proportional to the square of their separation. If this gravitational mass is to be the same as the inertial mass of (1.1), the proportionality constant (7 = 6.670 x 10“u m3-s'^-kg'2 The weight of a body, W, is the force with which it is attracted toward the earth, and may vary slightly over the earth’s surface, since the earth is not a perfect sphere of uniform density. Thus W — mg where g is the acceleration of free fall in vacuum. In practice, the mass of a body is measured by comparing its weight by means of a balance with that of known standards (m1/m2 = 4. Mechanical Work In mechanics, if the point of application of a force F moves, the force is said to do work. The amount of work done by a force F whose point of application moves a distance dr along the direction of the force is dw — F dr (1.3) If the direction of motion of the point of application is not the same as the direc­ tion of the force, but at an angle 0 to it, we have the situation shown in Fig. 1.1. The component of F in the direction of motion is F cos 0, and the element of work is dw=^F cos 6 dr (1.4) If we choose a set of Cartesian axes XYZ, the components of the force are *Defined by the mass of the international prototype, a platinum cylinder at the International Bureau of Weights and Measures at Sevres, near Paris. t Defined as duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels in the ground state of the cesium-133 atom. ^Defined as the length equal to 1 650 763.73 wavelengths in vacuum of the radiation corre­ sponding to the transition between the levels 2pi0 and 5d5 of the krypton-86 atom. Sec. 4 Mechanical work 5 dr FIG. 1.1 Definition of differential dw ~Fdr cos 0 element of work. Fx, F^, Fz and dw = Fxdx + Fy dy + Fz dz (1.5) For the case of a force that is constant in direction and magnitude, (1.3) can be integrated to give w = f F dr = F(rt — r0) J ro An example is the force acting on a body of mass m in the earth’s gravitational field. Over distances that are short compared to the diameter of the earth, this F = mg. To lift a body against earth’s gravitational attraction we must apply to it an external force equal to mg. What is the work done on a mass of 1 kg when it is lifted a distance of 1 m? w = mgrx = (l)(9.80665)(l) kg-m s'2-m = 9.80665 kg-m2-s"2 = 9.80665 newton metre (N-m) = 9.80665 joule (J) An application of (1.3) in which the force is not constant is to the stretching of a perfectly elastic spring. In accord with the law of Hooke, 1660, ut tensio sic vis : the restoring force is directly proportional to the extension, F=-Kr (1.6) where k is called the force constant of the spring. Hence, the work dw done on the spring to extend it by dr is dw — Kr dr Suppose the spring is stretched by a distance ru w = I Kr dr = ^-r\ Jo 2 The work done on the spring is taken by convention to be positive. In the general case, we can write the integral of (1.5) as w= C (Fxdx + Fydy + F,dz) (1.7) Ja The components of the force may vary from point to point along the curve fol­ lowed by the mass point. They are functions of the space coordinates x, y, z: Fx(x, y, z), Fy(x, y, z), and Fz(x, y, z). It is evident that the value of the integral depends upon the exact path or curve between the two limits a and b. It is called a line integral. 6 Physicochemical systems Chap, 1 5. Mechanical Energy In 1644, Rene Descartes declared that in the beginning God imparted to the uni­ verse a certain amount of motion in the form of vortices, and this motion endured eternally and could be neither increased nor decreased. For almost a century after the death of Descartes, a great controversy raged between his followers and those of Leibniz on the question of whether motion was conserved. As often happens, lack of precise definitions of the terms used prevented a meeting of minds. The word motion then usually designated what we now call momentum, In fact, the momentum in any given direction is conserved in collisions between elastic bodies. In 1669, Huygens discovered that if he multiplied each mass m by the square of its velocity v2, the sum of these products was conserved in all collisions between elastic bodies. Leibniz called mv2 the vis viva. In 1735, Jean Bernoulli asked himself what happened to the vis viva in an inelastic collision. He concluded that some of it was lost into some kind of vis mortua. In all mechanical systems operating without friction, the sum of vis viva and vis mortua was conserved at a constant value. In 1742, this idea was also clearly expressed by Emilie du Chatelet, who said that although it was difficult to follow the course of the vis viva in an inelastic collision, it must nevertheless be conserved in some way. The first to use the word ‘‘energy” appears to have been d’Alembert in the French Encyclopedic of 1785. “There is in a body in movement an effort or energie, which is not at all in a body at rest.” In 1787, Thomas Young called the vis viva the “actual energy” and the vis mortua the “potential energy.” The name “kinetic energy” for ^mv2 was introduced much later by William Thomson. We can give these developments a mathematical formulation starting with (1.3). Let us consider a particle at position r^ and apply to it a force F(r) that depends only on its position. In the absence of any other forces, the work done on the body in a finite displacement from rQ to rx is w=\'F(r)dr (1.8) J TO The integral over distance can be transformed to an integral over time: w= F(r) — dt — \ F{r)v dt J t0 at j to Introducing Newton’s Law of Force, (LI), we obtain dv CVx w= rn-r-v dt = m v dv J to dt Ko (1.9) iv = ^mv2x — ^mvl The kinetic energy is defined by Ek = ^mv2 Hence, w = P F(r) dr = Ekl - Eka (1.10) J to Sec. 6 Equilibrium 7 The work done on the body equals the difference between its kinetic energies in the final and the initial states. Since the force in (1.10) is a function of r alone, the integral defines another function of r which we can write as F(r) dr = -dU(r) or = (i-n) Thus, (1.10) becomes f' F(r) dr = U(rJ - = Ekl - Ek0 J r0 or Uo + EkQ = U,+ Ekl (1.12) The new function U(r) is the potential energy. The sum of the potential and the kinetic energies, U + Ek9 is the total mechanical energy of the body, and this sum evidently remains constant during the motion. Equation (1.12) has the form typical of an equation of conservation. It is a statement of the mechanical principle of the conservation of energy. For example, the gain in kinetic energy of a body falling in a vacuum is exactly balanced by an equal loss in potential energy. If a force depends on velocity as well as position, the situation is more complex. This would be the case if a body falls, not in a vacuum, but in a viscous fluid, such as air or water. The higher the velocity, the greater is the frictional or viscous resistance opposed to the gravitational force. We can no longer set F(r) = —dUjdr, and we can no longer obtain an equation such as (1.11), because the mechanical energy is no longer conserved. From the dawn of history it has been known that the frictional dissipation of energy is attended by the evolution of something called heat. We shall see later how it became possible to include heat among the ways of transforming energy, and in this way to obtain a new and more inclusive principle of the conservation of energy. It may be noted that whereas the kinetic energy is zero for a body at rest, there is no naturally defined zero of potential energy. Only differences in potential energy can be measured. Sometimes, however, a zero of potential energy is chosen by convention; an example is the choice U(r) = 0 for the gravitational potential energy when two bodies are an infinite distance apart. 6. Equilibrium The ordinary subjects for chemical experimentation are not individual particles of any sort but more complex systems, which may contain solids, liquids, and gases. A system is a part of the world separated from the rest of the world by definite boundaries. The world outside the system is called its surroundings. If the boundaries of the system do not permit any change to occur in the system as a consequence of a change in the surroundings, the system is said to be isolated. 8 Physicochemical systems Chap. 1 The experiments that we perform on a system are said to measure its properties, these being the attributes that enable us to describe it with all requisite complete­ ness. This complete description is said to define the state of the system. The idea of predictability enters here. Having once measured the properties of a system, we expect to be able to predict the behavior of a second system with the same set of properties from our knowledge of the behavior of the original. When a system shows no further tendency to change its properties with time, it is said to have reached a state of equilibrium. The condition of a system in equilibrium is reproducible and can be defined by a set of properties, which are functions of the state, i.e., which do not depend on the history of the system before it reached equilibrium. * A simple mechanical illustration will clarify the concept of equilibrium. Figure 1.2(a) shows three different equilibrium positions of a box resting on a table. In both positions A and C, the center of gravity of the box is lower than in any slightly displaced position, and if the box is tilted slightly it will tend to return spontaneously to its original equilibrium position. The gravitational potential energy of the box in positions A or C is at a minimum, and both positions represent stable equilibrium states. Yet it is apparent that position C is more stable than position A, and a certain tilt of A will suffice to push it over into C. In position A, therefore, the box is said to be in metastable equilibrium. Position B is also an equilibrium position, but it is a state of unstable equilibrium, as anyone who has *The specification of the state of a system that is not in equilibrium is a more difficult problem. It will require a larger number of variables and sometimes will not even be possible in practice. Sec. 7 The thermal properties of matter 9 tried to balance a chair on two legs will agree. The center of gravity of the box in B is higher than in any slightly displaced position, and the tiniest tilt will send the box into either position A or C. The potential energy at a position of unstable equilibrium is a maximum, and such a position could be realized only in the absence of any disturbing forces. These relations may be presented in more mathematical form by plotting in Fig. 1.2(b) the potential energy of the system as a function of the horizontal posi­ tion of the center of gravity in Fig. 1.2(a). Positions of stable equilibrium are seen to be minima in the curve, and the position of unstable equilibrium is represented by a maximum. Positions of stable and unstable equilibrium alternate in this way in any system. For an equilibrium position, the slope dUjdr of the curve for U vs. displacement is equal to zero, and one may write the equilibrium condition as Examination of the second derivative will indicate whether the equilibrium is stable or unstable: stab,e (< 0 unstable \ dr2 / Although these considerations have been presented in terms of a simple mechan­ ical model, similar principles will be found to apply in the more complex phys­ icochemical systems that we shall study. In addition to purely mechanical changes, such systems may undergo temperature changes, changes of state of aggregation, and chemical reactions. The problem of thermodynamics is to discover or invent new functions that will play the role in these more general systems which the potential energy plays in mechanics. 7. The Thermal Properties of Matter To specify precisely the state at equilibrium of a substance studied in the laboratory, we must give the numerical values of certain of its measured properties. Since there are equations that give relations between properties, it is not necessary to specify the values of each and every property to define exactly the state of a substance. In fact, if we ignore external fields of force (gravitational, electromagnetic) and take a gas or a liquid as the substance under consideration,* the exact specification of its state requires the values of only a few quantities. At present, we shall confine the problem to individual, pure substances, for which no composition variables are needed. To specify the state of a pure gas or liquid, we may first of all state the mass m of the substance. There are many properties of the substance that might be measured to define its state. In particular, we focus attention on three thermody­ namic variables: pressure P, volume V9 and temperature 6. If any two of these are ~*The properties of solids may depend in a rather complicated way on direction. 10 Physicochemical systems Chap. 1 fixed, we find as a matter of experimental fact that the value of the third will also be fixed, because of the existence of a relation between the variables. In other words, of the three variables of state, P, V, 6, only two are independent variables. Note particularly that we may describe the state of the substance entirely in terms of the two mechanical variables P and V, and not use the thermal variable 0 at all. The use of the pressure P as a variable to describe the state of a substance requires some care. In Fig. 1.3, consider a fluid contained in a cylinder with a FIG. 1.3 Definition of pressure in a fluid, neglecting gravitational field in the fluid. The force F represent­ ed by the weight includes also the force due to the earth's atmosphere. frictionless piston. We can calculate the pressure on the fluid by dividing the force on the piston by its area. (P = force/area). At equilibrium, this pressure will be uniform throughout the fluid, so that on any specified unit area in the fluid there will be a force P. A pressure is thus a stress that is uniform in all directions. In this analysis, we neglect the effect of the weight of the fluid itself. If we included the weight, there would be an extra force per unit area, increasing with the depth of the fluid and equal to the weight of the column of fluid above the given section. In the subsequent analysis, this effect of the weight will be neglected and we shall consider the pressure of a volume of fluid to be the same throughout. This simplification is what we mean by ‘‘ignoring the gravitational field.” If the fluid is not in equilibrium, we can still speak of the external pressure Pex on the piston, but this is clearly not a property of the state of the fluid itself. Until equilibrium is restored, the pressure may vary from point to point in the fluid and we cannot define its state by a single pressure P. The properties of a system can be classified as extensive or intensive. Extensive properties are additive; their value for the whole system is equal to the sum of their values for the individual parts. Sometimes they are called capacity factors. Examples Sec. 8 Temperature as a mechanical property 11 are the volume and the mass. Intensive properties, or intensity factors, are not additive. Examples are temperature and pressure. The temperature of any small part of a system in equilibrium is the same as the temperature of the whole. Before we use the temperature 0 as a physical quantity, we should consider how it can be measured quantitatively. The concept of temperature evolved from sensual perceptions of hotness and coldness. It was found that these perceptions could be correlated with the readings of thermometers based on the volume changes of liq­ uids. In 1631, the French physician Jean Rey used a glass bulb having a stem partly filled with water to follow the progress of fevers in his patients. In 1641, Ferdinand II, Grand Duke of Tuscany and founder of the Accademia del Cimento of Florence, invented an alcohol-in-glass thermoscope to which a scale was added by marking equal divisions between the volumes at “coldest winter cold” and “hottest summer heat.” A calibration based on two fixed points was introduced in 1688 by Dalence, who chose the melting point of snow as —10° and the melting point of butter as +10°. In 1694, Renaldi took the boiling point of water as the upper fixed point and the melting point of ice as the lower. To make the specifica­ tion of these fixed points exact, we must add the requirements that the pressure is maintained at 1 atmosphere (atm), and that the water in equilibrium with ice is saturated with air. Apparently Elvius, a Swede, in 1710, first suggested assigning the values 0° and 100° to these two points. They define the centigrade scale, officially called the Celsius scale, after a Swedish astronomer who used a similar system. 8. Temperature as a Mechanical Property The existence of a temperature function can be based on the fact that whenever two bodies are separately brought to equilibrium with a third body, they are then found to be in equilibrium with each other. We may choose one body (1) to be a pure fluid whose state is specified by Px and Kj and call it a thermometer, and use some property of the state of this body KJ to define a temperature scale. When any second body (2) is brought into equilibrium with the thermometer, the equilibrium value of 0fPi9 KJ measures its temperature. 01 = 0^,^) (1-13) Note that the temperature defined and measured in this way is defined entirely in terms of the mechanical properties of pressure and volume, which suffice to define the state of the pure fluids. We have left our sensory perceptions of hotness and coldness and reduced the concept of temperature to a mechanical concept. A simple example of (1.13) is a liquid thermometer, in which P1 is kept constant and the volume is used to measure the temperature. In other cases, electrical, magnetic, or optical properties can be used to define the temperature scale, since in every case the property Gx can be expressed as a function of the state of the pure fluid, fixed by specifying and K^ 12 Physicochemical systems Chap. 1 9. The Spring of the Air and Boyle's Law The mercury barometer was invented in 1643 by Evangelista Torricelli, a mathe­ matician who studied with Galileo in Florence. The height of the column under atmospheric pressure may vary from day to day over a range of several centimetres of mercury, but a standard atmosphere has been defined as a pressure unit equal to 101 325 newtons per square metre (N-m"2). Workers in the field of high pres­ sures often use the kilobar (kbar), 108 N m~2. At low pressures, one often uses the torr, which equals atm/760. * Robert Boyle and his contemporaries often referred to the pressure of a gas as the spring of the air. They knew that a volume of gas behaved mechanically like a spring. If you compress it in a cylinder with a piston, the piston recoils when the force on it is released. Boyle tried to explain the springiness of the air in terms of the corpuscular theories popular in his day. “Imagine the air,” said he, “to be such a heap of little bodies, lying one upon anothet, as may be resembled to a fleece of wool. For this... consists of many slender and flexible hairs, each of which may indeed, like a little spring, be still endeavoring to stretch itself out again.” In other words, Boyle supposed that the corpuscles of air were in close contact with one another, so that when air was compressed, the individual corpus­ cles were compressed like springs. This hypothesis was not correct. In 1660, Boyle published the first edition of his book New Experiments, Physico- Mechanical, Touching the Spring of the Air, and its Effects, in which he described observations made with a new vacuum pump he had constructed. He found that when the air surrounding the reservoir of a Torricellian barometer was evacuated, the mercury column fell. This experiment seemed to him to prove conclusively that the column was held up by the air pressure. Nevertheless, two attacks on Boyle’s work were immediately published, one by Thomas Hobbes, the famous political philosopher and author of Leviathan, and the other by a devout Aristotelian, Franciscus Linus. Hobbes based his criticism on the “philosophical impossibility of a vacuum.” (“A vacuum is nothing, and what is nothing cannot exist.”) Linus claimed that the mercury column was held up by an invisible thread, which fastened itself to the upper end of the tube. This theory seemed quite reasonable, he said, for anyone could easily feel the pull of the thread by covering the end of the barometer tube with his finger. In answer to these objections, Boyle included an appendix in the second edition of his book, published in 1662, in which he described an important new experiment. He used essentially the apparatus shown in Fig. 1.4. By addition of mercury to the open end of the J-shaped tube, the pressure could be increased on the gas in the closed end. Boyle observed that as the pressure increased, the volume of the gas proportionately decreased. The temperature of the gas was almost constant during these measurements. In modern terms, we would therefore state Boyle’s result thus: At constant temperature, the volume of a given sample of gas varies inversely as the pressure. In mathematical terms, this becomes P co 1 / V or *The torr differs from the conventional mmHg by less than 2 parts in 107. Sec. 10 The law of Gay-Lussac 13 Mercury column increased by pouring in mercury at A Shorter leg with scale FIG. 1.4 Boyle's J-tube, as used in an Initial level of ------ experiment in which the volume mercury of gas was halved when the pressure was doubled. P = C/V where C is a constant of proportionality. This relation is equivalent to PV = C (at constant 0) (1.14) Equation (1.14) is known as Boyle's Law. It is followed quite closely by many gases at moderate pressures, but the real behavior of gases may deviate greatly, especially at higher pressures. 10. The Law of Gay-Lussac The first detailed experiments on the variation with temperature of the volumes of gases at constant pressures were those published by Joseph Gay-Lussac from 1802 to 1808. Working with the “permanent gases,” such as nitrogen, oxygen, and hydrogen, he found that the different gases all showed the same dependence of V on 0. His results can be put into mathematical form as follows. We define a gas tem­ perature scale by assuming that the volume V varies linearly with the temperature 0. If Ko is the volume of a sample of gas at 0°C, we have V= K0(l +ao0) (1.15) The coefficient a0 is a thermal expansivity or coefficient of thermal expansion. * Gay-Lussac found a0 approximately equal to but in 1847, Regnault, with an improved experimental procedure, obtained a0 = With this value, (1.15) may be written as This relation is called the Law of Gay-Lussac. It states that a gas expands by of its volume at 0°C for each degree rise in temperature at constant pressure. *In Section 1.13, we define a somewhat different thermal expansivity, a. 14 Physicochemical systems Chap. 1 Careful measurements revealed that real gases do not obey exactly the laws of Boyle and Gay-Lussac. The variations are least when a gas is at high temperature and low pressure. Furthermore, the deviations vary from gas to gas; for example, helium obeys closely, whereas carbon dioxide is relatively disobedient. It is useful to introduce the concept of the ideal gas, a gas that follows these laws exactly. Since gases at low pressures—i.e., low densities—obey the gas laws most closely, we can often obtain the properties of ideal gases by extrapolation to zero pressure of measurements on real gases. Figure 1.5 shows the results of measurements of a0 on different gases at suc- FIG. 1.5 Extrapolation of thermal expansion coefficients to zero pressure. cessively lower pressures. Note that the scale is greatly expanded so that the maxi­ mum differences do not exceed about 0.5%. Within experimental uncertainty, the value found by extrapolation to zero pressure is the same for all these gases. This is the value of a0 for an ideal gas. The consensus of the best measurements gives a0 = 36.610 x IO"4 °C'1 or — = 273.15°C ± 0.02°C = To a0 0 Thus, the Law of Gay-Lussac for an ideal gas may be written K=r0(i + l) (i.i6) It is now possible and most convenient to define a new temperature scale with the temperature denoted by T and called the absolute temperature. The unit of temperature on this scale is called the kelvin, K.* Thus, T=0 + To In terms of T, the Law of Gay-Lussac, (1.16), becomes *It is read, for example, simply as “200 kelvin” or “200 K.” Sec. 12 Equation of state of an idea! gas 15 (1.17) Extensive work at very low temperatures, from 0 to 20 K, revealed serious inconveniences in the definition of a temperature scale based on two fixed points. Despite the most earnest attempts, it was evidently impossible to obtain a mea­ surement of the ice point (at which water and ice are in equilibrium under 1 atm pressure) accurate to within better than a few hundreths of a kelvin. Many years of careful work yielded results from 273.13 to 273.17 K. Therefore, in 1954, the Tenth Conference of the International Committee on Weights and Measures, meeting in Paris, defined a temperature scale with only one fundamental fixed point, and with an arbitrary choice of a universal constant for the temperature at this point. The point chosen was the triple point of water, at which water, ice, and water vapor are simultaneously in equilibrium. The tem­ perature of this point was set equal to 273.16 K. The value of the ice point then became 273.15 K. The boiling point of water is not fixed by convention, but is simply an experimental point to be measured with the best accuracy possible. 11. Definition of the Mole In accord with the latest recommendations of the International Union of Pure and Applied Chemistry, we shall consider the amount of substance n to be one of the basic physicochemical quantities. The SI unit of amount of substance is the mole (abbreviated mol). The mole is the amount of substance of a system containing as many elementary units as there are carbon atoms in exactly 0.012 kg of carbon-12. The elementary unit must be specified and may be an atom, a molecule, an ion, an electron, a photon, etc., or a specified group of such entities. Examples are as follows: 1. One mole of HgCl has a mass of 0.23604 kg. 2. One mole of Hg2Cl2 has a mass of 0.47208 kg. 3. One mole of Hg has a mass of 0.20059 kg. 4. One mole of Cu0 5Zn0.5 has a mass of 0.06446 kg. 5. One mole of Fe0 91 S has a mass of 0.08288 kg. 6. One mole of e~ has a mass of 5.4860 x 10"7 kg. 7. One mole of a mixture containing 78.09 mol % N2, 20.95 mol % O2, 0.93 mol % Ar, and 0.03 mol % CO2, has a mass of 0.028964 kg. 12. Equation of State of an Ideal Gas Any two of the three variables P, V, and T suffice to specify the state of a given amount of gas and to fix the value of the third variable. Equation (1.14) is an expression for the variation of P with V at constant T, and (1.17) is an expression for the variation of V with T at constant P. 16 Physicochemical systems Chap. 1 PV — const, (constant T) = const, (constant P) We can readily combine these two relations to give const. (1.18) It is evident that this expression contains both the others as special cases. The next problem is the evaluation of the constant in (1.18). The equation states that the product PV divided by T is always the same for all specified states of the gas; hence, if we know these values for any one state we can derive the value of the constant. Let us take this reference state to be an ideal gas at 1 atm pressure (Po) and 273.15 K. The volume under these conditions is 22 414 cm3 - mol"1 (V^ri). According to Avogadro’s Law (Sect. 4.2), this volume is the same for all ideal gases. If we have n moles of ideal gas in the reference state, (1.18) becomes % _ & _ (IStnOg^Wcma _ 82 057ncm,.atm.K-, The constant R is called the gas constant per mole. We often write this equation as PV=nRT (1.19) Equation (1.19) is called the equation of state of an ideal gas and is one of the most useful relations in physical chemistry. It contains the three gas laws: those of Boyle, Gay-Lussac, and Avogadro. We first obtained the gas constant R in units of cm3 atm K*1 mol*1. Note that cm3 atm has the dimensions of energy. Some convenient values of R in various units are summarized in Table 1.1. TABLE 1.1 Values of the Ideal Gas Constant R in Various Units (SI): J-K'i-mol-i 8.31431 cabK-1 ’mob1 1.98717 cm3 atm-K-1-mol-1 82.0575 batm^K^1 -mol-1 0.0820575 Equation (1.19) allows us to calculate the molar mass M of a gas from mea­ surements of its density. If a mass of gas m is weighed in a gas density bulb of volume V, the density p — m/V. The amount of gas n — m/M. Therefore, from (1.19), Sec. 13 The equation of state and PVT relationships 17 13. The Equation of State and PVT Relationships If P and V are chosen as independent variables, the temperature of a given amount n of a pure substance is some function of P and V. Thus, if Vm = V[n, T — f(P9Vm) (1.20) For any fixed value of T, this equation defines an isotherm of the substance under consideration. The state of a substance in thermal equilibrium can be fixed by specifying any two of the three variables, pressure, molar volume, and temperature. The third variable can then be found by solving (1.20). Equation (1.20) is a general form of the equation of state. If we do not care to specify a particular independent variable, it could be written as g(P9 Vm, T) = 0; for example, (PV — nRT) = 0. Geometrically considered, the state of a pure fluid in equilibrium can be rep­ resented by a point on a three-dimensional surface, described by the variables P9 V9 and T. Figure 1.6(a) shows such a PVT surface for an ideal gas. The isothermal lines connecting points at constant temperature are shown in Fig. 1.6(b), projected on the PV plane. The projection of lines of constant volume on the PT plane gives the isochores or isometrics shown in Fig. 1.6(c). Of course, for a nonideal gas, these would not be straight lines. Constant pressure lines are called isobars. The slope of an isobaric curve gives the rate of change of volume with tem­ perature at the constant pressure chosen. This slope is therefore written (dV/dT)P. It is a partial derivative because V is a function of the two variables T and P. The fractional change in V with T is a, the thermal expansivity. „ - 1 (dv\ (1-21) - V \3t 4 Note that a has the dimensions of T-1. In a similar way, the slope of an isothermal curve gives the variation of volume with pressure at constant temperature. We define P, the isothermal compressibility of a substance, as (1.22) The negative sign is introduced because an increase in pressure decreases the volume, so that (dV/dP)T is negative. The dimensions of P are those of P1. Since the volume is a function of both T and P, a differential change in volume can be written* (1-23) Equation (1.23) can be illustrated graphically by Fig. 1.7, which shows a section ♦W. A. Granville, P. F. Smith, W. R. Longley, Elements of Calculus (Boston: Ginn & Com­ pany, 1957), p. 445. The total differential of a function of several independent variables is the sum of the differential changes that would be caused by changing each variable separately. This is true because a change in one variable does not influence the change in another independent vari­ able. PRESSURE (b) FIG. 1.6 (a) PVT surface for an ideal gas. The solid lines are iso­ therms. the dashed lines are isobars, and the dotted lines are isometrics, (b) Projection of the PVT surface on PV plane, showing isotherms, (c) Projection of the PVT surface on the PT plane, showing isometrics. [After F. W. Sears, An Introduction to Thermodynamics (Cambridge, Mass.: Ad­ dison-Wesley, 1950).] 18 Sec. 13 The equation of state and PVT relationships 19 of the PVT surface plotted with V as the vertical axis. The area abed represents an infinitesimal element of surface area, cut out from the surface by planes parallel to the VT plane and the VP plane. Suppose we start with a state of the gas specified by point a, corresponding to values of the state variables specified by Va, Pa9 and Ta, Now suppose we change both P and T by infinitesimal amounts to P + dP and T + dT. The new state of the system is represented by the point c. The change in Kis dV= Vc-Va = (Vb - Va)+(VC - Vb) We see, however, that Vb — Va is the change in V that would occur if the P is kept constant and only the temperature changed. The slope of the line ab is, therefore, Ka Lim T-*0 Tb-Ta a (P const.) The infinitesimal change Vb — Va is, therefore, (dV/dT)P dT, In the same way, we can see that Vc — Vb is (0V/dP)T dP, Hence, the total change in V becomes the sum of these two partial changes, as shown in (1.23). For such infinitesimal changes as shown in Fig. (1.7), it makes no difference which partial change is considered first. We can derive an interesting relation between the partial differential coefficients. By solving (1.23), we have dp =___ * dV — dT ar (dV/dP)T (dV/dP)T ai But also, by analogy with (1.23), which is a general form, dT The coefficients of dT must be equal, so that (dP\ _ ~(dVldT}P _ a_ (1.24) \dT)v “ (dVjdP^ 0 The variation of P with T for any substance can therefore be readily calculated if we know a and p, An interesting example is suggested by a common laboratory accident, the breaking of a mercury-in-glass thermometer by overheating. If a thermometer is exactly filled with mercury at 50°C, what pressure will be developed within the thermometer if it is heated to 52°C? For mercury, under these condi­ 20 Physicochemical systems Chap. 1 tions, a = 1.8 x 10'4°C-1, P = 3.9 x 10-6 atm’1. Therefore, (dP/dT)^ = aj/i = 46 atm-°C”1. For AT= 2°C, AP = 92 atm. It is not surprising that even a little overheating will break the usual thermometer. 14. PVT Behavior of Real Gases The pressure, volume, temperature (PVT) relationships for gases, liquids, and solids would preferably all be succinctly summarized in the form of equations of state of the general type of (1.20). Only in the case of gases has there been much progress in the development of these state equations. They are obtained by correla­ tion of empirical PVT data, and also from theoretical considerations based on atomic and molecular structure. These theories are farthest advanced for gases, but more recent developments in the theory of liquids and solids give promise that suitable state equations may eventually be available in these fields also. The ideal gas equation PV = nRT describes the PVT behavior of real gases only to a first approximation. A convenient way of showing the deviations from ideality is to write, for the real gas, PV = znRT (1.25) The factor z is called the compressibility factor. It is equal to PVjnRT. For an ideal gas, z = 1, and departure from ideality will be measured by the deviation of the compressibility factor from unity. The extent of deviations from ideality de­ pends on the temperature and pressure, so that z is a function of T and P. Some compressibility-factor curves are shown in Fig. 1.8; these are determined from 0 200 400 600 800 1000 1200 FIG. 1.8 Compressibility factors z = PRESSURE, atm PV/nRT at 0°C. experimental measurements of the volumes of the substances at different pressures. (The data for NH3 and C2H4 at high pressures pertain to the liquid substances.) Sec. 15 Law of corresponding states 21 15. Law of Corresponding States Let us consider a liquid at some temperature and pressure at which it is in equi­ librium with its vapor. This equilibrium pressure is called the vapor pressure of the liquid. The liquid will be more dense than the vapor, and if we have a sample of the substance in a closed transparent tube, we can see a meniscus between liquid and vapor, indicating the coexistence of the two phases. Above a certain temperature, however, called the critical temperature Tc, only one phase exists, no matter how great the pressure applied to the system. A substance above its Tc is said to be in the fluid state. The pressure that would just suffice to liquefy the fluid at Tc is the critical pres­ sure Pc. The molar volume occupied by the substance at Tc and Pc is its critical volume Vc. These critical constants for various substances are collected in Table 1.2. TABLE 1.2 Critical Point Data and van der Waals Constants Formula Tc(K) Pc(atm) Kc(cm3 mol- 1) 10'^ 6(cm3 -mol-1) (cm6* atm’mol"2) He 5.3 2.26 61.6 3.41 23.7 h2 33.3 12.8 69.7 24.4 26.6 n2 126.1 33.5 90.0 139 39.1 CO 134.0 34.6 90.0 149 39.9 o2 154.3 49.7 74.4 136 31.8 c2h4 282.9 50.9 127.5 447 57.1 co2 304.2 72.8 94.2 359 42.7 nh3 405.6 112.2 72.0 417 37.1 h2o 647.2 217.7 55.44 546 30.5 Hg 1735.0 1036.0 40.1 809 17.0 The ratios of P, V, and T to the critical values Pc, Vc, and Tc are called the reduced pressure, volume, and temperature. These reduced variables may be written = (L26) In 1881, van der Waals pointed out that to a fairly good approximation, espe­ cially at moderate pressures, all gases would follow the same equation of state in terms of reduced variables, PR9 TR, VR—i.e., VR = f(PR, TR). He proposed to call this rule the Law of Corresponding States. If this “law” were true, the critical ratio PCVJRTC would be the same for all gases. Actually, as you can readily confirm from the data in Table 1.2, the ratio varies from about 3 to 5 for the common gases. 22 Physicochemical systems Chap. 1 Chemical engineers and other workers actively concerned with the properties of gases at elevated pressures have prepared extensive and useful graphs to show the variation of the compressibility factor z in (1.25) with P and T, and they have found that to a good approximation, even at fairly high pressures, z appears to be a universal function of PR and TR, z =f(PR> Tr} (1.27) This rule is illustrated in Fig. 1.9 for a number of different gases, where z = PV/nRT is plotted at various reduced temperatures, against the reduced pressure. At these moderate pressures, the fit is good to within about 1 %. [Gouq-Jen Su, /nd. Eng. Chem., 38, 803 (1946).] Sec. 17 The critical region 23 16. Equations of State for Gases If the equation of state is written in terms of reduced variables, as F(PR9 VR) = TR9 it is evident that it contains at least two independent constants, characteristic of the gas in question—e.g., Pc and Vc. Many equations of state, proposed on semi- empirical grounds, serve to represent the PVT data more accurately than does the ideal gas equation. Several of the best known of these also contain two added constants. For example, the equation of van der Waals, ^P+^V-nb) = nRT (1.28) and the equation of D. Berthelot, (p + - nB) = nRT (1.29) The van der Waals equation provides a reasonably good representation of the PVT data of gases in the range of moderate deviations from ideality. For example, consider the following values in liter atmospheres of the PV product for 1 mol of carbon dioxide at 40°C, observed experimentally and as calculated from van der Waals’ equation. We have written Vm = Vin for the volume per mole. P, atm 1 10 50 100 200 500 1100 PVm, obs. 25.57 24.49 19.00 6.93 10.50 22.00 40.00 PVm, calc. 25.60 24.71 19.75 8.89 14.10 29.70 54.20 The constants a and b are evaluated by fitting the equation to experimental PVT measurements, or from the critical constants of the gas. Some values for van der Waals’ a and b are included in Table 1.2. Berthelot’s equation is somewhat better than van der Waals’ at pressures not much above 1 atm, and is preferred for general use in this range. 17. The Critical Region The behavior of a gas in the neighborhood of its critical region was first studied by Thomas Andrews in 1869 in a classic series of measurements on carbon dioxide. Results of determinations by A. Michels of these PF isotherms around the critical temperature of 31.01 °C are shown in Fig. 1.10. Consider the isotherm at 30.4°C, which is below Tc. As the vapor is compressed, the PV curve first follows AB, which is approximately a Boyle’s Law isotherm. When the point B is reached, a meniscus appears and liquid begins to form. Further compression then occurs at constant pressure until the point C is reached, at which all the vapor has been converted into liquid. The curve CD is the isotherm of liquid carbon dioxide, its steepness indicating the low compressibility of the liquid. 24 Physicochemical systems Chap. 1 VOLUME FIG. 1.10 Isotherms of carbon dioxide near the critical point. [Michels, Blaisse, and Michels, Proc. Roy. Soc.A. 160, 367 (1937).] As isotherms are taken at successively higher temperatures, the points of dis­ continuity B and C are observed to approach each other gradually, until at 31.01°C they coalesce, and no appearance of a second phase is observable. This isotherm corresponds to the critical temperature of carbon dioxide. Isotherms above this temperature exhibit no formation of a second phase no matter how great the applied pressure. Above the critical temperature, there is no reason to draw any distinction between liquid and vapor, since there is a complete continuity of states. This may be demonstrated by following the path EFGH. The vapor at point E, at a tem­ perature below Tc, is warmed at constant volume to point F9 above Tc. It is then Sec. IS The van der Waals equation and liquefaction'of gases 25 compressed along the isotherm FG, and finally cooled at constant volume along GH. At the point H, below Tc, the carbon dioxide exists as a liquid, but at no point along this path are two phases, liquid and vapor, simultaneously present. One must conclude that the transformation from vapor to liquid occurs smoothly and continuously. 18. The van der Waals Equation and Liquefaction of Gases The van der Waals equation provides a reasonably accurate representation of the PVT data of gases under conditions that deviate only moderately from ideality. When we apply the equation to gases in states that depart greatly from ideality, we do not obtain a quantitative representation of the data, but we still get an interesting qualitative picture. A typical example is shown in Fig. 1.10, where the van der Waals isotherms, drawn as dashed lines, are compared with the experi­ mental isotherms for carbon dioxide in the neighborhood of the critical point. The van der Waals equation provides an adequate representation of the isotherms for the homogeneous vapor and even for the homogeneous liquid. As might be expected, the equation cannot represent the discontinuities arising during liquefaction. Instead of the experimental straight line, it exhibits a maximum and a minimum within the two-phase region. We note that as the temperature gradually approaches the critical temperature, the maximum and the minimum gradually approach each other. At the critical point itself, they have merged to become a point of inflection in the PV curve. The analytical condition for a maxi­ mum is that (dP/dV)T = 0 and (d2P/dV2)T 0. At the point of inflection, both the first and the second deriva­ tives vanish, (dP/dV)T = 0 = (d2P/dV2)T. According to van der Waals’ equation, therefore, the following three equations must be satisfied simultaneously at the critical point (T = Tc9 V — Vc, P = P^ for 1 mol of gas, n = 1: p PTc a c Vc - b V2C (^P\ -0- -RTC 2g (Kc - by + K’ (d2P\ _ n - 2RTc 6a \dv2)r (rc - by V* When these equations are solved for the critical constants, we find Values for the van der Waals constants and for R can be calculated from these equations. We prefer, however, to consider R as a universal constant, and to obtain the best fit by adjusting a and b only. Then (1.30) would yield the relation PCVJTC = 3R/8 for all gases. 26 Physicochemical systems Chap. 1 In terms of the reduced variables of state, PR, VR, and TR, one obtains from (1.30), The van der Waals equation then reduces to + = (1.31) A reduced equation of state similar to (1.31) can be obtained from an equation of state containing no more than three arbitrary constants, such as a, b, and R, provided it has an algebraic form capable of giving a point of inflection. Berthelot’s equation is often used in the following form, applicable at pressures of the order of 1 atm: + (1-32) where R = R(TC!PCVC}. 19. Other Equations of State To represent the behavior of gases with greater accuracy, especially at high pres­ sures or near their condensation temperatures, we must use expressions having more than two adjustable parameters. Consider, for example, a virial equation similar to that given by Kammerlingh-Onnes in 1901: _ 1 + B^n Q7X , WP , n 331 nRT “ V + V2 + V3 U 7 Here, B, C, D, etc., which are functions of temperature, are called the second, third, fourth etc., virial coefficients. Figure 1.11 shows the second virial coefficients B of several gases over a range of temperature. This B is an important property in theoretical calculations on imperfect gases. * The virial equation can be extended to as many terms as are needed to represent the experimental PVT data to any desired accuracy. It may also be extended to mixtures of gases, and in such cases gives important data on the effects of inter- molecular forces between the same and different molecules.f One of the best of the empirical equations is that proposed by Beattie and Bridgeman. J It contains five constants in addition to R, and fits the PKT data over a wide range of pressures and temperatures, even near the critical point, to within 0.5%. An equation with eight constants, which is based on a reasonable model of *See, for example, T. L. Hill, Introduction to Statistical Thermodynamics (Reading, Mass.: Addison-Wesley Publishing Co., Inc., 1960), Ch. 15. fAn example is the system methane-tetrafluoromethane, studied by D. R. Douslin, R. H. Harrison, R. T. Moore, J. Phys. Chern., 71, 3477 (1967). This paper illustrates the continuing interest in experimental work on fundamental properties of gases (see Table 1.3). IJ. A. Beattie and O. C. Bridgeman, Proc. Am. Acad. Arts. Sci., 63, 229 (1928); Sec. 20 Mixtures of ideal gases 27 FIG. 1.11 The second virial coefficients B of several gases as functions of temperature. dense fluids, has been described that reproduces the isotherms in the liquid region quite well.* 20. Mixtures of Ideal Gases A mixture can be specified by stating the amount of each component substance that it contains, n2,...,......... The total amount of all components is The composition of the mixture is then conveniently described by stating the mole fraction Xj of each component, Xj = (1.34) n YinJ Another method of specifying composition is the concentration, Cj = (1.35) *M. Benedict, G. W. Webb, L. C. Rubin, J. Chem. Phys., 10, 747 (1942). 28 Physicochemical systems Chap. 1 The SI unit of concentration is the mole per cubic metre, but the mole per cubic * decimetre is more often used. If the system under consideration is a mixture of gases, we can define the partial pressure Pj of any particular component as the pressure which that gas would exert if it occupied the total volume all by itself. If we know the concentra­ tion of a component gas in the mixture, we can find its partial pressure from its PKT data or equation of state. For nonideal gases, we would not expect the sum of these partial pressures to equal the total pressure of the mixture. Even if each gas individually behaves as an ideal gas with P= c.RT (1.36) it is possible that specific interactions between unlike gases would cause 2 Pj to differ from P. Thus, we require a separate definition of an ideal gas mixture as one in which P = P. + P2 + + Pc = S pj (137) This is called Dalton's Law of Partial Pressures. It simply represents a special kind of gas mixture. It is called a law for the historical reason that many gas mixtures at ordinary T and P follow it about as well as individual gases follow the ideal gas law. In the event that each gas individually behaves as an ideal gas, the system is an ideal mixture of ideal gases: P — RT(Ci + c2 + + cc) — RT T: ct p_ RT P y Since we have PJ = XjP (1.38) The partial pressure of each gas in an ideal mixture of ideal gases is equal to its mole fraction times the total pressure. 21. Mixtures of Nonideal Gases The PKT behavior of a mixture of gases at any given constant composition can be determined just as that for a single pure gas. The data can then be fitted to an equation of state. When such data are obtained for mixtures of different composi­ tions, we find that the parameters of the state equations depend on the composition of the mixture. The virial equation is most suitable for representing the PVT properties of gas *The liter (I) is defined as 10“3 m3 or 1 dm3. A solution with a concentration of (e.g.) 1.63 mol-dm-3 is often called a 1.63 molar solution. Sec. 22 The concepts of heat and heat capacity 29 mixtures, since theoretical relations between the coefficients can be obtained from statistical thermodynamics. For instance, the second virial coefficient Bm of a binary gas mixture is Bm = XjB" + 2X.X2Bl2 + X22B22 (1.39)* where Xt and X2 are mole fractions of the two components. The coefficient B12 represents the contribution to the second virial coefficient that is due to specific interactions between unlike gases. Table 1.3 is an example of precise data on second virial coefficients. TABLE 1.3 Second Virial Coefficients for Equimolar Mixtures of CH4 and CF4 Bi(CH4) B2(CF4) B12 K (cm3 - mol-1) (cm3 -mol-1) (cm3 mol-' 273.15 -53.35 -111.00 -62.07 298.15 -42.82 -88.30 -48.48 323.15 -34.23 -70.40 -37.36 348.15 -27.06 -55.70 -28.31 373.15 -21.00 -43.50 -20.43 423.15 -11.40 -24.40 -8.33 473.15 -4.16 -10.10 + 1.02 523.15 + 1.49 + 1.00 8.28 573.15 5.98 9.80 14.10 623.15 9.66 17.05 18.88 22. The Concepts of Heat and Heat Capacity The experimental observations that led to the concept of temperature also led to that of heat, but for a long time students did not clearly distinguish between these two concepts, often using the same name for both, calor or caloric. The beautiful work of Joseph Black on calorimetry, the measurement of heat changes, was published in 1803, four years after his death. In his Lectures on the Elements of Chemistry, he pointed out the distinction between the intensive factor, temperature, and the extensive factor, quantity of heat. Black showed that equi­ librium required an equality of temperature and did not imply that there was an equal “quantity of heat” in different bodies. He then proceeded to investigate the capacity for heat or the amount of heat needed to increase the temperature of different bodies by a given number of degrees. It was formerly a supposition that the quantities of heat required to increase the heat of different bodies by the same number of degrees were directly in proportion to the quantity of matter in each.... But very soon after I began to think on this subject (Anno 1760) I perceived that this opinion was a mistake, and that the quantities of heat which different kinds of matter must receive to reduce them to an equilibrium with one another, or to raise their temperatures by an equal number of *J. E. Lennard-Jones and W. R. Cook, Proc. Roy. Soc. (London), A 115, 334 (1927). 30 Physicochemical systems Chap. 1 degrees, are not in proportion to the quantity of matter of each, but in proportions widely different from this, and for which no general principle or reason can yet be assigned. In explaining his experiments, Black assumed that heat behaved as a s