Main
Physical Chemistry : A Modern Introduction, Second Edition

# Physical Chemistry : A Modern Introduction, Second Edition

*Davis, William M*

PrefaceAcknowledgmentsAuthorGuide for StudentsList of Special ExamplesWorld of Atoms and MoleculesIntroduction to Physical ChemistryTheory and Experiment in Physical ChemistryAtomic and Molecular EnergiesConfigurations, Entropy, and VolumeEnergy, Entropy, and TemperatureDistribution Law DerivationConclusionsPoint of Interest: James Clerk MaxwellExercisesBibliographyIdeal and Real GasesThe Ideal Gas LawsCollisions and PressureNonideal BehaviorThermodynamic State FunctionsEnergy and Thermodynamic RelationsConclusionsPoint of Interest: Intermolecular InteractionsExercisesBibliographyChanges of St. Read more...

Abstract: PrefaceAcknowledgmentsAuthorGuide for StudentsList of Special ExamplesWorld of Atoms and MoleculesIntroduction to Physical ChemistryTheory and Experiment in Physical ChemistryAtomic and Molecular EnergiesConfigurations, Entropy, and VolumeEnergy, Entropy, and TemperatureDistribution Law DerivationConclusionsPoint of Interest: James Clerk MaxwellExercisesBibliographyIdeal and Real GasesThe Ideal Gas LawsCollisions and PressureNonideal BehaviorThermodynamic State FunctionsEnergy and Thermodynamic RelationsConclusionsPoint of Interest: Intermolecular InteractionsExercisesBibliographyChanges of St

Abstract: PrefaceAcknowledgmentsAuthorGuide for StudentsList of Special ExamplesWorld of Atoms and MoleculesIntroduction to Physical ChemistryTheory and Experiment in Physical ChemistryAtomic and Molecular EnergiesConfigurations, Entropy, and VolumeEnergy, Entropy, and TemperatureDistribution Law DerivationConclusionsPoint of Interest: James Clerk MaxwellExercisesBibliographyIdeal and Real GasesThe Ideal Gas LawsCollisions and PressureNonideal BehaviorThermodynamic State FunctionsEnergy and Thermodynamic RelationsConclusionsPoint of Interest: Intermolecular InteractionsExercisesBibliographyChanges of St

Categories:
Chemistry\\Physical Chemistry

Year:
2011

Edition:
2nd ed

Publisher:
CRC Press

Language:
english

Pages:
513

ISBN 10:
1439897891

ISBN 13:
978-1-4398-9789-8

File:
PDF, 46.52 MB

Download (pdf, 46.52 MB)

Preview
- Checking other formats...

The file will be sent to your email address. It may take up to 1-5 minutes before you receive it.

The file will be sent to your Kindle account. It may takes up to 1-5 minutes before you received it.

Please note you need to add our

Please note you need to add our

**NEW**email**km@bookmail.org**to approved e-mail addresses. Read more.## You may be interested in

## Most frequently terms

quantum

^{558}molecules

^{372}molecule

^{337}molecular

^{323}particles

^{310}electron

^{265}physical chemistry

^{264}vibrational

^{263}harmonic

^{259}equilibrium

^{258}particle

^{244}momentum

^{239}hamiltonian

^{239}energies

^{232}operator

^{222}equations

^{214}atoms

^{203}wavefunction

^{202}mechanics

^{192}oscillator

^{191}electrons

^{189}transitions

^{177}matrix

^{173}orbital

^{163}magnetic

^{162}operators

^{162}atom

^{161}angular momentum

^{161}probability

^{153}rotational

^{152}wavefunctions

^{152}coordinates

^{148}quantum mechanical

^{143}quantum number

^{142}symmetry

^{140}vector

^{137}hydrogen

^{133}yields

^{133}harmonic oscillator

^{132}differential

^{130}thermodynamic

^{126}atomic

^{125}orbitals

^{125}appendix

^{123}approximation

^{121}entropy

^{120}dipole

^{118}nuclei

^{117}gases

^{117}coordinate

^{115}axis

^{114}derivative

^{113}dependence

^{111}diatomic

^{109}nuclear

^{109}coupling

^{107}mol

^{104}frequencies

^{102}radiation

^{101}spectroscopy

^{99}SECOND EDITION SECOND EDITION Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110824 International Standard Book Number-13: 978-1-4398-9789-8 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface�������������������������������������������������������������������������������������������������������������������������������������������� vii Acknowledgments�������������������������������������������������������������������������������������������������������������������������xi Authors����������������������������������������������������������������������������������������������������������������������������������������� xiii Guide for Students������������������������������������������������������������������������������������������������������������������������xv List of Special Examples������������������������������������������������������������������������������������������������������������ xvii 1. World of Atoms and Molecules............................................................................................1 1.1 Introduction to Physical Chemistry................................................................................ 1 1.2 Theory and Experiment in Physical Chemistry............................................................ 2 1.3 Atomic and Molecular Energies...................................................................................... 3 1.4 Configurations, Entropy, and Volume............................................................................ 7 1.5 Energy, Entropy, and Temperature................................................................................ 10 1.6 Distribution Law Derivation.......................................................................................... 13 1.7 Conclusions....................................................................................................................... 18 Point of Interest: James Clerk Maxwell............................................................................... 19 Exercises................................................................................................................................... 20 Bibliography.............................................................................................................................22 2. Ideal and Real Gases............................................................................................................. 23 2.1 The Ideal Gas Laws.......................................................................................................... 23 2.2 Collisions and Pressure................................................................................................... 27 2.3 Nonideal Behavior........................................................................................................... 33 2.4 Thermodynamic State Functions................................................................................... 35 2.5 Energy and Thermodynamic Relations........................................................................ 37 2.6 Conclusions....................................................................................................................... 45 Point of Interest: Intermolecular Interactions..................................................................... 46 Exercises................................................................................................................................... 48 Bibliography............................................................................................................................. 50 3. Changes of State.................................................................................................................... 51 3.1 Pressure–Volume Work................................................................................................... 51 3.2 Reversibility, Heat, and Work........................................................................................ 55 3.3 Entropy.............................................................................................................................. 62 3.4 The Laws of Thermodynamics......................................................................................65 3.5 Heat Capacities................................................................................................................ 68 3.6 Joule–Thomson Expansion............................................................................................. 73 3.7 Conclusions....................................................................................................................... 75 Point of Interest: Heat Capacities of Solids......................................................................... 76 Exercises................................................................................................................................... 78 Bibliography.............................................................................................................................80 4. Phases and Multicomponent Systems............................................................................... 81 4.1 Phases and Phase Diagrams........................................................................................... 81 4.2 The Chemical Potential................................................................................................... 86 4.3 Clapeyron Equation......................................................................................................... 89 iii iv Contents 4.4 First- and Second-Order Phase Transitions.................................................................. 93 4.5 Conclusions....................................................................................................................... 95 Point of Interest: Josiah Willard Gibbs................................................................................. 96 Exercises................................................................................................................................... 97 Bibliography............................................................................................................................. 99 5. Activity and Equilibrium of Gases and Solutions....................................................... 101 5.1 Activities and Fugacities of Gases............................................................................... 101 5.2 Activities of Solutions.................................................................................................... 106 5.3 Vapor Pressure Behavior of Solutions......................................................................... 108 5.4 Equilibrium Constants.................................................................................................. 111 5.5 Phase Equilibria Involving Solutions.......................................................................... 114 5.6 Conclusions..................................................................................................................... 118 Point of Interest: Gilbert Newton Lewis............................................................................ 119 Exercises................................................................................................................................. 121 Bibliography........................................................................................................................... 123 6. Chemical Reactions: Kinetics, Dynamics, and Equilibrium...................................... 125 6.1 Reaction of Atoms and Molecules............................................................................... 125 6.2 Collisions and Transport............................................................................................... 129 6.3 Rate Equations................................................................................................................ 135 6.4 Rate Laws for Complex Reactions............................................................................... 138 6.5 Temperature Dependence and Solvent Effects.......................................................... 142 6.6 Reaction Thermodynamics........................................................................................... 144 6.7 Electrochemical Reactions............................................................................................ 151 6.8 Conclusions..................................................................................................................... 157 Point of Interest: Galactic Reaction Chemistry................................................................. 158 Exercises................................................................................................................................. 160 Bibliography........................................................................................................................... 163 7. Vibrational Mechanics of Particle Systems................................................................... 165 7.1 Classical Particle Mechanics and Vibration............................................................... 165 7.2 Vibration in Several Degrees of Freedom................................................................... 170 7.3 Quantum Phenomena and Wave Character.............................................................. 176 7.4 Quantum Mechanical Harmonic Oscillator............................................................... 180 7.5 Harmonic Vibration of Many Particles....................................................................... 185 7.6 Conclusions..................................................................................................................... 187 Point of Interest: Molecular Force Fields........................................................................... 188 Exercises................................................................................................................................. 189 Bibliography........................................................................................................................... 191 8. Molecular Quantum Mechanics....................................................................................... 193 8.1 Quantum Mechanical Operators................................................................................. 193 8.2 Information from Wavefunctions................................................................................ 197 8.3 Multidimensional Problems and Separability........................................................... 203 8.4 Particles with Box and Step Potentials....................................................................... 206 8.5 Rigid Rotator and Angular Momentum..................................................................... 216 8.6 Coupling of Angular Momenta................................................................................... 224 8.7 Variation Theory............................................................................................................ 228 8.8 Perturbation Theory...................................................................................................... 232 Contents v 8.9 Conclusions..................................................................................................................... 238 Point of Interest: The Quantum Revolution...................................................................... 239 The Solvay Conference......................................................................................................... 239 Exercises................................................................................................................................. 241 Bibliography........................................................................................................................... 245 9. Vibrational–Rotational Spectroscopy............................................................................. 247 9.1 Molecular Spectroscopy and Transitions.................................................................... 247 9.2 Vibration and Rotation of a Diatomic Molecule........................................................254 9.3 Vibrational Anharmonicity and Spectra..................................................................... 260 9.4 Rotational Spectroscopy................................................................................................ 272 9.5 Harmonic Picture of Polyatomic Vibrations.............................................................. 276 9.6 Polyatomic Vibrational Spectroscopy......................................................................... 281 9.7 Conclusions..................................................................................................................... 285 Point of Interest: Laser Spectroscopy................................................................................. 286 Exercises................................................................................................................................. 287 Bibliography........................................................................................................................... 289 10. Electronic Structure............................................................................................................ 291 10.1 Hydrogen and One-Electron Atoms......................................................................... 291 10.2 Orbital and Spin Angular Momentum..................................................................... 297 10.3 Atomic Orbitals and Atomic States........................................................................... 301 10.4 Molecules and the Born–Oppenheimer Approximation........................................ 310 10.5 Antisymmetrization of Electronic Wavefunctions.................................................. 313 10.6 Molecular Electronic Structure.................................................................................. 317 10.7 Visible–Ultraviolet Spectra of Molecules.................................................................. 324 10.8 Properties and Electronic Structure........................................................................... 330 10.9 Conclusions................................................................................................................... 336 Point of Interest: John Clarke Slater................................................................................... 337 Exercises................................................................................................................................. 338 Bibliography...........................................................................................................................340 Advanced Texts and Monographs..................................................................................... 341 11. Statistical Mechanics..........................................................................................................343 11.1 Probability.....................................................................................................................343 11.1.1 Classical Behavior.............................................................................................345 11.2 Ensembles and Arrangements...................................................................................346 11.3 Distributions and the Chemical Potential................................................................ 347 11.3.1 High-Temperature Behavior............................................................................ 352 11.3.2 Low-Temperature Behavior............................................................................. 352 11.3.3 Dilute Behavior................................................................................................. 353 11.4 Molecular Partition Functions.................................................................................... 353 11.5 Thermodynamic Functions......................................................................................... 358 11.6 Heat Capacities............................................................................................................. 362 11.7 Conclusions................................................................................................................... 365 Point of Interest: Lars Onsager........................................................................................... 366 Exercises................................................................................................................................. 367 Bibliography........................................................................................................................... 369 vi Contents 12. Magnetic Resonance Spectroscopy.................................................................................. 371 12.1 Nuclear Spin States...................................................................................................... 371 12.2 Nuclear Spin–Spin Coupling..................................................................................... 378 12.3 Electron Spin Resonance Spectra............................................................................... 386 12.4 Extensions of Magnetic Resonance........................................................................... 391 12.5 Conclusions................................................................................................................... 393 Point of Interest: The NMR Revolution............................................................................. 394 Exercises................................................................................................................................. 396 Bibliography........................................................................................................................... 397 13. Introduction to Surface Chemistry.................................................................................. 399 13.1 Interfacial Layer and Surface Tension....................................................................... 399 13.2 Adsorption and Desorption....................................................................................... 402 13.3 Langmuir Theory of Adsorption............................................................................... 407 13.4 Temperature and Pressure Effects on Surfaces........................................................408 13.5 Surface Characterization Techniques........................................................................ 409 13.6 Conclusions................................................................................................................... 411 Point of Interest: Irving Langmuir..................................................................................... 412 Exercises................................................................................................................................. 413 Bibliography........................................................................................................................... 414 Appendix A: Mathematical Background............................................................................... 415 Appendix B: Molecular Symmetry......................................................................................... 437 Appendix C: Special Quantum Mechanical Approaches................................................... 453 Appendix D: Table of Integrals............................................................................................... 465 Appendix E: Table of Atomic Masses and Nuclear Spins.................................................. 469 Appendix F: Fundamental Constants and Conversion of Units....................................... 473 Appendix G: List of Tables....................................................................................................... 479 Appendix H: Points of Interest................................................................................................ 481 Appendix I: Atomic Masses and Percent Natural Abundance of Light Elements........483 Appendix J: Values of Constants............................................................................................. 485 Appendix K: The Greek Alphabet.......................................................................................... 487 Answers to Selected Exercises................................................................................................. 489 Preface This text has been designed and written especially for use in a one-year, two-course sequence in introductory physical chemistry. For semester-based courses, Chapters 1 through 6 can be covered in the first semester and Chapters 7 through 12 in the second. For quarter-based courses, Chapters 1 through 5 can be covered in the first quarter, Chapters 6 through 8 in the second quarter, and Chapters 9 through 12 in the third quarter. Chapter 13 has been written to enhance this edition and can be added to either semester or quarter as time permits. This text may also be used in one-semester surveys of physical chemistry if you select among the sections in each chapter. The text is organized in a way that minimizes extraneous material unnecessary to understand the fundamental concepts while focusing on a strong molecular approach to the subject. As you will see, this text has a novel approach. The ideas, organization, emphasis, and examples herein have evolved from the experience of teaching several different physical chemistry courses at several North American universities. Distinguishing Features of This Text A unifying molecular approach: The foremost goal of this text is to provide a unifying molecular view of the core elements of contemporary physical chemistry. This is done with a topically connected and focused development—in effect, a story line about molecules that leads one through the major areas of modern physical chemistry. At some places, this means a somewhat nontraditional organization of subtopics. The advantage is much improved retention of working knowledge of essential material. After finishing with this text, your students should have a good grasp of the concepts of physical chemistry and should be able to analyze problems and deal with new developments that occur during their careers. Seeing physical chemistry as a continuous story about molecular behavior helps accomplish that. Focus on core concepts: Throughout this text, fundamental issues are stressed and basic examples are selected rather than the myriad of applications often presented in other, more encyclopedic books. Physical chemistry need not appear as a large assortment of different, disconnected, and sometimes intimidating topics. Instead, students should see that physical chemistry provides a coherent framework for chemical knowledge, from the molecular level to the macroscopic level. That this text offers a streamlined introduction to the subject is apparent in the presentation of thermodynamics at the start of the text. As gas laws are first considered, a thorough yet concise development of real gases and equations of state is given. State functions are introduced in a global fashion. This organization offers students the strongest sophistication in the least amount of time. It prepares them for tackling more challenging topics. vii viii Preface Novel organization to foster student understanding: The first three chapters provide the foundation material for thermodynamics, always tying it to a molecular point of view. The approach in these chapters is to understand the behavior of thermodynamic systems and to express that in mathematical terms. This means, for instance, that gas kinetics is used as needed to understand pressure, reaction rates, and so on, rather than being collected as an isolated segment. A more usual organization considers the first law, then the second law, and then the third law of thermodynamics; however, that structure does not always bring out what thermodynamics is meant to explain, and it is not always an effective organization for remembering the material. It is the understanding of molecular behavior, such as that leading to chemical reactions, that is the focus of the development here. Most instructors will not find this organization much of a departure from a traditional approach, and yet the differences that do exist should benefit students. To streamline the presentation of quantum mechanics, notions that are more of historical interest than pedagogical value are removed. For example, the Bohr atom, important as it was in the development of quantum theory, was not correct. The photoelectric effect was part of the quantum story, but a detailed discussion is not essential to introducing the material. Also, a primary example, the one-dimensional oscillator, is introduced at the outset in order to have it serve as a continuing example as we build sophistication. The usual first problem, the particle in a box, is set aside for later because it simply is not as applicable as a model of chemical systems as the harmonic oscillator. This is another way to connect the new concepts to molecular behavior. It is easier to understand that molecules vibrate than to contemplate a potential becoming infinite at some point. Point of interest essays: Each chapter ends with a short Point of Interest. These essays discuss a selected set of the historical aspects of physical chemistry (to give students an appreciation of certain of the people whose clever and creative thinking have moved this discipline forward) as well as insights into modern applications and a few of the current areas of active research. These essays are set off from the technical story line; they are the roadside stops with interesting glimpses of individuals, revolutionary developments, and a few special areas for future study. Strong problem-solving emphasis: Working on exercises is a key to mastering all topics in physical chemistry. The end of each chapter provides numerous practice Exercises (mostly the “plug-and-chug” variety) as well as numerous Additional Exercises (which are more challenging and test your students’ understanding of concepts and ability to apply the material covered in the chapter). In addition, over two dozen special worked examples are interspersed with the topical development in the text. These special examples are boxed and usually on individual pages. They augment the in-line examples in the discussion but do not interrupt the flow of the material. They offer detail at the level of a chalkboard work-up of an exercise in a classroom. Finally, a number of exercises are included that are best handled using a spreadsheet of your choosing. These exercises are clearly identified. Mathematics review for students who have forgotten their calculus: Appendix A provides some quick review of (or even initial training in) the mathematics needed to follow the material in the text. In writing this book, it is assumed that students have completed two or three semesters of calculus and that they can differentiate simple functions easily and can form differentials. The material in Appendix A is meant to supplement mathematical preparation since, in our experience, that is often the biggest difficulty for students beginning their study of physical chemistry. It is strongly recommended that students review Appendix A as they begin to use this text. Preface ix Powerful streamlined development of group theory and advanced topics in quantum mechanics: Appendix B (Molecular Symmetry) and Appendix C (Special Quantum Mechanical Approaches) cover topics that many physical chemistry courses include and that could each in fact be their own chapter. However, they are not essential to the flow of the remaining material, and so these appear at the end for inclusion in a course at the instructor’s discretion. Acknowledgments WMD first and foremost thanks Clifford E. Dykstra for allowing him the privilege of revising an already excellent text into a second edition. Special thanks go to Barbara Glunn and Pat Roberson at Taylor & Francis for their help in bringing this textbook to fruition. My teaching style and ultimate interest in teaching physical chemistry is first and foremost influenced by my quantum chemistry professor at the University of Western Ontario, Dr. William J. Meath. I was astounded at how he could make such a complicated mathematical subject so fascinating. I can only strive to match his level of teaching ability. I would be remiss if I did not mention my graduate advisor at the University of Guelph, Dr. John D. Goddard. His patient guidance during my graduate days was always much appreciated. I must also thank my postdoc advisor at York University, Dr. Huw O. Pritchard, for allowing me the opportunity to teach my first lecture class and help me realize my passion for undergraduate teaching. Finally, I wish to thank my students, both at the University of Texas at Brownsville and my new home at Texas Lutheran University (TLU). Their comments, questions, and complaints have always helped me to refine my lectures and examples. A very special thanks goes out to the Fall 2009/Spring 2010 class of Chemistry 344/345 at TLU who used a draft of this book and helped to find errors and omissions in the text. You are all a continuing inspiration to teach. Comments from readers are most heartily encouraged and can be sent to me at wdavis@ tlu.edu. William M. Davis xi Authors William M. Davis received his BSc (honors) in chemistry from the University of Western Ontario, London, Ontario, Canada, and his MSc and PhD from the University of Guelph, Guelph, Ontario, Canada. He taught lecture and laboratory sections of general, physical, and inorganic chemistry at several Canadian universities before moving to Texas to take up a tenure-track position at The University of Texas at Brownsville, Texas, where he taught general, physical, inorganic, analytical, organic, and environmental chemistry for 10 years. In 2008, he moved to Texas Lutheran University, where he is currently associate professor and chair of chemistry and holds the George Kieffer Fellowship in Science. Dr. Davis’s research interests include application of computational and analytical chemistry techniques to systems of environmental and biochemical interest. Clifford E. Dykstra received a BS in chemistry and a BS in physics from the University of Illinois at Urbana–Champaign in 1973, and he received his PhD from the University of California, Berkeley in 1976. He joined the faculty at the University of Illinois at Urbana– Champaign in 1977. His research has focused on computational electronic structure theory with particular attention to molecular properties and weak intermolecular interaction. In 1990, he moved to Indiana University–Purdue University Indianapolis where he served as Associate Dean of Science (1992–1996) and was named Chancellor’s Professor (2001). He served as chair of the Department of Chemistry at Illinois State University from 2006 to 2009, and then returned to the University of Illinois at Urbana–Champaign. He serves as editor of the journal Computational and Theoretical Chemistry. xiii Guide for Students Physical chemistry is a required course in most undergraduate chemistry curricula and in most chemical engineering curricula. Many students in other areas, such as biochemistry or materials science, find it useful preparation. Students take physical chemistry with different aims, different objectives, and different backgrounds. In all cases, there are some ways to optimize the learning process. Here are some ideas for those using this text for undergraduate physical chemistry. Math background and proficiency are immensely important. Algebra, analytical geometry, and calculus are unavoidable in this subject. Before beginning your course, read the first four sections of Appendix A. This should provide a thorough review of the mathematics needed with the material in this text. If any of it seems to be more than a review, it may be worthwhile to study a mathematics text in that area. Next, familiarize yourself with the information available in the text. Each appendix is a source of information on which you may wish to draw throughout your course. In particular, look at Appendix D on units, since unit conversion can sometimes obscure the scientific concepts. As you get started going through the chapters of the text, try to anticipate upcoming subjects in your lecture sessions. Read chapter material on those subjects shortly before the lecture. In other words, read ahead. On a first reading, it is not necessary to strive for 100% comprehension, and you may choose to ignore a lot of the mathematical detail. Simply try to get the direction of the presentation and a general feel for the subtopics. Then, follow your instructor’s lecture closely, especially to see the areas that your instructor has selected to emphasize. After a lecture presentation on some topic or subtopic, do a thorough reading of the corresponding text sections. Attempt to follow all mathematical steps and go through the special boxed examples. At this stage, follow closely the examples where numerical values are used to obtain numerical results (i.e., make sure you can “plug-and-chug” the key formulas right away). The next recommended step in the learning process is one of the most important ones in physical chemistry—working exercises. A set of exercises are given at the end of each chapter. The first set includes the more straightforward problems, typically those that require applying a particular formula to achieve a numerical answer. These problems help solidify understanding of newly introduced quantities and functions. Answers for the majority of these problems can be found at the end of the text, and you can check your work right away. Realize that it is less advisable to simply look at and mull over a problem so as to convince yourself “I can do it” than it is to carry out the work in full. Actually working an exercise rather than merely “thinking a problem through” consistently builds better understanding and strengthens retention—retention you may value during that exam 3 weeks from now. Approach additional exercises in each chapter to develop a solid, working knowledge of the material. These problems may involve derivations, complicated calculations, analysis of new problems, and challenges that will call for a mastery of the subject material. As you work exercises, refer back to text material. Reread paragraphs if something is unclear. Consult other books, such as those listed in the bibliography at the end of each chapter. The concepts in physical chemistry may seem formidable. The first presentation xv xvi Guide for Students you see on some particular topic may not do the job, even if it should happen to be the best presentation around. Different individuals will generate different questions about any given topic. Checking alternate sources might offer a different perspective, a different approach, or a different derivation, and that may be what it takes for you to achieve understanding. Want to enhance your sophistication, understanding, and physical chemistry abilities even further? Then work with other students. Try to devise your own exercises with your own solutions, and then try to explain them to others. Nothing challenges the solidness of your knowledge as much as trying to explain it to others. Prepare for exams by quick rereads of chapters. Go over new terms (in bold when first used or defined) and the conclusions to each chapter. Rework problems you have already done. Learning and studying in a subject that is highly mathematical is different than in a subject that is wholly conceptual and qualitative. The sequence of seeing material, hearing it, looking at it in detail, and working with it through exercises seems to be the best learning process available. Obviously, this is not a passive learning approach. Few can grasp the richness and complexity of physical chemistry by simply listening to lectures. The guts of the material is not a stream of facts and some definitions. It is a physical-mathematical description of the world around, a description with many connected concepts, theories, formulas, and abstractions. You have to work with it to understand it all. What if you cannot do everything advised here? The best use of a limited amount of time is to focus on reading the chapters and studying the special examples. With any remaining time, work exercises, work exercises, and work exercises. Good luck! Physical chemistry tends to grab the interests of a good fraction of students who are required to take it as a course. The reason seems to be that physical chemistry is fundamental to chemistry. Nature has not made overly simple the structure of physical chemistry, and thus, there will be difficulties and frustrations for many in studying it. Many overcome initial frustrations and find clear sailing thereafter. From the effort to study the physics of molecules—physical chemistry—should come a sense of wonder, perhaps fascination, that mankind has obtained such incredible insight into the workings of things (atoms and molecules) that no human has ever directly seen. List of Special Examples 1.1 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 4.1 5.1 6.1 6.2 6.3 6.4 6.5 8.1 8.2 8.3 8.4 9.1 10.1 11.1 11.2 12.1 Energy Level Populations Most Probable Speed of Gas Particles Isothermal Compressibility Thermodynamic Relations Thermodynamic Compass Work in a Stepwise Gas Expansion ΔU for an Adiabatic Expansion ΔS for an Adiabatic Expansion ΔS of an Engine Cycle The Solid–Liquid Phase Boundary Fugacity and Activity of a Real Gas Integrated Rate Expression for an A + B Reaction Temperature Dependence of a Reaction Enthalpy ΔS and ΔG of a Reaction ΔG of an Electrochemical Cell Effect of Temperature on Cell Voltage Position Uncertainty for the Harmonic Oscillator Degenerate Energy Levels Particle in a Three-Dimensional Box Variational Treatment of a Quartic Oscillator Diatomic Molecule Vibrational Spectrum Diatomic Molecule Electronic Absorption Bands Products of Partition Functions of Independent Systems Internal Energy of an Ideal Diatomic Gas NMR Energy Levels of Methane xvii 1 World of Atoms and Molecules Physical chemistry is the study of the physical basis of phenomena related to the chemical composition and structure of substances. It has been pursued from two levels: the macroscopic and the molecular. The theories and laws of physical chemistry provide a rich, comprehensive view of the world of atoms and molecules that connects their nature with the macroscopic properties and phenomena of materials and substances. A starting point for an introduction to physical chemistry is the concept of energy levels in atoms and molecules, distributions among these energy levels, and something of familiar use in everyday life, temperature. 1.1 Introduction to Physical Chemistry Physical chemistry, or chemical physics, is an area of molecular science with boundaries that are still being enlarged. In many ways, it is at the core of chemical science because it is concerned, in part, with achieving the most detailed, quantitative view of molecules and of chemical phenomena. This means it covers the structure of molecules, starting from a description of electrons and nuclei and the nature of chemical bonds. It covers dynamics (the changes in a molecular system with time), and this includes chemical reactions. It also covers properties of assemblies of atoms and molecules. Beyond that, the subject deals with the properties and phenomena of gases, liquids, and solids. Surely, this is a subject with applications in every area of molecular science, and to study physical chemistry is to pursue a very fundamental understanding of chemistry. Because it is developed from basic physical laws, physical chemistry deals with most issues quantitatively and mathematically. Even the qualitative notions that emerge usually rely on mathematical arguments. Often the theories used in physical chemistry are presented most concisely as mathematical expressions, making mathematical sophistication advantageous. The mathematical basis of physical chemistry allows the derived theories and laws to be powerfully predictive tools in science. The modern atomic theory of matter is almost two centuries old. It was in the early nineteenth century that Dalton’s work (John Dalton, England, 1766–1844) advanced the proposal that matter is not continuously divisible and that there is some fundamental type of particle, the atom. The line of thought that began with the atomic theory of matter took its next major step in the early twentieth century when experiments pointed to the existence of subatomic particles. In a few more decades, it became clear that there are even smaller particles. Even today, the search for exotic subatomic particles continues. As matter is viewed using more and more powerful techniques, we can see that all matter is composed of discrete building blocks (particles) rather than continuous materials. In 1905, Einstein’s (Albert Einstein, 1879–1955) special theory of relativity connected the property of mass with energy with his now infamous equation E = mc2. This mass–energy 1 2 Physical Chemistry: A Modern Introduction connection makes it less surprising that scientists in the early twentieth century found that many strange observations could be explained if energy came in discrete packages or quanta. In other words, it is not only matter but also energy that comes in discrete building blocks in the tiny world of atoms and molecules. The problems that led to the hypothesis of quantization of energy involved the spectrum of atomic hydrogen, the photoelectric effect, the temperature dependence of the heat capacities of solids, and others. One after another, unexpected phenomena were explained by a quantum hypothesis, and this hypothesis eventually grew into what we now refer to as quantum mechanics. After one becomes familiar with quantum mechanics and its chemical implications, it is fascinating to look back at the early developments—they mark the start of a major scientific revolution. We know today that the constituents of atoms and molecules are electrons, neutrons, and protons. These constituents are particles, very small entities that have mass. They are so small and so light that they are beyond the limits of our own senses and experience; we cannot hold a single atom in our hands and look at it. Likewise, the mechanics of systems of such particles are outside everyday experience. Even so, there is a correspondence between our macroscopic world and the subatomic world, and we will analyze systems of particles in both worlds. The picture in the macroscopic world is generally referred to as a classical picture. It has been established by human perception and observation. The picture of small, light particles is termed a quantum picture because quantization of energy—the partitioning of energy into discrete blocks—is the distinction between this world and the macroscopic world. Because energy quanta tend to have such tiny amounts of energy, a macroscopic system involving numerous quanta appears to behave as if energy is continuous. Nonetheless, there are manifestations of quantum features that are detectable by macroscopic instruments and are quite recognizable. Many properties and qualities of substances, such as the temperature dependence of the pressure of gases, were well understood before the development of quantum theory. With the detailed molecular view obtained with quantum mechanical analysis, an even more fundamental basis for macroscopic chemical phenomena is at hand. 1.2 Theory and Experiment in Physical Chemistry The pursuit of understanding in most branches of science is a process of observation and analysis. In physical chemistry, laboratory experiments are the means for observation, which is to say that experiment is the means for probing and measuring. The analysis of the data may be carried out for different reasons. For one, it might use the data with some generally accepted theory so as to deduce some useful quantity that is not directly measurable. We will find, for instance, that bond lengths are not measured the way an object’s length is measured in our macroscopic world; instead they are often determined on the basis of measurements of energy changes in molecules. In this way, the established physical understanding provides the means for utilizing experimental information in examining molecular systems. A possible reason for carrying out an experiment and analyzing the data is to test one notion, concept, model, hypothesis, or theory, or else possibly to select from among several competing theories. If the data do not conform to what is anticipated by some particular theory, then its validity is challenged. In such a circumstance, one may devise a new notion, concept, or theory to better fit the data, or possibly reject one concept in favor of another. Whatever new idea emerges is then tested in still further experiments. World of Atoms and Molecules 3 Many problems in physical chemistry are analyzed with approximations or idealizations that make the mathematics of the analysis less complicated or that offer a more discernible physical picture. Experimental data and analysis offer a validation or a rejection of the approximation. There is a vital interplay between observation and understanding, or between experiment and theory. Of course, this is true throughout science, but in physical chemistry the interplay very much affects the way developments are viewed. The goal is always a physical understanding of how systems behave, and that means understanding molecules and their reactions in terms of physical laws, especially those laws familiar to us through our everyday experience. Ultimately, the knowledge is embodied in theories that at some point have been well tested by experiments. In many respects, a textbook presentation of physical chemistry is a presentation of theories, and yet, experiments are very much the basis for the story. We cannot properly explain our best physical picture of chemical and molecular behavior without knowing the means of observation (experiment). Therefore, the direction for this text is to present understanding (theories) integrated with the means for observation and measurement, though usually without detailed discussion of experimental techniques. An ideal introduction to the subject combines this grounding in theory with hands-on laboratory experience. 1.3 Atomic and Molecular Energies Our everyday experience tells us that energy can be stored continuously in mechanical systems. A child moving back and forth on a swing is a system with mechanical energy that can be set continuously. We can give a small push or a big push or anything in between. A baseball can be thrown at any desired velocity, subject only to the thrower’s ability, and thus, it can have any amount of kinetic energy. Any moving particle in our everyday world can be given as little or as great a kinetic energy desired without restriction. In the world of very small particles, such as atoms and molecules, the situation is different. Systems that are bound (connected together over time) store energy continuously but in a stepwise manner. Their energies are said to be quantized. Quantum mechanics, a subject for Chapters 8 through 10, deals with the mechanical behavior of systems whose size makes quantization of their mechanical energy a significant feature. Quantum mechanics may be regarded as, and may be shown to be, a more complete mechanical picture of our universe than the classical mechanics (e.g., Newton’s laws) that we use to analyze systems in our everyday world. That is, classical mechanics may be developed as a specialized type of mechanics corresponding to a heavy-particle limit of a quantum mechanical description. As well, classical mechanics may be regarded as an approximation to quantum mechanics, an approximation that is highly accurate for massive particles and for macroscopic systems. Both types of mechanics turn out to be useful in different ways in the understanding of atomic and molecular behavior. For now, it is not the full mechanical description that is of interest but rather one special concept, energy storage. Energy quantization means that a system can store energy only in certain fixed amounts. A harmonic oscillator is a standard, useful example. The harmonic oscillator system consists of a mass attached to a spring whose opposite end is connected to an infinitely heavy wall. Imagine a tennis ball attached to a lightweight spring hanging from a ceiling, and 4 Physical Chemistry: A Modern Introduction you have some idea of a harmonic oscillator system. Also imagine an atom attached by a chemical bond to a metal surface, and you are thinking about an analogous system in the small world of atoms and molecules. Pull the mass (tennis ball) away from its equilibrium position, its rest position, and release it. The system oscillates, which is to say that the mass moves back and forth. Such an oscillator system is harmonic if the spring has certain ideal properties, which are not yet of concern. As we pull the tennis ball and stretch the spring, we are adding energy (potential energy) to the mechanical system. We can stop stretching as desired, which means we can add any amount of energy. That does not hold for the atom attached to the metal surface. That system can accept (store) only certain specific increments of energy. Quantum mechanics dictates that we cannot add energy continuously to the tennis ball– spring system. This is not in conflict with our observations, however. For the tennis ball system, the energy is quantized into such small pieces that we cannot distinguish energy storage via numerous small pieces from continuous energy storage. Classical mechanics serves very well in describing the tennis ball system, though not the atomic system. This can be appreciated by applying without derivation one result of quantum mechanical analysis. It is that energy may be stored in a harmonic oscillator system in quanta equal in size to the fundamental constant known as Planck’s constant, h, times the frequency of oscillation, ν Equanta = hν (1.1) Planck’s constant is a very small fundamental constant with a value of 6.626069 × 10−34 J s. Thus, in our macroscopic, everyday world, the energy quanta are very tiny. The tennis ball suspended by a spring might have a frequency of around 1 s−1 (1 Hz). This would make the energy quanta on the order of 10−33 J relative to a total mechanical energy approaching 1 J given an initial displacement of several hundredths of a meter. The quanta are so small in relation to the behavior we can perceive that it is as if the stored energy varies continuously. In contrast, an atom vibrating against a metal surface to which it is bonded may have a frequency on the order of 1013 s−1. Then, the amount of energy that can be added or removed from the vibrational motion would be on the order of 10−20 J. This is much bigger than the quanta for the tennis ball system but still a very small amount of energy in our everyday world. It is, however, a large amount in the world of atoms and molecules. For instance, if the atom had a bond dissociation energy of 300 kJ mol−1, which is a representative value for chemical bonds, then on dividing this by Avogadro’s number (NA = 6.022142 × 1023), we would find that the required energy to break one bond is on the order of 10−18 J. This means that the vibrational quanta of energy of the atom system are sizable relative to chemical energies of bond breaking, the quanta being about 1% of the bond energy in this hypothetical case. Quantum mechanical analysis may be used for mechanical systems with numerous particles, including atomic and molecular systems. The analysis usually shows that there are many ways a system can exist, and associated with each way is a certain amount of stored energy. The distinct ways in which a quantum mechanical system can exist are referred to as quantum states. There is always a lowest energy state referred to as the ground state of the system. The ground state is not necessarily a state with zero energy; it simply corresponds to the lowest possible allowed energy for the system. Quantum states other than the ground state of a system are called excited states. There may be an infinite number of excited states. Quantum numbers are values used to 5 World of Atoms and Molecules distinguish or label the different states. Mostly, though not always, these are whole numbers, and they often arise in the course of the mathematics (differential equation solving) that goes along with a quantum mechanical analysis. It is possible for two or more states to have the same energy, in which case the states are said to be degenerate. The different energies that are possible are referred to as the energy levels of the system. Just as there can be an infinite number of states, there can be an infinite number of energy levels. It turns out that the energy levels of a number of model systems can be expressed in simple formulas involving the quantum numbers of the system. For example, the energies of a harmonic oscillator depend on the vibrational frequency, ν, and the quantum number, n. En = n + 1 hν 2 (1.2) To use this expression, we must know or must obtain the vibrational frequency of the specific system at hand, and we must know another result from the quantum mechanical analysis: The quantum number n can take on values of 0, 1, 2, 3, and so on to infinity. Notice that the lowest allowed energy is hν/2; this is the energy of the ground state. Thus, the ground state for a simple oscillator corresponds to the quantum number n being 0. The next lowest allowed energy for this system is 3hν/2, and this corresponds to the quantum number choice of n = 1. This is an energy step of hν up from the ground state energy. Likewise, the next step to the n = 2 level requires another hν in energy, and therefore, the size of the quanta that the oscillator may store is hν, as in Equation 1.1. Another common system in quantum mechanics is the so-called rigid rotator. The energy levels of a rotating linear molecule are given to good approximation by the following expression. EJ = BJ ( J + 1) (1.3) J is the quantum number associated with rotation, and the values it can take are the positive integers and zero. B is called the rotational constant and is specific to each molecule. It depends on the moment of inertia of the molecule and thereby on the bond lengths and atomic masses. Notice that this expression shows a different dependence on the quantum number than the dependence in Equation 1.2. The energy associated with molecular rotation increases quadratically with the quantum number J. Figure 1.1 illustrates the increasing energetic separations among rotational energy levels versus the uniform separation for a harmonic oscillator. Quantum mechanical analysis reveals that a rotating linear molecule can exist in several states with the same energy; another quantum number, M, distinguishes among the rotational states of the same energy. M is associated with the orientation of the angular momentum vector and does not affect the energy since the energy of a freely rotating body does not depend on orientation of the angular momentum vector. The number of states that may have the same energy is related to the quantum number J; it is simply the value 2J + 1. Thus, if J = 1, then there are three degenerate states for which the energy of the system is B(1)(2) = 2B. The total number of states of a given energy level is the degeneracy of the level. A diatomic molecule both vibrates and rotates. Strictly speaking, the motions are coupled, but to a good approximation the energies of the diatomic molecule are simply the 6 Physical Chemistry: A Modern Introduction cm–1 n=4 cm–1 cm–1 3,000 n=3 150 –20,000 n=3 n=2 J=3 2,000 n=2 100 –60,000 n=1 1,000 J=2 50 n=0 0 J=1 0 J=0 –100,000 0 n=1 FIGURE 1.1 The energy levels of an idealized harmonic oscillator (left), a rotating diatomic molecule (center), and the hydrogen atom (right). Each horizontal line is drawn according to the energy scale given by the vertical lines and following Equations 1.2 through 1.4, respectively. Notice that the oscillator levels are evenly spaced, whereas the rotational levels become increasingly separated with increasing energy. Also, notice the different energy scales. Typically, rotational energy levels are more closely spaced than vibrational energy levels, and vibrational energy levels are more closely spaced than the levels that develop from the electrons’ motions in an atom or molecule. sum of the rotational contributions following Equation 1.3 and the vibrational contributions following Equation 1.2. Thus, the overall energy levels depend on two quantum numbers, n and J. A final set of energy levels to consider are those of the hydrogen atom. The hydrogen atom is a two-particle system consisting of an electron and a much heavier particle, a proton. The electron “orbits” the proton, and the energies associated with that depend on one quantum number, n, in the following way. En = − R n2 (1.4) R is the Rydberg constant (1.09737 × 107 m−1). The hydrogen atom’s quantum number n can take on the values of all the positive integers; it obviously cannot be zero. The degeneracy of each energy level is n2. If n = 2, the energy is −R/4, which is one-quarter of the energy of the ground state, and the degeneracy of the level is 4. As seen in Figure 1.1, the energy levels of the hydrogen atom get closer together as the quantum number increases, and this behavior is different from that of both the harmonic oscillator energy levels and the rotational energy levels. Translation of a quantum mechanical particle in a container corresponds to a bound state with the energy quantized. Translation of a particle, such as an atom or molecule, in an unrestricted space (no walls and no potentials) is an unbound motion. The mechanics of such a free particle have certain similar elements in both the quantum and classical pictures. In both, the energy may vary continuously. In chemistry then, energy storage depends on the nature of the mechanical system and on whether or not there are bound states. We will revisit all of the quantum mechanical systems discussed earlier later in the book. 7 World of Atoms and Molecules 1.4 Configurations, Entropy, and Volume Depositing energy in a single atom or molecule depends on the existing energy level structure of the atom or molecule. But then, what happens in a very large collection of atoms and molecules? The answer to how energy is stored is an important property of a macroscopic substance, and it is one that reflects somewhat the behavior at the molecular level. To understand this, we shall consider statistical arguments in order to deal with the large numbers of particles found in a macroscopic system. This subfield is called statistical mechanics, a subject that will be covered in more detail in Chapter 11. The first step is to consider configurations or arrangements within systems. Ultimately, we will be concerned with arrangements of the particles in their individual energy levels, but we can start with a non-quantum mechanical example to illustrate the analysis. Consider three pairs of dice arranged on a table, all with a 6 showing. There is no energetic preference for this arrangement over any other, such as the arrangement with two 6s, a 5, a 4, and two 2s. The analogy with a molecular system is that the arrangements of dice are like the different degenerate quantum states. If the dice were collected from the table and dropped, we would be surprised if each had a 6 showing. Rather we would expect some “random” arrangement. If we picked up the dice a second time and threw them again, it would be just as much a surprise to see six 6s. Our experience tells us to expect some mix of 6s, 5s, 4s, 3s, 2s, and 1s on any throw. Thus, we find that there is a tendency toward certain arrangements even without any energy difference. Six 6s showing is very unlikely, but three 2s, a 4, a 5, and a 6, for instance, does not seem to be as rare. If we had first started with a throw yielding six 6s, subsequent throws would likely have taken us away from that arrangement and not returned us to it. The reasons for the behavior of the dice are not energetic but lie in statistics. Let us use the term state, but not exactly in the quantum mechanical sense of the last section, to refer to any specific arrangement of the dice. A state of the dice may be represented by a list of six integers, n1–n6, which tells how many dice have 1–6, respectively, showing. Thus, the initial state in our example is (n1, n2, n 3, n4, n 5, n6) = (0, 0, 0, 0, 0, 6). It is different from the state (1, 0, 0, 0, 0, 5) which would be five 6s and a 1. For the state (1, 0, 0, 0, 0, 5), the 1 could be showing on any of the six dice, and so there are six different ways for this state to exist. In contrast, there is only one way for the state (0, 0, 0, 0, 0, 6) to exist. We shall refer to these ways of existing as configurations or as microstates. For each possible state there are a specific number of configurations. Here is a tabulation of a few of them. State (6, 0, 0, 0, 0, 0) (0, 6, 0, 0, 0, 0) (1, 0, 0, 0, 0, 5) (5, 1, 0, 0, 0, 0) (4, 2, 0, 0, 0, 0) (4, 1, 1, 0, 0, 0) (1, 1, 1, 1, 1, 1) Number of Configurations 1 1 6 6 15 30 720 Meaning Six 1s showing occurs once Six 2s showing occurs once A 1 and five 6s occurs 6 times Five 1s and a 2 occurs 6 times Four 1s and two 2s occurs 15 times Four 1s, a 2, and a 3 occurs 30 times One of each occurs 720 times 8 Physical Chemistry: A Modern Introduction There is a simple formula that gives the number of configurations for all possible states. If N is the number of dice and C is the number of configurations, then C( N , n1 , n2 ,....) = N! n1! n2!... (1.5) C has been written as a function of N and of the integers that specify the state. We expect that each individual configuration is equally likely in any throw of the dice. But this means that the state (1, 1, 1, 1, 1, 1) is 720 times as likely to turn up as the state (0, 0, 0, 0, 0, 6). The dice throwing example is an illustration from common experience that systems tend to change from states with low C values to states with large values. They change from states with little probability to states with high probability. Ludwig Boltzmann (Austria, 1844–1906) associated this with the natural occurrence of spontaneous changes, arguing that all spontaneous processes go in a direction leading to a larger number of configurations. This is an important concept in chemistry because it associates a probability with a change. The melting of a cube of ice immersed in liquid water at room temperature and the lowering of the temperature of the liquid, then, comprise an occurrence with an overwhelmingly greater probability than the reverse spontaneous process of freezing some of the liquid water into a cube and raising the remaining liquid to room temperature. It is not that the latter cannot occur, but that it has a vanishingly small probability of occurring. A collection of molecules exists in any one configuration for only an instant as collisions and interactions among the molecules lead to changes that are the equivalent of continuously throwing the dice in our example. Thus, a system fluctuates in time, changing from one instantaneous configuration to another. However, for a large collection of molecules, one configuration tends to dominate. For a set of N dice, this is the configuration for which C in Equation 1.5 is a maximum. Entropy is a thermodynamic quantity that is related to this tendency of a system to be in the state of maximum probability. In the case of our hypothetical system of many dice, it must be a function of the C values, for they indicate the probability. For two noninteracting or separate systems, the overall tendency to be in the state of maximum probability (the total entropy) must be the same as the sum of their individual entropies. This requirement dictates the relationship between entropy, S, and the number of configurations, C, of a given state. First, though, we must realize that the number of configurations for combining two systems is not additive but is multiplicative. If system A has 3 possible configurations and system B has 4, then the number of configurations for the composite system AB is 12. This is because for every choice for the configuration of part A of the composite system (i.e., 1, 2, or 3), there are 4 choices for part B; in total, this gives 12 for the composite system. We may state the relationships between the composite and individual systems in the following way. CAB = CACB SAB = SA + SB Now, for the entropy to be a function of the configurations, that is, for S = f(C), the following relations must hold. SA = f (CA ) SB = f (CB ) SAB = f (CAB ) 9 World of Atoms and Molecules These relationships are satisfied if and only if S is proportional to the logarithm of C. Notice the effect of taking the logarithm of the expression for CAB. ln C AB = ln(C ACB ) = ln C A + ln CB With S taken to be proportional to ln C, the entropy for the composite system, SAB, is an additive result of contributions from A and B, as required. Absolute entropy is related to the number of configurations of a system, with the specific constant of proportionality being Boltzmann’s constant, k. S = k ln C (1.6) k is a fundamental physical constant with a value of 1.3806504 × 10−23 J K−1. Since ln C is dimensionless, then entropy has units of energy per unit of temperature. We shall examine why S has such units shortly, but we can appreciate that the tendency of a system to be in a state of maximum probability will be related to energy quantities and to temperature. It is perhaps worth noting at this point that we have not defined entropy on the basis of “randomness” or “disorder” as is commonly done. In most chemical systems, the state of maximum probability is directly related to the state of maximum “disorder.” However, entropy, in the strictest sense, should always be associated with probability. The number of configurations, C, of a given state represents a probability of being in that state. The bigger the value of C for the three pairs of dice, the greater the probability that a throw of the dice will leave them in that state. In some cases, the states of a system may be somewhat abstract or unknown, and yet statistical arguments can still be quite revealing and useful. For instance, consider a single molecule that is free to move in a box with dimensions l × l × l. If the width, height, and length of the box were suddenly doubled, would the entropy change? We expect that it would increase because the space available has increased and somehow the number of configurations should increase, even if we do not know exactly what the configurations are. To analyze this situation, we can consider the original box to have been divided into small boxes, each l/N × l/N × l/N, where N is an integer. A configuration is taken to consist of the particle being found in one of the boxes. For instance, if the original box were 10 m on a side, a choice of N = 10 would yield boxes 1 m on a side. There would be 1000 such small boxes within the original box’s volume. In the expanded box, where the sides have been doubled, there would have to be (2l/N)3 small boxes of the same size. Clearly, the ratio of the number of small boxes varies as the ratio of the volumes. (2l/N )3 (2l)3 Vexpanded = 3 = (l/N )3 l Voriginal If we consider the number of small boxes to be proportional to the number of configurations, then the change in absolute entropy from expanding the box must be related to the volume since V is directly proportional to the number of configurations C, that is, V ∝ C. ∆Sexpansion = Sexpanded − Soriginal = k ln(Cexpanded ) − k ln(Coriginal ) = k ln Cexpanded Coriginal = k ln Vexpanded Voriginal (1.7) 10 Physical Chemistry: A Modern Introduction This result has no dependence on N, the number of divisions we made along the sides of the box, and it is a general result. If there are n moles of the molecules in the volume instead of just one molecule, then there will be nNA times this entropy change, with NA being Avogadro’s number (i.e., the number of molecules in a mole). ∆Sexpansion = nN A k ln = nR ln Vexpanded Voriginal Vexpanded Voriginal (1.8) This is a thermodynamic relation for ideal gases composed of non-interacting particles. R, the constant introduced to replace the product of Boltzmann’s constant and Avogadro’s number, is called the gas constant. Its value is 8.314472 J mol−1 K−1. For a sample of gas particles free to move in some volume, the tendency to be in a state of maximum probability is related to the volume of the sample. 1.5 Energy, Entropy, and Temperature Entropy is related to the probability that a collection of atoms or molecules exists in a given state of a system. This relationship is used to understand the distribution of molecules among accessible molecular quantum states. That distribution is important because it ultimately serves to define temperature fundamentally. Of course, we all have a phenomenological understanding of temperature. As temperature increases around us, the mercury in a thermometer expands, and using a linear scale printed on the thermometer, we measure that increase while feeling warmer. The thermometer provides reproducible measurements and serves as a device for observation and experiment. However, it does not establish a basis for temperature change at the molecular level, only a way to reference temperature to the volume of a certain mercury sample. A more detailed basis for temperature can be developed by considering the distribution of atoms and molecules among their available states. If we know the probability that a system exists in a certain state i is the value Pi, this value can be used in place of the number of configurations in Equation 1.6. P and C must be proportional to each other, and so it is sufficient to know P even if we do not know the absolute numbers of configurations, C. Therefore, ∆S = k ln P2 P1 (1.9a) Dividing by k and taking the exponential of this equation gives P2 = P1e ∆S/k (1.9b) 11 World of Atoms and Molecules The probability of being in state 2 is related to the probability of being in state 1 via an exponential in the entropy difference, ΔS. Let us now postulate, or assume without any proof, that temperature, T, relates energy changes, ΔE, and entropy changes, ΔS, of a system via the equation ΔE = TΔS. Using a postulate is like trying to work a jigsaw puzzle by putting a puzzle piece in place with nothing connected and then fitting pieces around it. If the guess was right, we have solved the puzzle. If not, we have to remove the piece and use a different one in that spot. Postulates may be regarded as making a choice of a piece of a scientific puzzle. If everything else fits, that is, if there are no contradictions, then we can accept the postulate as valid. If something does not fit, we must reject the postulate. Here, we are postulating the existence of something we are calling temperature by presuming or guessing that it relates entropy and energy changes, two things we already understand at a fundamental level. In exploring the implications of this postulate, we look for contradictions with other established ideas or theories. Now consider a hypothetical system with one molecule embedded in a continuous medium (a heat bath) that is maintained at a constant temperature, T. We shall think of the molecule as component A and the surrounding heat bath as component B. One specific change in the system to consider is for the quantum state of the molecule to go from state 1 to state 2. With this comes a change in the energy of component A which is ΔE(A) = E2 − E1. For energy to be conserved, the energy change of the heat bath and the energy change of the molecule must sum to zero. ∆E( A) + ∆E(B) = 0 The entropy of the molecule does not change, assuming that all the states are nondegenerate, because there is only one configuration for each state. That is, ΔS(A) = 0. Therefore, the entropy change for the system is the same as ΔS(B), which is ΔE(B)/T according to our postulate. ∆S = ∆S(B) = ∆E(B) ∆E( A) E1 − E2 =− = T T T This can be related to the probabilities that the system exists in the two states being considered. ∆S = E1 − E2 P = k ln 2 T P1 Dividing by k and taking the exponential of this equation yields P2 e − E2/kT = P1 e − E1/kT (1.10) Since states 1 and 2 of this hypothetical system differ only in the quantum state of the molecule in component A, this law states that the probability of existing in a given quantum state for a molecule in contact with a heat bath diminishes exponentially with the energy 12 Physical Chemistry: A Modern Introduction of the state. Increasing the temperature raises the probability of existing in higher energy states. This implication of our postulate is experimentally confirmed, and the equation is one form of the Maxwell–Boltzmann distribution law. Temperature is thereby connected to the distributions found among the states of a system. For a sample of many molecules, under the assumption of no interaction between them, the probabilities are directly proportional to the numbers of molecules in each state, the populations, which we designate as n. For a large collection of A molecules at some temperature T, the number found in the ith energy level of an A molecule is ni = NPi, where N is the total number of A molecules. Is it possible for a large collection of molecules to have populations of molecular quantum states other than those dictated by the Maxwell– Boltzmann law? Yes. But in that event, the system is not at equilibrium, which means that it is not stable and is undergoing change. The distribution law holds for equilibrium conditions, and under those conditions, it can be used to determine the number of molecules in particular energy level states. A subtlety that we shall not yet consider in detail occurs when the quantum energy levels of the molecule are degenerate. In that case, the distribution law becomes P2 g 2e − E2/kT = P1 g1e − E1/kT (1.11) where g1 and g2 are the degeneracies of the respective levels. Equation 1.11 is a more general form of Equation 1.10. Notice that for systems in which the degeneracy happens to increase on moving up from one energy level to the next higher level, the gi factors in Equation 1.11 contribute to making the higher state more probable, thereby working against the exponentially diminishing energy factor. Example 1.1: Energy Level Populations Problem: Relative to the ground state population, find the populations of the n = 1 and n = 2 quantum states for an equilibrium system of identical harmonic oscillators at a temperature T = 100 K. The oscillator vibrational frequency, ν, is such that hν = 2.0 × 10−21 J. Solution: Since the populations are probability times the number of molecules, the ratio of the population of one state to the ground state population is the same as the corresponding ratio of the probabilities. To obtain that ratio, we use Equation 1.10 because the degeneracies of the harmonic oscillator energy levels all happen to be 1 (i.e., nondegenerate). All that is needed to complete the problem is the energy level expression for harmonic oscillators, and that is Equation 1.2. P1 e −3 hν/2 kT = − hν/2 kT = e − hν/kT P0 e P2 e −5 hν/2 kT = − hν/2 kT = e −2 hν/kT P0 e With T = 100 K, kT = 1.38 × 10−21 J. With the given value of hν, the following ratios are obtained. P1 = 0.23 and P0 P2 = 0.055 P0 13 World of Atoms and Molecules 1.6 Distribution Law Derivation The distribution law of Equation 1.11 was obtained by postulating that ΔE = TΔS. This section is aimed at achieving the same result but following a lengthier statistical argument that does not require making that postulate. The most probable arrangement for a system is the one with maximum entropy, and that means the state where ln C is a maximum, with C being the number of configurations. Let us maximize ln C and determine the distribution that results for a system of N non-interacting particles. An energy, εi, is associated with the ith quantum state of a particle. As we did in considering arrangements of dice, let us use a set of integers such that n0 is the number of particles in the lowest state, n1 is the number in the first excited state, and so on. The total energy of the system, E, is a fixed value given that no energy is being added or removed from the system. E is necessarily the sum of the energies of the individual particles. ∞ E= ∑n ε (1.12) i i i=0 Equation 1.12 is a constraint on the system. We are interested only in those arrangements (n0, n1, n2, …) that have this energy. Some other arrangement (n′0 , n1′ , n′2 , …) is of interest only if Equation 1.12 yields the same energy with that arrangement. This means that the ni are not freely adjustable. In addition, there is the constraint that the number of particles is the unchanging value, N. ∞ N= ∑n (1.13) i i =0 Now, the mathematical task is to maximize ln C, where C is obtained from Equation 1.5, under the constraints of Equations 1.12 and 1.13. Equation 1.5 involves factorials in N and ni, and this makes the logarithm of C a complicated function. However, there exists a very useful approximation for the logarithm of a factorial, Stirling’s approximation. 1 ln(2πN ) 2 ln N ! ≅ N ln N − N + (1.14) And if N is very large, the smallest term in Equation 1.14 may be dropped as a further approximation. ln N ! ≅ N ln N − N (1.15) Using Stirling’s approximation for large N and large ni, ln C is approximated in the following way. ln C ≅ N ln N − ∑ n ln n − N − ∑ n i i i i i (1.16a) 14 Physical Chemistry: A Modern Introduction The term in parentheses in Equation 1.16a is zero according to Equation 1.13, and so it may be deleted. ln C ≅ N ln N − ∑ n ln n i (1.16b) i i Now, in this form, we can proceed to maximize ln C. A good number of mathematical steps are required to accomplish this, and the result has general implications. When a function, f(x), is at a maximum or minimum value, the first derivative is zero, that is, ∂f(xmax)/∂x = 0. This is because the first derivative gives the slope of a line tangent to the function; at a maximum or minimum, the tangent should be a horizontal line, which is a line of zero slope (see Appendix A). To find the maximum of ln C, we need to find the choice of the set of ni values where all the first derivatives of ln C are zero. The values in this special set will be designated n′i to distinguish them from the variables ni. (This is the same as saying that x is the variable in f(x) but that xmax or x′ means the specific value at which f is a maximum.) We can use differentials to find the point at which the first derivatives are zero. The mathematical complication lies in the fact that Equations 1.12 and 1.13 are constraints; the ni values are not entirely independent. Since N is a constant, taking the differential of Equation 1.16b yields d(ln C ) = − ∑ (1 + ln n )dn = −∑ dn − ∑ ln n dn = − ∑ ln n dn i i i i i i i i i i (1.17) i since the sum of the dni is dN, which is zero. Likewise, taking the differentials of Equations 1.12 and 1.13 yields dE = 0 = ∑ ε dn (1.18) ∑ dn (1.19) i i i dN = 0 = i i The left-hand side is zero in both cases because N and E are constants. Something that is zero may be freely added or subtracted, and so it is possible to state that for any constant α and any constant β, d(ln C ) = − ∑ (ln n )dn − α∑ dn − β∑ ε dn i i i i i i i (1.20) i Though this step may seem arbitrary, it is the standard mathematical technique of using Lagrange multipliers to build external constraints into differential expressions. At the maximum of ln C, d(ln C) = 0. We will designate the ni values at that point as n′i , and then Equation 1.20 at the maximum of ln C is 0=− ∑ (ln n′)dn − α∑ dn − β∑ ε dn (1.21) ∑ (α + βε + ln n′)dn (1.22) i i i i =− i i i i i i i i 15 World of Atoms and Molecules This equation will be satisfied if each summation term in parentheses is independently zero. That is, ln ni’ = −α − βε i (1.23) Taking the exponential of both sides of Equation 1.23 gives an expression for n′i . n′i = e − α e −βεi (1.24a) α and β remain unknown at this point in the mathematical procedure. We can replace α in Equations 1.24a and b by using Equation 1.13; that is, we replace the factor involving α by using the expression for N when ln C is a maximum. N= ∑ n′ = ∑ e i i − α −βεi = e −α e i ∑e −βεi (1.24b) i Rearranging this equation gives N e −α = ∑ i e −βεi (1.25) The summation in the denominator is referred to as the partition function and will be designated by the symbol Q. Q= ∑e −βεi (1.26) i Therefore, n′i = N −βεi e Q (1.27) In effect, α has now been expressed in terms of β though β remains unknown. The effect of the partition function amounts to applying a common factor to each ni such that the sum of the ni ’s will be N. We can combine Equation 1.27 with the energy expression, Equation 1.12, to write E= ∑Qεe N i −βεi (1.28) i Noticing that dQ =− dβ ∑ε e i i −βεi (1.29) 16 Physical Chemistry: A Modern Introduction we have an alternate expression for the energy. E=− N dQ Q dβ (1.30) Hence, if we know Q or something proportional to Q, we can immediately find E. The development to this point is quite general, and Equations 1.28 and 1.30 can be applied anywhere. If we apply them to one specific problem that we understand or can analyze fully, we shall achieve a relation between β and E that is more explicit and better helps us understand and use β. An idealized problem that can be analyzed fully is that of a large number of non-quantum mechanical particles possessing only translational energy and free to move about without interaction between particles (a gas of non-interacting particles). For this system, we will be able to show that the energy is inversely proportional to β. We have already seen from Equation 1.28 that β determines the distribution, and so it seems to be a good choice for a thermodynamic variable. Is it the temperature? The answer is “no” because we expect energy to increase as temperature increases, not as temperature decreases. In other words, temperature could be the inverse of β to within some constant of proportionality, and that turns out to be the case. The energies of non-interacting gas atoms (particles) are associated with their individual translational motions. From a classical point of view, these energies are continuous because the velocities of the particles can be any value. From a quantum mechanical view, we can say that the kinetic energy of motion of each particle is not discrete. Under these conditions, the summation in Equation 1.26 needs only to be converted to a sum of infinitesimals, that is, converted to an integral, an integral over all the mechanical degrees of freedom. However, the energy depends only on the momenta of the particles because the kinetic energy of a free gas atom of mass m is ε atom = 1 px2 + py2 + pz2 2m ( ) (1.31) Integrating over only px, py, and pz, and not over the spatial degrees of freedom, yields in this case a value proportional to Q. We shall let C be the constant of proportionality. Thus, the integral expression for Q is ∞ ∞ ∞ Q=C ∫ ∫ ∫e −( px2 + py2 + pz2 )β/2 m (1.32) dpx dpy dpz −∞ −∞ −∞ This can be expressed as a product of integrals over the momentum components. ∞ ∞ ∫ Q=C e −∞ −βpx2 /2 m dpx ∫e −∞ −βpy2 /2 m ∞ dpy ∫e −∞ −βpz2 /2 m dpz (1.33) 17 World of Atoms and Molecules Evaluating these integrals (see Appendix D) yields 2mπ Q = C β 3/ 2 (1.34) Notice that the three integrals in Equation 1.33 have the same value, and this leads to the 3 in the exponent value of 3/2 in Equation 1.34. The derivative of Equation 1.34 can be used in Equation 1.30 to finally establish the relationship between the energy of the system and β. dQ 3 1 = −C(2mπ)3/2 dβ 2 β E= 5/2 (1.35) 3N 2 β This indicates that the total internal energy for this particular case is inversely proportional to β. As already derived, β is more than this for it generally dictates populations among available energy levels. Any new quantity we define that is directly related to β will serve as well as β, and one such choice is to define a new quantity, T, to be (kβ)−1, where k is Boltzmann’s constant. With that, T is what we usually refer to as temperature. If we replace β by its dependent quantity T in Equation 1.35, which is specific for a gas of non-interacting particles, then E= 3 3 NkT = nRT 2 2 (1.36) Likewise, substituting for β in Equation 1.27 yields the Maxwell–Boltzmann distribution law, which is a general result. ni = N −εi/kT e Q (1.37) Again, this indicates that the population diminishes exponentially with energy. Returning to the example of a gas of non-interacting particles, we note that the average energy per gas particle is the total energy divided by the number of particles. Using Equation 1.36, the average energy is E = 〈E〉 = E 3 = kT N 2 (1.38) where two designations, an overbar and 〈〉, have been used to identify the average or the mean of the value E. From the energy expression in Equation 1.31, we relate this average to the average of the square of the momentum. 18 Physical Chemistry: A Modern Introduction 1 3 px2 + py2 + pz2 = kT 2m 2 ( ) (1.39) Since the space for a moving free particle is isotropic (the same in each direction), the meansquared components of the momenta must be the same for each direction: 〈 px2 〉 = 〈 py2〉 = 〈 pz2 〉 . Using this expression with Equation 1.39 yields 〈 px2〉 = 〈 py2 〉 = 〈 pz2 〉 = mkT (1.40) Or, for velocities, v (recalling that px = mvx), 〈vx2 〉 = 〈vy2 〉 = 〈 vz2 〉 = kT m (1.41) Speed, s, is the magnitude of a velocity vector, and the square of the speed is the sum of the squares of the velocity components. Thus, 〈 s2 〉 = 3kT m (1.42) The square root of this value, 〈s2〉1/2, is the root-mean-squared speed of the non-interacting gas particles. The root-mean-squared speed increases as the square root of temperature. A manifestation of this is that as air warms, the speed at which sound may be transmitted increases because of the greater average speed of the particles. 1.7 Conclusions Physical chemistry is the exploration of the world of atoms and molecules in terms of fundamental physical laws; it is the physics of chemistry. It embraces properties and dynamics of isolated species and of bulk systems, and it seeks to provide a fundamental connection between both realms. Atoms and molecules are such small species that their mechanical behavior (i.e., vibrations, rotations, electron orbital motion, and so on) is governed by quantum mechanics. Atoms and molecules exist in different states with distinct energies. Statistical analysis of a large sample of quantum mechanical particles tells us that there are many ways in which they can be arranged among the available states. Equilibrium corresponds to existence largely in the most probable arrangement, and this condition is characterized by maximizing a quantity called entropy. At equilibrium, the distribution of particles among available states is dictated by the thermodynamic property of temperature following the Maxwell–Boltzmann distribution law. Temperature is not ad hoc but is fundamentally defined by the distribution of particles among energy levels in a condition of equilibrium. 19 World of Atoms and Molecules Point of Interest: James Clerk Maxwell James Clerk Maxwell James Clerk Maxwell is an interesting figure both in the development of physical chemistry, a subject at the boundary of his chosen field of physics, and in his execution of scientific discovery. He was born in Edinburgh, Scotland, on June 13, 1831, and he died in Cambridge, England, on November 5, 1879. He held professorships first in Aberdeen, Scotland, then at King’s College in London, and finally at Cambridge University. Despite his rather short life, he had a remarkable and lasting impact on physical science. Maxwell is perhaps most known for his work on electricity, magnetism, and the electromagnetic theory of light. This work began following his graduation from Cambridge in 1854 and included the development of the electromagnetic field relations known today as Maxwell’s equations. The use of analogy was a powerful tool in Maxwell’s hands, and it was something in which he strongly believed. His contributions to electromagnetic theory were aided by analogies Maxwell recognized between electric current, heat conduction, and fluid flow. These were analogies between mathematical relations more than similarities between the nature of different things. An interesting problem area that occupied Maxwell’s attention from 1855 to 1859 was the nature of the rings of Saturn. He showed that certain ideas of the time, such as that Saturn had solid or rigid rings, were not possible. Instead, the stability of the rings require that they consist of concentric circles of small objects, the orbital speed of each circle being dependent on its distance from the planet. In 1895, the differential rotation of the rings Maxwell predicted almost 40 years earlier was confirmed by observation. The study of Saturn’s rings led Maxwell to the problem of the motions of large numbers of colliding bodies, such as would be found in the rings. This in turn led him to the study of gas kinetics. Here he introduced the use of statistical methods, not for data analysis but for a description of the physical process. He recognized that there must be a distribution of velocities of gas particles, and by 1860 he had developed a statistical formula for that 20 Physical Chemistry: A Modern Introduction distribution (the first expression of what today we refer to as the Maxwell–Boltzmann distribution law). One idea of the time was that the pressure of a gas resulted from static repulsion between gas particles. Maxwell’s kinetic picture, on the other hand, correctly associated pressure with collisions of molecules with the walls of the vessel in which the gas is held. His theory led to successful predictions about viscosity, diffusion, and other properties of gases; and it launched the study of statistical mechanics. Many things were of interest to Maxwell, and he published papers on a wide range of topics (his first at the age of 14). He studied optics and optical properties of materials, and he developed the fish-eye lens. One of Maxwell’s interests and areas of investigation was color vision. He projected the first color photograph, and he explained color blindness as a deficiency in one or two of the three types of color receptors in the eye. He was interested in the stability of the earth’s atmosphere and its thermodynamics and in stress in building frameworks. Maxwell’s success, and the impact of his work on physical chemistry, came from his belief in the power of analogy and also from his interest in studying an assortment of problems. Exercises 1.1 The rotational constant, B, for carbon monoxide is 3.836 × 10−23 J. Use Equation 1.3 to find the energy of the lowest three rotational state levels (i.e., J = 0, 1, and 2) of a carbon monoxide molecule and find the degeneracies of each of these levels. 1.2 List the 15 configurations that are possible for three pair of dice in a state with two 6s and four 5s showing. 1.3 Apply Equation 1.5 to the arrangements of four dice to find the total probability that three and only three of the dice will have the same number showing. 1.4 Verify Equation 1.7 for an expansion where only one side of the original box is increased from l to 2l. 1.5 For a gas sample of non-interacting particles at T = 100 K, apply Equation 1.10 to find the ratio of probabilities for two states with energies E1 = 3.0 × 10−21 J and E2 = 9.0 × 10−21 J. 1.6 Obtain approximations to ln N! for N = 10, 1,000, and 100,000 by applying Equations 1.14 and 1.15. How significant are the differences? 1.7 Find the root-mean-squared speed of atoms of helium, neon, and argon at temperatures of 10, 300, and 500 K using Equation 1.42. 1.8 What is the mass of the particles of a hypothetical gas whose root-mean-squared speed at T = 300 K happens to match the speed of sound in air at 300 K (about 350 m s−1)? 1.9 Calculate the amount of energy required to ionize 1 mol of hydrogen atoms from their ground state. 1.10 What is the ratio of populations in the ground and second excited state of a harmonic oscillator with a vibrational frequency of 5.0 × 1012 Hz at a temperature of 300 K? 1.11 For a rigid rotator with a rotational constant of 8.5 cm−1, what is the temperature at which the population of the J = 2 rotational state is 50% of the J = 1 rotational state? 21 World of Atoms and Molecules 1.12 Use Equation 1.5 to find the number of configurations for all the states of a system consisting of three four-sided pyramidal bodies numbered 1 through 4 (i.e., foursided dice). 1.13 Make a plot of the number of configurations with all dice showing different numbers for a set of dice versus the number of dice in the set, with that number ranging from 1 to 6. 1.14 Given the following expression for the entropy change that accompanies a temperature change of a liquid, find the probability that a 1 kg sample of water initially at equilibrium at 300 K will be found to exist with half the molecules at 310 K and half at 290 K. T ∆S = nR ln 2 T1 1.15 Use the condition given in Exercise 1.11 and find a value for the temperature difference, ΔT, where half of the 1 kg sample of water is at a temperature of 300 + ΔT K and the other half is at 300 − ΔT K, such that the probability of this occurring is 1 in 100. 1.16 Test the two forms of Stirling’s approximation, Equations 1.14 and 1.15, by directly computing ln N! and comparing it with the results from the approximate expressions for N = 5, 10, 15, 20, and 25. 1.17 Consider a system of 10 identical harmonic oscillators. The state of the system is given by a list of integers (n0, n1, n2, …) that indicate the number of oscillators in their 0, 1, 2, 3, …, quantum states. Of course, the sum of these integers must be 10, the number of oscillators. If the system happens to be in the state (2, 1, 1, 5, 0, 1, 0, 0, …), make a list of at least five other states with the same energy; show that the constraints of Equations 1.12 and 1.13 are satisfied. Then, find a state where Equation 1.13 holds but Equation 1.12 does not. 1.18 Find the value of kT in joules at room temperature (about 300 K). Convert this value to cm−1 (see Appendix F). At this temperature, what must be the energy difference (in cm−1) between two nondegenerate molecular energy levels such that the relative population of the higher state to the lower state is 0.5? Does this energy difference seem typical of the separations between rotational states or between vibrational states? 1.19 The energy difference associated with excitation of the electronic structure of atoms and molecules is typically 20,000 cm−1 and above. Assuming no degeneracy of energy levels, find the ratio of the population of an excited state at 20,000 cm−1 to the population of the ground state at 300 K. Next, find the temperature at which this ratio becomes 0.1. 1.20 Using Equation 1.11, make a plot of the population of quantum states versus the quantum number, J, of a sample of a gas at 300 K given that J = 0, 1, 2, 3, …, the degeneracy is gJ = 2J + 1, and the energy in units of cm−1 (see Appendix F) is EJ = 20 cm −1 J ( J + 1) 1.21 Consider a system of N non-interacting particles trapped so that they are free to move in only two dimensions. Find Q for this system and then find an expression for the energy of the system. 22 Physical Chemistry: A Modern Introduction Bibliography Hill, T. L., An Introduction to Statistical Thermodynamics (Dover Publications, New York, 1987). This is a concise introductory level text. Ihde, A. J., The Development of Modern Chemistry (Dover Publications, New York, 1984). Kuhn, T. S., The Structure of Scientific Revolutions (University of Chicago Press, Chicago, IL, 1970). Servos, J. W., Physical Chemistry from Ostwald to Pauling (Princeton University Press, Princeton, NJ, 1990). Smith, E. B., Basic Chemical Thermodynamics, 5th edn. (World Scientific Books, Singapore, 2004). 2 Ideal and Real Gases Examination of the gaseous state of matter offers an excellent means for understanding certain basics of thermodynamics and for seeing connections with atomic and molecular level behavior. This chapter is concerned with explaining the differences between the behavior of real gases and the behavior of a hypothetical gas called an ideal gas because of its particularly simple behavior. The ideal gas is a model that under certain conditions can serve as a good approximation of real gas behavio