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FUNDAMENTAL CONSTANTS Constant Symbol Value Power of 10 Units Speed of light c 2.997 924 58* 10 m s−1 Elementary charge e 1.602 176 565 10−19 C Planck’s constant h 6.626 069 57 10 −34 Js ħ = h/2π 1.054 571 726 10−34 Js Boltzmann’s constant k 1.380 6488 10 J K−1 Avogadro’s constant NA 6.022 141 29 1023 mol−1 Gas constant R = NAk 8.314 4621 Faraday’s constant F = NAe 9.648 533 65 104 C mol−1 Electron me 9.109 382 91 10−31 kg Proton mp 1.672 621 777 10 −27 kg Neutron mn 1.674 927 351 10−27 kg Atomic mass constant 8 −23 J K−1 mol−1 Mass mu 1.660 538 921 10 kg Vacuum permeability μ0 4π* 10−7 J s2 C−2 m−1 Vacuum permittivity ε0 = 1/μ0c2 8.854 187 817 10−12 J−1 C2 m−1 4πε0 1.112 650 056 10 J−1 C2 m−1 Bohr magneton μB = eħ/2me 9.274 009 68 10−24 J T−1 Nuclear magneton μN = eħ/2mp 5.050 783 53 10 −27 J T−1 Proton magnetic moment µp 1.410 606 743 10−26 J T−1 g-Value of electron ge 2.002 319 304 −1.001 159 652 1010 C kg−1 2.675 222 004 108 C kg−1 a0 = 4πε0ħ /e me R = m e4/8h3cε 2 5.291 772 109 10 m 1.097 373 157 105 cm−1 hc R ∞ /e 13.605 692 53 −27 −10 Magnetogyric ratio Electron γe = −gee/2me Proton γp = 2µp/ħ Bohr radius Rydberg constant 2 ∞ e 2 0 −11 eV α = μ0e c/2h 7.297 352 5698 10 α−1 1.370 359 990 74 102 Stefan–Boltzmann constant σ = 2π5k4/15h3c2 5.670 373 10−8 Standard acceleration of free fall g 9.806 65* Gravitational constant G 6.673 84 Fine-structure constant 2 * Exact value. For current values of the constants, see the National Institute of Standards and Technology (NIST) website. −3 W m−2 K−4 m s−2 10−11 N m2 kg−2 Atkins’ PHYSICAL CHEMISTRY Eleventh edition Peter Atkins Fellow of Lincoln College, University of Oxford, Oxford, UK Julio de Paula Professor of Chemistry, Lewis & Clark College, Portland, Oregon, USA James Keeler Senior Lecturer in Chemistry and Fellow of Selwyn College, University of Cambridge, Cambridge, UK 1 3 Great Clarendon; Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Peter Atkins, Julio de Paula and James Keeler 2018 The moral rights of the author have been asserted Eighth edition 2006 Ninth edition 2009 Tenth edition 2014 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2017950918 ISBN 978–0–19–108255–9 Printed in Italy by L.E.G.O. S.p.A. Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. The cover image symbolizes the structure of the text, as a collection of Topics that merge into a unified whole. It also symbolizes the fact that physical chemistry provides a basis for understanding chemical and physical change. PREFACE Our Physical Chemistry is continuously evolving in response to users’ comments and our own imagination. The principal change in this edition is the addition of a new co-author to the team, and we are very pleased to welcome James Keeler of the University of Cambridge. He is already an experienced author and we are very happy to have him on board. As always, we strive to make the text helpful to students and usable by instructors. We developed the popular ‘Topic’ arrangement in the preceding edition, but have taken the concept further in this edition and have replaced chapters by Focuses. Although that is principally no more than a change of name, it does signal that groups of Topics treat related groups of concepts which might demand more than a single chapter in a conventional arrangement. We know that many instructors welcome the flexibility that the Topic concept provides, because it makes the material easy to rearrange or trim. We also know that students welcome the Topic arrangement as it makes processing of the material they cover less daunting and more focused. With them in mind we have developed additional help with the manipulation of equations in the form of annotations, and The chemist’s toolkits provide further background at the point of use. As these Toolkits are often relevant to more than one Topic, they also appear in consolidated and enhanced form on the website. Some of the material previously carried in the ‘Mathematical backgrounds’ has been used in this enhancement. The web also provides a number of sections called A deeper look. As their name suggests, these sections take the material in the text further than we consider appropriate for the printed version but are there for students and instructors who wish to extend their knowledge and see the details of more advanced calculations. Another major change is the replacement of the ‘Justifications’ that show how an equation is derived. Our intention has been to maintain the separation of the equation and its derivation so that review is made simple, but at the same time to acknowledge that mathematics is an integral feature of learning. Thus, the text now sets up a question and the How is that done? section that immediately follows develops the relevant equation, which then flows into the following text. The worked Examples are a crucially important part of the learning experience. We have enhanced their presentation by replacing the ‘Method’ by the more encouraging Collect your thoughts, where with this small change we acknowledge that different approaches are possible but that students welcome guidance. The Brief illustrations remain: they are intended simply to show how an equation is implemented and give a sense of the order of magnitude of a property. It is inevitable that in an evolving subject, and with evolving interests and approaches to teaching, some subjects wither and die and are replaced by new growth. We listen carefully to trends of this kind, and adjust our treatment accordingly. The topical approach enables us to be more accommodating of fading fashions because a Topic can so easily be omitted by an instructor, but we have had to remove some subjects simply to keep the bulk of the text manageable and have used the web to maintain the comprehensive character of the text without overburdening the presentation. This book is a living, evolving text. As such, it depends very much on input from users throughout the world, and we welcome your advice and comments. PWA JdeP JK vi 12 The properties of gases USING THE BOOK TO THE STUDENT For this eleventh edition we have developed the range of learning aids to suit your needs more closely than ever before. In addition to the variety of features already present, we now derive key equations in a helpful new way, through the How is that done? sections, to emphasize how mathematics is an interesting, essential, and integral feature of understanding physical chemistry. Innovative structure Short Topics are grouped into Focus sections, making the subject more accessible. Each Topic opens with a comment on why it is important, a statement of its key idea, and a brief summary of the background that you need to know. Notes on good practice Our ‘Notes on good practice’ will help you avoid making common mistakes. Among other things, they encourage conformity to the international language of science by setting out the conventions and procedures adopted by the International Union of Pure and Applied Chemistry (IUPAC). TOPIC 2A Internal energy ➤ Why do you need to know this material? The First Law of thermodynamics is the foundation of the discussion of the role of energy in chemistry. Wherever the generation or use of energy in physical transformations or chemical reactions is of interest, lying in the background are the concepts introduced by the First Law. ➤ What is the key idea? The total energy of an isolated system is constant. ➤ What do you need to know already? This Topic makes use of the discussion of the properties of gases (Topic 1A), particularly the perfect gas law. It builds on the definition of work given in The chemist’s toolkit 6. For the purposes of thermodynamics, the universe is divided into two parts, the system and its surroundings. The system is the part of the world of interest. It may be a reaction vessel, an engine, an electrochemical cell, a biological cell, and so on. The surroundings comprise the region outside the system and are where measurements are made. The type of system depends on the characteristics of the boundary that divides it from the For example, a closed system can expand and thereby raise a weight in the surroundings; a closed system may also transfer energy to the surroundings if they are at a lower temperature. An isolated system is a closed system that has neither mechanical nor thermal contact with its surroundings. 2A.1 Work, heat, and energy Although thermodynamics deals with observations on bulk systems, it is immeasurably enriched by understanding the molecular origins of these observations. (a) Operational definitions The fundamental physical property in thermodynamics is work: work is done to achieve motion against an opposing force (The chemist’s toolkit 6). A simple example is the process of raising a weight against the pull of gravity. A process does work if in principle it can be harnessed to raise a weight somewhere in the surroundings. An example of doing work is the expansion of a gas that pushes out a piston: the motion of the piston can in principle be used to raise a weight. Another example is a chemical reaction in a cell, which leads to an electric A note on good practice An allotrope is a particular molecular form of an element (such as O2 and O3) and may be solid, liquid, or gas. A polymorph is one of a number of solid phases of an element or compound. The number of phases in a system is denoted P. A gas, or a gaseous mixture, is a single phase (P = 1), a crystal of a sub- Resource section The Resource section at the end of the book includes a table of useful integrals, extensive tables of physical and chemical data, and character tables. Short extracts of most of these tables appear in the Topics themselves: they are there to give you an idea of the typical values of the physical quantities mentioned in the text. Checklist of concepts A checklist of key concepts is provided at the end of each Topic, so that you can tick off the ones you have mastered. Contents 1 Common integrals 862 866 2 Units 864 868 3 Data 865 869 Checklist of concepts ☐ 1. The physical state of a sample of a substance, its physical condition, is defined by its physical properties. ☐ 2. Mechanical equilibrium is the condition of equality of pressure on either side of a shared movable wall. Using the book vii PRESENTING THE MATHEMATICS How is that done? You need to understand how an equation is derived from reasonable assumptions and the details of the mathematical steps involved. This is accomplished in the text through the new ‘How is that done?’ sections, which replace the Justifications of earlier editions. Each one leads from an issue that arises in the text, develops the necessary mathematics, and arrives at the equation or conclusion that resolves the issue. These sections maintain the separation of the equation and its derivation so that you can find them easily for review, but at the same time emphasize that mathematics is an essential feature of physical chemistry. How is that done? 4A.1 Deducing the phase rule The argument that leads to the phase rule is most easily appreciated by first thinking about the simpler case when only one component is present and then generalizing the result to an arbitrary number of components. Step 1 Consider the case where only one component is present When only one phase is present (P = 1), both p and T can be varied independently, so F = 2. Now consider the case where two phases α and β are in equilibrium (P = 2). If the phases are in equilibrium at a given pressure and temperature, their chemical potentials must be equal: The chemist’s toolkits The chemist’s toolkit 2 The chemist’s toolkits, which are much more numerous in this edition, are reminders of the key mathematical, physical, and chemical concepts that you need to understand in order to follow the text. They appear where they are first needed. Many of these Toolkits are relevant to more than one Topic, and a compilation of them, with enhancements in the form of more information and brief illustrations, appears on the web site. The state of a bulk sample of matter is defined by specifying the values of various properties. Among them are: www.oup.com/uk/pchem11e/ Properties of bulk matter The mass, m, a measure of the quantity of matter present (unit: kilogram, kg). The volume, V, a measure of the quantity of space the sample occupies (unit: cubic metre, m3). The amount of substance, n, a measure of the number of specified entities (atoms, molecules, or formula units) present (unit: mole, mol). Annotated equations and equation labels We have annotated many equations to help you follow how they are developed. An annotation can take you across the equals sign: it is a reminder of the substitution used, an approximation made, the terms that have been assumed constant, an integral used, and so on. An annotation can also be a reminder of the significance of an individual term in an expression. We sometimes colour a collection of numbers or symbols to show how they carry from one line to the next. Many of the equations are labelled to highlight their significance. d(1/f )/dx = −(1/f 2)df/dx used twice Um(T) = Um(0) + NA 〈εV〉 2 CVV,m = V θV dN A 〈ε V 〉 d 1 eθ /T = Rθ V = R θ /T T dT dT eθ /T −1 (e −1)2 By noting that eθ into V V /T = (eθ V /2T 2 ) , this expression can be rearranged 2 CVV,m = Rf (T ) V θ V e −θ /2T f (T ) = T 1− e −θ /T V 2 V Vibrational contribution to CV,m Checklists of equations A handy checklist at the end of each topic summarizes the most important equations and the conditions under which they apply. Don’t think, however, that you have to memorize every equation in these checklists. Checklist of equations Property Equation Gibbs energy of mixing ΔmixG = nRT(xA ln xA + xB ln xB) Entropy of mixing ΔmixS = −nR(xA ln xA + xB ln xB) (13E.3) viii Using the book SET TING UP AND SOLVING PROBLEMS Brief illustrations A Brief illustration shows you how to use an equation or concept that has just been introduced in the text. It shows you how to use data and manipulate units correctly. It also helps you to become familiar with the magnitudes of quantities. Brief illustration 3B.1 When the volume of any perfect gas is doubled at constant temperature, Vf/Vi = 2, and hence the change in molar entropy of the system is ΔSm = (8.3145 J K−1 mol−1) × ln 2 = +5.76 J K−1 mol−1 Examples Worked Examples are more detailed illustrations of the application of the material, and typically require you to assemble and deploy the relevant concepts and equations. We suggest how you should collect your thoughts (that is a new feature) and then proceed to a solution. All the worked Examples are accompanied by Self-tests to enable you to test your grasp of the material after working through our solution as set out in the Example. Discussion questions Discussion questions appear at the end of every Focus, and are organised by Topic. These questions are designed to encourage you to reflect on the material you have just read, to review the key concepts, and sometimes to think about its implications and limitations. Exercises and problems Exercises and Problems are also provided at the end of every Focus and organised by Topic. Exercises are designed as relatively straightforward numerical tests; the Problems are more challenging and typically involve constructing a more detailed answer. The Exercises come in related pairs, with final numerical answers available online for the ‘a’ questions. Final numerical answers to the odd-numbered Problems are also available online. Example 1A.1 Using the perfect gas law In an industrial process, nitrogen gas is introduced into a vessel of constant volume at a pressure of 100 atm and a temperature of 300 K. The gas is then heated to 500 K. What pressure would the gas then exert, assuming that it behaved as a perfect gas? Collect your thoughts The pressure is expected to be greater on account of the increase in temperature. The perfect gas FOCUS 3 The Second and Third Laws Assume that all gases are perfect and that data refer to 298.15 K unless otherwise stated. TOPIC 3A Entropy Discussion questions D3A.1 The evolution of life requires the organization of a very large number of molecules into biological cells. Does the formation of living organisms violate the Second Law of thermodynamics? State your conclusion clearly and present detailed arguments to support it. At the end of every Focus you will find questions that span several Topics. They are designed to help you use your knowledge creatively in a variety of ways. D3A.3 Discuss the relationships between the various formulations of the Second Law of thermodynamics. Exercises E3A.1(a) Consider a process in which the entropy of a system increases by 125 J K−1 and the entropy of the surroundings decreases by 125 J K−1. Is the process spontaneous? E3A.1(b) Consider a process in which the entropy of a system increases by 105 J K−1 and the entropy of the surroundings decreases by 95 J K−1. Is the process spontaneous? E3A.2(a) Consider a process in which 100 kJ of energy is transferred reversibly and isothermally as heat to a large block of copper. Calculate the change in entropy of the block if the process takes place at (a) 0 °C, (b) 50 °C. E3A.2(b) Consider a process in which 250 kJ of energy is transferred reversibly and isothermally as heat to a large block of lead. Calculate the change in entropy of the block if the process takes place at (a) 20 °C, (b) 100 °C. gas of mass 14 g at 298 K doubles its volume in (a) an isothermal reversible expansion, (b) an isothermal irreversible expansion against pex = 0, and (c) an adiabatic reversible expansion. E3A.4(b) Calculate the change in the entropies of the system and the surroundings, and the total change in entropy, when the volume of a sample of argon gas of mass 2.9 g at 298 K increases from 1.20 dm3 to 4.60 dm3 in (a) an isothermal reversible expansion, (b) an isothermal irreversible expansion against pex = 0, and (c) an adiabatic reversible expansion. E3A.5(a) In a certain ideal heat engine, 10.00 kJ of heat is withdrawn from the hot source at 273 K and 3.00 kJ of work is generated. What is the temperature of cold sink? E3A.5(b) In an ideal heat engine the cold sink is at 0 °C. If 10.00 kJ of heat E3A.3(a) Calculate the change in entropy of the gas when 15 g of carbon dioxide is withdrawn from the hot source and 3.00 kJ of work is generated, at what temperature is the hot source? E3A.3(b) Calculate the change in entropy of the gas when 4.00 g of nitrogen is E3A.6(a) What is the efficiency of an ideal heat engine in which the hot source gas are allowed to expand isothermally from 1.0 dm3 to 3.0 dm3 at 300 K. 3 3 allowed to expand isothermally from 500 cm to 750 cm at 300 K. E3A.4(a) Calculate the change in the entropies of the system and the surroundings, and the total change in entropy, when a sample of nitrogen is at 100 °C and the cold sink is at 10 °C? E3A.6(b) An ideal heat engine has a hot source at 40 °C. At what temperature must the cold sink be if the efficiency is to be 10 per cent? Problems P3A.1 A sample consisting of 1.00 mol of perfect gas molecules at 27 °C is Integrated activities D3A.2 Discuss the significance of the terms ‘dispersal’ and ‘disorder’ in the context of the Second Law. expanded isothermally from an initial pressure of 3.00 atm to a final pressure of 1.00 atm in two ways: (a) reversibly, and (b) against a constant external pressure of 1.00 atm. Evaluate q, w, ΔU, ΔH, ΔS, ΔSsurr, and ΔStot in each case. P3A.2 A sample consisting of 0.10 mol of perfect gas molecules is held by a piston inside a cylinder such that the volume is 1.25 dm3; the external pressure is constant at 1.00 bar and the temperature is maintained at 300 K by a thermostat. The piston is released so that the gas can expand. Calculate (a) the volume of the gas when the expansion is complete; (b) the work done when the gas expands; (c) the heat absorbed by the system. Hence calculate ΔStot. P3A.3 Consider a Carnot cycle in which the working substance is 0.10 mol of perfect gas molecules, the temperature of the hot source is 373 K, and that of the cold sink is 273 K; the initial volume of gas is 1.00 dm3, which doubles over the course of the first isothermal stage. For the reversible adiabatic stages it may be assumed that VT 3/2 = constant. (a) Calculate the volume of the gas after Stage 1 and after Stage 2 (Fig. 3A.8). (b) Calculate the volume of gas after Stage 3 by considering the reversible adiabatic compression from the starting point. (c) Hence, for each of the four stages of the cycle, calculate the heat transferred to or from the gas. (d) Explain why the work done is equal to the difference between the heat extracted from the hot source and that deposited in the cold sink. (e) Calculate the work done over the cycle and hence the efficiency η. (f) Confirm that your answer agrees with the efficiency given by eqn 3A.9 and that your values for the heat involved in the isothermal stages are in accord with eqn 3A.6. P3A.4 The Carnot cycle is usually represented on a pressure−volume diagram (Fig. 3A.8), but the four stages can equally well be represented on temperature−entropy diagram, in which the horizontal axis is entropy and the vertical axis is temperature; draw such a diagram. Assume that the temperature of the hot source is Th and that of the cold sink is Tc, and that the volume of the working substance (the gas) expands from VA to VB in the first isothermal stage. (a) By considering the entropy change of each stage, derive an expression for the area enclosed by the cycle in the temperature−entropy diagram. (b) Derive an expression for the work done over the cycle. (Hint: The work done is the difference between the heat extracted from the hot source and that deposited in the cold sink; or use eqns 3A.7 and 3A.9) (c) Comment on the relation between your answers to (a) and (b). Using the book ix THERE IS A LOT OF ADDITIONAL MATERIAL ON THE WEB IMPAC T 1 …ON ENVIRONMENTAL SCIENCE: The gas laws and the weather 25 20 Altitude, h/km The biggest sample of gas readily accessible to us is the atmosphere, a mixture of gases with the composition summarized in Table 1. The composition is maintained moderately constant by diffusion and convection (winds, particularly the local turbulence called eddies) but the pressure and temperature vary with altitude and with the local conditions, particularly in the troposphere (the ‘sphere of change’), the layer extending up to about 11 km. 15 10 A DEEPER LOOK 2 The fugacity At various stages in the development of physical chemistry it is necessary to switch from a consideration of idealized systems to real systems. In many cases it is desirable to preserve the form of the expressions that have been derived for an idealized system. Then deviations from the idealized behaviour can be expressed most simply. For instance, the pressure-dependence of the molar Gibbs energy of a perfect gas is p ○ −− Gm = G m + RT ln −○− p (1a) In this expression, f1 is the fugacity when the pressure is p1 and f2 is the fugacity when the pressure is p2. That is, from eqn 3b, ∫ p2 p1 Vm d p = RT ln f2 f1 (4a) For a perfect gas, ∫ p2 p1 Vperfect ,mdp = RT ln p2 p1 (4b) ‘Impact’ sections Group theory tables ‘Impact’ sections show how physical chemistry is applied in a variety of modern contexts. They showcase physical chemistry as an evolving subject. www.oup.com/uk/pchem11e/ Comprehensive group theory tables are available to download. A deeper look Files containing molecular modelling problems can be downloaded, designed for use with the Spartan Student™ software. However they can also be completed using any modelling software that allows Hartree–Fock, density functional, and MP2 calculations. The site can be accessed at www.oup.com/ uk/pchem11e/. These online sections take some of the material in the text further and are there if you want to extend your knowledge and see the details of some of the more advanced derivations www.oup.com/uk/pchem11e/ Molecular modelling problems TO THE INSTRUC TOR We have designed the text to give you maximum flexibility in the selection and sequence of Topics, while the grouping of Topics into Focuses helps to maintain the unity of the subject. Additional resources are: Figures and tables from the book Lecturers can find the artwork and tables from the book in ready-to-download format. These may be used for lectures without charge (but not for commercial purposes without specific permission). Key equations Supplied in Word format so you can download and edit them. Lecturer resources are available only to registered adopters of the textbook. To register, simply visit www.oup.com/uk/pchem11e/ and follow the appropriate links. SOLUTIONS MANUALS Two solutions manuals have been written by Peter Bolgar, Haydn Lloyd, Aimee North, Vladimiras Oleinikovas, Stephanie Smith, and James Keeler. The Student’s Solutions Manual (ISBN 9780198807773) provides full solutions to the ‘a’ Exercises and to the oddnumbered Problems. The Instructor’s Solutions Manual provides full solutions to the ‘b’ Exercises and to the even-numbered Problems (available to download online for registered adopters of the book only). ABOUT THE AUTHORS Peter Atkins is a fellow of Lincoln College, Oxford, and was Professor of Physical Chemistry in the University of Oxford. He is the author of over seventy books for students and a general audience. His texts are market leaders around the globe. A frequent lecturer in the United States and throughout the world, he has held visiting professorships in France, Israel, Japan, China, Russia, and New Zealand. He was the founding chairman of the Committee on Chemistry Education of the International Union of Pure and Applied Chemistry and was a member of IUPAC’s Physical and Biophysical Chemistry Division. Photograph by Natasha Ellis-Knight. Julio de Paula is Professor of Chemistry at Lewis & Clark College. A native of Brazil, he received a B.A. degree in chemistry from Rutgers, The State University of New Jersey, and a Ph.D. in biophysical chemistry from Yale University. His research activities encompass the areas of molecular spectroscopy, photochemistry, and nanoscience. He has taught courses in general chemistry, physical chemistry, biophysical chemistry, inorganic chemistry, instrumental analysis, environmental chemistry, and writing. Among his professional honours are a Christian and Mary Lindback Award for Distinguished Teaching, a Henry Dreyfus Teacher-Scholar Award, and a Cottrell Scholar Award from the Research Corporation for Science Advancement. James Keeler is a Senior Lecturer in Chemistry at the University of Cambridge, and Walters Fellow in Chemistry at Selwyn College, Cambridge. He took his first degree at the University of Oxford and continued there for doctoral research in nuclear magnetic resonance spectroscopy. Dr Keeler is Director of Teaching for undergraduate chemistry, and teaches courses covering a range of topics in physical and theoretical chemistry. Photograph by Nathan Pitt, ©University of Cambridge. ACKNOWLEDGEMENTS A book as extensive as this could not have been written without significant input from many individuals. We would like to reiterate our thanks to the hundreds of people who contributed to the first ten editions. Many people gave their advice based on the tenth edition, and others, including students, reviewed the draft chapters for the eleventh edition as they emerged. We wish to express our gratitude to the following colleagues: Andrew J. Alexander, University of Edinburgh Stephen H. Ashworth, University of East Anglia Mark Berg, University of South Carolina Eric Bittner, University of Houston Melanie Britton, University of Birmingham Eleanor Campbell, University of Edinburgh Andrew P. Doherty, Queen’s University of Belfast Rob Evans, Aston University J.G.E. Gardeniers, University of Twente Ricardo Grau-Crespo, University of Reading Alex Grushow, Rider University Leonid Gurevich, Aalborg University Ronald Haines, University of New South Wales Patrick M. Hare, Northern Kentucky University John Henry, University of Wolverhampton Karl Jackson, Virginia Union University Carey Johnson, University of Kansas George Kaminski, Worcester Polytechnic Institute Scott Kirkby, East Tennessee State University Kathleen Knierim, University of Louisiana at Lafayette Jeffry Madura, University of Pittsburgh David H. Magers, Mississippi College Kristy Mardis, Chicago State University Paul Marshall, University of North Texas Laura R. McCunn, Marshall University Allan McKinley, University of Western Australia Joshua Melko, University of North Florida Yirong Mo, Western Michigan University Gareth Morris, University of Manchester Han J. Park, University of Tennessee at Chattanooga Rajeev Prabhakar, University of Miami Gavin Reid, University of Leeds Chad Risko, University of Kentucky Nessima Salhi, Uppsala University Daniel Savin, University of Florida Richard W. Schwenz, University of Northern Colorado Douglas Strout, Alabama State University Steven Tait, Indiana University Jim Terner, Virginia Commonwealth University Timothy Vaden, Rowan University Alfredo Vargas, University of Sussex Darren Walsh, University of Nottingham Collin Wick, Louisiana Tech University Shoujun Xu, University of Houston Renwu Zhang , California State University Wuzong Zhou, St Andrews University We would also like to thank Michael Clugston for proofreading the entire book, and Peter Bolgar, Haydn Lloyd, Aimee North, Vladimiras Oleinikovas, Stephanie Smith, and James Keeler for writing a brand new set of solutions. Last, but by no means least, we acknowledge our two commissioning editors, Jonathan Crowe of Oxford University Press and Jason Noe of OUP USA, and their teams for their assistance, advice, encouragement, and patience. BRIEF CONTENTS PROLOGUE 1 FOCUS 12 Magnetic resonance 487 FOCUS 1 The properties of gases 3 FOCUS 13 Statistical thermodynamics 531 FOCUS 2 The First Law 33 FOCUS 14 Molecular interactions 583 FOCUS 3 The Second and Third Laws 77 FOCUS 15 Solids 639 FOCUS 4 Physical transformations of pure substances FOCUS 16 Molecules in motion 689 FOCUS 17 Chemical kinetics 721 FOCUS 18 Reaction dynamics 779 FOCUS 19 Processes at solid surfaces 823 FOCUS 5 FOCUS 6 FOCUS 7 Simple mixtures Chemical equilibrium Quantum theory 119 141 203 235 Resource section FOCUS 8 Atomic structure and spectra 303 FOCUS 9 Molecular structure 341 FOCUS 10 Molecular symmetry 387 FOCUS 11 Molecular spectroscopy 417 1 2 3 4 Common integrals Units Data Character tables 862 864 865 895 Index899 FULL CONTENTS Conventionsxxv List of tables xxvi List of The chemist’s toolkitsxxviii List of material provided as A deeper lookxxix List of Impactsxxx PROLOGUE Energy, temperature, and chemistry FOCUS 1 The properties of gases TOPIC 1A The perfect gas 1 3 4 1A.1 Variables of state 4 (a) Pressure 4 (b) Temperature 5 1A.2 Equations of state 6 (a) The empirical basis 7 (b) Mixtures of gases 9 Checklist of concepts 10 Checklist of equations 10 TOPIC 1B The kinetic model 11 1B.1 The model 11 (a) Pressure and molecular speeds 12 (b) The Maxwell–Boltzmann distribution of speeds 13 (c) Mean values 15 1B.2 Collisions 17 (a) The collision frequency 17 (b) The mean free path 18 Checklist of concepts 18 Checklist of equations 18 TOPIC 1C Real gases 19 1C.1 Deviations from perfect behaviour 19 (a) The compression factor 20 (b) Virial coefficients (c) Critical constants 1C.2 The van der Waals equation 23 (a) Formulation of the equation 23 (b) The features of the equation 24 (c) The principle of corresponding states 34 (b) The molecular interpretation of heat and work 36 2A.2 The definition of internal energy 37 (a) Molecular interpretation of internal energy 37 (b) The formulation of the First Law 38 2A.3 Expansion work 38 (a) The general expression for work 39 (b) Expansion against constant pressure 39 (c) Reversible expansion 40 (d) Isothermal reversible expansion of a perfect gas 41 2A.4 Heat transactions 42 (a) Calorimetry 42 (b) Heat capacity 43 Checklist of concepts 45 Checklist of equations 45 TOPIC 2B Enthalpy 46 2B.1 The definition of enthalpy 46 (a) Enthalpy change and heat transfer 46 (b) Calorimetry 47 2B.2 The variation of enthalpy with temperature 48 (a) Heat capacity at constant pressure 48 (b) The relation between heat capacities 49 Checklist of concepts 50 Checklist of equations 50 TOPIC 2C Thermochemistry 2C.1 Standard enthalpy changes 51 51 (a) Enthalpies of physical change 51 (b) Enthalpies of chemical change 52 (c) Hess’s law 53 2C.2 Standard enthalpies of formation 54 2C.3 The temperature dependence of reaction enthalpies 55 2C.4 Experimental techniques 56 (a) Differential scanning calorimetry 56 (b) Isothermal titration calorimetry 57 20 Checklist of concepts 57 22 Checklist of equations 58 26 Checklist of concepts 27 Checklist of equations 27 FOCUS 2 The First Law (a) Operational definitions 33 TOPIC 2A Internal energy 34 2A.1 Work, heat, and energy 34 TOPIC 2D State functions and exact differentials 59 2D.1 Exact and inexact differentials 59 2D.2 Changes in internal energy 60 (a) General considerations 60 (b) Changes in internal energy at constant pressure 62 2D.3 Changes in enthalpy 63 2D.4 The Joule–Thomson effect 64 (a) The observation of the Joule–Thomson effect 64 (b) The molecular interpretation of the Joule–Thomson effect 65 Checklist of concepts 66 Checklist of equations 66 xvi Full Contents TOPIC 2E Adiabatic changes 67 2E.1 The change in temperature 67 2E.2 The change in pressure 68 Checklist of concepts 69 Checklist of equations 69 FOCUS 3 The Second and Third Laws 77 TOPIC 3A Entropy 78 3A.1 The Second Law 78 3A.2 The definition of entropy 80 (a) The thermodynamic definition of entropy 80 (b) The statistical definition of entropy 81 3A.3 The entropy as a state function 82 (a) The Carnot cycle 82 (b) The thermodynamic temperature 85 (c) The Clausius inequality 85 Checklist of concepts 86 Checklist of equations 87 TOPIC 3B Entropy changes accompanying specific processes 88 3E.2 Properties of the Gibbs energy 106 (a) General considerations 106 (b) The variation of the Gibbs energy with temperature 108 (c) The variation of the Gibbs energy with pressure 108 Checklist of concepts 110 Checklist of equations 110 FOCUS 4 Physical transformations of pure substances 119 TOPIC 4A Phase diagrams of pure substances 120 4A.1 The stabilities of phases 120 (a) The number of phases 120 (b) Phase transitions 120 (c) Thermodynamic criteria of phase stability 121 4A.2 Phase boundaries 122 (a) Characteristic properties related to phase transitions 122 (b) The phase rule 123 4A.3 Three representative phase diagrams 125 (a) Carbon dioxide 125 (b) Water 125 (c) Helium 126 Checklist of concepts 127 Checklist of equations 127 3B.1 Expansion 88 3B.2 Phase transitions 89 3B.3 Heating 90 3B.4 Composite processes 90 TOPIC 4B Thermodynamic aspects of phase transitions 128 Checklist of concepts 91 4B.1 The dependence of stability on the conditions 128 Checklist of equations 91 (a) The temperature dependence of phase stability 128 TOPIC 3C The measurement of entropy 92 (b) The response of melting to applied pressure 129 (c) The vapour pressure of a liquid subjected to pressure 130 131 3C.1 The calorimetric measurement of entropy 92 4B.2 The location of phase boundaries 3C.2 The Third Law 93 (a) The slopes of the phase boundaries 131 (a) The Nernst heat theorem 93 (b) The solid–liquid boundary 132 (b) Third-Law entropies 94 (c) The liquid–vapour boundary 132 (c) The temperature dependence of reaction entropy 95 (d) The solid–vapour boundary 134 Checklist of concepts 96 Checklist of concepts 134 Checklist of equations 96 Checklist of equations 134 TOPIC 3D Concentrating on the system 3D.1 The Helmholtz and Gibbs energies 97 97 (a) Criteria of spontaneity 97 (b) Some remarks on the Helmholtz energy 98 (c) Maximum work 98 (d) Some remarks on the Gibbs energy 99 FOCUS 5 Simple mixtures 141 TOPIC 5A The thermodynamic description of mixtures 143 5A.1 Partial molar quantities 143 (a) Partial molar volume 143 (b) Partial molar Gibbs energies 145 (e) Maximum non-expansion work 100 3D.2 Standard molar Gibbs energies 100 (c) The wider significance of the chemical potential 146 (a) Gibbs energies of formation 101 (d) The Gibbs–Duhem equation 146 (b) The Born equation 102 Checklist of concepts 103 5A.2 The thermodynamics of mixing 147 Checklist of equations 103 TOPIC 3E Combining the First and Second Laws 104 3E.1 Properties of the internal energy 104 (a) The Maxwell relations 104 (b) The variation of internal energy with volume 106 (a) The Gibbs energy of mixing of perfect gases 147 (b) Other thermodynamic mixing functions 149 5A.3 The chemical potentials of liquids 150 (a) Ideal solutions 150 (b) Ideal–dilute solutions 152 Checklist of concepts 153 Checklist of equations 154 Full Contents TOPIC 5B The properties of solutions 155 5B.1 Liquid mixtures 155 (a) Ideal solutions 155 (b) Excess functions and regular solutions 156 5B.2 Colligative properties 158 (a) The common features of colligative properties 158 (b) The elevation of boiling point 159 (c) The depression of freezing point 161 (d) Solubility 161 (e) Osmosis 162 Checklist of concepts 164 Checklist of equations 165 TOPIC 5C Phase diagrams of binary systems: liquids 166 5C.1 Vapour pressure diagrams 166 5C.2 Temperature–composition diagrams 168 (a) The construction of the diagrams 168 (b) The interpretation of the diagrams 169 5C.3 Distillation 170 (a) Simple and fractional distillation 170 (b) Azeotropes 171 (c) Immiscible liquids 172 5C.4 Liquid–liquid phase diagrams 172 (a) Phase separation 172 (b) Critical solution temperatures (c) The distillation of partially miscible liquids FOCUS 6 Chemical equilibrium 203 TOPIC 6A The equilibrium constant 204 6A.1 The Gibbs energy minimum 204 (a) The reaction Gibbs energy 204 (b) Exergonic and endergonic reactions 205 6A.2 The description of equilibrium 205 (a) Perfect gas equilibria 205 (b) The general case of a reaction 206 (c) The relation between equilibrium constants 209 (d) Molecular interpretation of the equilibrium constant 211 Checklist of equations 211 TOPIC 6B The response of equilibria to the conditions 212 6B.2 The response to temperature 213 (a) The van ’t Hoff equation 213 (b) The value of K at different temperatures 215 Checklist of concepts 216 Checklist of equations 216 TOPIC 6C Electrochemical cells 217 217 173 218 175 (a) Liquid junction potentials 218 Checklist of equations 176 TOPIC 5D Phase diagrams of binary systems: solids 177 177 (b) Notation 219 6C.3 The cell potential 219 (a) The Nernst equation 219 (b) Cells at equilibrium 221 6C.4 The determination of thermodynamic functions 5D.2 Reacting systems 178 Checklist of concepts 5D.3 Incongruent melting 179 Checklist of equations TOPIC 5E Phase diagrams of ternary systems 212 6B.1 The response to pressure 6C.2 Varieties of cells 176 Checklist of concepts 210 Checklist of concepts 6C.1 Half-reactions and electrodes Checklist of concepts 5D.1 Eutectics xvii 179 180 5E.1 Triangular phase diagrams 180 5E.2 Ternary systems 181 (a) Partially miscible liquids 181 (b) Ternary solids 182 Checklist of concepts 182 TOPIC 5F Activities TOPIC 6D Electrode potentials 221 223 223 224 6D.1 Standard potentials 224 (a) The measurement procedure 225 (b) Combining measured values 226 6D.2 Applications of standard potentials 226 (a) The electrochemical series 226 (b) The determination of activity coefficients 226 (c) The determination of equilibrium constants 227 183 Checklist of concepts 5F.1 The solvent activity 183 Checklist of equations 5F.2 The solute activity 183 (a) Ideal–dilute solutions 184 (b) Real solutes 184 (c) Activities in terms of molalities 185 5F.3 The activities of regular solutions 185 7A.1 Energy quantization 5F.4 The activities of ions 187 (a) Black-body radiation 237 (a) Mean activity coefficients 187 (b) Heat capacity 240 (b) The Debye–Hückel limiting law 187 (c) Atomic and molecular spectra 241 188 7A.2 Wave–particle duality 242 189 (a) The particle character of electromagnetic radiation 242 (b) The wave character of particles 244 (c) Extensions of the limiting law Checklist of concepts Checklist of equations 190 227 228 FOCUS 7 Quantum theory 235 TOPIC 7A The origins of quantum mechanics 237 237 xviii Full Contents Checklist of concepts 245 Checklist of concepts 290 Checklist of equations 245 Checklist of equations 290 TOPIC 7B Wavefunctions 246 7B.1 The Schrödinger equation 246 FOCUS 8 Atomic structure and spectra 303 7B.2 The Born interpretation 247 TOPIC 8A Hydrogenic atoms 304 (a) Normalization 248 8A.1 The structure of hydrogenic atoms 304 (b) Constraints on the wavefunction 249 (a) The separation of variables 304 (c) Quantization 250 (b) The radial solutions 305 Checklist of concepts 250 8A.2 Atomic orbitals and their energies 308 Checklist of equations 250 (a) The specification of orbitals 308 (b) The energy levels 308 (c) Ionization energies 309 TOPIC 7C Operators and observables 251 7C.1 Operators 251 (d) Shells and subshells 309 (a) Eigenvalue equations 251 (e) s Orbitals 310 (b) The construction of operators 252 (f) Radial distribution functions 311 (c) Hermitian operators 253 (g) p Orbitals 313 (d) Orthogonality 254 (h) d Orbitals 314 7C.2 Superpositions and expectation values 255 Checklist of concepts 314 7C.3 The uncertainty principle 257 Checklist of equations 315 7C.4 The postulates of quantum mechanics 259 Checklist of concepts Checklist of equations 260 TOPIC 8B Many-electron atoms 316 260 8B.1 The orbital approximation 316 8B.2 The Pauli exclusion principle 317 TOPIC 7D Translational motion 261 7D.1 Free motion in one dimension (a) Spin 317 261 (b) The Pauli principle 318 7D.2 Confined motion in one dimension 262 8B.3 The building-up principle 319 (a) The acceptable solutions 263 (a) Penetration and shielding 319 (b) The properties of the wavefunctions 264 (b) Hund’s rules 321 (c) The properties of the energy 265 (c) Atomic and ionic radii 323 266 (d) Ionization energies and electron affinities 324 266 8B.4 Self-consistent field orbitals 325 7D.3 Confined motion in two and more dimensions (a) Energy levels and wavefunctions (b) Degeneracy 267 Checklist of concepts 325 7D.4 Tunnelling 268 Checklist of equations 326 Checklist of concepts 271 Checklist of equations 272 TOPIC 8C Atomic spectra 327 8C.1 The spectra of hydrogenic atoms 327 TOPIC 7E Vibrational motion 273 8C.2 The spectra of many-electron atoms 328 7E.1 The harmonic oscillator 273 (a) Singlet and triplet terms 328 (a) The energy levels 274 (b) Spin–orbit coupling 329 275 (c) Term symbols 332 7E.2 Properties of the harmonic oscillator 277 (d) Hund’s rules 335 (a) Mean values 277 (b) The wavefunctions (e) Selection rules 278 Checklist of concepts Checklist of concepts 279 Checklist of equations Checklist of equations 280 (b) Tunnelling TOPIC 7F Rotational motion 7F.1 Rotation in two dimensions 281 281 (a) The solutions of the Schrödinger equation 283 (b) Quantization of angular momentum 284 7F.2 Rotation in three dimensions 285 FOCUS 9 Molecular structure PROLOGUE The Born–Oppenheimer approximation TOPIC 9A Valence-bond theory 335 336 336 341 343 344 9A.1 Diatomic molecules 344 346 (a) The wavefunctions and energy levels 285 9A.2 Resonance (b) Angular momentum 288 9A.3 Polyatomic molecules 346 (c) The vector model 288 (a) Promotion 347 (b) Hybridization 347 Full Contents xix Checklist of concepts 350 (e) The cubic groups 393 Checklist of equations 350 (f) The full rotation group 394 10A.3 Some immediate consequences of symmetry 394 (a) Polarity 394 TOPIC 9B Molecular orbital theory: the hydrogen molecule-ion 9B.1 Linear combinations of atomic orbitals 351 (b) Chirality 395 351 Checklist of concepts 395 (a) The construction of linear combinations 351 Checklist of operations and elements 396 (b) Bonding orbitals 353 (c) Antibonding orbitals 354 9B.2 Orbital notation TOPIC 10B Group theory 397 356 10B.1 The elements of group theory 397 Checklist of concepts 356 10B.2 Matrix representations 398 Checklist of equations 356 (a) Representatives of operations 398 (b) The representation of a group 399 (c) Irreducible representations 400 (d) Characters 401 401 TOPIC 9C Molecular orbital theory: homonuclear diatomic molecules 357 9C.1 Electron configurations 357 10B.3 Character tables (a) σ Orbitals and π orbitals 357 (a) The symmetry species of atomic orbitals 402 (b) The overlap integral 359 (b) The symmetry species of linear combinations of orbitals 403 (c) Period 2 diatomic molecules 360 (c) Character tables and degeneracy 404 9C.2 Photoelectron spectroscopy 362 Checklist of concepts 405 Checklist of concepts 363 Checklist of equations 405 Checklist of equations 364 TOPIC 9D Molecular orbital theory: heteronuclear diatomic molecules TOPIC 10C Applications of symmetry 10C.1 Vanishing integrals 406 406 365 (a) Integrals of the product of functions 407 9D.1 Polar bonds and electronegativity 365 (b) Decomposition of a representation 408 9D.2 The variation principle 366 10C.2 Applications to molecular orbital theory 409 (a) The procedure 367 (a) Orbital overlap 409 (b) The features of the solutions 369 (b) Symmetry-adapted linear combinations 409 Checklist of concepts 370 10C.3 Selection rules 411 Checklist of equations 370 TOPIC 9E Molecular orbital theory: polyatomic molecules 371 9E.1 The Hückel approximation 371 (a) An introduction to the method 371 (b) The matrix formulation of the method 372 9E.2 Applications 375 (a) π-Electron binding energy 375 (b) Aromatic stability 376 9E.3 Computational chemistry 377 (a) Semi-empirical and ab initio methods 378 (b) Density functional theory 379 (c) Graphical representations 379 Checklist of concepts 380 Checklist of equations 380 FOCUS 10 Molecular symmetry 387 TOPIC 10A Shape and symmetry 388 10A.1 Symmetry operations and symmetry elements 388 10A.2 The symmetry classification of molecules 390 (a) The groups C1, Ci, and Cs 392 (b) The groups Cn, Cnv, and Cnh 392 (c) The groups Dn, Dnh, and Dnd 393 (d) The groups Sn 393 Checklist of concepts 411 Checklist of equations 411 FOCUS 11 Molecular spectroscopy 417 TOPIC 11A General features of molecular spectroscopy 419 11A.1 The absorption and emission of radiation 420 (a) Stimulated and spontaneous radiative processes 420 (b) Selection rules and transition moments 421 (c) The Beer–Lambert law 421 11A.2 Spectral linewidths 423 (a) Doppler broadening 423 (b) Lifetime broadening 425 11A.3 Experimental techniques 425 (a) Sources of radiation 426 (b) Spectral analysis 426 (c) Detectors 428 (d) Examples of spectrometers 428 Checklist of concepts 429 Checklist of equations 429 TOPIC 11B Rotational spectroscopy 430 11B.1 Rotational energy levels 430 (a) Spherical rotors 432 xx Full Contents (b) Symmetric rotors 432 (c) Linear rotors 434 (d) Centrifugal distortion 434 11B.2 Microwave spectroscopy 435 (a) Selection rules 435 (b) The appearance of microwave spectra 436 TOPIC 11G Decay of excited states 470 11G.1 Fluorescence and phosphorescence 470 11G.2 Dissociation and predissociation 472 11G.3 Lasers 473 Checklist of concepts 474 11B.3 Rotational Raman spectroscopy 437 11B.4 Nuclear statistics and rotational states 439 FOCUS 12 Magnetic resonance 487 Checklist of concepts 441 Checklist of equations 441 TOPIC 12A General principles 488 TOPIC 11C Vibrational spectroscopy of diatomic molecules 442 11C.1 Vibrational motion 442 11C.2 Infrared spectroscopy 443 11C.3 Anharmonicity 444 (a) The convergence of energy levels 444 (b) The Birge–Sponer plot 445 11C.4 Vibration–rotation spectra 446 (a) Spectral branches 447 (b) Combination differences 448 11C.5 Vibrational Raman spectra 448 Checklist of concepts 449 Checklist of equations 450 TOPIC 11D Vibrational spectroscopy of polyatomic molecules 451 12A.1 Nuclear magnetic resonance 488 (a) The energies of nuclei in magnetic fields 488 (b) The NMR spectrometer 490 12A.2 Electron paramagnetic resonance 491 (a) The energies of electrons in magnetic fields 491 (b) The EPR spectrometer 492 Checklist of concepts 493 Checklist of equations 493 TOPIC 12B Features of NMR spectra 494 12B.1 The chemical shift 494 12B.2 The origin of shielding constants 496 (a) The local contribution 496 (b) Neighbouring group contributions 497 (c) The solvent contribution 498 12B.3 The fine structure 499 (a) The appearance of the spectrum 499 (b) The magnitudes of coupling constants 501 (c) The origin of spin–spin coupling 502 11D.1 Normal modes 451 11D.2 Infrared absorption spectra 452 (d) Equivalent nuclei 503 11D.3 Vibrational Raman spectra 453 (e) Strongly coupled nuclei 504 Checklist of concepts 454 12B.4 Exchange processes 505 Checklist of equations 454 TOPIC 11E Symmetry analysis of vibrational spectra 455 12B.5 Solid-state NMR 506 Checklist of concepts 507 Checklist of equations 508 TOPIC 12C Pulse techniques in NMR 509 11E.1 Classification of normal modes according to symmetry 455 11E.2 Symmetry of vibrational wavefunctions 457 12C.1 The magnetization vector 509 (a) Infrared activity of normal modes 457 (a) The effect of the radiofrequency field 510 (b) Raman activity of normal modes 458 (b) Time- and frequency-domain signals 511 (c) The symmetry basis of the exclusion rule 458 12C.2 Spin relaxation 513 458 (a) The mechanism of relaxation 513 (b) The measurement of T1 and T2 514 Checklist of concepts TOPIC 11F Electronic spectra 459 11F.1 Diatomic molecules 459 (a) Term symbols 459 (b) Selection rules 461 (c) Vibrational fine structure 462 (d) Rotational fine structure 465 11F.2 Polyatomic molecules 466 (a) d-Metal complexes 467 (b) π* ← π and π* ← n transitions 468 Checklist of concepts 469 Checklist of equations 469 12C.3 Spin decoupling 515 12C.4 The nuclear Overhauser effect 516 Checklist of concepts 518 Checklist of equations 518 TOPIC 12D Electron paramagnetic resonance 519 12D.1 The g-value 519 12D.2 Hyperfine structure 520 (a) The effects of nuclear spin 520 (b) The McConnell equation 521 (c) The origin of the hyperfine interaction 522 Full Contents xxi Checklist of concepts 523 (e) Residual entropies 565 Checklist of equations 523 Checklist of concepts 566 Checklist of equations 566 FOCUS 13 Statistical thermodynamics 531 TOPIC 13A The Boltzmann distribution 532 TOPIC 13F Derived functions 567 13F.1 The derivations 567 13A.1 Configurations and weights 532 13F.2 Equilibrium constants 570 (a) Instantaneous configurations 532 (a) The relation between K and the partition function 570 (b) The most probable distribution 533 (b) A dissociation equilibrium 570 (c) The values of the constants 535 (c) Contributions to the equilibrium constant 13A.2 The relative population of states 536 Checklist of concepts 573 Checklist of concepts 536 Checklist of equations 573 Checklist of equations 537 TOPIC 13B Molecular partition functions 538 13B.1 The significance of the partition function 538 571 FOCUS 14 Molecular interactions 583 TOPIC 14A The electric properties of molecules 585 13B.2 Contributions to the partition function 540 14A.1 Electric dipole moments (a) The translational contribution 540 14A.2 Polarizabilities 587 (b) The rotational contribution 542 14A.3 Polarization 588 (c) The vibrational contribution 546 (a) The frequency dependence of the polarization 588 (d) The electronic contribution 547 (b) Molar polarization 590 Checklist of concepts 548 Checklist of concepts 592 Checklist of equations 548 Checklist of equations 592 TOPIC 13C Molecular energies 549 TOPIC 14B Interactions between molecules 585 593 13C.1 The basic equations 549 14B.1 The interactions of dipoles 13C.2 Contributions of the fundamental modes of motion 550 (a) Charge–dipole interactions 593 (a) The translational contribution 550 (b) Dipole–dipole interactions 594 (b) The rotational contribution 550 (c) Dipole–induced dipole interactions 597 (c) The vibrational contribution 551 (d) Induced dipole–induced dipole interactions 597 (d) The electronic contribution 552 14B.2 Hydrogen bonding 598 (e) The spin contribution 552 14B.3 The total interaction 593 599 Checklist of concepts 553 Checklist of concepts 601 Checklist of equations 553 Checklist of equations 601 TOPIC 13D The canonical ensemble 554 13D.1 The concept of ensemble 554 (a) Dominating configurations 555 (b) Fluctuations from the most probable distribution 555 13D.2 The mean energy of a system 13D.3 Independent molecules revisited 13D.4 The variation of the energy with volume TOPIC 14C Liquids 14C.1 Molecular interactions in liquids 602 602 (a) The radial distribution function 602 (b) The calculation of g(r) 603 556 (c) The thermodynamic properties of liquids 604 556 14C.2 The liquid–vapour interface 605 557 (a) Surface tension 605 Checklist of concepts 558 (b) Curved surfaces 606 Checklist of equations 558 (c) Capillary action 606 14C.3 Surface films 608 TOPIC 13E The internal energy and the entropy 559 (a) Surface pressure 608 13E.1 The internal energy 559 (b) The thermodynamics of surface layers 609 (a) The calculation of internal energy 559 14C.4 Condensation 611 (b) Heat capacity 560 Checklist of concepts 612 13E.2 The entropy 561 Checklist of equations 612 (a) Entropy and the partition function 561 (b) The translational contribution 563 (c) The rotational contribution 563 14D.1 Average molar masses 613 (d) The vibrational contribution 564 14D.2 The different levels of structure 614 TOPIC 14D Macromolecules 613 xxii Full Contents 14D.3 Random coils 615 (a) Measures of size 615 (b) Constrained chains 618 (c) Partly rigid coils 618 14D.4 Mechanical properties 619 (a) Conformational entropy 619 (b) Elastomers 620 14D.5 Thermal properties 621 Checklist of concepts 622 Checklist of equations 622 TOPIC 14E Self-assembly 623 14E.1 Colloids 623 (a) Classification and preparation 623 (b) Structure and stability 624 (c) The electrical double layer 624 14E.2 Micelles and biological membranes 626 (a) The hydrophobic interaction 626 (b) Micelle formation 627 (c) Bilayers, vesicles, and membranes 628 Checklist of concepts 630 Checklist of equations 630 FOCUS 15 Solids TOPIC 15A Crystal structure 639 641 TOPIC 15D The mechanical properties of solids 666 Checklist of concepts 667 Checklist of equations 668 TOPIC 15E The electrical properties of solids 15E.1 Metallic conductors 669 669 15E.2 Insulators and semiconductors 670 15E.3 Superconductors 672 Checklist of concepts 673 Checklist of equations 673 TOPIC 15F The magnetic properties of solids 674 15F.1 Magnetic susceptibility 674 15F.2 Permanent and induced magnetic moments 675 15F.3 Magnetic properties of superconductors 676 Checklist of concepts 676 Checklist of equations 677 TOPIC 15G The optical properties of solids 678 15G.1 Excitons 678 15G.2 Metals and semiconductors 679 (a) Light absorption 679 (b) Light-emitting diodes and diode lasers 680 15G.3 Nonlinear optical phenomena Checklist of concepts 680 681 15A.1 Periodic crystal lattices 641 15A.2 The identification of lattice planes 643 FOCUS 16 Molecules in motion 689 (a) The Miller indices 643 (b) The separation of neighbouring planes 644 TOPIC 16A Transport properties of a perfect gas 690 Checklist of concepts 645 Checklist of equations 645 TOPIC 15B Diffraction techniques 646 15B.1 X-ray crystallography 646 (a) X-ray diffraction 646 (b) Bragg’s law 648 16A.1 The phenomenological equations 690 16A.2 The transport parameters 692 (a) The diffusion coefficient 693 (b) Thermal conductivity 694 (c) Viscosity 696 (d) Effusion (c) Scattering factors 649 Checklist of concepts (d) The electron density 649 Checklist of equations (e) The determination of structure 652 15B.2 Neutron and electron diffraction 654 Checklist of concepts 655 Checklist of equations 655 TOPIC 15C Bonding in solids 656 15C.1 Metals 656 (a) Close packing 656 (b) Electronic structure of metals 658 15C.2 Ionic solids 660 (a) Structure 660 (b) Energetics 661 15C.3 Covalent and molecular solids 663 Checklist of concepts 664 Checklist of equations 665 697 697 698 TOPIC 16B Motion in liquids 699 16B.1 Experimental results 699 (a) Liquid viscosity 699 (b) Electrolyte solutions 700 16B.2 The mobilities of ions 701 (a) The drift speed 701 (b) Mobility and conductivity 703 (c) The Einstein relations 704 Checklist of concepts 705 Checklist of equations 705 FOCUS 16C Diffusion 706 16C.1 The thermodynamic view 706 16C.2 The diffusion equation 708 (a) Simple diffusion 708 Full Contents xxiii (b) Diffusion with convection 710 (a) Stepwise polymerization 755 (c) Solutions of the diffusion equation 710 (b) Chain polymerization 756 16C.3 The statistical view 712 17F.3 Enzyme-catalysed reactions 758 Checklist of concepts 713 Checklist of concepts 761 Checklist of equations 714 Checklist of equations 761 FOCUS 17 Chemical kinetics 721 TOPIC 17A The rates of chemical reactions 723 17A.1 Monitoring the progress of a reaction TOPIC 17G Photochemistry 762 17G.1 Photochemical processes 762 17G.2 The primary quantum yield 763 723 17G.3 Mechanism of decay of excited singlet states 764 (a) General considerations 723 17G.4 Quenching 765 (b) Special techniques 724 17G.5 Resonance energy transfer 17A.2 The rates of reactions 725 Checklist of concepts 768 (a) The definition of rate 725 (b) Rate laws and rate constants 726 Checklist of equations 768 767 (c) Reaction order 727 (d) The determination of the rate law 728 FOCUS 18 Reaction dynamics 779 Checklist of concepts 729 Checklist of equations 730 TOPIC 18A Collision theory 780 18A.1 Reactive encounters 780 (a) Collision rates in gases 781 TOPIC 17B Integrated rate laws 731 17B.1 Zeroth-order reactions 731 17B.2 First-order reactions 731 17B.3 Second-order reactions 733 Checklist of concepts 736 Checklist of equations 736 TOPIC 17C Reactions approaching equilibrium 737 17C.1 First-order reactions approaching equilibrium 737 17C.2 Relaxation methods (b) The energy requirement 781 (c) The steric requirement 784 18A.2 The RRK model 785 Checklist of concepts 786 Checklist of equations 786 TOPIC 18B Diffusion-controlled reactions 18B.1 Reactions in solution 787 787 (a) Classes of reaction 787 738 (b) Diffusion and reaction 788 Checklist of concepts 740 18B.2 The material-balance equation 789 Checklist of equations 740 (a) The formulation of the equation 789 (b) Solutions of the equation 790 TOPIC 17D The Arrhenius equation 741 Checklist of concepts 790 17D.1 The temperature dependence of reaction rates 741 Checklist of equations 791 17D.2 The interpretation of the Arrhenius parameters 742 TOPIC 18C Transition-state theory 792 (a) A first look at the energy requirements of reactions 743 (b) The effect of a catalyst on the activation energy 744 18C.1 The Eyring equation Checklist of concepts 745 (a) The formulation of the equation 792 Checklist of equations 745 (b) The rate of decay of the activated complex 793 TOPIC 17E Reaction mechanisms 746 792 (c) The concentration of the activated complex 793 (d) The rate constant 794 17E.1 Elementary reactions 746 18C.2 Thermodynamic aspects 795 17E.2 Consecutive elementary reactions 747 (a) Activation parameters 795 17E.3 The steady-state approximation 748 (b) Reactions between ions 797 17E.4 The rate-determining step 749 18C.3 The kinetic isotope effect 798 17E.5 Pre-equilibria 750 Checklist of concepts 800 752 Checklist of equations 800 17E.6 Kinetic and thermodynamic control of reactions Checklist of concepts 752 Checklist of equations 752 TOPIC 17F Examples of reaction mechanisms 753 17F.1 Unimolecular reactions 753 17F.2 Polymerization kinetics 754 TOPIC 18D The dynamics of molecular collisions 18D.1 Molecular beams 801 801 (a) Techniques 801 (b) Experimental results 802 18D.2 Reactive collisions 804 (a) Probes of reactive collisions 804 xxiv Full Contents (b) State-to-state reaction dynamics 804 (d) The Temkin and Freundlich isotherms 837 18D.3 Potential energy surfaces 805 19B.2 The rates of adsorption and desorption 837 18D.4 Some results from experiments and calculations 806 (a) The precursor state 837 (a) The direction of attack and separation 807 (b) Adsorption and desorption at the molecular level 838 (b) Attractive and repulsive surfaces 808 (c) Mobility on surfaces 839 (c) Quantum mechanical scattering theory 808 Checklist of concepts 840 Checklist of concepts 809 Checklist of equations 840 Checklist of equations 809 TOPIC 18E Electron transfer in homogeneous systems TOPIC 19C Heterogeneous catalysis 19C.1 Mechanisms of heterogeneous catalysis 810 841 841 (a) Unimolecular reactions 841 18E.1 The rate law 810 (b) The Langmuir–Hinshelwood mechanism 842 18E.2 The role of electron tunnelling 811 (c) The Eley–Rideal mechanism 843 18E.3 The rate constant 812 19C.2 Catalytic activity at surfaces 813 Checklist of concepts Checklist of concepts 815 Checklist of equations Checklist of equations 815 18E.4 Experimental tests of the theory FOCUS 19 Processes at solid surfaces 823 TOPIC 19A An introduction to solid surfaces 824 19A.1 Surface growth 824 19A.2 Physisorption and chemisorption 825 19A.3 Experimental techniques 826 (a) Microscopy 827 (b) Ionization techniques 828 (c) Diffraction techniques 829 (d) Determination of the extent and rates of adsorption and desorption 830 Checklist of concepts 831 Checklist of equations 831 TOPIC 19B Adsorption and desorption 19B.1 Adsorption isotherms 832 832 (a) The Langmuir isotherm 832 (b) The isosteric enthalpy of adsorption 834 (c) The BET isotherm 835 TOPIC 19D Processes at electrodes 843 844 844 845 19D.1 The electrode–solution interface 845 19D.2 The current density at an electrode 846 (a) The Butler–Volmer equation 846 (b) Tafel plots 850 19D.3 Voltammetry 850 19D.4 Electrolysis 852 19D.5 Working galvanic cells 853 Checklist of concepts 854 Checklist of equations 854 Resource section 1 2 3 4 Common integrals Units Data Character tables 861 862 864 865 895 Index899 CONVENTIONS To avoid intermediate rounding errors, but to keep track of values in order to be aware of values and to spot numerical errors, we display intermediate results as n.nnn… and round the calculation only at the final step. Blue terms are used when we want to identify a term in an equation. An entire quotient, numerator/denominator, is coloured blue if the annotation refers to the entire term, not just to the numerator or denominator separately. LIST OF TABLES Table 1A.1 Pressure units Table 1B.1 ⦵ 4 Table 5F.1 Ionic strength and molality, I = kb/b The (molar) gas constant 14 Table 5F.2 Mean activity coefficients in water at 298 K 188 Table 1B.2 Collision cross-sections 17 Table 5F.3 Activities and standard states: a summary 189 Table 1C.1 Second virial coefficients, B/(cm3 mol−1) 21 Table 6C.1 Varieties of electrode 217 Table 1C.2 Critical constants of gases 23 Table 6D.1 Standard potentials at 298 K 224 Table 1C.3 van der Waals coefficients 23 Table 6D.2 The electrochemical series 227 Table 1C.4 Selected equations of state 25 Table 7E.1 The Hermite polynomials 275 Table 2A.1 Varieties of work 39 Table 7F.1 The spherical harmonics 286 Table 2B.1 Temperature variation of molar heat capacities, Cp,m/(J K−1 mol−1) = a + bT + c/T 2 Table 8A.1 49 Hydrogenic radial wavefunctions 306 Table 8B.1 Standard enthalpies of fusion and vaporization at the transition temperature Effective nuclear charge 320 52 Table 8B.2 Atomic radii of main-group elements, r/pm 323 Table 2C.2 Enthalpies of reaction and transition 52 Table 8B.3 Ionic radii, r/pm 324 Table 2C.3 Standard enthalpies of formation and combustion of organic compounds at 298 K Table 8B.4 First and second ionization energies 325 53 Table 8B.5 Electron affinities, E a/(kJ mol−1) 325 Standard enthalpies of formation of inorganic compounds at 298 K Table 9A.1 54 Some hybridization schemes 349 Table 9C.1 Standard enthalpies of formation of organic compounds at 298 K Overlap integrals between hydrogenic orbitals 359 54 Table 9C.2 Bond lengths 362 Table 9C.3 Bond dissociation energies 362 Table 9D.1 Pauling electronegativities 366 Table 2C.1 Table 2C.4 Table 2C.5 Table 2D.1 Table 2D.2 Table 3B.1 Expansion coefficients (α) and isothermal compressibilities (κT) at 298 K Inversion temperatures (TI), normal freezing (Tf ) and boiling (Tb) points, and Joule–Thomson coefficients (μ) at 1 atm and 298 K Standard entropies of phase transitions, ⦵ Δtrs S /(J K−1 mol−1), at the corresponding normal transition temperatures 62 63 89 188 Table 10A.1 The notations for point groups 390 Table 10B.1 The C2v character table 402 Table 10B.2 The C3v character table 402 Table 10B.3 The C4 character table 405 Table 11B.1 Moments of inertia 431 Table 11C.1 Properties of diatomic molecules 447 The standard enthalpies and entropies of vaporization of liquids at their boiling temperatures 89 Table 11F.1 Colour, frequency, and energy of light 459 Table 3C.1 Standard Third-Law entropies at 298 K 94 Table 11F.2 Table 3D.1 Standard Gibbs energies of formation at 298 K 101 Absorption characteristics of some groups and molecules 467 Table 3E.1 The Maxwell relations 105 473 Table 5A.1 Henry’s law constants for gases in water at 298 K Table 11G.1 Characteristics of laser radiation and their chemical applications 153 Freezing-point (Kf ) and boiling-point (Kb) constants Table 12A.1 Nuclear constitution and the nuclear spin quantum number 488 160 Table 12A.2 Nuclear spin properties 489 Table 3B.2 Table 5B.1 LIST OF TABLES xxvii Table 12D.1 Hyperfine coupling constants for atoms, a/mT 522 Table 17B.2 Kinetic data for second-order reactions 733 Table 13B.1 Rotational temperatures of diatomic molecules 544 Table 17B.3 Integrated rate laws 735 Table 13B.2 Symmetry numbers of molecules 545 Table 17D.1 Arrhenius parameters 741 Table 13B.3 Vibrational temperatures of diatomic molecules 547 Table 17G.1 Examples of photochemical processes 762 Table 14A.1 Dipole moments and polarizability volumes 585 Table 17G.2 Common photophysical processes 763 Table 14B.1 Interaction potential energies 597 Table 17G.3 Values of R0 for some donor–acceptor pairs 767 Table 14B.2 Lennard-Jones-(12,6) potential energy parameters Table 18A.1 Arrhenius parameters for gas-phase reactions 784 600 Table 14C.1 Surface tensions of liquids at 293 K 605 Table 18B.1 Arrhenius parameters for solvolysis reactions in solution 788 Table 14E.1 Micelle shape and the surfactant parameter 628 Table 19A.1 Maximum observed standard enthalpies of physisorption at 298 K 825 Table 15A.1 The seven crystal systems 642 Table 15C.1 The crystal structures of some elements 657 Table 19A.2 Standard enthalpies of chemisorption, ⦵ Δad H /(kJ mol−1), at 298 K 825 Table 15C.2 Ionic radii, r/pm 661 Table 19C.1 Chemisorption abilities 844 Table 15C.3 Madelung constants 662 Table 15C.4 Lattice enthalpies at 298 K, ΔHL/(kJ mol−1) 663 Table 19D.1 Exchange-current densities and transfer coefficients at 298 K 849 Table 15F.1 Magnetic susceptibilities at 298 K 675 Table 16A.1 Transport properties of gases at 1 atm 691 Table 16B.1 Viscosities of liquids at 298 K 699 Table 16B.2 Ionic mobilities in water at 298 K 702 Table 16B.3 Diffusion coefficients at 298 K, D/(10−9 m2 s −1) 704 Table 17B.1 Kinetic data for first-order reactions 732 Table A.1 Some common units 864 Table A.2 Common SI prefixes 864 Table A.3 The SI base units 864 Table A.4 A selection of derived units 864 Table 0.1 Physical properties of selected materials 866 Table 0.2 Masses and natural abundances of selected nuclides 867 LIST OF THE CHEMIST’S TOOLKITS Number Topic Title 1 1A Quantities and units 5 2 1A Properties of bulk matter 6 3 1B Momentum and force 11 4 1B Integration 14 5 1C Differentiation 22 6 2A Work and energy 35 7 2A The equipartition theorem 37 8 2A Electrical charge, current, power, and energy 43 9 2A Partial derivatives 44 10 3E Exact differentials 105 11 5A Measures of concentration 148 12 5B Series expansions 160 13 7A Electromagnetic radiation 237 14 7B Complex numbers 247 15 7C Integration by parts 254 16 7C Euler’s formula 256 17 7D Vectors 262 18 7E The classical harmonic oscillator 273 19 7F Cylindrical coordinates 281 20 7F Angular momentum 282 21 7F Spherical polar coordinates 286 22 8C The manipulation of vectors 330 23 9D Determinants 368 24 9E Matrices 373 25 9E Matrix methods for solving eigenvalue equations 375 26 11A Exponential and Gaussian functions 424 27 12B Dipolar magnetic fields 497 28 12C The Fourier transform 512 29 16B Electrostatics 702 30 17B Integration by the method of partial fractions 735 LIST OF MATERIAL PROVIDED AS A DEEPER LOOK Number Title 1 The Debye–Hückel theory 2 The fugacity 3 Separation of variables 4 The energy of the bonding molecular orbital of H2+ 5 Rotational selection rules 6 Vibrational selection rules 7 The van der Waals equation of state 8 The electric dipole–dipole interaction 9 The virial and the virial equation of state 10 Establishing the relation between bulk and molecular properties 11 The random walk 12 The RRK model 13 The BET isotherm LIST OF IMPAC TS Number Focus Title 1 1 . . .on environmental science: The gas laws and the weather 2 1 . . .on astrophysics: The Sun as a ball of perfect gas 3 2 . . .on technology: Thermochemical aspects of fuels and foods 4 3 . . .on engineering: Refrigeration 5 3 . . .on materials science: Crystal defects 6 4 . . .on technology: Supercritical ﬂuids 7 5 . . .on biology: Osmosis in physiology and biochemistry 8 5 . . .on materials science: Liquid crystals 9 6 . . .on biochemistry: Energy conversion in biological cells 10 6 . . .on chemical analysis: Species-selective electrodes 11 7 . . .on technology: Quantum computing 12 7 . . .on nanoscience: Quantum dots 13 8 . . .on astrophysics: The spectroscopy of stars 14 9 . . .on biochemistry: The reactivity of O2, N2, and NO 15 9 . . .on biochemistry: Computational studies of biomolecules 16 11 . . .on astrophysics: Rotational and vibrational spectroscopy of interstellar species 17 11 . . .on environmental science: Climate change 18 12 . . .on medicine: Magnetic resonance imaging 19 12 . . .on biochemistry and nanoscience: Spin probes 20 13 . . .on biochemistry: The helix–coil transition in polypeptides 21 14 . . .on biology: Biological macromolecules 22 14 . . .on medicine: Molecular recognition and drug design 23 15 . . .on biochemistry: Analysis of DNA by X-ray diffraction 24 15 . . .on nanoscience: Nanowires 25 16 . . .on biochemistry: Ion channels 26 17 . . .on biochemistry: Harvesting of light during plant photosynthesis 27 19 . . .on technology: Catalysis in the chemical industry 28 19 . . .on technology: Fuel cells PROLOGUE Energy, temperature, and chemistry 1–10 In this expression, k is Boltzmann’s constant (its value is listed inside the front cover), a universal constant (in the sense of having the same value for all forms of matter). Figure 2 shows the Boltzmann distribution for two temperatures: as the temperature increases higher energy states are populated at the expense of states lower in energy. According to the Boltzmann distribution, the temperature is the single parameter that governs the spread of populations over the available energy states, whatever their nature. Allowed energy states Population (a) Low temperature Electronic 100 Continuum 0.01 Vibration Ni ∝ e−εi/kT Figure 1 The relative energies of the allowed states of various kinds of atomic and molecular motion. Allowed energy states Energy Translation Rotation consists of large numbers of molecules, each of which is in one of its available energy states. The total number of molecules with a particular energy due to translation, rotation, vibration, and its electronic state is called the ‘population’ of that state. Most molecules are found in the lowest energy state, and higher energy states are occupied by progressively fewer molecules. The Boltzmann distribution gives the population, Ni, of any energy state in terms of the energy of the state, εi, and the absolute temperature, T: Energy Energy is a concept used throughout chemistry to discuss molecular structures, reactions, and many other processes. What follows is an informal first look at the important features of energy. Its precise definition and role will emerge throughout the course of this text. The transformation of energy from one form to another is described by the laws of thermodynamics. They are applicable to bulk matter, which consists of very large numbers of atoms and molecules. The ‘First Law’ of thermodynamics is a statement about the quantity of energy involved in a transformation; the ‘Second Law’ is a statement about the dispersal of that energy (in a sense that will be explained). To discuss the energy of individual atoms and molecules that make up samples of bulk matter it is necessary to use quantum mechanics. According to this theory, the energy associated with the motion of a particle is ‘quantized’, meaning that the energy is restricted to certain values, rather than being able to take on any value. Three different kinds of motion can occur: translation (motion through space), rotation (change of orientation), and vibration (the periodic stretching and bending of bonds). Figure 1 depicts the relative sizes and spacing of the energy states associated with these different kinds of motion of typical molecules and compares them with the typical energies of electrons in atoms and molecules. The allowed energies associated with translation are so close together in normal-sized containers that they form a continuum. In contrast, the separation between the allowed electronic energy states of atoms and molecules is very large. The link between the energies of individual molecules and the energy of bulk matter is provided by one of the most important concepts in chemistry, the Boltzmann distribution. Bulk matter Population (b) High temperature Figure 2 The relative populations of states at (a) low, (b) high temperature according to the Boltzmann distribution. 2 Prologue Energy, temperature, and chemistry The Boltzmann distribution, as well as providing insight into the significance of temperature, is central to understanding much of chemistry. That most molecules occupy states of low energy when the temperature is low accounts for the existence of compounds and the persistence of liquids and solids. That highly excited energy levels become accessible at high temperatures accounts for the possibility of reaction as one substance acquires the ability to change into another. Both features are explored in detail throughout the text. You should keep in mind the Boltzmann distribution (which is treated in greater depth later in the text) whenever considering the interpretation of the properties of bulk matter and the role of temperature. An understanding of the flow of energy and how it is distributed according to the Boltzmann distribution is the key to understanding thermodynamics, structure, and change throughout chemistry. FOCUS 1 The properties of gases A gas is a form of matter that fills whatever container it occupies. This Focus establishes the properties of gases that are used throughout the text. 1A The perfect gas This Topic is an account of an idealized version of a gas, a ‘perfect gas’, and shows how its equation of state may be assembled from the experimental observations summarized by Boyle’s law, Charles’s law, and Avogadro’s principle. 1A.1 Variables of state; 1A.2 Equations of state 1B The kinetic model A central feature of physical chemistry is its role in building models of molecular behaviour that seek to explain observed phenomena. A prime example of this procedure is the development of a molecular model of a perfect gas in terms of a collection of molecules (or atoms) in ceaseless, essentially random motion. As well as accounting for the gas laws, this model can be used to predict the average speed at which molecules move in a gas, and its dependence on temperature. In combination with the Boltzmann distribution (see the text’s Prologue), the model can also be used to predict the spread of molecular speeds and its dependence on molecular mass and temperature. 1B.1 The model; 1B.2 Collisions 1C Real gases The perfect gas is a starting point for the discussion of properties of all gases, and its properties are invoked throughout thermodynamics. However, actual gases, ‘real gases’, have properties that differ from those of perfect gases, and it is necessary to be able to interpret these deviations and build the effects of molecular attractions and repulsions into the model. The discussion of real gases is another example of how initially primitive models in physical chemistry are elaborated to take into account more detailed observations. 1C.1 Deviations from perfect behaviour; 1C.2 The van der Waals equation Web resources What is an application of this material? The perfect gas law and the kinetic theory can be applied to the study of phenomena confined to a reaction vessel or encompassing an entire planet or star. In Impact 1 the gas laws are used in the discussion of meteorological phenomena—the weather. Impact 2 examines how the kinetic model of gases has a surprising application: to the discussion of dense stellar media, such as the interior of the Sun. TOPIC 1A The perfect gas Table 1A.1 Pressure units* ➤ Why do you need to know this material? Equations related to perfect gases provide the basis for the development of many relations in thermodynamics. The perfect gas law is also a good first approximation for accounting for the properties of real gases. ➤ What is the key idea? The perfect gas law, which is based on a series of empirical observations, is a limiting law that is obeyed increasingly well as the pressure of a gas tends to zero. ➤ What do you need to know already? You need to know how to handle quantities and units in calculations, as reviewed in The chemist’s toolkit 1. You also need to be aware of the concepts of pressure, volume, amount of substance, and temperature, all reviewed in The chemist’s toolkit 2. The properties of gases were among the first to be established quantitatively (largely during the seventeenth and eighteenth centuries) when the technological requirements of travel in balloons stimulated their investigation. These properties set the stage for the development of the kinetic model of gases, as discussed in Topic 1B. Name Symbol Value pascal Pa 1 Pa = 1 N m−2, 1 kg m−1 s−2 bar bar 1 bar = 105 Pa atmosphere atm 1 atm = 101.325 kPa torr Torr 1 Torr = (101 325/760) Pa = 133.32… Pa millimetres of mercury mmHg 1 mmHg = 133.322… Pa pounds per square inch psi 1 psi = 6.894 757… kPa * Values in bold are exact. of pressure, the pascal (Pa, 1 Pa = 1 N m−2), is introduced in The chemist’s toolkit 1. Several other units are still widely used (Table 1A.1). A pressure of 1 bar is the standard pressure for reporting data; it is denoted p . If two gases are in separate containers that share a common movable wall (Fig. 1A.1), the gas that has the higher pressure will tend to compress (reduce the volume of) the gas that has lower pressure. The pressure of the high-pressure gas will fall as it expands and that of the low-pressure gas will rise as it is compressed. There will come a stage when the two pressures are equal and the wall has no further tendency to move. This condition of equality of pressure on either side of a movable wall is a state of mechanical equilibrium between the two gases. The pressure of a gas is therefore an indication of whether a container that contains the gas will be in mechanical equilibrium with another gas with which it shares a movable wall. ⦵ (a) 1A.1 Variables of state The physical state of a sample of a substance, its physical condition, is defined by its physical properties. Two samples of the same substance that have the same physical properties are in the same state. The variables needed to specify the state of a system are the amount of substance it contains, n, the volume it occupies, V, the pressure, p, and the temperature, T. (a) Pressure The origin of the force exerted by a gas is the incessant battering of the molecules on the walls of its container. The collisions are so numerous that they exert an effectively steady force, which is experienced as a steady pressure. The SI unit (b) (c) Movable High wall pressure Low pressure Equal pressures Equal pressures Low pressure High pressure Figure 1A.1 When a region of high pressure is separated from a region of low pressure by a movable wall, the wall will be pushed into one region or the other, as in (a) and (c). However, if the two pressures are identical, the wall will not move (b). The latter condition is one of mechanical equilibrium between the two regions. 1A The perfect gas The chemist’s toolkit 1 5 Quantities and units The result of a measurement is a physical quantity that is reported as a numerical multiple of a unit: physical quantity = numerical value × unit listed in Table A.2 in the Resource section. Examples of the use of these prefixes are: 1 nm = 10−9 m 1 ps = 10−12 s 1 µmol = 10−6 mol It follows that units may be treated like algebraic quantities and may be multiplied, divided, and cancelled. Thus, the expression (physical quantity)/unit is the numerical value (a dimensionless quantity) of the measurement in the specified units. For instance, the mass m of an object could be reported as m = 2.5 kg or m/kg = 2.5. In this instance the unit of mass is 1 kg, but it is common to refer to the unit simply as kg (and likewise for other units). See Table A.1 in the Resource section for a list of units. Although it is good practice to use only SI units, there will be occasions where accepted practice is so deeply rooted that physical quantities are expressed using other, non-SI units. By international convention, all physical quantities are represented by oblique (sloping) letters (for instance, m for mass); units are given in roman (upright) letters (for instance m for metre). Units may be modified by a prefix that denotes a factor of a power of 10. Among the most common SI prefixes are those Powers of units apply to the prefix as well as the unit they modify. For example, 1 cm3 = 1 (cm)3, and (10−2 m)3 = 10−6 m3. Note that 1 cm3 does not mean 1 c(m3). When carrying out numerical calculations, it is usually safest to write out the numerical value of an observable in scientific notation (as n.nnn × 10n). There are seven SI base units, which are listed in Table A.3 in the Resource section. All other physical quantities may be expressed as combinations of these base units. Molar concentration (more formally, but very rarely, amount of substance concentration) for example, which is an amount of substance divided by the volume it occupies, can be expressed using the derived units of mol dm−3 as a combination of the base units for amount of substance and length. A number of these derived combinations of units have special names and symbols. For example, force is reported in the derived unit newton, 1 N = 1 kg m s −2 (see Table A.4 in the Resource section). The pressure exerted by the atmosphere is measured with a barometer. The original version of a barometer (which was invented by Torricelli, a student of Galileo) was an inverted tube of mercury sealed at the upper end. When the column of mercury is in mechanical equilibrium with the atmosphere, the pressure at its base is equal to that exerted by the atmosphere. It follows that the height of the mercury column is proportional to the external pressure. The pressure of a sample of gas inside a container is measured by using a pressure gauge, which is a device with properties that respond to the pressure. For instance, a Bayard–Alpert pressure gauge is based on the ionization of the molecules present in the gas and the resulting current of ions is interpreted in terms of the pressure. In a capacitance manometer, the deflection of a diaphragm relative to a fixed electrode is monitored through its effect on the capacitance of the arrangement. Certain semiconductors also respond to pressure and are used as transducers in solid-state pressure gauges. to the Celsius scale of temperature. In this text, temperatures on the Celsius scale are denoted θ (theta) and expressed in degrees Celsius (°C). However, because different liquids expand to different extents, and do not always expand uniformly over a given range, thermometers constructed from different materials showed different numerical values of the temperature between their fixed points. The pressure of a gas, however, can be used to construct a perfect-gas temperature scale that is independent of the identity of the gas. The perfect-gas scale turns out to be identical to the thermodynamic temperature scale (Topic 3A), so the latter term is used from now on to avoid a proliferation of names. On the thermodynamic temperature scale, temperatures are denoted T and are normally reported in kelvins (K; not °K). Thermodynamic and Celsius temperatures are related by the exact expression (b) Temperature The concept of temperature is introduced in The chemist’s toolkit 2. In the early days of thermometry (and still in laboratory practice today), temperatures were related to the length of a column of liquid, and the difference in lengths shown when the thermometer was first in contact with melting ice and then with boiling water was divided into 100 steps called ‘degrees’, the lower point being labelled 0. This procedure led T/K = θ/°C + 273.15 Celsius scale [definition] (1A.1) This relation is the current definition of the Celsius scale in terms of the more fundamental Kelvin scale. It implies that a difference in temperature of 1 °C is equivalent to a difference of 1 K. Brief illustration 1A.1 To express 25.00 °C as a temperature in kelvins, eqn 1A.1 is used to write T/K = (25.00 °C)/°C + 273.15 = 25.00 + 273.15 = 298.15 6 1 The properties of gases The chemist’s toolkit 2 Properties of bulk matter The state of a bulk sample of matter is defined by specifying the values of various properties. Among them are: The mass, m, a measure of the quantity of matter present (unit: kilogram, kg). The volume, V, a measure of the quantity of space the sample occupies (unit: cubic metre, m3). The amount of substance, n, a measure of the number of specified entities (atoms, molecules, or formula units) present (unit: mole, mol). The amount of substance, n (colloquially, ‘the number of moles’), is a measure of the number of specified entities present in the sample. ‘Amount of substance’ is the official name of the quantity; it is commonly simplified to ‘chemical amount’ or simply ‘amount’. A mole is currently defined as the number of carbon atoms in exactly 12 g of carbon-12. (In 2011 the decision was taken to replace this definition, but the change has not yet, in 2018, been implemented.) The number of entities per mole is called Avogadro’s constant, NA; the currently accepted value is 6.022 × 1023 mol−1 (note that NA is a constant with units, not a pure number). The molar mass of a substance, M (units: formally kg mol−1 but commonly g mol−1) is the mass per mole of its atoms, its molecules, or its formula units. The amount of substance of specified entities in a sample can readily be calculated from its mass, by noting that n= m Amount of substance M A note on good practice Be careful to distinguish atomic or molecular mass (the mass of a single atom or molecule; unit: kg) from molar mass (the mass per mole of atoms or molecules; units: kg mol−1). Relative molecular masses of atoms and molecules, Mr = m/mu, where m is the mass of the atom or molecule and mu is the atomic mass constant (see inside front cover), are still widely called ‘atomic weights’ and ‘molecular weights’ even though they are dimensionless quantities and not weights (‘weight’ is the gravitational force exerted on an object). A sample of matter may be subjected to a pressure, p (unit: pascal, Pa; 1 Pa = 1 kg m−1 s−2), which is defined as the force, F, it is subjected to, divided by the area, A, to which that force is applied. Although the pascal is the SI unit of pressure, it is also common to express pressure in bar (1 bar = 105 Pa) or atmospheres (1 atm = 101 325 Pa exactly), both of which correspond to typical atmospheric pressure. Because many physical properties depend on the pressure acting on a sample, it is appropriate to select a certain value of the pressure to report their values. The standard pressure for report⦵ ing physical quantities is currently defined as p = 1 bar exactly. To specify the state of a sample fully it is also necessary to give its temperature, T. The temperature is formally a property that determines in which direction energy will flow as heat when two samples are placed in contact through thermally conducting walls: energy flows from the sample with the higher temperature to the sample with the lower temperature. The symbol T is used to denote the thermodynamic temperature which is an absolute scale with T = 0 as the lowest point. Temperatures above T = 0 are then most commonly expressed by using the Kelvin scale, in which the gradations of temperature are expressed in kelvins (K). The Kelvin scale is currently defined by setting the triple point of water (the temperature at which ice, liquid water, and water vapour are in mutual equilibrium) at exactly 273.16 K (as for certain other units, a decision has been taken to revise this definition, but it has not yet, in 2018, been implemented). The freezing point of water (the melting point of ice) at 1 atm is then found experimentally to lie 0.01 K below the triple point, so the freezing point of water is 273.15 K. Suppose a sample is divided into smaller samples. If a property of the original sample has a value that is equal to the sum of its values in all the smaller samples (as mass would), then it is said to be extensive. Mass and volume are extensive properties. If a property retains the same value as in the original sample for all the smaller samples (as temperature would), then it is said to be intensive. Temperature and pressure are intensive properties. Mass density, ρ = m/V, is also intensive because it would have the same value for all the smaller samples and the original sample. All molar properties, Xm = X/n, are intensive, whereas X and n are both extensive. p = 0, regardless of the size of the units, such as bar or pascal). However, it is appropriate to write 0 °C because the Celsius scale is not absolute. Note how the units (in this case, °C) are cancelled like numbers. This is the procedure called ‘quantity calculus’ in which a physical quantity (such as the temperature) is the product of a numerical value (25.00) and a unit (1 °C); see The chemist’s toolkit 1. Multiplication of both sides by K then gives T = 298.15 K. 1A.2 A note on good practice The zero temperature on the thermodynamic temperature scale is written T = 0, not T = 0 K. This scale is absolute, and the lowest temperature is 0 regardless of the size of the divisions on the scale (just as zero pressure is denoted Although in principle the state of a pure substance is specified by giving the values of n, V, p, and T, it has been established experimentally that it is sufficient to specify only three of these variables since doing so fixes the value of the fourth variable. Equations of state (1A.2) This equation states that if the values of n, T, and V are known for a particular substance, then the pressure has a fixed value. Each substance is described by its own equation of state, but the explicit form of the equation is known in only a few special cases. One very important example is the equation of state of a ‘perfect gas’, which has the form p = nRT/V, where R is a constant independent of the identity of the gas. The equation of state of a perfect gas was established by combining a series of empirical laws. (a) The empirical basis The following individual gas laws should be familiar: Boyle’s law: pV = constant, at constant n, T (1A.3a) Charles’s law: V = constant × T, at constant n, p (1A.3b) p = constant × T, at constant n, V (1A.3c) Avogadro’s principle: V = constant × n at constant p, T (1A.3d) Boyle’s and Charles’s laws are examples of a limiting law, a law that is strictly true only in a certain limit, in this case p → 0. For example, if it is found empirically that the volume of a substance fits an expression V = aT + bp + cp2, then in the limit of p → 0, V = aT. Many relations that are strictly true only at p = 0 are nevertheless reasonably reliable at normal pressures (p ≈ 1 bar) and are used throughout chemistry. Figure 1A.2 depicts the variation of the pressure of a sample of gas as the volume is changed. Each of the curves in the Pressure, p Increasing temperature, T 0 0 0 Volume, V Figure 1A.2 The pressure–volume dependence of a ﬁxed amount of perfect gas at different temperatures. Each curve is a hyperbola (pV = constant) and is called an isotherm. 7 Increasing temperature, T 1/Volume, 1/V Figure 1A.3 Straight lines are obtained when the pressure of a perfect gas is plotted against 1/V at constant temperature. These lines extrapolate to zero pressure at 1/V = 0. graph corresponds to a single temperature and hence is called an isotherm. According to Boyle’s law, the isotherms of gases are hyperbolas (a curve obtained by plotting y against x with xy = constant, or y = constant/x). An alternative depiction, a plot of pressure against 1/volume, is shown in Fig. 1A.3. The linear variation of volume with temperature summarized by Charles’s law is illustrated in Fig. 1A.4. The lines in this illustration are examples of isobars, or lines showing the variation of properties at constant pressure. Figure 1A.5 illustrates the linear variation of pressure with temperature. The lines in this diagram are isochores, or lines showing the variation of properties at constant volume. A note on good practice To test the validity of a relation between two quantities, it is best to plot them in such a way that they should give a straight line, because deviations from a straight line are much easier to detect than deviations from a curve. The development of expressions that, when plotted, give a straight line is a very important and common procedure in physical chemistry. 0 0 0 Extrapolation General form of an equation of state Extrapolation p = f(T,V,n) Volume, V That is, it is an experimental fact that each substance is described by an equation of state, an equation that interrelates these four variables. The general form of an equation of state is Pressure, p 1A The perfect gas Decreasing pressure, p Temperature, T Figure 1A.4 The variation of the volume of a ﬁxed amount of a perfect gas with the temperature at constant pressure. Note that in each case the isobars extrapolate to zero volume at T = 0, corresponding to θ = −273.15 °C. The properties of gases 0 0 Surface of possible states Pressure, p Decreasing volume, V Extrapolation Pressure, p 8 1 Volume, V Temperature, T Figure 1A.5 The pressure of a perfect gas also varies linearly with the temperature at constant volume, and extrapolates to zero at T = 0 (−273.15 °C). pe m Te Figure 1A.6 A region of the p,V,T surface of a ﬁxed amount of perfect gas. The points forming the surface represent the only states of the gas that can exist. The empirical observations summarized by eqn 1A.3 can be combined into a single expression: Isotherm pV = constant × nT Perfect gas law Pressure, p Isobar This expression is consistent with Boyle’s law (pV = constant) when n and T are constant, with both forms of Charles’s law (p ∝ T, V ∝ T) when n and either V or p are held constant, and with Avogadro’s principle (V ∝ n) when p and T are constant. The constant of proportionality, which is found experimentally to be the same for all gases, is denoted R and called the (molar) gas constant. The resulting expression pV = nRT ,T re tu ra V∝T p∝T A note on good practice Despite ‘ideal gas’ being the more common term, ‘perfect gas’ is preferable. As explained in Topic 5B, in an ‘ideal mixture’ of A and B, the AA, BB, and AB interactions are all the same but not necessarily zero. In a perfect gas, not only are the interactions all the same, they are also zero. The surface in Fig. 1A.6 is a plot of the pressure of a fixed amount of perfect gas against its volume and thermodynamic temperature as given by eqn 1A.4. The surface depicts the only possible states of a perfect gas: the gas cannot exist in states that do not correspond to points on the surface. The graphs in Figs. 1A.2 and 1A.4 correspond to the sections through the surface (Fig. 1A.7). ,T re Volume, V (1A.4) is the perfect gas law (or perfect gas equation of state). It is the approximate equation of state of any gas, and becomes increasingly exact as the pressure of the gas approaches zero. A gas that obeys eqn 1A.4 exactly under all conditions is called a perfect gas (or ideal gas). A real gas, an actual gas, behaves more like a perfect gas the lower the pressure, and is described exactly by eqn 1A.4 in the limit of p → 0. The gas constant R can be determined by evaluating R = pV/nT for a gas in the limit of zero pressure (to guarantee that it is behaving perfectly). pV = constant Isochore tu ra pe m Te Figure 1A.7 Sections through the surface shown in Fig. 1A.6 at constant temperature give the isotherms shown in Fig. 1A.2. Sections at constant pressure give the isobars shown in Fig. 1A.4. Sections at constant volume give the isochores shown in Fig. 1A.5. Example 1A.1 Using the perfect gas law In an industrial process, nitrogen gas is introduced into a vessel of constant volume at a pressure of 100 atm and a temperature of 300 K. The gas is then heated to 500 K. What pressure would the gas then exert, assuming that it behaved as a perfect gas? Collect your thoughts The pressure is expected to be greater on account of the increase in temperature. The perfect gas law in the form pV/nT = R implies that if the conditions are changed from one set of values to another, then because pV/nT is equal to a constant, the two sets of values are related by the ‘combined gas law’ p1V1 p2V2 Combined gas law (1A.5) = n1T1 n2T2 1A The perfect gas This expression is easily rearranged to give the unknown quantity (in this case p2) in terms of the known. The known and unknown data are summarized as follows: n p V T Initial Same 100 atm Same 300