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Atkins' Physical chemistry
Atkins' Physical chemistry
Peter Atkins, Julio de Paula
With its modern emphasis on the molecular view of physical chemistry, its wealth of contemporary applications, vivid fullcolor presentation, and dynamic new media tools, the thoroughly revised new edition is again the most modern, most effective fulllength textbook available for the physical chemistry classroom.
Available in Split Volumes
For maximum flexibility in your physical chemistry course, this text is now offered as a traditional text or in two volumes.
Volume 1: Thermodynamics and Kinetics; ISBN 1429231270
Volume 2: Quantum Chemistry, Spectroscopy, and Statistical Thermodynamics; ISBN 1429231262
Available in Split Volumes
For maximum flexibility in your physical chemistry course, this text is now offered as a traditional text or in two volumes.
Volume 1: Thermodynamics and Kinetics; ISBN 1429231270
Volume 2: Quantum Chemistry, Spectroscopy, and Statistical Thermodynamics; ISBN 1429231262
السنة:
2010
الطبعة:
9ed.
الناشر:
W. H. Freeman
اللغة:
english
الصفحات:
1010
ISBN 10:
1429218126
ISBN 13:
9781429218122
File:
PDF, 31.66 MB
تحميل (pdf, 31.66 MB)
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This page intentionally left blank General data and fundamental constants Quantity Symbol Value Power of ten Units Speed of light c 2.997 925 58* 108 m s−1 Elementary charge e 1.602 176 10−19 C Faraday’s constant F = NAe 9.648 53 104 C mol−1 10 −23 J K−1 Boltzmann’s constant k 1.380 65 Gas constant R = NAk 8.314 47 8.314 47 8.205 74 6.236 37 10 10−2 10 J K−1 mol−1 dm3 bar K −1 mol−1 dm3 atm K −1 mol−1 dm3 Torr K −1 mol−1 −2 Planck’s constant h $ = h/2π 6.626 08 1.054 57 10−34 10−34 Js Js Avogadro’s constant NA 6.022 14 10 23 mol−1 Atomic mass constant mu 1.660 54 10−27 kg Mass electron proton neutron me mp mn 9.109 38 1.672 62 1.674 93 10−31 10−27 10−27 kg kg kg ε 0 = 1/c 2μ 0 4πe0 μ0 8.854 19 10−12 J−1 C2 m−1 1.112 65 −10 10 J−1 C2 m−1 4π 10−7 J s2 C−2 m−1 (= T 2 J −1 m3) μ B = e$/2me μ N = e$/2mp ge a0 = 4πε0$2/mee 2 α = μ 0e 2c/2h α −1 c2 = hc/k σ = 2π5k 4/15h3c 2 R = mee 4/8h3cε 02 g G 9.274 01 5.050 78 2.002 32 10−24 10−27 J T −1 J T −1 5.291 77 10−11 m Vacuum permittivity Vacuum permeability Magneton Bohr nuclear g value Bohr radius Finestructure constant Second radiation constant Stefan–Boltzmann constant Rydberg constant Standard acceleration of free fall Gravitational constant −3 7.297 35 1.370 36 10 10 2 1.438 78 10−2 mK 5.670 51 10−8 W m−2 K −4 1.097 37 5 10 m s−2 9.806 65* 10−11 6.673 *Exact value The Greek alphabet Α, α Β, β Γ, γ Δ, δ Ε, ε Ζ, ζ alpha beta gamma delta epsilon zeta Η, η Θ, θ Ι, ι Κ, κ Λ, λ Μ, μ eta theta iota kappa lambda mu Ν, ν Ξ, ξ Π, π Ρ, ρ Σ, σ Τ, τ nu xi pi rho sigma tau Υ, υ Φ, φ Χ, χ Ψ, ψ Ω, ω cm−1 upsilon phi chi psi omega N m2 kg−2 This page intentionally left blank PHYSICAL CHEMISTRY This page intentionally left blank PHYSICAL CHEMISTRY Ninth Edition Peter Atkins Fellow of Lincoln College, University of Oxford, Oxford, UK Julio de Paula Professor of Chemistry, Lewis and Clark College, Portland, Oregon, USA W. H. Freeman and Company New York Physical Chemistry, Ninth Edition © 2010 by Peter Atkins and Julio de Paula All rights reserved ISBN: 1429218126 ISBN13: 9781429218122 Published in Great Britain by Oxford University Press This edition has been authorized by Oxford University Press for sale in the United States and Canada only and not for export therefrom. First printing. W. H. Freeman and Company 41 Madison Avenue New York, NY 10010 www.whfreeman.com Preface We have followed our usual tradition in that this new edition of the text is yet another thorough update of the content and its presentation. Our goal is to keep the book ﬂexible to use, accessible to students, broad in scope, and authoritative, without adding bulk. However, it should always be borne in mind that much of the bulk arises from the numerous pedagogical features that we include (such as Worked examples, Checklists of key equations, and the Resource section), not necessarily from density of information. The text is still divided into three parts, but material has been moved between chapters and the chapters themselves have been reorganized. We continue to respond to the cautious shift in emphasis away from classical thermodynamics by combining several chapters in Part 1 (Equilibrium), bearing in mind that some of the material will already have been covered in earlier courses. For example, material on phase diagrams no longer has its own chapter but is now distributed between Chapters 4 (Physical transformation of pure substances) and 5 (Simple mixtures). New Impact sections highlight the application of principles of thermodynamics to materials science, an area of growing interest to chemists. In Part 2 (Structure) the chapters have been updated with a discussion of contemporary techniques of materials science—including nanoscience—and spectroscopy. We have also paid more attention to computational chemistry, and have revised the coverage of this topic in Chapter 10. Part 3 has lost chapters dedicated to kinetics of complex reactions and surface processes, but not the material, which we regard as highly important in a contemporary context. To make the material more readily accessible within the context of courses, descriptions of polymerization, photochemistry, and enzyme and surfacecatalysed reactions are now part of Chapters 21 (The rates of chemical reactions) and 22 (Reaction dynamics)—already familiar to readers of the text—and a new chapter, Chapter 23, on Catalysis. We have discarded the Appendices of earlier editions. Material on mathematics covered in the appendices is now dispersed through the text in the form of Mathematical background sections, which review and expand knowledge of mathematical techniques where they are needed in the text. The review of introductory chemistry and physics, done in earlier editions in appendices, will now be found in a new Fundamentals chapter that opens the text, and particular points are developed as Brief comments or as part of Further information sections throughout the text. By liberating these topics from their appendices and relaxing the style of presentation we believe they are more likely to be used and read. The vigorous discussion in the physical chemistry community about the choice of a ‘quantum ﬁrst’ or a ‘thermodynamics ﬁrst’ approach continues. In response we have paid particular attention to making the organization ﬂexible. The strategic aim of this revision is to make it possible to work through the text in a variety of orders and at the end of this Preface we once again include two suggested paths through the text. For those who require a more thoroughgoing ‘quantum ﬁrst’ approach we draw attention to our Quanta, matter, and change (with Ron Friedman) which covers similar material to this text in a similar style but, because of the different approach, adopts a different philosophy. The concern expressed in previous editions about the level of mathematical ability has not evaporated, of course, and we have developed further our strategies for viii PREFACE showing the absolute centrality of mathematics to physical chemistry and to make it accessible. In addition to associating Mathematical background sections with appropriate chapters, we continue to give more help with the development of equations, motivate them, justify them, and comment on the steps. We have kept in mind the struggling student, and have tried to provide help at every turn. We are, of course, alert to the developments in electronic resources and have made a special effort in this edition to encourage the use of the resources on our website (at www.whfreeman.com/pchem). In particular, we think it important to encourage students to use the Living graphs on the website (and their considerable extension in the electronic book and Explorations CD). To do so, wherever we call out a Living graph (by an icon attached to a graph in the text), we include an interActivity in the ﬁgure legend, suggesting how to explore the consequences of changing parameters. Many other revisions have been designed to make the text more efﬁcient and helpful and the subject more enjoyable. For instance, we have redrawn nearly every one of the 1000 pieces of art in a consistent style. The Checklists of key equations at the end of each chapter are a useful distillation of the most important equations from the large number that necessarily appear in the exposition. Another innovation is the collection of Road maps in the Resource section, which suggest how to select an appropriate expression and trace it back to its roots. Overall, we have taken this opportunity to refresh the text thoroughly, to integrate applications, to encourage the use of electronic resources, and to make the text even more ﬂexible and uptodate. Oxford Portland P.W.A. J.de P. PREFACE Traditional approach Equilibrium thermodynamics Chapters 1–6 Chemical kinetics Chapters 20–22 Quantum theory and spectroscopy Chapters 7–10, 12–14 Special topics Statistical thermodynamics Chapters 11, 17–19, 23, and Fundamentals Chapters 15 and 16 Molecular approach Quantum theory and spectroscopy Chapters 7–10, 12–14 Statistical thermodynamics Chapters 15 and 16 Chemical kinetics Equilibrium thermodynamics Chapters 20–22 Chapters 1–6 Special topics Chapters 11, 17–19, 23, and Fundamentals This text is available as a customizable ebook. This text can also be purchased in two volumes. For more information on these options please see pages xv and xvi. ix About the book There are numerous features in this edition that are designed to make learning physical chemistry more effective and more enjoyable. One of the problems that make the subject daunting is the sheer amount of information: we have introduced several devices for organizing the material: see Organizing the information. We appreciate that mathematics is often troublesome, and therefore have taken care to give help with this enormously important aspect of physical chemistry: see Mathematics support. Problem solving—especially, ‘where do I start?’—is often a challenge, and we have done our best to help overcome this ﬁrst hurdle: see Problem solving. Finally, the web is an extraordinary resource, but it is necessary to know where to start, or where to go for a particular piece of information; we have tried to indicate the right direction: see About the Book Companion Site. The following paragraphs explain the features in more detail. Organizing the information Key points Justiﬁcations The Key points act as a summary of the main takehome message(s) of the section that follows. They alert you to the principal ideas being introduced. On ﬁrst reading it might be sufﬁcient simply to appreciate the ‘bottom line’ rather than work through detailed development of a mathematical expression. However, mathematical development is an intrinsic part of physical chemistry, and to achieve full understanding it is important to see how a particular expression is obtained. The Justiﬁcations let you adjust the level of detail that you require to your current needs, and make it easier to review material. 1.1 The states of gases Key points Each substance is described by an equation of state. (a) Pressure, force divided by area, provides a criterion of mechanical equilibrium for systems free to change their volume. (b) Pressure is measured with a barometer. (c) Through the Zeroth Law of thermodynamics, temperature provides a criterion of thermal equilibrium. The physical state of a sample of a substance, its physical condition, is deﬁned by its physical properties. Two samples of a substance that have the same physical properh h f f l ﬁ db These relations are called the Margules equations. Justiﬁcation 5.5 The Margules equations The Gibbs energy of mixing to form a nonideal solution is Δ mixG = nRT{xA ln aA + x B ln aB} Equation and concept tags The most signiﬁcant equations and concepts—which we urge you to make a particular effort to remember—are ﬂagged with an annotation, as shown here. , , p mental 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 p = f (T,V,n) General form of an equation of state (1.1) This relation follows from the derivation of eqn 5.16 with activities in place of mole fractions. If each activity is replaced by γ x, this expression becomes Δ mixG = nRT{xA ln xA + x B ln x B + xAln γA + x B ln γ B} Now we introduce the two expressions in eqn 5.64, and use xA + x B = 1, which gives Δ mixG = nRT{xA ln xA + x B ln xB + ξ xAx B2 + ξ x B x A2 } = nRT{xA ln xA + x B ln x B + ξ xAx B(xA + x B)} = nRT{xA ln xA + x B ln x B + ξ xAx B} as required by eqn 5.29. Note, moreover, that the activity coefﬁcients behave correctly for dilute solutions: γA → 1 as x B → 0 and γ B → 1 as xA → 0. At this point we can use the Margules equations to write the activity of A as 2 2 xi ABOUT THE BOOK Checklists of key equations Notes on good practice We have summarized the most important equations introduced in each chapter as a checklist. Where appropriate, we describe the conditions under which an equation applies. Science is a precise activity and its language should be used accurately. We have used this feature to help encourage the use of the language and procedures of science in conformity to international practice (as speciﬁed by IUPAC, the International Union of Pure and Applied Chemistry) and to help avoid common mistakes. Checklist of key equations Property Equation Comment Chemical potential Fundamental equation of chemica thermodynamics μJ = (∂G/∂nJ)p,T,n′ dG = Vdp − SdT + μAdnA + μBdnB + · · · G = nA μA + nB μB p ∑ n dμ = 0 Gibbs–Duhem equation J J Raoult’s law Henry’s law van’t Hoff equation Activity of a solvent Chemical potential Conversion to biological standard state Mean activity coefﬁcient μ = μ 7 + RT ln(p/p 7) ΔmixG = nRT(xA ln xA + x B ln x B) Δ mix S = −nR(xA ln xA + x B ln x B) Δ mix H = 0 pA = xA p*A pB = xB KB Π = [B]RT aA = pA /p*A μ J = μ J7 + RT ln aJ μ⊕(H+) = μ 7(H+) − 7RT ln 10 γ ± = (γ +pγ q−)1/(p+q) Ionic strength I = 12 Debye–Hückel limiting law Margules equation Lever rule log γ ± = −z+ z−  AI 1/2 ln γ J = ξ x J2 nα lα = nβ lβ Chemical potential of a gas Thermodynamic properties of mixing ∑ z (b /b ) 2 i i ( ) Answer The number of photons is J 7 i N= Perfect gas Perfect gases and ideal solutions E PΔt λPΔt = = hν h(c/λ) hc Substitution of the data gives True for ideal solutions; limiting law as xA → 1 True for ideal–dilute solutions; limiting law as xB → 0 Valid as [B] → 0 aA → xA as xA → 1 General form for a species J Deﬁnition Valid as I → 0 Model regular solution A note on good practice To avoid rounding and other numerical errors, it is best to carry out algebraic calculations ﬁrst, and to substitute numerical values into a single, ﬁnal formula. Moreover, an analytical result may be used for other data without having to repeat the entire calculation. N= (5.60 × 10−7 m) × (100 J s−1) × (1.0 s) = 2.8 × 1020 (6.626 × 10−34 J s) × (2.998 × 108 m s−1) Note that it would take the lamp nearly 40 min to produce 1 mol of these photons. Selftest 7.1 How many photons does a monochromatic (single frequency) infrared rangeﬁnder of power 1 mW and wavelength 1000 nm emit in 0.1 s? [5 × 1014] interActivities Road maps In many cases it is helpful to see the relations between equations. The suite of ‘Road maps’ summarizing these relations are found in the Resource section at the end of the text. Part 1 Road maps You will ﬁnd that many of the graphs in the text have an interActivity attached: this is a suggestion about how you can explore the consequences of changing various parameters or of carrying out a more elaborate investigation related to the material in the illustration. In many cases, the activities can be completed by using the online resources of the book’s website. Gas laws (Chapter 1) Compression factor Constant n, T Z = pVm /RT Constant n, p Yes pV = nRT p ∝ 1/V V∝T Constant n, V Perfect? Gas No Boyle’s law efore it is switched on, the t 20°C (293 K). When it is 000 K. The energy density tes nearly white light. • Charles’s law p∝T Vm = RT/p Molar volume Vc = 3b pVm = RT{1 + B /Vm + C/V 2m +...} Virial equation p = RT/(Vm – b) – a/V 2m van der Waals’ equation pc = a/27b 2 Zc = 3/8 Tc = 8a/27Rb Critical constants The First Law (Chapter 2) Impact sections Where appropriate, we have separated the principles from their applications: the principles are constant and straightforward; the applications come and go as the subject progresses. The Impact sections show how the principles developed in the chapter are currently being applied in a variety of modern contexts. IMPACT ON NANOSCIENCE I8.1 Quantum dots Nanoscience is the study of atomic and molecular assemblies with dimensions ranging from 1 nm to about 100 nm and nanotechnology is concerned with the incorporation of such assemblies into devices. The future economic impact of nanotechnology could be very signiﬁcant. For example, increased demand for very small digital electronic devices has driven the design of ever smaller and more powerful microprocessors. However, there is an upper limit on the density of electronic circuits that can be incorporated into siliconbased chips with current fabrication technologies. As the ability to process data increases with the number of components in a chip, it follows that soon chips and the devices that use them will have to become bigger if processing hile Rayleigh’s was not. The excites the oscillators of the l the oscillators of the ﬁeld the highest frequencies are s results in the ultraviolet oscillators are excited only o large for the walls to suphe latter remain unexcited. from the high frequency e energy available. eLouis Dulong and Alexis)V (Section 2.4), of a numwhat slender experimental ll monatomic solids are the ssical physics in much the diation. If classical physics fer that the mean energy of kT for each direction of disthe average energy of each tribution of this motion to ρ /{8π(kT)5/(hc)4} Gas laws (Chapter 1) 0 0.5 1 λkT/hc λ 1.5 2 The Planck distribution (eqn 7.8) accounts very well for the experimentally determined distribution of blackbody radiation. Planck’s quantization hypothesis essentially quenches the contributions of high frequency, short wavelength oscillators. The distribution coincides with the Rayleigh–Jeans distribution at long wavelengths. Fig. 7.7 interActivity Plot the Planck distribution at several temperatures and conﬁrm that eqn 7.8 predicts the behaviour summarized by Fig. 7.3. xii ABOUT THE BOOK Further information Mathematics support In some cases, we have judged that a derivation is too long, too detailed, or too different in level for it to be included in the text. In these cases, the derivations will be found less obtrusively at the end of the chapter. A brief comment Further information s in magnetic ﬁelds Further information 7.1 Classical mechanics pz Classical mechanics describes the behaviour of objects in terms of two equations. One expresses the fact that the total energy is constant in the absence of external forces; the other expresses the response of particles to the forces acting on them. c ﬁelds, which remove the degeneracy of the quantized resented on the vector model as vectors precessing at p (a) The trajectory in terms of the energy The velocity, V, of a particle is the rate of change of its position: V= dr Deﬁnition of velocity dt Deﬁnition of linear momentum moment m in a magnetic ﬁeld ; is equal to the py px (7.44) The velocity is a vector, with both direction and magnitude. (Vectors are discussed in Mathematical background 5.) The magnitude of the velocity is the speed, v. The linear momentum, p, of a particle of mass m is related to its velocity, V, by p = mV (7.45) Like the velocity vector, the linear momentum vector points in the direction of travel of the particle (Fig. 7.31). In terms of the linear The linear momentum of a particle is a vector property and points in the direction of motion. momentum, the total energy—the sum of the kinetic and potential energy—of a particle is E = Ek + V(x) = p2 2m + V(x) (7.46) Long tables of data are helpful for assembling and solving exercises and problems, but can break up the ﬂow of the text. The Resource section at the end of the text consists of the Road maps, a Data section with a lot of useful numerical information, and Character tables. Short extracts of the tables in the text itself give an idea of the typical values of the physical quantities being discussed. Table 1.6* van der Waals coeﬃcients (1.21a) quation is often written in (1.21b) (14.1) Fig. 7.31 Resource section van der Waals equation of state A topic often needs to draw on a mathematical procedure or a concept of physics; a brief comment is a quick reminder of the procedure or concept. 6 2 2 3 1 a/(atm dm mol ) b/(10 dm mol ) Ar 1.337 3.20 CO2 3.610 4.29 He 0.0341 2.38 Xe 4.137 5.16 * More values are given in the Data section. nduction and is measured in tesla, T; 1 T = G, is also occasionally used: 1 T = 104 G. A brief comment Scalar products (or ‘dot products’) are explained in Mathematical background 5 following Chapter 9. Mathematical background It is often the case that you need a more fullbodied account of a mathematical concept, either because it is important to understand the procedure more fully or because you need to use a series of tools to develop an equation. The Mathematical background sections are located between some chapters, primarily where they are ﬁrst needed, and include many illustrations of how each concept is used. MATHEMATICAL BACKGROUND 5 θ Vectors u A vector quantity has both magnitude and direction. The vector shown in Fig. MB5.1 has components on the x, y, and z axes with magnitudes vx, vy, and vz, respectively. The vector may be represented as V = vx i + vy j + vz k (MB5.1) where i, j, and k are unit vectors, vectors of magnitude 1, pointing along the positive directions on the x, y, and zaxes. The magnitude of the vector is denoted v or V and is given by v = (vx2 + vy2 + vz2)1/2 u v θ u v u+v 180° – θ θ (a) v v (b) (c) (a) The vectors u and V make an angle θ. (b) To add V to u, we ﬁrst join the tail of V to the head of u, making sure that the angle θ between the vectors remains unchanged. (c) To ﬁnish the process, we draw the resultant vector by joining the tail of u to the head of V. Fig. MB5.2 (MB5.2) Problem solving A brief illustration A brief illustration is a short example of how to use an equation that has just been introduced in the text. In particular, we show how to use data and how to manipulate units correctly. • A brief illustration The unpaired electron in the ground state of an alkali metal atom has l = 0, so j = 12 . Because the orbital angular momentum is zero in this state, the spin–orbit coupling energy is zero (as is conﬁrmed by setting j = s and l = 0 in eqn 9.42). When the electron is excited to an orbital with l = 1, it has orbital angular momentum and can give rise to a magnetic ﬁeld that interacts with its spin. In this conﬁguration the electron can have j = 32 or j = 12 , and the energies of these levels are E3/2 = 12 hcÃ{ 32 × 52 − 1 × 2 − 1 2 × 32 } = 12 hcÃ E1/2 = 12 hcÃ{ 12 × 32 − 1 × 2 − 1 2 × 32 } = −hcÃ The corresponding energies are shown in Fig. 9.30. Note that the baricentre (the ‘centre of gravity’) of the levels is unchanged, because there are four states of energy 12 hcÃ and two of energy −hcÃ. • ABOUT THE BOOK xiii Examples Discussion questions We present many worked examples throughout the text to show how concepts are used, sometimes in combination with material from elsewhere in the text. Each worked example has a Method section suggesting an approach as well as a fully worked out answer. The endofchapter material starts with a short set of questions that are intended to encourage reﬂection on the material and to view it in a broader context than is obtained by solving numerical problems. Discussion questions 9.1 Discuss the origin of the series of lines in the emission spectra of Example 9.2 Calculating the mean radius of an orbital hydrogen. What region of the electromagnetic spectrum is associated with each of the series shown in Fig. 9.1? Use hydrogenic orbitals to calculate the mean radius of a 1s orbital. 9.2 Describe the separation of variables procedure as it is applied to simplify Method The mean radius is the expectation value 9.3 List and describe the signiﬁcance of the quantum numbers needed to the description of a hydrogenic atom free to move through space. 冮 specify the internal state of a hydrogenic atom. 冮 9.4 Specify and account for the selection rules for transitions in hydrogenic 具r典 = ψ *rψ dτ = r ψ 2 dτ atoms. 9.5 Explain the signiﬁcance of (a) a boundary surface and (b) the radial distribution function for hydrogenic orbitals. We therefore need to evaluate the integral using the wavefunctions given in Table 9.1 and dτ = r 2dr sin θ dθ dφ. The angular parts of the wavefunction (Table 8.2) are normalized in the sense that π 2π 冮冮 0 Yl,ml  2 sin θ dθ dφ = 1 0 The integral over r required is given in Example 7.4. Answer With the wavefunction written in the form ψ = RY, the integration is ∞ π 具r典 = 2π 冮冮冮 0 0 ∞ 2 rR n,l Yl,ml  2r 2 dr sin θ dθ dφ = 0 冮rR 3 2 n,l dr 0 For a 1s orbital A Z D 3/2 R1,0 = 2 B E e−Zr/a0 C a0 F Hence 具r典 = 4Z 3 a30 ∞ 冮 re 0 3 −2Zr/a0 dr = 3a0 2Z their location in the periodic table. 9.7 Describe and account for the variation of ﬁrst ionization energies along Period 2 of the periodic table. Would you expect the same variation in Period 3? 9.8 Describe the orbital approximation for the wavefunction of a many electron atom. What are the limitations of the approximation? 9.9 Explain the origin of spin–orbit coupling and how it affects the appearance of a spectrum. 9.10 Describe the physical origins of linewidths in absorption and emission spectra. Do you expect the same contributions for species in condensed and gas phases? Exercises and Problems The core of testing understanding is the collection of endofchapter Exercises and Problems. The Exercises are straightforward numerical tests that give practice with manipulating numerical data. The Problems are more searching. They are divided into ‘numerical’, where the emphasis is on the manipulation of data, and ‘theoretical’, where the emphasis is on the manipulation of equations before (in some cases) using numerical data. At the end of the Problems are collections of problems that focus on practical applications of various kinds, including the material covered in the Impact sections. Exercises 9.1(a) Determine the shortest and longest wavelength lines in the Lyman series. Selftests 9.1(b) The Pfund series has n1 = 5. Determine the shortest and longest wavelength lines in the Pfund series. Each Example has a Selftest with the answer provided as a check that the procedure has been mastered. There are also a number of freestanding Selftests that are located where we thought it a good idea to provide a question to check your understanding. Think of Selftests as inchapter exercises designed to help you monitor your progress. 9.2(a) Compute the wavelength, frequency, and wavenumber of the n = 2 → n = 1 transition in He+. 9.2(b) Compute the wavelength, frequency, and wavenumber of the n = 5 → n = 4 transition in Li+2. 9.3(a) When ultraviolet radiation of wavelength 58.4 nm from a helium lamp is directed on to a sample of krypton, electrons are ejected with a speed of 1.59 Mm s−1. Calculate the ionization energy of krypton. 9.3(b) When ultraviolet radiation of wavelength 58.4 nm from a helium lamp is directed on to a sample of xenon, electrons are ejected with a speed of 1.79 Mm s−1. Calculate the ionization energy of xenon. [27a0/2Z] 9.12(a) What is the orbital angular momentum of an electron in the orbitals (a) 1s, (b) 3s, (c) 3d? Give the numbers of angular and radial nodes in each case. 9.12(b) What is the orbital angular momentum of an electron in the orbitals (a) 4d, (b) 2p, (c) 3p? Give the numbers of angular and radial nodes in each case. 9.13(a) Locate the angular nodes and nodal planes of each of the 2p orbitals of a hydrogenic atom of atomic number Z. To locate the angular nodes, give the angle that the plane makes with the zaxis. 9.13(b) Locate the angular nodes and nodal planes of each of the 3d orbitals of a hydrogenic atom of atomic number Z. To locate the angular nodes, give the angle that the plane makes with the zaxis. 9.14(a) Which of the following transitions are allowed in the normal electronic emission spectrum of an atom: (a) 2s → 1s, (b) 2p → 1s, (c) 3d → 2p? 9.14(b) Which of the following transitions are allowed in the normal electronic emission spectrum of an atom: (a) 5d → 2s (b) 5p → 3s (c) 6p → 4f? Problems* Numerical problems Selftest 9.4 Evaluate the mean radius of a 3s orbital by integration. 9.6 Outline the electron conﬁgurations of manyelectron atoms in terms of 9.1 The Humphreys series is a group of lines in the spectrum of atomic hydrogen. It begins at 12 368 nm and has been traced to 3281.4 nm. What are the transitions involved? What are the wavelengths of the intermediate transitions? 9.2 A series of lines in the spectrum of atomic hydrogen lies at 656.46 nm, 486.27 nm, 434.17 nm, and 410.29 nm. What is the wavelength of the next line in the series? What is the ionization energy of the atom when it is in the lower state of the transitions? 9.3 The Li2+ ion is hydrogenic and has a Lyman series at 740 747 cm−1, 877 924 cm−1, 925 933 cm−1, and beyond. Show that the energy levels are of the form −hcR/n2 and ﬁnd the value of R for this ion. Go on to predict the wavenumbers of the two longestwavelength transitions of the Balmer series of the ion and ﬁnd the ionization energy of the ion. the spectrum are therefore expected to be hydrogenlike, the differences arising largely from the mass differences. Predict the wavenumbers of the ﬁrst three lines of the Balmer series of positronium. What is the binding energy of the ground state of positronium? 9.9 The Zeeman effect is the modiﬁcation of an atomic spectrum by the application of a strong magnetic ﬁeld. It arises from the interaction between applied magnetic ﬁelds and the magnetic moments due to orbital and spin angular momenta (recall the evidence provided for electron spin by the Stern–Gerlach experiment, Section 8.8). To gain some appreciation for the socalled normal Zeeman effect, which is observed in transitions involving singlet states, consider a p electron, with l = 1 and ml = 0, ±1. In the absence of a magnetic ﬁeld, these three states are degenerate. When a ﬁeld of magnitude B is present, the degeneracy is removed and it is observed that the state with ml = +1 moves up in energy by μBB, the state with ml = 0 is unchanged, and the state with ml = −1 moves down in energy by μBB, where μB = e$/2me = 9.274 × 10−24 J T−1 is the Bohr magneton (see Section 13.1). Therefore, a Molecular modelling and computational chemistry Over the past two decades computational chemistry has evolved from a highly specialized tool, available to relatively few researchers, into a powerful and practical alternative to experimentation, accessible to all chemists. The driving force behind this evolution is the remarkable progress in computer xiv ABOUT THE BOOK technology. Calculations that previously required hours or days on giant mainframe computers may now be completed in a fraction of time on a personal computer. It is natural and necessary that computational chemistry ﬁnds its way into the undergraduate chemistry curriculum as a handson experience, just as teaching experimental chemistry requires a laboratory experience. With these developments in the chemistry curriculum in mind, the text’s website features a range of computational problems, which are intended to be performed with special software that can handle ‘quantum chemical calculations’. Speciﬁcally, the problems have been designed with the student edition of Wavefunction’s Spartan program (Spartan Student TM) in mind, although they could be completed with any electronic structure program that allows HartreeFock, density functional and MP2 calculations. It is necessary for students to recognize that calculations are not the same as experiments, and that each ‘chemical model’ built from calculations has its own strengths and shortcomings. With this caveat in mind, it is important that some of the problems yield results that can be compared directly with experimental data. However, most problems are intended to stand on their own, allowing computational chemistry to serve as an exploratory tool. Students can visit www.wavefun.com/cart/spartaned.html and enter promotional code WHFPCHEM to download the Spartan Student TM program at a special 20% discount. About the Book Companion Site The Book Companion Site to accompany Physical Chemistry 9e provides teaching and learning resources to augment the printed book. It is free of charge, and provides additional material for download, much of which can be incorporated into a virtual learning environment. The Book Companion Site can be accessed by visiting www.whfreeman.com/pchem Note that instructor resources are available only to registered adopters of the textbook. To register, simply visit www.whfreeman.com/pchem and follow the appropriate links. You will be given the opportunity to select your own username and password, which will be activated once your adoption has been veriﬁed. Student resources are openly available to all, without registration. For students Living graphs A Living graph can be used to explore how a property changes as a variety of parameters are changed. To encourage the use of this resource (and the more extensive Explorations in physical chemistry; see below), we have included a suggested interActivity to many of the illustrations in the text. Group theory tables Comprehensive group theory tables are available for downloading. For Instructors Artwork An instructor may wish to use the ﬁgures from this text in a lecture. Almost all the ﬁgures are available in electronic format and can be used for lectures without charge (but not for commercial purposes without speciﬁc permission). Tables of data All the tables of data that appear in the chapter text are available and may be used under the same conditions as the ﬁgures. Other resources Explorations in Physical Chemistry by Valerie Walters, Julio de Paula, and Peter Atkins Explorations in Physical Chemistry consists of interactive Mathcad® worksheets, interactive Excel® workbooks, and stimulating exercises. They motivate students to simulate physical, chemical, and biochemical phenomena with their personal computers. Students can manipulate over 75 graphics, alter simulation parameters, and solve equations, to gain deeper insight into physical chemistry. Explorations in Physical Chemistry is available as an integrated part of the eBook version of the text (see below). It can also be purchased on line at http://www.whfreeman.com/explorations. Physical Chemistry, Ninth Edition eBook The eBook, which is a complete online version of the textbook itself, provides a rich learning experience by taking full advantage of the electronic medium. It brings together a range of student resources alongside additional functionality unique to the eBook. The eBook also offers lecturers unparalleled ﬂexibility and customization options. The ebook can be purchased at www.whfreeman.com/pchem. Key features of the eBook include: • Easy access from any Internetconnected computer via a standard Web browser. • Quick, intuitive navigation to any section or subsection, as well as any printed book page number. • Living Graph animations. • Integration of Explorations in Physical Chemistry. • Text highlighting, down to the level of individual phrases. • A book marking feature that allows for quick reference to any page. • A powerful Notes feature that allows students or instructors to add notes to any page. • A full index. • Fulltext search, including an option to search the glossary and index. • Automatic saving of all notes, highlighting, and bookmarks. Additional features for instructors: • Custom chapter selection: Instructors can choose the chapters that correspond with their syllabus, and students will get a custom version of the eBook with the selected chapters only. xvi ABOUT THE BOOK COMPANION SITE • Instructor notes: Instructors can choose to create an annotated version of the eBook with their notes on any page. When students in their course log in, they will see the instructor’s version. • Custom content: Instructor notes can include text, web links, and images, allowing instructors to place any content they choose exactly where they want it. Volume 2: Physical Chemistry, 9e is available in two volumes! Chapter 13: Chapter 14: Chapter 15: Chapter 16: For maximum ﬂexibility in your physical chemistry course, this text is now offered as a traditional, full text or in two volumes. The chapters from Physical Chemistry, 9e, that appear each volume are as follows: Volume 1: Chapter 0: Chapter 1: Chapter 2: Chapter 3: Chapter 4: Chapter 5: Chapter 6: Chapter 20: Chapter 21: Chapter 22: Chapter 23: Thermodynamics and Kinetics (1429231270) Fundamentals The properties of gases The First Law The Second Law Physical transformations of pure substances Simple mixtures Chemical equilibrium Molecules in motion The rates of chemical reactions Reaction dynamics Catalysis Chapter 7: Chapter 8: Chapter 9: Chapter 10: Chapter 11: Chapter 12: Quantum Chemistry, Spectroscopy, and Statistical Thermodynamics (1429231262) Quantum theory: introduction and principles Quantum theory: techniques and applications Atomic structure and spectra Molecular structure Molecular symmetry Molecular spectroscopy 1: rotational and vibrational spectra Molecular spectroscopy 2: electronic transitions Molecular spectroscopy 3: magnetic resonance Statistical thermodynamics 1: the concepts Statistical thermodynamics 2: applications Chapters 17, 18, and 19 are not contained in the two volumes, but can be made available online on request. Solutions manuals As with previous editions, Charles Trapp, Carmen Giunta, and Marshall Cady have produced the solutions manuals to accompany this book. A Student’s Solutions Manual (978–1– 4292–3128–2) provides full solutions to the ‘b’ exercises and the oddnumbered problems. An Instructor’s Solutions Manual (978–1–4292–5032–0) provides full solutions to the ‘a’ exercises and the evennumbered problems. About the authors Professor Peter Atkins is a fellow of Lincoln College, University of Oxford, and the author of more than sixty 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, and New Zealand. He was the founding chairman of the Committee on Chemistry Education of the International Union of Pure and Applied Chemistry and a member of IUPAC’s Physical and Biophysical Chemistry Division. Julio de Paula is Professor of Chemistry at Lewis and Clark College. A native of Brazil, Professor de Paula 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, biophysical chemistry, and nanoscience. He has taught courses in general chemistry, physical chemistry, biophysical chemistry, instrumental analysis, and writing. 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 eight editions. Many people gave their advice based on the eighth edition of the text, and others reviewed the draft chapters for the ninth edition as they emerged. We would like to thank the following colleagues: Adedoyin Adeyiga, Cheyney University of Pennsylvania David Andrews, University of East Anglia Richard Ansell, University of Leeds Colin Bain, University of Durham Godfrey Beddard, University of Leeds Magnus Bergstrom, Royal Institute of Technology, Stockholm, Sweden Mark Bier, Carnegie Mellon University Robert Bohn, University of Connecticut Stefan Bon, University of Warwick Fernando Bresme, Imperial College, London Melanie Britton, University of Birmingham Ten Brinke, Groningen, Netherlands Ria Broer, Groningen, Netherlands Alexander Burin, Tulane University Philip J. Camp, University of Edinburgh David Cedeno, Illinois State University Alan Chadwick, University of Kent LiHeng Chen, Aquinas College Aurora Clark, Washington State University Nigel Clarke, University of Durham Ron Clarke, University of Sydney David Cooper, University of Liverpool Garry Crosson, University of Dayton John Cullen, University of Manitoba Rajeev Dabke, Columbus State University Keith Davidson, University of Lancaster Guy Dennault, University of Southampton Caroline Dessent, University of York Thomas DeVore, James Madison University Michael Doescher, Benedictine University Randy Dumont, McMaster University Karen Edler, University of Bath Timothy Ehler, Buena Vista University Andrew Ellis, University of Leicester Cherice Evans, The City University of New York Ashleigh Fletcher, University of Newcastle Jiali Gao, University of Minnesota Sophya Garashchuk, University of South Carolina in Columbia Benjamin Gherman, California State University Peter Grifﬁths, Cardiff, University of Wales Nick Greeves, University of Liverpool Gerard Grobner, University of Umeä, Sweden Anton Guliaev, San Francisco State University Arun Gupta, University of Alabama Leonid Gurevich, Aalborg, Denmark Georg Harhner, St Andrews University Ian Hamley, University of Reading Chris Hardacre, Queens University Belfast Anthony Harriman, University of Newcastle Torsten Hegmann, University of Manitoba Richard Henchman, University of Manchester Ulf Henriksson, Royal Institute of Technology, Stockholm, Sweden Harald Høiland, Bergen, Norway Paul Hodgkinson, University of Durham Phillip John, HeriotWatt University Robert Hillman, University of Leicester Pat Holt, Bellarmine University Andrew Horn, University of Manchester Ben Horrocks, University of Newcastle Rob A. Jackson, University of Keele Seogjoo Jang, The City University of New York Don Jenkins, University of Warwick Matthew Johnson, Copenhagen, Denmark Mats Johnsson, Royal Institute of Technology, Stockholm, Sweden Milton Johnston, University of South Florida Peter Karadakov, University of York Dale Keefe, Cape Breton University Jonathan Kenny, Tufts University Peter Knowles, Cardiff, University of Wales Ranjit Koodali, University Of South Dakota Evguenii Kozliak, University of North Dakota Krish Krishnan, California State University Peter Kroll, University of Texas at Arlington Kari Laasonen, University of Oulu, Finland Ian Lane, Queens University Belfast Stanley Latesky, University of the Virgin Islands Daniel Lawson, University of Michigan Adam Lee, University of York Donál Leech, Galway, Ireland Graham Leggett, University of Shefﬁeld Dewi Lewis, University College London Goran Lindblom, University of Umeä, Sweden Lesley Lloyd, University of Birmingham John Lombardi, City College of New York Zan LutheySchulten, University of Illinois at UrbanaChampaign Michael Lyons, Trinity College Dublin Alexander Lyubartsev, University of Stockholm Jeffrey Mack, California State University Paul Madden, University of Edinburgh Arnold Maliniak, University of Stockholm Herve Marand, Virginia Tech ACKNOWLEDGEMENTS Louis Massa, Hunter College Andrew Masters, University of Manchester Joe McDouall, University of Manchester Gordon S. McDougall, University of Edinburgh David McGarvey, University of Keele Anthony Meijer, University of Shefﬁeld Robert Metzger, University of Alabama Sergey Mikhalovsky, University of Brighton Marcelo de Miranda, University of Leeds Gerald Morine, Bemidji State University Damien Murphy, Cardiff, University of Wales David Newman, Bowling Green State University Gareth Parkes, University of Huddersﬁeld Ruben Parra, DePaul University Enrique PeacockLopez, Williams College NilsOla Persson, Linköping University Barry Pickup, University of Shefﬁeld Ivan Powis, University of Nottingham Will Price, University of Wollongong, New South Wales, Australia Robert Quandt, Illinois State University Chris Rego, University of Leicester Scott Reid, Marquette University Gavin Reid, University of Leeds Steve Roser, University of Bath David Rowley, University College London Alan Ryder, Galway, Ireland Karl Ryder, University of Leicester Stephen Saeur, Copenhagen, Denmark Sven Schroeder, University of Manchester Jeffrey Shepherd, Laurentian University Paul Siders, University of Minnesota Duluth Richard Singer, University of Kingston Carl Soennischsen, The Johannes Gutenberg University of Mainz Jie Song, University of Michigan David Steytler, University of East Anglia Michael Stockenhuber, NottinghamTrent University xix Sven Stolen, University of Oslo Emile Charles Sykes, Tufts University Greg Szulczewski, University of Alabama Annette Taylor, University of Leeds Peter Taylor, University of Warwick Jeremy Titman, University of Nottingham Jeroen VanDuijneveldt, University of Bristol Joop van Lenthe, University of Utrecht Peter Varnai, University of Sussex Jay Wadhawan, University of Hull Palle Waage Jensen, University of Southern Denmark Darren Walsh, University of Nottingham Kjell Waltersson, Malarden University, Sweden Richard Wells, University of Aberdeen Ben Whitaker, University of Leeds Kurt Winkelmann, Florida Institute of Technology Timothy Wright, University of Nottingham Yuanzheng Yue, Aalborg, Denmark David Zax, Cornell University We would like to thank two colleagues for their special contribution. Kerry Karaktis (Harvey Mudd College) provided many useful suggestions that focused on applications of the material presented in the text. David Smith (University of Bristol) made detailed comments on many of the chapters. We also thank Claire Eisenhandler and Valerie Walters, who read through the proofs with meticulous attention to detail and caught in private what might have been a public grief. Our warm thanks also go to Charles Trapp, Carmen Giunta, and Marshall Cady who have produced the Solutions manuals that accompany this book. Last, but by no means least, we would also like to thank our two publishers, Oxford University Press and W.H. Freeman & Co., for their constant encouragement, advice, and assistance, and in particular our editors Jonathan Crowe and Jessica Fiorillo. Authors could not wish for a more congenial publishing environment. This page intentionally left blank Summary of contents Fundamentals PART 1 Equilibrium 1 The properties of gases Mathematical background 1: Differentiation and integration 2 The First Law Mathematical background 2: Multivariate calculus 3 The Second Law 4 Physical transformations of pure substances 5 Simple mixtures 6 Chemical equilibrium PART 2 7 8 9 10 11 12 13 14 15 16 17 18 19 PART 3 20 21 22 23 1 17 19 42 44 91 94 135 156 209 Structure 247 Quantum theory: introduction and principles Mathematical background 3: Complex numbers Quantum theory: techniques and applications Mathematical background 4: Differential equations Atomic structure and spectra Mathematical background 5: Vectors Molecular structure Mathematical background 6: Matrices Molecular symmetry Molecular spectroscopy 1: rotational and vibrational spectra Molecular spectroscopy 2: electronic transitions Molecular spectroscopy 3: magnetic resonance Statistical thermodynamics 1: the concepts Statistical thermodynamics 2: applications Molecular interactions Materials 1: macromolecules and selfassembly Materials 2: solids Mathematical background 7: Fourier series and Fourier transforms 249 286 288 322 324 368 371 414 417 445 489 520 564 592 622 659 695 740 Change 743 Molecules in motion The rates of chemical reactions Reaction dynamics Catalysis 745 782 831 876 Resource section Answers to exercises and oddnumbered problems Index 909 948 959 This page intentionally left blank Contents Fundamentals F.1 F.2 F.3 F.4 F.5 F.6 F.7 Atoms Molecules Bulk matter Energy The relation between molecular and bulk properties The electromagnetic ﬁeld Units Exercises PART 1 Equilibrium 1 The properties of gases The perfect gas 1.1 1.2 I1.1 The states of gases The gas laws Impact on environmental science: The gas laws and the weather 1 1 2 4 6 7 9 10 Thermochemistry 2.7 I2.1 2.8 2.9 Standard enthalpy changes Impact on biology: Food and energy reserves Standard enthalpies of formation The temperature dependence of reaction enthalpies State functions and exact differentials 1.3 1.4 Molecular interactions The van der Waals equation Checklist of key equations Discussion questions Exercises Problems Mathematical background 1: Differentiation and integration 2 The First Law The basic concepts 2.1 2.2 2.3 2.4 2.5 I2.1 2.6 Work, heat, and energy The internal energy Expansion work Heat transactions Enthalpy Impact on biochemistry and materials science: Differential scanning calorimetry Adiabatic changes 65 70 71 73 74 Exact and inexact differentials Changes in internal energy The Joule–Thomson effect 74 75 79 Checklist of key equations Further information 2.1: Adiabatic processes Further information 2.2: The relation between heat capacities Discussion questions Exercises Problems 83 84 Mathematical background 2: Multivariate calculus MB2.1 Partial derivatives 91 2.10 2.11 2.12 13 17 19 19 19 23 MB2.2 Exact differentials 84 85 85 88 91 92 28 3 The Second Law Real gases 65 94 29 30 33 37 38 38 39 The direction of spontaneous change 3.1 3.2 I3.1 3.3 3.4 I3.2 42 44 44 45 47 49 53 56 62 63 The dispersal of energy Entropy Impact on engineering: Refrigeration Entropy changes accompanying speciﬁc processes The Third Law of thermodynamics Impact on materials chemistry: Crystal defects Concentrating on the system 3.5 3.6 The Helmholtz and Gibbs energies Standard molar Gibbs energies Combining the First and Second Laws 3.7 3.8 3.9 The fundamental equation Properties of the internal energy Properties of the Gibbs energy Checklist of key equations Further information 3.1: The Born equation Further information 3.2: The fugacity 95 95 96 103 104 109 112 113 113 118 121 121 121 124 128 128 129 xxiv CONTENTS Discussion questions Exercises Problems 4 Physical transformations of pure substances Phase diagrams 4.1 4.2 4.3 I4.1 The stabilities of phases Phase boundaries Three representative phase diagrams Impact on technology: Supercritical ﬂuids Thermodynamic aspects of phase transitions 4.4 4.5 4.6 The dependence of stability on the conditions The location of phase boundaries The Ehrenfest classiﬁcation of phase transitions Checklist of key equations Discussion questions Exercises Problems 5 Simple mixtures 130 131 132 135 6.2 135 137 140 142 The response of equilibria to the conditions 143 I6.2 143 146 149 Equilibrium electrochemistry 152 152 153 154 157 161 164 Liquid mixtures Colligative properties Impact on biology: Osmosis in physiology and biochemistry Phase diagrams of binary systems 5.6 5.7 5.8 5.9 I5.2 Vapour pressure diagrams Temperature–composition diagrams Liquid–liquid phase diagrams Liquid–solid phase diagrams Impact on materials science: Liquid crystals Activities 5.10 5.11 5.12 5.13 The solvent activity The solute activity The activities of regular solutions The activities of ions in solution Checklist of key equations Further information 5.1: The Debye–Hückel theory of ionic solutions Discussion questions Exercises Problems 6.3 6.4 6.5 6.6 6.7 6.8 6.9 I6.3 How equilibria respond to changes of pressure The response of equilibria to changes of temperature Impact on technology: Supramolecular chemistry Halfreactions and electrodes Varieties of cells The cell potential Standard electrode potentials Applications of standard potentials Impact on technology: Speciesselective electrodes 209 209 210 211 213 221 221 223 226 227 228 229 230 233 235 239 156 Partial molar quantities The thermodynamics of mixing The chemical potentials of liquids 5.4 5.5 I5.1 The Gibbs energy minimum Impact on biochemistry: Energy conversion in biological cells The description of equilibrium 6.1 I6.1 156 The properties of solutions Spontaneous chemical reactions 135 The thermodynamic description of mixtures 5.1 5.2 5.3 6 Chemical equilibrium 167 167 169 Checklist of key equations Discussion questions Exercises Problems PART 2 Structure 7 Quantum theory: introduction and principles 240 241 241 243 247 249 175 176 The origins of quantum mechanics 176 179 181 185 188 7.1 7.2 I7.1 190 190 191 194 195 Energy quantization Wave–particle duality Impact on biology: Electron microscopy The dynamics of microscopic systems 7.3 7.4 The Schrödinger equation The Born interpretation of the wavefunction Quantum mechanical principles 7.5 7.6 7.7 The information in a wavefunction The uncertainty principle The postulates of quantum mechanics 249 250 255 259 260 260 262 266 266 276 279 198 199 200 201 204 Checklist of key equations Further information 7.1: Classical mechanics Discussion questions Exercises Problems 280 280 283 283 284 CONTENTS Mathematical background 3: Complex numbers MB3.1 Deﬁnitions MB3.2 Polar representation MB3.3 Operations 286 8 Quantum theory: techniques and applications 288 Translational motion 8.1 8.2 I8.1 8.3 I8.2 A particle in a box Motion in two and more dimensions Impact on nanoscience: Quantum dots Tunnelling Impact on nanoscience: Scanning probe microscopy Vibrational motion 8.4 8.5 The energy levels The wavefunctions Rotational motion 8.6 8.7 8.8 Rotation in two dimensions: a particle on a ring Rotation in three dimensions: the particle on a sphere Spin 286 286 287 288 289 293 295 297 299 300 301 302 306 306 310 315 Checklist of key equations Discussion questions Exercises Problems 317 317 317 319 Mathematical background 4: Differential equations MB4.1 The structure of differential equations MB4.2 The solution of ordinary differential equations MB4.3 The solution of partial differential equations 322 9 Atomic structure and spectra The structure and spectra of hydrogenic atoms 9.1 9.2 9.3 The structure of hydrogenic atoms Atomic orbitals and their energies Spectroscopic transitions and selection rules 322 322 323 324 324 325 330 339 The structures of manyelectron atoms 340 The orbital approximation Selfconsistent ﬁeld orbitals 341 349 9.4 9.5 The spectra of complex atoms 9.6 9.7 9.8 9.9 9.10 I9.1 Linewidths Quantum defects and ionization limits Singlet and triplet states Spin–orbit coupling Term symbols and selection rules Impact on astrophysics: Spectroscopy of stars Checklist of key equations Further information 9.1: The separation of motion Further information 9.2: The energy of spin–orbit interaction Discussion questions Exercises Problems Mathematical background 5: Vectors MB5.1 Addition and subtraction MB5.2 Multiplication MB5.3 Differentiation 10 Molecular structure 363 363 364 365 368 368 369 369 371 The Born–Oppenheimer approximation 372 Valencebond theory 372 10.1 10.2 Homonuclear diatomic molecules Polyatomic molecules Molecular orbital theory 10.3 10.4 10.5 I10.1 The hydrogen moleculeion Homonuclear diatomic molecules Heteronuclear diatomic molecules Impact on biochemistry: The biochemical reactivity of O2, N2, and NO Molecular orbitals for polyatomic systems 10.6 10.7 10.8 The Hückel approximation Computational chemistry The prediction of molecular properties 372 374 378 378 382 388 394 395 395 401 405 Checklist of key equations Further information 10.1: Details of the Hartree–Fock method Discussion questions Exercises Problems 407 Mathematical background 6: Matrices MB6.1 Deﬁnitions MB6.2 Matrix addition and multiplication MB6.3 Eigenvalue equations 414 11 Molecular symmetry 350 The symmetry elements of objects 350 352 353 354 357 361 11.1 11.2 11.3 362 362 xxv Operations and symmetry elements The symmetry classiﬁcation of molecules Some immediate consequences of symmetry Applications to molecular orbital theory and spectroscopy 11.4 11.5 11.6 Character tables and symmetry labels Vanishing integrals and orbital overlap Vanishing integrals and selection rules 408 409 409 410 414 414 415 417 417 418 420 425 427 427 433 439 xxvi CONTENTS Checklist of key equations Discussion questions Exercises Problems 441 441 441 442 12 Molecular spectroscopy 1: rotational and vibrational spectra 445 General features of molecular spectroscopy 446 12.1 Experimental techniques 12.2 Selection rules and transition moments I12.1 Impact on astrophysics: Rotational and 446 447 vibrational spectroscopy of interstellar species The fates of electronically excited states 13.4 I13.2 13.5 13.6 Fluorescence and phosphorescence Impact on biochemistry: Fluorescence microscopy Dissociation and predissociation Laser action 503 503 507 507 508 Checklist of key equations Further information 13.1: Examples of practical lasers Discussion questions Exercises Problems 512 513 515 515 517 14 Molecular spectroscopy 3: magnetic resonance 520 447 449 The effect of magnetic ﬁelds on electrons and nuclei 520 Moments of inertia The rotational energy levels 12.5 Rotational transitions 12.6 Rotational Raman spectra 12.7 Nuclear statistics and rotational states 449 452 456 459 460 14.1 14.2 The energies of electrons in magnetic ﬁelds The energies of nuclei in magnetic ﬁelds 14.3 Magnetic resonance spectroscopy 521 522 523 Nuclear magnetic resonance 524 The vibrations of diatomic molecules 462 Pure rotation spectra 12.3 12.4 12.8 12.9 12.10 12.11 12.12 Molecular vibrations Selection rules Anharmonicity Vibration–rotation spectra Vibrational Raman spectra of diatomic molecules 462 464 465 468 469 The vibrations of polyatomic molecules 470 12.13 Normal modes 12.14 Infrared absorption spectra of polyatomic 471 molecules I12.2 Impact on environmental science: Climate change 12.15 Vibrational Raman spectra of polyatomic molecules 12.16 Symmetry aspects of molecular vibrations Checklist of key equations Further information 12.1: Spectrometers Further information 12.2: Selection rules for rotational and vibrational spectroscopy Discussion questions Exercises Problems 472 473 475 476 479 479 482 484 484 486 13 Molecular spectroscopy 2: electronic transitions 489 The characteristics of electronic transitions 489 13.1 13.2 13.3 I13.1 Measurements of intensity The electronic spectra of diatomic molecules The electronic spectra of polyatomic molecules Impact on biochemistry: Vision 490 491 498 501 14.4 14.5 14.6 14.7 The NMR spectrometer The chemical shift The ﬁne structure Conformational conversion and exchange processes Pulse techniques in NMR 14.8 14.9 I14.1 14.10 14.11 14.12 14.13 The magnetization vector Spin relaxation Impact on medicine: Magnetic resonance imaging Spin decoupling The nuclear Overhauser effect Twodimensional NMR Solidstate NMR Electron paramagnetic resonance 14.14 14.15 14.16 I14.2 The EPR spectrometer The gvalue Hyperﬁne structure Impact on biochemistry and nanoscience: Spin probes 525 526 532 539 540 540 542 546 548 548 550 551 553 553 553 555 557 Checklist of key equations Further information 14.1: Fourier transformation of the FID curve Discussion questions Exercises Problems 559 15 Statistical thermodynamics 1: the concepts 564 The distribution of molecular states 15.1 15.2 Conﬁgurations and weights The molecular partition function 559 559 560 561 565 565 568 The internal energy and the entropy 574 15.3 The internal energy 15.4 The statistical entropy I15.1 Impact on technology: Reaching very low 574 576 temperatures 578 The canonical partition function 579 The canonical ensemble The thermodynamic information in the partition function 15.7 Independent molecules 579 Checklist of key equations Further information 15.1: The Boltzmann distribution Further information 15.2: The Boltzmann formula Discussion questions Exercises Problems 585 585 587 588 588 590 16 Statistical thermodynamics 2: applications 592 15.5 15.6 581 582 CONTENTS xxvii 17.6 Repulsive and total interactions I17.2 Impact on materials science: Hydrogen storage 642 in molecular clathrates Gases and liquids 17.7 17.8 17.9 17.10 Molecular interactions in gases The liquid–vapour interface Surface ﬁlms Condensation 16.1 16.2 The thermodynamic functions The molecular partition function Using statistical thermodynamics 16.3 16.4 16.5 16.6 16.7 16.8 I16.1 Mean energies Heat capacities Equations of state Molecular interactions in liquids Residual entropies Equilibrium constants Impact on biochemistry: The helix–coil transition in polypeptides Checklist of key equations Further information 16.1: The rotational partition function of a symmetric rotor Discussion questions Exercises Problems 17 Molecular interactions Electric properties of molecules 17.1 17.2 17.3 17.4 Electric dipole moments Polarizabilities Polarization Relative permittivities 592 592 594 601 601 602 605 607 609 610 615 616 617 618 618 619 644 645 649 652 653 654 18 Materials 1: macromolecules and selfassembly 659 18.1 18.2 18.3 18.4 18.5 The different levels of structure Random coils The mechanical properties of polymers The electrical properties of polymers The structures of biological macromolecules Aggregation and selfassembly 18.6 18.7 Colloids Micelles and biological membranes 654 655 655 656 659 660 660 665 667 667 671 671 674 Determination of size and shape 677 Mean molar masses The techniques 678 680 18.8 18.9 Checklist of key equations Further information 18.1: Random and nearly random coils Discussion questions Exercises Problems 688 689 690 690 691 19 Materials 2: solids 695 622 Crystallography 695 622 625 626 628 19.1 19.2 19.3 19.4 19.5 19.6 19.7 I19.1 622 Interactions between molecules 631 17.5 Interactions between dipoles I17.1 Impact on medicine: Molecular recognition 631 and drug design 643 Checklist of key equations Further information 17.1: The dipole–dipole interaction Further information 17.2: The basic principles of molecular beams Discussion questions Exercises Problems Structure and dynamics Fundamental relations 643 640 Lattices and unit cells The identiﬁcation of lattice planes The investigation of structure Neutron and electron diffraction Metallic solids Ionic solids Molecular solids and covalent networks Impact on biochemistry: Xray crystallography of biological macromolecules 695 697 699 708 709 711 714 715 xxviii CONTENTS The properties of solids 19.8 19.9 I19.2 19.10 19.11 19.12 Mechanical properties Electrical properties Impact on nanoscience: Nanowires Optical properties Magnetic properties Superconductors Checklist of key equations Further information 19.1: Solid state lasers and lightemitting diodes Discussion questions Exercises Problems Mathematical background 7: Fourier series and Fourier transforms MB7.1 Fourier series MB7.2 MB7.3 Fourier transforms The convolution theorem 717 717 719 723 724 728 731 733 21 The rates of chemical reactions Empirical chemical kinetics 21.1 21.2 21.3 21.4 21.5 Experimental techniques The rates of reactions Integrated rate laws Reactions approaching equilibrium The temperature dependence of reaction rates Accounting for the rate laws 733 734 735 737 740 740 741 742 PART 3 Change 743 20 Molecules in motion 745 21.6 21.7 Elementary reactions Consecutive elementary reactions Examples of reaction mechanisms 21.8 21.9 21.10 I21.1 Unimolecular reactions Polymerization kinetics Photochemistry Impact on biochemistry: Harvesting of light during plant photosynthesis Checklist of key equations Discussion questions Exercises Problems 22 Reaction dynamics Molecular motion in gases 745 Reactive encounters 20.1 The kinetic model of gases I20.1 Impact on astrophysics: The Sun as a ball of 746 perfect gas Collisions with walls and surfaces The rate of effusion Transport properties of a perfect gas 752 753 754 755 22.1 22.2 22.3 20.2 20.3 20.4 Molecular motion in liquids 20.5 20.6 20.7 I20.2 Experimental results The conductivities of electrolyte solutions The mobilities of ions Impact on biochemistry: Ion channels Diffusion 20.8 20.9 20.10 20.11 The thermodynamic view The diffusion equation Diffusion probabilities The statistical view Checklist of key equations Further information 20.1: The transport characteristics of a perfect gas Discussion questions Exercises Problems Collision theory Diffusioncontrolled reactions The material balance equation Transition state theory 782 782 783 786 790 796 799 802 802 803 809 809 811 815 822 825 825 826 828 831 831 832 839 842 843 The Eyring equation Thermodynamic aspects 844 848 758 759 760 764 The dynamics of molecular collisions 851 766 The dynamics of electron transfer 856 766 770 772 773 22.9 Electron transfer in homogeneous systems 22.10 Electron transfer processes at electrodes I22.1 Impact on technology: Fuel cells 857 861 867 Checklist of key equations Further information 22.1: The Gibbs energy of activation of electron transfer Further information 22.2: The Butler–Volmer equation Discussion questions Exercises Problems 868 758 774 775 776 777 779 22.4 22.5 22.6 22.7 22.8 Reactive collisions Potential energy surfaces Some results from experiments and calculations 851 852 853 868 869 871 871 873 CONTENTS 23 Catalysis 876 I23.1 Impact on technology: Catalysis in the chemical industry Homogeneous catalysis 23.1 23.2 Features of homogeneous catalysis Enzymes Heterogeneous catalysis 23.3 23.4 23.5 23.6 23.7 The growth and structure of solid surfaces The extent of adsorption The rates of surface processes Mechanisms of heterogeneous catalysis Catalytic activity at surfaces 876 876 878 884 885 888 894 897 899 xxix Checklist of key equations Further information 23.1: The BET isotherm Discussion questions Exercises Problems Resource section Answers to exercises and oddnumbered problems Index 900 903 903 904 904 906 909 948 959 This page intentionally left blank List of impact sections Impact on astrophysics I9.1 I12.1 I20.1 Spectroscopy of stars Rotational and vibrational spectroscopy of interstellar species The Sun as a ball of perfect gas 361 447 752 Impact on biochemistry I2.1 I6.1 I10.1 I13.1 I13.2 I14.2 I16.1 I19.1 I20.2 I21.1 Differential scanning calorimetry Energy conversion in biological cells The biochemical reactivity of O2, N2, and NO Vision Fluorescence microscopy Spin probes The helix–coil transition in polypeptides Xray crystallography of biological macromolecules Ion channels Harvesting of light during plant photosynthesis 62 211 394 501 507 557 615 715 764 822 Impact on biology I2.2 I5.1 I7.1 Food and energy reserves Osmosis in physiology and biochemistry Electron microscopy 70 175 259 Impact on engineering I3.1 Refrigeration 103 Impact on environmental science I1.1 I12.2 The gas laws and the weather Climate change 28 473 Impact on materials science I3.2 I5.2 I17.2 Crystal defects Liquid crystals Hydrogen storage in molecular clathrates 112 188 643 xxxii LIST OF IMPACT SECTIONS Impact on medicine I14.1 I17.1 Magnetic resonance imaging Molecular recognition and drug design 546 640 Impact on nanoscience I8.1 I8.2 I19.2 Quantum dots Scanning probe microscopy Nanowires 295 299 723 Impact on technology I4.1 I6.2 I6.3 I15.1 I22.1 I23.1 Supercritical ﬂuids Supramolecular chemistry Speciesselective electrodes Reaching very low temperatures Fuel cells Catalysis in the chemical industry 142 226 239 578 867 900 Fundamentals Chemistry is the science of matter and the changes it can undergo. Physical chemistry is the branch of chemistry that establishes and develops the principles of the subject in terms of the underlying concepts of physics and the language of mathematics. It provides the basis for developing new spectroscopic techniques and their interpretation, for understanding the structures of molecules and the details of their electron distributions, and for relating the bulk properties of matter to their constituent atoms. Physical chemistry also provides a window on to the world of chemical reactions and allows us to understand in detail how they take place. In fact, the subject underpins the whole of chemistry, providing the principles in terms we use to understand structure and change and providing the basis of all techniques of investigation. Throughout the text we shall draw on a number of concepts, most of which should already be familiar from introductory chemistry. This section reviews them. In almost every case the following chapters will provide a deeper discussion, but we are presuming that we can refer to these concepts at any stage of the presentation. Because physical chemistry lies at the interface between physics and chemistry, we also need to review some of the concepts from elementary physics that we need to draw on in the text. F.1 Atoms Key points (a) The nuclear model is the basis for discussion of atomic structure: negatively charged electrons occupy atomic orbitals, which are arranged in shells around a positively charged nucleus. (b) The periodic table highlights similarities in electronic conﬁgurations of atoms, which in turn lead to similarities in their physical and chemical properties. (c) Monatomic ions are electrically charged atoms and are characterized by their oxidation numbers. Matter consists of atoms. The atom of an element is characterized by its atomic number, Z, which is the number of protons in its nucleus. The number of neutrons in a nucleus is variable to a small extent, and the nucleon number (which is also commonly called the mass number), A, is the total number of protons and neutrons, which are collectively called nucleons, in the nucleus. Atoms of the same atomic number but different nucleon number are the isotopes of the element. According to the nuclear model, an atom of atomic number Z consists of a nucleus of charge +Ze surrounded by Z electrons each of charge −e (e is the fundamental charge: see inside the front cover for its value and the values of the other fundamental constants). These electrons occupy atomic orbitals, which are regions of space where they are most likely to be found, with no more than two electrons in any one orbital. The atomic orbitals are arranged in shells around the nucleus, each shell being characterized by the principal quantum number, n = 1, 2, . . . . A shell consists of n2 F.1 Atoms F.2 Molecules F.3 Bulk matter F.4 Energy F.5 The relation between molecular and bulk properties (a) The Boltzmann distribution (b) Equipartition F.6 The electromagnetic ﬁeld F.7 Units Exercises 2 FUNDAMENTALS individual orbitals, which are grouped together into n subshells; these subshells, and the orbitals they contain, are denoted s, p, d, and f. For all neutral atoms other than hydrogen, the subshells of a given shell have slightly different energies. The sequential occupation of the orbitals in successive shells results in periodic similarities in the electronic conﬁgurations, the speciﬁcation of the occupied orbitals, of atoms when they are arranged in order of their atomic number, which leads to the formulation of the periodic table (a version is shown inside the back cover). The vertical columns of the periodic table are called groups and (in the modern convention) numbered from 1 to 18. Successive rows of the periodic table are called periods, the number of the period being equal to the principal quantum number of the valence shell, the outermost shell of the atom. The periodic table is divided into s, p, d, and f blocks, according to the subshell that is last to be occupied in the formulation of the electronic conﬁguration of the atom. The members of the d block (speciﬁcally the members of Groups 3–11 in the d block) are also known as the transition metals; those of the f block (which is not divided into numbered groups) are sometimes called the inner transition metals. The upper row of the f block (Period 6) consists of the lanthanoids (still commonly the ‘lanthanides’) and the lower row (Period 7) consists of the actinoids (still commonly the ‘actinides’). Some of the groups also have familiar names: Group 1 consists of the alkali metals, Group 2 (more speciﬁcally, calcium, strontium, and barium) of the alkaline earth metals, Group 17 of the halogens, and Group 18 of the noble gases. Broadly speaking, the elements towards the left of the periodic table are metals and those towards the right are nonmetals; the two classes of substance meet at a diagonal line running from boron to polonium, which constitute the metalloids, with properties intermediate between those of metals and nonmetals. A monatomic ion is an electrically charged atom. When an atom gains one or more electrons it becomes a negatively charged anion; when it loses one or more electrons it becomes a positively charged cation. The charge number of an ion is called the oxidation number of the element in that state (thus, the oxidation number of magnesium in Mg 2+ is +2 and that of oxygen in O2− is −2). It is appropriate, but not always done, to distinguish between the oxidation number and the oxidation state, the latter being the physical state of the atom with a speciﬁed oxidation number. Thus, the oxidation number of magnesium is +2 when it is present as Mg 2+, and it is present in the oxidation state Mg 2+. The elements form ions that are characteristic of their location in the periodic table: metallic elements typically form cations by losing the electrons of their outermost shell and acquiring the electronic conﬁguration of the preceding noble gas. Nonmetals typically form anions by gaining electrons and attaining the electronic conﬁguration of the following noble gas. F.2 Molecules Key points (a) Covalent compounds consist of discrete molecules in which atoms are linked by covalent bonds. (b) Ionic compounds consist of cations and anions in a crystalline array. (c) Lewis structures are useful models of the pattern of bonding in molecules. (d) The valenceshell electron pair repulsion theory (VSEPR theory) is used to predict the threedimensional structures of molecules from their Lewis structures. (e) The electrons in polar covalent bonds are shared unevenly between the bonded nuclei. A chemical bond is the link between atoms. Compounds that contain a metallic element typically, but far from universally, form ionic compounds that consist of cations and anions in a crystalline array. The ‘chemical bonds’ in an ionic compound F.2 MOLECULES are due to the Coulombic interactions (Section F.4) between all the ions in the crystal, and it is inappropriate to refer to a bond between a speciﬁc pair of neighbouring ions. The smallest unit of an ionic compound is called a formula unit. Thus NaNO3, consisting of a Na+ cation and a NO 3− anion, is the formula unit of sodium nitrate. Compounds that do not contain a metallic element typically form covalent compounds consisting of discrete molecules. In this case, the bonds between the atoms of a molecule are covalent, meaning that they consist of shared pairs of electrons. The pattern of bonds between neighbouring atoms is displayed by drawing a Lewis structure, in which bonds are shown as lines and lone pairs of electrons, pairs of valence electrons that are not used in bonding, are shown as dots. Lewis structures are constructed by allowing each atom to share electrons until it has acquired an octet of eight electrons (for hydrogen, a duplet of two electrons). A shared pair of electrons is a single bond, two shared pairs constitute a double bond, and three shared pairs constitute a triple bond. Atoms of elements of Period 3 and later can accommodate more than eight electrons in their valence shell and ‘expand their octet’ to become hypervalent, that is, form more bonds than the octet rule would allow (for example, SF6), or form more bonds to a small number of atoms (for example, a Lewis structure of SO42− with one or more double bonds). When more than one Lewis structure can be written for a given arrangement of atoms, it is supposed that resonance, a blending of the structures, may occur and distribute multiplebond character over the molecule (for example, the two Kekulé structures of benzene). Examples of these aspects of Lewis structures are shown in Fig. F.1. Except in the simplest cases, a Lewis structure does not express the threedimensional structure of a molecule. The simplest approach to the prediction of molecular shape is valenceshell electron pair repulsion theory (VSEPR theory). In this approach, the regions of high electron density, as represented by bonds—whether single or multiple—and lone pairs, take up orientations around the central atom that maximize their separations. Then the position of the attached atoms (not the lone pairs) is noted and used to classify the shape of the molecule. Thus, four regions of electron density adopt a tetrahedral arrangement; if an atom is at each of these locations (as in CH4), then the molecule is tetrahedral; if there is an atom at only three of these locations (as in NH3), then the molecule is trigonal pyramidal; and so on. The names of the various shapes that are commonly found are shown in Fig. F.2. In a reﬁnement of the theory, lone pairs are assumed to repel bonding pairs more strongly than bonding pairs repel each other. The shape a molecule then adopts, if it is not O C O H H O O N S – O OH H Expanded octet F F B F S S F F F F F F A note on good practice Some chemists use the term ‘molecule’ to denote the smallest unit of a compound with the composition of the bulk material regardless of whether it is an ionic or covalent compound and thus speak of ‘a molecule of NaCl’. We use the term ‘molecule’ to denote a discrete covalently bonded entity (as in H2O); for an ionic compound we use ‘formula unit’. F Incomplete octet F F F Hypervalent A collection of typical Lewis structures for simple molecules and ions. The structures show the bonding patterns and lone pairs and, except in simple cases, do not express the shape of the species. Fig. F.1 3 4 FUNDAMENTALS Linear The names of the shapes of the geometrical ﬁgures used to describe symmetrical polyatomic molecules and ions. Angular Trigonal planar Tetrahedral Square planar Trigonal bipyramidal Octahedral Fig. F.2 (a) (b) (a) The inﬂuences on the shape of the SF4 molecule according to the VSEPR model. (b) As a result the molecule adopts a bent seesaw shape. Fig. F.3 determined fully by symmetry, is such as to minimize repulsions from lone pairs. Thus, in SF4 the lone pair adopts an equatorial position and the two axial S–F bonds bend away from it slightly, to give a bent seesaw shaped molecule (Fig. F.3). Covalent bonds may be polar, or correspond to an unequal sharing of the electron pair, with the result that one atom has a partial positive charge (denoted δ +) and the other a partial negative charge (δ −). The ability of an atom to attract electrons to itself when part of a molecule is measured by the electronegativity, χ(chi), of the element. The juxtaposition of equal and opposite partial charges constitutes an electric dipole. If those charges are +Q and −Q and they are separated by a distance d, the magnitude of the electric dipole moment is μ = Qd. Whether or not a molecule as a whole is polar depends on the arrangement of its bonds, for in highly symmetrical molecules there may be no net dipole. Thus, although the linear CO2 molecule (which is structurally OCO) has polar CO bonds, their effects cancel and the molecule as a whole is nonpolar. F.3 Bulk matter Key points (a) The physical states of bulk matter are solid, liquid, or gas. (b) The state of a sample of bulk matter is deﬁned by specifying its properties, such as mass, volume, amount, pressure, and temperature. (c) The perfect gas law is a relation between the pressure, volume, amount, and temperature of an idealized gas. Bulk matter consists of large numbers of atoms, molecules, or ions. Its physical state may be solid, liquid, or gas: A solid is a form of matter that adopts and maintains a shape that is independent of the container it occupies. A liquid is a form of matter that adopts the shape of the part of the container it occupies (in a gravitational ﬁeld, the lower part) and is separated from the unoccupied part of the container by a deﬁnite surface. A gas is a form of matter that immediately ﬁlls any container it occupies. A liquid and a solid are examples of a condensed state of matter. A liquid and a gas are examples of a ﬂuid form of matter: they ﬂow in response to forces (such as gravity) that are applied. F.3 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). An extensive property of bulk matter is a property that depends on the amount of substance present in the sample; an intensive property is a property that is independent of the amount of substance. The volume is extensive; the mass density, ρ (rho), the mass of a sample divided by its volume, ρ = m/V, is intensive. 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’. The unit 1 mol is deﬁned as the number of carbon atoms in exactly 12 g of carbon12. 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 kilograms per mole but commonly grams per mole, 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 M (F.1) 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. A sample of gas exerts a pressure on the walls of its container because the molecules of gas are in ceaseless, random motion and exert a force when they strike the walls. The frequency of the collisions is normally so great that the force, and therefore the pressure, is perceived as being steady. Although pascal is the SI unit of pressure (Section F.6), 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. We shall see that, 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 reporting physical quantities is currently deﬁned as p 7 = 1 bar exactly. We shall see the role of the standard pressure starting in Chapter 2. 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 called kelvin (K). The Kelvin scale is 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. 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. The Kelvin scale is unsuitable for everyday A note on good practice Be careful to distinguish atomic or molecular mass (the mass of a single atom or molecule; units 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, are still widely called ‘atomic weights’ and ‘molecular weights’ even though they are dimensionless quantities and not weights (the gravitational force exerted on an object). Even IUPAC continues to use the terms ‘for historical reasons’. A note on good practice Note that we write T = 0, not T = 0 K. General statements in science should be expressed without reference to a speciﬁc set of units. Moreover, because T (unlike θ ) is absolute, the lowest point is 0 regardless of the scale used to express higher temperatures (such as the Kelvin scale or the Rankine scale). Similarly, we write m = 0, not m = 0 kg and l = 0, not l = 0 m. 5 6 FUNDAMENTALS measurements of temperature, and it is common to use the Celsius scale, which is deﬁned in terms of the Kelvin scale as θ/°C = T/K − 273.15 A note on good practice Although the term ‘ideal gas’ is almost universally used in place of ‘perfect gas’, there are reasons for preferring the latter term. In an ideal system (as will be explained in Chapter 5) the interactions between molecules in a mixture are all the same. In a perfect gas not only are the interactions all the same but they are in fact zero. Few, though, make this useful distinction. Deﬁnition of Celsius scale (F.2) Thus, the freezing point of water is 0°C and its boiling point (at 1 atm) is found to be 100°C (more precisely 99.974°C). Note that in this text T invariably denotes the thermodynamic (absolute) temperature and that temperatures on the Celsius scale are denoted θ (theta). The properties that deﬁne the state of a system are not in general independent of one another. The most important example of a relation between them is provided by the idealized ﬂuid known as a perfect gas (also, commonly, an ‘ideal gas’) pV = nRT Perfect gas equation (F.3) Here R is the gas constant, a universal constant (in the sense of being independent of the chemical identity of the gas) with the value 8.314 J K−1 mol−1. Equation F.3 is central to the development of the description of gases in Chapter 1. F.4 Energy Key points (a) Energy is the capacity to do work. (b) The total energy of a particle is the sum of its kinetic and potential energies. The kinetic energy of a particle is the energy it possesses as a result of its motion. The potential energy of a particle is the energy it possesses as a result of its position. (c) The Coulomb potential energy between two charges separated by a distance r varies as 1/r. Much of chemistry is concerned with transfers and transformations of energy, and it is appropriate to deﬁne this familiar quantity precisely: energy is the capacity to do work. In turn, work is deﬁned as motion against an opposing force. The SI unit of energy is the joule (J), with 1 J = 1 kg m2 s−2 (see Section F.7). A body may possess two kinds of energy, kinetic energy and potential energy. The kinetic energy, E k, of a body is the energy the body possesses as a result of its motion. For a body of mass m travelling at a speed v E k = 12 mv 2 Kinetic energy (F.4) The potential energy, Ep or more commonly V, of a body is the energy it possesses as a result of its position. No universal expression for the potential energy can be given because it depends on the type of force that the body experiences. For a particle of mass m at an altitude h close to the surface of the Earth, the gravitational potential energy is V(h) = V(0) + mgh Gravitational potential energy (F.5) where g is the acceleration of free fall (g = 9.81 m s−2). The zero of potential energy is arbitrary, and in this case it is common to set V(0) = 0. F.5 THE RELATION BETWEEN MOLECULAR AND BULK PROPERTIES 7 One of the most important forms of potential energy in chemistry is the Coulomb potential energy, the potential energy of the electrostatic interaction between two point electric charges. For a point charge Q1 at a distance r in a vacuum from another point charge Q2 V(r) = Q1Q2 4πε 0r Coulomb potential energy (F.6) It is conventional (as here) to set the potential energy equal to zero at inﬁnite separation of charges. Then two opposite charges have a negative potential energy at ﬁnite separations, whereas two like charges have a positive potential energy. Charge is expressed in coulombs (C), often as a multiple of the fundamental charge, e. Thus, the charge of an electron is −e and that of a proton is +e; the charge of an ion is ze, with z the charge number (positive for cations, negative for anions). The constant ε0 (epsilon zero) is the vacuum permittivity, a fundamental constant with the value 8.854 × 10−12 C2 J −1 m−1. In a medium other than a vacuum, the potential energy of interaction between two charges is reduced, and the vacuum permittivity is replaced by the permittivity, ε, of the medium. The permittivity is commonly expressed as a multiple of the vacuum permittivity ε = εr ε0 (F.7) with ε r the dimensionless relative permittivity (formerly, the dielectric constant). The total energy of a particle is the sum of its kinetic and potential energies Electronic 104 cm–1 1 cm The energy of a molecule, atom, or subatomic particle that is conﬁned to a region of space is quantized, or restricted to certain discrete values. These permitted energies are called energy levels. The values of the permitted energies depend on the characteristics of the particle (for instance, its mass) and the extent of the region to which it is conﬁned. The quantization of energy is most important—in the sense that the allowed energies are widest apart—for particles of small mass conﬁned to small regions of space. Consequently, quantization is very important for electrons in atoms and molecules, but usually unimportant for macroscopic bodies. For particles in containers of macroscopic dimensions the separation of energy levels is so small that for all practical purposes the motion of the particles through space—their translational motion—is unquantized and can be varied virtually continuously. As we shall see in detail in Chapter 7, quantization becomes increasingly important as we change focus from rotational to vibrational and then to electronic motion. The separation of rotational energy levels (in small molecules, about 10−23 J or 0.01 zJ, corresponding to about 0.01 kJ mol −1) is smaller than that of vibrational energy levels (about 10 kJ mol−1), which itself is smaller than that of electronic energy levels (about 10−18 J or 1 aJ, corresponding to about 103 kJ mol−1). Figure F.4 depicts these typical energy level separations. Continuum –1 Key points (a) The energy levels of conﬁned particles are quantized. (b) The Boltzmann distribution is a formula for calculating the relative populations of states of various energies. (c) The equipartition theorem provides a way to calculate the energy of some systems. 102–103 cm–1 F.5 The relation between molecular and bulk properties Rotational We make frequent use of the apparently universal law of nature that energy is conserved; that is, energy can neither be created nor destroyed. Although energy can be transferred from one location to another and transformed from one form to another, the total energy is constant. Vibrational (F.8) Translational E = E k + Ep Fig. F.4 The energy level separations (expressed as wavenumbers) typical of four types of system. A brief comment The uncommon but useful preﬁxes z (for zepto) and a (for atto) are explained in Section F.7 on the use of units. 8 FUNDAMENTALS T=∞ Energy T=0 The Boltzmann distribution of populations for a system of ﬁve energy levels as the temperature is raised from zero to inﬁnity. Fig. F.5 (a) The Boltzmann distribution The continuous thermal agitation that the molecules experience in a sample when T > 0 ensures that they are distributed over the available energy levels. One particular molecule may be in a state corresponding to a low energy level at one instant, and then be excited into a high energy state a moment later. Although we cannot keep track of the state of a single molecule, we can speak of the average numbers of molecules in each state. Even though individual molecules may be changing their states as a result of collisions, the average number in each state is constant (provided the temperature remains the same). The average number of molecules in a state is called the population of the state. Only the lowest energy state is occupied at T = 0. Raising the temperature excites some molecules into higher energy states, and more and more states become accessible as the temperature is raised further (Fig. F.5). The formula for calculating the relative populations of states of various energies is called the Boltzmann distribution and was derived by the Austrian scientist Ludwig Boltzmann towards the end of the nineteenth century. Although we shall derive and discuss this distribution in more detail in Chapter 15, at this point it is important to know that it gives the ratio of the numbers of particles in states with energies Ei and Ej as Ni −(Ei −Ej)/kT =e Nj Boltzmann distribution (F.9) where k is Boltzmann’s constant, a fundamental constant with the value k = 1.381 × 10−23 J K−1. This constant occurs throughout physical chemistry, often in a disguised (molar) form as the gas constant, for R = NAk (F.10) where NA is Avogadro’s constant. We shall see in Chapter 15 that the Boltzmann distribution provides the crucial link for expressing the macroscopic properties of bulk matter in terms of the behaviour of its constituent atoms. The important features of the Boltzmann distribution to bear in mind are: • The higher the energy of a state, the lower its population. Rotational Vibrational Electronic The Boltzmann distribution of populations for rotation, vibration, and electronic energy levels at room temperature. Fig. F.6 • The higher the temperature, the more likely it is that a state of high energy is populated. • More levels are signiﬁcantly populated if they are close together in comparison with kT (like rotational and translational states), than if they are far apart (like vibrational and electronic states). Figure F.6 summarizes the form of the Boltzmann distribution for some typical sets of energy levels. The peculiar shape of the population of rotational levels stems from the fact that eqn F.9 applies to individual states, and for molecular rotation the number of rotational states corresponding to a given energy increases with energy. Broadly speaking, the number of planes of rotation increases with energy. Therefore, although the population of each state decreases with energy, the population of the levels goes through a maximum. One of the simplest examples of the relation between microscopic and bulk properties is provided by kinetic molecular theory, a model of a perfect gas. In this model, it is assumed that the molecules, imagined as particles of negligible size, are in ceaseless, random motion and do not interact except during their brief collisions. Different speeds correspond to different kinetic energies, so the Boltzmann formula can be used to predict the proportions of molecules having a speciﬁc speed at a particular temperature. The expression giving the fraction of molecules that have a particular speed is called the Maxwell distribution, and has the features summarized in Fig. F.7. The Maxwell distribution, which is derived, speciﬁed, and discussed more fully in Chapter 20, can be used to show that the average speed, vmean, of the molecules depends on the temperature and their molar mass as A T D 1/2 vmean ∝ B E CMF (F.11) That is, the average speed increases as the squareroot of the temperature and decreases as the squareroot of the molar mass. Thus, the average speed is high for light molecules at high temperatures. The distribution itself gives more information than the average value. For instance, the tail towards high speeds is longer at high temperatures than at low, which indicates that at high temperatures more molecules in a sample have speeds much higher than average. Relative number of molecules F.6 THE ELECTROMAGNETIC FIELD Low temperature or high molecular mass Intermediate temperature or molecular High mass temperature or low molecular mass Speed, v (b) Equipartition The Boltzmann distribution can be used to calculate the average energy associated with each mode of motion of a molecule (as we shall see in detail in Chapters 15 and 16). However, for certain modes of motion (which in practice means translation of any molecule and the rotation of all except the lightest molecules) there is a short cut, called the equipartition theorem. This theorem (which is derived from the Boltzmann distribution) states: In a sample at a temperature T, all quadratic contributions to the total energy have the same mean value, namely 12 kT. 9 Equipartition theorem A ‘quadratic contribution’ simply means a contribution that depends on the square of the position or the velocity (or momentum). For example, because the kinetic energy of a body of mass m free to undergo translation in three dimensions is E k = 12 mv x2 + 12 mv y2 + 12 mv z2, there are three quadratic terms. The theorem implies that the average kinetic energy of motion parallel to the xaxis is the same as the average kinetic energy of motion parallel to the yaxis and to the zaxis. That is, in a normal sample (one at thermal equilibrium throughout), the total energy is equally ‘partitioned’ over all the available modes of motion. One mode of motion is not especially rich in energy at the expense of another. Because the average contribution of each mode is 12 kT, the average kinetic energy of a molecule free to move in three dimensions is 32 kT, as there are three quadratic contributions to the kinetic energy. We shall often use the equipartition theorem to make quick assessments of molecular properties and to judge the outcome of the competition of the ordering effects of intermolecular interactions and the disordering effects of thermal motion. F.6 The electromagnetic field Key point Electromagnetic radiation is characterized by its direction of propagation, its wavelength, frequency, and wavenumber, and its state of polarization. Light is a form of electromagnetic radiation. In classical physics, electromagnetic radiation is understood in terms of the electromagnetic ﬁeld, an oscillating electric and magnetic disturbance that spreads as a harmonic wave through empty space, the vacuum. The wave travels at a constant speed called the speed of light, c, which is about 3 × 108 m s−1. As its name suggests, an electromagnetic ﬁeld has two components, an electric ﬁeld that acts on charged particles (whether stationary or moving) and a magnetic ﬁeld that acts only on moving charged particles. The electromagnetic ﬁeld, The distribution of molecular speeds with temperature and molar mass. Note that the most probable speed (corresponding to the peak of the distribution) increases with temperature and with decreasing molar mass, and simultaneously the distribution becomes broader. Fig. F.7 interActivity (a) Plot different distributions by keeping the molar mass constant at 100 g mol−1 and varying the temperature of the sample between 200 K and 2000 K. (b) Use mathematical software or the Living graph applet from the text’s web site to evaluate numerically the fraction of molecules with speeds in the range 100 m s−1 to 200 m s−1 at 300 K and 1000 K. (c) Based on your observations, provide a molecular interpretation of temperature. 10 FUNDAMENTALS Wavelength, λ (a) Propagation like any periodic wave, is characterized by a wavelength, λ (lambda), the distance between the neighbouring peaks of the wave, and its frequency, ν (nu), the number of times in a given time interval at which its displacement at a ﬁxed point returns to its original value divided by the length of the time interval, normally in seconds (Fig. F.8). The frequency is measured in hertz, where 1 Hz = 1 s−1. The wavelength and frequency of an electromagnetic wave are related by λν = c Therefore, the shorter the wavelength, the higher the frequency. The characteristics of a wave are also reported by giving the wavenumber, # (nu tilde), of the radiation, where #= (b) (a) The wavelength, λ , of a wave is the peaktopeak distance. (b) The wave is shown travelling to the right at a speed c. At a given location, the instantaneous amplitude of the wave changes through a complete cycle (the six dots show half a cycle) as it passes a given point. The frequency, ν, is the number of cycles per second that occur at a given point. Wavelength and frequency are related by λν = c. Fig. F.8 A note on good practice You will hear people speaking of ‘a frequency of so many wavenumbers’. That is doubly wrong. First, wavenumber and frequency are two different physical observables. Second, wavenumber is a physical quantity, not a unit. The dimensions of wavenumber are 1/length and it is commonly reported in reciprocal centimetres, cm−1. (F.12) ν 1 = c λ (F.13) A wavenumber can be interpreted as the number of complete wavelengths in a given length. Wavenumbers are normally reported in reciprocal centimetres (cm−1), so a wavenumber of 5 cm−1 indicates that there are 5 complete wavelengths in 1 cm. A typical wavenumber of visible light is about 15 000 cm−1, corresponding to 15 000 complete wavelengths in each centimetre. The classiﬁcation of the electromagnetic ﬁeld according to its frequency and wavelength is summarized in Fig. F.9. Electromagnetic radiation is planepolarized if the electric and magnetic ﬁelds each oscillate in a single plane (Fig. F.10). The plane of polarization may be orientated in any direction around the direction of propagation with the electric and magnetic ﬁelds per