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Physical Chemistry Fourth Edition Robert J Silbey Class of 1942 Professor of Chemistry Massachusetts Institute of Technology Robert A Alberty Professor Emeritus of Chemistry Massachusetts Institute of Technology Moungi G Bawendi Professor of Chemistry Massachusetts Institute of Technology John Wiley & Sons, Inc ACQUISITIONS EDITOR Deborah Brennan SENIOR PRODUCTION EDITOR Patricia McFadden SENIOR MARKETING MANAGER Robert Smith SENIOR DESIGNER Kevin Murphy NEW MEDIA EDITOR Martin Batey This book was set in 10/12 Times Roman by Publication Services, Inc and printed and bound by Hamilton Printing The cover was printed by Lehigh Press, Inc This book is printed on acid-free paper.᭺ ϱ Copyright 2005 ᮊ John Wiley & Sons, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center Inc 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600 Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008 To order books or for customer service, call 1(800)-CALL-WILEY (225-5945) PREFACE The objective of this book is to make the concepts and methods of physical chemistry clear and interesting to students who have had a year of calculus and a year of physics The underlying theory of chemical phenomena is complicated, and so it is a challenge to make the most important concepts and methods understandable to undergraduate students However, these basic ideas are accessible to students, and they will find them useful whether they are chemistry majors, biologists, engineers, or earth scientists The basic theory of chemistry is presented from the viewpoint of academic physical chemists, but many applications of physical chemistry to practical problems are described One of the important objectives of a course in physical chemistry is to learn how to solve numerical problems The problems in physical chemistry help emphasize features in the underlying theory, and they illustrate practical applications There are two types of problems: problems that can be solved with a handheld calculator and COMPUTER PROBLEMS that require a personal computer with a mathematical application installed There are two sets of problems of the first type The answers to problems in the first set are given in the back of the textbook, and worked-out solutions to these problems are given in the Solutions Manual for Physical Chemistry The answers for the second set of problems are given in the Solutions Manual In the two sets of problems that can be solved using hand-held calculators, some problems are marked with an icon to indicate that they may be more conveniently solved on a personal computer with a mathematical program There are 170 COMPUTER PROBLEMS that require a personal computer with a mathematical application such as MathematicaTM , MathCadTM , MATLABTM , or MAPLETM installed The recent development of these mathematical applications makes it possible to undertake problems that were previously too difficult or too time consuming This is particularly true for two- and three-dimensional plots, integration and differentiation of complicated functions, and solving differential equations The Solutions Manual for Physical Chemistry provides MathematicaTM programs and printouts for the COMPUTER PROBLEMS The MathematicaTM solutions of the 170 COMPUTER PROBLEMS in digital form are available on the web at http://www.wiley.com/college/silbey They can be downloaded into a personal computer with MathematicaTM installed Students iv Preface can obtain Mathematica at a reduced price from Wolfram Research, 100 Trade Center Drive, Champaign, Illinois, 61820-7237 A password is required and will be available in the Solutions Manual, along with further information about how to access the Mathematica solutions in digital form Emphasis in the COMPUTER PROBLEMS has been put on problems that not require complicated programming, but make it possible for students to explore important topics more deeply Suggestions are made as to how to vary parameters and how to apply these programs to other substances and systems As an aid to showing how commands are used, there is an index in the Solutions Manual of the major commands used MathematicaTM plots are used in some 60 figures in the textbook The legends for these figures indicate the COMPUTER PROBLEM where the program is given These programs make it possible for students to explore changes in the ranges of variables in plots and to make calculations on other substances and systems One of the significant changes in the fourth edition is increased emphasis on the thermodynamics and kinetics of biochemical reactions, including the denaturation of proteins and nucleic acids In this edition there is more discussion of the uses of statistical mechanics, nuclear magnetic relaxation, nano science, and oscillating chemical reactions This edition has 32 new problems that can be solved with a hand-held calculator and 35 new problems that require a computer with a mathematical application There are 34 new figures and eight new tables Because the number of credits in physical chemistry courses, and therefore the need for more advanced material, varies at different universities and colleges, more topics have been included in this edition than can be covered in most courses The Appendix provides an alphabetical list of symbols for physical quantities and their units The use of nomenclature and units is uniform throughout the book SI (Syste`me International d’Unite´s) units are used because of their advantage as a coherent system of units That means that when SI units are used with all of the physical quantities in a calculation, the result comes out in SI units without having to introduce numerical factors The underlying unity of science is emphasized by the use of seven base units to represent all physical quantities HISTORY Outlines of Theoretical Chemistry, as it was then entitled, was written in 1913 by Frederick Getman, who carried it through 1927 in four editions The next four editions were written by Farrington Daniels In 1955, Robert Alberty joined Farrington Daniels At that time, the name of the book was changed to Physical Chemistry, and the numbering of the editions was started over The collaboration ended in 1972 when Farrington Daniels died It is remarkable that this textbook traces its origins back 91 years Over the years this book has profited tremendously from the advice of physical chemists all over the world Many physical chemists who care how their subject is presented have written to us with their comments, and we hope that will continue We are especially indebted to colleagues at MIT who have reviewed various sections and given us the benefit of advice These include Sylvia T Ceyer, Robert W Field, Carl W Garland, Mario Molina, Keith Nelson, and Irwin Oppenheim Preface The following individuals made very useful suggestions as to how to improve this fourth edition: Kenneth G Brown (Old Dominion University), Thandi Buthelez (Western Kentucky University), Susan Collins (California State University Northridge), John Gold (East Straudsburg University), Keith J Stine (University of Missouri–St Louis), Ronald J Terry (Western Illinois University), and Worth E Vaughan (University of Wisconsin, Madison) We are also indebted to reviewers of earlier editions and to people who wrote us about the third edition The following individuals made very useful suggestions as to how to improve the MathematicaTM solutions to COMPUTER PROBLEMS: Ian Brooks (Wolfram Research), Carl W David (U Connecticut), Robert N Goldberg (NIST), Mark R Hoffmann (University of North Dakota), Andre Kuzniarek (Wolfram Research), W Martin McClain (Wayne State University), Kathryn Tomasson (University of North Dakota), and Worth E Vaughan (University of Wisconsin, Madison) We are indebted to our editor Deborah Brennan and to Catherine Donovan and Jennifer Yee at Wiley for their help in the production of the book and the solutions manual We are also indebted to Martin Batey for making available the web site, and to many others at Wiley who were involved in the production of this fourth edition Cambridge, Massachusetts January 2004 Robert J Silbey Robert A Alberty Moungi G Bawendi v CONTENTS PART ONE THERMODYNAMICS Zeroth Law of Thermodynamics and Equations of State First Law of Thermodynamics 31 Second and Third Laws of Thermodynamics 74 Fundamental Equations of Thermodynamics 102 Chemical Equilibrium Phase Equilibrium 132 177 Electrochemical Equilibrium 218 Thermodynamics of Biochemical Reactions 254 PART TWO QUANTUM CHEMISTRY Quantum Theory 295 10 Atomic Structure 348 11 Molecular Electronic Structure 12 Symmetry 396 437 13 Rotational and Vibrational Spectroscopy 14 Electronic Spectroscopy of Molecules 15 Magnetic Resonance Spectroscopy 16 Statistical Mechanics 568 458 502 537 Contents PART THREE KINETICS 17 Kinetic Theory of Gases 613 18 Experimental Kinetics and Gas Reactions 19 Chemical Dynamics and Photochemistry 20 Kinetics in the Liquid Phase 641 686 724 PART FOUR MACROSCOPIC AND MICROSCOPIC STRUCTURES 21 Macromolecules 763 22 Electric and Magnetic Properties of Molecules 23 Solid-State Chemistry 24 Surface Dynamics 786 803 840 APPENDIX A Physical Quantities and Units B Values of Physical Constants 863 867 C Tables of Physical Chemical Data D Mathematical Relations E Greek Alphabet 868 884 897 F Useful Information on the Web 898 G Symbols for Physical Quantities and Their SI Units H Answers to the First Set of Problems INDEX 933 912 899 vii This page intentionally left blank 12.6 Identification of Point Groups of Molecules x Ϫx i y סϪy Ϫz z Ϫx x C2 y סϪ y z z x x xz y סϪy z z x Ϫx yz y סy z z 12.6 IDENTIFICATION OF POINT GROUPS OF MOLECULES A given molecule can have a number of symmetry operations The symmetry operations that apply to a given molecule in its equilibrium configuration form a mathematical group In order for a collection of operations to form a group, they must satisfy four requirements The operation that corresponds to the successive operation of two members of the group must be a member of the group This also applies to the square of an operation The identity operation E must be a member of the group The operations must be associative, that is, (AB )C סA(BC ) They not BA have to be commutative Thus it is possible that AB Each operation must have a unique inverse, that is, AAϪ1 סAϪ1 A סE The inverse operation AϪ1 is that which returns the object to its original position The groups of operations for molecules were developed by Schoenflies and are referred to as point groups because one point in the molecules is left unchanged by any operation; this point is not necessarily occupied by a nucleus It is useful to classify molecules according to their point groups, so there is a system of Schoenflies symbols for characterizing molecules For example, H2 O belongs to the C2v group The H2 O molecule has the operations C2 , v , and vЈ, and, of course, it has the operation E To make sure that these operations form a group, consider the multiplications in Table 12.2 All of the multiplications yield operations in the group, as required Note that the operation of reflection in one vertical symmetry plane (v ) followed by the operation of reflection in the other vertical symmetry plane (vЈ) is equivalent to a twofold rotation; that is, vЈv סC2 Similarly, the successive operations of C2 followed by v yield the same result as the vЈ operation (i.e., v C2 סvЈ) It happens that for this particular point group each of the operations is its own inverse; thus C2 C2 סE, v v סE, vЈvЈ סE , and E סE For operations in certain other point groups, this is not the case (e.g., C31 C32 סE ) 443 444 Chapter 12 Symmetry Table 12.2 Multiplication Tablea for the Group C2v Operation B Operation A E C 12 v Јv E C 12 v vЈ E C 12 v Јv C 12 E Јv v v Јv E C 12 vЈ v C 12 E a The table contains the products AB for the indicated operations Note that each column and each row has each symmetry operation represented only once The various types of point groups are defined below and are illustrated in Table 12.3 All molecules belong to one of the following point groups Point Group C1 Molecules with no symmetry other than the identity are in point group C1 An example is CHBrClF Point Group Cs Point group Cs is the group for molecules that only have a reflection plane An example is CH2 ClF Point Group Ci Molecules, such as 1,2-dibromo-1,2-dichloroethane, that have only a center of symmetry i belong to point group Ci Point Groups Cn Molecules possessing only an n-fold axis of rotation belong to a Cn point group Point Groups Cn v Molecules with an n-fold axis of rotation and n vertical mirror planes (which are necessarily colinear with the n-fold axis) belong in one of the Cnv point groups Point Groups Cn h Molecules with an n-fold axis and a plane of symmetry perpendicular to this axis belong to one of the Cnh point groups Such a plane is referred to as a horizontal mirror plane The C2h point group necessarily involves a center of symmetry as well Point Groups Dn Molecules with a Cn axis and a C2 axis perpendicular to this axis are in the Dn point group 445 12.6 Identification of Point Groups of Molecules Table 12.3 Common Schoenflies Point Groups with Examples Schoenflies Symbol Symmetry Elements Molecular Configuration H C1 Cs F H H H Ci D4h C F Cl Br Molecular Configuration F F D3h Cl E, Symmetry Elements Br C E Schoenflies Symbol E, 2C3 , 3C2 , h , 2S3 , 3v E, 2C4 , C2 , 2c 2Ј , 2C 2ЈЈ, i , 2S4 , h , 2v , 2d B 6 Cl F 32 7 Cl Cl Pt Cl Cl E, i D5h E, 2C5 , 2C 25 , 5C2 , h , 2S5 , 2S 25 , 5v D6h E, 2C6 , 2C3 , C2 , 3C Ј2 , 3C ЈЈ , i , 2S3 , 2S6 , h , 3d , 3v Dϱh E, 2Cϱ , ϱv , i , 2Sϱ , ϱC2 H Cl Ru Br C2 O E, C2 O H H H C2v E, C2 , v (xz ), vЈ (yz ) Cr H C Cl Cl H C H Cl C3v H D2d H H E, 2S4 , C2 , 2C 2Ј , 2d H C C Cl H C4v Cϱv E, 2C4 , C2 , 2v , 2d E, 2Cϱ , ϱv F O F Xe F H D3d F O C H C Cl E, 2C3 , 3v C D4d E, 2C3 , 3C2 , i , 2S6 , 3d E, 2S8 , 2C4 , 2S 38 , C2 , 4C 2Ј , 4d H H H H H H O C CO OC Mn CO OC OC CO Mn OC C CO O (continued) 446 Chapter 12 Table 12.3 Schoenflies Symbol Symmetry (continued) Symmetry Elements Molecular Configuration H C2h E, C2 , i , h Schoenflies Symbol Symmetry Elements D5d E, 2C5 , 2C 25 , 5C2 , i , 2S 310 , 2S10 , 5d Molecular Configuration Cl C C H Cl Fe H C3h E, C3 , C 23 , h , S3 , S 53 B H E, C2 (z ), C2 (y ), C2 (x ), i , xy , xz , yz Td O H H D2h H O O C H C Oh H E, 8C3 , 6C2 , 6C4 , 3C2 , i , 6S4 , 8S6 , 3h , 6d H H F F F S F F F H C H E, 8C3 , 3C2 , 6S4 , 6d Point Groups Dn d Molecules with a Cn axis, a perpendicular C2 axis, and a dihedral mirror plane are in the Dnd point group The dihedral mirror plane is colinear with the principal axis and bisects the two perpendicular C2 axes Point Groups Dn h As you may have guessed already, molecules in the Dnh point group have a horizontal mirror plane, that is, one perpendicular to the principal axis Point Groups Sn To be in one of the Sn point groups a molecule has to have an n-fold improper rotation axis Special Point Groups Linear molecules are either Cϱv or Dϱh Heteronuclear molecules, such as CO, are Cϱv because the molecular axis is an ϱ-fold axis, and they have an infinite number of vertical mirror planes Homonuclear diatomic molecules or polyatomics such as acetylene are Dϱh because the molecular axis is ϱ-fold, and there is an infinite number of perpendicular C2 axes since the molecule is symmetrical Tetrahedral molecules are Td The Th point group has all of the symmetry of a cube Octahedral molecules, such as SF6 , are Oh Molecules with the symmetry of an icosahedron or dodecahedron are Ih , and atoms with spherical symmetry are Kh Buckminsterfullerene,* is an example of Ih symmetry The regular icosahedron and dodecahedron also belong to this point group These molecules have the following symmetry operations: E , 12C5 , 12C52 , 20C5 , and 15C2 , i , 12S10 , 12S10 , *H W Kroto, J R Heath, S C O’Brien, R F Curl, and R E Smalley, Nature 318; 162–163 (1985) 12.7 What Symmetry Tells Us about Dipole Moments and Optical Activity 447 20S6 , 15 The number of symmetry operations h ס120 is the largest likely to be encountered, except for Cϱv and Dϱh , for which h סϱ The C2 , C3 , and C5 axes of C60 are shown in Fig 12.6 It is important to be able to identify the point group of a molecule so that group theory can be utilized in various applications to chemistry Fortunately, it is not necessary to identify all the symmetry operations of a molecule to identify its point group The most efficient way to proceed is to look for key symmetry elements in a prescribed sequence This sequence is illustrated by the flow chart in Fig 12.7 To use this flow chart look sequentially for the symmetries indicated by the perpendicular lines Then follow the right or left branch according to whether the particular kind of symmetry is present (“Yes”) or absent (“No”) The special groups are discussed in the previous paragraph If a molecule is not in one of these “special groups,” look for a principal axis of rotation If there is no axis of rotation, the molecule must belong to one of the low-symmetry nonrotational groups Cs Ci , or C1 If a molecule has one or more rotational axes, it is necessary to identify the principal axis of rotation 12.7 WHAT SYMMETRY TELLS US ABOUT DIPOLE MOMENTS AND OPTICAL ACTIVITY The presence of symmetry in a molecule can be used to determine when certain molecular properties will be zero For example, certain symmetry groups preclude “Special Groups” No Yes Cn No Yes σh D C∞ν D∞h No Yes i nC2’s No Yes Td Oh Ih Cn Cs No Yes C1 Ci σh D σh D No Yes Figure 12.6 Twofold, threefold, and fivefold axes of C60 [From F Chung and S Sternberg, Am Sci 81:56 (1993).] No Yes σd’s no σ ν’s no No Yes No Yes Dnh Cnh S2n No Yes Cnν Cn Dn Dnd S2n Fig 12.7 Flow chart for systematically determining the point group of a molecule (With permission from R L Carter, Molecular Symmetry and Group Theory, ᮊWiley, Hoboken, NJ, 1998.) 448 Chapter 12 Symmetry the possibility of a dipole moment or optical activity We consider these two examples in this section The dipole moment (Section 11.8) is a vector quantity that is not affected either in direction or in magnitude by any symmetry operation of the molecule Therefore, the dipole moment vector must be contained in each of the symmetry elements Consequently, molecules that possess dipole moments belong only to the point groups Cn , Cs , and Cnv The presence or absence of a dipole moment therefore tells something about the symmetry of a molecule For example, carbon dioxide and water might have structures corresponding to a symmetrical linear molecule, to an unsymmetrical linear molecule, or to a bent molecule The dipole moments recorded in Table 22.2 show that carbon dioxide in its ground electronic state has zero moment; therefore, the molecule must be symmetrical and linear If it were unsymmetrical or bent, there would have been a permanent dipole moment On the other hand, water in its ground electronic state has a pronounced dipole moment and cannot have the symmetrical linear structure A molecule with a center of symmetry cannot have a dipole moment If a molecule and its mirror image cannot be superimposed, it is potentially optically active Since a rotation followed by a reflection always converts a righthanded object to a left-handed object, an Sn axis guarantees that a molecule cannot exist in separate left- and right-handed forms All improper rotation axes (Sn ), including a mirror plane ( סS1 ) and center of symmetry (i סS2 ), convert a right-handed object into a left-handed object (i.e., produce a mirror image of the original object), whereas all proper rotation axes (Cn ) leave a right-handed object unchanged in this respect Hence, only molecules that have no improper symmetry elements can be optically active In a molecule in which internal rotation can take place (e.g., ethane or H2 O2 ) it is possible to have optically active conformations, but in a gas or solution these conformers are so rapidly interconverted that optical isomers cannot be resolved Comment: These discussions of the symmetry of molecules have added a new class of operators to the operators of quantum mechanics The operators we encountered earlier operated on molecular wavefunctions Symmetry operators operate on points in molecules, but they are related to the operators of quantum mechanics Thus questions of commutability and noncommutability between these two types of operators arise Unfortunately, we cannot follow up on this (but see the advanced textbooks in the reference list); it is important to know that the Hamiltonian operator for a molecule must be invariant under (commute with) all the symmetry operations of the molecule 12.8 SPECIAL TOPIC: MATRIX REPRESENTATIONS We have seen several examples of products of operations in connection with Table 12.2 These products and the effects of these operations on a point in a molecule can be given an actual algebraic significance by writing the symmetry operations as matrices (see Appendix D.8) For example, the effect of the inversion operation is to convert the point with coordinates x1 , y1 , and z1 to 12.8 Special Topic: Matrix Representations point x2 , y2 , and z2 , as shown in Fig 12.4 The new coordinates are given by the equations x2 סϪx1 ם0y1 ם0z1 y2 ס0x1 Ϫ y1 ם0z1 z2 ס0x1 ם0y1 Ϫ z1 (12 1) (12 2) (12 3) These relations are expressed in matrix notation by x2 0 Ϫ1 y2 ס Ϫ1 0 Ϫ1 z2 x1 Ϫx y1 סϪy z1 Ϫz (12 4) Thus, the transformation matrix of i is R (i ) ס Ϫ1 0 Ϫ1 0 Ϫ1 (12 5) The identity operation and the other three operations of the C2v point group are represented by R (E ) ס0 R (C2 ) ס Ϫ1 0 0 0 Ϫ1 0 (12 6) (12 7) 0 R (v ) ס0 Ϫ1 0 (12 8) Ϫ1 0 0 (12 9) R (vЈ) ס R(E ) is the identity matrix (see Appendix D) When these matrices are multiplied by each other the results are the same as when the operations are multiplied by each other, as shown in Table 12.2 Therefore, these matrices are referred to as representatives of their respective operations in the C2v point group The group multiplication table can be reproduced by matrix multiplications of the matrix representatives The set of four matrices is referred to as a representation of the C2v point group In Section 11.2, we used symmetry to classify wavefunctions of the hydrogen molecule ion If (x, y, z ) ( סϪx, Ϫy, Ϫz ), the wavefunction has even parity and is designated with the subscript g for gerade Now we observe that i (x, y, z ) ( סϪx, Ϫy, Ϫz ), so that the eigenvalue is ם1 For an odd-parity wavefunction, the eigenvalue is Ϫ1 and the wavefunction is designated by subscript u for ungerade A symmetry operation that applies to a molecule will commute with the electronic Hamiltonian operator, and the electronic wavefunction is an eigenfunction of this symmetry operation The effect of the operations of the C2v point group on the px orbital are shown in Fig 12.8 449 450 Chapter 12 Symmetry z z x + C2 x + – – (a) y z x + y z – σ (xz) x + – E (b) y z + y z x σ (yz) x + – – (c) Figure 12.8 Effects of operations of the C2v group on the px orbital Example 12.3 Multiplication of symmetry operations Multiply the transformation matrix for C2 by the transformation matrix for v and identify the operation that corresponds with the product See Appendix D 0 Ϫ1 0 Ϫ1 0 Ϫ1 0 Ϫ1 0 ס 0 0 Thus, the product yields the transformation matrix for vЈ so that v C2 סvЈ as shown in Table 12.2 Although we found this representation by considering the effect of the operations on a point in three-dimensional space, the notion of a representation is more general Any set of numbers of matrices that have the same multiplication table as the operations in the group form a representation of the group There are an infinite number of such representations of a group, but there are a finite number that are, in a mathematical sense, more fundamental than the others These are called irreducible representations The representation we found for the C2v point group (equations 12.6–12.9) is not irreducible It is, in fact, reducible to three different irreducible representations because the matrices are diagonal In this case the diagonal elements themselves form an irre- 12.8 Special Topic: Matrix Representations ducible representation For example, if we take the xx elements of 12.6–12.9 we have R (E ) ם ס1 R (v ) ם ס1 R (vЈ) סϪ1 R (C ) סϪ R (v ) סϪ R (vЈ) ם ס1 (12 11) R (C2 ) ם ס1 R (v ) ם ס1 R (vЈ) ם ס1 (12 12) R (C2 ) סϪ1 (12 10) The yy elements give us R (E ) ם ס1 and the zz elements R (E ) ם ס1 These are representations because their multiplication table is identical to that of the C2v group operations themselves Another representation is R (E ) ם ס1 R (C2 ) ם ס1 R ( v ) סϪ R (vЈ) סϪ1 (12 13) These turn out to be all the irreducible representations of C2v Example 12.4 The multiplication table of the representation is the same as that of the group Show that the representation given in equation 12.16 has the same multiplication table as the operations in the group C2v (Table 12.2) From the multiplication table of the R’s of equation 12.16: R(E ) R(C2 ) R(v ) R( Јv ) R(E ) R(C2 ) R(v ) R( Јv ) Ϫ1 Ϫ1 Ϫ1 Ϫ1 1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 If we compare these numbers with those we would obtain by replacing the operations in Table 12.2 by the R’s of equation 12.10, we see that they are identical In the case of the C2v group, as for all commutative groups, all the irreducible representations are one-dimensional (i.e., numbers) Many groups have higherdimensional irreducible representations (e.g., D6h , Td ), and then the matrices in the representation have that dimension Example 12.5 Matrix representation of a rotation To examine the effect of rotation in the xy plane, we rotate about the z axis by an angle Then the x and y coordinates of a point change in the following way: ͫͬ ͫ X Y ס ͬ ͫ cos x cos Ϫ y sin ס x sin םy cos sin Ϫ sin cos ͬͫ ͬ x y 451 452 Chapter 12 Symmetry y using matrix multiplication This can be seen in Fig 12.9 and using some simple trigonometry by the following argument The initial x, y coordinates can be written as x סr cos , y סr sin A rotation by brings these to X סr cos( ס ) םr cos cos Ϫ r sin sin Y סr sin( ס ) םr cos sin םr sin cos (X,Y) (x, y) θ φ proving the formula above What are the matrices representing operations C3 and C32 ? The C3 operation implies ס120 Њ and C32 implies ס240 Њ Thus, the matrices for these are x Figure 12.9 Effect of a rotation about the z axis on the coordinates of a point in the xy plane Ϫ Ί3 Ϫ1 C3 ס2 Ί3 C32 Ϫ1 Ϫ1 ס Ϫ Ί3 Ί3 Ϫ1 Note that when the matrix for C3 is squared, the matrix for C32 results, as it should Finally, we note that since the z component of a point does not change for a rotation about the z axis, we can write the matrices in three-dimensional notation; for example, Ϫ1 C3 סΊ3 12.9 Ϫ Ί3 Ϫ1 0 SPECIAL TOPIC: CHARACTER TABLES Often we not work with the matrix representations of a group themselves, but with the character of the representation, which is defined as the sum of the diagonal elements(trace)ofthematrixrepresentation.Forone-dimensionalrepresentations, then, the character and the representation are identical The character table for the C2v group is given in Table 12.4 On the left is the label of the irreducible representation (A1 , A2 , B1 , B2 ) and on the right are examples of functions of x, y, and z that transformliketheserepresentationsundertheoperationsofthe C2v group.Forexample, consider the function x Operating on x with the elements of the C2v group gives Ex C2 x v x vЈx ( ס1)x ( סϪ1)x (12 14) (12 15) (12 16) (12 17) ( ס1)x ( סϪ1)x By comparing this with Table 12.4, we see that x can be labeled B1 Table 12.4 Character Table for the C2v Group C2v E C2 v (xz ) vЈ (yz ) A1 A2 B1 B2 1 1 1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 Ϫ1 z, z , x , y xy x, xz y, yz 12.9 Special Topic: Character Tables Example 12.6 Symmetry properties of the operations of C2v Verify the symmetry properties of all the functions on the right-hand side of Table 12.4 Consider the functions y and z Under the operation of C2v , they transform in the following way: Ey ( ס1)y Ez ( ס1)z C2 y ( סϪ1)y C2 z ( ס1)z v y ( סϪ1)y v z ( ס1)z vЈy ם( ס1)y vЈz ( ס1)z so that y transforms like B2 and z like A1 Composite functions such as yz can be found from the transformations of y and z alone, so that, for example, v (yz ) ( סϪ1)(ם1)yz Symmetry operations may also be applied to wavefunctions, and thus wavefunctions can be classified by their symmetry properties For example, consider the effects of the operators of the C2v group on p orbitals The effect of the C2 operation on a px orbital is illustrated in Fig 12.8a This is summarized by C2 px סϪ1px (12 18) where Ϫ1 indicates sign reversal Now what are the effects of the reflection operations (xz ) and (yz )? As shown in Fig 12.8b, reflection in the xz plane has no effect, and so (xz )px ס1px (12 19) As shown in Fig 12.8c, reflection in the yz plane causes a sign change so that (yz )px סϪ1px (12 20) The identity operation E has no effect, so the number that represents this operation is ם1 Thus, the px orbital can be labeled B1 in the C2v point group Now consider the effects of the operators of the C2v group on a py orbital Epy C2 py (xz )py (yz )py ם ס1py סϪ1py סϪ1py ם ס1py (12 21) (12 22) (12 23) (12 24) Therefore, py can be labeled B2 in the C2v group The importance of symmetry (or group theory) to chemical problems lies in the fact that if the symmetry of a molecule is that of a given point group, then the wavefunctions must transform like one of the irreducible representations of that group Thus, the electronic wavefunctions for H2 O (in its C2v equilibrium geometry) can be labeled A1 , A2 , B1 , or B2 as in Table 12.4 Furthermore, various operations, such as the Hamiltonian or the dipole moment operation, also transform like particular irreducible representations of the point group From the transformation properties (i.e., symmetry labels) of the wavefunctions and operations, we can derive rules that tell us when certain integrals involving those operations equal zero For example, the probability 453 454 Chapter 12 Symmetry amplitude that a molecule in electronic eigenstate i will absorb a photon and end up in state f depends on the integral Ύ d ء f i (12 25) (see Section 14.1) The vector operator has three components, x , y , z , which transform like x, y, and z in the molecular symmetry group The above integral will be zero unless the integrand is unchanged when any of the symmetry operations of the group is applied to it Suppose the molecule belongs to the C2v group and i transforms like A1 (or z) and f transforms like B2 (or y) Then the component of the integral with x will vanish because the integrand will then transform as zyx (A2 ) and therefore will change sign under v (see Table 12.4) On the other hand, the y component of the integral will not necessarily vanish because the integrand then transforms as zy or A1 , which is unchanged when any operation is applied to it Finding out which integrals vanish for symmetry reasons greatly decreases the amount of work we have to in solving problems, and it yields general rules (selection rules) for spectroscopy and other areas of physical chemistry Such chemical applications are the subject of many of the books listed at the end of the chapter The programmed introduction to chemical applications by Vincent is especially recommended Example 12.7 Symmetry operations applied to molecular wavefunctions So far we have applied symmetry operations to points and atomic wavefunctions, but we can also apply them to molecular wavefunctions (a) Apply the inversion operation i to ם Hם in its ground state What is the eigenvalue? (b) Apply the inversion operation i to H2 in its first excited state What is the eigenvalue? (a) For ground-state Hם , ig 1s ( ס1)g 1s because the molecule is symmetrical about its center The eigenvalue is ם1 (b) For Hם in the first excited state, iu 1s ( סϪ1)u 1s because the wavefunction has the opposite sign on the other side of the center of the molecule The eigenvalue is Ϫ1 Ⅲ Five Key Ideas in Chapter 12 The symmetry of a molecule can be described in terms of five types of symmetry elements and the corresponding operations The symmetry operations operate on molecules and on their wavefunctions The operations for a molecule are associative, but they not have to be commutative Molecules that possess dipole moments belong only to the point groups Cn , Cs , and Cnv An Sn axis guarantees that a molecule cannot exist in separate left- and righthanded forms Matrices provide representations of point groups; that is, matrices can be devised that have the same properties as the various symmetry operations Problems REFERENCES P F Bernath, Spectra of Atoms and Molecules Oxford, UK: Oxford Press, 1995 R L Carter, Molecular Symmetry and Group Theory Hoboken, NJ: Wiley, 1998 F A Cotton, Chemical Applications of Group Theory Hoboken, NJ: Wiley, 1990 B E Douglas and C E Hollingsworth, Symmetry in Bonding and Spectra New York: Academic, 1985 R L Flurry, Symmetry Groups Englewood Cliffs, NJ: Prentice-Hall, 1980 S F A Kettle, Symmetry and Structure.Hoboken, NJ: Wiley, 1985 A Vincent, Molecular Symmetry and Group Theory Hoboken, NJ: Wiley, 1977 P H Wolton, Beginning Group Theory for Chemistry Oxford, UK: Oxford University Press, 1998 PROBLEMS For problems 12.1–12.14, list the Schoenflies symbol and symmetry elements for each molecule 12.1 12.6 CHClBr(CH3 ) H H2 S CH3 S 12.2 C H H Cl Cl Cl PCl3 12.7 Br IF5 F P I Cl 12.3 66 64 H2 O H2O 12.4 F trans-[CrBr2 (H2 O)4 ]( םignore the H’s) 3+ 77 OH2 12.8 C6 H12 (cyclohexane) Cl 12.9 B2 H6 Br OH2 Cr F Br gauche-CH2 ClCH2 Cl H Cl H H H H C6 H3 Br3 (1,3,5-tribromobenzene) H Br Br H H Br H 12.10 H B B H 12.5 F F H C10 H8 (naphthalene) (planar) H 455 456 Chapter 12 Symmetry 12.11 C5 H8 (spiropentane) 12.22 Consider the three distinct isomers of dichlorobenzene To which symmetry group does each belong? Which can have a dipole moment? (triangles ? to each other) 12.12 C4 H4 S (thiophene) H H S H H (planar) 12.23 There are 10 distinct isomers of dichloronaphthalene, C10 H6 Cl2 Two of them not have a dipole moment List these two and find the symmetry group to which each belongs 12.24 Some of the excited electronic states of acetylene are cisbent and some are trans-bent What is the symmetry group of these structures? (Cis-bent means that the hydrogens bend toward one another, while trans-bent means they bend away from one another.) For the molecules in problems 12.25–12.39, give the Schoenflies symbol and symmetry elements 12.25 C8 H8 12.13 C6 H4 Cl2 (p-dichlorobenzene) H Cl C H C C C C H 12.14 Cl H C C 12.26 HCOOH (formic acid) O trans-CFClBrCFClBr Cl C F Br H C Br (planar) Cl F 12.27 UO2 F35Ϫ F 4F 12.15 The symmetry elements for the staggered form of ethane are given in Table 12.3, and it is in the D3d point group What are the elements for the eclipsed form of ethane (this is the sterically hindered form), and what is the point group? F 12.16 Construct the operation multiplication table for the point group C2h 12.17 List the operations associated with the S6 elements and their equivalents, if any How many distinct operations are produced? 12.18 Consider the three distinct isomers of dichloroethylene, C2 H2 Cl2 To which symmetry group does each belong? Which can have a permanent dipole moment? 12.19 The first excited singlet state of ethylene is twisted so that the two hydrogens and carbon on one side are in a plane perpendicular to the plane containing the other three atoms To which symmetry group does it belong? Does it have a dipole moment? 12.20 Which of the molecules in Problems 12.1–12.14 can have a permanent dipole moment? 12.21 Which of the molecules in Problems 12.1–12.14 can be optically active? H O O U 33 F7 F O 12.28 C14 H10 (phenanthrene) 12.29 Fe(CN)36Ϫ 6 NC N C N C Fe C N 33 C 7 CN 7 N Problems 12.30 (HNBCl)3 457 12.37 Ni(CO)4 Cl H N B Cl B O C H N Ni (planar) B N OC Cl H 12.31 C10 H16 (adamantane) C H C C C C Cl H Cl O 12.33 HOCl O Cl 12.34 (HNBH)3 H H N B H B H 12.39 Pt(Br)4 (NH3 )2 (ignore the H’s) [tetrabromodiammineplatinum(IV)] N (planar) B H H H3 N Br Br Pt Br N H3 Br 12.40 Construct the operation multiplication table for the point group C3v 12.41 In some of the excited states of benzene, the molecule is “stretched” so that the hexagon is elongated What is the symmetry group of the molecule in such a state? H N H H (planar) H Br C C C 12.32 C6 H2 O2 Cl2 (2,5-dichloroquinone) O CO 12.38 H3 CCH2 Br C C C O 12.42 What is the symmetry group of HD? Can it have a dipole moment? Computer Problems 12.35 C6 H3 (C6 H5 )3 (1,3,5-triphenylbenzene) 12.A Show that the matrix product of the C3 operation and the C32 operation is equal to the identity operation E A matrix for the C3 operation is given in Example 12.5 (nonplanar) 12.C Use Mathematica to show the shapes of a tetrahedron, an octahedron, a dodecahedron, an icosohedron, and a bucky ball 12.36 CH3 Cl (methyl chloride) Cl C H H 12.B The multiplication table for the group C2v is given in Table 12.2, and the application of these operations to the water molecule is discussed in connection with this table Since matrices for these operations are given in equations 12.6–12.9, verify that vЈv סC2 , v C2 סvЈ, and C2 C2 סE , where these symbols refer to operations H 12.D The following matrix transforms a point (x, y, z ) through a cunterclocksise rotation about the z axis through an angle : Apply this matrix to a point at (1, 1, 1) for the angles 0, /4, /2, 3 /4, 3 /2, and 2 ... ס8. 314 51 J K 1 mol 1 /4 .18 4 J cal 1 ס 1. 987 22 cal K 1 mol 1 Since the liter is 10 Ϫ3 m3 and the bar is 10 Pa, R ( ס8. 314 51 Pa m3 K 1 mol 1 ) (10 L mϪ3 ) (10 Ϫ5 bar Pa 1 ) ס0.083 14 5 L... 1. 5 Real Gases and the Virial Equation Table 1. 1 Second and Third Virial Coefficients at 298 .15 K Gas B /10 Ϫ6 m3 mol 1 H2 He N2 O2 Ar CO 14 .1 11. 8 Ϫ4.5 16 .1 15 .8 Ϫ8.6 C /10 12 m6 molϪ2 350 12 1... intentionally left blank Zeroth Law of Thermodynamics and Equations of State 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 8 1. 9 1. 10 1. 11 State of a System The Zeroth Law of Thermodynamics The Ideal Gas Temperature