P H Y S I C A L C H E M I S T R Y

A M O L E C U L A R A P P R O A C H

D o n a l d A . M c Q u a r r i e

U N I V E R S I T Y O F C A L I F O R N I A , D A V I S

j o h n D . S i m o n

G e o r g e B . G e l l e r P r o f e s s o r o f C h e m i s t r y

D U K E U N I V E R S I T Y

~

U n i v e r s i t y S c i e n c e B o o k s

S a u s a l i t o , C a l i f o r n i a

. .

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c;Lf/.3 M t1'5

University Science Books ,~ 55D Gate Five Road

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Production manager: Susanna Tadlock Manuscript editor: Ann McGuire Designer: Robert Ishi Illustrator: John Choi Compositor: Eigentype Printer & Binder: Edwards Brothers, Inc.

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Copyright 1997 by University Science Books

Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, University Science Books.

Library of Congress Cataloging-in-Publication Data

McQuarrie, Donald A. (Donald Allen) Physical chemistry : a molecular approach I Donald A.

McQuarrie, John D. Simon. p. em.

Includes bibliographical references and index. ISBN 0-935702-99-7 I. Chemistry, Physical and theoretical. I. Simon, John

D. (John Douglas), 1957- . II. Title. QD453.2.M394 1997 541-dc21 97-142

Printed in the United States of America 10987654321

CIP

P r e f a c e x v i i

T o t h e S t u d e n t x v 1 1

T o t h e I n s t r u c t o r x i x

A c k n o w l e d g m e n t x x 1 1 1

C H A P T E R 1 I T h e D a w n o f t h e Q u a n t u m T h e o r y

1 - 1 . B l a c k b o d y R a d i a t i o n C o u l d N o t B e E x p l a i n e d b y C l a s s i c a l P h y s i c s 2

C o n t e n t s

1 - 2 . P l a n c k U s e d a Q u a n t u m H y p o t h e s i s t o D e r i v e t h e B l a c k b o d y R a d i a t i o n L a w 4

1 - 3 . E i n s t e i n E x p l a i n e d t h e P h o t o e l e c t r i c E f f e c t w i t h a Q u a n t u m H y p o t h e s i s 7

1 - 4 . T h e H y d r o g e n A t o m i c S p e c t r u m C o n s i s t s o f S e v e r a l S e r i e s o f L i n e s 1 0

1 - 5 . T h e R y d b e r g F o r m u l a A c c o u n t s f o r A l l t h e L i n e s i n t h e H y d r o g e n A t o m i c S p e c t r u m 1 3

1 - 6 . L o u i s d e B r o g l i e P o s t u l a t e d T h a t M a t t e r H a s W a v e l i k e P r o p e r t i e s 1 5

1 - 7 . d e B r o g l i e W a v e s A r e O b s e r v e d E x p e r i m e n t a l l y 1 6

1 - 8 . T h e B o h r T h e o r y o f t h e H y d r o g e n A t o m C a n B e U s e d t o D e r i v e t h e R y d b e r g

F o r m u l a 1 8

1 - 9 . T h e H e i s e n b e r g U n c e r t a i n t y P r i n c i p l e S t a t e s T h a t t h e P o s i t i o n a n d t h e M o m e n t u m

o f a P a r t i c l e C a n n o t B e S p e c i f i e d S i m u l t a n e o u s l y w i t h U n l i m i t e d P r e c i s i o n 2 3

P r o b l e m s 2 5

M A T H C H A P T E R A I C o m p l e x N u m b e r s 3 1

P r o b l e m s 3 5

C H A P T E R 2 I T h e C l a s s i c a l W a v e E q u a t i o n 3 9

2 - 1 . T h e O n e - D i m e n s i o n a l W a v e E q u a t i o n D e s c r i b e s t h e M o t i o n o f a V i b r a t i n g S t r i n g 3 9

2 - 2 . T h e W a v e E q u a t i o n C a n B e S o l v e d b y t h e M e t h o d o f S e p a r a t i o n o f V a r i a b l e s 4 0

2 - 3 . S o m e D i f f e r e n t i a l E q u a t i o n s H a v e O s c i l l a t o r y S o l u t i o n s 4 4

2 - 4 . T h e G e n e r a l S o l u t i o n t o t h e W a v e E q u a t i o n I s a S u p e r p o s i t i o n o f N o r m a l M o d e s 4 6

2 - 5 . A V i b r a t i n g M e m b r a n e I s D e s c r i b e d b y a T w o - D i m e n s i o n a l W a v e E q u a t i o n 4 9

P r o b l e m s 5 4

M A T H C H A P T E R 8 I P r o b a b i l i t y a n d S t a t i s t i c s 6 3

P r o b l e m s 7 0

v

PHYSICAL CHEMISTRY

CHAPTER l I The Schrodinger Equation and a Particle In a Box 73 3-1. The Schri:idinger Equation Is the Equation for Finding the Wave Function

of a Particle 73 3-2. Classical-Mechanical Quantities Are Represented by Linear Operators in

Quantum Mechanics 75 3-3. The Schri:idinger Equation Can Be Formulated As an Eigenvalue Problem 77 3-4. Wave Functions Have a Probabilistic Interpretation 80 3-5. The Energy of a Particle in a Box Is Quantized 81 3-6. Wave Functions Must Be Normalized 84 3-7. The Average Momentum of a Particle in a Box Is Zero 86 3-8. The Uncertainty Principle Says That upux > h/2 88 3-9. The Problem of a Particle in a Three-Dimensional Box Is a Simple Extension

of the One-Dimensional Case 90 Problems 96

MATHCHAPTER C I Vectors 105 Problems 11 3

CHAPTER 4 I Some Postulates and General Principles of Quantum Mechanics 115

4-1. The State of a System Is Completely Specified by Its Wave Function 115 4-2. Quantum-Mechanical Operators Represent Classical-Mechanical Variables 118 4-3. Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators 122 4-4. The Time Dependence of Wave Functions Is Governed by the Time-Dependent

Schri:idinger Equation 125 4-5. The Eigenfunctions of Quantum Mechanical Operators Are Orthogonal 127 4-6. The Physical Quantities Corresponding to Operators That Commute Can Be Measured

Simultaneously to Any Precision 131 Problems 134

/Z")MATHCHAPTER D I Spherical Coordinates {.__;Y Problems 153

147

CHAPTER 5 I The Harmonic Oscillator and the Rigid Rotator: Two Spectroscopic Models 157

5-1. A Harmonic Oscillator Obeys Hooke's Law 157 -s=2. The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the

Reduced Mass of the Molecule 161 5-3. The Harmonic-Oscillator Approximation Results from the Expansion of an Internuclear

Potential Around Its Minimum 163 >K 5-4. The Energy Levels of a Quantum-Mechanical Harmonic Oscillator Are Ev = hw(v + ~)

with v=O, 1, 2, ... 166 5-5. The Harmonic Oscillator Accounts for the Infrared Spectrum of a Diatomic

Molecule 167 /5-6. The Harmonic-Oscillator Wave Functions Involve Hermite Polynomials /.5-7. Hermite Polynomials Are Either Even or Odd Functions 1 72 B The Energy Levels of a Rigid Rotator Are E = h2 J(J + 1)/21 173 / vi

169

C o n t e n t s

/~The R i g i d R o t a t o r I s a M o d e l f o r a R o t a t i n g D i a t o m i c M o l e c u l e 1 7 7

P r o b l e m s 1 7 9

C H A P T E R 6 I T h e H y d r o g e n A t o m 1 9 1

6 - 1 . T h e S c h r o d i n g e r E q u a t i o n f o r t h e H y d r o g e n A t o m C a n B e S o l v e d E x a c t l y 1 9 1

6 - 2 . T h e W a v e F u n c t i o n s o f a R i g i d R o t a t o r A r e C a l l e d S p h e r i c a l H a r m o n i c s 1 9 3

6 - 3 . T h e P r e c i s e V a l u e s o f t h e T h r e e C o m p o n e n t s o f A n g u l a r M o m e n t u m C a n n o t B e

M e a s u r e d S i m u l t a n e o u s l y 2 0 0

6 - 4 . H y d r o g e n A t o m i c O r b i t a l s D e p e n d u p o n T h r e e Q u a n t u m N u m b e r s 2 0 6

6 - 5 . s O r b i t a l s A r e S p h e r i c a l l y S y m m e t r i c 2 0 9

6 - 6 . T h e r e A r e T h r e e p O r b i t a l s f o r E a c h V a l u e o f t h e P r i n c i p a l Q u a n t u m N u m b e r ,

n ~ 2 2 1 3

6 - 7 . T h e S c h r o d i n g e r E q u a t i o n f o r t h e H e l i u m A t o m C a n n o t B e S o l v e d E x a c t l y 2 1 9

P r o b l e m s 2 2 0

M A T H C H A P T E R E I D e t e r m i n a n t s 2 3 1

P r o b l e m s 2 3 8

C H A P T E R 7 I A p p r o x i m a t i o n M e t h o d s 2 4 1

~ 7 - 1 . T h e V a r i a t i o n a l M e t h o d P r o v i d e s a n U p p e r B o u n d t o t h e G r o u n d - S t a t e E n e r g y

o f a S y s t e m 2 4 1

7 - 2 . A T r i a l F u n c t i o n T h a t D e p e n d s L i n e a r l y o n t h e V a r i a t i o n a l P a r a m e t e r s L e a d s

t o a S e c u l a r D e t e r m i n a n t 2 4 9

7 - 3 . T r i a l F u n c t i o n s C a n B e L i n e a r C o m b i n a t i o n s o f F u n c t i o n s T h a t A l s o C o n t a i n

V a r i a t i o n a l P a r a m e t e r s 2 5 6

~ 7 - 4 . P e r t u r b a t i o n T h e o r y E x p r e s s e s t h e S o l u t i o n t o O n e P r o b l e m i n T e r m s o f A n o t h e r

P r o b l e m S o l v e d P r e v i o u s l y 2 5 7

P r o b l e m s 2 6 1

C H A P T E R 8 I M u l t i e l e c t r o n A t o m s 2 7 5

8 - 1 . A t o m i c a n d M o l e c u l a r C a l c u l a t i o n s A r e E x p r e s s e d i n A t o m i c U n i t s 2 7 5

8 - 2 . B o t h P e r t u r b a t i o n T h e o r y a n d t h e V a r i a t i o n a l M e t h o d C a n Y i e l d E x c e l l e n t R e s u l t s

f o r H e l i u m 2 7 8

8 - 3 . H a r t r e e - F o c k E q u a t i o n s A r e S o l v e d b y t h e S e l f - C o n s i s t e n t F i e l d M e t h o d 2 8 2

8 - 4 . A n E l e c t r o n H a s a n I n t r i n s i c S p i n A n g u l a r M o m e n t u m 2 8 4

8 - 5 . W a v e F u n c t i o n s M u s t B e A n t i s y m m e t r i c i n t h e I n t e r c h a n g e o f A n y T w o E l e c t r o n s 2 8 5

8 - 6 . A n t i s y m m e t r i c W a v e F u n c t i o n s C a n B e R e p r e s e n t e d b y S l a t e r D e t e r m i n a n t s 2 8 8

8 - 7 . H a r t r e e - F o c k C a l c u l a t i o n s G i v e G o o d A g r e e m e n t w i t h E x p e r i m e n t a l D a t a 2 9 0

8 - 8 . A T e r m S y m b o l G i v e s a D e t a i l e d D e s c r i p t i o n o f a n E l e c t r o n C o n f i g u r a t i o n 2 9 2

@ j r h e A l l o w e d V a l u e s o f J a r e L + S , L + S - 1 , 0 0 0 , I L - S l 2 9 6

8 - 1 0 . H u n d ' s R u l e s A r e U s e d t o D e t e r m i n e t h e T e r m S y m b o l o f t h e G r o u n d

E l e c t r o n i c S t a t e 3 0 1

- ! 8 - 1 1 . A t o m i c T e r m S y m b o l s A r e U s e d t o D e s c r i b e A t o m i c S p e c t r a 3 0 2

P r o b l e m s 3 0 8

C H A P T E R 9 I T h e C h e m i c a l B o n d : D i a t o m i c M o l e c u l e s 3 2 3

v i i

PHYSICAL CHEMISTRY

@-1. The Born-Oppenheimer Approximation Simplifies the Schrodinger Equation for Molecules 323

9-2. Hi Is the Prototypical Species of Molecular-Orbital Theory 325 9-3. The Overlap Integral Is a Quantitative Measure of the Overlap of Atomic Orbitals

Situated on Different Atoms 327 9-4. The Stability of a Chemical Bond Is a Quantum-Mechanical Effect 329 9-5. The Simplest Molecular Orbital Treatment of Hi Yields a Bonding Orbital and

an Antibonding Orbital 333 9-6. A Simple Molecular-Orbital Treatment of H2 Places Both Electrons in a

Bonding Orbital 336 9-7. Molecular Orbitals Can Be Ordered According to Their Energies 336 9-8. Molecular-Orbital Theory Predicts That a Stable Diatomic Helium Molecule

Does Not Exist 341 9-9. Electrons Are Placed into Molecular Orbitals in Accord with the Pauli

Exclusion Principle 342 9-10. Molecular-Orbital Theory Correctly Predicts That Oxygen Molecules

Are Paramagnetic 344 9-11. Photoelectron Spectra Support the Existence of Molecular Orbitals 346 9-12. Molecular-Orbital Theory Also Applies to Heteronuclear Diatomic Molecules 346 9-13. An SCF-LCAO-MO Wave Function Is a Molecular Orbital Formed from a Linear

Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently 349

9-14. Electronic States of Molecules Are Designated by Molecular Term Symbols 355 9-15. Molecular Term Symbols Designate the Symmetry Properties of Molecular

Wave Functions 358 9-16. Most Molecules Have Excited Electronic States 360 Problems 362

CHAPTER 1 0 I Bonding In Polyatomic Molecules 371 10-1. Hybrid Orbitals Account for Molecular Shape 371 10-2. Different Hybrid Orbitals Are Used for the Bonding Electrons and the Lone Pair

Electrons in Water 378 10-3. Why is BeH2 Linear and H20 Bent? 381 10-4. Photoelectron Spectroscopy Can Be Used to Study Molecular Orbitals 387 10-5. Conjugated Hydrocarbons and Aromatic Hydrocarbons Can Be Treated

by a n-Eiectron Approximation 390 10-6. Butadiene Is Stabilized by a Delocalization Energy 393 Problems 3 99

CHAPTER 11 I Computational Quantum Chemistry 411 11-1. Gaussian Basis Sets Are Often Used in Modern Computational Chemistry 411 11-2. Extended Basis Sets Account Accurately for the Size and Shape of Molecular

Charge Distributions 41 7 11-3. Asterisks in the Designation of a Basis Set Denote Orbital Polarization Terms 422 11-4. The Ground-State Energy of H2 can be Calculated Essentially Exactly 425 11-5. Gaussian 94 Calculations Provide Accurate Information About Molecules 427 Problems 434

MATHCHAPTER F I Matrices 441 Problems 448

viii

C o n t e n t s

C H A P T E R 1 2 I G r o u p T h e o r y : T h e E x p l o i t a t i o n o f S y m m e t r y 4 5 3

1 2 - 1 . T h e E x p l o i t a t i o n o f t h e S y m m e t r y o f a M o l e c u l e C a n B e U s e d t o S i g n i f i c a n t l y S i m p l i f y

N u m e r i c a l C a l c u l a t i o n s 4 5 3

1 2 - 2 . T h e S y m m e t r y o f M o l e c u l e s C a n B e D e s c r i b e d b y a S e t o f S y m m e t r y E l e m e n t s 4 5 5

1 2 - 3 . T h e S y m m e t r y O p e r a t i o n s o f a M o l e c u l e F o r m a G r o u p 4 6 0

1 2 - 4 . S y m m e t r y O p e r a t i o n s C a n B e R e p r e s e n t e d b y M a t r i c e s 4 6 4

1 2 - 5 . T h e C

3

v P o i n t G r o u p H a s a T w o - D i m e n s i o n a l I r r e d u c i b l e R e p r e s e n t a t i o n 4 6 8

1 2 - 6 . T h e M o s t I m p o r t a n t S u m m a r y o f t h e P r o p e r t i e s o f a P o i n t G r o u p I s I t s

C h a r a c t e r T a b l e 4 7 1

1 2 - 7 . S e v e r a l M a t h e m a t i c a l R e l a t i o n s I n v o l v e t h e C h a r a c t e r s o f I r r e d u c i b l e

R e p r e s e n t a t i o n s 4 7 4

1 2 - 8 . W e U s e S y m m e t r y A r g u m e n t s t o P r e d i c t W h i c h E l e m e n t s i n a S e c u l a r D e t e r m i n a n t

E q u a l Z e r o 4 8 0

1 2 - 9 . G e n e r a t i n g O p e r a t o r s A r e U s e d t o F i n d L i n e a r C o m b i n a t i o n s o f A t o m i c O r b i t a l s T h a t

A r e B a s e s f o r I r r e d u c i b l e R e p r e s e n t a t i o n s 4 8 4

P r o b l e m s 4 8 9

( . ; C H A P T E R 1 3 I M o l e c u l a r S p e c t r o s c o p y 4 9 5

1 3 - 1 . D i f f e r e n t R e g i o n s o f t h e E l e c t r o m a g n e t i c S p e c t r u m A r e U s e d t o I n v e s t i g a t e D i f f e r e n t

M o l e c u l a r P r o c e s s e s 4 9 5

1 3 - 2 . R o t a t i o n a l T r a n s i t i o n s A c c o m p a n y V i b r a t i o n a l T r a n s i t i o n s 4 9 7

1 3 - 3 . V i b r a t i o n - R o t a t i o n I n t e r a c t i o n A c c o u n t s f o r t h e U n e q u a l S p a c i n g o f t h e L i n e s i n t h e

P a n d R B r a n c h e s o f a V i b r a t i o n - R o t a t i o n S p e c t r u m 5 0 1

1 3 - 4 . T h e L i n e s i n a P u r e R o t a t i o n a l S p e c t r u m A r e N o t E q u a l l y S p a c e d 5 0 3

1 3 - 5 . O v e r t o n e s A r e O b s e r v e d i n V i b r a t i o n a l S p e c t r a 5 0 4

1 3 - 6 . E l e c t r o n i c S p e c t r a C o n t a i n E l e c t r o n i c , V i b r a t i o n a l , a n d R o t a t i o n a l I n f o r m a t i o n 5 0 7

1 3 - 7 . T h e F r a n c k - C o n d o n P r i n c i p l e P r e d i c t s t h e R e l a t i v e I n t e n s i t i e s o f V i b r o n i c

T r a n s i t i o n s 5 1 1

1 3 - 8 . T h e R o t a t i o n a l S p e c t r u m o f a P o l y a t o m i c M o l e c u l e D e p e n d s U p o n t h e P r i n c i p a l

M o m e n t s o f I n e r t i a o f t h e M o l e c u l e 5 1 4

1 3 - 9 . T h e V i b r a t i o n s o f P o l y a t o m i c M o l e c u l e s A r e R e p r e s e n t e d b y N o r m a l

C o o r d i n a t e s 5 1 8

1 3 - 1 0 . N o r m a l C o o r d i n a t e s B e l o n g t o I r r e d u c i b l e R e p r e s e n t a t i o n s o f M o l e c u l a r

P o i n t G r o u p s 5 2 3

1 3 - 1 1 . S e l e c t i o n R u l e s A r e D e r i v e d f r o m T i m e - D e p e n d e n t P e r t u r b a t i o n T h e o r y 5 2 7

1 3 - 1 2 . T h e S e l e c t i o n R u l e i n t h e R i g i d R o t a t o r A p p r o x i m a t i o n I s / " ; . ] = 1 5 3 1

1 3 - 1 3 . T h e H a r m o n i c - O s c i l l a t o r S e l e c t i o n R u l e I s / " ; . v = 1 5 3 3

1 3 - 1 4 . G r o u p T h e o r y I s U s e d t o D e t e r m i n e t h e I n f r a r e d A c t i v i t y o f N o r m a l

C o o r d i n a t e V i b r a t i o n s 5 3 5

P r o b l e m s 5 3 7

C H A P T E R 1 4 I N u c l e a r M a g n e t i c R e s o n a n c e S p e c t r o s c o p y 5 4 7

1 4 - 1 . N u c l e i H a v e I n t r i n s i c S p i n A n g u l a r M o m e n t a 5 4 8

1 4 - 2 . M a g n e t i c M o m e n t s I n t e r a c t w i t h M a g n e t i c F i e l d s 5 5 0

1 4 - 3 . P r o t o n N M R S p e c t r o m e t e r s O p e r a t e a t F r e q u e n c i e s B e t w e e n 6 0 M H z a n d

7 5 0 M H z 5 5 4

1 4 - 4 . T h e M a g n e t i c F i e l d A c t i n g u p o n N u c l e i i n M o l e c u l e s I s S h i e l d e d 5 5 6

1 4 - 5 . C h e m i c a l S h i f t s D e p e n d u p o n t h e C h e m i c a l E n v i r o n m e n t o f t h e N u c l e u s 5 6 0

1 4 - 6 . S p i n - S p i n C o u p l i n g C a n L e a d t o M u l t i p l e t s i n N M R S p e c t r a 5 6 2

i x

PHYSICAL CHEMISTRY

14-7. Spin-Spin Coupling Between Chemically Equivalent Protons Is Not Observed 570 14-8. Then+ 1 Rule Applies Only to First-Order Spectra 573 14-9. Second-Order Spectra Can Be Calculated Exactly Using the Variational Method 576 Problems 585

CHAPTER 15 I Lasers, Laser Spectroscopy, and Photochemistry 591 15-1. Electronically Excited Molecules Can Relax by a Number of Processes 592 15-2. The Dynamics of Spectroscopic Transitions Between the Electronic States of Atoms

Can Be Modeled by Rate Equations 595 15-3. A Two-Level System Cannot Achieve a Population Inversion 601 15-4. Population Inversion Can Be Achieved in a Three-Level System 603 15-5. What Is Inside a Laser? 604 15-6. The Helium-Neon Laser is an Electrical-Discharge Pumped, Continuous-Wave,

Gas-Phase Laser 609 15-7. High-Resolution Laser Spectroscopy Can Resolve Absorption Lines That Cannot Be

Distinguished by Conventional Spectrometers 613 15-8. Pulsed Lasers Can Be Used to Measure the Dynamics of Photochemical

Processes 61 4 Problems 62 0

MATHCHAPTER G I Numerical Methods 627 Problems 634

CHAPTER 16 I The Properties of Gases 637 16-1. All Gases Behave Ideally lfThey Are Sufficiently Dilute 637 16-2. The van der Waals Equation and the Redlich-Kwong Equation Are Examples

of Two-Parameter Equations of State 642 16-3. A Cubic Equation of State Can Describe Both the Gaseous and Liquid States 648 16-4. The van der Waals Equation and the Redlich-Kwong Equation Obey the Law

of Corresponding States 655 16-5. Second Virial Coefficients Can Be Used to Determine Intermolecular Potentials 658 16-6. London Dispersion Forces Are Often the Largest Contribution to the r-6 Term in the

Lennard-jones Potential 665 16-7. The van der Waals Constants Can Be Written in Terms of Molecular Parameters 670 Problems 67 4

MATHCHAPTER H I Partial Differentiation 683 Problems 689

~ The Boltzmann Factor and Partition Functions 693 17-1. The Boltzmann Factor Is One of the Most Important Quantities in the Physical

Sciences 694 17-2. The Probability That a System in an Ensemble Is in the State j with Energy E.(N, V)

1 Is Proportional toe -Ej(N.Vl/kBT 696

17-3. We Postulate That the Average Ensemble Energy Is Equal to the Observed Energy of a System 698

17-4. The Heat Capacity at Constant Volume Is the Temperature Derivative of the Average Energy 702

17-5. We Can Express the Pressure in Terms of a Partition Function 704

X

C o n t e n t s

1 7 - 6 . T h e P a r t i t i o n F u n c t i o n o f a S y s t e m o f I n d e p e n d e n t , D i s t i n g u i s h a b l e M o l e c u l e s

I s t h e P r o d u c t o f M o l e c u l a r P a r t i t i o n F u n c t i o n s 7 0 7

1 7 - 7 . T h e P a r t i t i o n F u n c t i o n o f a S y s t e m o f I n d e p e n d e n t , I n d i s t i n g u i s h a b l e A t o m s o r

M o l e c u l e s C a n U s u a l l y B e W r i t t e n a s [ q ( V , T ) ] N 1 N ! 7 0 8

1 7 - 8 . A M o l e c u l a r P a r t i t i o n F u n c t i o n C a n B e D e c o m p o s e d i n t o P a r t i t i o n F u n c t i o n s

f o r E a c h D e g r e e o f F r e e d o m 7 1 3

P r o b l e m s 7 1 6

M A T H C H A P T E R I I S e r i e s a n d L i m i t s 7 2 3

P r o b l e m s 7 2 8

C H A P T E R 1 8 I P a r t i t i o n F u n c t i o n s a n d I d e a l G a s e s 7 3 1

1 8 - 1 . T h e T r a n s l a t i o n a l P a r t i t i o n F u n c t i o n o f a n A t o m i n a M o n a t o m i c I d e a l G a s I s

( 2 : r r m k B T I h

2

)

3

1

2

V 7 3 1

1 8 - 2 . M o s t A t o m s A r e i n t h e G r o u n d E l e c t r o n i c S t a t e a t R o o m T e m p e r a t u r e 7 3 3

1 8 - 3 . T h e E n e r g y o f a D i a t o m i c M o l e c u l e C a n B e A p p r o x i m a t e d a s a S u m

o f S e p a r a t e T e r m s 7 3 7

1 8 - 4 . M o s t M o l e c u l e s A r e i n t h e G r o u n d V i b r a t i o n a l S t a t e a t R o o m T e m p e r a t u r e 7 4 0

1 8 - 5 . M o s t M o l e c u l e s A r e i n E x c i t e d R o t a t i o n a l S t a t e s a t O r d i n a r y T e m p e r a t u r e s 7 4 3

1 8 - 6 . R o t a t i o n a l P a r t i t i o n F u n c t i o n s C o n t a i n a S y m m e t r y N u m b e r 7 4 6

1 8 - 7 . T h e V i b r a t i o n a l P a r t i t i o n F u n c t i o n o f a P o l y a t o m i c M o l e c u l e I s a P r o d u c t o f H a r m o n i c

O s c i l l a t o r P a r t i t i o n F u n c t i o n s f o r E a c h N o r m a l C o o r d i n a t e 7 4 9

1 8 - 8 . T h e F o r m o f t h e R o t a t i o n a l P a r t i t i o n F u n c t i o n o f a P o l y a t o m i c M o l e c u l e D e p e n d s u p o n

t h e S h a p e o f t h e M o l e c u l e 7 5 2

1 8 - 9 . C a l c u l a t e d M o l a r H e a t C a p a c i t i e s A r e i n V e r y G o o d A g r e e m e n t w i t h

E x p e r i m e n t a l D a t a 7 5 4

P r o b l e m s 7 5 7

C H A P T E R 1 9 I T h e F i r s t L a w o f T h e r m o d y n a m i c s 7 6 5

1 9 - 1 . A C o m m o n T y p e o f W o r k i s P r e s s u r e - V o l u m e W o r k 7 6 6

1 9 - 2 . W o r k a n d H e a t A r e N o t S t a t e F u n c t i o n s , b u t E n e r g y I s a S t a t e F u n c t i o n 7 6 9

1 9 - 3 . T h e F i r s t L a w o f T h e r m o d y n a m i c s S a y s t h e E n e r g y I s a S t a t e F u n c t i o n 7 7 3

1 9 - 4 . A n A d i a b a t i c P r o c e s s I s a P r o c e s s i n W h i c h N o E n e r g y a s H e a t I s T r a n s f e r r e d 7 7 4

1 9 - 5 . T h e T e m p e r a t u r e o f a G a s D e c r e a s e s i n a R e v e r s i b l e A d i a b a t i c E x p a n s i o n 7 7 7

1 9 - 6 . W o r k a n d H e a t H a v e a S i m p l e M o l e c u l a r I n t e r p r e t a t i o n 7 7 9

1 9 - 7 . T h e E n t h a l p y C h a n g e I s E q u a l t o t h e E n e r g y T r a n s f e r r e d a s H e a t i n a

C o n s t a n t - P r e s s u r e P r o c e s s I n v o l v i n g O n l y P - V W o r k 7 8 0

1 9 - 8 . H e a t C a p a c i t y I s a P a t h F u n c t i o n 7 8 3

1 9 - 9 . R e l a t i v e E n t h a l p i e s C a n B e D e t e r m i n e d f r o m H e a t C a p a c i t y D a t a a n d H e a t s

o f T r a n s i t i o n 7 8 6

1 9 - 1 0 . E n t h a l p y C h a n g e s f o r C h e m i c a l E q u a t i o n s A r e A d d i t i v e 7 8 7

1 9 - 1 1 . H e a t s o f R e a c t i o n s C a n B e C a l c u l a t e d f r o m T a b u l a t e d H e a t s o f F o r m a t i o n 7 9 1

1 9 - 1 2 . T h e T e m p e r a t u r e D e p e n d e n c e o f t ; . , H I s G i v e n i n T e r m s o f t h e H e a t

C a p a c i t i e s o f t h e R e a c t a n t s a n d P r o d u c t s 7 9 7

P r o b l e m s 8 0 0

M A T H C H A P T E R J I T h e B i n o m i a l D i s t r i b u t i o n a n d S t i r l i n g ' s

A p p r o x i m a t i o n 8 0 9

P r o b l e m s 8 1 4

X I

r

CHAPTER 20 I Entropy and the Second Law of Thermodynamics 817

PHYSICAL CHEMISTRY

20-1. The Change of Energy Alone Is Not Sufficient to Determine the Direction of a Spontaneous Process 817

20-2. Nonequilibrium Isolated Systems Evolve in a Direction That Increases Their Disorder 819

20-3. Unlike qrev' Entropy Is a State Function 821 20-4. The Second Law of Thermodynamics States That the Entropy of an Isolated System

Increases as a Result of a Spontaneous Process 825 20-5. The Most Famous Equation of Statistical Thermodynamics Is S = k8 In W 829 20-6. We Must Always Devise a Reversible Process to Calculate Entropy Changes 833 20-7. Thermodynamics Gives Us Insight into the Conversion of Heat into Work 838 20-8. Entropy Can Be Expressed in Terms of a Partition Function 840 20-9. The Molecular Formula S = k8 In W Is Analogous to the Thermodynamic Formula

dS = 8qrevfT 843 Problems 844

CHAPTER 21 I Entropy and the Third Law of Thermodynamics 853 21-1. Entropy Increases with Increasing Temperature 853 21-2. The Third Law of Thermodynamics Says That the Entropy of a Perfect Crystal

Is Zero at 0 K 855 21-3. t.1,,S = t.1,,H/Ttrs at a Phase Transition 857 21-4. The Third Law of Thermodynamics Asserts That C P ---+ 0 as T ---+ 0 858 21-5. Practical Absolute Entropies Can Be Determined Calorimetrically 859 21-6. Practical Absolute Entropies of Gases Can Be Calculated from Partition Functions 861 21-7. The Values of Standard Entropies Depend upon Molecular Mass and Molecular

Structure 865 21-8. The Spectroscopic Entropies of a Few Substances Do Not Agree with the

Calorimetric Entropies 868 21-9. Standard Entropies Can Be Used to Calculate Entropy Changes of Chemical

Reactions 869 Problems 870

CHAPTER 22 I Helmholtz and Gibbs Energies 881 22-1. The Sign of the Helmholtz Energy Change Determines the Direction of a Spontaneous

Process in a System at Constant Volume and Temperature 881 22-2. The Gibbs Energy Determines the Direction of a Spontaneous Process for a System

at Constant Pressure and Temperature 884 22-3. Maxwell Relations Provide Several Useful Thermodynamic Formulas 888 22-4. The Enthalpy of an Ideal Gas Is Independent of Pressure 893 22-5. The Various Thermodynamic Functions Have Natural Independent Variables 896 22-6. The Standard State for a Gas at Any Temperature Is the Hypothetical Ideal Gas

at One Bar 899 22-7. The Gibbs-Helmholtz Equation Describes the Temperature Dependence of the

Gibbs Energy 901 22-8. Fugacity Is a Measure of the Non ideality of a Gas 905 Problems 91 0

CHAPTER 23 I Phase Equilibria 925 xii

C o n t e n t s

2 3 - 1 . A P h a s e D i a g r a m S u m m a r i z e s t h e S o l i d - L i q u i d - G a s B e h a v i o r o f a S u b s t a n c e 9 2 6

2 3 - 2 . T h e G i b b s E n e r g y o f a S u b s t a n c e H a s a C l o s e C o n n e c t i o n t o I t s P h a s e D i a g r a m 9 3 3

2 3 - 3 . T h e C h e m i c a l P o t e n t i a l s o f a P u r e S u b s t a n c e i n T w o P h a s e s i n E q u i l i b r i u m

A r e E q u a l 9 3 5

2 3 - 4 . T h e C l a u s i u s - C l a p e y r o n E q u a t i o n G i v e s t h e V a p o r P r e s s u r e o f a S u b s t a n c e A s a

F u n c t i o n o f T e m p e r a t u r e 9 4 1

2 3 - 5 . C h e m i c a l P o t e n t i a l C a n B e E v a l u a t e d f r o m a P a r t i t i o n F u n c t i o n 9 4 5

P r o b l e m s 9 4 9

C H A P T E R 2 4 I C h e m i c a l E q u i l i b r i u m 9 6 3

2 4 - 1 . C h e m i c a l E q u i l i b r i u m R e s u l t s w h e n t h e G i b b s E n e r g y I s a M i n i m u m w i t h R e s p e c t

t o t h e E x t e n t o f R e a c t i o n 9 6 3

2 4 - 2 . A n E q u i l i b r i u m C o n s t a n t I s a F u n c t i o n o f T e m p e r a t u r e O n l y 9 6 7

2 4 - 3 . S t a n d a r d G i b b s E n e r g i e s o f F o r m a t i o n C a n B e U s e d t o C a l c u l a t e E q u i l i b r i u m

C o n s t a n t s 9 7 0

2 4 - 4 . A P l o t o f t h e G i b b s E n e r g y o f a R e a c t i o n M i x t u r e A g a i n s t t h e E x t e n t o f R e a c t i o n

I s a M i n i m u m a t E q u i l i b r i u m 9 7 2

2 4 - 5 . T h e R a t i o o f t h e R e a c t i o n Q u o t i e n t t o t h e E q u i l i b r i u m C o n s t a n t D e t e r m i n e s t h e

D i r e c t i o n i n w h i c h a R e a c t i o n W i l l P r o c e e d 9 7 4

2 4 - 6 . T h e S i g n o f , ! c , } J A n d N o t T h a t o f b . , G o D e t e r m i n e s t h e D i r e c t i o n o f R e a c t i o n

S p o n t a n e i t y 9 7 6

2 4 - 7 . T h e V a r i a t i o n o f a n E q u i l i b r i u m C o n s t a n t w i t h T e m p e r a t u r e I s G i v e n b y t h e

V a n ' t H o f f E q u a t i o n 9 7 7

2 4 - 8 . W e C a n C a l c u l a t e E q u i l i b r i u m C o n s t a n t s i n T e r m s o f P a r t i t i o n F u n c t i o n s 9 8 1

2 4 - 9 . M o l e c u l a r P a r t i t i o n F u n c t i o n s a n d R e l a t e d T h e r m o d y n a m i c D a t a A r e

E x t e n s i v e l y T a b u l a t e d 9 8 5

2 4 - 1 0 . E q u i l i b r i u m C o n s t a n t s f o r R e a l G a s e s A r e E x p r e s s e d i n T e r m s o f P a r t i a l

F u g a c i t i e s 9 9 2

2 4 - 1 1 . T h e r m o d y n a m i c E q u i l i b r i u m C o n s t a n t s A r e E x p r e s s e d i n T e r m s o f A c t i v i t i e s 9 9 4

P r o b l e m s 9 9 8

C H A P T E R 2 5 I T h e K i n e t i c T h e o r y o f G a s e s 1 0 1 1

2 5 - 1 . T h e A v e r a g e T r a n s l a t i o n a l K i n e t i c E n e r g y o f t h e M o l e c u l e s i n a G a s I s D i r e c t l y

P r o p o r t i o n a l t o t h e K e l v i n T e m p e r a t u r e 1 0 1 1

2 5 - 2 . T h e D i s t r i b u t i o n o f t h e C o m p o n e n t s o f M o l e c u l a r S p e e d s A r e D e s c r i b e d b y a

G a u s s i a n D i s t r i b u t i o n 1 0 1 6

2 5 - 3 . T h e D i s t r i b u t i o n o f M o l e c u l a r S p e e d s I s G i v e n b y t h e M a x w e l l - B o l t z m a n n

D i s t r i b u t i o n 1 0 2 2

2 5 - 4 . T h e F r e q u e n c y o f C o l l i s i o n s T h a t a G a s M a k e s w i t h a W a l l i s P r o p o r t i o n a l t o I t s

N u m b e r D e n s i t y a n d t o t h e A v e r a g e M o l e c u l a r S p e e d 1 0 2 6

2 5 - 5 . T h e M a x w e l l - B o l t z m a n n D i s t r i b u t i o n H a s B e e n V e r i f i e d E x p e r i m e n t a l l y 1 0 2 9

2 5 - 6 . T h e M e a n F r e e P a t h I s t h e A v e r a g e D i s t a n c e a M o l e c u l e T r a v e l s

B e t w e e n C o l l i s i o n s 1 0 3 1

2 5 - 7 . T h e R a t e o f a G a s - P h a s e C h e m i c a l R e a c t i o n D e p e n d s U p o n t h e R a t e o f C o l l i s i o n s

i n w h i c h t h e R e l a t i v e K i n e t i c E n e r g y E x c e e d s S o m e C r i t i c a l V a l u e 1 0 3 7

P r o b l e m s 1 0 3 9

C H A P T E R 2 6 I C h e m i c a l K i n e t i c s 1 : R a t e L a w s 1 0 4 7

2 6 - 1 . T h e T i m e D e p e n d e n c e o f a C h e m i c a l R e a c t i o n I s D e s c r i b e d b y a R a t e L a w 1 0 4 8

x i i i

PHYSICAL CHEMISTRY

26-2. Rate Laws Must Be Determined Experimentally 1051 26-3. First-Order Reactions Show an Exponential Decay of Reactant Concentration

with Time 1054 26-4. The Rate Laws for Different Reaction Orders Predict Different Behaviors for the

Time-Dependent Reactant Concentration 1058 26-5. Reactions Can Also Be Reversible 1 062 26-6. The Rate Constants of a Reversible Reaction Can Be Determined Using

Relaxation Methods 1 062 26-7. Rate Constants Are Usually Strongly Temperature Dependent 1071 26-8. Transition-State Theory Can Be Used to Estimate Reaction Rate Constants 1075 Problems 1 079

CHAPTER 27 I Chemical Kinetics II: Reaction Mechanisms 1091 27-1. A Mechanism is a Sequence of Single-Step Chemical Reactions called

Elementary Reactions 1 092 27-2. The Principle of Detailed Balance States That when a Complex Reaction is at

Equilibrium, the Rate of the Forward Process Is Equal to the Rate of the Reverse Process for Each and Every Step of the Reaction Mechanism 1093

27-3. When Are Consecutive and Single-Step Reactions Distinguishable? 1196 27-4. The Steady-State Approximation Simplifies Rate Expressions by Assuming

That d[l]/dt = 0, where lis a Reaction Intermediate 1101 27-5. The Rate Law for a Complex Reaction Does Not Imply a Unique Mechanism 1103 27-6. The Lindemann Mechanism Explains How Unimolecular Reactions Occur 1108 27-7. Some Reaction Mechanisms Involve Chain Reactions 1113 27-8. A Catalyst Affects the Mechanism and Activation Energy of a Chemical Reaction 1116 27-9. The Michaelis-Menten Mechanism Is a Reaction Mechanism for Enzyme

Catalysis 1119 Problems 1123

CHAPTER 28 I Gas-Phase Reaction Dynamics 1139 28-1. The Rate of a Bimolecular Gas-Phase Reaction Can Be Calculated Using Hard-Sphere

Collision Theory and an Energy-Dependent Reaction Cross Section 1139 28-2. A Reaction Cross Section Depends upon the Impact Parameter 1144 28-3. The Rate Constant for a Gas-Phase Chemical Reaction May Depend on the

Orientations of the Colliding Molecules 1147 28-4. The Internal Energy of the Reactants Can Affect the Cross Section of a Reaction 1148 28-5. A Reactive Collision Can Be Described in a Center-of-Mass Coordinate System 1149 28-6. Reactive Collisions Can Be Studied Using Crossed Molecular Beam Machines 1154 28-7. The Reaction F(g) + D2 (g) =? DF(g) + D(g) Can Produce Vibrationally Excited

DF(g) Molecules 1156 28-8. The Velocity and Angular Distribution of the Products of a Reactive Collision

Provide a Molecular Picture of the Chemical Reaction 1158 28-9. Not All Gas-Phase Chemical Reactions Are Rebound Reactions 1165

28-10. The Potential-Energy Surface for the Reaction F(g) + D2(g) =? DF(g) + D(g) Can Be Calculated Using Quantum Mechanics 1168

Problems 11 71

CHAPTER 29 I Solids and Surface Chemistry 1181 29-1. The Unit Cell Is the Fundamental Building Block of a Crystal 1181

xiv

C o n t e n t s

2 9 - 2 . T h e O r i e n t a t i o n o f a L a t t i c e P l a n e I s D e s c r i b e d b y I t s M i l l e r I n d i c e s 1 1 8 1

2 9 - 3 . T h e S p a c i n g B e t w e e n L a t t i c e P l a n e s C a n B e D e t e r m i n e d f r o m X - R a y D i f f r a c t i o n

M e a s u r e m e n t s 1 1 9 1

2 9 - 4 . T h e T o t a l S c a t t e r i n g I n t e n s i t y I s R e l a t e d t o t h e P e r i o d i c S t r u c t u r e o f t h e E l e c t r o n

D e n s i t y i n t h e C r y s t a l 1 1 9 8

2 9 - 5 . T h e S t r u c t u r e F a c t o r a n d t h e E l e c t r o n D e n s i t y A r e R e l a t e d b y a F o u r i e r

T r a n s f o r m 1 2 0 3

2 9 - 6 . A G a s M o l e c u l e C a n P h y s i s o r b o r C h e m i s o r b t o a S o l i d S u r f a c e 1 2 0 5

2 9 - 7 . I s o t h e r m s A r e P l o t s o f S u r f a c e C o v e r a g e a s a F u n c t i o n o f G a s P r e s s u r e a t

C o n s t a n t T e m p e r a t u r e 1 2 0 7

2 9 - 8 . T h e L a n g m u i r I s o t h e r m C a n B e U s e d t o D e r i v e R a t e L a w s f o r S u r f a c e - C a t a l y z e d

G a s - P h a s e R e a c t i o n s 1 2 1 3

2 9 - 9 . T h e S t r u c t u r e o f a S u r f a c e i s D i f f e r e n t f r o m T h a t o f a B u l k S o l i d 1 2 1 7

2 9 - 1 0 . T h e R e a c t i o n B e t w e e n H

2

( g ) a n d N

2

( g ) t o M a k e N H

3

( g ) C a n B e

S u r f a c e C a t a l y z e d 1 2 1 9

P r o b l e m s 1 2 2 1

A n s w e r s t o t h e N u m e r i c a l P r o b l e m s 1 2 3 7

I l l u s t r a t i o n C r e d i t s 1 2 5 7

I n d e x 1 2 5 9

X V

P r e f a c e

T o t h e S t u d e n t

Y o u a r e a b o u t t o b e g i n y o u r s t u d y o f p h y s i c a l c h e m i s t r y . Y o u m a y h a v e b e e n t o l d t h a t

p h y s i c a l c h e m i s t r y i s t h e m o s t d i f f i c u l t c h e m i s t r y c o u r s e t h a t y o u w i l l t a k e , o r y o u

m a y h a v e e v e n s e e n t h e b u m p e r s t i c k e r t h a t s a y s " H o n k i f y o u p a s s e d P C h e r n . " T h e

a n x i e t y t h a t s o m e s t u d e n t s b r i n g t o t h e i r p h y s i c a l c h e m i s t r y c o u r s e h a s b e e n e l o q u e n t l y

a d d r e s s e d b y t h e B r i t i s h p r o f e s s o r E . B r i a n S m i t h i n t h e p r e f a c e o f h i s i n t r o d u c t o r y

t e x t , B a s i c C h e m i c a l T h e r m o d y n a m i c s , O x f o r d U n i v e r s i t y P r e s s :

_ - - - . . T h e f i r s t t i m e I h e a r d a b o u t C h e m i c a l T h e r m o d y n a m i c s w a s w h e n a s e c o n d -

y e a r u n d e r g r a d u a t e b r o u g h t m e t h e n e w s i n m y f r e s h m a n y e a r . H e t o l d a

s p i n e - c h i l l i n g s t o r y o f e n d l e s s l e c t u r e s w i t h a l m o s t t h r e e h u n d r e d n u m b e r e d

e q u a t i o n s , a l l o f w h i c h , i t a p p e a r e d , h a d t o b e c o m m i t t e d t o m e m o r y a n d

r e p r o d u c e d i n e x a c t l y t h e s a m e f o r m i n s u b s e q u e n t e x a m i n a t i o n s . N o t o n l y d i d

t h e s e e q u a t i o n s c o n t a i n a l l t h e n o r m a l a l g e b r a i c s y m b o l s b u t i n a d d i t i o n t h e y

w e r e l i b e r a l l y s p r i n k l e d w i t h s t a r s , d a g g e r s , a n d c i r c l e s s o a s t o s t r e t c h e v e n

t h e m o s t p o w e r f u l o f m i n d s . F e w w o u l d w i s h t o d e n y t h e m i n d - i m p r o v i n g a n d

i n d e e d c h a r a c t e r - b u i l d i n g q u a l i t i e s o f s u c h a s u b j e c t ! H o w e v e r , m a n y y o u n g

c h e m i s t s h a v e m o r e u r g e n t p r e s s u r e s o n t h e i r t i m e . "

W e c e r t a i n l y a g r e e w i t h t h i s l a s t s e n t e n c e o f P r o f e s s o r S m i t h ' s . T h e f a c t i s , h o w e v e r ,

t h a t e v e r y y e a r t h o u s a n d s u p o n t h o u s a n d s o f s t u d e n t s t a k e a n d p a s s p h y s i c a l c h e m i s t r y ,

a n d m a n y o f t h e m r e a l l y e n j o y i t . Y o u m a y b e t a k i n g i t o n l y b e c a u s e i t i s r e q u i r e d

b y y o u r m a j o r , b u t y o u s h o u l d b e a w a r e t h a t m a n y r e c e n t d e v e l o p m e n t s i n p h y s i c a l

c h e m i s t r y a r e h a v i n g a m a j o r i m p a c t i n a l l t h e a r e a s o f s c i e n c e t h a t a r e c o n c e r n e d w i t h

t h e b e h a v i o r o f m o l e c u l e s . F o r e x a m p l e , i n b i o p h y s i c a l c h e m i s t r y , t h e a p p l i c a t i o n o f

b o t h e x p e r i m e n t a l a n d t h e o r e t i c a l a s p e c t s o f p h y s i c a l c h e m i s t r y t o b i o l o g i c a l p r o b l e m s

h a s g r e a t l y a d v a n c e d o u r u n d e r s t a n d i n g o f t h e s t r u c t u r e a n d r e a c t i v i t y o f p r o t e i n s a n d

n u c l e i c a c i d s . T h e d e s i g n o f p h a r m a c e u t i c a l d r u g s , w h i c h h a s s e e n g r e a t a d v a n c e s i n

r e c e n t y e a r s , i s a d i r e c t p r o d u c t o f p h y s i c a l c h e m i c a l r e s e a r c h .

T r a d i t i o n a l l y , t h e r e a r e t h r e e p r i n c i p a l a r e a s o f p h y s i c a l c h e m i s t r y : t h e r m o d y n a m -

i c s ( w h i c h c o n c e r n s t h e e n e r g e t i c s o f c h e m i c a l r e a c t i o n s ) , q u a n t u m c h e m i s t r y ( w h i c h

c o n c e r n s t h e s t r u c t u r e s o f m o l e c u l e s ) , a n d c h e m i c a l kinetic~ ( w h i c h c o n c e r n s t h e r a t e s X V I I

xviii PHYSICAL CHEMISTRY

of chemical reactions). Many physical chemistry courses begin with a study of ther-modynamics, then discuss quantum chemistry, and treat chemical kinetics last. This order is a reflection of the historic development of the field. Today, however, physical chemistry is based on quantum mechanics, and so we begin our studies with this topic. We first discuss the underlying principles of quantum mechanics and then show how they can be applied to a number of model systems. Many of the rules you have learned in general chemistry and organic chemistry are a natural result of the quantum theory. In organic chemistry, for example, you learned to assign molecular structures using infrared spectra and nuclear magnetic resonance spectra, and in Chapters 13 and 14 we explain how these spectra are governed by the quantum-mechanical properties of molecules.

Your education in chemistry has trained you to think in terms of molecules and their interactions, and we believe that a course in physical chemistry should reflect this viewpoint. The focus of modem physical chemistry is on the molecule. Cur-rent experimental research in physical chemistry uses equipment such as molecular beam machines to study the molecular details of gas-phase chemical reactions, high vacuum machines to study the structure and reactivity of molecules on solid inter-faces, lasers to determine the structures of individual molecules and the dynamics of chemical reactions, and nuclear magnetic resonance spectrometers to learn about the structure and dynamics of molecules. Modem theoretical research in physical chemistry uses the tools of classical mechanics, quantum mechanics, and statistical mechanics along with computers to develop a detailed understanding of chemical phenomena in terms of the structure and dynamics of the molecules involved. For example, computer calculations of the electronic structure of molecules are providing fundamental insights into chemical bonding and computer simulations of the dynamical interaction between molecules and proteins are being used to understand how proteins function.

In general chemistry, you learned about the three laws of thermodynamics and were introduced to the quantities, enthalpy, entropy, and the Gibbs energy (formerly called the free energy). Thermodynamics is used to describe macroscopic chemical systems. Armed with the tools of quantum mechanics, you shall learn that thermody-namics can be formulated in terms of the properties of the atoms and molecules that make up macroscopic chemical systems. Statistical thermodynamics provides a way to describe thermodynamics at a molecular level. You shall see that the three laws of thermodynamics can be explained simply and beautifully in molecular terms. We be-lieve that a modem introduction to physical chemistry should, from the outset, develop the field of thermodynamics from a molecular viewpoint.

Our treatment of chemical kinetics, which constitutes the last five chapters, devel-ops an understanding of chemical reactions from a molecular viewpoint. For example, we have devoted more than half of the chapter of gas-phase reactions (Chapter 28) to the reaction between a fluorine atom and a hydrogen molecule to form a hydrogen fluoride molecule and a hydrogen atom. Through our study of this seemingly simple reaction, many of the general molecular concepts of chemical reactivity are revealed. Again, quantum chemistry provides the necessary tools to develop a molecular understanding of the rates and dynamics of chemical reactions.

P r e f a c e

P e r h a p s t h e m o s t i n t i m i d a t i n g a s p e c t o f p h y s i c a l c h e m i s t r y i s t h e l i b e r a l u s e o f

m a t h e m a t i c a l t o p i c s t h a t y o u m a y h a v e f o r g o t t e n o r n e v e r l e a r n e d . A s p h y s i c i s t s s a y

a b o u t p h y s i c s , p h y s i c a l c h e m i s t r y i s d i f f i c u l t w i t h m a t h e m a t i c s ; i m p o s s i b l e w i t h o u t i t .

Y o u m a y n o t h a v e t a k e n a m a t h c o u r s e r e c e n t l y , a n d y o u r u n d e r s t a n d i n g o f t o p i c s s u c h

a s d e t e r m i n a n t s , v e c t o r s , s e r i e s e x p a n s i o n s , a n d p r o b a b i l i t y m a y s e e m a b i t f u z z y a t

t h i s t i m e . I n o u r y e a r s o f t e a c h i n g p h y s i c a l c h e m i s t r y , w e h a v e o f t e n f o u n d i t h e l p f u l t o

r e v i e w m a t h e m a t i c a l t o p i c s b e f o r e u s i n g t h e m t o d e v e l o p t h e p h y s i c a l c h e m i c a l t o p i c s .

C o n s e q u e n t l y , w e h a v e i n c l u d e d a s e r i e s o f t e n c o n c i s e r e v i e w s o f m a t h e m a t i c a l t o p i c s .

W e r e a l i z e t h a t n o t e v e r y o n e o f t h e s e s o - c a l l e d r e v i e w s m a y a c t u a l l y b e a r e v i e w f o r

y o u . E v e n i f s o m e o f t h e t o p i c s a r e n e w t o y o u ( o r s e e m t h a t w a y ) , w e d i s c u s s o n l y

t h e m i n i m u m a m o u n t t h a t y o u n e e d t o k n o w t o u n d e r s t a n d t h e s u b s e q u e n t p h y s i c a l

c h e m i s t r y . W e h a v e p o s i t i o n e d t h e s e r e v i e w s s o t h a t t h e y i m m e d i a t e l y p r e c e d e t h e f i r s t

c h a p t e r t h a t u s e s t h e m . B y r e a d i n g t h e s e r e v i e w s f i r s t ( a n d d o i n g t h e p r o b l e m s ! ) , y o u

w i l l b e a b l e t o s p e n d l e s s t i m e w o r r y i n g a b o u t t h e m a t h , a n d m o r e t i m e l e a r n i n g t h e

p h y s i c a l c h e m i s t r y , w h i c h i s , a f t e r a l l , y o u r g o a l .

T o t h e I n s t r u c t o r

T h i s t e x t e m p h a s i z e s a m o l e c u l a r a p p r o a c h t o p h y s i c a l c h e m i s t r y . C o n s e q u e n t l y , u n l i k e

m o s t o t h e r p h y s i c a l c h e m i s t r y b o o k s , w e d i s c u s s t h e p r i n c i p l e s o f q u a n t u m m e c h a n i c s

f i r s t a n d t h e n u s e t h e s e i d e a s e x t e n s i v e l y i n o u r s u b s e q u e n t d e v e l o p m e n t o f t h e r m o d y -

n a m i c s a n d k i n e t i c s . F o r e x a m p l e , f r o m t h e C o n t e n t s y o u w i l l s e e t h a t c h a p t e r s t i t l e d

T h e B o l t z m a n n F a c t o r a n d P a r t i t i o n F u n c t i o n s ( C h a p t e r 1 7 ) a n d P a r t i t i o n F u n c t i o n s

a n d I d e a l G a s e s ( C h a p t e r 1 8 ) c o m e b e f o r e T h e F i r s t L a w o f T h e r m o d y n a m i c s ( C h a p -

t e r 1 9 ) . T h i s a p p r o a c h i s p e d a g o g i c a l l y s o u n d b e c a u s e w e t r e a t o n l y e n e r g y , p r e s s u r e ,

a n d h e a t c a p a c i t y ( a l l m e c h a n i c a l p r o p e r t i e s t h a t t h e s t u d e n t s h a v e d e a l t w i t h i n t h e i r

g e n e r a l c h e m i s t r y a n d p h y s i c s c o u r s e s ) i n C h a p t e r s 1 7 a n d 1 8 . T h i s a p p r o a c h a l l o w s

u s t o i m m e d i a t e l y g i v e a m o l e c u l a r i n t e r p r e t a t i o n t o t h e t h r e e l a w s o f t h e r m o d y n a m -

i c s a n d t o m a n y t h e r m o d y n a m i c r e l a t i o n s . T h e m o l e c u l a r i n t e r p r e t a t i o n o f e n t r o p y i s

a n o b v i o u s e x a m p l e ( a n i n t r o d u c t i o n t o e n t r o p y w i t h o u t a m o l e c u l a r i n t e r p r e t a t i o n i s

s t r i c t l y f o r p u r i s t s a n d n o t f o r t h e f a i n t o f h e a r t ) , b u t e v e n t h e c o n c e p t s o f w o r k a n d h e a t

i n t h e F i r s t L a w o f T h e r m o d y n a m i c s h a v e a n i c e , p h y s i c a l , m o l e c u l a r i n t e r p r e t a t i o n i n

t e r m s o f e n e r g y l e v e l s a n d t h e i r p o p u l a t i o n s .

T e c h n o l o g i c a l a d v a n c e s d u r i n g t h e p a s t f e w d e c a d e s h a v e c h a n g e d t h e f o c u s o f

p h y s i c a l c h e m i c a l r e s e a r c h a n d t h e r e f o r e s h o u l d a f f e c t t h e t o p i c s c o v e r e d i n a m o d e m

p h y s i c a l c h e m i s t r y c o u r s e . V e r y f e w p h y s i c a l c h e m i s t s t o d a y m e a s u r e s u c h q u a n t i t i e s

a s v a p o r p r e s s u r e s , m u l t i c o m p o n e n t p h a s e d i a g r a m s , e l e c t r o d e p o t e n t i a l s , o r t h e t h e r -

m o d y n a m i c p r o p e r t i e s o f i o n s i n s o l u t i o n . T o i n t r o d u c e t h e t y p e o f p h y s i c a l c h e m i c a l

r e s e a r c h t h a t i s c u r r e n t l y b e i n g d o n e , w e h a v e i n c l u d e d c h a p t e r s s u c h a s C o m p u t a t i o n a l

Q u a n t u m C h e m i s t r y ( C h a p t e r 1 1 ) , G r o u p T h e o r y ( C h a p t e r 1 2 ) , N u c l e a r M a g n e t i c R e s -

o n a n c e S p e c t r o s c o p y ( C h a p t e r 1 4 ) , L a s e r s , L a s e r S p e c t r o s c o p y , a n d P h o t o c h e m i s t r y

( C h a p t e r 1 5 ) , a n d G a s - P h a s e R e a c t i o n D y n a m i c s ( C h a p t e r 2 8 ) . T h e i n c l u s i o n o f n e w

t o p i c s n e c e s s i t a t e d t h e o m i s s i o n o f s o m e t o p i c s t h a t a r e t r a d i t i o n a l l y c o v e r e d i n a p h y s -

i c a l c h e m i s t r y c o u r s e . T h e c h o i c e o f w h a t t o p i c s t o d e l e t e w a s d i f f i c u l t , b u t w e h a d

X I X

XX PHYSICAL CHEMISTRY

to make such decisions because our publisher was reluctant to publish a 2000 page physical chemistry book.

For example, we have no chapters on the thermodynamics of solutions, which usually discuss Raoult's law and deviations from it, leading to the introduction of activities and their associated standard states. As important as this material might be, some topics must be eliminated to make room for new material. Furthermore, an informal survey we made over the past year or so has assured us that most instructors teaching physical chemistry for the first few times are not comfortable with Raoult's law standard states and Henry's law standard states, and they don't really enjoy teaching the thermodynamics of solutions. Another classical topic that we have eliminated is electrochemistry. This topic, which traditionally constitutes a chapter in physical chemistry books, is now amply covered in analytical chemistry courses. Some other traditional topics that have been omitted are the Gibbs phase rule, the Carnot cycle, Debye-Hiickel theory, and ternary or more complicated phase diagrams. Should a modern course in physical chemistry reflect material that increasingly few physical chemists use in their research and that many have not seen other than in the courses they took years ago? We think not. Although such information is undoubtedly important, especially in fields such as environmental chemistry and geological chemistry, these fields are growing to such an extent that specialized courses are being offered that address the essential physical chemical topics and their applications in much more detail than the necessarily cursory treatment they receive in a standard course in physical chemistry.

Some instructors have raised the question about whether these omitted topics are on the GRE exam and whether their students would be disadvantaged. The answer to this question is "No." In fact, one of us was recently on the GRE Chemistry Board for eight years and was Chair for the last two of them. More than ten years ago, the exam was rather heavy in classical thermodynamics, but in the past decade, the committee made a serious effort to significantly de-emphasize the thermodynamics in favor of quantum chemistry and spectroscopy. The need for this book became clear as a result of the GRE Chemistry Board meetings.

Keeping in mind that our purpose is to teach the next generation of chemists, the quantities, units, and symbols used in this text are those presented in the 1993 International Union of Pure and Applied Chemistry (IUPAC) publication Quantities, Units, and Symbols in Physical Chemistry by Ian Mills et al. (Blackwell Scientific Publications, Oxford). Our decision to follow the IUPAC recommendations means that some of the symbols, units, and standard states presented in this book may differ from those used in the literature and older textbooks and may be unfamiliar to some instructors. In some instances, we took a while to come to grips with the new notation and units, but it turned out that indeed there was an underlying logic to their use, and we found that it was actually worth the effort to become facile with them.

A unique feature of this text is the introduction of ten so-called MathChapters, which are short reviews of the mathematical topics that are used in subsequent chapters. Some of the topics covered that should be familiar to most students are complex numbers, vectors, spherical coordinates, determinants, partial derivatives, and Taylor and Maclaurin series. Some topics that may be new are probability, matrices (used

P r e f a c e

o n l y i n t h e c h a p t e r o n g r o u p t h e o r y ) , n u m e r i c a l m e t h o d s , a n d b i n o m i a l c o e f f i c i e n t s .

I n e a c h c a s e , h o w e v e r , t h e d i s c u s s i o n s a r e b r i e f , e l e m e n t a r y , a n d s e l f - c o n t a i n e d . A f t e r

r e a d i n g e a c h M a t h C h a p t e r a n d d o i n g t h e p r o b l e m s , a s t u d e n t s h o u l d b e a b l e t o f o c u s o n

t h e f o l l o w i n g p h y s i c a l c h e m i c a l m a t e r i a l r a t h e r t h a n h a v i n g t o c o p e w i t h t h e p h y s i c a l

c h e m i s t r y a n d t h e m a t h e m a t i c s s i m u l t a n e o u s l y . W e b e l i e v e t h a t t h i s f e a t u r e g r e a t l y

e n h a n c e s t h e p e d a g o g y o f o u r t e x t .

T o d a y ' s s t u d e n t s a r e c o m f o r t a b l e w i t h c o m p u t e r s . I n t h e p a s t f e w y e a r s w e h a v e

s e e n h o m e w o r k a s s i g n m e n t s t u r n e d i n f o r w h i c h s t u d e n t s u s e d p r o g r a m s s u c h a s M a t h -

C a d a n d M a t h e m a t i c a t o s o l v e p r o b l e m s , r a t h e r t h a n p e n c i l a n d p a p e r . D a t a o b t a i n e d

i n l a b o r a t o r y c o u r s e s a r e n o w g r a p h e d a n d f i t t o f u n c t i o n s u s i n g p r o g r a m s s u c h a s

E x c e l , L o t u s l 2 3 , a n d K a l e i d a g r a p h . A l m o s t a l l s t u d e n t s h a v e a c c e s s t o p e r s o n a l c o m -

p u t e r s , a n d a m o d e m c o u r s e i n t h e p h y s i c a l s c i e n c e s s h o u l d e n c o u r a g e s t u d e n t s t o

t a k e a d v a n t a g e o f t h e s e t r e m e n d o u s r e s o u r c e s . A s a r e s u l t , w e h a v e w r i t t e n a n u m b e r

o f o u r p r o b l e m s w i t h t h e u s e o f c o m p u t e r s i n m i n d . F o r e x a m p l e , M a t h C h a p t e r G

i n t r o d u c e s t h e N e w t o n - R a p h s o n m e t h o d f o r s o l v i n g h i g h e r - o r d e r a l g e b r a i c e q u a t i o n s

a n d t r a n s c e d e n t a l e q u a t i o n s n u m e r i c a l l y . T h e r e i s n o r e a s o n n o w a d a y s t h a t c a l c u l a t i o n s

i n a p h y s i c a l c h e m i s t r y c o u r s e s h o u l d b e l i m i t e d t o s o l v i n g q u a d r a t i c e q u a t i o n s a n d

o t h e r a r t i f i c i a l e x a m p l e s . S t u d e n t s s h o u l d g r a p h d a t a , e x p l o r e e x p r e s s i o n s t h a t f i t e x -

p e r i m e n t a l d a t a , a n d p l o t f u n c t i o n s t h a t d e s c r i b e p h y s i c a l b e h a v i o r . T h e u n d e r s t a n d i n g

o f p h y s i c a l c o n c e p t s i s g r e a t l y e n h a n c e d b y e x p l o r i n g t h e p r o p e r t i e s o f r e a l d a t a . S u c h

e x e r c i s e s r e m o v e t h e a b s t r a c t n e s s o f m a n y t h e o r i e s a n d e n a b l e s t u d e n t s t o a p p r e c i a t e

t h e m a t h e m a t i c s o f p h y s i c a l c h e m i s t r y s o t h a t t h e y c a n d e s c r i b e a n d p r e d i c t t h e p h y s i c a l

b e h a v i o r o f c h e m i c a l s y s t e m s .

X X I

A c k n o w l e d g m e n t s

M a n y p e o p l e h a v e c o n t r i b u t e d t o t h e w r i t i n g o f t h i s b o o k . W e t h a n k o u r c o l l e a g u e s ,

P a u l B a r b a r a , J a m e s T . H y n e s , V e r o n i c a V a i d a , J o h n C r o w e l l , A n d y K u m m e l , R o b e r t

C o n t i n e t t i , A m i t S i n h a , J o h n W e a r e , K i m B a l d r i d g e , J a c k K y t e , a n d B i l l T r o g l e r f o r

s t i m u l a t i n g d i s c u s s i o n s o n t h e t o p i c s t h a t s h o u l d b e i n c l u d e d i n a m o d e m p h y s i c a l

c h e m i s t r y c o u r s e , a n d o u r s t u d e n t s , B a r y B o l d i n g , P e i j u n C o n g , R o b e r t D u n n , S c o t t

F e l l e r , S u s a n F o r e s t , J e f f G r e a t h o u s e , K e r r y H a n s o n , B u l a n g L i , a n d S u n n e y X i e f o r

r e a d i n g p o r t i o n s o f t h e m a n u s c r i p t a n d m a k i n g m a n y h e l p f u l s u g g e s t i o n s . W e a r e e s p e -

c i a l l y i n d e b t e d t o o u r s u p e r b r e v i e w e r s , M e r v H a n s o n , J o h n F r e d e r i c k , A n n e M e y e r s ,

G e o r g e S h i e l d s , a n d P e t e r R o c k , w h o r e a d a n d c o m m e n t e d o n t h e e n t i r e m a n u s c r i p t ;

t o H e a t h e r C o x , w h o a l s o r e a d t h e e n t i r e m a n u s c r i p t , m a d e n u m e r o u s i n s i g h t f u l s u g -

g e s t i o n s , a n d d i d e v e r y p r o b l e m i n t h e c o u r s e o f p r e p a r i n g t h e a c c o m p a n y i n g S o l u t i o n

M a n u a l ; t o C a r o l e M c Q u a r r i e , w h o s p e n t m a n y h o u r s i n t h e l i b r a r y a n d o n t h e i n t e r n e t

l o o k i n g u p e x p e r i m e n t a l d a t a a n d b i o g r a p h i c a l d a t a i n o r d e r t o w r i t e a l l t h e b i o g r a p h i c a l

s k e t c h e s ; a n d t o K e n n e t h P i t z e r a n d K a r m a B e a l f o r s u p p l y i n g u s w i t h s o m e c r i t i c a l

b i o g r a p h i c a l d a t a . W e a l s o t h a n k S u s a n n a T a d l o c k f o r c o o r d i n a t i n g t h e e n t i r e p r o j e c t ,

B o b I s h i f o r d e s i g n i n g w h a t w e t h i n k i s a b e a u t i f u l - l o o k i n g b o o k , J a n e E l l i s f o r c o m -

p e t e n t l y d e a l i n g w i t h m a n y o f t h e p r o d u c t i o n d e t a i l s , J o h n C h o i f o r c r e a t i v e l y h a n d l i n g

a l l t h e a r t w o r k , A n n M c G u i r e f o r a v e r y h e l p f u l c o p y e d i t i n g o f t h e m a n u s c r i p t , a n d o u r

p u b l i s h e r , B r u c e A r m b r u s t e r , f o r e n c o u r a g i n g u s t o w r i t e o u r o w n b o o k a n d f o r b e i n g

a n e x e m p l a r y p u b l i s h e r a n d a g o o d f r i e n d . L a s t , w e t h a n k o u r w i v e s , C a r o l e a n d D i a n e ,

b o t h o f w h o m a r e c h e m i s t s , f o r b e i n g g r e a t c o l l e a g u e s a s w e l l a s g r e a t w i v e s .

x x i i i

~----------------------------------------------------------------------------------------------------------------------

H:::)VO~ddV ~Vln:::)310W V

A~lSIW3 H:J lV:JISAHd

Max Planck was born in Kiel, Germany (then Prussia) on April23, 1858, and died in 1948. He showed early talent in both music and science. He received his Ph.D. in theoretical physics in 1879 at the University of Munich for his dissertation on the second law of thermodynamics. He joined the faculty of the University of Kiel in 1885, and in 1888 he was appointed director of the Institute of Theoretical Physics, which was formed for him at the University of Berlin, where he remained until 1926. His application of thermodynamics to physical chemistry won him an early international reputation. Planck was president of the Kaiser Wilhelm Society, later renamed the Max Planck Society, from 1930 until 1937, when he was forced to retire by the Nazi government. Planck is known as the father of the quantum theory because of his theoretical work on blackbody radiation at the end of the 1890s, during which time he introduced a quantum hypothesis to achieve agreement between his theoretical equations and experimental data. He maintained his interest in thermodynamics throughout his long career in physics. Planck was awarded the Nobel Prize in physics in 1918 "in recognition of services he rendered to the advancement of physics by his discovery of energy quanta." Planck's personal life was clouded by tragedy. His two daughters died in childbirth, one son died in World War I, and another son was executed in World War II for his part in the assassination attempt on Hitler in 1944.

C H A P T E R

1

T h e D a w n o f t h e Q u a n t u m T h e o r y

T o w a r d t h e e n d o f t h e n i n e t e e n t h c e n t u r y , m a n y s c i e n t i s t s b e l i e v e d t h a t a l l t h e f u n -

d a m e n t a l d i s c o v e r i e s o f s c i e n c e h a d b e e n m a d e a n d l i t t l e r e m a i n e d b u t t o c l e a r u p a

f e w m i n o r p r o b l e m s a n d t o i m p r o v e e x p e r i m e n t a l m e t h o d s t o m e a s u r e p h y s i c a l r e s u l t s

t o a g r e a t e r n u m b e r o f d e c i m a l p l a c e s . T h i s a t t i t u d e w a s s o m e w h a t j u s t i f i e d b y t h e

g r e a t a d v a n c e s t h a t h a d b e e n m a d e u p t o t h a t t i m e . C h e m i s t s h a d f i n a l l y s o l v e d t h e

s e e m i n g l y i n s u r m o u n t a b l e p r o b l e m o f a s s i g n i n g a s e l f - c o n s i s t e n t s e t o f a t o m i c m a s s e s

t o t h e e l e m e n t s . S t a n i s l a o C a n n i z z a r o ' s c o n c e p t o f t h e m o l e c u l e , w h i l e i n i t i a l l y c o n t r o -

v e r s i a l , w a s t h e n w i d e l y a c c e p t e d . T h e g r e a t w o r k o f D m i t r i M e n d e l e e v h a d r e s u l t e d

i n a p e r i o d i c t a b l e o f t h e e l e m e n t s , a l t h o u g h t h e u n d e r l y i n g r e a s o n s t h a t s u c h p e r i o d i c

b e h a v i o r o c c u r r e d i n n a t u r e w e r e n o t u n d e r s t o o d . F r i e d r i c h K e k u l e h a d s o l v e d t h e

g r e a t c o n t r o v e r s y c o n c e r n i n g t h e s t r u c t u r e o f b e n z e n e . T h e f u n d a m e n t a l s o f c h e m i c a l

r e a c t i o n s h a d b e e n e l u c i d a t e d b y S v a n t e A r r h e n i u s , a n d t h e r e m a i n i n g w o r k s e e m e d t o

c o n s i s t p r i m a r i l y o f c a t a l o g i n g t h e v a r i o u s t y p e s o f c h e m i c a l r e a c t i o n s .

I n t h e r e l a t e d f i e l d o f p h y s i c s , N e w t o n i a n m e c h a n i c s h a d b e e n e x t e n d e d b y C o m t e

J o s e p h L a g r a n g e a n d S i r W i l l i a m H a m i l t o n . T h e r e s u l t i n g t h e o r y w a s a p p l i e d t o p l a n -

e t a r y m o t i o n a n d c o u l d a l s o e x p l a i n o t h e r c o m p l i c a t e d n a t u r a l p h e n o m e n a s u c h a s

e l a s t i c i t y a n d h y d r o d y n a m i c s . C o u n t R u m f o r d a n d J a m e s J o u l e h a d d e m o n s t r a t e d t h e

e q u i v a l e n c e o f h e a t a n d w o r k , a n d i n v e s t i g a t i o n s b y S a d i C a m o t r e s u l t e d i n t h e f o r -

m u l a t i o n o f w h a t i s n o w e n t r o p y a n d t h e s e c o n d l a w o f t h e r m o d y n a m i c s . T h i s w o r k

w a s f o l l o w e d b y J o s i a h G i b b s ' c o m p l e t e d e v e l o p m e n t o f t h e f i e l d o f t h e r m o d y n a m i c s .

S h o r t l y , s c i e n t i s t s w o u l d d i s c o v e r t h a t t h e l a w s o f p h y s i c s w e r e a l s o r e l e v a n t t o t h e u n -

d e r s t a n d i n g o f c h e m i c a l s y s t e m s . T h e i n t e r f a c e b e t w e e n t h e s e t w o s e e m i n g l y u n r e l a t e d

d i s c i p l i n e s f o r m e d t h e m o d e m f i e l d o f p h y s i c a l c h e m i s t r y , t h e t o p i c o f t h i s b o o k . I n

f a c t , G i b b s ' s t r e a t m e n t o f t h e r m o d y n a m i c s i s s o i m p o r t a n t t o c h e m i s t r y t h a t i t i s t a u g h t

i n a f o r m t h a t i s e s s e n t i a l l y u n c h a n g e d f r o m G i b b s ' s o r i g i n a l f o r m u l a t i o n .

T h e r e l a t e d f i e l d s o f o p t i c s a n d e l e c t r o m a g n e t i c t h e o r y w e r e u n d e r g o i n g s i m i l a r

m a t u r a t i o n . T h e n i n e t e e n t h c e n t u r y w i t n e s s e d a c o n t i n u i n g c o n t r o v e r s y a s t o w h e t h e r

l i g h t w a s w a v e l i k e o r p a r t i c l e l i k e . M a n y d i v e r s e a n d i m p o r t a n t o b s e r v a t i o n s w e r e

u n i f i e d b y J a m e s C l e r k M a x w e l l i n a s e r i e s o f d e c e p t i v e l y s i m p l e - l o o k i n g e q u a t i o n s 1

2 Chapter 1 I The Dawn of the Quantum Theory

that bear his name. Not only did Maxwell's predictions of the electromagnetic behavior of light unify the fields of optics with electricity and magnetism, but their subsequent experimental demonstration by Heinrich Hertz in 1887 appeared to finally demonstrate that light was wavelike. The implications of these fields to chemistry would not be appreciated for several decades but are now important aspects of the discipline of physical chemistry, particularly in spectroscopy.

The body of these accomplishments in physics is considered the development of what we now call classical physics. Little did scientists realize in that justifiably heady era of success that the fundamental tenets of how the physical world works were to be shortly overturned. Fantastic discoveries were about to revolutionize not only physics, chemistry, biology, and engineering but would have significant effects on technology and politics as well. The early twentieth century saw the birth of the theory of relativity and quantum mechanics. The first, due to the work of Albert Ein-stein alone, completely altered scientist's ideas of space and time and was an extension of the classical ideas to include high velocities and astronomical distances. Quantum mechanics, the extension of classical ideas into the behavior of subatomic, atomic, and molecular species, on the other hand, resulted from the efforts of many creative scientists over several decades. To date, the effect of relativity on chemical systems has been limited. Although it is important in understanding electronic properties of heavy atoms, it does not play much of a role in molecular structure and reactivity and so is not generally taught in physical chemistry. Quantum mechanics, however, forms the foundation upon which all of chemistry is built. Our current understanding of atomic structure and molecular bonding is cast in terms of the fundamental prin-ciples of quantum mechanics and no understanding of chemical systems is possible without knowing the basics of this current theory of matter. For this reason, we begin this book with several chapters that focus on the fundamental principles of quantum mechanics. We then follow with a discussion of chemical bonding and spectroscopy, which clearly demonstrate the influence that quantum mechanics has had on the field of chemistry.

Great changes in science are spurred by observations and new creative ideas. Let us go back to the complacent final years of the nineteenth century to see just what were the events that so shook the world of science.

1-1. Blackbody Radiation Could Not Be Explained by Classical Physics

The series of experiments that revolutionized the concepts of physics had to do with the radiation given off by material bodies when they are heated. We all know, for instance, that when the burner of an electric stove is heated, it first turns a dull red and progressively becomes redder as the temperature increases. We also know that as a body is heated even further, the radiation becomes white and then blue as its temperature continues to increase. Thus, we see that there is a continual shift of the color of a heated body from red through white to blue as the body is heated to higher

1 - 1 . B l a c k b o d y R a d i a t i o n C o u l d N o t B e E x p l a i n e d b y C l a s s i c a l P h y s i c s

t e m p e r a t u r e s . I n t e r m s o f f r e q u e n c y , t h e r a d i a t i o n e m i t t e d g o e s f r o m a l o w e r f r e q u e n c y

t o a h i g h e r f r e q u e n c y a s t h e t e m p e r a t u r e i n c r e a s e s , b e c a u s e r e d i s i n a l o w e r f r e q u e n c y

r e g i o n o f t h e s p e c t r u m t h a n i s b l u e . T h e e x a c t f r e q u e n c y s p e c t r u m e m i t t e d b y t h e

b o d y d e p e n d s o n t h e p a r t i c u l a r b o d y i t s e l f , b u t a n i d e a l b o d y , w h i c h a b s o r b s a n d e m i t s

a l l f r e q u e n c i e s , i s c a l l e d a b l a c k b o d y a n d s e r v e s a s a n i d e a l i z a t i o n f o r a n y r a d i a t i n g

m a t e r i a l . T h e r a d i a t i o n e m i t t e d b y a b l a c k b o d y i s c a l l e d b l a c k b o d y r a d i a t i o n .

A p l o t o f t h e i n t e n s i t y o f b l a c k b o d y r a d i a t i o n v e r s u s f r e q u e n c y f o r s e v e r a l t e m p e r -

a t u r e s i s g i v e n i n F i g u r e 1 . 1 . M a n y t h e o r e t i c a l p h y s i c i s t s t r i e d t o d e r i v e e x p r e s s i o n s

c o n s i s t e n t w i t h t h e s e e x p e r i m e n t a l c u r v e s o f i n t e n s i t y v e r s u s f r e q u e n c y , b u t t h e y w e r e

a l l u n s u c c e s s f u l . I n f a c t , t h e e x p r e s s i o n t h a t i s d e r i v e d a c c o r d i n g t o t h e l a w s o f n i n e -

t e e n t h c e n t u r y p h y s i c s i s

8 n k T

d p ( v , T ) = p ( T ) d v = - -

3

B - v

2

d v

v c

( 1 . 1 )

w h e r e P v ( T ) d v i s t h e r a d i a n t e n e r g y d e n s i t y b e t w e e n t h e f r e q u e n c i e s v a n d v + d v a n d

h a s u n i t s o f j o u l e s p e r c u b i c m e t e r ( J m -

3

) . I n E q u a t i o n 1 . 1 , T i s t h e a b s o l u t e t e m p e r -

a t u r e , a n d c i s t h e s p e e d o f l i g h t . T h e q u a n t i t y k B i s c a l l e d t h e B o l t z m a n n c o n s t a n t a n d

i s e q u a l t o t h e i d e a l g a s c o n s t a n t R d i v i d e d b y t h e A v o g a d r o c o n s t a n t ( f o r m e r l y c a l l e d

A v o g a d r o ' s n u m b e r ) . T h e u n i t s o f k B a r e J K -

1

p a r t i c l e - ' , b u t p a r t i c l e - ' i s u s u a l l y n o t

e x p r e s s e d . ( A n o t h e r c a s e i s t h e A v o g a d r o c o n s t a n t , 6 . 0 2 2 x 1 0

2 3

p a r t i c l e m o l -

1

, w h i c h

w e w i l l w r i t e a s 6 . 0 2 2 x 1 0

2 3

m o l -

1

; t h e u n i t " p a r t i c l e " i s n o t e x p r e s s e d . ) E q u a t i o n 1 . 1

c a m e f r o m t h e w o r k o f L o r d R a y l e i g h a n d J . H . J e a n s a n d i s c a l l e d t h e R a y l e i g h - J e a n s

l a w . T h e d a s h e d l i n e i n F i g u r e 1 . 1 s h o w s t h e p r e d i c t i o n o f t h e R a y l e i g h - J e a n s l a w .

F I G U R E 1 . 1

V J

. . . .

~

: : I

; > . .

. . . . .

e l l

. . . . .

. . . .

~

. . . . .

e l l

- . . .

; > . .

. . . .

-

V J

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1 ! . )

. . . .

~

. . . . . .

6 0 0 0 K

0 5

1 0 I S 2 0

v I 1 0

1 4

s -

1

S p e c t r a l d i s t r i b u t i o n o f t h e i n t e n s i t y o f b l a c k b o d y r a d i a t i o n a s a f u n c t i o n o f f r e q u e n c y f o r

s e v e r a l t e m p e r a t u r e s . T h e i n t e n s i t y i s g i v e n i n a r b i t r a r y u n i t s . T h e d a s h e d l i n e i s t h e p r e d i c t i o n

o f c l a s s i c a l p h y s i c s . A s t h e t e m p e r a t u r e i n c r e a s e s , t h e m a x i m u m s h i f t s t o h i g h e r f r e q u e n c i e s

a n d t h e t o t a l r a d i a t e d e n e r g y ( t h e a r e a u n d e r e a c h c u r v e ) i n c r e a s e s s i g n i f i c a n t l y . N o t e t h a t t h e

h o r i z o n t a l a x i s i s l a b e l e d b y v / 1 0

1 4

s -

1

. T h i s n o t a t i o n m e a n s t h a t t h e d i m e n s i o n l e s s n u m b e r s

o n t h a t a x i s a r e f r e q u e n c i e s d i v i d e d b y 1 0

1 4

s -

1

W e s h a l l u s e t h i s n o t a t i o n t o l a b e l c o l u m n s i n

t a b l e s a n d a x e s i n f i g u r e s b e c a u s e o f i t s u n a m b i g u o u s n a t u r e a n d a l g e b r a i c c o n v e n i e n c e .

3

4 Chapter 1 I The Dawn of the Quantum Theory

Note that the Rayleigh-Jeans law reproduces the experimental data at low frequencies. At high frequencies, however, the Rayleigh-Jeans law predicts that the radiant energy density diverges as v2 Because the frequency increases as the radiation enters the ul-traviolet region, this divergence was termed the ultraviolet catastrophe, a phenomenon that classical physics could not explain theoretically. This was the first such failure to explain an important naturally occuring phenomenon and therefore is of great his-torical interest. Rayleigh and Jeans did not simply make a mistake or misapply some of the ideas of physics; many other people reproduced the equation of Rayleigh and Jeans, showing that this equation was correct according to the physics of the time. This result was somewhat disconcerting and many people struggled to find a theoretical explanation of blackbody radiation.

1-2. Planck Used a Quantum Hypothesis to Derive the Blackbody Radiation Law

The first person to offer a successful explanation of blackbody radiation was the German physicist Max Planck in 1900. Like Rayleigh and Jeans before him, Planck assumed that the radiation emitted by the blackbody was caused by the oscillations of the electrons in the constituent particles of the material body. These electrons were pictured as oscillating in an atom much like electrons oscillate in an antenna to give off radio waves. In these "atomic antennae", however, the oscillations occur at a much higher frequency; hence, we find frequencies in the visible, infrared, and ultraviolet regions rather than in the radio-wave region of the spectrum. Implicit in the derivation of Rayleigh and Jeans is the assumption that the energies of the electronic oscillators responsible for the emission of the radiation could have any value whatsoever. This assumption is one of the basic assumptions of classical physics. In classical physics, the variables that represent observables (such as position, momentum, and energy) can take on a continuum of values. Planck had the great insight to realize that he had to break away from this mode of thinking to derive an expression that would reproduce experimental data such as those shown in Figure 1.1. He made the revolutionary assumption that the energies of the oscillators were discrete and had to be proportional to an integral multiple of the frequency or, in equation form, that E = nh v, where E is the energy of an oscillator, n is an integer, h is a proportionality constant, and v is the frequency. Using this quantization of energy and statistical thermodynamic ideas that we will cover in Chapter 17, Planck derived the equation

8nh v3dv dp(v, T) = p (T)dv = - 3 h fk T

v ce"B-1 (1.2)

All the symbols except h in Equation 1.2 have the same meaning as in Equation 1.1. The only undetermined constant in Equation 1.2 ish. Planck showed that this equation gives excellent agreement with the experimental data for all frequencies and temperatures if h has the value 6.626 x 10-34 joule-seconds (Js). This constant is now one of the most famous and fundamental constants of physics and is called the Planck constant.

1 - 2 . P l a n c k U s e d a Q u a n t u m H y p o t h e s i s t o D e r i v e t h e B l a c k b o d y R a d i a t i o n L a w

E q u a t i o n 1 . 2 i s k n o w n a s t h e P l a n c k d i s t r i b u t i o n l a w f o r b l a c k b o d y r a d i a t i o n . F o r

s m a l l f r e q u e n c i e s , E q u a t i o n s 1 . 1 a n d 1 . 2 b e c o m e i d e n t i c a l ( P r o b l e m 1 - 4 ) , b u t t h e

P l a n c k d i s t r i b u t i o n d o e s n o t d i v e r g e a t l a r g e f r e q u e n c i e s a n d , i n f a c t , l o o k s l i k e t h e

c u r v e s i n F i g u r e 1 . 1 .

E X A M P L E 1 - 1

S h o w t h a t p v ( T ) d v i n b o t h E q u a t i o n s 1 . 1 a n d 1 . 2 h a s u n i t s o f e n e r g y p e r u n i t v o l u m e ,

J

- 3

i l l .

S O L U T I O N : T h e u n i t s o f T a r e K , o f k B a r e J - K -

1

, o f h a r e J - s , o f v a n d d v a r e s -

1

,

a n d o f c a r e m - s -

1

. T h e r e f o r e , f o r t h e R a y l e i g h - J e a n s l a w ( E q u a t i o n 1 . 1 ) ,

8 n k T

d p ( v , T ) = p ( T ) d v = - -

3

B - v

2

d v

v c

(

J K -

1

) ( K )

( s - 1 ) 2 ( s - 1 ) = J - m - 3

( m s - 1 ) 3

F o r t h e P l a n c k d i s t r i b u t i o n ( E q u a t i o n 1 . 2 ) ,

8 n h v

3

d v

d p ( v , T ) = p ( T ) d v = - 3 - h f k T

" c e " s - 1

( J s ) ( s - 1 ) 3 ( s - 1 ) -

~ = J m 3

( m s - 1 ) 3

T h u s , w e s e e t h a t p v ( T ) d v , t h e r a d i a n t e n e r g y d e n s i t y h a s u n i t s o f e n e r g y p e r u n i t

v o l u m e .

E q u a t i o n 1 . 2 e x p r e s s e s t h e P l a n c k d i s t r i b u t i o n l a w i n t e r m s o f f r e q u e n c y . B e c a u s e

w a v e l e n g t h ( A . ) a n d f r e q u e n c y ( v ) a r e r e l a t e d b y A . v = c , t h e n d v = - c d A . j A .

2

, a n d w e

c a n e x p r e s s t h e P l a n c k d i s t r i b u t i o n l a w i n t e r m s o f w a v e l e n g t h r a t h e r t h a n f r e q u e n c y

( P r o b l e m 1 - 1 0 ) :

S n h c d A .

d p ( A , T ) = p " A ( T ) d ) . . = ) . : 5 e h c / A k

8

T _ 1

( 1 . 3 )

T h e q u a n t i t y p ' " ( T ) d A . i s t h e r a d i a n t e n e r g y d e n s i t y b e t w e e n ) . . a n d ) . . + d A . . T h e i n t e n s i t y

c o r r e s p o n d i n g t o E q u a t i o n 1 . 3 i s p l o t t e d i n F i g u r e 1 . 2 f o r s e v e r a l v a l u e s o f T .

W e c a n u s e E q u a t i o n 1 . 3 t o j u s t i f y a n e m p i r i c a l r e l a t i o n s h i p k n o w n a s t h e W i e n

d i s p l a c e m e n t l a w . T h e W i e n d i s p l a c e m e n t l a w s a y s t h a t i f ) . . m a x i s t h e w a v e l e n g t h a t

w h i c h p ' " ( T ) i s a m a x i m u m , t h e n

0 ' . ) . . 7 J ' T ; ( J r F l -

A m a x T = 2 : 9 0 X 1 0 -

3

m K

, \ \ _ ;

B y d i f f e r e n t i a t i n g p ' " ( T ) w i t h r e s p e c t t o A , w e c a n s h o w ( P r o b l e m 1 - 5 ) t h a t

h e

A m a x T = 4 . 9 6 5 k B

( 1 . 4 )

( 1 . 5 )

5

6

"' ......

-s:: ::I >-. .....

Clj .....

......

-..0 .....

Clj ......

>-. ......

-"' s:: v ......

s:: .......

0

FIGURE 1.2

'

Visible:

500 1000 Alnm

Infrared

1500 2000

The distribution of the intensity of the radiation emitted by a blackbody versus wavelength for various temperatures. As the temperature increases, the total radiation emitted (the area under the curve) increases.

in accord with the Wein displacement law. Using the modem values of h, c, and kB given inside the front cover, we obtain 2.899 x 10-3 mK for the right side of Equation 1.5, in excellent agreement with the experimental value given in Equation 1.4.

The theory of blackbody radiation is used regularly in astronomy to estimate the surface temperatures of stars. Figure 1.3 shows the electromagnetic spectrum of the sun measured at the earth's upper atmosphere. A comparison of Figure 1.3 with Figure 1.2 suggests that the solar spectrum can be described by a blackbody at approximately 6000 K. If we estimate A. max from Figure 1.3 to be 500 nm, then the Wein displacement law (Equation 1.4) gives the temperature of the surface of the sun to be

2.90 x 10-3 mK T = = 5800 K

500 x 10-9 m

The star Sirius, which appears blue, has a surface temperature of about 11 000 K ( cf. Problem 1-7).

Certainly Planck's derivation of the blackbody distribution law was an impres-sive feat. Nevertheless, Planck's derivation and, in particular, his assumption that the energies of the oscillators have to be an integral multiple of h v was not accepted by most scientists at the time and was considered simply an arbitrary derivation. Most believed that in time a satisfactory derivation would be found that obeyed the laws of classical physics. In a sense, Planck's derivation was little more than a curiosity. Just a few years later, however, in 1905, Einstein used the very same idea to explain the photoelectric effect.

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