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CHMY 373 2019 – Problem Set 7
Due: Friday Mar. 8
Reading: Chapter 20.1‐20.9, Simon & McQuarrie
Note: for statistical thermodynamics problems, you may use the following numbers
of configurational microstates depending on the phase of the system ideal gas, ,
liquid, , or solid, :
,
!
,
!
,
1. Communal Entropy. The difference in entropy upon melting of a solid is commonly
referred to as the “communal entropy,” and for a crystalline solid is predominantly
due to the difference in configurational entropy of the resulting liquid denoted by
compared to the initial solid denoted by :
∆
∆
→
,
,
Use statistical thermodynamics, by assuming that the molar or molecular volume
of the solid and liquid are approximately equal. Hint: you will need to employ
Stirling’s formula Equation J.7 in Simon & McQuarrie .
2. Hard Sphere Gas. Determine the expression for the change in entropy upon
isothermal expansion of a hard sphere gas from to , in the following two ways:
a. Use classical thermodynamics, starting with the equation of state for a hard
sphere gas:
where is the molar excluded volume of the hard sphere gas equivalent to
the coefficient for a van der Waals fluid .
b. Use statistical thermodynamics, by assuming that the change in entropy is
entirely due to the change in the number of configurational microstates of
the system. As above, use as the molar excluded volume of the hard sphere
gas has dimensions of volume per mole not per molecule . Note: you will
need to modify the given expression for the number of configurational
microstates of an ideal gas to one for a hard sphere gas.
3. Entropy of Mixing on a Crystalline Lattice. What is the change in entropy, ∆
,
associated with mixing two elements ○ and on a solid lattice of lattice sites?
Both pure elements exhibit the same crystal structure as the resulting alloy, and
there are no interactions between any sites the entropy change is purely
configurational . The number of ○ atoms is ○ and the number of atoms is .
Initial State
Final State
4. Entropy of Mixing of Ideal Gases. What is the change in entropy, ∆
, associated
with mixing two ideal gases of different kinds ○ and by simply removing the
barrier between them? You may assume that the entropy change is purely
configurational. The number of ○ molecules is ○ and the number of molecules is
and , and
, as shown.
. The volumes of their initial containers are
Initial State
,
Final State
,
,
5. Entropy of Reversible Mixing of Ideal Gases. What is the change in entropy, ∆
,
associated with mixing two ideal gases of different kinds ○ and reversibly ? This
can be accomplished in a special container with two semi‐permeable pistons that
each only allow only one type of gas to pass through, as shown, that can be slowly
moved apart. The number of ○ molecules is ○ and the number of molecules is .
The volumes of their initial containers are and , and
, as shown.
Solve using classical thermodynamics, not statistical thermodynamics in this case.
Initial State
,
,
Final State
,
piston permeable only to ○
piston permeable only to
**Hint: compare the work done by each gas on the opposite respective piston as the
pistons are moved apart and recall the thermodynamic definition of entropy . Each
gas can only perform reversible PV‐type work on the piston which it cannot diffuse
through. The temperature of the container is held constant, so the change in internal
energy is also constant throughout this process these are ideal gases .
6. Comment on the answers to Problems 3‐5 by comparing them each as a single,
simple equation in the following form:
∆
○,
In this equation, is some function only of ○ , , and any physical constants e.g.,
. The variables ○ and are defined in terms of the total number of atoms, ,
as:
○
○
○
Note: not much work is required here, the idea is simply to rewrite the answers to
Problems 3‐5 in the same terms for comparison purposes. If your answers were
already written in these terms, simply rewrite them and comment on the
similarities and/or differences.
..
PHYSICAL CHEMISTRY
A MOLECULAR APPROACH
Donald A. McQuarrie
UNIVERSITY OF CALIFORNIA, DAVIS
john D. Simon
George B. Geller Professor of Chemistry
DUKE UNIVERSITY
~
University Science Books
Sausalito, California
J
c;Lf/.3
Mt1'5
University Science Books
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Fax: (415) 332-5393
Production manager: Susanna Tadlock
Manuscript editor: Ann McGuire
Designer: Robert Ishi
Illustrator: John Choi
Compositor: Eigentype
Printer & Binder: Edwards Brothers, Inc.
This book is printed on acid-free paper.
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
CIP
Printed in the United States of America
10987654321
Contents
Preface xvii
To the Student xv11
To the Instructor xix
Acknowledgment
xx111
CHAPTER 1 I The Dawn of the Quantum Theory
1-1.
1-2.
1-3.
1-4.
1-5.
1-6.
1-7.
1-8.
Blackbody Radiation Could Not Be Explained by Classical Physics 2
Planck Used a Quantum Hypothesis to Derive the Blackbody Radiation Law 4
Einstein Explained the Photoelectric Effect with a Quantum Hypothesis 7
The Hydrogen Atomic Spectrum Consists of Several Series of Lines 10
The Rydberg Formula Accounts for All the Lines in the Hydrogen Atomic Spectrum
Louis de Broglie Postulated That Matter Has Wavelike Properties 15
de Broglie Waves Are Observed Experimentally 16
The Bohr Theory of the Hydrogen Atom Can Be Used to Derive the Rydberg
Formula 18
1-9. The Heisenberg Uncertainty Principle States That the Position and the Momentum
of a Particle Cannot Be Specified Simultaneously with Unlimited Precision 23
13
Problems 25
MATHCHAPTER A I Complex Numbers
Problems
31
35
CHAPTER 2 I The Classical Wave Equation
39
2-1. The One-Dimensional Wave Equation Describes the Motion of a Vibrating String 39
2-2. The Wave Equation Can Be Solved by the Method of Separation of Variables 40
2-3. Some Differential Equations Have Oscillatory Solutions 44
2-4. The General Solution to the Wave Equation Is a Superposition of Normal Modes
2-5. A Vibrating Membrane Is Described by a Two-Dimensional Wave Equation 49
Problems 54
MATHCHAPTER
Problems
8 I Probability and Statistics
70
v
63
46
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
Problems
105
11 3
CHAPTER 4 I Some Postulates and General Principles of
Quantum Mechanics 115
The State of a System Is Completely Specified by Its Wave Function 115
Quantum-Mechanical Operators Represent Classical-Mechanical Variables 118
Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators 122
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
4-1.
4-2.
4-3.
4-4.
/Z")MATHCHAPTER D I Spherical Coordinates
{.__;Y
Problems
147
153
CHAPTER 5
I The Harmonic Oscillator and the Rigid Rotator:
Two Spectroscopic Models
157
5-1. A Harmonic Oscillator Obeys Hooke's Law 157
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 169
/.5-7. Hermite Polynomials Are Either Even or Odd Functions 1 72
The Energy Levels of a Rigid Rotator Are E = h 2 J(J + 1)/21 173
-s=2.
/
B
vi
Contents
/~The Rigid Rotator Is a Model for a Rotating Diatomic Molecule
Problems
177
1 79
CHAPTER
6 I The Hydrogen Atom
191
6-1. The Schrodinger Equation for the Hydrogen Atom Can Be Solved Exactly 191
6-2. The Wave Functions of a Rigid Rotator Are Called Spherical Harmonics 193
6-3. The Precise Values of the Three Components of Angular Momentum Cannot Be
Measured Simultaneously 200
6-4. Hydrogen Atomic Orbitals Depend upon Three Quantum Numbers 206
6-5. s Orbitals Are Spherically Symmetric 209
6-6. There Are Three p Orbitals for Each Value of the Principal Quantum Number,
n ~ 2 213
6-7. The Schrodinger Equation for the Helium Atom Cannot Be Solved Exactly 219
Problems 220
MATHCHAPTER
E I Determinants
231
Problems 238
CHAPTER 7
I Approximation Methods
241
~ 7-1. The Variational Method Provides an Upper Bound to the Ground-State Energy
of a System 241
7-2. A Trial Function That Depends Linearly on the Variational Parameters Leads
to a Secular Determinant 249
7-3. Trial Functions Can Be Linear Combinations of Functions That Also Contain
Variational Parameters 256
~ 7-4. Perturbation Theory Expresses the Solution to One Problem in Terms of Another
Problem Solved Previously 257
Problems 2 61
CHAPTER
8 I Multielectron Atoms
275
8-1. Atomic and Molecular Calculations Are Expressed in Atomic Units 275
8-2. Both Perturbation Theory and the Variational Method Can Yield Excellent Results
for Helium 278
8-3. Hartree-Fock Equations Are Solved by the Self-Consistent Field Method 282
8-4. An Electron Has an Intrinsic Spin Angular Momentum 284
8-5. Wave Functions Must Be Antisymmetric in the Interchange of Any Two Electrons 285
8-6. Antisymmetric Wave Functions Can Be Represented by Slater Determinants 288
8-7. Hartree-Fock Calculations Give Good Agreement with Experimental Data 290
8-8. A Term Symbol Gives a Detailed Description of an Electron Configuration 292
@jrhe Allowed Values of J are L + S, L + S- 1,
IL- Sl 296
8-10. Hund's Rules Are Used to Determine the Term Symbol of the Ground
Electronic State 301
-! 8-11. Atomic Term Symbols Are Used to Describe Atomic Spectra 302
Problems 308
0
CHAPTER
0
0,
9 I The Chemical Bond: Diatomic Molecules
vii
323
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 H 2 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 BeH 2 Linear and H 2 0 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 H 2 can be Calculated Essentially Exactly 425
11-5. Gaussian 94 Calculations Provide Accurate Information About Molecules 427
Problems 434
MATHCHAPTER F I Matrices
Problems
441
448
viii
Contents
CHAPTER
12 I Group Theory: The Exploitation of Symmetry
453
12-1. The Exploitation of the Symmetry of a Molecule Can Be Used to Significantly Simplify
Numerical Calculations 453
12-2. The Symmetry of Molecules Can Be Described by a Set of Symmetry Elements 455
12-3. The Symmetry Operations of a Molecule Form a Group 460
12-4. Symmetry Operations Can Be Represented by Matrices 464
12-5. The C 3v Point Group Has a Two-Dimensional Irreducible Representation 468
12-6. The Most Important Summary of the Properties of a Point Group Is Its
Character Table 471
12-7. Several Mathematical Relations Involve the Characters of Irreducible
Representations 474
12-8. We Use Symmetry Arguments to Predict Which Elements in a Secular Determinant
Equal Zero 480
12-9. Generating Operators Are Used to Find Linear Combinations of Atomic Orbitals That
Are Bases for Irreducible Representations 484
Problems 489
(.§;
CHAPTER
13 I Molecular Spectroscopy
495
13-1. Different Regions of the Electromagnetic Spectrum Are Used to Investigate Different
Molecular Processes 495
13-2. Rotational Transitions Accompany Vibrational Transitions 497
13-3. Vibration-Rotation Interaction Accounts for the Unequal Spacing of the Lines in the
P and R Branches of a Vibration-Rotation Spectrum 501
13-4. The Lines in a Pure Rotational Spectrum Are Not Equally Spaced 503
13-5. Overtones Are Observed in Vibrational Spectra 504
13-6. Electronic Spectra Contain Electronic, Vibrational, and Rotational Information 507
13-7. The Franck-Condon Principle Predicts the Relative Intensities of Vibronic
Transitions 511
13-8. The Rotational Spectrum of a Polyatomic Molecule Depends Upon the Principal
Moments of Inertia of the Molecule 514
13-9. The Vibrations of Polyatomic Molecules Are Represented by Normal
Coordinates 518
13-10. Normal Coordinates Belong to Irreducible Representations of Molecular
Point Groups 523
13-11. Selection Rules Are Derived from Time-Dependent Perturbation Theory 527
13-12. The Selection Rule in the Rigid Rotator Approximation Is/";.]= ±1 531
13-13. The Harmonic-Oscillator Selection Rule Is /";.v = ±1 533
13-14. Group Theory Is Used to Determine the Infrared Activity of Normal
Coordinate Vibrations 535
Problems 537
CHAPTER
14 I Nuclear Magnetic Resonance Spectroscopy
547
14-1. Nuclei Have Intrinsic Spin Angular Momenta 548
14-2. Magnetic Moments Interact with Magnetic Fields 550
14-3. Proton NMR Spectrometers Operate at Frequencies Between 60 MHz and
750 MHz 554
14-4. The Magnetic Field Acting upon Nuclei in Molecules Is Shielded 556
14-5. Chemical Shifts Depend upon the Chemical Environment of the Nucleus 560
14-6. Spin-Spin Coupling Can Lead to Multiplets in NMR Spectra 562
ix
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
Problems
G I Numerical Methods
627
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
Problems
H I Partial Differentiation
683
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
Contents
17-6. The Partition Function of a System of Independent, Distinguishable Molecules
Is the Product of Molecular Partition Functions 707
17-7. The Partition Function of a System of Independent, Indistinguishable Atoms or
Molecules Can Usually Be Written as [q(V, T)]N 1N! 708
17-8. A Molecular Partition Function Can Be Decomposed into Partition Functions
for Each Degree of Freedom 713
Problems 716
MATHCHAPTER I
Problems
I Series and Limits
723
728
CHAPTER 18
I Partition Functions and Ideal Gases
731
18-1. The Translational Partition Function of an Atom in a Monatomic Ideal Gas Is
(2:rrmkB T I h 2 ) 312 V 731
18-2. Most Atoms Are in the Ground Electronic State at Room Temperature 733
18-3. The Energy of a Diatomic Molecule ...