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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 ,~ 55D Gate Five Road . ~'',.~Sausalito, CA 94965 ~~~....,:~ · 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 ...

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