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(BQ) Part 1 book Physical chemistry has contents: Gases and the zeroth law of thermodynamics, the first law of thermodynamics, the second and third laws of thermodynamics, free energy and chemical potential, introduction to chemical equilibrium, equilibria in single component systems,...and other contents.
Physical Chemistry Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Physical Chemistry David W Ball Cleveland State University Australia • Canada • Mexico • Singapore • Spain United Kingdom • United States Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part This is an electronic version of the print textbook Due to electronic rights restrictions, some third party may be suppressed Edition review has deemed that any suppressed content does not materially affect the over all learning experience The publisher reserves the right to remove the contents from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate format, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Chemistry Editor: Angus McDonald Assistant Editor: Karoliina Tuovinen Technology Project Manager: Ericka Yeoman Marketing Manager: Julie Conover Marketing Assistant: Mona Weltmer Advertising Project Manager: Stacey Purviance Signing Representative: Shelly Tommasone Project Manager, Editorial Production: Jennie Redwitz Editorial Assistant: Lauren Raike Print/Media Buyer: Karen Hunt Permissions Editor: Joohee Lee Production Service: Robin Lockwood Productions Text Designer: Susan Schmidler Photo Researcher: Linda L Rill Copy Editor: Anita Wagner Illustrator: Lotus Art Cover Designer: Larry Didona Cover Images: Teacher and blackboard: CORBIS; STM image of DNA: Lawrence Berkeley Lab/Photo Researchers; Molecular model: Kenneth Eward/BioGrafx/Photo Researchers Compositor: ATLIS Graphics and Design Text and Cover Printer: Quebecor World/Taunton COPYRIGHT © 2003 Brooks/Cole, a division of Thomson Learning, Inc Thomson LearningTM is a trademark used herein under license Brooks/Cole—Thomson Learning 511 Forest Lodge Road Pacific Grove, CA 93950 USA ALL RIGHTS RESERVED No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means—graphic, electronic, or mechanical, including but not limited to photocopying, recording, taping, Web distribution, information networks, or information storage and retrieval systems—without the written permission of the publisher Asia Thomson Learning Shenton Way #01-01 UIC Building Singapore 068808 Printed in the United States of America 06 05 04 03 02 Australia Nelson Thomson Learning 102 Dodds Street South Melbourne, Victoria 3205 Australia For more information about our products, contact us at: Thomson Learning Academic Resource Center 1-800-423-0563 For permission to use material from this text, contact us by: Phone: 1-800-730-2214 Fax: 1-800-730-2215 Web: http://www.thomsonrights.com Library of Congress Control Number: 2002105398 Canada Nelson Thomson Learning 1120 Birchmount Road Toronto, Ontario M1K 5G4 Canada Europe/Middle East/Africa Thomson Learning High Holborn House 50/51 Bedford Row London WC1R 4LR United Kingdom ISBN 0-534-26658-4 Latin America Thomson Learning Seneca, 53 Colonia Polanco 11560 Mexico D.F Mexico Spain Paraninfo Thomson Learning Calle/Magallanes, 25 28015 Madrid, Spain Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part For my father Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part Contents Preface xv Gases and the Zeroth Law of Thermodynamics 1.1 Synopsis 1.2 System, Surroundings, and State 1.3 The Zeroth Law of Thermodynamics 1.4 Equations of State 1.5 Partial Derivatives and Gas Laws 1.6 Nonideal Gases 10 1.7 More on Derivatives 18 1.8 A Few Partial Derivatives Defined 20 1.9 Summary 21 Exercises 22 The First Law of Thermodynamics 24 2.1 Synopsis 24 2.2 Work and Heat 24 2.3 Internal Energy and the First Law of Thermodynamics 32 2.4 State Functions 33 2.5 Enthalpy 36 2.6 Changes in State Functions 38 2.7 Joule-Thomson Coefficients 42 2.8 More on Heat Capacities 46 2.9 Phase Changes 50 2.10 Chemical Changes 53 2.11 Changing Temperatures 58 2.12 Biochemical Reactions 60 2.13 Summary 62 Exercises 63 vii Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part viii CONTENTS The Second and Third Laws of Thermodynamics 66 3.1 Synopsis 66 3.2 Limits of the First Law 66 3.3 The Carnot Cycle and Efficiency 68 3.4 Entropy and the Second Law of Thermodynamics 72 3.5 More on Entropy 75 3.6 Order and the Third Law of Thermodynamics 79 3.7 Entropies of Chemical Reactions 81 3.8 Summary 85 Exercises 86 Free Energy and Chemical Potential 89 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Synopsis 89 Spontaneity Conditions 89 The Gibbs Free Energy and the Helmholtz Energy 92 Natural Variable Equations and Partial Derivatives 96 The Maxwell Relationships 99 Using Maxwell Relationships 103 Focusing on ⌬G 105 The Chemical Potential and Other Partial Molar Quantities 108 4.9 Fugacity 110 4.10 Summary 114 Exercises 115 Introduction to Chemical Equilibrium 118 5.1 Synopsis 118 5.2 Equilibrium 119 5.3 Chemical Equilibrium 121 5.4 Solutions and Condensed Phases 129 5.5 Changes in Equilibrium Constants 132 5.6 Amino Acid Equilibria 135 5.7 Summary 136 Exercises 138 Equilibria in Single-Component Systems 141 6.1 Synopsis 141 6.2 A Single-Component System 145 6.3 Phase Transitions 145 6.4 The Clapeyron Equation 148 6.5 The Clausius-Clapeyron Equation 152 6.6 Phase Diagrams and the Phase Rule 154 6.7 Natural Variables and Chemical Potential 159 6.8 Summary 162 Exercises 163 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part CONTENTS Equilibria in Multiple-Component Systems 166 7.1 Synopsis 166 7.2 The Gibbs Phase Rule 167 7.3 Two Components: Liquid/Liquid Systems 169 7.4 Nonideal Two-Component Liquid Solutions 179 7.5 Liquid/Gas Systems and Henry’s Law 183 7.6 Liquid/Solid Solutions 185 7.7 Solid/Solid Solutions 188 7.8 Colligative Properties 193 7.9 Summary 201 Exercises 203 Electrochemistry and Ionic Solutions 206 8.1 Synopsis 206 8.2 Charges 207 8.3 Energy and Work 210 8.4 Standard Potentials 215 8.5 Nonstandard Potentials and Equilibrium Constants 218 8.6 Ions in Solution 225 8.7 Debye-Hückel Theory of Ionic Solutions 230 8.8 Ionic Transport and Conductance 234 8.9 Summary 237 Exercises 238 Pre-Quantum Mechanics 241 9.1 Synopsis 241 9.2 Laws of Motion 242 9.3 Unexplainable Phenomena 248 9.4 Atomic Spectra 248 9.5 Atomic Structure 251 9.6 The Photoelectric Effect 253 9.7 The Nature of Light 253 9.8 Quantum Theory 257 9.9 Bohr’s Theory of the Hydrogen Atom 262 9.10 The de Broglie Equation 267 9.11 Summary: The End of Classical Mechannics 269 Exercises 271 10 Introduction to Quantum Mechanics 273 10.1 10.2 10.3 10.4 10.5 Synopsis 273 The Wavefunction 274 Observables and Operators 276 The Uncertainty Principle 279 The Born Interpretation of the Wavefunction; Probabilities 281 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part ix x CONTENTS 10.6 Normalization 283 10.7 The Schrödinger Equation 285 10.8 An Analytic Solution: The Particle-in-a-Box 288 10.9 Average Values and Other Properties 293 10.10 Tunneling 296 10.11 The Three-Dimensional Particle-in-a-Box 299 10.12 Degeneracy 303 10.13 Orthogonality 306 10.14 The Time-Dependent Schrödinger Equation 308 10.15 Summary 309 Exercises 311 11 Quantum Mechanics: Model Systems and the Hydrogen Atom 315 11.1 Synopsis 315 11.2 The Classical Harmonic Oscillator 316 11.3 The Quantum-Mechanical Harmonic Oscillator 318 11.4 The Harmonic Oscillator Wavefunctions 324 11.5 The Reduced Mass 330 11.6 Two-Dimensional Rotations 333 11.7 Three-Dimensional Rotations 341 11.8 Other Observables in Rotating Systems 347 11.9 The Hydrogen Atom: A Central Force Problem 352 11.10 The Hydrogen Atom: The Quantum-Mechanical Solution 353 11.11 The Hydrogen Atom Wavefunctions 358 11.12 Summary 365 Exercises 367 12 Atoms and Molecules 370 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 Synopsis 370 Spin 371 The Helium Atom 374 Spin Orbitals and the Pauli Principle 377 Other Atoms and the Aufbau Principle 382 Perturbation Theory 386 Variation Theory 394 Linear Variation Theory 398 Comparison of Variation and Perturbation Theories 402 Simple Molecules and the Born-Oppenheimer Approximation 403 12.11 Introduction to LCAO-MO Theory 405 12.12 Properties of Molecular Orbitals 409 12.13 Molecular Orbitals of Other Diatomic Molecules 410 12.14 Summary 413 Exercises 416 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 11.10 The Hydrogen Atom: The Quantum-Mechanical Solution Table 11.4 355 Complete wavefunctions for hydrogen-like atomsa mᐉ ⌿n,ᐉ,mᐉ n ᐉ 0 0 Ϫ1 2 ϩ1 0 Ϫ1 3 1 Ϫ2 Ϫ1 3 ϩ1 ϩ2 Z3 ᎏaᎏ e 2Z Zr ᎏᎏᎏᎏ 2 Ϫ ᎏᎏe a a 2Z Zr ᎏᎏᎏᎏ ᎏᎏe sin и e a a 2Z Zr ᎏᎏᎏᎏ ᎏᎏe cos a a 2Z Zr ᎏᎏᎏᎏ ᎏᎏe sin и e a a 18Zr 2Zr 3Z ᎏᎏᎏᎏ 27 Ϫ ᎏᎏ ϩ ᎏᎏe a a 243 a Zr Zr Z ᎏᎏᎏᎏ ᎏᎏ6 Ϫ ᎏᎏe sin и e a a 81 a Zr Zr 2Z ᎏᎏᎏᎏ ᎏᎏ6 Ϫ ᎏᎏe cos a a 81 a Zr Zr Z ᎏᎏᎏᎏ ᎏᎏ6 Ϫ ᎏᎏe sin и e a a 81 a Z r Z ᎏᎏᎏᎏ ᎏᎏe sin и e a 162 a Z r Z ᎏᎏᎏᎏ ᎏᎏe sin cos и e a 81 a Z r 6Z ᎏᎏᎏᎏ ᎏᎏe (3 cos Ϫ 1) a 486 a Z r Z ᎏᎏᎏᎏ ᎏᎏe sin cos и e a 81 a Z r Z ᎏᎏᎏᎏ ᎏᎏe sin и e a 162 a 1/2 ϪZr/a 3 1/2 ϪZr/2a 3 1/2 ϪZr/2a Ϫi 3 1/2 ϪZr/2a 3 1/2 ϪZr/2a i 3 1/2 ϪZr/3a 1/2 ϪZr/3a Ϫi 3 1/2 ϪZr/3a 3 1/2 ϪZr/3a i 3 1/2 2 3 ϪZr/3a Ϫ2i 2 1/2 2 ϪZr/3a Ϫi 1/2 2 3 ϪZr/3a 2 1/2 2 ϪZr/3a i 3 1/2 2 ϪZr/3a 2i 4⑀0ប2 a ϭ ᎏᎏ e2 a For convenience, several of the first few wavefunctions are listed in Table 11.4 along with their respective n, ᐉ, and mᐉ quantum numbers Each characteristic set (n, ᐉ, mᐉ) refers to a specific wavefunction It is easy to show that for any n, the total number of possible wavefunctions having that value of n is n2 (This will increase by a factor of when we include the spin of the electron, but that will be considered in Chapter 12.) The eigenvalue for energy also has an analytic solution It is e4 E ϭ Ϫᎏ ᎏ 8⑀20h2n2 (11.63) The energy is negative here This is due to the convention that the interaction of oppositely charged particles contributes to a decrease in energy (Conversely, the repulsion of similarly charged particles would be positive in energy.) An energy of zero corresponds to the proton and electron at infinite distance from each other (so that the potential energy is zero) and having no kinetic energy with respect to each other The energy depends on a collection of constants— the charge on the electron, e , the reduced mass of the hydrogen atom , the permittivity of free space ⑀0, Planck’s constant h—and the integer n The energy depends on the index n; n is a quantum number and the total energy is quantized The energy of the hydrogen atom does not depend on the quantum numbers ᐉ or mᐉ, only on n The index n is therefore called the principal Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 356 C H A P T E R 11 Quantum Mechanics: Model Systems and the Hydrogen Atom quantum number Because n2 wavefunctions have the same quantum number n, the degeneracy of each energy state of the hydrogen atom is n2 (Again, this will change by a factor of 2.) Each set of wavefunctions having the same value for the principal quantum number is said to define a shell Example 11.21 Calculate the energy values for the first three shells of the hydrogen atom The reduced mass of the hydrogen atom is 9.104 ϫ 10Ϫ31 kg Solution Values are substituted into equation 11.63 for n ϭ 1, 2, and 3: (1.602 ϫ 10Ϫ19 C)4(9.104 ϫ 10Ϫ31 kg) E ϭ Ϫ ᎏᎏᎏᎏᎏᎏ 8[8.854 ϫ 10Ϫ12 C2/(Jиm)]2(6.626 ϫ 10Ϫ34 Jиs)212 (1.602 ϫ 10Ϫ19 C)4(9.104 ϫ 10Ϫ31 kg) E ϭ Ϫ ᎏᎏᎏᎏᎏᎏ 8[8.854 ϫ 10Ϫ12 C2/(Jиm)]2(6.626 ϫ 10Ϫ34 Jиs)222 (1.602 ϫ 10Ϫ19 C)4(9.104 ϫ 10Ϫ31 kg) E ϭ Ϫ ᎏᎏᎏᎏᎏᎏ 8[8.854 ϫ 10Ϫ12 C2/(Jиm)]2(6.626 ϫ 10Ϫ34 Jиs)232 These expressions give E (n ϭ 1) ϭ 2.178 ϫ 10Ϫ18 J E (n ϭ 2) ϭ 5.445 ϫ 10Ϫ19 J E (n ϭ 3) ϭ 2.420 ϫ 10Ϫ19 J where it can easily be shown that the units are joules: C4иkgиJ2иm2 kgиm2 C4иkg ᎏ ᎏ ϭ ᎏ ᎏ ϭ ᎏ ᎏϭJ s2 [C2/(Jиm)]2(Jиs)2 C4иJ2иs2 Remember that spectroscopy measures the changes in energy between two states Quantum mechanics can also be used to determine a change in energy, ⌬E, for the hydrogen atom: e4 e4 E(n1) Ϫ E(n2) ϵ ⌬E ϭ Ϫᎏ ᎏ 2ᎏ 2 Ϫ Ϫᎏ 8⑀0h n1 8⑀20h2n22 where the principal quantum numbers n1 and n2 are used to differentiate between the two energy levels involved A little algebraic rearranging yields e4 1 ⌬E ϭ ᎏ2ᎏ2 ᎏᎏ2 Ϫ ᎏᎏ2 8⑀0h n2 n1 (11.64) This is the same form of equation that Balmer got by considering the spectrum of hydrogen, and that Bohr got by assuming quantized angular momentum! In fact, the collection of constants multiplying the quantum number expression is familiar: e4 (1.602 ϫ 10Ϫ19 C)4(9.104 ϫ 10Ϫ31 kg) ᎏ2ᎏ2 ϭ ᎏᎏᎏᎏᎏ 8⑀0h 8[8.854 ϫ 10Ϫ12 C2/(Jиm]2(6.626 ϫ 10Ϫ34 Jиs)2 ϭ 2.178 ϫ 10Ϫ18 J Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 11.10 The Hydrogen Atom: The Quantum-Mechanical Solution 357 which is easily shown to be, in units of wavenumbers and to four significant figures, e4 ᎏ2ᎏ2 ϭ 109,700 cmϪ1 8⑀0h This is the Rydberg constant, RH, from the hydrogen atom spectrum.* Quantum mechanics therefore predicts the experimentally determined hydrogen atom spectrum At this point, quantum mechanics predicts everything that Bohr’s theory did and more, and so supersedes the Bohr theory of the hydrogen atom Since the spherical harmonics are part of the hydrogen atom’s wavefunctions, it should come as no surprise that the total angular momentum and the z component of the total angular momentum are also observables that have known analytic and quantized values They are ˆ L 2⌿n,ᐉ,mᐉ ϭ ᐉ(ᐉ ϩ 1)ប2⌿n,ᐉ,mᐉ Lˆz ⌿n,ᐉ,mᐉ ϭ mᐉប⌿n,ᐉ,mᐉ so that the quantized values for total angular momentum are ͙ᐉ(ᐉ ෆ1) ϩ ෆប and for the z component are mᐉប The quantum number ᐉ is called the angular momentum quantum number The mᐉ quantum number is the z-component angular momentum quantum number, sometimes called the magnetic quantum number due to the differing behavior of wavefunctions having different mᐉ values in a magnetic field (another topic for later) The angular momentum of the hydrogen atom (due mostly to the electron) is quantized, as Bohr assumed However, the exact values of the quantized angular momentum are slightly different than what Bohr assumed It was not possible to know this in 1913, however, and though ultimately incorrect, Bohr’s theory should be remembered as a crucial step in the right direction This treatment of the hydrogen atom is also applicable to any atom that has only one electron In cases of other atoms, the nuclear charge is different and the overall atom itself has a charge The atomic number, Z, and the reduced mass are the only changes in any of the equations from above (and the reduced mass approaches the mass of the electron as the nucleus gets larger) The Schrödinger equation for these hydrogen-like ions is Ϫប2 Ѩ Ѩ Ѩ Ѩ Ѩ2 ϪZe2 ᎏᎏ ᎏ2ᎏ ᎏᎏ r2ᎏᎏ ϩ ᎏ ᎏ ᎏ2 ϩ ᎏᎏ ⌿ ᎏ ᎏᎏ sin ᎏᎏ ϩ ᎏ ᎏ 2 r Ѩr Ѩr r sin Ѩ Ѩ r sin Ѩ 4⑀0r ϭ E⌿ (11.65) Ά ΄ ΅ · where Z shows up only in the potential energy The only other major change is in the expression for the quantized energy of these ions, which now has the form Z2e4 E ϭ Ϫᎏ ᎏ 8⑀20h2n2 (11.66) The wavefunctions themselves also have a Z dependence on them Table 11.4 gives the complete wavefunctions with their Z dependence already included (In our previous treatment of the hydrogen atom, Z was 1.) The angular momenta observables have the same forms as given above Spectra of the hydrogen-like ions, which have been observed experimentally, are as simple as that *Using modern values of the fundamental constants and to eight significant figures, RH ϭ 109,677.58 cmϪ1 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 358 C H A P T E R 11 Quantum Mechanics: Model Systems and the Hydrogen Atom of the hydrogen atom The transitions appear at different wavelengths of light, however Example 11.22 Predict the wavelength of light emitted by an excited Li2ϩ ion (Z ϭ 3) as an electron goes from the n ϭ state to the n ϭ state Use the mass of the electron in place of the reduced mass (this imparts a very minor 0.008% error in the calculation) Solution We can use an expression for ⌬E similar to the one in equation 11.64, with addition of the Z2 term: Z2e4 1 ⌬E ϭ ᎏ2ᎏ ᎏᎏ Ϫ ᎏᎏ2 8⑀0h2 n22 n1 For n2 ϭ and n1 ϭ 4: 1 32(1.602 ϫ 10Ϫ19 C)4(9.104 ϫ 10Ϫ31 kg) ⌬E ϭ ᎏᎏᎏᎏᎏᎏ ᎏᎏ2 Ϫ ᎏᎏ2 Ϫ12 2 Ϫ34 8[8.854 ϫ 10 C /(J иm)] (6.626 ϫ 10 Jиs) ⌬E ϭ 3.677 ϫ 10Ϫ18 J Using E ϭ h and c ϭ as conversions, we can determine the wavelength of the photon having this energy: ϭ 54.0 nm This wavelength is in the vacuum ultraviolet region of the spectrum 11.11 The Hydrogen Atom Wavefunctions Let us take a closer look at the wavefunctions themselves to finish this chapter Each wavefunction of a hydrogen atom is called an orbital As mentioned, the energy of an electron in an orbital (that is, an electron having its motion described by a particular wavefunction) is dependent only on the principal quantum number n and a collection of physical constants Each group of wavefunctions having the same value of the quantized energy defines a shell Each shell has a degeneracy of n2 Each group of same-ᐉ wavefunctions (for every ᐉ there are 2ᐉ ϩ wavefunctions, having different values of mᐉ) constitutes a subshell In hydrogen and hydrogen-like atoms, all of the subshells within each shell have the same energy This is illustrated in Figure 11.17 In labeling shells and subshells in hydrogen-like atoms (and other atoms, as we will see), we make use of the quantum numbers n and ᐉ The numerical value of the principal quantum number is used in the labeling, and for ᐉ a letter designation is used: ᐉ Letter designation s p d f g Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 11.11 The Hydrogen Atom Wavefunctions Eϭ0 nϭ5 nϭ4 nϭ3 ϭ1 ϭ2 ϭ0 ϭ1 ϭ0 ϭ1 ϭ3 E ϭ Ϫ4,389 cm Ϫ1 E ϭ Ϫ12,193 cm Ϫ1 ϭ2 E ϭ Ϫ27,434 cm Ϫ1 Energy nϭ2 ϭ0 E ϭ Ϫ6,858 cm Ϫ1 359 nϭ1 ϭ0 E ϭ Ϫ109,737 cm Ϫ1 Figure 11.17 The energy level diagram for a hydrogen atom, showing the n and ᐉ quantum numbers for the levels The quantized energy levels are labeled Degenerate wavefunctions are shown Orbitals are designated by pairing the value of the principal quantum number and the letter representing the value of ᐉ: 1s, 2s, 2p, 3s, 3p, 3d, and so forth A numerical subscript can be used to label the mᐉ values of the individual orbitals: 2pϪ1, 2p0, 2pϩ1, and so on Since the value of n restricts the value of ᐉ, the first shell has only an s subshell (because ᐉ can only be 0) The second shell has only s and p subshells (because ᐉ can only be or 1), and so forth These restrictions are due to the nature of the mathematical solution of the Schrödinger equation Example 11.23 What are the possible subshells in the n ϭ shell? How many orbitals are in each subshell? Do not include mᐉ labels Solution For n ϭ 5, ᐉ can be 0, 1, 2, 3, or Each subshell has 2ᐉ ϩ orbitals In tabular form: n, ᐉ Label No of Orbitals 5, 5, 5, 5, 5, 5s 5p 5d 5f 5g Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 360 C H A P T E R 11 Quantum Mechanics: Model Systems and the Hydrogen Atom Wavefunctions for hydrogen-like systems, determined by quantum numbers, can be labeled with those quantum numbers Therefore, it is common to see ⌿1s, ⌿3d, and so on As can be seen from Table 11.4, wavefunctions having a nonzero value for mᐉ have an imaginary exponential function part This means that the overall wavefunction is a complex function In cases where completely real functions are desired, it is useful to define real wavefunctions as linear combinations of the complex wavefunctions, taking advantage of Euler’s theorem For example: ⌿2px ϵ ᎏᎏ(⌿2pϩ1 ϩ ⌿2pϪ1) ͙2ෆ i ⌿2py ϵ Ϫᎏᎏ(⌿2pϩ1 Ϫ ⌿2pϪ1) ͙2ෆ ͉R ͉ Distance from nucleus Figure 11.18 A plot of the square of the ra- dial function of ⌿1s versus distance from the nucleus for the hydrogen atom It suggests that the electron has a maximum probability of existing at the nucleus 4r 2͉R ͉ Most probable distance ϭ a ϭ Bohr radius Distance from nucleus (Å) Figure 11.19 A plot of 4r 2͉R͉2 for ⌿1s versus distance from the nucleus The 4r contribution accounts for the spherical symmetry of the 1s wavefunction about the nucleus By looking at the probability of existence in spherical shells rather than straight away from the nucleus, we get a more realistic picture of the expected behavior of an electron in a hydrogen atom (11.67) The p wavefunctions defined like this are real, not complex, and so are easier to work with in many situations Real wavefunctions for d, f, and other orbitals are defined similarly These nonimaginary wavefunctions are not eigenfunctions of Lˆz any longer, since they are composed of parts that have different eigenvalues of mᐉ They are still eigenfunctions of the energy and total angular momentum, however (In fact, it is only because the original wavefunctions are degenerate that we are able to take linear combinations, like those in equation 11.67.) The behavior of the wavefunctions in space raises some interesting points Every s-type orbital has spherical symmetry, since there is no angular dependence in the wavefunction Because the probability of an electron existing at any point in space is related to ͉⌿͉2 or, in this case, ͉R͉2, the probability of an s electron existing in space is spherically symmetric also Starting from the nucleus and moving out along a straight line, one can plot the probability of the electron having a certain value of r versus the radial distance r itself Such a plot for ⌿1s is shown in Figure 11.18 This plot shows the surprising conclusion that the radius of maximum probability occurs at the nucleus, that is, where r ϭ This analysis is a little misleading From a spherical polar viewpoint, there is very little volume of space close to the nucleus, because for all values of and a small value of r sweeps out a very tiny sphere The total probability of the electron existing in such a small volume of space should be small However, as the radius increases, the spherical volume swept out by the spherically symmetric wavefunction gets larger and larger, and one would expect an increase in probability that the electron will be located at greater distances from the nucleus Instead of considering the electron probability along a straight line out from the nucleus, consider the electron probability on a spherical surface around the nucleus, each spherical surface getting larger and larger Mathematically, the spherical surface corresponds not to ͉R͉2, but 4r2͉R͉2 A plot of 4r2͉R͉2 versus r for ⌿1s is shown in Figure 11.19 The probability starts at zero (a consequence of the “zero” volume at the nucleus), increases to a maximum, then decreases toward zero as the radius gets larger and approaches infinity Quantum mechanics shows that an electron doesn’t have a specific distance from the nucleus Instead, it can have a range of distances having differing probabilities It does have a most probable distance It can be shown mathematically that the value of r at the most probable distance is 4ប2⑀0 rmax ϭ ᎏᎏ ϵa e2 a ϭ 0.529 Å Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part (11.68) (11.69) 11.11 The Hydrogen Atom Wavefunctions 361 where a is the same constant previously defined for the R functions Since it is defined as a group of constants, a itself is a constant and has units of length (shown in equation 11.69 in units of Å) The constant a is called the Bohr radius This most probable distance is exactly the same distance that an electron of Bohr’s theory would have in its first orbit Quantum mechanics does not constrain the distance of the electron from the nucleus as did Bohr’s theory But it does predict that the distance Bohr calculated for the electron in its lowest energy state is in fact the most probable distance of the electron from the nucleus (It is sometimes written a0, which is defined similarly but uses the mass of the electron instead of the reduced mass of the hydrogen atom The difference is very slight.) Example 11.24 a What is the probability that an electron in the ⌿1s orbital of hydrogen will be within a radius of 2.00 Å from the nucleus? b Calculate a similar probability, but now for an electron within 0.250 Å of a Be3ϩ nucleus Solution a For a normalized wavefunction, the probability P is equal to ͵ b P ϭ ⌿*⌿ d a where a and b are the limits of the space being considered For the hydrogen atom, this becomes the three-dimensional expression ͵ ͵ 2 P ϭ ᎏ3ᎏ d и sin d и a0 ͵ 2.00 Å r2eϪ2r/a dr where the wavefunction in terms of the Bohr radius a has already been squared and the expression has been separated into three integrals The two angular integrals we have done before, and the integral over r can be found in Appendix The expression becomes Ϫr2a ra2 a3 P ϭ ᎏ3ᎏ и 2 и eϪ2r/a ᎏᎏ Ϫ ᎏᎏ Ϫ ᎏᎏ ͉2.00Å 2 a ΄ ΅ If the value of a in units of angstroms, 0.529 Å, is used in the above expression, then the 2.00-Å limit can be used directly because the quantities are expressed in the same units Substituting and evaluating the expression at the limits: P ϭ ᎏᎏ и 2 и 2[(5.201 ϫ 10Ϫ4)(1.337841)Å3 Ϫ (0.529 Å)3 (1)(Ϫ3.701 ϫ 10Ϫ2)Å3] Note that the Å3 units cancel from the expression, and the probability is unitless (as it should be) Evaluating this expression, we find that P ϭ 0.981, or 98.1% This example shows that the electron has a 98.1% probability of being within 2.00 Å, or slightly under Bohr radii, from the nucleus You might want to compare this with Figure 11.19, where the total probability is represented by the area under the curve Finally, note that this implies a 1.9% chance that the electron is farther than 2.00 Å from the nucleus Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 362 C H A P T E R 11 Quantum Mechanics: Model Systems and the Hydrogen Atom b For the Be3ϩ nucleus, the solution to the problem is along similar lines but now the nuclear charge for the beryllium atom must be included explicitly For Z ϭ 4, the integrals being evaluated are ͵ ͵ 2 43 P ϭ ᎏ3ᎏ d и sin d и a0 ͵ 0.250Å r 2eϪ8r/a dr Note that the upper limit on the r integral is now 0.250 Å These expressions integrate to yield Ϫr 2a ra2 64 a3 P ϭ ᎏ3ᎏ и 2 и eϪ8r/a ᎏ ᎏ Ϫ ᎏᎏ Ϫ ᎏᎏ ͉0.250Å 32 a 256 ΄ ΅ which yields 64 P ϭ ᎏᎏ и 2 и 2[(2.28 ϫ 10Ϫ2)(Ϫ0.00690)Å3 (0.529 Å)3 ϭ Ϫ (1)(Ϫ5.78 ϫ 10Ϫ4)Å3] P ϭ 728 or 72.8% This is to be expected, since the larger nuclear charge pulls the single electron in closer to the nucleus Therefore, there is a 72ϩ% probability of finding a ⌿1s electron within 0.250 Å of a Be3ϩ nucleus Radial probability plots for ⌿2s, ⌿2p, ⌿3s, ⌿3p, ⌿3d, are shown in Figure 11.20 For each wavefunction having quantum numbers n and ᐉ, there are n Ϫ ᐉ Ϫ points along a spherical radius where the probability of finding an electron becomes exactly zero These points are nodes Specifically, these are radial nodes, since we are considering the total electron probability at a spherical shell at each value of the radius Although s subshells are spherically symmetric, individual p, d, f, subshells are not and have angular dependence There are several ways of conveying the angular dependence of subshells One common way is to draw an outline within which the probability of the electron’s appearance is 90% It is easiest to use the real form of the wavefunctions to illustrate this behavior Figure 11.21 shows the 90% boundary surfaces of real (that is, nonimaginary) p and d subshells of hydrogen It is these angular distributions of the subshells that lend the “dumbbell” and “rosette” descriptions to the p and d orbitals There are several things to note about these plots First, for each orbital, different axes are used to illustrate the plot, which means that the orbitals point in different directions in space even though they look very similar Each section of the plots is labeled with a plus or a minus to indicate the sign of the wavefunction in that region Next, for each p orbital there is one plane that is tangent to all electron probability As an example, for the pz orbital, the xy plane is the plane of exactly zero electron probability For the px orbital, the yz plane has zero electron probability For the d orbitals, there are two planes where the electron probability is zero These are examples of angular nodes (also called nodal planes or nodal surfaces) Figure 11.22 shows some of the angular nodes for p and d orbitals For the dz2 orbital, the nodal surface is a two-dimensional cone For quantum number ᐉ, there will be ᐉ angular nodes Combining angular nodes with radial nodes, there will be a total of n Ϫ nodes (both radial and angular) for any wavefunction ⌿n,ᐉ Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 4r 2͉R (2 p)͉ 4r 2͉R (2s)͉ 2 Distance from nucleus (Å) Distance from nucleus (Å) 4r 2͉R (3s)͉ 10 Distance from nucleus (Å) 15 4r 2͉R (3p)͉ 4r 2͉R (3d )͉ 0 10 Distance from nucleus (Å) 15 10 Distance from nucleus (Å) 15 Figure 11.20 Plots of 4r 2͉R͉2 versus distance for other hydrogen atom wavefunctions, as la- beled There is a simple relationship between the quantum numbers and the number of radial nodes 363 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part + – + – + – pz py px + – + – + + – + – + + – – – + – – + + dxz dyz dx Ϫ y dxy dz2 Figure 11.21 The 90% boundary plots for the real forms of p and d wavefunctions The spe- cific label on the p or d orbital depends on the direction the orbital takes in 3-D space + – + – + – pz py px + – + – + – + + dxz – – + – + – dyz – + dxy + dx Ϫ y – + dz2 Figure 11.22 Nodal planes for p and d orbitals Each p orbital has one nodal plane Each d orbital has two nodal planes For the dz2 orbital, the nodal planes are represented by a single conical surface 364 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 11.12 Summary 365 Example 11.25 a What is the average value of the total angular momentum for ⌿3p for the hydrogen atom? b Is there an easier way to determine this value? Solution a The square of the total angular momentum is defined, so we will assume that the average angular momentum is the square root of the squared angular momentum Therefore, we need to find ͗L͘ ϭ ͙͗L ෆ ͘ To this, we need to determine the average value ͗L2͘ This is done with the following expression: ͗L2͘ ϭ ͵ ⌿* ˆL ⌿ 3p 3p d It might seem at first that we may have to use the long, complete form of ⌿3p and the long, complete form of ˆ L 2; we don’t, though Since ⌿3p is an eigen2 function of ˆ L , we can substitute the eigenvalue for the operator in the integral above Since the eigenvalue is ᐉ(ᐉ ϩ 1)ប2, the integral above becomes ͗L2͘ ϭ ͵ ⌿* [ᐉ(ᐉ ϩ 1)ប ] ⌿ 3p 3p d where the constants are multiplied with the wavefunctions instead of any function-changing operation occurring Multiplicative constants are moved outside the integral sign, so the above expression becomes ͗L2͘ ϭ ᐉ(ᐉ ϩ 1)ប2 ͵ ⌿* ⌿ 3p 3p d The wavefunction is normalized, so the integral is simply Therefore, ͗L2͘ ϭ ᐉ(ᐉ ϩ 1)ប2 and using the value of ᐉ ϭ for a p orbital, we can determine the average value of the total angular momentum ͗L͘ as ͗L͘ ϭ ͙͗L ෆ ͘ ϭ ͙ᐉ(ᐉ ෆ1)ប ϩ ෆ2 ϭ ͙2ෆប ϭ 1.491 ϫ 10Ϫ34 Jиs b The easier way is to realize that L2 is a quantized observable for the 3p wavefunction of a hydrogen atom The average value is equal to the quantized value This will not always be the case for average values (see exercises 11.58 or 11.60, for example) 11.12 Summary With the solution of the hydrogen atom, the list of analytically solvable systems to be considered here is complete Planck’s quantum theory of light described blackbody radiation, and now the simplicity of the spectrum of the hydrogen (and hydrogen-like ions) is adequately explained by quantum mechanics We will find in the next chapter that although an exact analytic understanding of the behavior of electrons in larger atoms is not forthcoming, quantum mechanics does provide the tools for a numerical solution to larger Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 366 C H A P T E R 11 Quantum Mechanics: Model Systems and the Hydrogen Atom systems, like molecules (for smaller molecules in practice, and for larger molecules at least in theory) Because this is far more than the classical theories of chemistry and physics could provide, quantum mechanics is accepted as the superior theory of the behavior of matter at the electronic level The postulates of quantum mechanics allow for some seemingly unusual and unexpected behavior—like tunneling, quantized angular momenta, and “fuzzy” electron orbitals But so far, the predictions of quantum mechanics have been borne out when examined experimentally That is the true test of a theory Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part E X E R C I S E S F O R C H A P T E R 1 11.2 Classical Harmonic Oscillator 11.4 Harmonic Oscillator Wavefunctions 11.1 Convert 3.558 mdyn/Å into units of N/m 11.12 Show that ⌿2 and ⌿3 for the harmonic oscillator are orthogonal 11.2 A swinging pendulum has a frequency of 0.277 Hz and a mass of 500.0 kg Calculate the force constant for this harmonic oscillator 11.3 An object having mass m at some height above the ground h has a gravitational potential energy of mgh, where g is the acceleration due to gravity (ϳ9.8 m/s2) Explain why objects moving back and forth under the influence of gravity (like a clock’s pendulum) can be treated as harmonic oscillators (Hint: see equation 11.1.) 11.3 Quantum-Mechanical Harmonic Oscillator 11.4 In equation 11.6, in order to properly subtract the two terms in parentheses on the left, they must have the same units overall Verify that 2mE/ប2 and ␣2x have the same units Use standard SI units for x (position/distance) Do the same for the two terms in parentheses in equation 11.11 11.5 Verify that the three substitutions mentioned in the text yield equation 11.6 11.6 Verify that the second derivative of ⌿ given by equation 11.8 gives equation 11.9 11.7 Derive equation 11.16 from the equation immediately preceding it 11.8 Show that the energy separation between any two adjacent energy levels for an ideal harmonic oscillator is h, where is the classical frequency of the oscillator 11.13 Substitute ⌿1 into the complete expression for the Hamiltonian operator of an ideal harmonic oscillator and show that E ϭ ᎏ32ᎏh 11.14 Calculate ͗px͘ for ⌿0 and ⌿1 for a harmonic oscillator Do the values you calculate make sense? 11.15 Use the expression for ⌿1 in equations 11.17 and normalize the wavefunction Use the integral defined for the Hermite polynomials in Table 11.2 Compare your answer with the wavefunction defined by equation 11.19 11.16 Simply using arguments based on odd or even functions, determine whether the following integrals involving harmonic oscillator wavefunctions are identically zero, are not identically zero, or are indeterminate If indeterminate, state why ϩؕ ϩؕ ͵ ⌿*⌿ dx (b) ͵ ⌿* ˆx ⌿ dx (c) ͵ ⌿*xˆ ⌿ dx, where ˆ x ϭˆ x иˆ x (d) ͵ ⌿*⌿ dx (e) ͵⌿*⌿ dx ˆ⌿ dx, where ˆ (f) ͵ ⌿*V V is some undefined potential energy (a) Ϫؕ ϩؕ Ϫؕ ϩؕ Ϫؕ ϩؕ Ϫؕ 1 1 2 Ϫؕ 3 function 11.9 (a) For a pendulum having a classical frequency of 1.00 sϪ1, what is the energy difference in J between quantized energy levels? (b) Calculate the wavelength of light that must be absorbed in order for the pendulum to go from one level to another (c) Can you determine in what region of the electromagnetic spectrum such a wavelength belongs? (d) Comment on your results for parts a and b based on your knowledge of the state of science in the early twentieth century Why wasn’t the quantum mechanical behavior of nature noticed? 11.17 Determine the value(s) of x for the classical turning point of a harmonic oscillator in terms of k and n There may be other constants in the expression you derive 11.10 (a) A hydrogen atom bonded to a surface is acting as a harmonic oscillator with a classical frequency of 6.000 ϫ 1013 sϪ1 What is the energy difference in J between quantized energy levels? (b) Calculate the wavelength of light that must be absorbed in order for the hydrogen atom to go from one level to another (c) Can you determine in what region of the electromagnetic spectrum such a wavelength belongs? (d) Comment on your results for parts a and b based on your knowledge of the state of science in the early twentieth century 11.19 Reduced mass is not reserved only for atomic systems A solar system or a planet/satellite system, for example, can have its behavior described by first determining its reduced mass If the mass of Earth is 2.435 ϫ 1024 kg and that of the moon is 2.995 ϫ 1022 kg, what is the reduced mass of the Earth-moon system? (This is not to imply any support of a planetary model for atoms!) 11.11 The O–H bond in water vibrates at a frequency of 3650 cmϪ1 What wavelength and frequency (in sϪ1) of light would be required to change the vibrational quantum number from n ϭ to n ϭ 4, assuming O–H acts as a harmonic oscillator? 11.5 Reduced Mass 11.18 Compare the mass of the electron, me, with (a) the reduced mass of a hydrogen atom; (b) the reduced mass of a deuterium atom (deuterium ϭ 2H); (c) the reduced mass of a carbon-12 atom having a ϩ5 charge, that is, C5ϩ Suggest a conclusion to the trend presented by parts a–c 11.20 (a) Calculate the expected harmonic-oscillator frequency of vibration for carbon monoxide, CO, if the force constant is 1902 N/m (b) What is the expected frequency of 13 CO, assuming the force constant remains the same? 11.21 An O–H bond has a frequency of 3650 cmϪ1 Using equation 11.27 twice, set up a ratio and determine the expected frequency of an O–D bond, without calculating the Exercises for Chapter 11 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 367 force constant D ϭ deuterium (2H) Assume that the force constant remains the same 11.6 2-D Rotations 11.22 Why can’t the quantized values of the 2-D angular momentum be used to determine the mass of a rotating system, like classical angular momentum can? 11.23 Show that ⌿3 of 2-D rotational motion has the same normalization constant as ⌿13 by normalizing both wavefunctions 11.24 What are the energies and angular momenta of the first five energy levels of benzene in the 2-D rotational motion approximation? Use the mass of the electron and a radius of 1.51 Å to determine I 11.25 A 25-kg child is on a merry-go-round/calliope, going around and around in a large circle that has a radius of meters The child has an angular momentum of 600 kgиm2/s (a) From these facts, estimate the approximate quantum number for the angular momentum the child has (b) Estimate the quantized amount of energy the child has in this situation How does this compare to the child’s classical energy? What principle does this illustrate? 11.26 Using Euler’s identity, rewrite the first four 2-D rotational wavefunctions in terms of sine and cosine 11.27 (a) Using the expression for the energy of a 2-D rigid rotor, construct the expression for the energy difference between two adjacent levels, E(m ϩ 1) Ϫ E(m) (b) For HCl, E(1) Ϫ E(0) ϭ 20.7 cmϪ1 Calculate E(2) Ϫ E(1), assuming HCl acts as a 2-D rigid rotor (c) This energy difference is determined experimentally as 41.4 cmϪ1 How good would you say a 2-D model is for this system? 11.34 A 3-D rotational wavefunction has the quantum number ᐉ equal to and a moment of inertia of 4.445 ϫ 10Ϫ47 kgиm2 What are the possible numerical values of (a) the energy; (b) the total angular momentum; (c) the z component of the total angular momentum? 11.35 (a) Using the expression for the energy of a 3-D rigid rotor, construct the expression for the energy difference between two adjacent levels, E(ᐉ ϩ 1) Ϫ E(ᐉ) (b) For HCl, E(1) Ϫ E(0) ϭ 20.7 cmϪ1 Calculate E(2) Ϫ E(1), assuming HCl acts as a 3-D rigid rotor (c) This energy difference is determined experimentally as 41.4 cmϪ1 How good would you say a 3-D model is for this system? 11.36 See Example 11.17, regarding the “spherical” C60 molecule Assuming the electrons in this molecule are experiencing 3-D rotations, calculate the wavelength of light necessary to cause a transition from state ᐉ ϭ to ᐉ ϭ and from ᐉ ϭ to ᐉ ϭ Compare your answers with experimentally measured absorptions at wavelengths of 328 and 256 nm How good is this model for describing C60’s electronic absorptions? 11.37 In exercise 11.36 regarding C60, what are the numerical values of the total angular momenta of the electron for each state having quantum number ᐉ? What are the z components of the angular momentum for each state? 11.38 Draw graphical representations (see Figure 11.15) of the possible values for ᐉ and mᐉ for the first four energy levels of the 3-D rigid rotor What are the degeneracies of each state? 11.39 What is a physical explanation of the difference between a particle having the 3-D rotational wavefunction ⌿3,2 and an identical particle having the wavefunction ⌿3,Ϫ2? 11.28 Derive equation 11.35 from 11.34 11.9, 11.10, & 11.11 Hydrogen-Like Atoms 11.7 & 11.8 3-D Rotations 11.40 List the charges on hydrogen-like atoms whose nuclei are of the following elements (a) lithium, (b) carbon, (c) iron, (d) samarium, (e) xenon, (f) francium, (g) uranium, (h) seaborgium 11.29 Use trigonometry to verify the relationships between the Cartesian and spherical polar coordinates as given in equation 11.40 11.30 Why can’t the square root of equation 11.45 be taken analytically? (Hint: consider how you would have to take the square root of the right side of the equation Can it be done?) 11.41 Calculate the electrostatic potential energy V between an electron and a proton if the electron is at a distance of Bohr radius (0.529 Å) from the proton Be careful that the correct units are used! 11.31 For both 2-D and 3-D rotations, the radius of the particle’s motion is kept constant Consider a nonzero, constant potential energy acting on the particle Show that the form of the Schrödinger equation in equation 11.46 would be equivalent to its form if V were identically zero (Hint: use the idea that Enew ϭ E Ϫ V.) 11.42 Using Newton’s law of gravity and the relationship between force and potential energy, the gravitational potential energy can be written as 11.32 Can you evaluate ͗r ͘ for the spherical harmonic Y 2Ϫ2? Why or why not? Use the masses of the electron and the proton and the gravitational constant G ϭ 6.673 ϫ 10Ϫ11 Nиm2/kg2 to show that the gravitational potential energy is negligible compared to the electrostatic potential energy at a distance of Bohr radius 11.33 Using the complete form of ⌿3,Ϫ2 (where ᐉ ϭ and mᐉ ϭ Ϫ2) for 3-D rotations (get the Legendre polynomial from Table 11.3) and the complete forms of the operators, evaluate the eigenvalues of (a) L2, (b) Lz, (c) E Do not use the analytic expressions for the observables Instead, operate on ⌿3,Ϫ2 with the appropriate operators and see that you get the proper eigenvalue equation From the eigenvalue equation, determine the value of the observable 368 m m2 V ϭ ϪG ᎏ1ᎏ r 11.43 Show that for constant r and V ϭ 0, equation 11.56 becomes equation 11.46 (Hint: you will have to apply the chain rule of differentiation to the derivatives in the second term of equation 11.56.) Exercises for Chapter 11 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 11.44 Calculate the difference between the Bohr radius defined as a and the Bohr radius defined as a0 11.45 To four significant figures, the first four lines in the Balmer series in the hydrogen atom (n2 ϭ 2) spectrum appear at 656.5, 486.3, 434.2, and 410.3 nm (a) From these numbers, calculate an average value of RH, the Rydberg constant (b) At what wavelengths would similar transitions appear for Heϩ? 11.46 What would the wavelengths of the Balmer series for deuterium be? 11.47 Construct an energy level diagram showing all orbitals for the hydrogen atom up to n ϭ 5, labeling each orbital with its appropriate quantum numbers How many different orbitals are in each shell? 11.48 What are the values of E, L, and Lz for an F8ϩ atom whose electron has the following wavefunctions, listed as ⌿n,ᐉ,mᐉ? (a) ⌿1,0,0 (b) ⌿3,2,2 (c) ⌿2,1,Ϫ1 (d) ⌿9,6,Ϫ3 11.49 Why does the wavefunction ⌿4,4,0 not exist? Similarly, why does a 3f subshell not exist? (See exercise 11.48 for notation definition.) 11.50 Calculate the total electronic energy of a mole of hydrogen atoms Calculate the total electronic energy of a mole of Heϩ atoms What accounts for the difference in the two total energies? 11.51 What is the probability of finding an electron in the 1s orbital within 0.1 Å of a hydrogen nucleus? 11.52 What is the probability of finding an electron in the 1s orbital within 0.1 Å of an Ne9ϩ nucleus? Compare your answer to the answer to exercise 11.51 and justify the difference 11.53 State how many radial, angular, and total nodes are in each of the following hydrogen-like wavefunctions (a) ⌿2s (b) ⌿3s (c) ⌿3p (d) ⌿4f (e) ⌿6g (f) ⌿7s 11.56 Show that rmax is given by equation 11.68 for ⌿1s Take the derivative of 4r 2⌿2 with respect to r, set it equal to zero, and solve for r 11.57 Use the forms of the wavefunctions in Table 11.4 to determine the explicit forms for the 2px and 2py nonimaginary wavefunctions 11.58 Evaluate ͗Lz͘ for 3px Compare it to the answer in Example 11.25, and explain the difference in the answers 11.59 Using equations 11.67 as an example, what would the combinations for the five real 3d wavefunctions be? Use Table 11.4 to assist you 11.60 Evaluate ͗r͘ for ⌿1s (assume that the operator rˆ is defined as “multiplication by the coordinate r ”) Why does ͗r͘ not equal 0.529 Å for ⌿1s? In this case, d ϭ 4r dr Symbolic Math Exercises 11.61 Graph the first five wavefunctions for the harmonic oscillators and their probabilities Superimpose these graphs on the potential energy function for a harmonic oscillator and numerically determine the x values of the classical turning points What is the probability that an oscillator will exist beyond the classical turning points? Do plots of the probability begin to show a distribution as expected by the correspondence principle? 11.62 Construct three-dimensional plots of the first three families of spherical harmonics Can you identify the values of and that correspond to nodes? 11.63 Set up and evaluate numerically the integral that shows that Y 11 and Y 1Ϫ1 are orthogonal 11.64 Plot the 90% surfaces of the hydrogen atom 2s and 2p angular wavefunctions in 3-D space Can you identify nodes in your graphic? 11.54 Illustrate that the hydrogen wavefunctions are orthogonal by evaluating ͐⌿* 2s⌿1s d over all space 11.55 Verify the specific value of a, the Bohr radius, by using the values of the various constants and evaluating equation 11.68 Exercises for Chapter 11 Copyright 2011 Cengage Learning All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 369 ... Theory 446 13 .11 Hybrid Orbitals 450 13 .12 Summary 456 Exercises 457 14 Rotational and Vibrational Spectroscopy 4 61 14 .1 14.2 14 .3 14 .4 14 .5 14 .6 14 .7 14 .8 14 .9 14 .10 14 .11 14 .12 Synopsis 4 61 Selection... 504 14 .17 Rotational-Vibrational Spectroscopy 506 14 .18 Raman Spectroscopy 511 14 .19 Summary 514 Exercises 515 15 Introduction to Electronic Spectroscopy and Structure 519 15 .1 15.2 15 .3 15 .4 15 .5... Schrödinger Equation 308 10 .15 Summary 309 Exercises 311 11 Quantum Mechanics: Model Systems and the Hydrogen Atom 315 11 .1 Synopsis 315 11 .2 The Classical Harmonic Oscillator 316 11 .3 The Quantum-Mechanical