Table Of Content

Values of Some Physical Constants Constant Symbol Value 27 Atomic mass constant 1.660 5402 x 10- kg 23 I Avogadro constant 6.022 1367 x 10 mol- 24 I Bohr magneton 9.274 0154 x 10- J . T- 11 Bohr radius 5.291 772 49 x 10- m 23 J Boltzmann constant 1.380 658 x 10- J . K- I 0.695 038 cm- 31 Electron rest mass m 9.109 3897 x 10- kg e 11 3 Gravitational constant G 6.672 59 x 10- m . kg-I. S-2 I Molar gas constant R 8.314 510 J . K- . mol-I 3 I I 0.083 1451 dm • bar . K- • mol- 3 1 0.082 0578 dm • atm· K- • mol-I Molar volume, ideal gas (one bar, 0° C) 22.711 08 L . mol 1 (one atm, O°C) 22.414 09 L . mol-I 27 I Nuclear magneton J-l = enj2m 5.050 7866 x 10- J . T- N p 12 2 I I Permittivity of vacuum So 8.854 187 816 x 10- C • J- • m- 10 2 I I 4nso 1.112 650 056 x 10- C • J- • m- 34 Planck constant h 6.626 0755 x 10- J . s 34 n 1.054 572 66 x 10- J . s 19 Proton charge e 1.602 177 33 x 10- C 8 1 Proton magnetogyric ratio Yp 2.675 221 28 x 10 S-I • T- 27 Proton rest mass m 1.672 6231 x 10- kg p 4 2 18 Rydberg constant (Bohr) R = m e j8s h2 2.179 8736 x 10- J 00 e 0 I 109 737.31534 cm- 1 Rydberg constant (exptl) RH 109 677.581 cm- Speed of light in vacuum c 299 792 458 m . s I (defined) 5 3 2 8 2 4 Stefan-Boltzmann constant a = 2n k~/ 15h c 5.670 51 x 10- J . m- . K- . S-I Conversion Factors for Energy Units joule kJ . mol-I eV 20 18 1 joule 1 6.022 137 x 10 6.241 506 x 10 1 kJ . mol-I 1.660 540 x 10-21 1 1.036 427 x 10-2 19 leV 1.602 177 x 10- 96.4853 1 18 1 Eh 4.359 748 x 10- 2625.500 27.2114 1 cm- I 1.986 447 x 10-23 1.196 266 x 10-2 1.239 842 x 10-4 34 13 15 1 Hz 6.626 076 x 10- 3.990 313 x 10- 4.135 669 x 10- Some Mathematical Formulas sin(x ± y) = sin x cos y ± cosxsin y cos(x ± y) =cosxcosy =f sin x siny' sin x sin y = ~ cos(x - y) - 4c os(x + y) cos x cosy =4 cos(x - y) + 4c os(x + y) sin x cos y = 4s in(x + y) + 4s in(x - y) e±ix = cosx ± i sin x ix ix . e _ e- cosx = ---- slnx = ---- 2 2i eX + e-X eX _ e-X coshx = . sinhx = --- 2 2 1 1 2 j(x) = j(a) + j'(a)(x - a) + ~ j"(a) (x - al + - j"'(a) (x - a)3 + ... 21 ' 3! " , x 2 x 3 X4 X e =l+x+-+-+-+··· 2! 3! 4! 2 6 x X4 x COSX = 1 - - + - - - + ... 2! 4! 6! 3 S 7 x X x sinx = x - - + - - - + ... 3! S! 7! 2 3 . x x . X4 In(1 +x) = x - "2 + 3" - 4 + ... -1<x<1 1 2 3 4 2 --=l+x+x +x +x + ... x <1 I-x . n _ 1 ± . ± n(n - 1) 2.± n(n - l)(n - 2) 3 (1 ± X ) - nx x x + ... .' 2! 3! (n positive integer) (n positive integer) 00 . , <!' 2n+l. - ax2d n. x e 'X=-- (n positive integer) n 1 1o , 2a + a. nnx·. mnx La nnx mnx a sIn sm = cos cos = -~ Lo a .a 0 a a 2 nm a n1fx. mnx 0 cos sm = (m/and n integers) Lo a a ",.'~~ plane polar coordinates: x = r cos e O<r<oo y = r sin e 0<8<2n - - dr = rdrde 2 2 a a \72=_+_ 2 ax ay2 2 2 2 1 a ( a) 1 a a 1 a 1 a = r ar r ar + r2 ae2 = ar2 + r ar + r2 ae2 spherical coordinates: x = r sin e cos ¢ 0 < r < 00 y = r sin e sin ¢ 0 < e < n z = r cos e 0 < ¢ < 2n dr = r2 sin edrded¢ 2 2 2 a a a \72=_+_+_ 2 2 ax ay2 az = ~~ (r2~) + 1 ~ (sine~) + 1 ~ 2 r2 ar ar r2 sin e ae ae r2 sin e a¢2 lf cosn e sin e de = 11 xndx = 0 if n is odd lo -I 2 if n is even n+1 n 3 2 {If cos e sin e de = l' xn (1 - x )dx = 0 if n is odd 10 -I 4 if n is even (n + l)(n + 3) SI Prefixes Fraction Prefix Symbol Multiple Prefix Symbol 1 10- deci d 10 deka da 2 2 10- centi c 10 hecto h 3 3 10- milli m 10 kilo k 6 . 6 10- mlcro J-L 10 mega M 9 9 . 10- nano n 10 gIg a G 12 . 12 10- PICO P 10 tera T 15 15 10- femto f 10 peta P 18 18 10- atto a 10 exa E Pressure Conversion Factors Pa bar atm torr 1 Pa = 1 10-5 9.869 23 x 10-6 7.500 62 x 10-3 1 bar = 105 1 0.986 923 750.062 1 atm = 1.013 25 x 105 1.013 25 1 760 3 3 1 torr = 133.322 1.333 22 x 10- 1.315 79 x 10- 1 Some Commonly Used Non-SI Units Unit Quantity Symbol SI value 10 Angstrom length A 10- m = 100 pm 6 Micron length J-L 10- m Calorie energy cal 4.184 J (defined) 30 Debye dipole moment D 3.3356 x 10- C . m 4 Gauss magnetic field strength G 10- T Greek Alphabet Alpha A ct Iota I l Rho P p Beta B fJ Kappa K K Sigma b a Gamma r y Lambda A A Tau T r Delta ~ 8 Mu M /-l Upsilon Y v Epsilon E E Nu N v Phi <l> ¢ ,.... Zeta Z ~ Xi Co. ~ Chi X X Eta H Y} Omicron 0 0 Psi \11 1/1 Theta e (J Pi n ]f Omega Q w I cm- Hz 17 22 33 2.293 710 x 10 5.034 11 x 10 1.509 189 x 10 4 12 3.808 798 x 10- 83.5935 2.506 069 x 10 2 14 3.674 931 x 10- 8065.54 2.417 988 x 10 1 2.194 7463 x 105 6.579 684 x 1015 6 10 4.556 335 x 10- 1 2.997 925 x 10 16 11 1.519 830 x 10- 3.335 64 x 10- 1 PHYSICAL CH EMISTRY A MOLECULAR APPROACH 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 www.uscibooks.com University Science Books www.uscibooks.com 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 Allan) Physical chemistry : a molecular approach / Donald A. McQuarrie, John D. Simon. p. cm. Includes bibliographical references and index. I ISBN 978-0-935702-99-6 I 1. 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 10 9 Contents Preface xvi i .. To the Student XVII To the Instructor XIX Our Web Site xxi Acknowledgment XXIII CHAPTER 1 / The Dawn of the Quantum Theory 1 1-1. Blackbody Radiation Could Not Be Explained by Classical Physics 2 1-2. Planck Used a Quantum Hypothesis to Derive the Blackbody Radiation Law 4 1-3. Einstein Explained the Photoelectric Effect with a Quantum Hypothesis 7 1-4. The Hydrogen Atomic Spectrum Consists of Several Series of Lines 10 1-5. The Rydberg Formula Accounts for All the Lines in the Hydrogen Atomic Spectrum 13 1-6. Louis de Broglie Postulated That Matter Has Wavelike Properties 15 1-7. de Broglie Waves Are Observed Experimentally 16 1-8. 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 Problems 25 MATHCHAPTER A / Complex Numbers 31 Problems 35 CHAPTER 2 / 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 46 2-5. A Vibrating Membrane Is Described by a Two-Dimensional Wave Equation 49 Problems 54 MATHCHAPTER B / Probability and Statistics 63 Problems 70 v PHYSICAL CHEMISTRY CHAPTER 3 / The Schrodinger Equation and a Particle In a Box 73 3-1. The Schrodinger 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 Schrodinger 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 Fu nctions Must Be Normal ized 84 3-7. The Average Momentum of a Particle in a Box Is Zero 86 3-8. The Uncertainty Principle Says That apa > n/2 88 x 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 / Vectors 105 Problems 11 3 CHAPTER 4 / 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 Schrodinger 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 1 34 MATHCHAPTER D / Spherical Coordinates 147 Problems 1 53 CHAPTER 5 / The Harmonic Oscillator and the Rigid Rotator: Two Spectroscopic Models 157 5-1. A Harmonic Oscillator Obeys Hooke's Law 157 5-2. 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 5-4. The Energy Levels of a Quantum-Mechanical Harmonic Oscillator Are Ev = Tzw(v + !) with v = 0, 1, 2, ... 1 66 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 172 5-8. The Energy Levels of a Rigid Rotator Are E = Tz2 J (J + 1) /21 1 73 VI