www.TheSolutionManual.com Physical Chemistry www.TheSolutionManual.com Third Edition www.TheSolutionManual.com Physical Chemistry Robert G Mortimer Professor Emeritus Rhodes College Memphis, Tennessee AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier www.TheSolutionManual.com Third Edition Cover Design: Eric DeCicco Cover Image: © iStockphoto Elsevier Academic Press 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper ∞ Copyright © 2008, Elsevier Inc All rights reserved Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: permissions@elsevier.co.uk You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress Catalog-in-Publishing Data Mortimer, Robert G Physical chemistry / Robert G Mortimer – 3rd ed p cm Includes bibliographical references and index ISBN 978-0-12-370617-1 (hardcover : alk paper) Chemistry, Physical and theoretical I Title QD453.2.M67 2008 541–dc22 2008007675 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-12-370617-1 For information on all Elsevier Academic Press publications visit our Web site at www.books.elsevier.com Printed in Canada 08 09 10 www.TheSolutionManual.com No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher www.TheSolutionManual.com To my wife, Ann, and to my late father, William E Mortimer, who was responsible for my taking my first chemistry course www.TheSolutionManual.com Contents List of Numerical Tables in Appendix A Inside front cover Information Tables Inside back cover Preface xv Acknowledgments Part Chapter xvii Thermodynamics and the Macroscopic Description of Physical Systems The Behavior of Gases and Liquids 1.1 Introduction 1.2 Systems and States in Physical Chemistry 12 1.3 Real Gases 21 1.4 The Coexistence of Phases and the Critical Point 27 Chapter Work, Heat, and Energy: The First Law of Thermodynamics 39 2.1 Work and the State of a System 40 2.2 Heat 51 2.3 Internal Energy: The First Law of Thermodynamics 55 2.4 Calculation of Amounts of Heat and Energy Changes 60 2.5 Enthalpy 74 2.6 Calculation of Enthalpy Changes of Processes without Chemical Reactions 81 2.7 Calculation of Enthalpy Changes of a Class of Chemical Reactions 86 2.8 Calculation of Energy Changes of Chemical Reactions 94 Chapter The Second and Third Laws of Thermodynamics: Entropy 105 3.1 The Second Law of Thermodynamics and the Carnot Heat Engine 106 vii www.TheSolutionManual.com Periodic Table Inside front cover viii Contents 3.3 3.4 3.5 The Mathematical Statement of the Second Law: Entropy 114 The Calculation of Entropy Changes 121 Statistical Entropy 133 The Third Law of Thermodynamics and Absolute Entropies 139 Chapter The Thermodynamics of Real Systems 151 4.1 Criteria for Spontaneous Processes and for Equilibrium: The Gibbs and Helmholtz Energies 152 4.2 Fundamental Relations for Closed Simple Systems 158 4.3 Additional Useful Thermodynamic Identities 167 4.4 Gibbs Energy Calculations 175 4.5 Multicomponent Systems 182 4.6 Euler’s Theorem and the Gibbs–Duhem Relation 188 Chapter Phase Equilibrium 199 5.1 The Fundamental Fact of Phase Equilibrium 200 5.2 The Gibbs Phase Rule 202 5.3 Phase Equilibria in One-Component Systems 205 5.4 The Gibbs Energy and Phase Transitions 215 5.5 Surfaces in One-Component Systems 222 5.6 Surfaces in Multicomponent Systems 230 Chapter The Thermodynamics of Solutions 237 6.1 Ideal Solutions 238 6.2 Henry’s Law and Dilute Nonelectrolyte Solutions 6.3 Activity and Activity Coefficients 258 6.4 The Activities of Nonvolatile Solutes 267 6.5 Thermodynamic Functions of Nonideal Solutions 6.6 Phase Diagrams of Nonideal Mixtures 282 6.7 Colligative Properties 292 248 275 Chapter Chemical Equilibrium 303 7.1 Gibbs Energy Changes and the Equilibrium Constant 304 7.2 Reactions Involving Gases and Pure Solids or Liquids 310 7.3 Chemical Equilibrium in Solutions 315 7.4 Equilibria in Solutions of Strong Electrolytes 328 7.5 Buffer Solutions 331 7.6 The Temperature Dependence of Chemical Equilibrium The Principle of Le Châtelier 335 7.7 Chemical Equilibrium and Biological Systems 343 Chapter The Thermodynamics of Electrochemical Systems 351 8.1 The Chemical Potential and the Electric Potential 352 8.2 Electrochemical Cells 354 8.3 Half-Cell Potentials and Cell Potentials 361 8.4 The Determination of Activities and Activity Coefficients of Electrolytes 371 8.5 Thermodynamic Information from Electrochemistry 374 www.TheSolutionManual.com 3.2 1372 N See Newton Nanomaterials, 1205–1206 Nanotubes, 1205–1206 Naphthalene, in ideal solution, 246–247, 247f NDDO See Neglect of diatomic differential overlap Negative deviation, from Raoult’s law, 248 Neglect of diatomic differential overlap (NDDO), for computation chemistry, 906 Nernst equation, 378 activation overpotential and, 605 for cells without liquid junction, 359–360 for concentration cells, 367 for electrochemical cells, 358 for pH measurement, 372–373 Nernst, Walther Hermann, 139, 358 Nernst’s distribution law, 251 Nernst’s heat theorem, 139 Net ionic equation, for electrolyte reactions, 318 Net rate of chemical reactions, 486, 507 in fast reaction study, 518 Net work, 156 of surface area, 224 Neutral atoms diagonal rule for, 810, 810f exceptions to, 811, 811t orbital energies in, 808–809, 809f Neutrino, discovery of, 57 Newton (N), 8, 622 Newton, Sir Isaac, 8, 452, 620 Newton’s law of viscous flow, 444f, 452–453 Newton’s laws of motion, 622–623 in classical mechanics, 622–623, 1267–1268 first law, 387, 1267 gas kinetic theory and, 387–389, 388f second law, 387–388, 388f, 1267 third law, 389, 1267 dilute gas pressure, 412 Nitrogen electronic spectra of, 974, 974f photoelectron spectrum of, 992 NMR See Nuclear magnetic resonance Nodal surfaces, 743 of hydrogen molecule ion, 826 for hydrogen-like orbital, 746, 746t of orbital regions, 747, 748f Node, of classical waves, 629 NOESY See Nuclear Overhauser Effect Spectroscopy Nonadiabatic processes, entropy changes for, 120–121, 120f Noncompetitive inhibitor, 580 Nondegenerate, energy level, 670 Nonelectrolyte solutes, in dilute solution, 237 Nonelectronic states, of diatomic molecules, 919–929 rigid rotor, 921–922, 921f rotation and vibration, 922–929, 922f, 926f Nonequilibrium electrochemical cell, 565 Nonequilibrium electrochemistry, 595–608 electrochemical cells with finite currents, 596–599, 598f electrolytic cells near equilibrium, 596, 596f overpotential, 600–608 rates of electrode processes, 599–600 Nonequilibrium states, 442–443 driving forces and linear laws of, 445 equation of continuity, 447–448, 447f Fick’s law of diffusion, 446–447 Fick’s second law of diffusion, 449–452, 450–451f Fourier’s law of heat conduction, 445–446 Poiseuille’s equation, 453–457, 454f rates of, variables for, 444–445, 444f Stoke’s law, 457–458 Nonequilibrium steady states, entropy changes of, 128–129, 129f Nonequilibrium thermodynamics See Irreversible thermodynamics Nonideal gas, activity and activity coefficient of, 260 equation of state for, 424 work on, 44–45 Nonideal solutions enthalpy of, 277–279 entropy of, 277–278 Gibbs energy of, 276–277 partial molar quantities in, 275–276 phase diagrams of, 282–290 liquid-vapor, 282–284f, 282–285 solid-liquid, 285–288, 285–288f solid-liquid with compounds, 288–289f, 288–290 three-component, 290, 290f solute thermodynamic properties with, 279–280 Non-Newtonian fluids, 453 Nonpolar covalent bonds, 881 Nonpolarized electrode See Ideal depolarized electrode Nonsimple systems, equilibrium criteria for, 157 Nonspecifically adsorbed ions, 597 Nonvolatile solutes, activities of, 267–274 Debye–Hückel theory, 270–274 Gibbs–Duhem integration, 267–270 Normal coordinates, 938 Normal melting temperature, 28 Normal modes, of vibration, 937–939, 939f Normalization of hydrogen atom orbitals, 749–751 of LCAOMOs, 837 of Φ Functions, 731–732 of probability distributions, 385, 394 of wave functions, 697–698 Normalization integral, 1290 Normalized, spherical harmonic functions, 733, 734t www.TheSolutionManual.com Multicomponent systems (cont.) phase equilibrium of phase diagrams for, 237 surfaces in, 230–234 Multielectron atoms electron wave function for, 763 electron-electron repulsion in, 763, 789 energy levels of, 789 helium-like atom, 764–766, 764f angular momentum, 774–780 excited states, 772–773 ground state, 768–771 spectra of, 960 zero-order approximation for, 763–786 indistinguishability of electrons, 766–768 lithium atom, 781–783 more than three electrons, 784–785 Pauli exclusion principle, 766–768 summary for, 786 Multiple bonds double bonds, 878–879 fifth-order bonds, 880 fourth-order bonds, 880 triple bonds, 879–880 valence-bond descriptions of, 883 Multiple integrals, 1239–1240f, 1239–1241 Multiplet splitting See Spin-spin splitting Multiplication table, for water, 899, 899t Index nth-order reactions, 493–494 half-life, 496 integrated rate laws, 495, 495f method of initial rates, 497, 497f rate law for, 493 n-type semiconductor, 1174 Nuclear g factor, 1008 Nuclear magnetic dipole, 1008–1010 Nuclear magnetic resonance (NMR), 1014–1022, 1032 diamagnetic shielding, 1015–1017, 1016f, 1020 impurity and solvent effects, 1021, 1021f magnetic dipole transition and, 951 NMR with other nuclei, 1022 saturation of a signal, 1021–1022 spectrum, 1016, 1016f spin-spin coupling and spin-spin splitting, 1017–1020, 1019–1020f two-dimensional, 1028–1032, 1028–1033f Nuclear magneton, 1008–1009 Nuclear motion, Hamiltonian for, 919 Nuclear Overhauser Effect Spectroscopy (NOESY), 1028, 1032 Nucleic acids, 1197 Nucleons, 930 Null matrix, 1251 Number density, in dilute gas, 413 Number fraction, of polymer molecules, 592 Number-average molar mass, of polymers, 592–593 Number-average molecular mass, 1200 Numerical mathematics, Oblate symmetric top, 934 Octet rule, 877 Odd function, 661 in Fick’s second law of diffusion, 451 Odd parity, 932 Ohm, Georg Simon, 475 Ohm’s law, 475, 482 electrical resistance, 1180–1181 overpotential and, 606 One mole of reaction, 87 One-body forces, 419–420 One-component ideal gas, partial molar quantities in, 186–187 One-component system intensive variables of, 205 partial molar quantities in, 185–186 phase equilibrium in, 205–213 Clapeyron equation, 208–210 Clausius–Clapeyron equation, 210–212 vapor pressure and total pressure, 212–213 surfaces in, 222–229 energy attributed to, 222–224 Laplace equation, 227–229 surface tension, 224–227, 225–226f One-phase fluid system, 3, 13 Onnes, Heike Kamerlingh, 1183 Operator See Mathematical operator Operator algebra, 685–687 Oppenheimer, J Robert, 824 Opsin, in rhodopsin, 983, 984f Optical activity, group theory and, 903 Optical density See Absorbance Optical rotatory dispersion, 993–996, 994–995f Orbital angular momentum of electron, 725, 742, 755 of helium, 775 intrinsic angular momentum vs., 756 Orbital energies in benzene, 888, 888f in neutral atoms, 808–809, 809f Orbital regions in benzene, 889, 889f of beryllium hydride, 869, 869f of hybridized orbitals, 854–855, 855–856f for LCAOMOs, 834–835, 835f, 844, 844f Orbital wave function, 765 electronic transitions and, 978 self-consistent field method for, 789, 798–799 Orbitals, 725 of helium-like atom, 779–780, 779t of hydrogen molecule ion, 825–826, 826f of hydrogen-like atom, 741–748 normalization, 749–751 qualitative properties of, 743–747, 744f, 746–747f radial distribution function, 752, 752f of lithium, 782–783 regions of, 747–748, 748f Slater determinant, 782–783 Order-disorder transition, 217 Oregonator, BZ reaction and, 589 Original Gibbs–Duhem relation, 190, 196 Orthogonal matrix, 1251 Orthogonality, of functions, 693, 1253–1254 Oscillatory chemical reactions, 565, 587–589, 615 Osmometer, 297, 297f Osmotic coefficient, of solvent, 269 Osmotic pressure, 297–298, 297f Outer Helmholtz plane, of ions, 597 Overlap integral, 837, 1290 Overpotential, 600–608 activation, 602–604 concentration, 600–602, 600f, 602f Oxidation half-reaction of electrochemical cells, 355, 355f of hydrogen electrode, 361, 362f Oxidation potentials, reduction potentials vs., 364 Oxygen molecules, probability distribution molecular speeds, 408, 408f velocity component, 399, 399f p subshell, 742 Pair potential energy function, 420 Pairwise intermolecular potential energy, 1147 Parallel band, of vibrational spectra, 977 Parameter α, molecular partition function and, 1055–1056 Parameter β, molecular partition function and, 1056 Pariser–Pople–Parr method, of computational chemistry, 905–906 Partial derivatives, 14–15, 623 second, 16 variables related to, 17–19 Partial differential equation, 449, 631 Partial fractions for autocatalysis, 586 for rate law integration, 501 Partial molar enthalpy of ideal gases, 186 method of intercepts for, 193 in nonideal solutions, 276 Partial molar entropy of ideal gases, 186 method of intercepts for, 193 in nonideal solutions, 275–276 Partial molar Gibbs energy chemical potential and, 184 method of intercepts for, 193 Partial molar quantities experimental determination of, 191–194 method of intercepts, 192–194, 193–194f in multicomponent systems, 184 www.TheSolutionManual.com 1373 Index 1374 Perpetual motion machine of the second kind, 106 Perrin, Jean Baptiste, 469 Perturbation method, 789, 799, 819, 1283–1287 degenerate case, 803–805, 1285–1287 helium ground state application of, 800–802, 800f nondegenerate case, 1283–1285 variation method vs., 802 Perturbation theory degenerate, 803–805, 1285–1287 time-dependent, for electric dipole transitions, 951–953 Petit, Alexis Thérèse, 77 Pfaffian differential equations, 1245, 1245f Pfaffian form, 1238 pH measurement cell for, 372–373, 373f glass electrode for, 373–374, 373f pH meters, 374 Phase(s), 27, 200, 203 coexistence of, 3, 27–35, 27f counting of, 203–204 equilibrium of, 199–235 fundamental fact of, 200–202 Gibbs energy and phase transitions, 215–221 Gibbs phase rule, 202–204 multicomponent system surfaces, 230–234 one-component system surfaces, 222–229 in one-component systems, 205–213 transitions of, 27, 27f Phase diagram(s), 27, 27f Clapeyron equation for, 208–210 Clausius and Clausius–Clapeyron equation and, 199 with DSC, 287–288 Gibbs phase rule for, 199 of helium, 207, 207f of ideal solutions pressure-composition, 243–245, 244f temperature-composition, 245–246, 245–246f two-component, 243 for nonideal mixtures, 282–290 liquid-vapor, 282–284f, 282–285 solid-liquid, 285–288, 285–288f solid-liquid with compounds, 288–289f, 288–290 three-component, 290, 290f for phase equilibrium, 237 of two-component ideal solutions, 243 of water, 206, 206f Phase equilibrium, 199–235, 234 fundamental fact of, 200–202 Gibbs phase rule, 202–204 of multicomponent systems phase diagrams, 237 surfaces, 230–234 one-component system surfaces, 222–229 energy attributed to, 222–224 Laplace equation, 227–229 surface tension, 224–227, 225–226f in one-component systems, 205–213 Clapeyron equation, 208–210 Clausius–Clapeyron equation, 210–212 vapor pressure and total pressure, 212–213 Phase transitions, 27, 27f absolute entropy and, 141 classification of, 216–218 Clausius–Clapeyron equation and, 211–212 entropy changes of irreversible, 127–128 reversible, 123–124 first-order, 216 Gibbs energy and, 215–221 liquid-vapor transition critical point, 218–219 Maxwell equal-area construction, 219–220, 219f temperature dependence, 220–221 heat transfer during, 54 second-order, 217 Phenomenological law, Fourier’s law of heat conduction, 445–446 Φ Functions in Born–Oppenheimer approximation, 825 in relative Schrödinger equation, 731–732 zero value in, 745–746, 746f Phlogiston, 55 Phonons, 1165 Phosphoenolpyruvic acid (PEP), hydrolysis of, 344–345 Phosphorescence, 978–980, 980f Photoacoustic spectroscopy, 993 Photochemical chain reactions, 558, 981 initiation of, 523 Photochemical reaction, 981 Photochemistry, 949, 951, 981–983, 982f www.TheSolutionManual.com Partial molar quantities (cont.) in nonideal solutions, 275–276 in one-component ideal gas, 186–187 in one-component system, 185–186 Partial molar volume, of nonideal solutions, 276 Partial pressure of dilute gas pressure, 415 of one-component ideal gas, 187 Partial vapor pressure of benzene and toluene, 240, 240f of hydrochloric acid, 329–330, 330f Raoult’s law, 238 Particle in a box, 663–673 free particle in one dimension, 670–672 free particle in three dimensions, 672–673 probability density, 705 one-dimensional, 663–666, 664f classical vs quantum mechanics, 664 equations for, 664–665 probability density, 701–702, 701–702f solution for, 665–666, 666f uncertainty product, 711–712, 712t Schrödinger equation and de Broglie waves, 664 specification of state of, 668–669 in three-dimensions, 669–670, 669f, 1276–1278 time-dependent wave function for, 668 Pascal, 8, 622 Pascal, Blaise, 8, 622 Path-independent, line integral, 48 Pauli exclusion principle, 763, 766–768, 785, 1011 Pauli, Wolfgang, 768 Pauling, Linus, 862 PEP See Phosphoenolpyruvic acid Period, of oscillations, 626 Periodic boundary conditions, 1175 Periodic function, 1253 Periodic motion, of harmonic oscillator, 626, 626f Periodic table of elements, 789, 813–818, 815–816f, 820 Peritectic point, 289–290 Permeability, 1002, 1275 Permittivity, 270, 1275 Perpendicular band, of vibrational spectra, 977, 977f Perpetual motion machine of the first kind, 57 Index hydrogen reaction with other halogens, 559–560 laws of, 558–559 Photoelectric effect, Einstein’s theory of, 645 Photoelectron spectroscopy, 991–993, 992f Photons, 645 Bohr frequency rule, 949–950 einsteins of, 559, 951, 981 electric dipole transitions and, 953 in photochemistry, 558 Physical adsorption, Langmuir isotherm for, 569 pKa , 332 Planck, Max Karl Ernst Ludwig, 139, 643 Planck-Einstein relation, 645, 649 photon energy and, 950 Schrödinger equation vs., 661 Planck’s constant, for blackbody radiation, 643 Planck’s theory of blackbody radiation, 641–645, 643f Rayleigh–Jeans theory, 641–642, 642f Stefan–Boltzmann law, 641 Plane polarized electromagnetic wave, 1275 Plane wave, 630–631 Point groups molecular assignment to, 901, 902f of symmetry operators, 899–902, 901t, 902f Point symmetry operators, 828 Point-mass particle(s) of dilute gas model system, 386, 436 state of, 387 Poiseuille, Jean Leonard Marie, 455 Poiseuille’s equation, 453–457, 454f derivation of, 453–455, 454f Reynolds number, 456–457 Poisson equation, of electrostatics, 597 Polanyi, Michael, 1109 Polar covalent bond, 856 Polarizability, of molecule, 986–987 Polarization, of electrochemical cell, 595 Polarized light, circularly, 994–995f Polyatomic molecules electronic structure of, 867–912 ammonia molecule, 875 applications of symmetry to molecular orbitals, 894–895, 895t beryllium hydride and sp hybrid orbitals, 867–912 boron hydride and sp2 hybrid orbitals, 871–872, 872f computational chemistry, 904–911 delocalized bonding, 885–890 free-electron molecular orbital method, 892–893 groups of symmetry operators, 896–903 methane molecule, 873–875, 874f multiple bonds, 878–880 summary for, 912 water molecule, 875–876f, 875–878 rotation of, 933–937, 934f, 945–947 spectra of, 975–978 electronic, 978 microwave, 975 vibrational, 976–977, 977f valence-bond description of, 881–884 vibrations of, 937–939, 937f, 939f, 945–947 Polycrystalline, 1154 Polyesterification reactions, 592, 592f Polymerization kinetics, 590–594, 592f, 594f degree of, 591–592 mass fractions, 593, 594f number-average molar mass, 592–593 Polymers, 589 formation of, 565, 590, 1194–1197 reaction kinetics of, 565, 589–594, 592f, 594f in solution, 1198–1200 Polymorphism, 200 of water, 206–207 Polyprotic acid, 324 ionization of water with, 324–325 Position vector, 387, 390, 390f in classical mechanics, 621 in gas kinetic theory, 387 sum of, 390, 390f Positive deviation, from Raoult’s law, 248, 249f Potential energy of electron, 648 of free particle, 670 of gas kinetic theory, 389, 392–393, 418–421 intermolecular forces, 420–421, 420–421f one-body forces, 419–420 of harmonic oscillator, 627, 627f of hydrogen atom, 726 of internuclear repulsion, 824 of liquids, 434 of magnetic dipoles, 1004 pairwise intermolecular, 1147 particle force and, 623 in quantum mechanics, 688 surfaces, 1107–1110, 1108f, 1109f symmetry operator and, 897 vibrational, 926, 926f Potential well, for particle in a box, 663 Potentiometer, for electrochemical cells, 356 Predictable case Heisenburg uncertainty principle, 713–714 statistical case distinguishing from, 705–707 wave functions, 698–699 Preexponential factor, in Arrhenius relation, 533 Pressure of dilute gas, 1084–1085 at equilibrium, 36 of hard-sphere gas, 424 molar volume and temperature vs., 29, 30f, 32–33, 33f partial of dilute gas pressure, 415 of one-component ideal gas, 187 at phase transitions, 27–28, 27f rate constant dependence on, 487 reduced, 33 vapor pressure and total, 212–213 Pressure virial coefficients, 22 Pressure virial equation of state, 22 Pressure-composition phase diagrams of ideal solutions, 243–245, 244f of nonideal mixtures, liquid-vapor, 282–283f Pressure-jump method, for fast reaction experimental, 516–520, 518f Primitive lattice, 1155t, 1156, 1156f Principle axes, 934 Principle moments, 934 Principle of detailed balance, 583–585, 583f Principle of Le Châtelier, 303, 337–340, 348 buffer solution, 340 enzyme molecules and, 345 gas-phase reaction, 338 reactant or product addition with, 339–340 reaction coupling and, 345 statement of, 337–338 Principle of microscopic reversibility, 584 Principle of superposition, 669 for wave equation, 635–636, 636f Principle quantum number, 739 Probability density, 394, 700–705, 701–704f of antisymmetrized wave function, 768 of electron, 758 www.TheSolutionManual.com 1375 Index 1376 Prolate symmetric top, 934 Protease, 575–576 Proteins, 1197 Proust, Joseph, Pseudo first-order reaction, 504 Pseudo second-order reaction, 504 P-type semiconductor, 1174 Pyruvate kinase, 345–346 q See Heat Quantum chemistry software, for computational chemistry, 909–911 Quantum harmonic oscillator, 674–679 Quantum mechanical operator, for mechanical variables, 689 Quantum mechanics, 619–620 classical mechanics vs., 620 correspondence principle of, 702 particle encounters, 767 De Broglie waves, 654–656 mathematics used in, 1275–1281 operators of, 688–695 eigenfunction of coordinate, 695 hermitian, 692–694 linear, 692 particle in a box, 663–673 free particle in one dimension, 670–672 free particle in three dimensions, 672–673 one-dimensional, 663–666, 664f Schrödinger equation and de Broglie waves, 666–667 specification of state of, 668–669 in three-dimensions, 669–670, 669f, 1276–1278 time-dependent wave function, 668 postulates of, 683–722 fifth, 683, 717–720, 722 first, 683–684, 721 fourth, 683, 696–710, 721–722 second, 683–684, 721 summary for, 721–722 third, 683–695, 721 uncertainty principle of Heisenburg, 711–716 quantum harmonic oscillator, 674–679 Schrödinger equation, 657–663 eigenvalue equations, 662–663 in three dimensions, 661–662 time-dependent, 659–661 time-independent, 658–659 of spectroscopic transitions, 951–955, 954–955f summary for, 680–681 Quantum numbers, 643, 648, 774 of hydrogen atoms, 739 translational, 916 types of, 739 Quantum statistical mechanics molecular partition function calculation of, 1064–1075 probability distribution, 1055–1063 postulates of, 1042–1043 probability distribution for dilute gas, 1047–1054 molecular partition function, 1055–1063 for molecular states, 1039–1077 of simple model system, 1040–1046 summary for, 1077 Quantum theory, 619, 640–650 atomic nature of matter, 640–641 Bohr’s theory of hydrogen atom, 646–650 Einstein’s theory of photoelectric effect, 645 introduction to, 620 Planck’s theory of blackbody radiation, 641–645 summary for, 652 wave-particle duality of light, 646 Quantum yield, of photochemical reaction, 559, 981 Quasi-equilibrium approximation See Equilibrium approximation Quasi-equilibrium process See Reversible process Quasi-static process See Reversible process Quasi-steady-state approximation See Steady-state approximation Quenched, reaction, 982–983 R See Resistance Radial distribution function, 751 for dense gases and liquids, 1148 for hydrogen orbitals, 752, 752f Radial factor in hydrogen atom wave function, 736–740, 744f, 760 in relative Schrödinger equation, 730 Radiation theory, for unimolecular reactions, 545 Radiationless transition, 780 Radioactive nuclides, decay of, 490–491 Raman active, 987–988 www.TheSolutionManual.com Probability density (cont.) for excited state, 773 of harmonic oscillator, 703–704, 703–704f for helium, 794–795 orbital region, 747 of particle in one-dimensional box, 701–702, 701–702f of particle in three dimensions, 705 symmetric, 767 for two particles, 769–771 Probability distribution, 385, 394 average, 1043–1044, 1045t Boltzmann, 942 canonical, 1122–1124 different expression for, 1116–1117 for dilute gas, 1047–1054 average distribution, 1048 most probable distribution, 1048–1054 of gas kinetic theory derivation of, 395–397, 395f identification of parameter b, 398–404, 399f, 404f macrostates and microstates, 384–386 velocity, 394–404 molecular partition function and, 1055–1063 definition of, 1055 of a dilute monatomic gas, 1057–1063, 1059f parameter α and, 1055–1056 parameter β and, 1056 probability distribution and, 1055–1063 of molecular speeds, 383, 405–410, 437 equation for, 407 mean-square speed, 409 results of, 408, 408f spherical polar coordinates for, 405–406, 406f for molecular states, 1043–1046 normalized, 399 standard deviation of, 399–400, 399f three-dimensional, 403–404, 404f Probability equations, 700–701 Probability of reaction, in collision theory, 537 Product concentration, chemical reactions dependence on, 485 of mathematical group, 898 of two operators, 686 Product partition function, corrections to, 1070–1072 Products of inertia, 933–934 Index Raman, Chandrasekhara Venkata, 985 Raman shift, 986 Raman spectrophotometer, 985, 985f Raman spectroscopy, 949, 985–991, 985f, 997 process of, 985–987 rotational, 987 vibrational, 987–991, 989–990f Random walk, Fick’s second law of diffusion and, 451–452 Randomness, disorder vs., 137–138 Raoult, Francois Marie, 238 Raoult’s law, 237–240, 299 deviation from, 248–249, 249f dilute solutions and, 256–257 ideal solution and, 238–239 nearly ideal solution and, 240, 240f osmotic pressure and, 297 vapor pressure lowering and, 296 Rate constant apparent, 504 of chemical reactions, 487 reverse, 508 temperature dependence of, 487, 533–539 Rate law(s), 487 of autocatalysis, 586–587 of chemical reactions, 485, 521, 523–562 first-order, 489–490 nth-order, 493 second-order, 491 zero-order, 494 with definite orders, 487, 1106 determination of, 485, 521 integration of, 499–502 of photochemistry, 559 reaction mechanisms and, 540–553 deduction of, 549–553 determination of, 540–541 Lindemann mechanism, 545–547 more than two step mechanisms, 547–549, 1245–1246 rate-limiting step approximation, 541–543 steady-state approximation, 543–545 temperature dependence, 553 for reaction order determination, 494–495, 495f Rate of adsorption, in heterogenously catalyzed reactions, 572 Rate of change of Gibbs energy per mole of reaction, 305 Rate of collisions, in collision theory, 535–536, 535f Rate of desorption, in heterogenously catalyzed reactions, 572 Rate of flow, Poiseuille’s equation for, 453–455 Rate of reaction, 486 See also Chemical reaction rates Rate of shear in liquids, 471 viscous flow, 445 Newton’s law of, 453 Rate-limiting step approximation, for rate law approximation, 523 with reaction mechanisms, 541–543 Rayleigh, Baron See Strutt, John William Rayleigh–Jeans theory, of blackbody radiation, 641–642, 642f Reactant concentration chemical reactions dependence on, 485 in reaction mechanism, 511–512, 511f forward reactions with more than one, 499–505 method of initial rates, 502–504 method of isolation, 504–505 rate law integration, 499–502 forward reactions with one, 488–497 first-order, 489–491 nth-order, 493–494 reaction order determination, 494–497, 495f, 497f second-order, 491–493 zero-order, 494 Reaction coordinate in activation overpotential, 602–603, 603f in potential energy surfaces, 1107–1108, 1108f Reaction cross section, in collision theory, 525 Reaction diameter in bimolecular processes, 529 for hydrogen and hydroxide ions, 531 Reaction kinetics, of polymer formation, 565, 589–594, 592f, 594f Reaction mechanisms autocatalysis, 585–587 catalysis, 566–580 enzyme, 575–580, 576f, 578–579f heterogenous, 566–572 homogenous, 573–575 chain reactions, 556–561 branching-chain, 560–561 hydrogen reaction with halogens, 559–560 photochemical, 558 for chemical reactions, 510–512, 511f, 521, 561 concentration in, 511–512, 511f equations for, 510–511 competing of, 583–585, 583f elementary processes in gases and, 523–527 collision theory of bimolecular processes, 523–526 collision theory of termolecular gas-phase reactions, 527 collision theory of unimolecular processes, 527 general statement for, 527 elementary processes in liquids and, 527–532 bimolecular, 528–532 termolecular and unimolecular, 532 experimental molecular study of, 608–614 detection of reactive intermediates, 609–610 molecular beam reactions, 610–611f, 610–614, 613f nonequilibrium electrochemistry, 595–608 electrochemical cells with finite currents, 596–599 overpotential, 600–608 rates of electrode processes, 599–600 polymerization kinetics, 589–594, 592f, 594f principle of detailed balance, 583–585, 583f rate laws and, 540–553 deduction of, 549–553 determination of, 540–541 Lindemann mechanism, 545–547 more than two step mechanisms, 547–549, 1245–1246 rate-limiting step approximation, 541–543 steady-state approximation, 543–545 temperature dependence, 553 summary of, 561–562, 615 Reactive intermediate in chemical reaction, 510, 524 detection of, 609–610, 615 in steady-state approximations, 543–544 Reciprocal identity, 1236 Recursion relation, 1195–1196 www.TheSolutionManual.com 1377 Index 1378 Resonance, valence-bond method and, 885–886 Response time, of measuring instrument, 412 Retinal, in rhodopsin, 983, 984f Retinol, 983, 984f Reverse rate of chemical reactions, 486 constant, 508 Reverse reactions in competing reactions, 514 inclusion in rate determination, 507–509 rate of, 486 Reversible adiabat, 68, 68f Reversible electrode, 600 Reversible process(es), 16, 42 adiabatic entropy changes, 117–119, 118f for heat and energy change calculation, 67–71, 68f cyclic, line integral, 114–115, 115f enthalpy changes for, 81–84, 85f entropy changes of closed system of isothermal, 121–122 phase changes, 123–124 temperature changes, 124–126 of fluid closed system, 41–43, 43f heat in, ideal gas, 62–63 line integral of, 114–115 work in, 43–45 Reversible voltage, of electrochemical cells, 356 Reynolds number, 456 Poiseuille’s equation, 456–457 Stoke’s law, 458 Rhodopsin, in rod cells, 983, 984f Ribozymes, 575–576 Richards, Theodor William, 139 Right-hand rule, 1002 Right-polarized radiation, 994, 995f Rigid rotor, 921–922, 921f Rigid rotor-harmonic oscillator approximation, 923–924 Rod cells, in vision, 983 Rohrer, Heinrich, 1205 Roothaan, Clemons C J., 798 Root-mean-square value, 451 Rotation of diatomic molecules, 922–929, 922f, 926f population of, 943–944 operators, 829 of polyatomic molecules, 933–937, 934f, 945–947 Rotational energy of dilute gas, statistical thermodynamics, 1091 in internal energy, 59 Rotational partition functions, calculation of diatomic gases, 1066–1069, 1067f polyatomic gases, 1073 Rotational transitions spectroscopy of, 949, 961–964, 996–997 Raman, 987 vibrational transitions and, 965–966, 966f Rotational wave function, 924, 932 Rubber elasticity of, 1200–1204, 1206 molecular theory of, 1203–1204 thermodynamics of, 1201–1202 Rule of mutual exclusion, 988, 997 Rumford, Count, 55 Russell–Saunders approximation for angular momentum approximation, 763, 774–775 for multielectron atom energy levels, 789 Russell–Saunders term symbols, 775 Rutherford, Ernest, 641 Rydberg, Johannes Robert, 647 Rydberg’s constant, for spectral lines, 646–647, 647f Rydberg’s formula, 649 S See Entropy s See Second s subshell, 742 Sackur–Tetrode equation, 1096 Salt bridge, 366–367, 366f Saturated, ideal solution, 246–247 Scalar function, gradient of, 442 Scalar product, 40, 40f, 390, 390f, 1241–1243 Scalar quantity, 390, 390f, 621 Scanning tunneling microscope (STM), 1205 Scattering states, of energy eigenvalues, 738 Schoenflies symbols, of point groups, 899–901, 901t Schottky effect, 1101 Schrödinger equation, 653, 657–663, 680 Born–Oppenheimer approximation for, 823–824, 824f de Broglie waves and, 657, 666–667 electronic, 825 Hamiltonian operator, 662 www.TheSolutionManual.com Redlich–Kwong equation of state, 23, 23t for fluid surface, 30f, 31 Reduced coordinate distribution function, 1148 Reduced mass in harmonic oscillator, 627 for hydrogen atom, 727 in multicomponent collisions, 431 Reduced pressure, 33–35, 34f Reduced temperature, 34–35, 34f Reduced volume, 33 Reducible representation, 1294–1295 Reduction half-reaction of calomel electrode, 361, 362f of electrochemical cells, 355, 355f Reduction potentials, oxidation potentials vs., 364 Reference electrode, for electrode study, 602, 602f Reflection operators, 828 Refractive index, 994 Relative coordinates, 726, 1272 Relative Hamiltonian, of hydrogen atom, 727 Relative momentum, for hydrogen atom, 727 Relative motion, energy eigenvalue, 725 Relative Schrödinger equation angular momentum and, 729–736 the Φ functions, 731–732 first separation of variables, 730 the Θ functions, 733 second separation of variables, 730–731 values, 733–736, 735f spherical polar coordinates, 725 Relative speed, in molecular collisions, 427 Relative viscosity, 1198–1199 Relaxation effect, in electrolyte solutions, 479 Relaxation techniques, for fast reaction study, 515–520 flash photolysis method, 516, 517f shock-tube method, 515–516 temperature-jump and pressure-jump methods, 516–520, 518f Relaxation time in chemical reactions, 508 in fast reaction study, 518 in first-order reaction, 490 Representative elements, 814, 815f Resistance (R), in Ohm’s law, 475 Resistivity, 475 Resolution, in spectroscopy, 958 Resonance energy, 886 Index for hydrogen atom, 725–726 relative angular momentum values, 733–736, 735f the Φ functions, 731–732 first separation of variables, 730 the Θ functions, 733 second separation of variables, 730–731 spherical polar coordinates, 725 in three dimensions, 661–662 time-dependent, 659–661, 680 for particle in a box, 668 second postulate, 684 wave functions, 708–709 time-independent, 680 for atoms with more than three electrons, 785 free particle in one dimension, 671–672 free particle in three dimensions, 672–673, 728–729, 728f harmonic oscillator, 674–679, 1278–1280 for hydrogen atom, 727–728, 759 nonrigorous derivation of, 658–659 particle in a one-dimensional box, 663–666, 664f particle in a three-dimensional box, 669–670, 669f, 1276–1278 wave functions, 707 zero-order helium, 765–766 lithium, 781–782 Schrödinger, Erwin, 657 Second (s), 8, 621 Second harmonic See First overtone Second law of thermodynamics See Thermodynamics, second law of Second moment, 1195 Second order equation, 624 Second overtone, 966 Second partial derivative, 16 Second postulate, of quantum mechanics, 683–684, 721 Second-order phase transitions, 217, 218f Second-order reactions, 491–493 collision theory and, 536 first-order vs., 493, 493f gaseous bimolecular elementary process, 526 half-life of, 492, 496 method of initial rates, 497, 497f pseudo, 504 rate law for, 491, 495, 495f Second-order spectrum, NMR, 1018 Secular equation, 804, 887, 1287, 1290 Sedimentation coefficient, 472 Sedimentation, of liquids, 472–473 Sedimentation speed, 472 Selection rules for electric dipole transitions, 953 for electronic spectra, 972–973 for hydrogen atom, 959 for multielectron atoms, 960, 962 for Raman spectroscopy, 990–991 for vibrational spectra, 964, 976 Self-consistent field, 798 Self-consistent field method for ground state of helium atom, 797–798 Hartree–Fock–Roothaan method, 798–799 for hydrogen molecule, 840, 840t for lithium atom, 807 for orbital wave function, 789, 819 overview of, 796 Self-diffusion, in hard-sphere gas, 460–464, 461f analysis of, 461, 461f diffusion coefficient, 463 mean free path, 462–463 mean speed, 462 Semiconductors, crystalline solids as, 1172–1174, 1173–1174f Semiempirical methods of computation chemistry, 904 Hückel method, 887 Separation constant, of flexible string, 632 Separation of variables, 631 Sequential mechanism, of chemical reactions, 524 Shearing flow, in liquids, 471 Shell, of electrons, 742 Shielding diamagnetism, 1015 of variation function for helium atom, 764f, 793–795 Shock-tube method for fast reaction study, 515–516 reactive intermediate detection with, 609 SI units See International System of Units Siemens, conductivity, 476 Significant digits, 10 Silver, 286–287, 286–287f Similarity transformation, 1295 Simon, Franz Eugen, 139 Simon, Pierre, 227, 449 Simple lattice, 1155t, 1156, 1156f Simple system, 42 chemical potential of, 184 equilibrium state of, 155–157 Gibbs energy, 155–156, 155f Helmholtz energy, 156 fundamental relations for, 158–166, 195 Gibbs energy of, 182–183 internal energy of, 183–184 maximum work by, 156–157 Maxwell relations of, 159–166 two-phase, 200–201, 200f Single covalent bond, 839 Singular, matrix, 1250 Singularity, in phase transition, 217, 217f Sir Francis See Simon, Franz Eugen Slater determinant, 782 for atoms with more than three electrons, 785 for beryllium hydride, 869–870 for lithium atom, 782–783 Slater, John C., 782, 798 Slater-type orbitals (STOs), 798 computation of, 907–908 of water, 894–895, 895t Smalley, Richard E., 1205 Soave equation of state, 23, 23t Sodium chloride, unit cells of, 1154 Solid solutions, ideal solutions as, 243 Solid-liquid phase diagrams, 285–288, 285–288f with compounds, 288–289f, 288–290 Solids activities of, 259 chemical reactions, involving gases, 310–313 electrical resistance in, 1179–1184 superconductivity, 1183–1184 equilibrium constant of, 312–313 Gibbs energy calculations for, 178, 195 heat capacity of, 76–77 standard state of, 87 structure of, 1154–1161, 1155f, 1206 crystal systems and Bravais lattices, 1154–1158, 1155t, 1156–1157f crystal vibrations, 1162–1170 crystalline solid electronic structures, 1171–1179 X-ray diffraction and Miller indices, 1158–1161, 1159f, 1161f www.TheSolutionManual.com 1379 Index 1380 Solvent standard state, 263 sp hybrid orbitals, 868–871, 912 sp2 hybrid orbitals, 871–872, 872f, 912 sp3 hybrid orbitals, 912 of ammonia, 875 of methane, 873–875, 874f of water, 875–877, 875f Space orbital, 757 Specific heat capacity (CP ), 52 heat capacity and, 170–172 Specific rotation, 995–996 Specific viscosity, 1198–1199 Specifically adsorbed ions, 597 Specificity, of enzymes, 576 Spectra of atoms, 959–960, 959f hydrogen, 959–960, 959f multielectron, 960 for ESR, 1012–1013, 1013f of microwave radiation, 962, 963f of NMR, 1016, 1016f rotational, of diatomic molecules, 961–964 Spectral lines, 646 in absorption spectroscopy, 957–958, 957f in emission spectroscopy, 954 fundamental band of, 965–966, 966f intensity of, 964 Spectral radiant emittance, of blackbody, 641, 642f Spectrophotometer, 955–956, 955f Spectroscopic transitions, quantum mechanics of, 951–955, 954–955f Spectroscopy, 949 absorption, 951–958, 955–957f circular dichroism and optical rotatory dispersion, 993–996, 994–995f emission, 951–955, 954–955f ESR, 1010–1013, 1013f, 1032 Fourier transform infrared, 969–971, 969f NMR spectroscopy, 1024–1032 infrared, 969–971, 969f Raman spectroscopy vs., 988 microwave of polyatomic molecules, 975 Raman spectroscopy vs., 988 NMR, 1014–1022, 1032 photoacoustic, 993 photoelectron, 991–993 Raman, 949, 985–991, 985f, 997 process of, 985–987 rotational, 987 vibrational, 987–991, 989–990f reactive intermediate detection with, 609 resolution in, 958 of rotational transitions, 949, 961–964 summary for, 996–997 of vibrational transitions, 949 Specular collision, 411 Speed, 391 in classical mechanics, 621 of traveling waves, 637 Spheres, closest packing of, 1156–1158, 1157f Spherical harmonic functions, 733, 760 normalized, 733, 734t Spherical polar coordinates hydrogen atom orbitals in, 750, 750f infinitesimal volume element in, 1239–1241, 1240f for molecular speed distribution, 405–406, 406f for relative Schrödinger equation, 725, 729–730, 730f Spherical top, 934, 936 Spin of electrons, 725, 760 directions of, 757, 757f magnetic dipole, 1007–1008 of nucleus, 930–932 magnetic dipole, 1008–1010 Spin decoupling, 1021 Spin function, 757–758 Spin orbital, 757 diagonal rule for, 810, 810f exceptions to, 811, 811t Spin-echo technique, for NMR, 1026–1027f, 1026–1028 Spin-lattice relaxation time, 1025 Spin-orbit coupling, 758 Spin-spin coupling of NMR, 1017–1020, 1019–1020f patterns of, 1019 removal of, 1021 Spin-spin coupling constant, 1017 Spin-spin relaxation time, 1025, 1026f spin-echo technique for, 1026–1027f, 1026–1028 Spin-spin splitting impurity and solvent effects, 1021, 1021f of NMR, 1017–1020, 1019–1020f patterns of, 1019, 1019f Spontaneous emission, of radiation, 953 www.TheSolutionManual.com Solute(s), 249 activity of, 315 colligative properties of, 292–298 boiling point elevation, 295–296 freezing point depression, 292–295 osmotic pressure, 297–298, 297f vapor pressure lowering, 296 distribution between two solvents, 251–252 equilibrium constant of, 316, 348 nonvolatile, activities of, 267–274 stoichiometric molality of, 269 thermodynamic properties for, 279–280 Solute standard state, 263 Solution(s), 237 activities and activity coefficients in, 260–264 chemical equilibrium in, 315–326 electrolyte solutes, 318–320 water ionization, 320–326 colligative properties of, 292–298 boiling point elevation, 295–296 freezing point depression, 292–295 osmotic pressure, 297–298, 297f vapor pressure lowering, 296 composition of, 252–255 ideal, 237 of ions, electrical conductivity of, 441 nonelectrolyte solutes in dilute, Henry’s law, 237 nonideal phase diagrams of, 282–290 thermodynamic functions of, 275–280 polymers in, 1198–1200 thermodynamics of, 237–299 activity and activity coefficients, 258–266 colligative properties, 292–298 Henry’s law and dilute nonelectrolyte solutions, 248–257 ideal solutions, 238–247 nonideal solutions, 275–280 phase diagrams, 282–290 summary of, 299 Solvation effect, in electrolyte solutions, 479 Solvent(s), 249 activity of, 315 enthalpy change of fusion of, 293 equilibrium constant of, 316 NMR and, 1021, 1021f osmotic coefficient of, 269 Raoult’s law and, 256–257 solute distribution between two, 251–252 Index Spontaneous processes, in closed systems, 152–154, 195 Square integrable, function, 1256 Square matrix, 1249 Square well potential, for intermolecular forces, 421, 421f Standard deviation, 399, 705 of a function, 400 of probability distribution, 399–400, 399f variance of, 705 Standard normal distribution, 399, 706 Standard pressure, for Gibbs energy, 175 Standard state activity and, 259 chemical potential, 265 for entropy, 142–143 for Gibbs energy, 175 Henry’s law, 250 of liquid or solid, 87, 142, 178 molality, 253 solute, 263 solvent, 263 Standard state cell voltage, 358 Standard-state cell potential, 362–363f, 362–364 Standard-state enthalpy change of formation, for enthalpy calculation, 87–89 Standard-state Gibbs energy change, 306, 310 for electrolyte solutes, 318 Standard-state pressure, 1096 Standard-state reaction enthalpy change for, 87 Gibbs energy change for, 316, 343 Standing wave, 629, 630f, 635f traveling waves and, 638 Stark–Einstein law of photochemical equivalence, 558, 981 State(s) law of corresponding, 33–35 macro and micro, 133 molar enthalpy and, 87 in physical chemistry, 12–19, 13f of point-mass particles, 387 of a system, 13–14 in classical mechanics, 621 measurements and determination, 717–720 work and, 40–50 wave function and, 668 State functions, 13–14, 620 See also Wave function entropy, 114 internal energy, 58 State point, in state space, 42 State space of closed fluid system, 42 curve in, 42, 43f reversible adiabat, 68 State variables, 13–14 Stationary state, of wave function, 697 Statistical case Heisenburg uncertainty principle, 713–714 position measurements for, 699 predictable case distinguishing from, 705–707 wave functions, 698–699, 707–708 Statistical electron correlation, 795 Statistical entropy, 133–138, 148, 1082 interpretation of, 137–138 lattice gas and, 134, 134f microstate coordination calculation, 134–135 thermodynamic entropy and, 136–137, 1117–1118 third law of thermodynamics and, 144–145 Statistical mechanics See also Classical statistical mechanics; Equilibrium statistical mechanics heat in, 1117 postulates of, 1042–1043, 1122 work in, 1117 Statistical thermodynamics, 1081–1118 of dilute gas, 1082–1088 activated complex theory, 1106–1115 chemical equilibrium in, 1101–1105 chemical potential, 1087–1088, 1099–1100 electronic energy, 1090–1091 enthalpy, 1086 entropy, 1082–1084, 1095–1098 Gibbs energy, 1088 heat capacity, 1086, 1093–1095 Helmholtz energy, 1087, 1098–1099 internal energy, 1089 pressure, 1084–1085 rotational energy, 1091 translational energy, 1090 vibrational energy, 1091–1092 working equations for, 1089–1100 miscellaneous topics in, 1116–1118 heat and work interpretation, 1117 probability distribution alternate expression, 1116–1117 statistical entropy and thermodynamic entropy, 1117–1118 summary for, 1118 Steady state, nonequilibrium, entropy changes of, 128–129, 129f Steady-state approximation for oscillatory chemical reactions, 588 for rate law approximation, 523 with reaction mechanisms, 543–545 Steam distillation, 285 Steam engine, simple, 106–107, 107f Stefan–Boltzmann law, for blackbody radiation, 641 Stereochemical specificity, of enzymes, 576 Steric factor, in collision theory, 538 Stern layer See Compact layer Stimulated emission, of radiation, 953 Stirling’s approximation, 135, 1052–1053 STM See Scanning tunneling microscope Stoichiometric coefficients, in chemical equations, 87 Stoichiometric molality of hydrochloric acid, 329–330 of solute, 269 Stokes, George Gabriel, 457 Stoke’s law, 457–458 Stokes lines, 986, 988 Stopped-flow method, for fast reaction study, 515, 516f STOs See Slater-type orbitals Strong acids, 318–319 Strong electrolytes, 318 chemical equilibrium in, 328–331 Structure of liquids, 1184–1188, 1206 classical statistical mechanics approach, 1184–1185, 1185f computer simulations of, 1187–1188 solid-like models, 1186–1187 of solids, 1154–1161, 1155f, 1206 crystal systems and Bravais lattices, 1154–1158, 1155t, 1156–1157f crystal vibrations, 1162–1170 crystalline solid electronic structures, 1171–1179 X-ray diffraction and Miller indices, 1158–1161, 1159f, 1161f Strutt, John William, 642 Sublimation, 27–28, 27f www.TheSolutionManual.com 1381 Index 1382 potential energy and, 897 for water, 899, 899t Symmetry species, 1299 Symmetry-adapted basis functions, 1299 System(s), closed fluid, work done on, 41–43 in physical chemistry, 12–19, 13f simple, description of, 42 state of, 13–14 in classical mechanics, 621 measurements and determination, 717–720 work and, 40–50 thermodynamics of, 151–196 types of, 12–13 Tafel equation, for overpotential, 607 Taylor series, for freezing point depression, 293–294 Temperature Carnot cycle and, 111 cell voltage dependence on, 368–369 chemical equilibrium dependence on, 335–340 chemical reactions dependence on, 485, 524–525 nonelementary, 553 diffusion dependence on, 470–472 gas-phase reaction dependence on, 523 Gibbs energy dependence on, 178–180, 220–221 gradient, for heat conduction, 445 heat transfer during change of, 51–53 molar concentration, molality and, 254 pressure vs molar volume and, 29, 30f, 32–33, 33f rate constant dependence on, 487, 533–539 reduced, 34 thermodynamic, 110–113 viscosity dependence on, 470–472 Temperature-composition phase diagrams of ideal solutions, 245–246, 245–246f of nonideal mixtures liquid-vapor, 282–283, 282–284f solid-liquid, 285–286, 285–286f, 288, 288f solid-liquid with compounds, 288–289, 288–289f Temperature-jump method, for fast reaction experimental, 516–520, 518f Tension force, 1201 Term symbols, for homonuclear diatomic molecules, 847–848 Terminals, of electrochemical cells, 354 Termolecular process, 524 gas-phase reactions, 527 liquids, 532 Tesla, Nikola, 1002 Tetrahedral angle, 874, 874f Tetramethyl silane, for NMR, 1016 Theorem of equipartition of energy, 1142–1143 Theoretical plate, of still, 246, 246f Thermal conduction, in hard-sphere gas, 464–465 Thermal conductivity, 445 Thermal equilibrium, 204 Thermodynamic control, for competing reactions, 514 Thermodynamic energy See Internal energy Thermodynamic entropy, statistical entropy and, 136–137, 1117–1118 Thermodynamic equation of state heat capacity and, 170 Maxwell relations, 167 Thermodynamic functions in canonical ensembles, 1128–1130 classical, 1141–1146 of nonideal solutions, 275–280 Thermodynamic limit, 1048–1049 Thermodynamic probability, of macrostates, 133–134, 1082 Thermodynamic temperature, 111 ideal gas temperature vs., 111–112 zeroeth law and, 110–113 Thermodynamics, 39 chemical equilibrium, 303–348 of electrochemical systems, 351–378 activities and activity coefficients of electrolytes, 371–374 chemical potential and electric potential, 352–353 electrochemical cells, 354–361 half-cell potentials and cell potentials, 361–369 summary for, 378 thermodynamic information from electrochemistry, 374–376 first law of, 39–102 for closed simple systems, 158 energy change calculations with chemical reactions, 94–100 www.TheSolutionManual.com Subshell configuration Aufbau principle and, 809–810f, 809–812, 811t orbital wave functions, 772 Subshells, 742 Sum of two operators, 685 Superconductivity, 1183–1184 Supercooled systems entropy changes of, 127–128 phase transitions in, 215–216, 215–216f Supercritical fluids, 28 Superheated systems cavity vapor pressure in, 229 entropy changes of, 127–128 phase transitions in, 215–216, 215–216f Surface concentration, 233 Surface entropy, 232 Surface excess, 231 Surface Gibbs energy, 232, 235 Surface tension in multicomponent system, 232–233 in one-component system, 224–227, 225–226f Surfaces in multicomponent systems, 230–234 in one-component system, 222–229 energy attributed to, 222–224 Laplace equation, 227–229 tension of, 224–227, 225–226f one-component vs two-component systems, 230–231, 231f Surfactant, 233 Surroundings, 13 Svedburg, 472 Svedburg, Theodor, 472 Symmetric matrix, 1251 Symmetric spin factor, 812 Symmetric stretch, 939 Symmetric top, 934 Symmetry element, 828 Symmetry number, 936, 1068 Symmetry operations, 897 on wave functions, 930–932 Symmetry operators, 897 function operation of, 830–832 group theory, 898–899 elementary applications of, 902–903 for molecular orbitals, 827–829, 829f, 894–895, 895t point groups and Schoenflies symbols, 899–902, 901t, 902f Index enthalpy, 74–80 enthalpy calculations with chemical reactions, 86–93 enthalpy calculations without chemical reactions, 81–85, 85f equation for, 57 heat, 51–54 heat and energy calculations, 60–72 internal energy, 55–59 summary of, 101–102 work and state of system, 40–50 of real systems, 151–196 criteria for spontaneous processes and equilibrium, 152–157, 195 Euler’s theorem and Gibbs–Duhem relation, 188–194, 196 fundamental relations for closed simple systems, 158–166, 195 Gibbs energy calculations, 175–180 multicomponent systems, 182–187, 195 summary for, 195–196 thermodynamic identities, 167–172 of rubber, 1201–1202 second law of, 106–113, 147 Carnot engine, 106–110, 107–109f for closed simple systems, 158 mathematical statement of, 105, 114–121 physical statements of, 105–106 of solutions, 237–299 activity and activity coefficients, 258–266 colligative properties, 292–298 Henry’s law and dilute nonelectrolyte solutions, 248–257 ideal solutions, 238–247 nonideal solutions, 275–280 phase diagrams, 282–290 summary of, 299 third law of, 105, 139–146, 148 absolute entropies, 140–141 absolute zero unattainability, 139–140, 140f chemical reaction entropy changes, 141–144 entropy standard state, 142–143 statistical entropy and, 144–145 Trouton’s rule, 145–146 zeroeth law of, 110–113 Thermoplastic polymers, 589 Theta solvent, polymers in, 1198 Θ Functions in relative Schrödinger equation, 733 zero value in, 745, 746f Third law of thermodynamics See Thermodynamics, third law of Third postulate, of quantum mechanics, 683–695, 721 Third-order reactions, gaseous termolecular elementary process, 527 Thixotropic fluids See Non-Newtonian fluids Thompson, Benjamin See Count Rumford Thomson, Joseph John, 640 Thomson, William, 78, 106 Three-component systems, phase diagrams of, 290, 290f Threshold wavelength, 645 Tie line, 28–29, 29f in fluid pressure graphs, 30–32, 30f, 33f Time correlation functions, for liquid transport processes, 1193 Time, uncertainty relation of, 715–716 Time-dependent perturbation theory, for electric dipole transitions, 951–953 Time-of-flight selector, for molecular beam reactions, 610, 610f Toluene liquid-vapor pressure-composition, 244, 244f partial vapor pressure, 240, 240f Torr, Totally symmetric representation, 1296 Trace, of square matrix, 1250–1251, 1296 Tracer diffusion, 461 Trajectory calculations, for activated complex theory, 1108–1109 Transition dipole moment, rotational spectra of, 961–964 Transition elements, 814, 815f Transition state, 1108, 1108f Transition temperature, superconductivity, 1183 Translational energy of dilute gas, statistical thermodynamics, 1090 in internal energy, 59 spacing of, 917–918 Translational factor, 916, 931 Translational partition functions, 1057–1063, 1059f calculation of, diatomic gases, 1065–1066 Translational quantum numbers, 916–918 Translational states, of atoms, 916–918 Transmission coefficient, 1111 Transmission grating, for light dispersion, 954, 954f Transmittance, with spectrophotometer, 956 Transmitted pressure, in closed fluid system, 42 Transport processes, 441–482 in dilute gases, 441 driving forces and linear laws, 445 electrical conduction in electrolyte solutions, 475–481 equation of continuity, 447–448, 447f Fick’s law of diffusion, 446–447 Fick’s second law of diffusion, 449–452, 450–451f Fourier’s law of heat conduction, 445–446 gas kinetic theory of, 460–466 self-diffusion, 460–464, 461f thermal conduction, 464–465 viscosity, 465–466 in liquids, 441, 467–473 approximate theories, 1188–1193 Brownian motion, 469 diffusion, 1189–1192, 1189f friction, 468–469 more advanced theories, 1193 sedimentation, 472–473 temperature dependence of diffusion and viscosity, 470–472 viscosity, 1192 macroscopic description of nonequilibrium states, 442–443 Newton’s law of viscous flow, 444f, 452–453 Poiseuille’s equation, 453–457, 454f principles of, 441 Stoke’s law, 457–458 summary for, 482 variables for rates of, 444–445, 444f Transpose, of matrix, 1251 Transverse relaxation time, 1025, 1026f Traveling waves, 629, 630f, 636–638 Trial solution, 624, 631 Triple bonds, hybrid orbitals of, 879–880 Triple point, 27–28, 27f, 206, 260f Trivial solution, 1287 Trough, of classical waves, 629 Trouton’s rule, 145–146 Vapor pressure and, 212 Tunneling, 677, 703–704, 704f www.TheSolutionManual.com 1383 Index 1384 U See Internal energy Ultraviolet catastrophe, 642 Ultraviolet radiation electronic energy level transitions and, 949 photoelectron spectroscopy, 991–993, 992f Uncertainty broadening, 716 Uncertainty principle of Heisenburg See Heisenburg uncertainty principle Uncertainty product, 712 for different energy states, 713 of free particle, 714–715 for particle in one-dimensional box, 711–712, 712t Uncertainty relation, 715–716 Undetermined multipliers, 1052 Ungerade, eigenfunction, 832 Uniform harmonic motion, of harmonic oscillator, 626, 626f Unimolecular process gases, 524, 527 Lindemann mechanism for, 545–547 liquids, 532 Unit cells, of crystal lattice, 1154 Unitary matrix, 1251 United atom, 826 Universe entropy of, 151 heat death of, 121 Unperturbed energy eigenvalue, 799 Unperturbed wave function, 799 Unsöld’s theorem, 817 Upper consolute point, in phase diagram, 283–284 Upper critical solution point See Upper consolute point Upper triangular matrix, 1251 Urease, 575–576 V See Voltage Valence shell, chemical properties and, 816 Valence shell electron pair repulsion (VSEPR), 877–880 Valence-bond method, 823, 866 for heteronuclear diatomic molecules, 863–865, 866 for homonuclear diatomic molecules, 849–850, 866 for hydrogen molecule, 840–842, 840t resonance and, 885–886 Valence-bond wave function, for benzene, 885–886 Valence-bonds, 881–884 Valence-state ionization potential (VSIP), in extended Hückel method, 905 Van der Waals equation of state, 21, 30f, 31–34 Van der Waals gas, molar heat capacity of, 70 van’t Hoff equation, 297, 336 van’t Hoff, Jacobus Henricus, 297 Vapor phase, 200, 216, 216f Vapor pressure, 206, 206f electrolyte solutes, activity and activity coefficient determination, 371 of hydrochloric acid, 328–329 Laplace equation and, 228 lowering, 296 partial of benzene and toluene, 240, 240f of hydrochloric acid, 329–330, 330f Raoult’s law, 238 total pressure and, 212–213 Trouton’s rule and, 212 Vaporization, 27–28, 27f, 28 See also Phase transitions entropy changes of, 145–146 latent heat of, 54 Variable-change identity, 1086 Variables, types of, 5, 13–14 Variance, of standard deviation, 705 Variation energy, 790 Variation function See Variation trial function Variation method, 790–791 extended Hückel method, 905 for ground state energies, 789 helium application of, 791–793 perturbation method vs., 802 shielding, 793–795 Variation theorem, 790 extended, 803 Variation trial function, 790–791 for helium, 793 Vector addition, 1241, 1242f of angular momentum, 775, 776f Vector derivatives, 1243–1244 Vector product, 1243 Vectors, 40, 40f, 1241–1244, 1242f Velocity in classical mechanics, 621 of flexible string, 633–634 of point-mass particles, 387 probability distribution of, 394–404 derivation of, 395–397, 395f identification of parameter b, 398–404, 399f, 404f Velocity space, 391, 391f spherical polar coordinates in, 405–406, 406f spherical shell in, 407, 407f Velocity vector, 391, 391f Vibration(s) of diatomic molecules, 922–929, 922f, 926f population of, 944–945 normal modes of, 937–939, 939f of polyatomic molecules, 937–939, 937f, 939f, 945–947 Vibrational energy of dilute gas, statistical thermodynamics, 1091–1092 potential, 926, 926f Vibrational partition functions, calculation of diatomic gases, 1069–1070 polyatomic gases, 1074–1075 Vibrational spectra, of polyatomic molecules, 976–977, 977f Vibrational states, four harmonic oscillators, 1041, 1042t Vibrational transitions rotation transitions and, 965–966, 966f spectroscopy of, 949, 997 Raman, 987–991, 989–990f Vibrational wave function, 924 Vibration-rotation spectra, of diatomic molecules, 964–968, 966–967f Virial coefficients, 22, 1148–1149 Virial equation of state, 22, 1148 Virial theorem, of mechanics, 751 Viscosity See also Viscosity coefficient of hard-sphere gas, 465–466 of liquids, 1192 temperature dependence of, 470–472 types of, 1198–1199 Viscosity coefficient hard-sphere gas, 465 in Newton’s law of viscous flow, 452 for non-Newtonian fluids, 453 www.TheSolutionManual.com Turbulent flow, 444, 444f Turning point, harmonic oscillator and, 627 Turnover rate, with enzyme catalysis, 578 Two-body problem, 1272–1274 Two-component ideal solutions, 243–244 Two-dimensional graph, Two-phase simple system, 200–202, 200f Index 1385 Index w See Work Wall collisions, 416–417 Water, 875–876f, 875–878 correlation diagram for, 876–877, 876f dielectric constant of, 270 ionization of, 320–326 molar Gibbs energy of, 215–216, 215–216f multiplication table for, 899, 899t phase diagram of, 206, 206f sp3 hybrid orbitals of, 875–877, 875f symmetry operators for, 894–895, 895t, 899, 899t Waterston, John James, 387 Wave, 629 Wave function(s) antisymmetrized, 768 classes of, 668–669 coordinate, 660 for diatomic helium, 843 for diatomic molecules, 882–883 of flexible string, 633 derivation of, 1268–1269, 1269f of free particle, 714–715 for helium-like atoms, 765–766, 777–780, 779t for hydrogen atom central force system, 726–729 radial factor, 736–740, 744f linear combination, 708–709 nodes and energy of, 667 normalization of, 697–698 orbitals, 725, 741 excited states, 772–773 ground states, 768–771 of hydrogen-like atoms, 741–748 position measurements for, 699–700 predictable case and statistical case, 698–699 distinguishing, 705–707 for quantum harmonic oscillator, 676 rotational, 924 Schrödinger equation, 662–663 state and, 668 symmetry operations on, 930–932 time-dependent, 659 for free particle in one dimension, 672 of hydrogen atom, 753–755 for particle in a box, 668 for traveling wave, 636–637 two electrons, 766–767, 776–780, 779t for variation method, 790 vibrational, 924 zero-order, 799 Wave numbers, 928, 962 Wave theory, of electromagnetic radiation, 638–639, 639f Wavelength of classical waves, 629 for flexible string, 634 of traveling waves, 637–638 Wave-particle duality of light, 646 of particles, 656 Weak acid, 319 conjugate base of, 325 ionization of water with, 323–324 Weak base, 325 conjugate acid of, 325 ionization of water with, 325–326 Wien’s law, for blackbody radiation, 643 Work (w), 40–50 on closed fluid system, 41–43 in closed simple system, 156–157 compression, 42, 47 constant-pressure processes, 45–46 displacement with, 41, 41f enthalpy and, 83–84 expansion processes, 46–47 as inexact differential, 48–50 mechanical, 40–41 in nonsimple system, 157 reversible, 43–45 in statistical mechanics, 1117 Work function See Helmholtz energy X-ray diffraction, of solids, 1158–1161, 1159f, 1161f Zeigler-Natta catalyst, for rubber formation, 589 Zero differential overlap, 906 Zero matrix, 1251 Zeroeth law of thermodynamics See Thermodynamics, zeroeth law of Zero-order approximation for diatomic helium, 843 for helium-like atom, 765 angular momentum of helium atom, 774–780 excited state, 772–773 ground state, 768–771, 794, 794f probability density, 769–771 for hydrogen molecule, 839–840 for lithium atom, 781–783 for lithium hydride, 851–852 for more than three electrons, 784–785 for multielectron atoms, 763–786 excited states of helium atom, 772–773 Pauli exclusion principle, 766–768 summary for, 786 Zero-order energy eigenvalue, 799 Zero-order Hamiltonian, 765, 799, 952 Zero-order reactions heterogenously catalyzed, 570 rate law for, 494 test for half-life, 496 integrated rate laws, 495, 495f method of initial rates, 497, 497f Zero-order wave function, 799 degeneracy of, 804 Zone refining, 285–286 Zustandsumme, 1056, 1125 www.TheSolutionManual.com Viscosity-average molecular mass, 1200 Viscous flow, 441–442, 482 driving force for, 445 of hard-sphere gas, 465–466 laminar flow and turbulent flow for, 444, 444f Newton’s law of viscous flow, 444f, 452–453 Poiseuille’s equation for, 453 Stoke’s law, 457–458 Visible radiation, electronic energy level transitions and, 949 Vision, 983–984 Volt, 352 Volta, Alessandro Guiseppi Antonio Anastasio, 352 Voltage (V ), in Ohm’s law, 475 Volume in energy change of formation calculation, 94–95 of gases, graphical representation, 24–25, 24f reduced, 33 Volumetric behavior, of gas or liquid, 6, 35 VSEPR See Valence shell electron pair repulsion Vulcanization, 1201 www.TheSolutionManual.com ... homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Library of Congress Catalog-in-Publishing Data Mortimer, Robert G Physical chemistry / Robert G Mortimer. .. containing Robert G Mortimer, Mathematics for Physical Chemistry, 3rd ed., Academic Press, San Diego, CA, U.S.A., 2005; James R Barrante, Applied Mathematics for Physical Chemistry, 3rd ed., Pearson... www.TheSolutionManual.com This is the third edition of a physical chemistry textbook designed for a two-semester undergraduate physical chemistry course The physical chemistry course is often the first