Physical Chemistry: Sixth Edition PDF

lev38627_fm.qxd 4/9/08 12:32 PM Page i PHYSICAL CHEMISTRY lev38627_fm.qxd 4/9/08 12:32 PM Page ii lev38627_fm.qxd 4/9/08 12:32 PM Page iii PHYSICAL CHEMISTRY Sixth Edi...

lev38627_fm.qxd 4/9/08 12:32 PM Page i PHYSICAL CHEMISTRY lev38627_fm.qxd 4/9/08 12:32 PM Page ii lev38627_fm.qxd 4/9/08 12:32 PM Page iii PHYSICAL CHEMISTRY Sixth Edition Ira N. Levine Chemistry Department Brooklyn College City University of New York Brooklyn, New York lev38627_fm.qxd 4/9/08 12:32 PM Page iv PHYSICAL CHEMISTRY, SIXTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020. Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved. Previous editions © 2002, 1995, 1988, 1983, and 1978. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on recycled, acid-free paper containing 10% postconsumer waste. 1 2 3 4 5 6 7 8 9 0 QPD/QPD 0 9 8 ISBN 978–0–07–253862–5 MHID 0–07–253862–7 Publisher: Thomas Timp Senior Sponsoring Editor: Tamara L. Hodge Director of Development: Kristine Tibbetts Senior Developmental Editor: Shirley R. Oberbroeckling Marketing Manager: Todd L. Turner Project Coordinator: Melissa M. Leick Senior Production Supervisor: Sherry L. Kane Senior Designer: David W. Hash Cover Designer: Ron E. Bissell, Creative Measures Design Inc. Supplement Producer: Melissa M. Leick Compositor: ICC Macmillan Inc. Typeface: 10.5/12 Times Printer: Quebecor World Dubuque, IA Library of Congress Cataloging-in-Publication Data Levine, Ira N. Physical chemistry / Ira N. Levine. -- 6th ed. p. cm. Includes index. ISBN 978–0–07–253862–5 --- ISBN 0–07–253862–7 (hard copy : alk. paper) 1. Chemistry, Physical and theoretical. I. Title. QD453.3.L48 2009 541-- dc22 2008002821 www.mhhe.com lev38627_fm.qxd 4/9/08 12:32 PM Page v To the memory of my mother and my father lev38627_fm.qxd 4/9/08 12:32 PM Page vi Table of Contents Preface xiv Chapter 1 THERMODYNAMICS 1 1.1 Physical Chemistry 1 1.2 Thermodynamics 3 1.3 Temperature 6 1.4 The Mole 9 1.5 Ideal Gases 10 1.6 Differential Calculus 17 1.7 Equations of State 22 1.8 Integral Calculus 25 1.9 Study Suggestions 30 1.10 Summary 32 Chapter 2 THE FIRST LAW OF THERMODYNAMICS 37 2.1 Classical Mechanics 37 2.2 P-V Work 42 2.3 Heat 46 2.4 The First Law of Thermodynamics 47 2.5 Enthalpy 52 2.6 Heat Capacities 53 2.7 The Joule and Joule–Thomson Experiments 55 2.8 Perfect Gases and the First Law 58 2.9 Calculation of First-Law Quantities 62 2.10 State Functions and Line Integrals 65 2.11 The Molecular Nature of Internal Energy 67 2.12 Problem Solving 70 2.13 Summary 73 Chapter 3 THE SECOND LAW OF THERMODYNAMICS 78 3.1 The Second Law of Thermodynamics 78 3.2 Heat Engines 80 3.3 Entropy 85 3.4 Calculation of Entropy Changes 87 3.5 Entropy, Reversibility, and Irreversibility 93 3.6 The Thermodynamic Temperature Scale 96 3.7 What Is Entropy? 97 3.8 Entropy, Time, and Cosmology 103 3.9 Summary 104 vi lev38627_fm.qxd 4/9/08 12:32 PM Page vii vii Chapter 4 MATERIAL EQUILIBRIUM 109 Table of Contents 4.1 Material Equilibrium 109 4.2 Entropy and Equilibrium 110 4.3 The Gibbs and Helmholtz Energies 112 4.4 Thermodynamic Relations for a System in Equilibrium 115 4.5 Calculation of Changes in State Functions 123 4.6 Chemical Potentials and Material Equilibrium 125 4.7 Phase Equilibrium 129 4.8 Reaction Equilibrium 132 4.9 Entropy and Life 134 4.10 Summary 135 Chapter 5 STANDARD THERMODYNAMIC FUNCTIONS OF REACTION 140 5.1 Standard States of Pure Substances 140 5.2 Standard Enthalpy of Reaction 141 5.3 Standard Enthalpy of Formation 142 5.4 Determination of Standard Enthalpies of Formation and Reaction 143 5.5 Temperature Dependence of Reaction Heats 151 5.6 Use of a Spreadsheet to Obtain a Polynomial Fit 153 5.7 Conventional Entropies and the Third Law 155 5.8 Standard Gibbs Energy of Reaction 161 5.9 Thermodynamics Tables 163 5.10 Estimation of Thermodynamic Properties 165 5.11 The Unattainability of Absolute Zero 168 5.12 Summary 169 Chapter 6 REACTION EQUILIBRIUM IN IDEAL GAS MIXTURES 174 6.1 Chemical Potentials in an Ideal Gas Mixture 175 6.2 Ideal-Gas Reaction Equilibrium 177 6.3 Temperature Dependence of the Equilibrium Constant 182 6.4 Ideal-Gas Equilibrium Calculations 186 6.5 Simultaneous Equilibria 191 6.6 Shifts in Ideal-Gas Reaction Equilibria 194 6.7 Summary 198 Chapter 7 ONE-COMPONENT PHASE EQUILIBRIUM AND SURFACES 205 7.1 The Phase Rule 205 7.2 One-Component Phase Equilibrium 210 7.3 The Clapeyron Equation 214 7.4 Solid–Solid Phase Transitions 221 lev38627_fm.qxd 4/9/08 12:32 PM Page viii viii Table of Contents 7.5 Higher-Order Phase Transitions 225 7.6 Surfaces and Nanoparticles 227 7.7 The Interphase Region 227 7.8 Curved Interfaces 231 7.9 Colloids 234 7.10 Summary 237 Chapter 8 REAL GASES 244 8.1 Compression Factors 244 8.2 Real-Gas Equations of State 245 8.3 Condensation 247 8.4 Critical Data and Equations of State 249 8.5 Calculation of Liquid–Vapor Equilibria 252 8.6 The Critical State 254 8.7 The Law of Corresponding States 255 8.8 Differences Between Real-Gas and Ideal-Gas Thermodynamic Properties 256 8.9 Taylor Series 257 8.10 Summary 259 Chapter 9 SOLUTIONS 263 9.1 Solution Composition 263 9.2 Partial Molar Quantities 264 9.3 Mixing Quantities 270 9.4 Determination of Partial Molar Quantities 272 9.5 Ideal Solutions 275 9.6 Thermodynamic Properties of Ideal Solutions 278 9.7 Ideally Dilute Solutions 282 9.8 Thermodynamic Properties of Ideally Dilute Solutions 283 9.9 Summary 287 Chapter 10 NONIDEAL SOLUTIONS 294 10.1 Activities and Activity Coefficients 294 10.2 Excess Functions 297 10.3 Determination of Activities and Activity Coefficients 298 10.4 Activity Coefficients on the Molality and Molar Concentration Scales 305 10.5 Solutions of Electrolytes 306 10.6 Determination of Electrolyte Activity Coefficients 310 10.7 The Debye–Hückel Theory of Electrolyte Solutions 311 10.8 Ionic Association 315 10.9 Standard-State Thermodynamic Properties of Solution Components 318 10.10 Nonideal Gas Mixtures 321 10.11 Summary 324 lev38627_fm.qxd 4/9/08 12:32 PM Page ix ix Chapter 11 REACTION EQUILIBRIUM IN NONIDEAL SYSTEMS 330 Table of Contents 11.1 The Equilibrium Constant 330 11.2 Reaction Equilibrium in Nonelectrolyte Solutions 331 11.3 Reaction Equilibrium in Electrolyte Solutions 332 11.4 Reaction Equilibria Involving Pure Solids or Pure Liquids 337 11.5 Reaction Equilibrium in Nonideal Gas Mixtures 340 11.6 Computer Programs for Equilibrium Calculations 340 11.7 Temperature and Pressure Dependences of the Equilibrium Constant 341 11.8 Summary of Standard States 343 11.9 Gibbs Energy Change for a Reaction 343 11.10 Coupled Reactions 345 11.11 Summary 347 Chapter 12 MULTICOMPONENT PHASE EQUILIBRIUM 351 12.1 Colligative Properties 351 12.2 Vapor-Pressure Lowering 351 12.3 Freezing-Point Depression and Boiling-Point Elevation 352 12.4 Osmotic Pressure 356 12.5 Two-Component Phase Diagrams 361 12.6 Two-Component Liquid–Vapor Equilibrium 362 12.7 Two-Component Liquid–Liquid Equilibrium 370 12.8 Two-Component Solid–Liquid Equilibrium 373 12.9 Structure of Phase Diagrams 381 12.10 Solubility 381 12.11 Computer Calculation of Phase Diagrams 383 12.12 Three-Component Systems 385 12.13 Summary 387 Chapter 13 ELECTROCHEMICAL SYSTEMS 395 13.1 Electrostatics 395 13.2 Electrochemical Systems 398 13.3 Thermodynamics of Electrochemical Systems 401 13.4 Galvanic Cells 403 13.5 Types of Reversible Electrodes 409 13.6 Thermodynamics of Galvanic Cells 412 13.7 Standard Electrode Potentials 417 13.8 Liquid-Junction Potentials 421 13.9 Applications of EMF Measurements 422 13.10 Batteries 426 13.11 Ion-Selective Membrane Electrodes 427 13.12 Membrane Equilibrium 429 13.13 The Electrical Double Layer 430 13.14 Dipole Moments and Polarization 431 13.15 Bioelectrochemistry 435 13.16 Summary 436 lev38627_fm.qxd 4/9/08 12:32 PM Page x x Table of Contents Chapter 14 KINETIC THEORY OF GASES 442 14.1 Kinetic–Molecular Theory of Gases 442 14.2 Pressure of an Ideal Gas 443 14.3 Temperature 446 14.4 Distribution of Molecular Speeds in an Ideal Gas 448 14.5 Applications of the Maxwell Distribution 457 14.6 Collisions with a Wall and Effusion 460 14.7 Molecular Collisions and Mean Free Path 462 14.8 The Barometric Formula 465 14.9 The Boltzmann Distribution Law 467 14.10 Heat Capacities of Ideal Polyatomic Gases 467 14.11 Summary 469 Chapter 15 TRANSPORT PROCESSES 474 15.1 Kinetics 474 15.2 Thermal Conductivity 475 15.3 Viscosity 479 15.4 Diffusion and Sedimentation 487 15.5 Electrical Conductivity 493 15.6 Electrical Conductivity of Electrolyte Solutions 496 15.7 Summary 509 Chapter 16 REACTION KINETICS 515 16.1 Reaction Kinetics 515 16.2 Measurement of Reaction Rates 519 16.3 Integration of Rate Laws 520 16.4 Finding the Rate Law 526 16.5 Rate Laws and Equilibrium Constants for Elementary Reactions 530 16.6 Reaction Mechanisms 532 16.7 Computer Integration of Rate Equations 539 16.8 Temperature Dependence of Rate Constants 541 16.9 Relation Between Rate Constants and Equilibrium Constants for Composite Reactions 546 16.10 The Rate Law in Nonideal Systems 547 16.11 Unimolecular Reactions 548 16.12 Trimolecular Reactions 550 16.13 Chain Reactions and Free-Radical Polymerizations 551 16.14 Fast Reactions 556 16.15 Reactions in Liquid Solutions 560 16.16 Catalysis 564 16.17 Enzyme Catalysis 568 16.18 Adsorption of Gases on Solids 570 16.19 Heterogeneous Catalysis 575 16.20 Summary 579 lev38627_fm.qxd 4/9/08 12:32 PM Page xi xi Chapter 17 QUANTUM MECHANICS 590 Table of Contents 17.1 Blackbody Radiation and Energy Quantization 591 17.2 The Photoelectric Effect and Photons 593 17.3 The Bohr Theory of the Hydrogen Atom 594 17.4 The de Broglie Hypothesis 595 17.5 The Uncertainty Principle 597 17.6 Quantum Mechanics 599 17.7 The Time-Independent Schrödinger Equation 604 17.8 The Particle in a One-Dimensional Box 606 17.9 The Particle in a Three-Dimensional Box 610 17.10 Degeneracy 612 17.11 Operators 613 17.12 The One-Dimensional Harmonic Oscillator 619 17.13 Two-Particle Problems 621 17.14 The Two-Particle Rigid Rotor 622 17.15 Approximation Methods 623 17.16 Hermitian Operators 627 17.17 Summary 630 Chapter 18 ATOMIC STRUCTURE 637 18.1 Units 637 18.2 Historical Background 637 18.3 The Hydrogen Atom 638 18.4 Angular Momentum 647 18.5 Electron Spin 649 18.6 The Helium Atom and the Spin–Statistics Theorem 650 18.7 Total Orbital and Spin Angular Momenta 656 18.8 Many-Electron Atoms and the Periodic Table 658 18.9 Hartree–Fock and Configuration-Interaction Wave Functions 663 18.10 Summary 666 Chapter 19 MOLECULAR ELECTRONIC STRUCTURE 672 19.1 Chemical Bonds 672 19.2 The Born–Oppenheimer Approximation 676 19.3 The Hydrogen Molecule Ion 681 19.4 The Simple MO Method for Diatomic Molecules 686 19.5 SCF and Hartree–Fock Wave Functions 692 19.6 The MO Treatment of Polyatomic Molecules 693 19.7 The Valence-Bond Method 702 19.8 Calculation of Molecular Properties 704 19.9 Accurate Calculation of Molecular Electronic Wave Functions and Properties 708 19.10 Density-Functional Theory (DFT) 711 19.11 Semiempirical Methods 717 19.12 Performing Quantum Chemistry Calculations 720 19.13 The Molecular-Mechanics (MM) Method 723 lev38627_fm.qxd 4/9/08 12:32 PM Page xii xii Table of Contents 19.14 Future Prospects 727 19.15 Summary 727 Chapter 20 SPECTROSCOPY AND PHOTOCHEMISTRY 734 20.1 Electromagnetic Radiation 734 20.2 Spectroscopy 737 20.3 Rotation and Vibration of Diatomic Molecules 743 20.4 Rotational and Vibrational Spectra of Diatomic Molecules 750 20.5 Molecular Symmetry 756 20.6 Rotation of Polyatomic Molecules 758 20.7 Microwave Spectroscopy 761 20.8 Vibration of Polyatomic Molecules 763 20.9 Infrared Spectroscopy 766 20.10 Raman Spectroscopy 771 20.11 Electronic Spectroscopy 774 20.12 Nuclear-Magnetic-Resonance Spectroscopy 779 20.13 Electron-Spin-Resonance Spectroscopy 793 20.14 Optical Rotatory Dispersion and Circular Dichroism 794 20.15 Photochemistry 796 20.16 Group Theory 800 20.17 Summary 811 Chapter 21 STATISTICAL MECHANICS 820 21.1 Statistical Mechanics 820 21.2 The Canonical Ensemble 821 21.3 Canonical Partition Function for a System of Noninteracting Particles 830 21.4 Canonical Partition Function of a Pure Ideal Gas 834 21.5 The Boltzmann Distribution Law for Noninteracting Molecules 836 21.6 Statistical Thermodynamics of Ideal Diatomic and Monatomic Gases 840 21.7 Statistical Thermodynamics of Ideal Polyatomic Gases 851 21.8 Ideal-Gas Thermodynamic Properties and Equilibrium Constants 854 21.9 Entropy and the Third Law of Thermodynamics 858 21.10 Intermolecular Forces 861 21.11 Statistical Mechanics of Fluids 866 21.12 Summary 870 Chapter 22 THEORIES OF REACTION RATES 877 22.1 Hard-Sphere Collision Theory of Gas-Phase Reactions 877 22.2 Potential-Energy Surfaces 880 22.3 Molecular Reaction Dynamics 887 lev38627_fm.qxd 4/9/08 12:32 PM Page xiii xiii 22.4 Transition-State Theory for Ideal-Gas Reactions 892 Table of Contents 22.5 Thermodynamic Formulation of TST for Gas-Phase Reactions 902 22.6 Unimolecular Reactions 904 22.7 Trimolecular Reactions 906 22.8 Reactions in Solution 906 22.9 Summary 911 Chapter 23 SOLIDS AND LIQUIDS 913 23.1 Solids and Liquids 913 23.2 Polymers 914 23.3 Chemical Bonding in Solids 914 23.4 Cohesive Energies of Solids 916 23.5 Theoretical Calculation of Cohesive Energies 918 23.6 Interatomic Distances in Crystals 921 23.7 Crystal Structures 922 23.8 Examples of Crystal Structures 928 23.9 Determination of Crystal Structures 931 23.10 Determination of Surface Structures 937 23.11 Band Theory of Solids 939 23.12 Statistical Mechanics of Crystals 941 23.13 Defects in Solids 946 23.14 Liquids 947 23.15 Summary 951 Bibliography 955 Appendix 959 Answers to Selected Problems 961 Index 967 lev38627_fm.qxd 4/9/08 12:32 PM Page xiv Preface This textbook is for the standard undergraduate course in physical chemistry. In writing this book, I have kept in mind the goals of clarity, accuracy, and depth. To make the presentation easy to follow, the book gives careful definitions and expla- nations of concepts, full details of most derivations, and reviews of relevant topics in mathematics and physics. I have avoided a superficial treatment, which would leave students with little real understanding of physical chemistry. Instead, I have aimed at a treatment that is as accurate, as fundamental, and as up-to-date as can readily be pre- sented at the undergraduate level. LEARNING AIDS Physical chemistry is a challenging course for many students. To help students, this book has many learning aids: Each chapter has a summary of the key points. The summaries list the specific kinds of calculations that students are expected to learn how to do. 3.9 SUMMARY We assumed the truth of the Kelvin–Planck statement of the second law of ther- modynamics, which asserts the impossibility of the complete conversion of heat to work in a cyclic process. From the second law, we proved that dqrev /T is the differ- ential of a state function, which we called the entropy S. The entropy change in a process from state 1 to state 2 is ⌬S ⫽ 兰21 dqrev /T, where the integral must be eval- uated using a reversible path from 1 to 2. Methods for calculating ⌬S were dis- cussed in Sec. 3.4. We used the second law to prove that the entropy of an isolated system must increase in an irreversible process. It follows that thermodynamic equilibrium in an isolated system is reached when the system’s entropy is maximized. Since isolated systems spontaneously change to more probable states, increasing entropy corre- sponds to increasing probability p. We found that S ⫽ k ln p ⫹ a, where the Boltzmann constant k is k ⫽ R/NA and a is a constant. Important kinds of calculations dealt with in this chapter include: Calculation of ⌬S for a reversible process using dS ⫽ dqrev /T. Calculation of ⌬S for an irreversible process by finding a reversible path between the initial and final states (Sec. 3.4, paragraphs 5, 7, and 9). Calculation of ⌬S for a reversible phase change using ⌬S ⫽ ⌬H/T. Calculation of ⌬S for constant-pressure heating using dS ⫽ dqrev /T ⫽ (CP /T) dT. Calculation of ⌬S for a change of state of a perfect gas using Eq. (3.30). Calculation of ⌬S for mixing perfect gases at constant T and P using Eq. (3.33). Since the integral of dqrev /T around any reversible cycle is zero, it follows (Sec. 2.10) that the value of the line integral 兰21 dqrev /T is independent of the path be- tween states 1 and 2 and depends only on the initial and final states. Hence dqrev /T is the differential of a state function. This state function is called the entropy S: dqrev dS K closed syst., rev. proc. (3.20)* Equations that students should memorize T are marked with an asterisk. These are the The entropy change on going from state 1 to state 2 equals the integral of (3.20): fundamental equations and students are cau- 2 冮 dqrev tioned against blindly memorizing unstarred ¢S ⫽ S2 ⫺ S1 ⫽ closed syst., rev. proc. (3.21)* T 1 equations. xiv lev38627_fm.qxd 4/9/08 12:32 PM Page xv xv A substantial number of worked-out examples are included. Most examples are Preface followed by an exercise with the answer given, to allow students to test their understanding. EXAMPLE 2.6 Calculation of ⌬H CP,m of a certain substance in the temperature range 250 to 500 K at 1 bar pres- sure is given by CP,m ⫽ b ⫹ kT, where b and k are certain known constants. If n moles of this substance is heated from T1 to T2 at 1 bar (where T1 and T2 are in the range 250 to 500 K), find the expression for ⌬H. Since P is constant for the heating, we use (2.79) to get 2 T2 冮 nC 冮 1b ⫹ kT 2 dT ⫽ n1bT ⫹ 12kT 2 2 ` T2 ¢H ⫽ qP ⫽ P,m dT ⫽ n T1 T1 1 ¢H ⫽ n 3b1T2 ⫺ T1 2 ⫹ 1 2 2 k1T 2 ⫺ T 21 2 4 Exercise Find the ⌬H expression when n moles of a substance with CP,m ⫽ r ⫹ sT1/2, where r and s are constants, is heated at constant pressure from T1 to T2. [Answer: nr(T2 ⫺ T1) ⫹ 23ns(T 3/2 2 ⫺ T 1 ).] 3/2 A wide variety of problems are included. As well as being able to do calculational problems, it is important for students to have a good conceptual understanding of the material. To this end, a substantial number of qualitative questions are in- cluded, such as True/False questions and questions asking students to decide whether quantities are positive, negative, or zero. Many of these questions result from misconceptions that I have found that students have. A solutions manual is available to students. Although physical chemistry students have studied calculus, many of them Integral Calculus have not had much experience with sci- Frequently one wants to find a function y(x) whose derivative is known to be a certain function f(x); dy/dx ⫽ f(x). The most general function y that satisfies this equation is ence courses that use calculus, and so called the indefinite integral (or antiderivative) of f(x) and is denoted by 兰 f(x) dx. have forgotten much of what they learned. This book reviews relevant If dy>dx ⫽ f 1x2 then y ⫽ 冮 f 1x2 dx (1.52)* portions of calculus (Secs. 1.6, 1.8, and The function f (x) being integrated in (1.52) is called the integrand. 8.9). Likewise, reviews of important topics in physics are included (classical mechanics in Sec. 2.1, electrostatics in Sec. 13.1, electric dipoles in Sec. 13.14, and magnetic fields in Sec. 20.12.) Section 1.9 discusses effective study methods. 1.9 STUDY SUGGESTIONS A common reaction to a physical chemistry course is for a student to think, “This looks like a tough course, so I’d better memorize all the equations, or I won’t do well.” Such a reaction is understandable, especially since many of us have had teachers who emphasized rote memory, rather than understanding, as the method of instruction. Actually, comparatively few equations need to be remembered (they have been marked with an asterisk), and most of these are simple enough to require little effort at conscious memorization. Being able to reproduce an equation is no guarantee of being able to apply that equation to solving problems. To use an equation properly, one must understand it. Understanding involves not only knowing what the symbols stand for but also knowing when the equation applies and when it does not apply. Everyone knows the ideal-gas equation PV ⫽ nRT, but it’s amazing how often students will use lev38627_fm.qxd 4/9/08 12:32 PM Page xvi xvi Preface Section 2.12 contains advice on how to solve problems in physical chemistry. 2.12 PROBLEM SOLVING Trying to learn physical chemistry solely by reading a textbook without working prob- lems is about as effective as trying to improve your physique by reading a book on body conditioning without doing the recommended physical exercises. If you don’t see how to work a problem, it often helps to carry out these steps: 1. List all the relevant information that is given. 2. List the quantities to be calculated. 3. Ask yourself what equations, laws, or theorems connect what is known to what is unknown. 4. Apply the relevant equations to calculate what is unknown from what is given. The derivations are given in full detail, so that students can readily follow them. The assumptions and approximations made are clearly stated, so that students will be aware of when the results apply and when they do not apply. Many student errors in thermodynamics result from the use of equations in situa- tions where they do not apply. To help prevent this, important thermodynamic equations have their conditions of applicability listed alongside the equations. Systematic listings of procedures to calculate q, w, ¢U, ¢H, and ¢S (Secs. 2.9 and 3.4) for common kinds of processes are given. Detailed procedures are given for the use of a spreadsheet to solve such problems as fitting data to a polynomial (Sec. 5.6), solving simultaneous equilibria (Sec. 6.5), doing linear and nonlinear least-squares fits of data (Sec. 7.3), using an equation of state to calculate vapor pressures and molar volumes of liquids and vapor in equilibrium (Sec. 8.5), and computing a liquid–liquid phase diagram by minimization of G (Sec. 12.11). 154 Chapter 5 A B C D E F Standard Thermodynamic Functions of Reaction 1 CO Cp polynomial fit a b c d 2 T/K Cp Cpfit 28.74 -0.00179 1.05E-05 -4.29E-09 Figure 5.7 3 298.15 29.143 29.022 4 400 29.342 29.422 y = -4.2883E-09x3 + 1.0462E-05x2 - Cubic polynomial fit to C°P,m of 5 500 29.794 29.923 CO C P, m 1.7917E-03x + 2.8740E+01 CO(g). 6 600 30.443 30.504 7 700 31.171 31.14 36 8 800 31.899 31.805 34 9 900 32.577 32.474 10 1000 33.183 33.12 32 11 1100 33.71 33.718 30 12 1200 34.175 34.242 13 1300 34.572 34.667 28 14 1400 34.92 34.967 0 500 1000 1500 15 1500 35.217 35.115 Although the treatment is an in-depth one, the mathematics has been kept at a rea- sonable level and advanced mathematics unfamiliar to students is avoided. The presentation of quantum chemistry steers a middle course between an exces- sively mathematical treatment that would obscure the physical ideas for most un- dergraduates and a purely qualitative treatment that does little beyond repeat what students have learned in previous courses. Modern ab initio, density functional, semiempirical, and molecular mechanics methods are discussed, so that students can appreciate the value of such calculations to nontheoretical chemists. lev38627_fm.qxd 4/9/08 12:32 PM Page xvii xvii IMPROVEMENTS IN THE SIXTH EDITION Preface Students often find that they can solve the problems for a section if they work the problems immediately after studying that section, but when they are faced with an exam that contains problems from a few chapters, they have trouble. To give prac- tice on dealing with this situation, I have added review problems at the ends of Chapters 3, 6, 9, 12, 16, 19, and 21, where each set of review problems covers about three chapters. REVIEW PROBLEMS R3.1 For a closed system, give an example of each of the fol- R3.2 State what experimental data you would need to look up lowing. If it is impossible to have an example of the process, to calculate each of the following quantities. Include only the state this. (a) An isothermal process with q ⫽ 0. (b) An adia- minimum amount of data needed. Do not do the calculations. batic process with ⌬T ⫽ 0. (c) An isothermal process with (a) ⌬U and ⌬H for the freezing of 653 g of liquid water at 0°C ⌬U ⫽ 0. (d) A cyclic process with ⌬S ⫽ 0. (e) An adiabatic and 1 atm. (b) ⌬S for the melting of 75 g of Na at 1 atm and its process with ⌬S ⫽ 0. ( f ) A cyclic process with w ⫽ 0. normal melting point. (c) ⌬U and ⌬H when 2.00 mol of O2 gas One aim of the new edition is to avoid the increase in size that usually occurs with each new edition and that eventually produces an unwieldy text. To this end, Chapter 13 on surfaces was dropped. Some of this chapter was put in the chapters on phase equilibrium (Chapter 7) and reaction kinetics (Chapter 16), and the rest was omitted. Sections 4.2 (thermodynamic properties of nonequilibrium systems), 10.5 (models for nonelectrolyte activity coefficients), 17.19 (nuclear decay), and 21.15 (photoelectron spectroscopy) were deleted. Some material formerly in these sections is now in the problems. Several other sections were shortened. The book has been expanded and updated to include material on nanoparticles (Sec. 7.6), carbon nanotubes (Sec. 23.3), polymorphism in drugs (Sec. 7.4), diffusion-controlled enzyme reactions (Sec. 16.17), prediction of dihedral angles (Sec. 19.1), new functionals in density functional theory (Sec. 19.10), the new semiempirical methods RM1, PM5, and PM6 (Sec. 19.11), the effect of nuclear spin on rotational-level degeneracy (Sec. 20.3), the use of protein IR spectra to follow the kinetics of protein folding (Sec. 20.9), variational transition-state theory (Sec. 22.4), and the Folding@home project (Sec. 23.14). ACKNOWLEDGEMENTS The following people provided reviews for the sixth edition: Jonathan E. Kenny, Tufts University; Jeffrey E. Lacy, Shippensburg University; Clifford LeMaster, Boise State University; Alexa B. Serfis, Saint Louis University; Paul D. Siders, University of Minnesota, Duluth; Yan Waguespack, University of Maryland, Eastern Shore; and John C. Wheeler, University of California, San Diego. Reviewers of previous editions were Alexander R. Amell, S. M. Blinder, C. Allen Bush, Thomas Bydalek, Paul E. Cade, Donald Campbell, Gene B. Carpenter, Linda Casson, Lisa Chirlian, Jefferson C. Davis, Jr., Allen Denio, James Diamond, Jon Draeger, Michael Eastman, Luis Echegoyen, Eric Findsen, L. Peter Gold, George D. Halsey, Drannan Hamby, David O. Harris, James F. Harrison, Robert Howard, Darrell Iler, Robert A. Jacobson, Raj Khanna, Denis Kohl, Leonard Kotin, Willem R. Leenstra, Arthur Low, John P. Lowe, Jack McKenna, Howard D. Mettee, Jennifer Mihalick, George Miller, Alfred Mills, Brian Moores, Thomas Murphy, Mary Ondrechen, Laura Philips, Peter Politzer, Stephan Prager, Frank Prochaska, John L. Ragle, James Riehl, lev38627_fm.qxd 4/9/08 12:32 PM Page xviii xviii Preface Roland R. Roskos, Sanford Safron, Thedore Sakano, Donald Sands, George Schatz, Richard W. Schwenz, Robert Scott, Paul Siders, Agnes Tenney, Charles Trapp, Michael Tubergen, George H. Wahl, Thomas H. Walnut, Gary Washington, Michael Wedlock, John C. Wheeler, Grace Wieder, Robert Wiener, Richard E. Wilde, John R. Wilson, Robb Wilson, Nancy Wu, Peter E. Yankwich, and Gregory Zimmerman. Helpful suggestions for this and previous editions were provided by Thomas Allen, Fitzgerald Bramwell, Dewey Carpenter, Norman C. Craig, John N. Cooper, Thomas G. Dunne, Hugo Franzen, Darryl Howery, Daniel J. Jacob, Bruno Linder, Madan S. Pathania, Jay Rasaiah, J. L. Schrieber, Fritz Steinhardt, Vicki Steinhardt, John C. Wheeler, Grace Wieder, and my students. Professor Wheeler’s many com- ments over the years are especially appreciated. I thank all these people for the considerable help they provided. The help I received from the developmental editor Shirley Oberbroeckling and the project coordinator Melissa Leick at McGraw-Hill is gratefully acknowledged. I welcome any suggestions for improving the book that readers might have. Ira N. Levine [email protected] lev38627_ch01.qxd 2/20/08 11:38 AM Page 1 C H A P T E R Thermodynamics 1 CHAPTER OUTLINE 1.1 Physical Chemistry 1.2 Thermodynamics 1.1 PHYSICAL CHEMISTRY 1.3 Temperature Physical chemistry is the study of the underlying physical principles that govern the properties and behavior of chemical systems. 1.4 The Mole A chemical system can be studied from either a microscopic or a macroscopic viewpoint. The microscopic viewpoint is based on the concept of molecules. The 1.5 Ideal Gases macroscopic viewpoint studies large-scale properties of matter without explicit use of 1.6 Differential Calculus the molecule concept. The first half of this book uses mainly a macroscopic viewpoint; the second half uses mainly a microscopic viewpoint. 1.7 Equations of State We can divide physical chemistry into four areas: thermodynamics, quantum chemistry, statistical mechanics, and kinetics (Fig. 1.1). Thermodynamics is a macro- 1.8 Integral Calculus scopic science that studies the interrelationships of the various equilibrium properties of a system and the changes in equilibrium properties in processes. Thermodynamics 1.9 Study Suggestions is treated in Chapters 1 to 13. 1.10 Summary Molecules and the electrons and nuclei that compose them do not obey classical mechanics. Instead, their motions are governed by the laws of quantum mechanics (Chapter 17). Application of quantum mechanics to atomic structure, molecular bond- ing, and spectroscopy gives us quantum chemistry (Chapters 18 to 20). The macroscopic science of thermodynamics is a consequence of what is hap- pening at a molecular (microscopic) level. The molecular and macroscopic levels are related to each other by the branch of science called statistical mechanics. Statistical mechanics gives insight into why the laws of thermodynamics hold and allows calcu- lation of macroscopic thermodynamic properties from molecular properties. We shall study statistical mechanics in Chapters 14, 15, 21, 22, and 23. Kinetics is the study of rate processes such as chemical reactions, diffusion, and the flow of charge in an electrochemical cell. The theory of rate processes is not as well developed as the theories of thermodynamics, quantum mechanics, and statistical mechanics. Kinetics uses relevant portions of thermodynamics, quantum chemistry, and statistical mechanics. Chapters 15, 16, and 22 deal with kinetics. The principles of physical chemistry provide a framework for all branches of chemistry. Figure 1.1 Statistical Quantum Thermodynamics mechanics chemistry The four branches of physical chemistry. Statistical mechanics is the bridge from the microscopic approach of quantum chemistry to the macroscopic approach of thermodynamics. Kinetics uses Kinetics portions of the other three branches. lev38627_ch01.qxd 2/20/08 11:38 AM Page 2 2 Chapter 1 Organic chemists use kinetics studies to figure out the mechanisms of reactions, Thermodynamics use quantum-chemistry calculations to study the structures and stabilities of reaction intermediates, use symmetry rules deduced from quantum chemistry to predict the course of many reactions, and use nuclear-magnetic-resonance (NMR) and infrared spectroscopy to help determine the structure of compounds. Inorganic chemists use quantum chemistry and spectroscopy to study bonding. Analytical chemists use spec- troscopy to analyze samples. Biochemists use kinetics to study rates of enzyme- catalyzed reactions; use thermodynamics to study biological energy transformations, osmosis, and membrane equilibrium, and to determine molecular weights of biological molecules; use spectroscopy to study processes at the molecular level (for example, in- tramolecular motions in proteins are studied using NMR); and use x-ray diffraction to determine the structures of proteins and nucleic acids. Environmental chemists use thermodynamics to find the equilibrium composition of lakes and streams, use chemical kinetics to study the reactions of pollutants in the atmosphere, and use physical kinetics to study the rate of dispersion of pollutants in the environment. Chemical engineers use thermodynamics to predict the equilibrium composition of reaction mixtures, use kinetics to calculate how fast products will be formed, and use principles of thermodynamic phase equilibria to design separation procedures such as fractional distillation. Geochemists use thermodynamic phase diagrams to un- derstand processes in the earth. Polymer chemists use thermodynamics, kinetics, and statistical mechanics to investigate the kinetics of polymerization, the molecular weights of polymers, the flow of polymer solutions, and the distribution of conforma- tions of a polymer molecule. Widespread recognition of physical chemistry as a discipline began in 1887 with the founding of the journal Zeitschrift für Physikalische Chemie by Wilhelm Ostwald with J. H. van’t Hoff as coeditor. Ostwald investigated chemical equilibrium, chemi- cal kinetics, and solutions and wrote the first textbook of physical chemistry. He was instrumental in drawing attention to Gibbs’ pioneering work in chemical thermody- namics and was the first to nominate Einstein for a Nobel Prize. Surprisingly, Ostwald argued against the atomic theory of matter and did not accept the reality of atoms and molecules until 1908. Ostwald, van’t Hoff, Gibbs, and Arrhenius are generally regarded as the founders of physical chemistry. (In Sinclair Lewis’s 1925 novel Arrowsmith, the character Max Gottlieb, a medical school professor, proclaims that “Physical chemistry is power, it is exactness, it is life.”) In its early years, physical chemistry research was done mainly at the macroscopic level. With the discovery of the laws of quantum mechanics in 1925–1926, emphasis began to shift to the molecular level. (The Journal of Chemical Physics was founded in 1933 in reaction to the refusal of the editors of the Journal of Physical Chemistry to publish theoretical papers.) Nowadays, the power of physical chemistry has been greatly increased by experimental techniques that study properties and processes at the molecular level and by fast computers that (a) process and analyze data of spec- troscopy and x-ray crystallography experiments, (b) accurately calculate properties of molecules that are not too large, and (c) perform simulations of collections of hun- dreds of molecules. Nowadays, the prefix nano is widely used in such terms as nanoscience, nano- technology, nanomaterials, nanoscale, etc. A nanoscale (or nanoscopic) system is one with at least one dimension in the range 1 to 100 nm, where 1 nm ⫽ 10⫺9 m. (Atomic diameters are typically 0.1 to 0.3 nm.) A nanoscale system typically contains thou- sands of atoms. The intensive properties of a nanoscale system commonly depend on its size and differ substantially from those of a macroscopic system of the same composition. For example, macroscopic solid gold is yellow, is a good electrical con- ductor, melts at 1336 K, and is chemically unreactive; however, gold nanoparticles of lev38627_ch01.qxd 2/20/08 11:38 AM Page 3 3 radius 2.5 nm melt at 930 K, and catalyze many reactions; gold nanoparticles of 100 nm Section 1.2 radius are purple-pink, of 20 nm radius are red, and of 1 nm radius are orange; gold Thermodynamics particles of 1 nm or smaller radius are electrical insulators. The term mesoscopic is sometimes used to refer to systems larger than nanoscopic but smaller than macro- scopic. Thus we have the progressively larger size levels: atomic → nanoscopic → mesoscopic → macroscopic. 1.2 THERMODYNAMICS Thermodynamics We begin our study of physical chemistry with thermodynamics. Thermodynamics (from the Greek words for “heat” and “power”) is the study of heat, work, energy, and the changes they produce in the states of systems. In a broader sense, thermodynamics studies the relationships between the macroscopic properties of a system. A key prop- erty in thermodynamics is temperature, and thermodynamics is sometimes defined as the study of the relation of temperature to the macroscopic properties of matter. We shall be studying equilibrium thermodynamics, which deals with systems in equilibrium. (Irreversible thermodynamics deals with nonequilibrium systems and rate processes.) Equilibrium thermodynamics is a macroscopic science and is inde- pendent of any theories of molecular structure. Strictly speaking, the word “molecule” is not part of the vocabulary of thermodynamics. However, we won’t adopt a purist attitude but will often use molecular concepts to help us understand thermodynamics. Thermodynamics does not apply to systems that contain only a few molecules; a sys- tem must contain a great many molecules for it to be treated thermodynamically. The term “thermodynamics” in this book will always mean equilibrium thermodynamics. Thermodynamic Systems The macroscopic part of the universe under study in thermodynamics is called the system. The parts of the universe that can interact with the system are called the surroundings. For example, to study the vapor pressure of water as a function of temperature, we might put a sealed container of water (with any air evacuated) in a constant-temperature bath and connect a manometer to the container to measure the pressure (Fig. 1.2). Here, the system consists of the liquid water and the water vapor in the container, and the surroundings are the constant-temperature bath and the mercury in the manometer. Figure 1.2 A thermodynamic system and its surroundings. lev38627_ch01.qxd 2/20/08 11:38 AM Page 4 4 Chapter 1 An open system is one where transfer of matter between system and surroundings Thermodynamics can occur. A closed system is one where no transfer of matter can occur between sys- tem and surroundings. An isolated system is one that does not interact in any way with its surroundings. An isolated system is obviously a closed system, but not every closed system is isolated. For example, in Fig. 1.2, the system of liquid water plus water vapor in the sealed container is closed (since no matter can enter or leave) but not isolated (since it can be warmed or cooled by the surrounding bath and can be compressed or expanded by the mercury). For an isolated system, neither matter nor energy can be transferred between system and surroundings. For a closed system, energy but not matter can be transferred between system and surroundings. For an open system, both matter and energy can be transferred between system and surroundings. A thermodynamic system is either open or closed and is either isolated or non- isolated. Most commonly, we shall deal with closed systems. Walls A system may be separated from its surroundings by various kinds of walls. (In Fig. 1.2, the system is separated from the bath by the container walls.) A wall can be either rigid or nonrigid (movable). A wall may be permeable or impermeable, where by “impermeable” we mean that it allows no matter to pass through it. Finally, a wall may be adiabatic or nonadiabatic. In plain language, an adiabatic wall is one that does not conduct heat at all, whereas a nonadiabatic wall does conduct heat. However, we have not yet defined heat, and hence to have a logically correct devel- opment of thermodynamics, adiabatic and nonadiabatic walls must be defined without reference to heat. This is done as follows. W Suppose we have two separate systems A and B, each of whose properties are ob- served to be constant with time. We then bring A and B into contact via a rigid, imper- meable wall (Fig. 1.3). If, no matter what the initial values of the properties of A and B A B are, we observe no change in the values of these properties (for example, pressures, vol- umes) with time, then the wall separating A and B is said to be adiabatic. If we gener- ally observe changes in the properties of A and B with time when they are brought in con- tact via a rigid, impermeable wall, then this wall is called nonadiabatic or thermally Figure 1.3 conducting. (As an aside, when two systems at different temperatures are brought in Systems A and B are separated by contact through a thermally conducting wall, heat flows from the hotter to the colder sys- a wall W. tem, thereby changing the temperatures and other properties of the two systems; with an adiabatic wall, any temperature difference is maintained. Since heat and temperature are still undefined, these remarks are logically out of place, but they have been included to clarify the definitions of adiabatic and thermally conducting walls.) An adiabatic wall is an idealization, but it can be approximated, for example, by the double walls of a Dewar flask or thermos bottle, which are separated by a near vacuum. In Fig. 1.2, the container walls are impermeable (to keep the system closed) and are thermally conducting (to allow the system’s temperature to be adjusted to that of the surrounding bath). The container walls are essentially rigid, but if the interface between the water vapor and the mercury in the manometer is considered to be a “wall,” then this wall is movable. We shall often deal with a system separated from its surroundings by a piston, which acts as a movable wall. A system surrounded by a rigid, impermeable, adiabatic wall cannot interact with the surroundings and is isolated. Equilibrium Equilibrium thermodynamics deals with systems in equilibrium. An isolated system is in equilibrium when its macroscopic properties remain constant with time. A non- isolated system is in equilibrium when the following two conditions hold: (a) The system’s macroscopic properties remain constant with time; (b) removal of the system lev38627_ch01.qxd 2/20/08 11:38 AM Page 5 5 from contact with its surroundings causes no change in the properties of the system. Section 1.2 If condition (a) holds but (b) does not hold, the system is in a steady state. An exam- Thermodynamics ple of a steady state is a metal rod in contact at one end with a large body at 50°C and in contact at the other end with a large body at 40°C. After enough time has elapsed, the metal rod satisfies condition (a); a uniform temperature gradient is set up along the rod. However, if we remove the rod from contact with its surroundings, the tempera- tures of its parts change until the whole rod is at 45°C. The equilibrium concept can be divided into the following three kinds of equilib- rium. For mechanical equilibrium, no unbalanced forces act on or within the system; hence the system undergoes no acceleration, and there is no turbulence within the sys- tem. For material equilibrium, no net chemical reactions are occurring in the system, nor is there any net transfer of matter from one part of the system to another or be- tween the system and its surroundings; the concentrations of the chemical species in the various parts of the system are constant in time. For thermal equilibrium between a system and its surroundings, there must be no change in the properties of the system or surroundings when they are separated by a thermally conducting wall. Likewise, we can insert a thermally conducting wall between two parts of a system to test whether the parts are in thermal equilibrium with each other. For thermodynamic equilibrium, all three kinds of equilibrium must be present. Thermodynamic Properties What properties does thermodynamics use to characterize a system in equilibrium? Clearly, the composition must be specified. This can be done by stating the mass of each chemical species that is present in each phase. The volume V is a property of the system. The pressure P is another thermodynamic variable. Pressure is defined as the magnitude of the perpendicular force per unit area exerted by the system on its sur- roundings: P ⬅ F>A (1.1)* where F is the magnitude of the perpendicular force exerted on a boundary wall of area A. The symbol ⬅ indicates a definition. An equation with a star after its number should be memorized. Pressure is a scalar, not a vector. For a system in mechanical equilibrium, the pressure throughout the system is uniform and equal to the pressure of the surroundings. (We are ignoring the effect of the earth’s gravitational field, which causes a slight increase in pressure as one goes from the top to the bottom of the sys- tem.) If external electric or magnetic fields act on the system, the field strengths are thermodynamic variables; we won’t consider systems with such fields. Later, further thermodynamic properties (for example, temperature, internal energy, entropy) will be defined. An extensive thermodynamic property is one whose value is equal to the sum of its values for the parts of the system. Thus, if we divide a system into parts, the mass of the system is the sum of the masses of the parts; mass is an extensive property. So is volume. An intensive thermodynamic property is one whose value does not depend on the size of the system, provided the system remains of macroscopic size—recall nanoscopic systems (Sec. 1.1). Density and pressure are examples of intensive prop- erties. We can take a drop of water or a swimming pool full of water, and both sys- tems will have the same density. If each intensive macroscopic property is constant throughout a system, the sys- tem is homogeneous. If a system is not homogeneous, it may consist of a number of homogeneous parts. A homogeneous part of a system is called a phase. For example, if the system consists of a crystal of AgBr in equilibrium with an aqueous solution of AgBr, the system has two phases: the solid AgBr and the solution. A phase can con- sist of several disconnected pieces. For example, in a system composed of several lev38627_ch01.qxd 2/20/08 11:38 AM Page 6 6 Chapter 1 AgBr crystals in equilibrium with an aqueous solution, all the crystals are part of the Thermodynamics same phase. Note that the definition of a phase does not mention solids, liquids, or gases. A system can be entirely liquid (or entirely solid) and still have more than one phase. For example, a system composed of the nearly immiscible liquids H2O and CCl4 has two phases. A system composed of the solids diamond and graphite has two phases. A system composed of two or more phases is heterogeneous. The density r (rho) of a phase of mass m and volume V is r ⬅ m>V (1.2)* Figure 1.4 plots some densities at room temperature and pressure. The symbols s, l, and g stand for solid, liquid, and gas. Suppose that the value of every thermodynamic property in a certain thermody- namic system equals the value of the corresponding property in a second system. The systems are then said to be in the same thermodynamic state. The state of a thermodynamic system is defined by specifying the values of its thermodynamic prop- erties. However, it is not necessary to specify all the properties to define the state. Specification of a certain minimum number of properties will fix the values of all other properties. For example, suppose we take 8.66 g of pure H2O at 1 atm (atmosphere) pressure and 24°C. It is found that in the absence of external fields all the remaining properties (volume, heat capacity, index of refraction, etc.) are fixed. (This statement ignores the possibility of surface effects, which are considered in Chapter 7.) Two thermodynamic systems each consisting of 8.66 g of H2O at 24°C and 1 atm are in the same thermodynamic state. Experiments show that, for a single-phase system con- taining specified fixed amounts of nonreacting substances, specification of two addi- tional thermodynamic properties is generally sufficient to determine the thermody- namic state, provided external fields are absent and surface effects are negligible. A thermodynamic system in a given equilibrium state has a particular value for each thermodynamic property. These properties are therefore also called state functions, since their values are functions of the system’s state. The value of a state function depends only on the present state of a system and not on its past history. It doesn’t matter whether we got the 8.66 g of water at 1 atm and 24°C by melting ice Figure 1.4 and warming the water or by condensing steam and cooling the water. Densities at 25°C and 1 atm. The scale is logarithmic. 1.3 TEMPERATURE Suppose two systems separated by a movable wall are in mechanical equilibrium with each other. Because we have mechanical equilibrium, no unbalanced forces act and each system exerts an equal and opposite force on the separating wall. Therefore each system exerts an equal pressure on this wall. Systems in mechanical equilibrium with each other have the same pressure. What about systems that are in thermal equilibrium (Sec. 1.2) with each other? Just as systems in mechanical equilibrium have a common pressure, it seems plausible that there is some thermodynamic property common to systems in thermal equilibrium. This property is what we define as the temperature, symbolized by u (theta). By definition, two systems in thermal equilibrium with each other have the same temper- ature; two systems not in thermal equilibrium have different temperatures. Although we have asserted the existence of temperature as a thermodynamic state function that determines whether or not thermal equilibrium exists between systems, we need experimental evidence that there really is such a state function. Suppose that we find systems A and B to be in thermal equilibrium with each other when brought in contact via a thermally conducting wall. Further suppose that we find systems B and lev38627_ch01.qxd 2/20/08 11:38 AM Page 7 7 C to be in thermal equilibrium with each other. By our definition of temperature, we Section 1.3 would assign the same temperature to A and B (uA ⫽ uB) and the same temperature to Temperature B and C (uB ⫽ uC). Therefore, systems A and C would have the same temperature (uA ⫽ uC), and we would expect to find A and C in thermal equilibrium when they are brought in contact via a thermally conducting wall. If A and C were not found to be in thermal equilibrium with each other, then our definition of temperature would be invalid. It is an experimental fact that: Two systems that are each found to be in thermal equilibrium with a third sys- tem will be found to be in thermal equilibrium with each other. This generalization from experience is the zeroth law of thermodynamics. It is so called because only after the first, second, and third laws of thermodynamics had been for- mulated was it realized that the zeroth law is needed for the development of thermody- namics. Moreover, a statement of the zeroth law logically precedes the other three. The zeroth law allows us to assert the existence of temperature as a state function. Having defined temperature, how do we measure it? Of course, you are familiar with the process of putting a liquid-mercury thermometer in contact with a system, waiting until the volume change of the mercury has ceased (indicating that thermal equilibrium between the thermometer and the system has been reached), and reading the thermometer scale. Let us analyze what is being done here. To set up a temperature scale, we pick a reference system r, which we call the thermometer. For simplicity, we choose r to be homogeneous with a fixed composi- tion and a fixed pressure. Furthermore, we require that the substance of the ther- mometer must always expand when heated. This requirement ensures that at fixed pressure the volume of the thermometer r will define the state of system r uniquely— two states of r with different volumes at fixed pressure will not be in thermal equilib- rium and must be assigned different temperatures. Liquid water is unsuitable for a thermometer since when heated at 1 atm, it contracts at temperatures below 4°C and expands above 4°C (Fig. 1.5). Water at 1 atm and 3°C has the same volume as water at 1 atm and 5°C, so the volume of water cannot be used to measure temperature. Liquid mercury always expands when heated, so let us choose a fixed amount of liquid mercury at 1 atm pressure as our thermometer. We now assign a different numerical value of the temperature u to each different volume Vr of the thermometer r. The way we do this is arbitrary. The simplest approach is to take u as a linear function of Vr. We therefore define the temperature to be u ⬅ aVr ⫹ b, where Vr is the volume of a fixed amount of liquid mercury at 1 atm Figure 1.5 pressure and a and b are constants, with a being positive (so that states which are ex- perienced physiologically as being hotter will have larger u values). Once a and b are Volume of 1 g of water at 1 atm specified, a measurement of the thermometer’s volume Vr gives its temperature u. versus temperature. Below 0°C, the water is supercooled (Sec. 7.4). The mercury for our thermometer is placed in a glass container that consists of a bulb connected to a narrow tube. Let the cross-sectional area of the tube be A, and let the mercury rise to a length l in the tube. The mercury volume equals the sum of the mercury volumes in the bulb and the tube, so u ⬅ aVr ⫹ b ⫽ a1Vbulb ⫹ Al 2 ⫹ b ⫽ aAl ⫹ 1aVbulb ⫹ b 2 ⬅ cl ⫹ d (1.3) where c and d are constants defined as c ⬅ aA and d ⬅ aVbulb ⫹ b. To fix c and d, we define the temperature of equilibrium between pure ice and liq- uid water saturated with dissolved air at 1 atm pressure as 0°C (for centigrade), and we define the temperature of equilibrium between pure liquid water and water vapor at 1 atm pressure (the normal boiling point of water) as 100°C. These points are called the ice point and the steam point. Since our scale is linear with the length of the mer- cury column, we mark off 100 equal intervals between 0°C and 100°C and extend the marks above and below these temperatures. lev38627_ch01.qxd 2/20/08 11:38 AM Page 8 8 Chapter 1 Having armed ourselves with a thermometer, we can now find the temperature of Thermodynamics any system B. To do so, we put system B in contact with the thermometer, wait until thermal equilibrium is achieved, and then read the thermometer’s temperature from the graduated scale. Since B is in thermal equilibrium with the thermometer, B’s tem- perature equals that of the th