This page intentionally left blank ATMOSPHERIC THERMODYNAMICS Elementary Physics and Chemistry This textbook presents a uniquely integrated approach in linking both physics and chemistry to the study of atmospheric thermodynamics. The book begins by explaining the classical laws of thermodynamics, and discusses Gibbs energy and the elementary kinetic theory of gases with special applications to the atmosphere. Individual chapters focus on various fluid systems, including vapor pressure over flat and curved surfaces of pure liquids and solutions, and examine the vertical dependence of temperature and pressure for environmental sounding and moving air parcels. Recognizing the increasing importance of chemistry in the meteorological and climate sciences, a chapter is devoted to chemical thermodynamics and contains an overview of photochemistry. Although students are expected to have some background knowledge of calculus, general chemistry and classical physics, the book provides set-aside refresher boxes as useful reminders. It contains over 100 diagrams and graphs to supplement the discussions. It also contains a similar number of worked examples and exercises, with solutions included at the end of the book. It is ideal for a single-semester advanced course on atmospheric thermodynamics, and will prepare students for higher-level synoptic and dynamics courses. gerald r. north received a Ph.D. in Physics from the University of Wisconsin in 1966, and has been a Distinguished Professor of Atmospheric Sciences at Texas A&M University for over 20 years. His notable research career includes receiving the Outstanding Publication Award, National Center for Atmospheric Research in 1975, the Exceptional Scientific Achievement Medal for NASA in 1983, and the Jule G. Charney Award from the American Meteorological Society in 2008. tatiana l. erukhimova received a Ph.D. in Physics from the Institute of Applied Physics, Russian Academy of Sciences, in 1999, and is now a Lecturer in the Department of Physics at Texas A&M University. Her areas of research include large-scale and mesoscale atmospheric transport and mixing, atmospheric wave dynamics, atmospheric ozone, and remote sensing. ATMOSPHERIC THERMODYNAMICS Elementary Physics and Chemistry GERALD R. NORTH AND TATIANA L. ERUKHIMOVA Texas A & M University, USA CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521899635 © G. North and T. Erukhimova 2009 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2009 ISBN-13 978-0-511-51677-1 eBook (EBL) ISBN-13 978-0-521-89963-5 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. We both wish to dedicate this book to our families, who have kindly endured our long preoccupation with the project. Contents Preface page xi 1 Introductory concepts 1 1.1 Units 2 1.2 Earth, weight and mass 3 1.3 Systems and equilibrium 7 1.4 Constraints 12 1.5 Intensive and extensive quantities 13 1.6 System boundaries 14 1.7 Thermodynamics and atmospheric science 16 Notes 17 Notation and abbreviations 18 Problems 18 2 Gases 20 2.1 Ideal gas basics 20 2.2 Distribution of velocities 29 2.3 Flux of molecules striking a wall 35 2.4 Moles, etc. 36 2.5 Dalton’s Law 39 Notes 41 Notation and abbreviations 41 Problems 42 3 The First Law of Thermodynamics 44 3.1 Reversible and irreversible work 47 3.2 Heating a system 48 3.3 Ideal gas results 51 3.4 Enthalpy 61 3.5 Standard enthalpy of fusion and vaporization 63 vii viii Contents Notes 65 Notation and abbreviations 65 Problems 66 4 The Second Law of Thermodynamics 69 4.1 Entropy 73 4.2 The Second Law of Thermodynamics 76 4.3 Systems and reversibility 77 4.4 Additivity of entropy 78 4.5 Extremum principle 82 4.6 Entropy summary 87 4.7 Criteria for equilibrium 88 4.8 Gibbs energy 89 4.9 Multiple components 93 Notes 95 Notation and abbreviations 96 Problems 96 5 Air and water 99 5.1 Vapor pressure 99 5.2 Saturation vapor pressure 102 5.3 Van der Waals equation 103 5.4 Multiple phase systems 105 5.5 Phase boundaries 107 5.6 Clausius–Clapeyron equation 109 5.7 Integration of the Clausius–Clapeyron equation 110 5.8 Mixing air and water 112 5.9 Wet-bulb temperature, LCL 117 5.10 Equilibrium vapor pressure over a curved surface 121 5.11 Isobaric mixing of air parcels 130 Notes 131 Notation and abbreviations 131 Problems 132 6 Profiles of the atmosphere 134 6.1 Pressure versus height 134 6.2 Slope of the dry adiabat 138 6.3 Geopotential height and thickness 141 6.4 Archimedes’ Principle 143 6.5 Stability 145 6.6 Vertical oscillations 147 Contents ix 6.7 Where is the LCL? 149 6.8 Slope of a moist adiabat 152 6.9 Lifting moist air 154 6.10 Moist static energy 158 6.11 Profiles of well-mixed layers 158 Notes 161 Notation and abbreviations 161 Problems 162 7Thermodynamiccharts163 7.1 Areas and energy 164 7.2 Skew T diagram 166 7.3 Chart exercises 170 7.4 Stability problem: example sounding 178 7.5 Convective available potential energy (CAPE) 181 Notation and abbreviations 186 Problems 186 8 Thermochemistry 191 8.1 Standard enthalpy of formation 192 8.2 Photochemistry 194 8.3 Gibbs energy for chemical reactions 199 8.4 Elementary kinetics 201 8.5 Equilibrium constant 206 8.6 Solutions 211 Notes 217 Notation and abbreviations 217 Problems 218 9 The thermodynamic equation 220 9.1 Scalar and vector fields 225 9.2 Pressure gradient force 228 9.3 Surface integrals and flux 230 9.4 Conduction of heat 232 9.5 Two-dimensional divergence 234 9.6 Three-dimensional divergence 237 9.7 Divergence theorem 239 9.8 Continuity equation 240 9.9 Material derivative 242 9.10 Thermodynamic equation 243 9.11 Potential temperature form 245 x Contents 9.12 Contributions to DQM/Dt 245 Notes 246 Notation and abbreviations 246 Problems 247 Appendix A Units and numerical values of constants 249 Appendix B Notation and abbreviations 252 Appendix CAnswers for selected problems 258 Bibliography 263 Index 265 Preface This book is intended as a text for undergraduates in the atmospheric sciences. The students are expected to have some calculus, general chemistry and classical physics background although we provide a number of refreshers for those who might have less experience or need reminders. Our students have also had a survey of the atmospheric sciences in a qualitative course at freshman level. The primary aim of the book is to prepare the student for the synoptic and dynamics courses that follow. We intend that the student gain some understanding of thermodynamics as it applies to the elementary systems of interest in the atmospheric sciences. A major goal is for the students to gain some facility in making straightforward calculations. We have taught the material in a semester course, but in a shorter course some material can be omitted without regrets later in the book. The book ends with two chapters that are independent of one another: Chapter 8 on thermochemistry and Chapter 9 on the thermodynamic equation. This book is the result of teaching an introductory atmospheric thermodynamics course to sophomores and juniors at Texas A&M University. Several colleagues have taught the course using earlier versions of the notes and we gratefully acknowledge Professors R. L. Panetta, Ping Yang, and Don Collins as well as the students for their many helpful comments. In addition, we have received useful comments on the chemistry chapter from Professors Sarah Brooks, Gunnar Schade, and Renyi Zhang. We also thank Professor Kenneth Bowman for many fruitful discussions. We are grateful for financial support provided by the Harold J. Haynes Endowed Chair in Geosciences. xi 1 Introductory concepts The atmosphere is a compressible fluid, and the description of such a form of matter is usually unfamiliar to students who are just completing calculus and classical mechanics as part of a standard university physics course. To complicate matters the atmosphere is composed of not just a single ingredient, but several ingredients, including different (mostly nonreactive) gases and particles in suspension (aerosols). Some of the ingredients change phase (primarily water) and there is an accompanying exchange of energy with the environment. The atmosphere also interacts with its lower boundary which acts as a source (and sink) of friction, thermal energy, water vapor, and various chemical species. Electromagnetic radiation enters and leaves the atmosphere and in so doing it warms and cools layers of air, interacting selectively with different constituents in different wavelength bands. Meteorology is concerned with describing the present state of the atmosphere (temperature, pressure, winds, humidity, precipitation, cloud cover, etc.) and in predicting the evolution of these primary variables over time intervals of a few days. The broader field of atmospheric science is concerned with additional themes such as climate (statistical summaries of weather), air chemistry (its present, future, and history), atmospheric electricity, atmospheric optics (across all wavelengths), aerosols and cloud physics. Both the present state of the bulk atmosphere and its evolution are determined by Newton’s laws of mechanics as they apply to such a compressible fluid.