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Understanding Non-equilibrium Thermodynamics G. Lebon • D. Jou • J. Casas-Vázquez Understanding Non-equilibrium Thermodynamics Foundations, Applications, Frontiers 123 Prof. Dr. José Casas-Vázquez Universitat Autònoma de Barcelona Dept. Fisica - Edifici Cc Grup Fisica Estadistica Bellaterra 08193 Catalonia, Spain Jose.Casas@uab.es Prof. Dr. David Jou Universitat Autònoma de Barcelona Dept. Fisica - Edifici Cc Grup Fisica Estadistica Bellaterra 08193 Catalonia, Spain David.Jou@uab.es Prof. Dr. Georgy Lebon Université de Liège Dept. d’Astrophysique Géophysique et Océanographie B5 Sart Tilman Liege 4000 Belgium g.lebon@ulg.ac.be Cover image: Thermal radiation leaving the Earth, seen by the EOS-Terra satellite (NASA). Image from www.visibleearth.nasa.gov. Owner: NASA. ISBN: 978-3-540-74251-7 e-ISBN: 978-3-540-74252-4 Library of Congress Control Number: 2007935452 c 2008 Springer-Verlag Berlin Heidelberg
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Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com Preface Our time is characterized by an explosion of information and by an acceleration of knowledge. A book cannot compete with the huge amount of data available on the Web. However, to assimilate all this information, it is necessary to structure our knowledge in a useful conceptual framework. The purpose of the present work is to provide such a structure for students and researchers interested by the current state of the art of non-equilibrium thermodynamics. The main features of the book are a concise and critical presentation of the basic ideas, illustrated by a series of examples, selected not only for their pedagogical value but also for the perspectives offered by recent technological advances. This book is aimed at students and researchers in physics, chemistry, engineering, material sciences, and biology. We have been guided by two apparently antagonistic objectives: generality and simplicity. To make the book accessible to a large audience of nonspecialists, we have decided about a simplified but rigorous presentation. Emphasis is put on the underlying physical background without sacrificing mathematical rigour, the several formalisms being illustrated by a list of examples and problems. All over this work, we have been guided by the formula: “Get the more from the less”, with the purpose to make a maximum of people aware of a maximum of knowledge from a minimum of basic tools. Besides being an introductory text, our objective is to present an overview, as general as possible, of the more recent developments in non-equilibrium thermodynamics, especially beyond the local equilibrium description. This is partially a terra incognita, an unknown land, because basic concepts as temperature, entropy, and the validity of the second law become problematic beyond the local equilibrium hypothesis. The answers provided up to now must be considered as partial and provisional, but are nevertheless worth to be examined. Chapters 1 and 2 are introductory chapters in which the main concepts underlying equilibrium thermodynamics and classical non-equilibrium thermodynamics are stated. The basic notions are discussed with special emphasis on these needed later in this book. V VI Preface Several applications of classical non-equilibrium thermodynamics are presented in Chaps. 3 and 4. These illustrations have not been chosen arbitrarily, but keeping in mind the perspectives opened by recent technological advancements. For instance, advances in material sciences have led to promising possibilities for thermoelectric devices; localized intense laser heating used to make easier the separation of molecules has contributed to a revival of interest in thermodiffusion; chemical reactions are of special interest in biology, in relation with their coupling with active transport across membranes and recent developments of molecular motors. The purpose of Chaps. 5 and 6 is to discuss two particular aspects of classical non-equilibrium thermodynamics which have been the subject of active research during the last decades. Chapter 5 is devoted to finite-time thermodynamics whose main concern is the competition between maximum efficiency and maximum power and its impact on economy and ecology. This classical subject is treated here in an updated form, taking into account the last technological possibilities and challenges, as well as some social concerns. Chapter 6 deals with instabilities and pattern formation; organized structures occur in closed and open systems as a consequence of fluctuations growing far from equilibrium under the action of external forces. Patterns are observed in a multitude of our daily life experiences, like in hydrodynamics, biology, chemistry, electricity, material sciences, or geology. After introducing the mathematical theory of stability, several examples of ordered structures are analysed with a special attention to the celebrated Bénard cells. Chapters 1–6 may provide a self-consistent basis for a graduate introductory course in non-equilibrium thermodynamics. In the remainder of the book, we go beyond the framework of the classical description and spend some time to address and compare the most recent developments in non-equilibrium thermodynamics. Chapters 7–11 will be of interest for students and researchers, who feel attracted by new scientific projects wherein they may be involved. This second part of the book may provide the basis for an advanced graduate or even postgraduate course on the several trends in contemporary thermodynamics. The coexistence of several schools in non-equilibrium thermodynamics is a reality; it is not a surprise in view of the complexity of most macroscopic systems and the fact that some basic notions as temperature and entropy are not univocally defined outside equilibrium. To appreciate this form of multiculturalism in a positive sense, it is obviously necessary to know what are the foundations of these theories and to which extent they are related. A superficial inspection reveals that some viewpoints are overlapping but none of them is rigorously equivalent to the other. A detailed and complete understanding of the relationship among the diverse schools turns out to be not an easy task. The first difficulty stems from the fact that each approach is associated with a certain insight, we may even say an intuition or feeling that is sometimes rather difficult to apprehend. Also some unavoidable differences in the terminology and the notation do not facilitate the communication. Another Preface VII factor that contributes to the difficulty to reaching a mutual comprehension is that the schools are not frozen in time: they evolve as a consequence of internal dynamics and by contact with others. Our goal is to contribute to a better understanding among the different schools by discussing their main concepts, results, advantages, and limitations. Comparison of different viewpoints may be helpful for a deeper comprehension and a possible synthesis of the many faces of the theory. Such a comparative study is not found in other textbooks. One problem was the selection of the main representative ones among the wealth of thermodynamic formalisms. Here we have focused our attention on five of them: extended thermodynamics (Chap. 7), theories with internal variables (Chap. 8), rational thermodynamics (Chap. 9), Hamiltonian formulation (Chap. 10), and mesoscopic approaches (Chap.11). In each of them, we have tried to save the particular spirit of each theory. It is clear that our choice is subjective: we have nevertheless been guided not only by the pedagogical aspect and/or the impact and universality of the different formalisms, but also by the fact that we had to restrict ourselves. Moreover, it is our belief that a good comprehension of these different versions allows for a better and more understandable comprehension of theories whose opportunity was not offered to be discussed here. The common points shared by the theories presented in Chaps. 7–11 are not only to get rid of the local equilibrium hypothesis, which is the pillar of the classical theory, but also to propose new phenomenological approaches involving non-linearities, memory and non-local effects, with the purpose to account for the technological requirements of faster processes and more miniaturized devices. It could be surprising that the book is completely devoted to macroscopic and mesoscopic aspects and that microscopic theories have been widely omitted. The reasons are that many excellent treatises have been written on microscopic theories and that we decided to keep the volume of the book to a reasonable ratio. Although statistical mechanics appears to be more fashionable than thermodynamics in the eyes of some people and the developments of microscopic methods are challenging, we hope to convince the reader that macroscopic approaches, like thermodynamics, deserve a careful attention and are the seeds of the progress of knowledge. Notwithstanding, we remain convinced that, within the perspectives of improvement and unification, it is highly desirable to include as many microscopic results as possible into the macroscopic framework. Chapters 7–11 are autonomous and self-consistent, they have been structured in such a way that they can be read independently of each other and in arbitrary order. However, it is highly recommended to browse through all the chapters to better apprehend the essence and the complementarity of the diverse theories. At the end of each chapter is given a list of problems. The aim is not only to allow the reader to check his understanding, but also to stimulate his interest to solve concrete situations. Some of these problems have been VIII Preface inspired by recent papers, which are mentioned, and which may be consulted for further investigation. More technical and advanced parts are confined in boxes and can be omitted during a first reading. We acknowledge many colleagues, and in particular M. Grmela (Montreal University), P.C Dauby and Th. Desaive (Liège University), for the discussions on these and related topics for more than 30 years. We also appreciate our close collaborators for their help and stimulus in research and teaching. Drs. Vicenç Méndez and Vicente Ortega-Cejas deserve special gratitude for their help in the technical preparation of this book. We also acknowledge the finantial support of the Dirección General de Investigación of the Spanish Ministry of Education under grants BFM2003-06003 and FIS200612296-C02-01, and of the Direcció General de Recerca of the Generalitat of Catalonia, under grants 2001 SGR 00186 and 2005 SGR 00087. Liège-Bellaterra, March 2007 G. Lebon, D. Jou, J. Casas-Vázquez Contents 1 Equilibrium Thermodynamics: A Review . . . . . . . . . . . . . . . . . 1.1 The Early History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Scope and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Fundamental Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Zeroth Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The First Law or Energy Balance . . . . . . . . . . . . . . . . . . 1.3.3 The Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 The Third Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Gibbs’ Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Fundamental Relations and State Equations . . . . . . . . . 1.4.2 Euler’s Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Gibbs–Duhem’s Relation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Some Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 The Basic Problem of Equilibrium Thermodynamics . . 1.5 Legendre Transformations and Thermodynamic Potentials . . . 1.5.1 Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Thermodynamic Potentials and Extremum Principles . 1.6 Stability of Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Stability of Single Component Systems . . . . . . . . . . . . . . 1.6.2 Stability Conditions for the Other Thermodynamic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Stability Criterion of Multi-Component Mixtures . . . . . 1.7 Equilibrium Chemical Thermodynamics . . . . . . . . . . . . . . . . . . . 1.7.1 General Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . 1.7.2 Heat of Reaction and van’t Hoff Relation . . . . . . . . . . . . 1.7.3 Stability of Chemical Equilibrium and Le Chatelier’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 5 5 6 8 14 14 15 16 16 17 18 19 20 21 24 24 27 27 29 30 31 32 34 34 IX X 2 Contents Classical Irreversible Thermodynamics . . . . . . . . . . . . . . . . . . . 2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Local Equilibrium Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Entropy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Minimum Entropy Production Principle . . . . . . . . . . . . . 2.6 Applications to Heat Conduction, Mass Transport, and Fluid Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Heat Conduction in a Rigid Body . . . . . . . . . . . . . . . . . . 2.6.2 Matter Diffusion Under Isothermal and Isobaric Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Limitations of the Classical Theory of Irreversible Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 38 39 41 44 50 51 3 Coupled Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Electrical Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Thermoelectric Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Phenomenological Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Efficiency of Thermoelectric Generators . . . . . . . . . . . . . 3.3 Thermodiffusion: Coupling of Heat and Mass Transport . . . . . 3.4 Diffusion Through a Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Entropy Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Phenomenological Relations . . . . . . . . . . . . . . . . . . . . . . . 3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 70 72 72 76 79 83 83 85 87 4 Chemical Reactions and Molecular Machines . . . . . . . . . . . . . 4.1 One Single Chemical Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Coupled Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 General Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Cyclical Chemical Reactions and Onsager’s Reciprocal Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Efficiency of Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Chemical Reactions and Mass Transport: Molecular Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Autocatalytic Reactions and Diffusion: Morphogenesis . . . . . . 4.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 92 96 96 102 108 109 Finite-Time Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Finite-Time Carnot Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Curzon–Ahlborn’s Model: Heat Losses . . . . . . . . . . . . . . 5.1.2 Friction Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Economical and Ecological Constraints . . . . . . . . . . . . . . . . . . . . 113 114 115 120 122 5 54 54 59 60 63 65 97 100