Table Of Content

Equilibrium Between Phases of Matter Equilibrium Between Phases of Matter Phenomenology and Thermodynamics H.A.J. Oonk Utrecht University, The Netherlands and M.T. Calvet University of Barcelona, Spain AC.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress. ISBN978-1-4020-6123-3(HB) ISBN978-1-4020-6408-1(e-book) PublishedbySpringer, P.O.Box17,3300AADordrecht,TheNetherlands. www.springer.com Printedonacid-freepaper AllRightsReserved ©2008Springer Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedin anyformorbyanymeans,electronic,mechanical,photocopying,microfilming,recording orotherwise,withoutwrittenpermissionfromthePublisher,withtheexception ofanymaterialsuppliedspecificallyforthepurposeofbeingentered andexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. TABLE OF CONTENTS INTRODUCTION vii LIST OF FREQUENTLY USED SYMBOLS xi LEVEL 0: an introduction to phase diagrams § 001 equilibrium 3 § 002 variables 10 § 003 the rules of the game 19 § 004 pure substances 28 § 005 binary and ternary systems 41 § 006 distribution and separation 61 § 007 chemical equilibrium 76 LEVEL 1: an introduction to thermodynamics and phase theory § 101 differential expressions 89 § 102 work heat energy 93 § 103 heat capacity and enthalpy 97 § 104 the ideal gas, expansion and compression 101 § 105 chemical energy 107 § 106 entropy 113 § 107 characteristic functions 121 § 108 Gibbs energy and equilibrium 128 § 109 data and tables 139 § 110 pure substances 151 § 111 chemical reactions and equilibrium 168 vi Table of contents LEVEL 2: phase theory: the thermodynamics of equilibrium between phases § 201 mixtures and partial quantities 183 § 202 the open system, chemical potentials 190 § 203 change to molar quantities, molar Gibbs energy 197 § 204 the ideal mixture 206 § 205 non-ideal behaviour, excess functions 211 § 206 magic formulae 222 § 207 dilute solutions 231 § 208 the solvent laws 237 § 209 the solute laws 245 § 210 ideal equilibria 252 § 211 non-ideal systems - geometrically 263 § 212 non-ideal systems - analytically 279 § 213 non-ideal systems - numerically 299 SOLUTIONS OF EXERCISES level 0 319 level 1 330 level 2 356 REFERENCES 377 SUBJECT INDEX 389 SUBSTANCES AND SYSTEMS INDEX 401 INTRODUCTION About the book, the project Equilibrium Between Phases of Matter – Phenomenology and Thermodynamics is a textbook, in which the phenomenology, the thermodynamic theory, and the practical use of phase diagrams are presented in three levels that diverge in nature – in particular as regards the role of thermodynamics. The book has been written from a chemical and geological teaching background. Each of the three levels of the book is representative of a particular course in a curriculum. Level 0: an introduction to phase diagrams The philosophy behind the ground level is that most of the characteristics of equilibrium between phases can be understood without the use of thermodynamics, realizing that, in a common-sense manner, the experimental observations on equilibria and spontaneous changes, and elementary notions about interactions, indicate the way to go. In spite of all this, the central figure in level zero, right from the beginning, is the chemical potential – a concept firmly rooted in thermodynamics. Equilibrium conditions in terms of chemical potentials, and the variables necessary to define a system in equilibrium are, are the basic elements of the system formulation. The first three sections deal with the characteristics of a system in equilibrium; the variables necessary to define the system; the conditions for equilibrium and the rules – the phase rule and the lever rule. Pure substances and their forms are the subject of the fourth section. In the fifth section the phenomenology of binary phase diagrams is developed in terms of the nature of the interaction between the entities of the components; and, subsequently ternary phase diagrams are developed from the diagrams of the binary subsystems. Much of the use of binary phase diagrams is based on the fact that phases in equilibrium have different compositions – sixth section. The last section is devoted to homogeneous and heterogeneous chemical equilibria. Level 1: an introduction to thermodynamics and phase theory The intermediate level is an introduction to classical thermodynamics - culminating in the Gibbs energy as the arbiter in equilibrium matters - followed by the thermodynamic treatment of equilibrium between phases for a number of elementary cases. The thermodynamics part starts with differential expressions; work, heat, and energy; heat capacity and enthalpy. After the ideal gas and its expansion and compression; and a section on chemical energy; entropy and Gibbs energy make their appearance – in particular their role in spontaneous changes and matters of equilibrium. In the last sections the principles of equilibrium are applied to pure substances – where molar Gibbs energy has the status of chemical potential – and to homogeneous and heterogeneous chemical equilibria. viii Introduction Level 2: phase theory: the thermodynamics of equilibrium between phases In the upper level the step is made to mixtures with their ‘puzzling’ partial molar properties, revealing the identity between chemical potential and partial Gibbs energy. After the introduction of ideal mixtures, a ‘magic formula’ for the Gibbs energy of mixing is used to explain the phenomenon of demixing, and to derive the properties of the ideal dilute solution. Ideality and non-ideality, in a certain sense, give rise to the existence of two sub-levels. In the first sub-level the function recipes of the chemical potentials are such that explicit relationships can be formulated for systems where the phases are pure substances and/or ideal mixtures or ideal dilute solutions. The second sub-level, the last part of the work is devoted to systems where the phases are non-ideal mixtures, whether or not, in combination with phases of fixed composition – and where the leading role of partial Gibbs energy, read the chemical potential, is taken over by the integral Gibbs energy. A priori and a posteriori To assist the reader in assessing her/his own level of understanding the ins and outs of equilibrium between phases, each of the sections is provided with an a priori, in which the subject is introduced, and an a posteriori, in which the main conclusions are summarized. Besides, owing to the level structure of the work, there is a certain, small overlap between parts of different levels. Or, in other terms, some overlap has not been kept out of the authors their way. Exercises Exercises are given at the end of each section, and the solutions of most of them at the end of the book. Their main function, of course, is to interest and stimulate the reader or a group of students in the class room. Occasionally, exercises are used as a vehicle to introduce issues that are not treated in the main text. Follow-up Equilibrium Between Phases of Matter – Phenomenology and Thermodynamics, is the first volume of a two-volume project. The second volume, by M.H.G. Jacobs and H.A.J. Oonk, Equilibrium Between Phases of Matter – Thermodynamic Analysis and Prediction, more than the first, has a postgraduate and professional character – and is written from a research background in materials science and geophysics. In terms of thermodynamic properties, the distinction between the two volumes is in the role of the temperature and pressure derivatives of the Gibbs energy. In the first volume a dominating role is played by the Gibbs energy itself and its first derivatives, which are entropy and volume. The three together account for most of the phenomenology of equilibrium between phases – but not all of it; and one Introduction ix can think of retrograde phenomena. And when it comes to the real thermophysical properties of a given system, and the prediction of its behaviour under circumstances far from ambient temperature and pressure, one is completely lost without the incorporation/availability of the second and higher- order derivatives. About languages Writing a text in a foreign language is a delicate undertaking – in the sense that one cannot express oneself as clearly and linguistically correctly as one would like to. It is also extra time-consuming: the time needed to consult dictionaries and books of synonyms is commensurate with the time needed for the real work. The manuscript of the senior author’s 1981 book Phase Theory was read and corrected by Dr Philip Spencer, who was rather satisfied with the English. On the other hand, an American reviewer of the book, found the English somewhat clumsy. The authors of this book have been working together for more than twenty years as participants in the REALM (Réseau Européen sur les Alliages Moléculaires) – a European network on molecular alloys, as may be clear. French is the REALM’s official language – in the family circle one of the authors speaks Catalan, and the other Dutch. Anyhow, dear reader, the message is clear. Please, enjoy the science and wink at the linguistic shortcomings. Not only communication in terms of real languages, but also the communication between thermodynamicists from different branches of materials science has its typical difficulties. Just to give an idea, in certain fields a prominent role is played by activities and fugacities, whereas in other fields these terms virtually are non- existent. And, of course, the relationship a physical chemist has with thermodynamics is quite different from the one built up by a geologist. A chemist creates his own systems out of pure chemicals, and having the phases of his own choice. The geologist, on the other hand, focuses his attention to stones created by nature and often having a multitude of phases, which, sometimes, reflect the history of the material. The first volume of Equilibrium Between Phases of Matter has been written in the tradition of the Dutch School - in the footsteps of Josiah Willard Gibbs’s disciples J.D. van der Waals, H.W. Bakhuis Roozeboom, F.A.H. Schreinemakers, J.J. van Laar, J.L. Meijering, and others. In this tradition a prominent role is given to chemical potentials – and, when it comes to deal with deviation from ideal behaviour, the use of activities and fugacities is avoided. x Introduction Acknowledgements We should like to take the opportunity of expressing our gratitude to Mercedes Aguilar, Jan den Boesterd, Ingrid van Rooijen, and Aloys Lurvink for fine artwork; to Petra van Steenbergen, Hermine Vloemans, and Tamsin Kent for most valuable suggestions related to publishing; and to Koos Blok for thorough bibliographic research. We are grateful to our students, who, through their remarks and probing questions, have kept us alert – thermodynamics is tricky business after all. Dedication We dedicate this book to our spouses and the other members of our families, and to our friends, who all, every now and then, remind us of the fact that equilibrium between faces is also a matter of great importance. Utrecht, The Netherlands, Harry A.J. Oonk Barcelona, Spain M. Teresa Calvet Spring 2007 LIST OF FREQUENTLY USED SYMBOLS Latin letters A Helmholtz energy; magnitude parameter of AB(cid:300) model A first component in binary and ternary system α activity a, b, c system-dependent parameters B general for substance; second component in binary and ternary system B asymmetry parameter in AB(cid:300) model C third component in ternary system c number of components d ordinary differential C heat capacity at constant pressure P C heat capacity at constant volume V E electromotive force e electric charge f variance, number of degrees of freedom; fugacity; activity coefficient G Gibbs (free) energy g acceleration of free fall; parameter in excess Gibbs energy H enthalpy h parameter in excess enthalpy; altitude ln natural logarithm LN defined as LN(X) = (1-X)ln(1-X) + XlnX M set of variables necessary to define an equilibrium system, and number of elements in it; molar mass m mass; molality N set of equilibrium conditions, and number of elements in it N Avogadro’s number Av n amount of substance P pressure p number of phases Q, q heat R gas constant S entropy s parameter in excess entropy T thermodynamic temperature t Celsius temperature U energy V volume W, w work W number of configurations X variable in general; mole fraction; mole fraction of second component