Phase Transformations

Transcription

1 CHEMICAL ENGINEERING SERIES CHEMICAL THERMODYNAMICS SET Volume 5 Phase Transformations Michel Soustelle

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3 Phase Transformations

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5 Chemical Thermodynamics Set coordinated by Michel Soustelle Volume 5 Phase Transformations Michel Soustelle isle WILEY

6 First published 5 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 7-37 St George's Road London S W 9 4EU UK John Wiley & Sons, Inc. III River Street Hoboken, NJ 73 USA ISTE Ltd 5 The rights of Michel Soustelle to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 988. Library of Congress Control Number: British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN

7 Contents Preface... Notations and Symbols... xi xv Chapter. Phase Transformations of Pure Substances..... Standard state: standard conditions of a transformation..... Classification and general properties of phase transformations First-order transformations and the Clapeyron relation Second-order transformations Liquid vapor transformations and equilibrium states Method of two equations of state, using the Clapeyron equation Gibbs energy and fugacity method Unique equation of state method The region of the critical point and spinodal decomposition Microscopic modeling Liquid vapor equilibrium in the presence of an inert gas Solid vapor transformations and equilibriums Macroscopic treatment Microscopic treatment Transformations and solid liquid equilibria Macroscopic treatment Microscopic treatment... 3

8 vi Phase Transformations.6. Diagram for the pure substance and properties of the triple point Allotropic and polymorphic varieties of a solid Enantiotropy Monotropy Transition from enantiotropy to monotropy and vice versa Mesomorphic states... 4 Chapter. Properties of Equilibria Between Binary Phases Classification of equilibria between the phases of binary systems General properties of two-phase binary systems Equilibrium conditions for two-phase binary systems Conditions of evolution of a two-phase binary system Graphical representation of two-phase binary systems Gibbs energy graphs Phase diagram in the mono- and bi-phase zones Isobaric cooling curves Isobaric representation of three-phase binary systems Gibbs energy curve Isobaric phase diagram in tri-phase regions Isobaric cooling curves with tri-phase zones Isothermal phase diagrams Composition/composition curves Activity of the components and consequences of Raoult s and Henry s laws Chapter 3. Equilibria Between Binary Condensed Phases Equilibria between phases of the same nature: liquid liquid or solid solid Thermodynamics of demixing Demixing in the case of low reciprocal solubilities Demixing of strictly-regular solutions Liquid solid systems... 84

9 Contents vii 3... Thermodynamics of the equilibria between a liquid phase and a solid phase Isobaric phase diagrams of equilibria between a solid and a liquid Solidus and liquidus in the vicinity of the pure substance Equilibria between two solids with two polymorphic varieties of the solid Applications of solid liquid equilibria Solubility of a solid in a liquid: Schröder Le Châtelier law Determination of molar mass by cryometry Membrane equilibria osmotic pressure Thermodynamics of osmotic pressure Osmotic pressure of infinitely-dilute solutions: the Van t the Hoff law Application of osmotic pressure to the determination of the molar mass of polymers Osmotic pressure of strictly-regular solutions Osmotic pressure and the osmotic coefficient... Chapter 4. Equilibria Between Binary Fluid Phases Thermodynamics of liquid vapor equilibrium in a binary system Liquid vapor equilibrium in perfect solutions far from the critical conditions Partial pressures and total pressure of a perfect solution Isothermal diagram of a perfect solution Isobaric diagram of a perfect solution Phase composition curve Liquid gas equilibria in ideal dilute solutions Diagrams of the liquid vapor equilibria in real solutions Total miscibility in the liquid phase Partial miscibility in the liquid phase, heteroazeotropes Thermodynamics of liquid vapor azeotropy Relation between the pressure of the azeotrope and the activity coefficients of the liquid phase at the azeotropic composition... 9

10 viii Phase Transformations Relation between the activity coefficient and the temperature of the azeotrope Liquid vapor equilibria and models of solutions Liquid vapor equilibria in strictly-regular solutions Liquid vapor equilibrium in associated solutions Liquid vapor equilibria in the critical region Applications of liquid vapor equilibria Solubility of a gas in a liquid Determination of molar masses by tonometry Determination of molar masses by ebulliometry Continuous rectification or fractional distillation Chapter 5. Equilibria Between Ternary Fluid Phases Representation of the composition of ternary systems Symmetrical representation of the Gibbs triangle Dissymmetrical representation of the right triangle Representation of phase equilibria Isothermal projections Conjugate points and conodes Isopleth sections Equilibria in liquid phases with miscibility gaps Representation of the miscibility gap Sharing in liquid liquid systems Application of sharing between two liquids to solvent extraction Liquid vapor systems Isothermal and isopleth sections (boiling and dew) Distillation trajectories Systems with two distillation fields Systems with three distillation fields Examples of applications of ternary diagrams between fluid phases Treatment of argentiferous lead Purity of oil products: aniline point Obtaining concentrated ethyl alcohol Chapter 6. Equilibria Between Condensed Ternary Fluid Phases Solidification of a ternary system with total miscibility in the liquid state and in the solid state Solidification of a ternary system with no miscibility and with a ternary eutectic... 9

11 Contents ix 6... Invariant transformations of a liquid solid ternary system Representations of the ternary system with no miscibility in the solid state Lowering of the melting point of a binary system by the addition of a component Slope at the ternary eutectic Ternary systems with partial miscibilities in the solid state and ternary eutectic Solidification of ternary systems with definite compounds Ternary system with a binary definite compound binary with congruent melting Generalization to the case of a ternary compound and of multiple definite compounds Definite compound with incongruent melting: quasi-peritectic transformation A peritectic transformation in one binary system and total miscibility in the other two The ternary peritectic transformation... 7 Bibliography... 9 Index...

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13 Preface This book an in-depth examination of chemical thermodynamics is written for an audience of engineering undergraduates and Masters students in the disciplines of chemistry, physical chemistry, process engineering, materials, etc., and doctoral candidates in those disciplines. It will also be useful for researchers at fundamental or applied-research labs dealing with issues in thermodynamics during the course of their work. These audiences will, during their undergraduate degree, have received a grounding in general and chemical thermodynamics which all science students are normally taught, and will therefore be familiar with the fundamentals. This may includes the principles and basic functions of thermodynamics, and the handling of phase and chemical equilibrium states, essentially in an ideal medium, usually for fluid phases, in the absence of electrical fields and independently of any surface effects. This set of books, positioned somewhere between an introduction to the subject and a research paper, offers a detailed examination of chemical thermodynamics that is necessary in the various disciplines relating to chemical or material sciences. It lays the groundwork for students to go and read specialized publications in their different areas. It constitutes a series of reference books that touch on all of the concepts and methods. It discusses both scales of modeling: microscopic (by statistical thermodynamics) and macroscopic, and illustrates the link between them at every step. These models are then used in the study of solid, liquid and gaseous phases, either of pure substances or comprising several components.

14 xii Phase Transformations The various volumes of the set will deal with the following topics: phase modeling tools: application to gases; modeling of liquid phases; modeling of solid phases; chemical equilibrium states; phase transformations; electrolytes and electrochemical thermodynamics; thermodynamics of surfaces, capillary systems and phases of small dimensions. Appendices in each volume give an introduction to the general methods used in the text, and offer additional mathematical tools and some data. This series owes a great deal to the feedback, comments and questions from all my students at the École nationale supérieure des mines (engineering school) in Saint Étienne who have endured my lecturing in thermodynamics for many years. I am very grateful to them, and also thank them for their stimulating attitude. This work is also the fruit of numerous discussions with colleagues who teach thermodynamics in the largest establishments particularly in the context of the group Thermodic, founded by Marc Onillion. My thanks go to all of them for their contributions and conviviality. This fifth volume is devoted to the study of transformations between phases of pure substances and of binary and ternary systems. Chapter examines equilibria between the phases of a pure substance. Following a presentation of the different types of transformations particularly of the first and second order we study, in turn, at the macroscopic and microscopic scale, liquid vapor, liquid solid and solid vapor equilibria. Solid-solid equilibria are examined for monotropy and enantiotropy. Finally, mesomorphic states are described. Chapter is devoted to the study of the general properties of phase equilibria in binary systems. The isobaric and isothermal modes of representation are discussed, as are composition curves. The chapter closes with the representation

15 Preface xiii of the activities of the components in two-phase systems, with the particular cases of Raoult s and Henry s laws being examined. In Chapter 3, we study the equilibria of binary systems between condensed phases. The thermodynamics of demixing is studied, before being applied to strictly-regular solutions. Liquid solid equilibria, with or without demixing, of one or both of the phases, define eutectics and peritectics, with the reactions corresponding to each of these. Solid solid equilibria with polymorphic varieties are studied and compared to the previous equilibria, with the reactions at the eutectoid and peritectoid points. Two applications of liquid solid equilibria are examined through the lens of the Schröder Le Châtelier solubility lens and cryoscopy. Finally, the thermodynamics of equilibria of membranes is studied, with an application to strictly-regular solutions. Chapter 4 is given over to equilibria of binary liquid vapor phases. These systems are studied far from and near to the critical conditions. Applications for different types of solution are presented. The thermodynamics of azeotropy is examined in detail, and then applied to strictly-regular solutions. A certain number of applications of these equilibria are presented, such as tonometry, ebulliometry and continuous fractional distillation. In Chapter 5, we look at ternary systems with modes of representation, before going on to study systems where no solid phase is present. The presence of miscibility gaps in liquid phases is applied to solvent extraction, and particularly to the treatment of argentiferous lead ores. Liquid vapor equilibria lead us to distillation fields and to binary and ternary azeotropes. The chapter concludes with the application of the charts to alcohol purification and determination of the aniline point. Finally, Chapter 6 describes ternary systems involving solid phases. The concepts of ternary solidus and liquidus, and the existence of binary and ternary eutectics, are analyzed. Definite compounds, binary or ternary and exhibiting congruent or incongruent melting are represented, and the different means of solidification, with one, two or three solid phases, are examined. Michel SOUSTELLE February 6

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17 Notations and Symbols { gas} pure, {{ gas }} in a mixture, ( liquid ) pure, (( )) solid pure, solid in solution A: affinity A, B, : components of a mixture C: concentration C i : C V, C P : D : D(T/Θ D ): molar concentration (or molarity) of component i liquid in solution, specific heat capacity at constant volume and constant pressure, respectively dielectric constant of the medium Debye function d: distance between two liquid molecules d i S: internal entropy production E: energy of the system E : Eb: E p : internal energy associated with a reaction at the temperature of K balance equation set of variables with p intensive variables chosen to define a system

18 xvi Phase Transformations F: Helmholtz energy F : molar excess Helmholtz energy mix m xs i F : partial molar excess Helmholtz energy of component i mix i F : partial molar mixing Helmholtz energy of component i F i: F m : free energy, partial molar Helmholtz energy of component i molar Helmholtz energy F: faraday (unit) f i : fugacity of component i in a gaseous mixture f i : molar Helmholtz energy of pure component i f or f i : fugacity of a pure gas i G : excess Gibbs energy xs m xs i G : partial excess molar Gibbs energy of component i G, G i, [G]: Gibbs energy, partial molar Gibbs energy of i, generalized Gibbs energy G : m molar Gibbs energy G : molar Gibbs energy of mixing mix m g: osmotic coefficient g i : molar Gibbs energy of pure component i g(r): g * : radial distribution function molar Gibbs energy of gas i at pressure of atmosphere in a mixture H T : standard molar enthalpy of formation at temperature T H, H i: enthalpy, partial molar enthalpy of i H: Hamiltonian

19 Notations and Symbols xvii H : molar enthalpy xs m H : enthalpy of mixing mix m xs i H : partial molar excess enthalpy of component i mix i H : partial molar enthalpy of mixing of component i h: Planck s constant h i : molar enthalpy of pure component i J : i partial molar value of J relative to component i J : value of mixing of J relative to component i mix i J : partial molar mixing value of J relative to component i mix i J * i : value of J relative to component i in a perfect solution J * i : partial molar value of J relative to component i in a perfect solution j i : value of J for the pure component i in the same state of segregation ( Tr ) i : K constant of equilibrium change for the phase transition Tr for component i K AX : Ki( αβ ) : K d : ( c) r : solubility product of the solid AX coefficient of sharing of the compound i between the α and β phases dissociation constant K equilibrium constant relative to concentrations K equilibrium constant relative to fugacity values ( f ) r : K equilibrium constant relative to partial pressure values ( P) r :

20 xviii Phase Transformations (I) (II) K r, K r, K r, K (III) r : equilibrium constant or equilibrium constant in convention (I), in convention (II), in convention (III) K s : solubility product k B : Boltzmann s constant L t : latent heat accompanying the transformation t M: molar mass M: magnetic moment or Madelung s constant m : s mass of solute s in grams per kg of solvent m: total mass m i : mass of component i N: number of components of a solution or a mixture of gases or involved in a reaction N a : Avogadro s number N A : number of molecules of component A N C : number of elements in the canonical ensemble n i : number of objects i in the system with energy ε i or number of moles of component i n: total number of moles in a solution or a mixture n (α) : <n>: total number of moles in an α phase average number of gaps neighboring a molecule in a liquid P : critical pressure of the mixture mix c P: pressure of a gas P sublimating vapor pressure of component i ( subl ) i : ( vap) P i P i : saturating vapor pressure of component i P : relative pressure of the mixture mix r P c : critical pressure

21 Notations and Symbols xix P i : partial pressure of component i p: number of external physical variables Q: heat involved Q a : Q P : Q r : Q V : R : reaction quotient in terms of activities heat of transformation at constant pressure; reaction quotient in terms of partial pressures reaction quotient of transformation r transformation heat at constant volume reaction rate R: perfect gas constant R: distillation reflux rate S : molar entropy of mixing mix m xs i S : partial excess molar entropy of component i mix i S : partial molar entropy of mixing of component i S: oversaturation of a solution S i: entropy or partial molar entropy of i S : excess molar entropy xs m s i : molar entropy of pure component i T: temperature T : critical temperature of mixture mix c T * : second-order transition temperature T : relative temperature of the mixture mix r T (Az) : T c : T F : boiling point of the azeotrope critical temperature Fermi temperature

22 xx Phase Transformations T i ( Boil) : T i(f) : T s : T v : boiling point of pure i fusion (melting) point of pure i sublimation temperature vaporization temperature U : excess molar internal energy xs m U : molar internal energy of mixing mix m xs i U : excess partial molar internal energy of component i mix i U : partial molar internal energy of mixing of component i U, U i : U m : u i : internal energy, partial molar internal energy of i molar internal energy molar internal energy of pure component i V, V i : V c : V G : V m : v D : volume, partial molar volume of i critical volume Gibbs variance molar volume Duhem variance v i : molar volume of pure component i v m : v M : w i : molecular volume molar volume of solid at melting point mass fraction of component i x () k : molar fraction of component k in the phase x, y, z: coordinates of a point in space x i : <y>: molar fraction of component i in a solution average value of y

23 Notations and Symbols xxi Y i and X i : y i : intensive and extensive conjugate values molar fraction of component i in a gaseous phase α: relative volatility Γ: characteristic function γ : γ : γ i : activity coefficient of the component regardless of the reference state used activity coefficient of a solvent activity coefficient of the species i γ i(i) : activity coefficient of component i in the pure-substance reference γ i(ii) : activity coefficient of component i in the infinitely-dilutesolution reference γ i(iii) : activity coefficient of component i in the molar-solution reference γ s : Δ r (A): activity coefficient of a solute value of A associated with transformation r ε : Wagner interaction coefficient j i Θ D : Θ E : Debye vibrational temperature Einstein vibrational temperature μ i : chemical potential of compoent i, (L) μ i, (G) μ i : chemical potential of component i in the liquid and gaseous states, respectively ν : k ( ρ ) algebraic stoichiometric number of component A k in reaction ρ ξ: reaction extent ρ: density

24 xxii Phase Transformations Φ : practical osmotic coefficient; expansion pressure Φ : coefficient of fugacity of component i in a gaseous mixture i φ: number of phases φ : coefficient of fugacity of gas i in a mixture i φ or φ i : coefficient of fugacity of a pure gas χ i: χ T : Π : calorimetric coefficient relative to variable x i coefficient of compressibility at temperature T osmotic pressure

25 Phase Transformations of Pure Substances In this chapter we shall examine the transformations undergone by a definite pure compound, with no chemical alteration. These transformations belong to the category of phase transitions, or phase transformations. Hence, this chapter excludes the transformations of isomerization and decompositions which are accompanied by a chemical alteration i.e. a modification of the molecule... Standard state: standard conditions of a transformation The standard state of a substance at temperature T is defined as the state of the pure substance at that temperature, at a pressure of bar and as its stable state of aggregation in these conditions (solid, liquid or gas). If the substance is a gas, its behavior is perfect. If the pure substance is a crystalline solid, its stable state of aggregation determines the crystalline system. A transformation takes place in standard conditions if it occurs with the components in their standard state in the final state. This means that at the start of the reaction, the substance is in a non-standard state, which is unstable, and that it returns to its stable standard state at the end. Thus, only the state of aggregation may possibly have been modified by the transformation. This is indeed a phase transformation, in that the state of aggregation defines a phase. Phase Transformations, First Edition. Michel Soustelle. ISTE Ltd 5. Published by ISTE Ltd and John Wiley & Sons, Inc.

26 Phase Transformations.. Classification and general properties of phase transformations Phase transitions are classified into different types. The advantage of classifying definite compound transformations depending on their order lies in the fact that a series of relations characterizes each order. The first classification of these transitions is attributable to Ehrenfest, in which we say that a transformation is of order n when at least an n th derivative of the characteristic function in relation to its canonical variables undergoes a discontinuity for certain values of those variables, when the derivatives of order less than n are continuous. The most important are the first- and second-order derivatives which, by their discontinuities, respectively give us the first- and second-order transformations. This classification has been abandoned, because it does not allow for the possibility of divergences other than discontinuities of derivatives of the Gibbs energy. However, numerous models allow for such divergences within the thermodynamic limit i.e. when the system s dimensions increase indefinitely. For example, a derivative such as the specific heat capacity (second derivative of the Gibbs energy in relation to temperature) is thought to be divergent during the ferromagnetic transition. The current classification still distinguishes between first- and secondorder transformations, but the definitions used are different. First-order transformations are those which involve latent heat. In other words, they are transformations accompanied by an associated enthalpy value. During the course of these transformations the system absorbs or emits a certain fixed amount of energy, which is usually fairly large; as that energy cannot be transferred instantaneously between the system and the external environment, the transformations take place over extended periods of time, during which not all parts of the system undergo the transformation at the same moment. During these transformations the systems are heterogeneous, meaning that at each moment of the transformation they involve the simultaneous presence of multiple phases, and therefore interfaces between those phases. Such is the case with numerous transitions between the solid, liquid and gaseous phases.

27 Phase Transformations of Pure Substances 3 H V V S G T T P T T Figure.. Shape of the curves representative of the different values during a first-order transformation Figure. shows the profiles of the modifications of first-order transformation functions; discontinuity of H (associated enthalpy), volume with temperature and pressure, entropy with temperature and continuity but with a change in the slope of the Gibbs energy. For simplicity s sake, curved segments have been represented by linear segments which do not accurately reflect reality. Second-order transformations are continuous-phase transitions: there is no latent heat associated with the transition, the system is homogeneous, the transition takes place within the phase at every point and no interfaces manifest themselves. These transformations are also known as continuous transformations. There are a number of substances which exhibit secondorder transitions; let us cite the following examples: the transition between the two forms I and II of liquid helium; ferromagnetic substances whose transition point is the Curie temperature, at which they cease to manifest ferromagnetism; certain alloys which exhibit so-called order disorder transitions in a fixed composition: - superconductive substances at the point of disappearance of the property, - certain crystals, such as ammonium salts which, at a low temperature, undergo what is known as a lambda transformation. Unlike with the first-order phase-transition, in a second-order transition, the two states are not localized separately in space, and thus there is no interface: the two states constitute a single, unique phase. These transitions were examined in Volume 3 of this set [SOU 5c].

28 4 Phase Transformations S H V V G T T T P T α χ C P T T T Figure.. Shape of the different functions in a second-order transformation The transition between a superfluid, liquid and gas near to the critical point is a second-order phase-transition. Figure. shows the profile of the variations of the different functions at the transition point. To simplify, the curves have been represented by linear segments. We see the continuity of entropy with temperature but with a change of slope. The same is true of enthalpy and volume. The curve of Gibbs energy takes place with no change in slope and the variations in expansion coefficient, isothermal compressibility and specific heat capacity at constant pressure exhibit discontinuities at the transition.... First-order transformations and the Clapeyron relation According to the definition given by Ehrenfest, at least one first derivative of the characteristic function in relation to a variable undergoes a discontinuity, though the characteristic function itself (the zero-order derivative) remains constant. We first examine it in the case of a system with two physical variables: temperature and pressure. The characteristic function is the Gibbs energy, such that: G = S T [.a] and: G = V P [.b]

29 Phase Transformations of Pure Substances 5 The first-order transformation is characterized by a discontinuity of entropy or volume at a given pressure and temperature, at which the system can exist in two states: and. Such is the case, in particular, with phase changes of pure substances (melting, vaporization, sublimation, and polymorphic transformation of a solid). Let us now establish the Clapeyron relation, which governs these firstorder transformations. Consider states and, both stable simultaneously in the same thermodynamic conditions (i.e. same pressure and temperature). The Gibbs energy is constant, which is expressed by the following: G = G [.] If the temperature and pressure are modified by an infinitesimal amount, the Gibbs energy takes on the new value G + dg, and the new continuity condition is written thus: G + dg = G + dg [.3] which, in view of equation [.], gives us: dg = dg [.4] Relation [.4] is simply the application of the general equilibrium condition to the balance equation of the phase change (stoichiometric coefficients equal to + and -). However, in state, the differential of the Gibbs energy is: dg = S dt + V dp [.5] A similar expression can be written for state. The continuity condition [.4] becomes: S dt + V dp= S dt + V dp [.6]

30 6 Phase Transformations and: For simplicity s sake, let us set the differences: r V = V V [.7a] r S = S S [.7b] These are the changes in volume and entropy associated with the transformation. By feeding these values back into expression [.6], we find: SdT VdP= [.8] r r This relation is known as the Clapeyron equation. In particular, it shows that if one of the intensive values conjugal to one of the variables (e.g. the entropy) undergoes a discontinuity, the intensive value conjugal to the other variable (in this case the volume) also undergoes a discontinuity, and these two discontinuities are linked to one another by the Clapeyron equation. Often, the term latent heat L of the transformation between states and is used to denote the amount of heat involved in reversible conditions, and therefore: L= T r S [.9] Hence, there is equivalence between the first order as understood by Ehrenfest and the first order in the new classification. Transformations which are accompanied by latent heat exhibit a discontinuity of entropy. By applying the Clapeyron equation [.8], we obtain: d P rs L = = [.] dt V T V r r The Clapeyron equation can be generalized to apply to a system defined by the set of p physical variables Z i, whose conjugal extensive variables are X. The characteristic function, then, is a function. By thinking about i

31 Phase Transformations of Pure Substances 7 that function in the same way as we did with the Gibbs energy, we find the result: ε iir ( Xi ) dzi = [.] i ε ii is a switch whose value is + if the value Z i is intensive and - if Z i is extensive. Thus, we generalize the Clapeyron equation, which shows that if a first derivative of the characteristic function i undergoes a discontinuity, there is at least one other first derivative which also undergoes a discontinuity. The magnitudes of the different discontinuities are linked by equation [.].... Second-order transformations Second-order transformations cannot result from the Clapeyron equations, because the first derivatives are continuous.... Ehrenfest equations According to Ehrenfest s definition, we are dealing with a second-order transformation if a second derivative of the characteristic function is discontinuous, with that function and its first derivative being continuous. According to the new definition, it is sufficient for entropy to be continuous (no latent heat). Ehrenfest established the relations mirroring Clapeyron s relations in second-order transformations. In the case of chemical systems (choosing the variables P and T as before) we write the continuity of the Gibbs energy between states and, at equilibrium in the same conditions of pressure and temperature: G = G [.] The continuity of the two first derivatives is expressed by the two relations: G G = P P [.3a]

32 8 Phase Transformations and: G G = T T [.3b] If we modify the Gibbs energy by an infinitesimal amount, the first derivative in relation to pressure assumes the value: G G + d P P The new continuity condition becomes: G d G G d G + = + P P P P [.4] and from this we get the condition for the derivative in relation to pressure: G G d d = P P [.5] Similarly, we write the continuity of the derivative in relation to temperature: G G d d = T T [.6] By replacing the first derivatives with volume and entropy, equations [.5] and [.6] take the form: G G V V V V d d = dp+ dt dp+ dt = P P P T P T G G S S S S d d = dp dt dp dt = T T P T P T [.7] [.8]

33 Phase Transformations of Pure Substances 9 However, for reasons of symmetry of the characteristic matrix, we have: S V = P T [.9] By substituting into relation [.8], this continuity condition becomes: G G V S V S d d = dp dt dp dt = T T T T T T [.] Remember the meanings of the derivatives of volume and entropy in relation to pressure and temperature, which are respectively linked to the expansion coefficient β, the isothermal compressibility coefficient χ and the specific heat capacity at constant pressure C P : V T V P = V β = Vχ [.a] [.b] C p S = T T [.c] If we feed these definitions back into relations [.7] and [.], the continuity conditions become: χ VdP+ α VdT + χ VdP α VdT = [.] CP CP VαdP+ dt + VαdP dp= [.3] T T From relation [.], we deduce: d P rα = [.4] dt χ r

34 Phase Transformations From relation [.3], we can extract: d P rcp = [.5] dt VT α r By combining equations [.4] and [.5], we obtain: TV ( α) = χ C [.6] r r r P Equations [.4], [.5] and [.6] constitute the Ehrenfest equations. They express the relations linking the discontinuities of the expansion coefficient, the compression coefficient and the specific heat capacity, with one another and with the derivative of the pressure in relation to temperature. As was the case with the Clapeyron equation [.] the Ehrenfest equations can be generalized to apply to other pairs of variables Z i and Y i. Such is the case, for example, with the ferro-paramagnetic transformation involving temperature/entropy and magnetic moment/magnetic field pairs.... Landau theory Landau theory aims to offer a general description of second-order transformations, disregarding the peculiarities applying to any specific system. Indeed, the manifestation of order suggests that there is a certain degree of universal behavior. Given that the transformation takes place without any spectacular results, it can be considered that the effects (discontinuity or indetermination of second derivatives of the Gibbs energy or another property) associated with this type of transformation depend essentially on a universal value known as the order parameter (or parameters), whose exact definition depends on the physical nature of the transition. Thus, the occurrence of the transformation manifests as the continuous evolution of that parameter.... Symmetry and order parameters Phases before and after a transition often have different symmetries, although this is not always the case.

35 Phase Transformations of Pure Substances For instance, in the transition between a crystalline solid and a fluid (liquid or gas), the substance moves from a solid state to a liquid state. The solid state is stable at a low temperature, is ordered and less symmetrical, because the properties, whilst they are periodic, change from one point in the solid to another, and differ with direction (anisotropy). The liquid is stable at a higher temperature, less ordered but more symmetrical, with properties that are more homogeneous from one point to another. We say that the solid fluid transition breaks the symmetry. The ferromagnetic paramagnetic transition breaks the symmetry, because of the disappearance of magnetic domains containing aligned magnetic moments in the low-temperature state. The breaking of symmetry is important in the behavior of a phasetransition. Landau noted that it was not possible to find a continuous and derivable function between states having different symmetry. This explains why it is not possible to have a critical point in a solid fluid transformation. Transformations which break symmetry are necessarily either first- or second-order. In general, the most symmetrical phase is stable at a high temperature. The breaking of symmetry necessitates the involvement of additional variables to describe the state of the system. These are the (intensive) variables, which are the order parameters introduced by Landau. Table. gives a few examples of order parameters. We can posit that for the stable phase at high temperature, less ordered and more symmetrical, the order parameter is null (s = ), whereas it is different to zero in the stable phase at a lower temperature, which is more ordered and less symmetrical (s ). Transition Solid fluid Ferro paramagnetic Order disorder transition in alloys Gas liquid Order parameter Fraction of liquid phase Magnetization M Fraction of sites correctly occupied Difference between the densities of the liquid and the gas Table.. Examples of order parameters

36 Phase Transformations NOTE.. It is possible to show that we cannot define order parameters if there is no breaking of symmetry, but Landau nevertheless introduced a pseudo-order parameter in these cases, as seen from the last row in Table. for the gas liquid transition, which does not break symmetry.... Second-order transitions according to Landau Consider a phase-transition with breaking of symmetry and therefore an order parameter. At a given temperature, the state of stable equilibrium corresponds to a value of the order parameter which renders the Gibbs energy G(s, T) minimal. In the vicinity of the transition temperature T * at which the order parameter takes the values of, that parameter is small, and thus we can take the first terms of the MacLaurin expansion of the Gibbs energy in the form: G( s, T) = G ( T) + A( T) s+ A ( T) s + A ( T) s + A ( T) s +... [.7] In this expression, G ( ) T is the Gibbs energy of the stable phase at high temperature: for T > T * GT ( ) must be minimal for s = and A (T) > ; for T < T *, GT ( ) must be minimal for s and A (T) <. The simplest choice fulfilling these conditions is: A T = a T T where a > [.8] * ( ) ( ) For a second-order transition, we can show that the coefficients of the odd-powered terms are null ( A( T) = A3( T) = ) and that A 4 ( T ) is a positive constant ( A4( T) = a4 > ), and therefore: GsT (, ) = G( T) + a( T T) s + as [.9] * 4 4 The value of the order parameter is the solution to the equation which renders a zero value of the derivative of the function [.9]: GsT (, ) * 3 = a( T T ) s+ 4a4s = s [.3]

37 Phase Transformations of Pure Substances 3 for T > T *, the only solution is s = ; for T < T *, there are three solutions: a maximum at s = and two minima for: s =± s =± a ( T T)/a [.3] * 4 In the vicinity of the temperature T*, the order parameter behaves like * ( T T), so the curve s(t) has a vertical tangent at T = T * (see Figure.3). s T/T * Figure.3. Variation of the order parameter with changing temperature Figure.4 illustrates the three modes of variation of G(s) depending on the temperature value. G G G-G G G G-G G G G-G s s -s s s s T>T* * T=T* * T<T* * Figure.4. Variations of the Gibbs energy with the order parameter We can see that, although at a given pressure a first-order transformation occurs at a very specific temperature, a second-order transformation takes place in a temperature range limited to a so-called critical temperature, which is no longer a transition temperature.

38 4 Phase Transformations G G s = s = T * First order T T * Second order T Figure.5. Comparison of the variations in the Gibbs energy with temperature of the first- and second-order transformations This can be seen clearly in Figure.5, which compares the Gibbs energies of a system undergoing a first-order transformation, with a rupture of the slope at the transition T, and a system undergoing a second-order transformation in the temperature domain of variation of the order parameter between and, and ending at the critical temperature T *. If we examine the variation of specific heat capacity at the critical temperature during the second-order transition, taking account of relations [.7] and [.3], we obtain: in the vicinity of the critical temperature (slightly above it), s = and: C P G = T T * T= T + ε [.3] in the vicinity of the critical temperature (slightly below it), s and: G CP = T + a T * T * a T= T ε 4 [.33] which gives us the discontinuity of the specific heat capacity: C t P * T = a [.34] a 4

39 Phase Transformations of Pure Substances Critical exponents To represent the behavior of the physical properties in the vicinity of the critical temperature of a second-order transformation, we usually use power laws dependent on the temperature. These laws are practically the same regardless of the substance undergoing the transition in question. These variations are written as: for the order parameter: * s T T β [.35] for the specific heat capacity: * CP T T δ [.36] for the isothermal compressibility: * T T γ χ [.37] These exponents are known as the critical exponents. For example, the constant α is the critical exponent associated with the specific heat capacity. Because the transition has no latent heat, it is imperative that δ be less than, otherwise the enthalpy would no longer be continuous with the temperature. NOTE.. By virtue of the Ehrenfest equation [.6], a relationship exists between the critical exponents associated with the specific heat capacity, the expansion coefficient and the isothermal compressibility Limitation of Landau s model In actual fact, Landau s model proves true only for a limited number of transitions. This limit is probably due to the fact that this theory is an approximation, which does not take into account fluctuations of the order parameter in the vicinity of the transition, where the value of that parameter is of a similar order of magnitude to its fluctuations. There is a more elaborate model, developed by Ginsburg and Landau, which incorporates terms of fluctuations in the development of the Gibbs energy given by relation [.7].

40 6 Phase Transformations.3. Liquid vapor transformations and equilibrium states The liquid vapor system of a pure substance A obeys the balance equation: (A) = {A} [R.] This balance equation highlights that the gas is considered to be pure in its phase. Such a system has a Gibbs variance equal to, meaning that there is only one intensive variable which needs to be established in order to define the equilibrium. The gas pressure known as the saturating vapor pressure is a function of the temperature. It is a first-order transformation i.e. it is accompanied by a latent heat of vaporization, and therefore discontinuities of the first derivatives of the Gibbs energy (volume, entropy and enthalpy). There are a number of different ways in which we can approach the study of a liquid vapor phase-change and define the equilibrium conditions. Below we examine four such methods..3.. Method of two equations of state, using the Clapeyron equation On the basis of the Clapeyron equation, supposing the gaseous phase to vap be pure and therefore having the saturating vapor pressure P A, the equilibrium of the liquid-to-gas transition is written as: vap d PA h h dt = T = T v v v v v v ( { A} ( A) ) [.38] To explain the equilibrium conditions, it is helpful to feed the equations of state for the liquid and the gas back into expression [.38]. Very frequently, when the pressure is moderate, the molar volume of the liquid is negligible in comparison to that of the gas, so we can assume that v >> v, and use only the equation of state of the gas in relation [.38], { A} ( A) using the SRK equation of state, for example, to represent the vapor phase. Obviously, it is not acceptable to neglect the molar volume of the liquid when conditions are quite far removed from the critical conditions.

41 Phase Transformations of Pure Substances 7 The enthalpy of vaporization is positive, because vaporization is an endothermic phenomenon. As the molar volume of the gas is always greater than that of the liquid, the slope of the curve P(T) is positive. We frequently encounter the perfect gas approximation as the equation of state of the gas, by writing: v { A} RT = [.39] P vap A which gives us the new form of the Clapeyron equation: d P P vap A vap A vh = dt [.4] RT By integrating between two temperatures T and T, where the respective vap vap saturating vapor pressures are P and P, and ignoring the variations in A() A( ) enthalpy with changing temperature, we obtain: P ln P T T vap A( ) vh = vap A R () [.4] Remembering that the boiling point is defined as the temperature at which the saturating vapor pressure is bar, then if the reference pressure P is chosen as equal to bar, we find: vap PA vap vh ln = ln P A = P R Teb T [.4] This relation uses standard enthalpy, because boiling takes place in standard conditions. The curve representing the logarithm of the saturating vapor pressure as a function of the inverse of the temperature, then, is a straight line whose slope is equal to the opposite of the standard enthalpy of vaporization (positive) (Figure.6).

42 8 Phase Transformations Figure.6. Variations of the logarithm of the saturating vapor pressure as a function of the inverse of the temperature It must never be forgotten that this relation [.4] is founded on the hypothesis that the gas behavior is perfect, which is a very significant approximation, because clearly, for example, in the presence of the liquid, the pressure in an enclosed volume does not depend on the quantity of gas, as suggested by the Boyle Mariotte law. However, relation [.4] is still accepted, because we are very far from the critical conditions..3.. Gibbs energy and fugacity method We use the de Donder equilibrium condition. By making the affinity equal to zero, in view of the influence of pressure on the Gibbs energy of the liquid, we find: vap * * fa vap A= g{ } + g A ( A) = g{ A} + g( A) RTln + v (A) ( PA P ) = [.43] P If we ignore the liquid volume term, the fugacity of saturation i.e. the fugacity at equilibrium becomes: vap * f g A { A} g ( A) vg = exp = exp P RT RT [.44] Again, assuming usage conditions approximately corresponding to the perfect gas law, the fugacity is replaced by the pressure, which gives us the following equilibrium condition: P P vap A = exp RT g v [.45]

43 Phase Transformations of Pure Substances 9 By writing the Gibbs energy of vaporization as a function of the corresponding enthalpy and entropy, we obtain: vap PA vg vh vs ln = = + [.46] P RT RT R If we compare the two equivalent expressions [.46] and [.4], we find: vh T eb s = [.47] R v This relation [.47] merely expresses the fact that the liquid and the vapor are in a state of equilibrium at the boiling point ( g = ). eb.3.3. Unique equation of state method We now choose a unique equation of state, applicable both to the liquid and the gas. For instance, if we choose a cubic equation in a volume, such as the van der Waals equation, we can show that with these equations, the isothermal curve P(v ), depending on the temperature domain, presents two or no positive extrema, and the first extremum is inevitably a minimum, because both parts of the isotherm of the liquid and the gas, in order to yield P/ V <. stable phases, must be decreasing ( ) Let us trace such an isotherm at temperature T, which is represented by the curve a abedd in Figure.7. As soon as the two phases are present simultaneously, the Gibbs variance falls to. If the temperature is static, then so too is the pressure, and the curve needs to be replaced by a horizontal line ad. Thus, in the curve, we can distinguish five regions: between a and a: the material is completely liquid with the molar ( l) volume v in a; between d and d : the material is entirely gaseous with the molar ( g ) volume v in d;

44 Phase Transformations between a and b: the liquid phase is stable, but a curve of the Gibbs energy would show that the liquid is less stable than the gas; nonetheless, we can conserve liquid which is metastable; P a a T Vaporization Branche de vaporisation branch Liquide T c a b Domaine Supercritical supercritique domain C e c K Spinode d Branche Liquefaction de liquéfaction branch T c T d Vapeur Vapor A I J D I J D v (a) v (b) v (b) v (c) v (e) v (d) v Figure.7. Clapeyron diagram of a pure substance, showing an isotherm and the critical isotherm between b and e: the phase is not stable because the diagonal thermodynamic coefficient P/ V is negative; between e and d: the gaseous phase is stable, but a curve illustrating the Gibbs energy would show that the gas is less stable than the liquid; however, it remains gaseous until a germination facilitates the transformation; the gas is metastable. On an isotherm with two extrema (see Figure.7), the two areas between the horizontal and each of the extrema are equal. Indeed, at thermodynamic equilibrium, the integral of the function P(v ) along the closed line abcdea must be null, because it represents the variation of the internal energy on a loop. However, the internal energy is a state function, and thus we must have: c d c { A} { A} ( A) ( A) a c d c a P dv + P dv + P dv + P dv = [.48]

45 Phase Transformations of Pure Substances Hence: a d ( ( A) { A} ) { A} ( A) c c ( ) P P dv + P P dv = [.49] From this, we deduce: a c ( ( ) { }) d = A A { A} ( A) c d ( ) P P V P P dv [.5] This demonstrates the equality of the areas. Thus, this property enables us to position the horizontal platform on an isotherm. At every point K on the segment ad, the matter occupies the volume: ( ) { } V = x v + x v = x v v + v [.5] K { A} { A} ( A) ( A) ( A) ( A) { A} A The fractions x { A} and x ( A) respectively represent the fraction of gas and the fraction of liquid at point K, with their sum being equal to. From relation [.5], we deduce: x Thus: x VK v{ A} ( A) = v( A) v{ A} ( A) Kd = [.5] ad This expression is the lever rule, which we can use on a diagram to evaluate the molar fraction of the liquid at K by the ratio between the lengths of the segments Kd and ad The region of the critical point and spinodal decomposition If we alter the temperature, the point a follows a branch AC called the vaporization branch. Similarly, the point d follows a branch DC known as the liquefaction branch. These two branches intersect at point C, which is