THE GIBBS PHASE RULE

For a given redox couple, the potential of an intercalation electrode considered as solution of guest A in the host lattice (H) is provided by the classical thermodynamic law  

In a closed system at equilibrium, the Gibbs phase rule states that the relation between the number of degrees of freedom, /, and the number of independent components, c, is given by  

The cathode can be treated as a binary system (c = 2) consisting of Li and the host. Since the temperature and the pressure are kept constant in the experiments, the degrees of freedom reduce to  

If only one phase exists in a particle, p=l and/= 1 therefore, the potential has a degree of freedom and varies with the lithium concentration. On the other hand, if the particles contain two phases (Fig. 3.3a), p = 2 so that/=0, in which case no intensive variable has a degree of freedom, meaning the cell potential cannot change it is a constant, A wide voltage plateau is observed in the composition range i x /Jj as shown in Fig. 3.3b  

A general feature is the fact that the insertion or deinsertion process for the 1aJ 4P04 (M = Fe, Co, Ni) olivine materials is a two-phase process at temperature 

The Gibbs phase rule gives the number of independent intensive variables in a simple system that can have several phases and several components. The equilibrium thermodynamic state of a one-phase simple system with c components is specified by the values of c -F 2 thermodynamic variables, at least one of which must be an extensive variable. All other equilibrium variables are dependent variables. The intensive state is the state of the system so far as only intensive variables are concerned. Changing the size of the system without changing any intensive variables does not affect the intensive state. Intensive variables cannot depend on extensive variables, so specification of the 

We denote the number of phases in a system by p. In counting phases, we count only regions that are different in their intensive properties from other regions. For example, liquid water and crushed ice make up a two-phase system, just like a system of liquid water and a single ice cube. The components of a system are substances whose amounts can be varied independently. The number of components is equal to the number of chemical species present minus the number of relations that constrain the amounts of the species. There are three principal types of constraints (1) relations due to chemical equilibrium (2) a relation due to a requirement of electrical neutrality (which we always assume to exist) and (3) relations due to the way the system was prepared (such as a specification that two substances are in their stoichiometric ratio). 

If the phases in a multiphase simple system of p phases and c components are separated from each other so that they cannot equilibrate, there are c + 1 independent intensive variables for each phase, a total of p c + 1) variables. Now place the phases in contact with each other, open them to each other, and allow them to equilibrate. There are three aspects of equilibrium. Thermal equilibrium implies that all phases have the same temperature, mechanical equilibrium implies that all phases have the same pressure, and phase equilibrium implies that the chemical potential of every substance has the same value in every phase. 

Each new equality turns one variable into a dependent variable. Specifying that one variable has the same value in two phases means one equality, specifying that one variable has the same value in three phases means two equalities, and so on. There are — 1 equalities for one variable and p phases. The number of variables that have equal values in all phases is c + 2 (T, P and the chemical potentials of c components), for a total of p — l)(c + 2) equalities. This means that /, the number of independent intensive variables after equilibration of all phases, is equal to 

This equation is the phase rule of Gibbs. The number of independent intensive variables denoted by / is called the number of degrees of freedom or the variance. Try not to be confused by the fact that the term variance is also used for the square of a standard deviation of a distribution. 

The Gibbs phase rule reveals the relationship between the number of freely chosen variables or degrees of freedom /, the number of components c and the number of phases p. 

In figure 6.5 three areas are indicated S (solid), L (liquid) and G (gas, vapour) in which only ice, water and water vapour occur respectively. In the S-field the following rule holds / = 

1 + 2 - 1 = 2. This means that there are two degrees of freedom. We have to record both pressure and temperature in order to define a system, i.e. in order to be able to indicate a point (= system) in the graph. We are free to choose the pressure and temperature within certain values without changing anything in the system. No water nor water vapour will be formed. 

Now let us have a closer look at the division between the L and the G area. With those combinations of pressure and temperature the system contains both water and water vapour, so one component and two phases / =l + 2- 2 = l. Now there is only question of one degree of freedom. When we select a temperature and want to retain both water and water vapour, the system will determine the accompanying pressure. After all, the new combination of P and T must also be found on the line. 

Finally, the point where the three lines meet, the so-called triple point a system in which the S, L and G phase occur. In that case /=l+2-3 = 0. We cannot choose any degree of freedom. The system determines at which unique combination of temperature and pressure the three phases can occur. 

The Gibbs phase rule describes how many state variables of a multiphase system may vary independently (degrees of freedom of 

F the degrees of freedom of the system at equilibrium (chosen from the state variables pressure, temperature, concentration of each component in each phase) number of variables describing the state of the system which may be varied independently without disturbing the system equilibrium 

P number of phases of the heterogeneous system (one gas phase, one or more liquid phases depending on the miscibility of the components, one — at mixed crystal forming - or several solid phases according the number of crystal types) K number of components the minimum number of components of the system forming the phases which must be independently declared 

F = 0 invariant system F= 1 monovariant system F=2 divariant system, etc. 

The phase rule is important for thermal separation processes as, if certain process parameters are choosen, it establishes which state variables are cogently fixed at an arbitrarily adjusted phase equilibrium (Table 1-5). 

Before proving the phase mle (7.6), we list some elementary guidelines and caveats regarding the definitions (7.7a-c)  

Attention to these points will avoid many common misconceptions and misuses of phase mle concepts. 

Let us now proceed to prove the Gibbs phase mle (7.6). For each phase a, the total possible intensities are c + 2 in number (namely, T, P,. ..p,c), 

However, this total number/max is subject to the equilibrium conditions between phases, namely, 

Subtraction of (7.11) from (7.9) therefore leads to the independent number of intensities (/mdpt), namely, 

When two phases are brought into contact with each other, a redistribution of the components of each phase normally takes place—species evaporate, condense, dissolve, or precipitate until a state of equilibrium is reached in which the temperatures and pressures of both phases are the same and the composition of each phase no longer changes with time. 

Suppose you have a closed vessel containing three components A, B, and C distributed between gas and liquid phases, and you wish to describe this system to someone else in sufficient detail for that person to duplicate it exactly. Specifying the system temperature and pressure, the masses of each phase, and two mass or mole fractions for each phase would certainly be sufficient however, these variables are not all independent—once some of them are specified, others are fixed by nature and, in some cases, may be calculated from physical properties of the system components. 

The variables that describe the condition of a process system fall into two categories extensive variables, which depend on the size of the system, and intensive variables, which do not. Mass and volume are examples of extensive variables intensive variables include temperature, pressure, density and specific volume, and mass and mole fractions of individual system components in each phase. 

The number of intensive variables that can be specified independently for a system at equilibrium is called the degrees of freedom of the system. Let 

The relationship among DF, FT, and c is given by the Gibbs phase rule. If no reactions occur among the system components, the phase rule is 

The construction of phase diagrams—as well as some of the principles governing the conditions for phase equilibria—are dictated by laws of thermodynamics. One of these is the Gibbs phase rule, proposed by the nineteenth-century physicist J. Willard Gibbs. This rule represents a criterion for the number of phases that coexist within a system at equilibrium and is expressed by the simple equation 

Consider the case of single-phase fields on the phase diagram (e.g., a, ji, and liquid regions). Because only one phase is present, P = 1 and 

This means that to completely describe the characteristics of any alloy that exists within one of these phase fields, we must specify two parameters—composition and temperature, which locate, respectively, the horizontal and vertical positions of the alloy on the phase diagram. 

Figpre 9.23 Enlarged copper-rich section of the Cn-Ag phase diagram in which the Gibbs phase rule for the coexistence of two phases (a and L) is demonstrated. Once the composition of either phase Ca or C, ) or the temperatnre (Ti) is spedfied, values for the two remaining parameters are established by construction of the appropriate tie hne. 

For binary systems, when three phases are present, there are no degrees of freedom because 

In Chapter 6, we introduced some important concepts that we can apply to systems at equilibrium. The Clapeyron equation, the Clausius-Clapeyron equation, and the Gibbs phase rule are tools that are used to understand the establishment and changes of systems at equilibrium. However, so far we have considered only systems that have a single chemical component. This is very limiting, because most chemical systems of interest have more than one chemical component. They are multiple-component systems. 

We will consider multiple-component systems in two ways. One way will be to extend some of the concepts of Chapter 6. We will do that only in a limited fashion. The other way will be to build on the ideas of Chapter 6 and develop new ideas (and equations) that apply to multiple-component systems. This will be our main approach. 

We start by extending the Gibbs phase rule to multiple-component systems, in its most general form. We will confine our development of multiple-component systems to relatively simple ones, having two or three components at most. However, the ideas we will develop are generally applicable, so there will be little need to consider more complicated systems here. One example of a simple two-component system is a mixture of two liquids. We will consider that, as well as the characteristics of the vapor phase in equilibrium with the liquid. This will lead into a more detailed study of solutions, where different phases (solid, liquid, and gas) will act as either the solute or solvent. 

The equilibrium behavior of solutions can be generalized by statements like Henry s law or Raoulfs law, and can be understood in terms of activity rather than concentration. Changes in certain properties of all solutions can be understood simply in terms of the number of solvent and solute particles. These properties are called coUigative properties. 

Throughout the chapter, we will introduce new ways of graphically representing the behavior of multi-component systems in an efficient visual way. New ways of drawing phase diagrams, some simple and some complex, will be presented. 

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The general thermodynamic treatment of binary systems which involve the incorporation of an electroactive species into a solid alloy electrode under the assumption of complete equilibrium was presented by Weppner and Huggins [19-21], Under these conditions the Gibbs Phase Rule specifies that the electrochemical potential varies with composition in the single-phase regions of a binary phase diagram, and is composition-independent in two-phase regions if the temperature and total pressure are kept constant. 

It was shown some time ago that one can also use a similar thermodynamic approach to explain and/or predict the composition dependence of the potential of electrodes in ternary systems [22-25], This followed from the development of the analysis methodology for the determination of the stability windows of electrolyte phases in ternary systems [26]. In these cases, one uses isothermal sections of ternary phase diagrams, the so-called Gibbs triangles, upon which to plot compositions. In ternary systems, the Gibbs Phase Rule tells us... 

Having phases together in equilibrium restricts the number of thermodynamic variables that can be varied independently and still maintain equilibrium. An expression known as the Gibbs phase rule relates the number of independent components C and number of phases P to the number of variables that can be changed independently. This number, known as the degrees of freedom f is equal to the number of independent variables present in the system minus the number of equations of constraint between the variables. 

The concept of chemical potentials, the equilibrium criterion involving chemical potentials, and the various relationships derived from it (including the Gibbs phase rule derived in Chapter 5) can be used to explain the effect of pressure and temperature on phase equilibria in both a qualitative and quantitive way. 

While the Gibbs phase rule provides for a qualitative explanation, we can apply the Clapeyron equation, derived earlier [equation (5.71)], in conjunction with studying the temperature and pressure dependences of the chemical potential, to explain quantitatively some of the features of the one-component phase diagram. 

We will be looking at first-order phase transitions in a mixture so that the Clapeyron equation, as well as the Gibbs phase rule, apply. We will describe mostly binary systems so that C = 2 and the phase rule becomes... 

The international temperature scale is based upon the assignment of temperatures to a relatively small number of fixed points , conditions where three phases, or two phases at a specified pressure, are in equilibrium, and thus are required by the Gibbs phase rule to be at constant temperature. Different types of thermometers (for example, He vapor pressure thermometers, platinum resistance thermometers, platinum/rhodium thermocouples, blackbody radiators) and interpolation equations have been developed to reproduce temperatures between the fixed points and to generate temperature scales that are continuous through the intersections at the fixed points. 

The framework for constructing such multi-component equilibrium models is the Gibbs phase rule. This rule is valid for a system that has reached equilibrium and it states that... 

When a reversible transition from one monolayer phase to another can be observed in the 11/A isotherm (usually evidenced by a sharp discontinuity or plateau in the phase diagram), a two-dimensional version of the Gibbs phase rule (Gibbs, 1948) may be applied. The transition pressure for a phase change in one or both of the film components can be monitored as a function of film composition, with an ideally miscible system following the relation (12). A completely immiscible system will not follow this ideal law, but will... 

The application of n additional thermodynamic potentials (of electric, magnetic or other origin) implies that the Gibbs phase rule must be rewritten to take these new potentials into account ... 

For a given set of constraints (for example temperature, pressure and overall composition), the algorithm identifies the phases present and the relative amounts of these phases, as well as the mole fraction of all the components in all phases. The global minimum evidently must obey the Gibbs phase rule, and not all phases need to be present at the global equilibrium. 

For three-component (C = 3) or ternary systems the Gibbs phase rule reads Ph + F = C + 2 = 5. In the simplest case the components of the system are three elements, but a ternary system may for example also have three oxides or fluorides as components. As a rule of thumb the number of independent components in a system can be determined by the number of elements in the system. If the oxidation state of all elements are equal in all phases, the number of components is reduced by 1. The Gibbs phase rule implies that five phases will coexist in invariant phase equilibria, four in univariant and three in divariant phase equilibria. With only a single phase present F = 4, and the equilibrium state of a ternary system can only be represented graphically by reducing the number of intensive variables. 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

Let us now include an additional component to the Fe-0 system considered above, for instance S, which is of relevance for oxidation of FeS and for hot corrosion of Fe. In the Fe-S-0 system iron sulfides and sulfates must be taken into consideration in addition to the iron oxides and pure iron. The number of components C is now 3 and the Gibbs phase rule reads Ph + F = C + 2 = 5, and we may have a maximum of four condensed phases in equilibrium with the gas phase. A two-dimensional illustration of the heterogeneous phase equilibria between the pure condensed phases and the gas phase thus requires that we remove one degree of... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

Equation (6.42) introduces a new independent variable of the system the mean curvature c = (c1 +C2). This variable must be taken into account in the Gibbs phase rule, which now reads F + Ph = C + 2 + 1. The number of degrees of freedom (F) of a two-phase system (Ph = 2) with a curved interface is given by... 

Figure 16. Two-step discharge curve of a Mn02 electrode in aqueous solution showing the influence of one- and two-phase discharge reaction mechanisms on the shape of the discharge curve. The different shapes of the discharge curves can be explained with the help of the Gibbs phase rule.

When applying the Gibbs phase rule, it must be remembered that the choice of components is not arbitrary the number of components is the minimum number compatible with the compositional limits of the system. 

Cistola, D. P, Hamilton, J. A., Jackson, D., and Small, D. M. (1988). Ionization and phase-behavior of fatty-acids in water. Application of the Gibbs phase rule. Biochemistry, 27, 1881-8. 

In discussing chemical systems, one must be aware of the rules which determine the chemical species that are permitted to occur for a given set of conditions. The basic rule governing systems which are considered to be in thermodynamic equilibrium was first stated by J. Willard Gibbs as early as 1876. The Gibbs phase rule relates the physical state of a mixture with the chemical species of which it is composed and is given in its simplest form as... 

The identification of the superconducting phase YBagCug-O7 g provides an example in which knowledge of thermodynamics, i.e. the Gibbs phase rule and the theory of equilibrium phase diagrams coupled with X-ray diffraction techniques led to success. Further, the use of databases that can now be easily accessed and searched on-line provided leads to a preliminary structure determination. The procedures outlined here are among the basic approaches used in solid state chemistry research, but by no means are they the only ones. Clearly the results from other analytical techniques such as electron microscopy and diffraction, thermal... 

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