THE PHASE RULE

The phase rale by the mathematician and physicist Gibbs was commented by Fuller in this way  

The number of phases that can coexist at equilibrium, P, is specihed by the (Gibbs) phase mle  

In a one-component, or unitary, system, only one substance is present. In this case, the phase rule becomes  

That is, the region over which the single phase is stable has to be specified in terms of two variables. These are taken as temperature, normally specified in degrees centigrade, and pressure, specified in atmospheres (1 atm = 1.01325 x 10 Pa). 

Along the phase boundary separating ice and water - the liquid-solid freezing curve - the number of phases present, solid and liquid, is two (P = 2), so, from Equation (S1.2), F = 1. Thus, this phase boundary (and aU phase boundaries in a one-component system) is characterised by one degree of freedom. This means that one variable, either pressure or temperature, can be changed and two phases in equilibrium can still be found. However, the two variables are closely connected. If the pressure is changed then the temperature must also change, by exactly the amount specified in 

The phase rule for a binary system is given by Equation (SI.4)  

Consider a closed system, containing k nonreacting components distributed over l phases, at equilibrium at some temperature T and pressure P. 

The number of variables for determining the intensive state of the system are  

To establish the number of equations we note that, since the fiigacity of any component i in a given phase must equal its value in the other phases, we have ( -1) equations for each component, or k 4 -1) for all of them. 

The degrees of freedom, F, i.e. the number of variables that must be specified for the complete determination of the system, is then given by  

1 represents the Phase Rule and, as its development indicates, it deals with the intensive properties of the system (T, P, composition, and partial molar properties). 

The framework within which condensed phases are studied is provided by the phase rule, which gives the number of independent thermodynamic variables required to determine completely the state of the system. This number, called the number of degrees of freedom or the variance of the system, will be denoted by F. The number of phases present (that is, the number of homogeneous, physically distinct parts) will be denoted by P, and the number of independently variable chemical constituents will be called C. By independently variable constituents, we mean those whose concentrations are not determined by the concentrations of other constituents through chemical-equilibrium equations or other subsidiary conditions. The phase rule states that 

In order to prove equation (51), it is convenient to consider first the case in which N chemical components are present in each of the P phases and there are no chemical reactions or subsidiary conditions. In writing the thermodynamic properties, the phase will be identified by superscripts and the species will be identified by subscripts. If p, 7, and are known for i 1. and for y = 1. P, then the state of the system is completely determined (except for the total mass of material in each phase, knowledge of which is seldom wanted). Since 

Let us consider vapor-liquid (or vapor-solid) equilibria for binary mixtures. For the sake of simplicity it will be assumed that all gases are ideal. In addition to the vapors of each component of the condensed phase, the gas will be assumed to contain a completely insoluble constituent, the partial pressure p of which may be adjusted so that the total pressure of the system, p, assumes a prescribed value. Therefore, C = 3, P = 2, and, according to equation (51), F = 3. Let us study the dependence of the equilibrium vapor pressures of the two soluble species p and P2 on their respective mass fractions in the condensed phase X and X2 at constant temperature and at constant total pressure. Since it is thus agreed that T and p are fixed, only one remaining variable [say X ( = l — 2)] is at our disposal p, P2 and the total vapor pressure p = p + p2 will depend only on X.  

The functional relationship between p- and X depends on the nature of the material. If the condensed mixture is ideal with respect to species 1, then equation (31) is valid for species 1. Since / p. for ideal gases, equation (31) reduces to RaoulPs law,  

It can be shown thermodynamically that the Pi-X curve determines the shape of the pj-X curve (and therefore the p -X curve). Since p and T are constant, if equation (8) is applied to the (binary) condensed phase, it is found that 

In this chapter we will consider not only the traditional Gibbs phase rule, but how it becomes modified or extended when aqueous solutes are included in the phase compositions. We then have a look at buffered systems, which are essentially an application of the phase rule. 

As indicated earlier, the state of a pure homogeneous fluid is fixed whenever two intensive tliemiodynamie properties are set at definite values. In contrast, when two phases are in equilibrium, the state of the system is fixed when only a single property is speeified. For example, a mixture of steam and liquid water in equilibrium at 101.325 kPa ean exist only at 373.15 K (100°C). It is impossible to change the temperature without also ehanging the pressure if vapor and liquid are to eontinue to exist in equilibrium. 

For any system at equilibrium, the number of independent variables that must be arbitrarily fixed to establish its intensive state is given by the celebrated phase mle of J. Willard 

who deduced it by theoretical reasoning in 75. It is presented here without proof in the form applicable to nonreacting systems  

The intensive state of a system at equilibrium is established when its temperature, pressure, and tire compositions of all phases are fixed. These are tlrerefore phase-rale variables, but tliey are not all independent. The phase rale gives the number of variablesfrom tliis set wliich must be arbitrarily specified to fix all remaining phase-rale variables. 

The phase-rale variables are intensive properties, wliich are independent of the extent of the system and of the individual phases. Tlius the phase rale gives the same information for a large system as for a small one and for different relative amounts of the phases present. Moreover, only the compositions of the individual phases are phase-rale variables. Overall or total compositions are not phase-rale variables when more tlian one phase is present. 

We now seem to have two types of components. For example, for the system NaCl-H2O, we have either the two traditional components NaCl and H2O, which allow us to describe the bulk composition of all phases in this system, or we have the four basis speciesNa.+, Cl-, H+, and H20, which allow us to describe not only the compositions of the phases but also the concentration of all dissolved species in the system. Traditional components and basis species are simply different choices of components, which have different purposes and different descriptive powers. We need more basis species because they are called upon to provide more information. 

Readers familiar with Morel and Hering (1993) will recognize that what we have termed traditional components are Morel and Hering s recipes , and what we refer to as basis species they call simply components. 

The Phase Rule links the number of components and the number of phases present at equilibrium to something called degrees of freedom. The number of degrees of freedom possessed by a system is the number of properties of the system which must be specified to specify or fix completely the equilibrium state of the system. This number is of some importance to modelers, as it is the number of pieces of information about a system which must be supplied to a modeling program before it can begin. 

The concept of degrees of freedom is perhaps most easily seen in mathematical terms. It boils down to the fact that if we have a mathematical system consisting of n unknown variables, we need n relationships or equations among the variables in order to determine all the variables. If there are less than n relationships or equations, then there are some degrees of freedom we have to supply some information before we can solve for all the variables in the system. 

The Phase Rule is simply an application of this fundamental principle to chemical systems. It can be derived from fundamental thermodynamic equations in the manner described above, but we will describe it here in more intuitive terms. 

For a system consisting of C components and P phases in equilibrium the number of intensive variables required to specify the state of the system (that is the number of degrees of freedom F) is given by the Gibbs phase rule  

In this equation R represents the number of restrictions imposed on the system. While the value for isothermal, isobaric or isochoric changes are obvious, the restrictions imposed by chemical reactions are often more subtle. For example, liquid water will exist as a mixture of H2O, and OH  

This makes it clear that 3 phases can have the same composition only in the special case of a pure component that is at the triple point. For an azeotrope Paz = 2, then F= 1, and a line always results irrespective of the number of 

The second special case applies to the critical state. Here the P phases that become identical at the critical state are considered separately from the P phases that behave normally. In this case, the additional restriction is P = 2Pc 1, and the number of degrees of freedom becomes  

Consequently, at a critical point (P= 0) is unique for a pure component while, for a binary mixture, a critical line (P= 1) is the simplest case and critical endpoints (Pno= F Fnc = 2) are unique. Since P cannot be negative  

Polymorphic structures of molecular crystals are different phases of a particular molecular entity. To understand the formation of those phases and relationships between them we make use of the classic tools of the Phase Rule, and of thermodynamics and kinetics. In this chapter we will review the thermodynamics in the context of its relevance to polymorphism and explore a number of areas in which it has proved useful in understanding the relationship between polymorphs and polymorphic behaviour. This will be followed by a summary of the role of kinetic factors in detecting the growth of polymorphic forms. We will then provide some guidelines for presenting and comparing the structural aspects of different polymorphic structures, with particular emphasis on those that are dominated by hydrogen bonds. 

It is beyond the scope of this book to provide a comprehensive review of the thermodynamics of molecular crystals (Stall et al. 1969). The field was very adequately covered in the comprehensive chapter by Westrum and McCullough (1963), a work which has stood the test of time remarkably well, and can serve as an excellent resource on this general subject. An earlier useful reference is the chapter on polymorphism in the classic book by Tammann (1926). 

The Phase Rule was first formulated by Gibbs (1876, 1878) on the basis of thermodynamic principles and then applied to physical chemistry by Roozeboom (1911). As with so much of chemistry, the apparently absolute physical principles stated in the Phase Rule must be tempered by real chemical situations (Dunitz 1991 Brittain 

1999c Alper 1999 Jensen 2001). However, in order to establish a working language, it is necessary to define terms and then indicate the difficulties that may arise in the practical use of those definitions. 

The number of components is the minimum number of independent species required to define the composition of all of the phases in the system. The simplest example usually cited to demonstrate the concept of components is that of water, which can exist in various equilibria involving the solid, liquid, and gas. In such a system there is one component. Likewise for acetic acid, even though it associates into dimers in the solid, liquid, and gaseous state, the composition of each phase can be expressed in terms of the acetic acid molecule and this is the only component. The important point for such a system is that the monomer-dimer equilibrium is established very rapidly, that is, faster than the time required to determine, say, the vapour pressure. In the cases in which the equilbrium between molecular species is established more slowly than the time required for a physical measurement, the vapour pressure, for example, will no longer be a function only of temperature, but also of the composition of the mixture, and the definition of a component acquires a kinetic aspect. 

A phase is that portion of a system which is homogeneous. A complicated system can have more than one solid phase corresponding to different crystal structure variations (e.g. pleomorphic forms of CaCXf calcite, aragonite and vaterite). Sometimes it can also have more than one liquid phase but usually contains only one gaseous phase. We can have a system having a total of p phases with c components (i.e. substances) present. Thus many of the important phase systems are more complicated than the simple one-component systems so far discussed. 

We first discuss what is meant by degree of freedom and how we can specify the composition of a given mulitcomponent phase. We define independent and dependent variables and the Phase Rule itself. The latter, which is quite general, applies to multicomponent systems and links the number of components (c), the number of phases (p) and the number of degrees of freedom (/) required for complete specification of a system in terms of temperature, pressure and composition. 

The number of degrees of freedom (= f) of a thermodynamic or other system is the minimum number of variables needed completely define the system at equilibrium. Knowledge of these variables enable the system to reproduced in all 

Here there are no degrees of freedom (f = 0) since there is only one unique value of temperature (7jp) and pressure (/Jlp), corresponding to a single (triple) point at which solid, liquid and gas (i.e. three phases hence p = 3) co-exist together. Thus at the triple point  

Here we can see an example of a quite general principle that  

The number of components of a system is the minimum number of chemieal compounds required to express the composition of any phase. In the system water opper sulphate, for instanee, five different chemical compounds can exist, viz. CUSO4 SHiO, CUSO4 3H2O, CUSO4 H2O, CUSO4 and H2O but for the purpose of applying the Phase Rule there are considered to be only two components, CUSO4 and H2O, because the eomposition of eaeh phase can be expressed by the equation 

A phase is a homogeneous part of a system. Thus any heterogeneous system comprises two or more phases. Any mixture of gases or vapours is a one-phase system. Mixtures of two or more completely miscible liquids or solids are also one-phase systems, but mixtures of two partially miscible liquids or a heterogeneous mixture of two solids are two-phase systems, and so on. 

The three variables that can be considered in a system are temperature, pressure and concentration. The number of these variables that may be changed in magnitude without changing the number of phases present is called the number of degrees of freedom. In the equilibrium system water-ice-water vapour C = 1, P = 3, and from the Phase Rule, P = 0. Therefore in this system there are no degrees of freedom no alteration may be made in either temperature or pressure (concentration is obviously not a variable in a one-component system) without a change in the number of phases. Such a system is called invariant . 

Summarizing, it may be said that the physical nature of a system can be expressed in terms of phases, and that the number of phases can be changed by altering one or more of three variables temperature, pressure or concentration. The chemical nature of a system can be expressed in terms of components, and the number of components is fixed for any given system. 

These two relations determine T and p completely. No other information is necessary for the description of the state of the system. Such a system is invariant it has no degrees of freedom. Table 12.1 shows the relation between the number of degrees of freedom and the number of phases present for a one-component system. The table suggests a rule relating the number of degrees of freedom, F, to the number of phases, P, present. 

It would be helpful to have a simple rule by which we can decide how many independent variables are required for the description of the system. Particularly in the study of systems in which many components and many phases are present, any simplification of the problem is welcome. 

We begin by finding the total conceivable number of intensive variables that would be needed to describe the state of the system containing C components and P phases. These are listed in Table 12.2. Each equation that connects these variables implies that one 

Composition variables (in each phase the mole fraction of each component PC 

In each phase there is a relation between the mole fractions  

The reader will have noted that in the equilibria considered so far, temperature was assumed to be constant or fixed. The wider question of how many such variables have to be prescribed to define a particular state of equilibrium was not addressed. It was tacitly assumed that, once a temperature and the concentration or pressure in one phase was chosen, a unique composition in the second phase was automatically assured. This approach worked without difficulty in the simple equilibria we have considered so far, but becomes somewhat tenuous when more complex systems are to be dealt with. What is required here is a formalism that will tell us exactly how many variables have to be fixed to assure a unique state of equilibrium. That formalism is provided by the Gibbsian phase rule, which states that the number of variables F to be prescribed equals the difference of the number of components and phases C - P, plus 2. Thus, 

which is also referred to as the degrees of freedom of the system, is a measure of the latitude we have in arbitrarily assigning values to the independent variables of the system. Let us see how this rule is applied in practice. 

we can fix, for example, temperature and the vapor composition, and expect the system to set its own values of liquid composition as well as total pressure. If, on the other hand, we choose to prescribe pressure and vapor composition, the system will respond with a particular temperature (i.e., its boiling point), as well as a particular liquid composition. A third possibility is to fix both temperature and total pressure, in which case the system will set its own values of both liquid and vapor composition. All three cases are encoxmtered in practice and are expressed in terms of appropriate phase diagrams, which are taken up below. 

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While the phase rule requires tliree components for an unsymmetrical tricritical point, theory can reduce this requirement to two components with a continuous variation of the interaction parameters. Lindli et al (1984) calculated a phase diagram from the van der Waals equation for binary mixtures and found (in accord with figure A2.5.13 that a tricritical point occurred at sufficiently large values of the parameter (a measure of the difference between the two components). 

At this point the system has throe phases (CUSO4 CuS04,Hj0 HjO vapour) and the number of components is two (anhydrous salt water). Hence by the phase rule, F + F = C + 2, t.e., 3+F = 2 + 2, or F=l. The system is consequently univariant, in other words, only one variable, e.g., temperature, need be fixed to define the system completely the pressure of water vapour in equilibrium with CUSO4 and CuS04,Hj0 should be constant at constant temperature. 

However, as given by group renormalization theory (45), the values of the universal exponents depend on the (thermodynamic) dimensionality of the system. For four dimensions (as required by the phase rule for the existence of tricritical points), the exponents have classical values. This means the values are multiples of 1/2. The dimensions of the volume of tietriangles are (31)... 

Two-phase equiUbria may be soHd—Hquid, Hquid—vapor, or soHd—vapor. As is evident from the phase rule of Gibbs, two-phase equiUbria are pressure-dependent ... 

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

The phase rule specifies the number of intensive properties of a system that must be set to estabUsh all other intensive properties at fixed values (3), without providing information about how to calculate values for these properties. The field of appHed engineering thermodynamics has grown out of the need to assign numerical values to thermodynamic properties within the constraints of the phase rule and fundamental laws. In the engineering disciplines there is a particular demand for physical properties, both for pure fluids and mixtures, and for phase equiUbrium data (4,5). 

The intensive state of a PVT system is established when its temperature and pressure and the compositions of all phases are fixed. However, for equihbrium states these variables are not aU independent, and fixing a hmited number of them automaticaUy estabhshes the others. This number of independent variables is given by the phase rule, and is called the number of degrees of freedom of the system. It is the number of variables which may be arbitrarily specified and which must be so specified in order to fix the intensive state of a system at equihbrium. This number is the difference between the number of variables needed to characterize the system and the number of equations that may be written connecting these variables. 

For a system containing N chemical species distributed at equihbrium among 7C phases, the phase-rule variables are temperature and pressure, presumed uniform throughout the system, and N — mole fraciions in each phase. The number of these variables is 2 -t- (V — 1)7T. The masses of the phases are not phase-rule variables, because they have nothing to do with the intensive state of the system. 

The equations that may be written connecting the phase-rule variables are ... 

The total number of independent equations is therefore (tt — )N + r In their fundamental forms these equations relate chemical potentials, which are functions of temperature, pressure, and composition, the phase-rule variables. Since the degrees of freedom of the system F is the difference between the number of variables and the number of equations. 

The two degrees of freedom for this system may be satisfied by setting T and P, or T and t/j, or P and a-j, or Xi and i/i, and so on, at fixed values. Thus, for equilibrium at a particular T and P, this state (if possible at all) exists only at one liquid and one vapor composition. Once the two degrees of freedom are used up, no further specification is possible that would restrict the phase-rule variables. For example, one cannot m addition require that the system form an azeotrope (assuming this possible), for this requires Xi = i/i, an equation not taken into account in the derivation of the phase rule. Thus, the requirement that the system form an azeotrope imposes a special constraint and reduces the number of degrees of freedom to one. 

Since the phase rule treats only the intensive state of a system, it apphes to both closed and open systems. Duhem s theorem, on the other hand, is a nJe relating to closed systems only For any closed system formed initially from given masses of preseribed ehemieal speeies, the equilibrium state is completely determined by any two propeities of the system, provided only that the two propeities are independently variable at the equilibrium state The meaning of eom-pletely determined is that both the intensive and extensive states of the system are fixed not only are T, P, and the phase compositions established, but so also are the masses of the phases. 

The general XT E problem involves a multicomponent system of N constituent species for which the independent variables are T, P, N — 1 liquid-phase mole fractions, and N — 1 vapor-phase mole fractions. (Note that Xi = 1 and y = 1, where x, and y, represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to estabhsh the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by siiTUiltaneous solution of the N equihbrium relations ... 

The phase rule permits only two variables to be specified arbitrarily in a binaiy two-phase system at equilibrium. Consequently, the cui ves in Fig. 13-27 can be plotted at either constant temperature or constant pressure but not both. The latter is more common, and data in Table 13-1 are for that case. The y-x diagram can be plotted in either mole, weight, or volume frac tions. The units used later for the phase flow rates must, of course, agree with those used for the equilibrium data. Mole fractious, which are almost always used, are appfied here. 

Check, using the phase rule, that three phases can coexist only at a point (the eutectic point) in the lead-tin system at constant pressure. If you have trouble, revise the phase rule on p. 327. 

The prehistory of the phase rule, the steps taken by Gibbs and the crucial importance of the rule in understanding phase equilibria, are outlined in an article published in a German journal to mark its centenary (Pelzow and Henig 1977). 

The phase rule is a mathematical expression that describes the behavior of chemical systems in equilibrium. A chemical system is any combination of chemical substances. The substances exist as gas, liquid, or solid phases. The phase rule applies only to systems, called heterogeneous systems, in which two or more distinct phases are in equilibrium. A system cannot contain more than one gas phase, but can contain any number of liquid and solid phases. An alloy of copper and nickel, for example, contains two solid phases. The rule makes possible the simple correlation of very large quantities of physical data and limited prediction of the behavior of chemical systems. It is used particularly in alloy preparation, in chemical engineering, and in geology. 

Step 2. Determine the vapor pressure in the evaporator. According to the phase rule, for a mixture of two components (propane and butane) it is necessary to establish two variables of the liquid-vapor system in the evaporator to completely define the system and fix the value of all other variables. The assumed liquid mol fraction and a temperature of 0°F is known. The... 

The material in this section is divided into three parts. The first subsection deals with the general characteristics of chemical substances. The second subsection is concerned with the chemistry of petroleum it contains a brief review of the nature, composition, and chemical constituents of crude oil and natural gases. The final subsection touches upon selected topics in physical chemistry, including ideal gas behavior, the phase rule and its applications, physical properties of pure substances, ideal solution behavior in binary and multicomponent systems, standard heats of reaction, and combustion of fuels. Examples are provided to illustrate fundamental ideas and principles. Nevertheless, the reader is urged to refer to the recommended bibliography [47-52] or other standard textbooks to obtain a clearer understanding of the subject material. Topics not covered here owing to limitations of space may be readily found in appropriate technical literature. 

Gibb s Phase Rule. The phase rule derived by W. J. Gibbs applies to multiphase equilibria in multicomponent systems, in the absence of chemical reactions. It is written as... 

The term ff denotes the number of independent phase variables that should be specified in order to establish all of the intensive properties of each phase present. The phase variables refer to the intensive properties of the system such as temperature (T), pressure (P), composition of the mixture (e.g., mole fractions, x ), etc. As an example, consider the triple point of water at which all three phases—ice, liquid water, and water vapor—coexist in equilibrium. According to the phase rule,... 

Assume that at the isothermal temperature of interest the following stable condensed phases (solid or liquid) can be formed M, MO, MS, MSO4. From the Phase Rule it is clear that the maximum number of condensed phases in contact with each other can be three, in addition to the gaseous phase (SO2 and O2). Following the suggestion of Kellog and Basu , the... 

For the purpose of classifying heterogeneous equilibria we shall make use of a very general law, called the Phase Rule of Willard Gibbs (1876), the proof of which is deferred to a later chapter. 

This is a consequence of the phase rule there are two components in three phases, hence the number of degrees of freedom is 2, so that when the temperature and pressure are fixed, the composition of each layer is also defined. 

If two salts which do not react chemically to produce a double salt (contact with a quantity of solvent insufficient for complete dissolution, the composition of the solution is independent of the proportions of the two solids and is definite at a fixed temperature, as we see from the phase rule ... 

A surface of separation between two phases is called a specific surface of separation, and in considering the statesof such systems it is evident that every specific surface constitutes a new independent variable. If there are n components in r phases with x specific surfaces, the Phase Rule will therefore read ... 

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