Phase rule

The phase rule as has been pointed out in the preceding paragraph deals with the behavior of heterogeneous systems at equilibria. It essentially includes three special terms. These are (i) number of phases in the system (P) (ii) the number of components for the system (C) and (iii) the number of degrees of freedom available to the system (F). A system for the present purpose could be any substance or combination of substances, which is set apart from its surroundings or other substances, such that its equilibrium state may be studied. The simplest way to express the rule in the form of an equation combining the three terms is as follows  

The coverage thus far has provided an account of the usefulness of phase rule to classify equilibria and to establish the number of independent variables or degrees or of freedom available in a specific situation. In the following paragraphs the equilibria used in mass transfer are analyzed in terms of phase rule in the case of leaching, drying and crystallisation. 

Two situations are found in leaching. In the first, the solvent available is more than sufficient to solubilize all the solute, and, at equilibrium, all the solute is in solution. There are, then, two phases, the solid and the solution. The number of components is 3, and F = 3. The variables are temperature, pressure, and concentration of the solution. All are independently variable. In the second case, the solvent available is insufficient to solubilize all the solute, and the excess solute remains as a solid phase at equilibrium. Then the number of phases is 3, and F = 2. The variables are pressure, temperature and concentration of the saturated solution. If the pressure is fixed, the concentration depends on the temperature. This relationship is the ordinary solubility curve. 

In drying a water-wet solid, free liquid water may or may not be present. If it is, there are in all three phases-vapour, liquid and solid- and three components, so F = 2. At constant 

As far as crystallization is concerned, there are two components, solvent and solute, and F = C = 2. The solid phase is pure, and variables are concentrations, temperature, and pressure. Fixing one, the pressure, leaves either concentration or temperature as an independent variable. The relationship between temperature and concentration is the usual solubility curve. 

We continue to consider the system containing 1/ components and 7 phases (Gibbs, 1928 Munster, 1969). It is supposed to be in equilibrium, so conditions 8 hold, and in the case of infinitely small variations of T, P, p, the equalities 

This expression can be regarded as a system of 7 equations with respect to the y + 2) unknowns dT, dP, and dp,. Hence, the number of independent variations in temperature, pressure, and chemical potentials (provided the phase number is kept) is 

This equation is called Gibbs phase rule. It defines the number of thermodynamic degrees of freedom /, i.e. the number of variables T, P, pi which can arbitrarily be varied without breaking the equilibrium among 7 phcises. When / = 0, the equilibrium is referred to eis non-variant, with / = 1 it is monovariant, when / = 2 it is bivariant, and with / 3 it is polyvariant. 

Equilibria of the sort just discussed can be quantified in terms of the (Gibbs) phase mle (see Further Reading at the end of this chapter). The number of phases that can coexist at equilibrium, P, is specified by 

When the system (e.g., a sealed ampoule containing Ag, Ag20, and O2 gas) contains three phases, P = 3 and Eq. (7.3b) gives 

The system is said to have a variance of one it is univariant. This means that under conditions in which three phases are in equihbrium, just one thermodynamic parameter needs to be given in order to specify the state of the system. In this case the most important parameters are temperature and oxygen partial pressure because in the Ag-O system the phases all have fixed compositions. Thus, knowledge of 

The situation is different in the case of a nonstoichiometric oxide MO, in equilibrium with the vapor phase. In this case, if the composition of the oxide varies slightly, a second solid phase does not appear. Over the composition range of the nonstoichiometric oxide, the number of phases, P, will be two, the nonstoichiometric oxide MO and 02, gas, and Eq. (7.3b) gives 

This means that two thermodynamic parameters can now vary freely while the system still remains in equilibrium, and the system is said to be bivariant. In this case, the composition becomes an important parameter and must be added to the partial pressure and temperature. Thus, the partial pressure over a nonstoichiometric oxide in a sealed tube will no longer depend solely upon the temperature but on the composition as well (Fig. 7.6) (see also Section 6.8.2). 

The subject matter is introduced by a short exposition of the Gibbs phase rule in Sec. 8.2. Unary component systems are discussed in Sec. 8.3. Binary and ternary systems are addressed in Secs. 8.4 and 8.5, respectively. Sec. 8.6 makes the connection between free energy, temperature, and composition, on one hand, and phase diagrams, on the other. 

As noted above, phase diagrams are equilibrium diagrams. J. W. Gibbs showed that the condition for equilibrium places constraints on the degrees of freedom F that a system may possess. This constraint is embodied in the phase rule which relates F to the number of phases P present and the number of components C 

The number of phases P is the number of physically distinct and, in principle, mechanically separable portions of the system. One of the easiest and least ambiguous methods to identify a phase is by analyzing its X-ray diffraction pattern — every phase has a unique pattern with peaks that occur at very well defined angles (see Chap. 4). For solid solutions and nonstoichiometric compounds, the situation is more complicated the phases still have a unique X-ray diffraction pattern, but the angles at which the peaks appear depend on composition. 

In the liquid state, the number of phases is much more limited than in the solid state, since for the most part liquid solutions are single-phase (alcohol and water are a common example). However, in some systems, most notably the silicates, liquid-liquid immiscibility results in the presence of two or more phases (e.g., oil and water). The gaseous state is always considered one phase because gases are miscible in all proportions. 

The number of components C is the minimum number of constituents needed to fully describe the compositions of all the phases present. When one is dealing with binary systems, then perforce the number of components is identical to the number of elements present. Similarly, in ternary systems, one would expect C to be 3. There are situations, however, when C is only 2. For example, for any binary join in a ternary phase diagram the number of components is 2, since one element is common. 

In chapter 3 we saw that carbon is found in nature as diamond and graphite with the same chemical composition C, but with completely different crystal structures. Diamond and graphite are two different solid phases or modifications which can convert into each other under the influence of certain temperatures and and which can exist simultaneously. 

Phase rule studies and describes the occurence of modifications and states of aggregation of pure substances or in mixtures in closed systems as well as the changes which occur in those systems when the pressure, temperature and composition of these substances in the system change. The behaviour of many pure substances and mixtures has thus been studied and recorded in diagrams. These diagrams constitute a vital aid for any scientist studying the development of materials, e.g. ceramics. 

In phase rule the concept phase has a wider meaning than state of aggregation , as appears from the following examples. A homogeneous system, e.g. an alcohol - water mixture, consists of one liquid phase and has one state of aggregation (figure 6.1). The circumstances described 

As can be seen in figure 6.1 and 6. 2 a heterogeneous system exhibits interfaces or planes of separation whereas a homogeneous one does not. By increasing the temperature of the alcohol - water system, we can create a vapour phase next to the liquid phase (figure 6.3). Vapours are completely miscible which means that at most one vapour phase can occur in a system. 

A system containing a solution of sodium chloride and sand exhibits one liquid and one solid phase (figure 6.4). 

As we have seen in Sec. 6-3, a heterogeneous system in equilibrium has certain restraints imposed upon it. In Sec. 6-3, we assumed that surface contributions were negligible and therefore that the extensive properties E, S, and V could be considered to be sums of contributions from the bulk phases. In this approximation and under the assumptions that the only external force acting on the system is a uniform normal pressure and that the interphase surfaces are deformable, permeable to all components, and heat-conducting, the equilibrium restraints imposed upon a system of v phases and r independent components are 

The intensive properties of phase a are. determined by / x/ ,. . . , 1 if we assume that the only external force is a uniform 

The relation (9-2) is known as the phase rule of Gibbs, and/is usually called the number of degrees of freedom of the system. It is important to note that the phase rule in the form (9-2) holds only under the following assumptions  

The phase rule can be derived in an alternative fashion from a consideration of the Gibbs-Duhem equations (6-60) for the phases of the heterogeneous system and the restrictions imposed on the variations of the intensive variables and/ = 1,. . . , r a = l,. . . v, 

Equations (9-8) [or (9-9)] are a system of v differential equations relating the r + 2 intensive variables p . . . , p T, and p. The phase rule [Eq. (9-2)] is an immediate consequence of this system of differential equations. 

This difference is further reinforced by application of the phase mle to the equihbrium between two strictly defined polymorphs of a compound or the equihbrium between a compound and a corresponding solvate of that compound. In the former case, there is only one component (in the phase rule sense—the compound). There are two phases (the two polymorphs) and, therefore, there is only one degree of freedom for equihbrium between two polymorphs by application of the phase rule equation 

Crystallization of Organic Compounds An Industrial Perspective. By H.-H. Tung, E. L. Paul, M. Midler, and J. A. McCauley 

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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). 

Closs G L and Czeropski M S 1977 Amendment of the CIDNP phase rules. Radical pairs leading to triplet states J. Am. Chem. Soc. 99 6127-8... 

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. 

Gibbsitic [14762-49-3] Gibbs-Kelvin equation Gibbs phase rule Gibbs s phase rule Gibbs s theorem Gibbs-Thomson equation... 

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). 

This is Gibbs s phase rule. It specifies the number of independent intensive variables that can and must be fixed in order to estabbsh the intensive equibbrium state of a system and to render an equibbrium problem solvable. 

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 mass-balance restrictions are the C balances written for the C components present in the system. (Since we will only deal with non-reactive mixtures, each chemical compound present is a phase-rule component.) An alternative is to write (C — 1) component balances and one overall mass balance. 

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. 

The problem presented to the designer of a gas-absorption unit usually specifies the following quantities (1) gas flow rate (2) gas composition, at least with respect to the component or components to be sorbed (3) operating pressure and allowable pressure drop across the absorber (4) minimum degree of recoverv of one or more solutes and, possibly, (5) the solvent to be employed. Items 3, 4, and 5 may be subject to economic considerations and therefore are sometimes left up to the designer. For determining the number of variables that must be specified in order to fix a unique solution for the design of an absorber one can use the same phase-rule approach described in Sec. 13 for distillation systems. 

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). 

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