Phase Stability Relations in Invariant Systems - American Chemical

J. Phys. Chem. B 2005, 109, 12133-12144

12133

Phase Stability Relations in Invariant Systems Ilie Fishtik* Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609-2280 ReceiVed: January 14, 2005; In Final Form: April 27, 2005

Except for the trivial case of one-component systems, the conventional Schreinemakers phase stability analysis in invariant systems is shown to be thermodynamically and stoichiometrically inconsistent in that the partition of the stability relations into contributions coming from univariant subsystems is formulated only qualitatively. Although the stability relations in invariant systems are essentially additive (i.e., the stability relations in invariant system may be partitioned into a sum contributions coming from univariant subsystems), the quantitative form of this partition has never been considered. On the basis of a new approach to the stability of chemical species in multiple chemical reaction systems that has been recently developed by us (Fishtik, I. J. Phys. Chem. B, 2005, 109, 3851), we show how the stability relations in invariant systems may be uniquely partitioned into contributions coming from univariant reactions. This finding provides a simple algorithm for the construction of various types of thermodynamically consistent stability diagrams.

Introduction Graphical methods of phase stability analysis in complex heterogeneous systems that are subjects to the Gibbs phase rule have been used for years in various areas. Because of a lack of self-consistent thermodynamic data for many phases, a general graphical analysis of the possible topologies of the phase boundaries in systems comprising a large number of components and phases may be the only way to get insights into the stability relations. Such problems are routinely solved by employing a general thermodynamic and geometric approach developed almost 100 years ago by Schreinemakers1 and further improved by Morey,2 Niggli,3 Korzhinsky,4 and Zen.5 A particularly detailed and clear discussion of the Schreinemakers method has been presented by Zen.5 The set of rules established by Schreinemakers for an invariant system (i.e., a system for which the number of degrees of freedom is equal to zero) allows the construction of a general P, T diagram based, in fact, on a purely stoichiometric analysis. If thermodynamic data are available, the stability diagram can be easily made quantitative. For more complex systems (e.g., systems with several invariant points (so-called multisystems)), the Schreinemakers analysis becomes exceedingly complicated.6-8 The Schreinemakers method is essentially a graphical, semiquantitative variant of the so-called stoichiometric approach to chemical reaction equilibrium, that is, the approach based on the substitution of the mass balance conditions with a set of chemical reactions.9,10 Briefly, the essence of the Schreinemakers analysis may be formulated as follows. First, it employs the fundamental principle of chemical thermodynamics according to which a single chemical reaction is thermodynamically favorable (spontaneous, feasible) or unfavorable depending on the sign of the Gibbs free energy change ∆G of the reaction under a given set of conditions (e.g., temperature, pressure, species activities). Thus, if ∆G < 0, the reaction should proceed to the right; that is, the reactants are unstable, while the products are stable. On the contrary, if ∆G > 0, the reaction should proceed to the left; that is, the reactants are stable, while the products are unstable. When ∆G ) 0, reactants and products coexist. Hence, for two independent variables, for instance, the * E-mail: [email protected].

equilibrium line ∆G ) 0 separates the variable space into a stable field for reactants and a stable field for products. Next, Schreinemakers method generalizes this simple thermodynamic principle by postulating that stability relations in multiple chemical reaction systems may be obtained by summing up the stability relations of individual univariant subsystems. Finally, a set of rules are introduced aimed at eliminating the metastable extensions of the univariant reactions. While the Schreinemakers idea of partitioning the stability relations in an invariant system into contributions associated with univariant subsystems is a correct and brilliant guess, the exact mathematical form of this partition, as far as we are aware, has never been discussed. Recently,11 we developed a general thermodynamic and stoichiometric approach to the stability of chemical species in multiple chemical reaction systems. The approach is based on a new definition of the species stability, referred to as the overall stability.12 Our approach not only provides an algorithm for evaluation of the species stabilities but also clearly shows how the stability relations may be partitioned into a sum of contributions coming from a special class of stoichiometrically unique reactions. Here, we apply this general thermodynamic and stoichiometric method to the analysis of phase stability relations in invariant systems. In particular, we present a quantitative and unique partition of the stability relations in an invariant system into contributions coming from univariant reactions. This finding proves that Schreinemakers method, except for the trivial case of onecomponent systems, is thermodynamically inconsistent. Although our approach is general, for the sake of simplicity, in this paper we limit our analysis to the case of invariant systems. The general case will be considered elsewhere. Notation and Definitions We consider an invariant system comprising n components Ci (i ) 1, 2, ..., n) and n + 2 pure phases Bk (k ) 1, 2, ..., n, n + 1, n + 2). Let G h k(X, Y) (k ) 1, 2, ..., n, n + 1, n + 2) be the partial Gibbs free energy of the phases viewed as a function of any two variables X and Y from the conventional list of thermodynamic variables, for example, temperature T, pressure P, and composition (activities) a1, a2, ..., an, an+1, an+2. Further, let ik(i ) 1, 2, ..., n; k ) 1, 2, ..., n, n + 1, n + 2) be the

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12134 J. Phys. Chem. B, Vol. 109, No. 24, 2005

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number of components Ci (i ) 1, 2, ..., n) in the phase Bk (k ) 1, 2, ..., n, n + 1, n + 2). It is thus convenient to define the vectors

C ) (C1, C2, ..., Cn)T

(1)

B ) (B1, B2, ..., Bn, Bn+1, Bn+2)T

(2)

h 2, ..., G h n, G h n+1, G h n+2)T G h ) (G h 1, G

(3)

[

We define further the formula matrix9

11 21 ‚‚‚ E) n1 n+1,1 n+2,1 so that

12 22 ‚‚‚ n2 n+1,2 n+2,2

‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚

1n 2n ‚‚‚ nn n+1,2 n+2,n

]

(4)

(5)

It is assumed that the components are linearly independent, that is, rank E ) n. From linear algebra, it follows that in an invariant system comprising n components and n + 2 phases the number of linearly independent reactions is equal to n + 2 - rank E ) 2. Let an arbitrary set of two linearly independent reactions be

G ) νB ) 0

(6)

where G is the reaction vector

G ) (F1, F2)T

(7)

and ν is the stoichiometric matrix

[

]

(8)

where νjk (j ) 1, 2; k ) 1, 2, ..., n, n + 1, n + 2) is the stoichiometric coefficient of phase Bk in reaction Fj (j ) 1, 2). The reactions (i.e., the stoichiometric matrix ν) are normally generated by solving the mass-balance conditions

νE ) 0

(9)

Let ∆G be the vector of Gibbs free energy changes of the reactions G

∆G ) (∆G1, ∆G2)

T

(10)

From chemical thermodynamics, it is known that10

∆G ) νG h

11 21 ‚‚‚ k-1,1 k+1,1 ‚‚‚ n1 n+1,1 n+2,1

F′(Bk) )

B ) EC

ν ν ‚‚‚ ν1n ν1,n+1 ν1,n+2 ν ) ν11 ν12 21 22 ‚‚‚ ν2n ν2,n+1 ν2,n+2

tioned into a sum of contributions associated with RERs. Let F′(Bk) be the invariant reaction in which the phase Bk is absent. According to the RERs formalism, the general equation of F′(Bk) is given by

(11)

Next, we define and generate a set of n + 2 univariant reactions among phases Bk (k ) 1, 2, ..., n, n + 1, n + 2), each of them involving no more than n + 1 phases. In other words, the univariant reactions are minimal or “shortest” in the sense that, if one of the phases is omitted from a univariant reaction, there is no way to balance the reaction involving the remaining phases. Such reactions are often used in petrology and are usually labeled by employing the absent phases convention.5 It may be noticed that the univariant reactions are also response reactions (RERs).13 The latter were shown to be intimately related to chemical thermodynamics in that the fundamental equations of chemical thermodynamics can be uniquely parti-

12 22 ‚‚‚ k-1,2 k+1,2 ‚‚‚ n1 n+1,2 n+2,2

B1 ‚‚‚ 1n B2 ‚‚‚ 2n ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ k-1,n Bk-1 ‚‚‚ k+1,n Bk+1 ) 0; ‚‚‚ ‚‚‚ ‚‚‚ Bn ‚‚‚ nn ‚‚‚ n+1,n Bn+1 ‚‚‚ n+2,n Bn+2 k ) 1, 2, ..., n, n + 1, n + 2 (12)

This equation has been also deduced by Korzhinsky4 from different considerations. An alternative (and equivalent) way to generate the univariant reactions is via the linear combination of any two linearly independent reactions F1 and F2. Thus, according to the RERs formalism,13 the equation of the univariant reaction that does not involve the phase Bk is given by n+2

|

|

ν F F′′(Bk) ) ν1k F1 ) 2k 2

ν1k

ν1iBi ∑ i)1

n+2

ν2k

∑ i)1

ν2iBi

|

1k ∑ | ν2k i)1

n+2

)

ν

|

ν1i ν1i Bi ) 0 (13)

It is seen that whenever i ) k two columns in the determinant are equal, and hence, the determinant, or the stoichiometric coefficient of the phase Bk, is equal to zero, and consequently, the phase Bk is not involved in F′′(Bk). When generating a complete list of univariant reactions according to eqs 12 or 13, it appears that the stoichiometric coefficients are not always equal to the smallest integers. It is, therefore, convenient to consider univariant reactions F(Bk) in which stoichiometric coefficients are the smallest integers, that is

F′(Bk) ) γ′F(Bj)F(Bk) F′′(Bk) ) γ′F(Bk)F(Bk) where γ′F(Bj) and γ′′F(Bk) are constants. In conventional thermodynamic applications, the constants γ′F(Bj) and γ′′F(Bk) are usually neglected. In the theory developed below, however, these constants cannot be canceled. Since the univariant reactions are stoichiometrically unique, the ratio γ′F(Bj)/γ′′F(Bk) is constant and independent of k. Further, the absolute values of γ′F(Bj) and γ′′F(Bk) are not important. For this reason, it is convenient to present the constants γ′F(Bj) and γ′′F(Bk) as

γ′F(Bj) ) γ′γF(Bk)

(14)

γ′F(Bk) ) γ′′γF(Bk)

(15)

where γF(Bk) are the smallest integers. In this case, the constants γ′ and γ′′ may be neglected. Thus, each univariant reaction F(Bk) is characterized by a fundamental stoichiometric quantity γF(Bk) that is referred to as the stoichiometric factor of F(Bk). As shown below, in all applications, γF(Bk) is present as γF(Bj)2, and hence, the sign of γF(Bk) does not matter. In cases where several

Phase Stability Relations in Invariant Systems

J. Phys. Chem. B, Vol. 109, No. 24, 2005 12135

univariant reactions are stoichiometrically identical (degenerate systems), their stoichiometric factors should be summed up as γF(Bj)2. Finally, let ∆GF(Bj) (j ) 1, 2, ..., n, n + 1, n + 2) be the Gibbs free energy changes of the univariant reactions F(Bj) (j ) 1, 2, ..., n, n + 1, n + 2). These may be evaluated by employing an equation similar to eq 12

∆GF(Bj) )

1 γ′F(Bj)

11 21 ‚‚‚ k-1,1 k+1,1 ‚‚‚ n1 n+1,1 n+2,1

12 G h1 ‚‚‚ 1n 22 G h2 ‚‚‚ 2n ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ k-1,2 ‚‚‚ k-1,n G h k-1 k+1,2 ‚‚‚ k+1,n G h k+1 ; ‚‚‚ ‚‚‚ ‚‚‚ ‚‚‚ n1 G hn ‚‚‚ nn n+1,2 ‚‚‚ n+1,n G h n+1 n+2,2 ‚‚‚ n+2,n G h n+2 k ) 1, 2, ..., n, n + 1, n + 2 (16)

or, equivalently, by employing an equation similar to eq 13

∆GF(Bk) )

|

|

1 ν1k ∆G1 ; k ) 1, 2, ..., n, n + 1, n + 2 γ′F(Bk) ν2k ∆G2 (17)

Schreinemakers Phase Stability Analysis Within the conventional Schreinemakers stability analysis, a stability diagram of an invariant system contains an invariant point of n + 2 phases, n + 2 univariant assemblages each of n + 1 phases, and n + 1 divariant assemblages each of n phases. The invariant point is determined by solving simultaneously the system of equations describing the condition of chemical equilibrium

∆G1(X, Y) ) 0 ∆G2(X, Y) ) 0

(18)

The univariant assemblages are located on the univariant curves expressing the individual conditions of chemical equilibrium for univariant reactions, that is

∆GF(Bk)(X, Y) ) 0; k ) 1, 2, ..., n, n + 1, n + 2

(19)

The divariant assemblages are further determined by applying a set of rules known as the Schreinemakers rules. The latter are based on the so-called fundamental axiom formulated as follows:5 “When two divariant assemblages, each of n phases, meet along a univariant curve of n + 1 phase, then on one side of the invariant curve the divariant assemblage I is relatively less metastable than assemblage II, whereas on the other side of the curve assemblage II is relatively less metastable than assemblage I.” Except for the trivial case of one-component (n ) 1) systems, however, the fundamental axiom violates the condition of chemical equilibrium in the system. Interestingly, in his famous paper,5 Zen acknowledged in a footnote the difficulties in applying the fundamental axiom. Thus, he observed the following inadequacy of the Schreinemakers stability analysis: “Because the two invariant reaction equations that bound a given divariant assemblage are different, they involve different amounts of the phases of this divariant assemblage. Consequently, the Gibbs free energy surface of this assemblage at the two boundaries cannot be directly compared.” Zen called this inadequacy an “apparent inconsistency” and tried to rationalize it.

A more careful analysis of the problem, however, reveals that the fundamental axiom cannot be rationalized in principle, thus making the entire Schreinemakers stability analysis thermodynamically inconsistent. Here, we face a quite common situation when a simple thermodynamic principle strictly valid for a single reaction system is naively thought to be applicable in multiple chemical reaction systems too (i.e., is assumed to be additive). Clearly, the fundamental axiom is always true for a separate univariant (one reaction) subsystem of n + 1 phases. Basically, the conventional Schreinemakers stability analysis postulates that the phase stability relations in an invariant system comprising n + 2 phases and described by two independent univariant reactions may be deduced by summing up the (correct!) stability relations of n + 2 univariant subsystems each comprising n + 1 phases and employing additionally a set of rules that are non-thermodynamic in character. A rigorous thermodynamic analysis that is presented below shows that the Schreinemakers postulate is qualitatively correct, that is, the stability relations in an invariant system may be indeed partitioned into a sum of contributions associated with univariant subsystems. This partition, however, is not trivial and involves an additional factor that is stoichiometric in nature. As is wellknown, Schreinemakers stability analysis neglects the stoichiometric factor. Thus, stoichiometric coefficients in the univariant reactions are often dropped when performing a Schreinemakers stability analysis. As a consequence, failure to appropriately account for the stoichiometric factor results in serious errors. In fact, on a conventional Schreinemakers stability diagram of an invariant system, there is only one point that is thermodynamically consistent, and this is the invariant point. Indeed, the invariant point is determined by solving the correct form of the chemical equilibrium conditions (i.e., eq 18). The divariant assemblages as well as their stability fields in invariant systems where the stoichiometric coefficients of the phases in the univariant reactions are different from 1, however, are wrong. Let us illustrate the effect of stoichiometry with the help of a simple example. Consider the three-component (n ) 3) invariant system anorthite (CaAl2Si2O8) (An), grossular (Ca3Al2Si3O12) (Gr), kyanite (Al2SiO5) (Ky), quartz (SiO2) (Q), and wollastonite (CaSiO3) (Wo). As shown below, from a total of n + 2 ) 5 univariant reactions, only 4 are stoichiometrically distinct

F(An) ) -Gr + Ky - Q + 3Wo ) 0 F(Gr) ) F(Q) ) -An + Ky + Wo ) 0 F(Ky) ) -An + Gr + Q - 2Wo ) 0 F(Wo) ) -3An + Gr + 2Ky + Q ) 0 The conventional P, T diagram for this system is presented in Figure 1. Four stable divariant assemblages I, II, III, and IV on this diagram are separated by univariant curves expressing the equilibrium conditions of individual univariant reactions

h Gr(T, P) + G h Ky(T, P) - G h Q(T, P) + ∆GF(An)(T, P) ) - G 3G h Wo(T, P) ) 0 (20) ∆GF(Gr)(T, P) ) - G h An(T, P) + G h Ky(T, P) + G h Wo(T, P) ) 0 (21) h An(T, P) + G h Gr(T, P) + G h Q(T, P) ∆GF(Ky)(T, P) ) - G 2G h Wo(T, P) ) 0 (22) h An(T, P) + G h Gr(T, P) + ∆GF(Wo)(T, P) ) - 3G h Q(T, P) ) 0 (23) 2G h Ky(T, P) + G


Fishtik systems can be avoided by employing a completely new approach. The latter has been presented in detail earlier.11 Briefly, the main idea behind this approach may be summarized as follows. We consider an initial state of the system in which the phases have a certain value of the partial Gibbs free energy G h k (k ) 1, 2, ..., n, n + 1, n + 2). The phases are allowed further to react at constant P and T according to eq 6, thus arriving at an equilibrium state of the system in which the partial Gibbs free energies of the phases are equal to their equilibrium values G h eq k (k ) 1, 2, ..., n, n + 1, n + 2). Let us define the vectors T G h eq ) (G h eq h eq h eq 1 ,G 2 , ..., G n )

(24)

T h1 - G h eq h2 - G h eq hn - G h eq Σ)G h -G h eq ) (G 1 ,G 2 , ..., G n ) (25)

Figure 1. Conventional P-T diagram for the invariant system anorthite (CaAl2Si2O8) (An), grossular (Ca3Al2Si3O12) (Gr), kyanite (Al2SiO5) (Ky), quartz (SiO2) (Q), and wollastonite (CaSiO3) (Wo).

In order for the P, T diagram to be thermodynamically consistent, the intersections of the Gibbs free energy surfaces of the four assemblages should coincide with eqs 20-23. Consider, first, the Gibbs free energy surfaces of assemblages I and II

GI(T, P) ) 3G h An(T, P) + G h Gr(T, P) + G h Q(T, P) h Ky(T, P) + 2G h Gr(T, P) + 2G h Q(T, P) GII(T, P) ) 2G Here, the numbers represent the total number of moles of the species obtained according to the equations of the univariant reactions that bound the stability field of divariant assemblages. The intersection of these two surfaces is given by

GII(T, P) - GI(T, P) ) - 3G h An(T, P) + G h Gr(T, P) + 2G h Ky(T, P) + G h Q(T, P) ) 0 As can be seen, this intersection coincides with the equilibrium condition of the respective univariant reaction, F(Wo), and hence, the boundary between assemblages I and II is thermodynamically consistent. Next, consider the Gibbs free energy surfaces of assemblages II and III.

h Ky(T, P) + 2G h Gr(T, P) + 2G h Q(T, P) GII(T, P) ) 2G h Ky(T, P) + 4G h Wo(T, P) GIII(T, P) ) 2G The intersection between these two surfaces is

GIII(T, P) - GII(T, P) ) - 2G h Gr(T, P) - 2G h Q(T, P) + 4G h Wo(T, P) ) 0 It is seen that the equation of the intersection differs from the equation describing the univariant curve that separates assemblages II and III on the diagram, that is, ∆GF(An)(T, P) ) 0 (eq 20). In other words, this intersection violates the equilibrium condition of the respective univariant reaction, F(An). A similar analysis shows that the intersections between the Gibbs free energy surfaces of assemblages III and IV as well as IV and I also violate the equilibrium conditions of the respective univariant reactions. Notice that all of these thermodynamic inconsistencies are stoichiometric in nature. A New Approach to the Phase Stability Analysis The above-mentioned thermodynamic inconsistencies of the conventional Schreinemakers phase stability analysis in invariant

The function Σk ) G hk - G h eq k (k ) 1, 2, ..., n, n + 1, n + 2) is directly related to the stability of phase Bk and is referred to as hk - G h eq oVerall phase stability. More specifically, if Σk ) G k > 0, phase Bk has an excess of Gibbs free energy in the initial state as compared to the equilibrium state, and hence, the Gibbs free energy of phase Bk decreases during the reaction. On the hk - G h eq contrary, if Σk ) G k < 0, phase Bk has a deficit of Gibbs free energy in the initial state as compared to the equilibrium state, and the partial Gibbs free energy of phase Bk increases during the reaction. Thus, the stability/instability criterion of a phase Bk in an invariant system is defined as (a) Phase Bk is unstable (is consumed) if Σk ) G hk > 0. G h eq k hk - G h eq (b) Phase Bk is stable (is formed) if Σk ) G k < 0. eq (c) Phase Bk is at equilibrium if Σk ) G hk - G h k ) 0. The overall phase stability vector Σ is given by

Σ ) νT(ννT)-1∆G

(26)

A brief proof of this result is presented in the Appendix. It should be noted that, although the stoichiometric matrix ν is generated arbitrarily, the overall phase stability vector Σ is unique in that Σ is independent of the choice of ν.14 Substituting eqs 8 and 10 into eq 26 and performing the respective matrix operations gives

1 Σ k ) [(ν1k∆ν22 - ν2k∆ν12)∆G1 + ∆ (ν2k∆ν11 - ν1k∆ν12)∆G2] k ) 1, 2, ..., n, n + 1, n + 2 (27) where n+2

∆ν11 )

ν1r2 ∑ r)1

(28)

n+2

∆ν22 )

ν2r2 ∑ r)1

(29)

n+2

∆ν12 )

|

ν1rν2r ∑ r)1

∆ν ∆ν ∆ ) ∆ν11 ∆ν12 12 22

|

(30) (31)

The overall phase stabilities defined above have a remarkable physicochemical meaning. Namely, the overall phase stabilities



may be partitioned into a linear sum of contributions coming from univariant reactions according to11

Σk )

1 D

n+2

γF(B )2νF(B ),k∆GF(B ) ∑ j)1 j

j

j

k ) 1, 2, ..., n, n + 1, n + 2 (32)

where νF(Bj),k is the stoichiometric coefficient of phase Bk in the univariant reaction F(Bj) and

D)

1 n+2 n+2

γF(B )2νF(B ),k2 ∑∑ j)1

2 k)1

j

j

(33)

Notice that D, eq 33, is necessarily a positive quantity. Equation 32 may be regarded as an exact formulation of the Schreinemakers postulate. Namely, it shows explicitly how the overall phase stabilities may be partitioned into contributions coming from a complete set of univariant reactions. Each of these contributions has a rigorous mathematical form, that is, it is a product of a purely stoichiometric term, γF(Bj)2νF(Bj),k, and a thermodynamic term, ∆GF(Bj). The quantity D in eq 33 is a purely stoichiometric function and may be treated as a unique normalization factor for the overall phase stabilities. In this respect, the overall phase stabilities represent a complete stoichiometric and thermodynamic balance in the system. On contrast, the conventional Schreinemakers analysis is based on a qualitative formulation of the additivity principle that neglects the effect of system’s stoichiometry, for example, the stoichiometric coefficients νF(Bj),k of the species and the stoichiometric factors γF(Bj)2 of the univariant reactions. Properties of the Overall Phase Stabilities in Invariant Systems The overall phase stability approach provides a new and thermodynamically consistent algorithm for the evaluation of phase stability boundaries in invariant systems. We notice, first, that the overall stabilities of all phases are analytical linear functions of the Gibbs free energy changes of the linearly independent reactions (i.e., Σk(∆G1, ∆G2)). Obviously, if ∆G1(X, Y) and ∆G2(X, Y) are known, then the overall phase stabilities may be viewed as analytical functions of any thermodynamic variables X and Y (i.e., Σk(X, Y)). However, even if ∆G1(X, Y) and ∆G2(X, Y) are unknown, a detailed analysis of Σk(∆G1, ∆G2) can lead to a series of useful results concerning the general topology of the phase diagram. In what follows, we will focus our attention on the analysis of the overall phase stabilities in the coordinates ∆G1 and ∆G2. Because ∆G1 and ∆G2 are independent variables, we formally allow them to take values between -∞ and +∞. The straight line Σk(∆G1, ∆G2) ) 0, hereafter referred to as the overall stability line, divides the space (∆G1, ∆G2) into a stability (Σk(∆G1, ∆G2) < 0), equilibrium (Σk(∆G1, ∆G2) ) 0) and instability (Σk(∆G1, ∆G2) > 0) field (Figure 2). We also define a straight line extension from the invariant point such that the overall stabilities of two phases Bq and Bs at any point on this line are equal. This line is called a predominance line extension, and its equation is Σq(∆G1, ∆G2) - Σs(∆G1, ∆G2) ) 0. Next, we point out two important properties of the overall phase stabilities in invariant systems that will be used later to draw overall stability diagrams. Property 1. OVerall stability lines in an inVariant system either intersect at the inVariant point or coincide.

Figure 2. Overall stability line, stability field, and a closed path (circle) around the invariant point.

The proof of this property is as follows. Consider the overall stability lines of two phases, say, Bq and Bs. According to eq 27, these are

(ν1q∆ν22 - ν2q∆ν12)∆G1 + (ν2q∆ν11 - ν1q∆ν12)∆G2 ) 0 (34) (ν1s∆ν22 - ν2s∆ν12)∆G1 + (ν2s∆ν11 - ν1s∆ν12)∆G2 ) 0 (35) These overall stability lines will intersect at the invariant point, ∆G1 ) ∆G2 ) 0, only if

|

|

ν1q∆ν22 - ν2q∆ν12 ν2q∆ν11 - ν1q∆ν12 ν1s∆ν22 - ν2s∆ν12 ν2s∆ν11 - ν1s∆ν12 * 0

(36)

Equation 36 can be readily rearranged into

|

||

| |

|

∆ν11 ∆ν12 ν1q ν1s ν1q ν1s ∆ν12 ∆ν22 ν2q ν2s ) ∆ ν2q ν2s * 0

(37)

Since ∆ is necessarily a positive quantity, the overall stability lines will intersect at the invariant point only if

|

|

ν1q ν1s ν2q ν2s * 0

(38)

ν1q ν1s ν2q ν2s ) ν1qν2s - ν2qν1s ) 0

(39)

If

|

|

the stability lines coincide. Indeed, from eq 39, we have

ν2s )

ν2qν1s ν1q

(40)

Substituting eq 40 into eq 35, we obtain precisely eq 34. Property 2. The oVerall phase stability on a closed path (circle) around the inVariant point passes through a maximum and a minimum.


Fishtik

The easiest way to prove this statement is to use the polar coordinates (Figure 2)

∆G1 ) r cos θ

(41)

∆G2 ) r sin θ

(42)

where r is the radial coordinate and θ is the angular coordinate. Substituting eqs 41 and 42 into eq 27 and assuming, for simplicity, that r ) ∆, we have

Σk ) (ν1k∆ν22 - ν2k∆ν12) cos θ + (ν2k∆ν11 - ν1k∆ν12) sin θ (43) Now, it is a matter of elementary algebra to show that this function for 0 e θ e 2π passes through a maximum and a minimum. At the maximum, the overall phase stability is positive, and consequently, the maximum represents the point of highest instability. On the contrary, at the minimum, the overall phase stability is negative, and hence, the minimum represents the point of highest stability (Figure 2). Overall Stability Diagrams On the basis of the properties of the overall phase stabilities outlined above, two different types of overall stability diagram may be easily constructed. Namely, we can plot the stability fields of all phase assemblages that obey the Gibbs phase rule. Diagrams of this type are referred to as the overall stability diagrams. Alternatively, we can construct a more restrictive type of diagram by plotting only predominance lines. This type of diagram involves only stability fields in which the overall stabilities of individual phases are dominant and are referred to as the predominance stability diagrams. The overall stability diagrams may be constructed as follows. First, the overall stability lines Σk(∆G1, ∆G2) ) 0 are plotted in the coordinates ∆G1 and ∆G2 thus delineating the stability fields of each phase. Because the stability lines are straight, the stability sector for each phase does not exceed 180°. Obviously, the stability fields of the phases will inevitably overlap. That is, the overall stability lines divide the diagram into stability fields involving, concomitantly, several phases. If the number of phases bound by two overall stability line extensions is equal to or less than the number of phases allowed by the Gibbs phase rule for a divariant assemblage (i.e., n), then the respective phase assemblage is stable. If the number of phases between two extensions of the stability lines exceeds n, then this phase assemblage violates the Gibbs phase rule and should be eliminated. The elimination may be done by substituting the overall stability line extensions that bound the unstable assemblage by a predominance line extension. In doing this, it is necessary to ensure that the stability fields of the two adjacent phases concomitantly decrease. This procedure will be illustrated below in details with the help of examples. The predominance stability diagrams may be constructed by finding in a combinatorial manner the predominance line extensions Σq(∆G1, ∆G2) - Σs(∆G1, ∆G2) ) 0 between all possible pairs of phases. More easily, the predominance stability diagrams may be deduced by directly plotting the overall phase stabilities in polar coordinates according to eq 43. It may be noticed that according to Schreinemakers stability analysis the predominance stability diagrams cannot be constructed in principle. Examples The theoretical developments presented above are next illustrated with the help of three simple examples. These

examples should not be considered as a rigorous discussion of the phase stability relations in real systems. Rather, our goal is to illustrate in detail the differences between the Schreinemakers and overall phase stability analysis. Example 1. As a first example, consider the trivial case of the equilibrium in a one-component (n ) 1), three-phase (n + 2 ) 3) invariant system. For this simple system, we want to show that the Schreinemakers and overall phase stability analysis coincide. According to the overall stability approach, the starting point is a set of two linearly independent reactions involving phases B1, B2, and B3. Let us choose these as

F1 ) -B1 + B2 ) 0

∆G1

F2 ) -B2 + B3 ) 0

∆G2

Thus, the respective stoichiometric matrix and Gibbs free energy changes vector are

ν)

[

]

[ ]

∆G -1 1 0 and ∆G ) ∆G1 0 -1 1 2

Substituting ν and ∆G into eq 26 and performing the matrix operations results in the following overall phase stabilities

1 ΣB1 ) (-2∆G1 - ∆G2) 3 1 ΣB2 ) (∆G1 - ∆G2) 3 1 ΣB3 ) (∆G1 + 2∆G2) 3 The corresponding overall stability lines of the phases are

ΣB1: 2∆G1 + ∆G2 ) 0 ΣB2: ∆G1 - ∆G2 ) 0 ΣB3: ∆G1 + 2∆G2 ) 0 and are plotted in Figure 3(I). Observe that the overall stability lines intersect at the invariant point. As expected, the stability fields of the phases overlap. That is, the overall stability lines divide the space (∆G1, ∆G2) into six different fields. Three of them involve one phase, while the other three involve two phases. According to Gibbs phase rule, however, only one phase can be stable in a divariant field. Therefore, the three divariant fields each involving two phases should be eliminated. This may be done by substituting the overall stability line extensions bounding the fields involving two phases with a predominance line extension. For instance, the overall stability line extensions bounding the overlap of the stability fields of phases B1 and B2 is substituted with the predominance line extension ΣB1 ΣB2 ) 0, or ∆G1 ) 0. Similarly, the predominance line extension separating phases B2 and B3 is given by ΣB2 - ΣB3 ) 0, or ∆G2 ) 0, while the predominance line extension between phases B1 and B3 is given by ΣB1 - ΣB3 ) 0, or ∆G1 + ∆G2 ) 0. These considerations are graphically illustrated in Figure 3(II), and the final stability diagram is shown in Figure 3(III). Taking into account that ∆G1 + ∆G2 is just the Gibbs free energy change of the univariant reaction between phases B1 and B3, that is


F1 ) -B1 + B2 ) 0

∆G1

F2 ) -B2 + B3 ) 0

∆G2

F3 ) -B1 + B3 ) 0


∆G3 ) ∆G1 + ∆G3

it follows immediately that the overall stability diagram is identical with the conventional diagram. It should be noticed that the stability diagram for a onecomponent invariant system is essentially a predominance diagram. This is clearly seen from Figure 4 in which the overall phase stabilities are plotted in polar coordinates around the invariant point along a circle of radius equal to 3. Under these conditions, according to eq 43, the overall stabilities for r ) 3 are

ΣB1 ) -2 cos θ - sin θ ΣB2 ) cos θ - sin θ ΣB3 ) cos θ + 2 sin θ The interrelation between Figure 3(III) and Figure 4 is selfexplanatory. Example 2. Next, let us analyze the overall stability relations in a three-component (n ) 3) invariant system: anorthite (CaAl2Si2O8) (An), grossular (Ca3Al2Si3O12) (Gr), kyanite (Al2SiO5) (Ky), quartz (SiO2) (Q), and wollastonite (CaSiO3) (Wo) and compare the results with the stability relations obtained according to Schreinemakers analysis (Figure 1). A set of two linearly independent reactions in this system is selected as

F1 ) -3An + Gr + 2Ky + Q ) 0 F2 ) -An + Ky + Wo ) 0

∆G1 ∆G2

Hence, the stoichiometric matrix and Gibbs free energy changes vector are

ν)

[

-3 1 2 1 0 -1 0 1 0 1

]

[ ]

∆G ∆G ) ∆G1 2

Substituting these relations into eq 26 and performing the matrix operations gives

1 ΣAn ) - ∆G1 5 ΣGr ) ΣQ ) ΣKy )

1 (3∆G1 - 5∆G2) 20

1 (∆G1 + 5∆G2) 20

1 ΣWo ) (-∆G1 + 3∆G2) 4 Observe that the overall stabilities of Gr and Q coincide. The corresponding overall stability lines of the phases are

ΣAn: ΣGr,ΣQ: ΣKy: ΣWo:

∆G1 ) 0 3∆G1 - 5∆G2 ) 0 ∆G1 + 5∆G2 ) 0

-∆G1 + 3∆G2 ) 0

and are plotted in Figure 5(I). As expected, the stability fields

Figure 3. Construction of the overall stability diagram in a onecomponent invariant system. (I) Stability lines. (II) Elimination of the unstable fields. (III) Final overall stability diagram.

overlap. Moreover, in all divariant assemblages, except one, the number of phases is equal to or less than 3. Consequently, all of the phase assemblages, except one, are stable. The assemblage that is bound by the overall stability line extensions ΣWo ) 0


Figure 4. Overall phase stabilities in a one-component invariant system in polar coordinates.

Fishtik

Figure 6. Overall phase stabilities in polar coordinates in the invariant system anorthite (CaAl2Si2O8) (An), grossular (Ca3Al2Si3O12) (Gr), kyanite (Al2SiO5) (Ky), quartz (SiO2) (Q), and wollastonite (CaSiO3) (Wo).

extension ΣWo - ΣGr ) 0. The final overall stability diagram is shown in Figure 5(II). By comparing Figures 2 and 5(II), it is seen that the Schreinemakers and overall stability relations qualitatively differ in that the overall stability diagram comprises seven stable phase assemblages compared to only four stable assemblages according to the Schreinemakers analysis. The predominance diagram for this system may be constructed on the basis of the evaluation of the overall phase stabilities in polar coordinates (Figure 6) around the invariant point along a circle of radius equal to 20, that is

ΣAn ) -4 cos θ ΣGr ) ΣQ ) 3 cos θ - 3 sin θ ΣKy ) cos θ + 5 sin θ ΣWo ) -5 cos θ + 15 sin θ As can be seen from Figure 6, depending on the Gibbs free energy changes of the univariant reactions, all phases can be dominant. Interestingly, the absolute value of the overall stability of Wo is much higher than that of the overall stability of other phases. Figure 6 may be easily translated into a predominance diagram assuming that the dominant phases are those that have the highest overall stabilities, while their intersections represent the predominance lines. For instance, the predominance line separating the stability fields of An and Wo is

1 1 ΣAn - ΣWo ) - ∆G1 - (-∆G1 + 3∆G2) ) 5 4 1 - (∆G1 - 15∆G2) ) 0 20 or Figure 5. Construction of the overall stability diagram in the invariant system anorthite (CaAl2Si2O8) (An), grossular (Ca3Al2Si3O12) (Gr), kyanite (Al2SiO5) (Ky), quartz (SiO2) (Q), and wollastonite (CaSiO3) (Wo). (I) Elimination of the four-phase unstable assemblage Gr-QKy-Wo. (II) Final overall stability diagram.

and ΣGr ) ΣQ ) 0, however, involves four phases, Gr, Q, Ky, and Wo, and violates the Gibbs phase rule. Therefore, this assemblage needs to be eliminated by substituting the bounding overall stability line extensions with the predominance line

∆G1 - 15∆G2 ) 0 The final predominance diagram is presented in Figure 7. A deeper insight into the stability relations may be obtained by partitioning the overall phase stabilities into contributions coming from univariant reactions. As explained above, a complete list of univariant reactions may be generated starting either from formula matrix or from an arbitrary set of linearly


J. Phys. Chem. B, Vol. 109, No. 24, 2005 12141 TABLE 1: Complete List of Univariant Reactions and Stoichiometric Factors in the Invariant System Anorthite (CaAl2Si2O8) (An), Grossular (Ca3Al2Si3O12) (Gr), Kyanite (Al2SiO5) (Ky), Quartz (SiO2) (Q), and Wollastonite (CaSiO3) (Wo) 2 γF(B j)

F(Wo) ) -3An + Gr + 2Ky + Q ) 0 F(Gr) ) -An + Ky + Wo ) 0 F(Ky) ) -An + Gr + Q - 2Wo ) 0 F(An) ) -Gr + Ky - Q + 3Wo ) 0

1 2 1 1

ometrically identical. Therefore, the stoichiometric factors of these two univariant reactions may be combined into one as γF(Gr)2 + γF(Q)2 ) 1 + 1 ) 2. The complete list of univariant reactions along with their stoichiometric factors is presented in Table 1. According to eqs 32 and 33, the overall phase stabilities may be partitioned into contributions associated with univariant reactions as follows Figure 7. Predominance stability diagram for the invariant system anorthite (CaAl2Si2O8) (An), grossular (Ca3Al2Si3O12) (Gr), kyanite (Al2SiO5) (Ky), quartz (SiO2) (Q), and wollastonite (CaSiO3) (Wo).

independent reactions. Consider, for instance, the generation of the univariant reactions from formula matrix

[

CaO Al2O3 SiO2

E)

1 3 0 0 1

1 1 1 0 0

2 3 1 1 1

]

An Gr Ky Q Wo

According to eq 12, the univariant reactions are

F′(Wo) )

1 3 0 0

F′(Q) )

F′(Ky) )

1 3 0 1

F′(Gr) )

F′(An) )

3 0 0 1

1 1 1 0

2 3 1 1

An Gr Ky Q

1 3 0 1

1 1 1 0

2 3 1 1

1 1 0 0

2 3 1 1

An Gr Q Wo

1 0 0 1

1 1 0 0

2 1 1 1

1 1 0 0

3 1 1 1

Gr Ky Q Wo

An Gr Ky Wo

An Ky Q Wo

) -3An + Gr + 2Ky + Q ) 0

) -An + Ky + Wo ) 0

) -An + Gr + Q - 2Wo ) 0

) -An + Ky + Wo ) 0

) -Gr + Ky - Q + 3Wo ) 0

Because the stoichiometric coefficients in all univariant reactions are equal to the smallest integers, their stoichiometric factors are all equal to 1, that is, F′(Wo) ) F(Wo), F′(Q) ) F(Q), F′(Ky) ) F(Ky), F′(Gr) ) F(Gr), and F′(An) ) F(An). Further, two univariant reactions, namely, F(Gr) and F(Q), are stoichi-

ΣAn )

1 [-3∆GF(Wo) - 2∆GF(Gr) - ∆GF(Ky)] 20

ΣGr ) ΣQ )

1 [∆GF(Wo) + ∆GF(Ky) - ∆GF(An)] 20

ΣKy )

1 [2∆GF(Wo) + 2∆GF(Gr) + ∆GF(An)] 20

ΣWo )

1 [2∆GF(Gr) - 2∆GF(Ky) + 3∆GF(An)] 20

where ∆GF(Wo), ∆GF(Gr), ∆GF(Ky), and ∆GF(An) are the Gibbs free energy changes of F(Wo), F(Gr), F(Ky), and F(An) and 20 is the value of the quantity D, eq 33. These equations show how the overall stabilities of phases are explicitly related to the stoichiometry and thermodynamics of univariant reactions and may be used to rationalize the phase stability relations. Example 3. As a last example, we consider a more realistic, n ) 5 component system CaO-MgO-SiO2-CO2-H2O comprising n + 2 ) 7 species, namely, CO2, H2O, calcite CaCO3 (Cc), diopside CaMgSi2O6 (Di), dolomite CaMg(CO3)2 (Do), qurtz SiO2(Q), and tremolite Ca2Mg5Si8O22(OH)2 (Tr). Various aspects of the phase stability relations in this system have been recently discussed15 in a more general context. For simplicity, it is assumed that CO2 and H2O form a fluid phase, while the remaining species are pure solid phases. In general, this system is not invariant, since the number of degrees of freedom is equal to 1. However, the conventional phase diagram constructed in the coordinates T-x (mole fraction of CO2 in the fluid phase) at P ) const (P ) 3 kbar) and presented in Figure 8 shows that, in fact, both CO2 and H2O are at equilibrium. Alternatively, CO2 and H2O may be considered to be present in excess. As a result, the system may be regarded as a pseudo-ternary, invariant system. The numerical calculations below were performed by employing the Holland and Powell data set.16 To construct the overall stability diagram, we first generate the overall phase stabilities from an arbitrary set of reactions, say

F1 ) Di - Do - 2Q + 2CO2 ) 0 ∆G1 ) ∆G01 + 2RT ln fCO2 F2 ) -3Cc + 4Di + Do - Tr + CO2 + H2O ) 0 ∆G2 ) ∆G03 + RT ln fCO2 + RT ln fH2O


Fishtik

Figure 8. Conventional T-xCO2 diagram at P ) 3 kbar for the invariant system calcite CaCO3(Ca), diopside CaMgSi2O6 (Di), dolomite CaMg(CO3)2 (Do), quartz SiO2 (Q), and tremolite Ca2Mg5Si8O22(OH)2 (Tr).

Figure 9. Overall T-xCO2 diagram at P ) 3 kbar for the invariant system calcite CaCO3(Ca), diopside CaMgSi2O6 (Di), dolomite CaMg(CO3)2 (Do), quartz SiO2 (Q), and tremolite Ca2Mg5Si8O22(OH)2 (Tr).

where fCO2 and fH2O are the fugacities of CO2 and H2O, respectively. Hence

TABLE 2: Complete List of Univariant Reactions and Stoichiometric Factors in the Invariant System Calcite CaCO3(Ca), Diopside CaMgSi2O6 (Di), Dolomite CaMg(CO3)2 (Do), Quartz SiO2 (Q), and Tremolite Ca2Mg5Si8O22(OH)2 (Tr)

Cc Di Do Q Tr 0 1 -1 -2 0 ν) -3 4 1 0 -1

[

( )(

∆G01 + RT ln fCO2

]

∆G ∆G ) ∆G1 ) ∆G02 + RT ln fH2O + RT ln fCO2 2

2 γF(B j)

)

Observe that CO2 and H2O were eliminated from the stoichiometric matrix, because these species were assumed to be at equilibrium, or in excess. In other words, the overall stabilities of CO2 and H2O are assumed equal to 0. For this reason, the evaluation of phase stabilities employing the Gibbs free energy minimization is problematic, since the Gibbs free energy minimization should include all species. This assumption is not an impediment of the overall stability analysis. Thus, both CO2 and H2O may be readily considered nonequilibrium species. Rather, this assumption is accepted here only to obtain an overall stability diagram that is comparable with the conventional stability diagram. The overall phase stabilities are obtained by substituting the stoichiometric matrix and Gibbs free energy vector into eq 26. This gives

ΣCc )

1 (∆G1 - 2∆G2) ) 17 1 (∆G01 - 2∆G02 - RT ln fCO2 - 2RT ln fH2O) 17

ΣDi )

1 (5∆G1 + 7∆G2) ) 51 1 (5∆G01 + 7∆G02 + 12RT ln fCO2 + 7RT ln fH2O) 51

ΣDo )

1 (-10∆G1 + 3∆G2) ) 51 1 (-10∆G01 + 3∆G02 - 7RT ln fCO2 + 3RT ln fH2O) 51

ΣQ )

ΣTr )

2 (-9∆G1 + ∆G2) ) 51 1 (-9∆G01 + ∆G02 - 8RT ln fCO + RT ln fH2O) 51 1 (∆G1 - 2∆G2) ) 51 1 (∆G01 - 2∆G02 - RT ln fCO2 - 2RT ln fH2O) 51

F(Cc) ) F(Tr) ) Di - Do - 2Q + 2CO2 ) 0 F(Q) ) -3Cc + 4Di + Do - Tr + CO2 + H2O ) 0 F(Do) ) -3Cc + 5Di - 2Q - Tr + 3CO2 + H2O ) 0 F(Di) ) 3Cc - 5Do - 8Q + Tr + 7CO2 - H2O ) 0

10 4 1 1

The final overall stability diagram constructed according to the rules outlined above is presented in Figure 9. A comparison between Figure 8 and Figure 9 reveals that the conventional and overall stability diagrams differ substantially. Thus, the overall stability diagram predicts seven stable assemblages, while the conventional stability diagram predicts only four stable assemblages. Interestingly, the conventional stability diagram violates the Schreinemakers 180° rule. Indeed, as can be seen from Figure 8, the stability fields of Do and Q exceed 180°. On the contrary, the phase stability fields on the overall stability diagram never exceed 180°. The distinct feature of the overall stability diagram is that the stability relations among phases may be evaluated numerically at any point on the diagram as well as rationalized in terms of contributions coming from univariant reactions. As an example, consider the point T ) 800 K and xCO2 ) 0.98. At this point, the overall stability diagram (Figure 9) predicts that the only stable phase is Di. On the contrary, the conventional stability diagram predicts that, under the same conditions, Do is also stable (Figure 8). The advantage of the overall stability approach is that it provides a reasonable physicochemical explanation of the phase stability relations by partitioning the overall stabilities into a sum of contributions coming from univariant reactions. A complete set of univariant reactions and their stoichiometric factors in this system generated according to eq 12 is presented in Table 2. Let us use these data to explain why at the point T ) 800 K and xCO2 ) 0.98 Di is stable, while Do is unstable. The overall stabilities of Di and Do, according to eq 32 and Table 2, may be partitioned into contributions coming from univariant reactions as

ΣDi )

1 (10∆GF(Cc) + 16∆GF(Qz) + 5∆GF(Do)) 153

ΣDo )

1 (-10∆GF(Cc) + 4∆GF(Qz) - 5∆GF(Di)) 153

where 153 is the value of D, eq 33. Now, for T ) 800 K and xCO2 ) 0.98, the Gibbs free energy of the univariant reactions are ∆GF(Cc) ) -11.7 kJ/mol, ∆GF(Q) ) -10.9 kJ/mol, ∆GF(Do) ) -22.6 kJ/mol, and ∆GF(Di) ) -36.0 kJ/mol. It is



immediately seen that all three univariant reactions involving Di, namely, F(Cc), F(Q), and F(Do), are thermodynamically favorable toward Di formation, thus making Di stable. On the other hand, the formation of Do is thermodynamically favorable only in one univariant reaction, namely, F(Q). This stability, however, is overcompensated by two other univariant reactions, F(Cc) and F(Di), that are thermodynamically unfavorable toward formation of Do, thus making Do unstable overall.

Subtracting eq A1 from eq 11 and taking into account eq 25, we have

νΣ ) ∆G

(A2)

The stability vector Σ is obtained by minimizing the product ΣTΣ subject to the constraints given by eq A2. Employing the method of Lagrange’s undetermined multipliers, we minimize the Lagrangean function

Discussion and Concluding Remarks

F ) Σ TΣ + λT(νΣ - ∆G)

The conventional phase stability analysis is essentially based on the assumption that phase stability relations are additive in the sense that phase stability relations that are valid for one chemical reaction system are assumed to be valid in a multiple chemical reaction system too. As applied to invariant systems, the Schreinemakers analysis implies that phase stability relations may be generated by simply summing up phase stability relations of individual univariant subsystems. This approach, although qualitatively correct, results in thermodynamic and stoichiometric inconsistencies and ultimately, except for the trivial case of one-component systems, in erroneous phase stability relations. In this work, the stability relations in invariant systems were analyzed by employing a new approach. The main idea behind this new approach is to define and evaluate a quantitative measure of the stability. It appears that a natural choice of the stability measure of chemical species in a reacting system is just the change in the partial Gibbs free energy of the species when the system is moving from a given initial state to an equilibrium state. Such a definition of the species stability, referred to as overall stability, implies the Gibbs free energy changes of an arbitrary set of linear reactions as independent variables rather than the conventional thermodynamic variables (e.g., P and T). When these considerations are coupled with an optimization procedure similar to that of the least-squares method, the result is a linear analytical function of the overall stability in terms of the Gibbs free energy changes of the linear independent reactions. The overall species stabilities are further shown to have the remarkable property of being partitioned into a sum of contributions associated with univariant reactions. This rigorous thermodynamic and stoichiometric result shows that, although the overall phase stability relations in invariant systems are additive, the partition of the overall phase stabilities into contributions coming from univariant reactions is not trivial. Thus, besides the natural and expected dependence of the overall phase stabilities on the Gibbs free energy changes of univariant reactions, there is an additional, purely stoichiometric contribution that, so far, has been completely neglected. Most importantly, however, the overall stability approach provides a simple algorithm to construct various types of thermodynamically and stoichiometrically consistent phase diagrams. In this work, we focused our attention on the phase stability relations in invariant systems. Even for these rather simple systems, the results are new and unusual. In the meantime, it is clear that really interesting action is to be expected in more complex systems (i.e., multisystems). Work along this line is in progress and will be reported elsewhere. Appendix

(A3)

with respect to Σ and λ. This gives

{

2ΣT + λTν ) 0 νΣ ) ∆G

(A4)

Equations A4 are further solved as follows. Let

Σ ) (Σ ′, Σ ′′)T

(A5)

ν ) (ν′, ν′′)

(A6)

Σ′ ) (Σ1, Σ2)T

(A7)

Σ′′ ) (Σ3, ...,Σn, Σn+1, Σn+2)T

(A8)

such that

[

ν ν v′ ) ν11 ν12 21 22

[

]

(A9)

ν ‚‚‚ ν1n ν1,n+1 ν1,n+2 ν′′ ) ν13 23 ‚‚‚ ν2n ν2,n+1 ν2,n+ 2

]

(A10)

Without loss of generality, it is assumed that ν′ is nonsingular. Thus, eqs A4 may be written as

2Σ′T + λTν′ ) 0

(A11)

2Σ′′T + λTν′′ ) 0

(A12)

ν′Σ′ + ν′′Σ′′ ) ∆G

(A13)

Now, since ν′ is nonsingular, eq A11 may be solved for λ

λ ) -2(ν′)-1Σ′

(A14)

Substituting eq A14 into eq A12 results in

Σ′′ ) ν′′T(ν′T)-1Σ′

(A15)

Substituting further eq A15 into eq A13, we have

[ν′ + ν′′ν′′T(ν′)-1]Σ′ ) ∆G

(A16)

Consider next the square matrix

(

∆ν ∆ν g ) ννT ) ∆ν11 ∆ν12 12 22

)

(A17)

Here, we present a brief proof of eq 26. Obviously, at T equilibrium, the vector G h eq ) (G h eq h eq h eq 1 , G 2 , ,,,, G n ) , eq 24, must satisfy the relation

where ∆ν11,∆ν22, and ∆ν12 are given by eqs 28-30. The determinant of g is necessarily a positive value, and therefore, the inverse of g exists. Taking into account eqs A9 and A10, matrix g may be presented as

νG h eq ) 0

g ) ν′ν′T + ν′′ν′′T

(A1)


Fishtik

or

ν′′ν′′T ) g - ν′ν′T

(A18)

Substituting eq A18 into eq A16 and solving for Σ′, we obtain

Σ′ ) ν′Tg-1∆G ) ν′T(ννT) - 1∆G

(A19)

and from eq A19 and A15

Σ′′ ) ν′′Tg-1∆G ) ν′′T(ννT)-1∆G

(A20)

Equation 26 is readily obtained by substituting eqs A19 and A20 into eq A5. References and Notes (1) Schreinemakers, F. A. H. Mono- and divariant equilibria. Proc. K. Ned. Akad. Wet. 1915-1925, 18-28 (29 separate articles in the series).

(2) Morey, G. W. In The phase rule and heterogeneous equilibrium; Donnan, F. G., Haar, A., Eds.; Yale University Press: New Haven, CT, 1936; Vol 1. (Thermodynamics), pp 233-293. (3) Niggli, P. Rocks and mineral deposits (English translation); W. H. Freeman and Co.: San Francisco, 1954. (4) Korzhinsky, D. S. Physicochemical basis of analysis of the paragenesis of minerals (English translation); Consultants Bureau, Inc.: New York, 1957. (5) Zen, E.-A. U.S. Geol. SurV. Bull. 1966, 1225, 1. (6) Stout, J. H.; Guo, Q. Am. J. Sci. 1994, 294, 337. (7) Hillert, M. Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis; Cambridge University Press: New York, 1998. (8) Hu, J.; Yin, H.; Duan, Z. J. Metamorph. Geol. 2004, 22, 413. (9) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithms; Wiley & Sons: New York, 1982. (10) Sandler, S. I. Chemical and Engineering Thermodynamics; Wiley & Sons: New York, 1999. (11) Fishtik, I. J. Phys. Chem. B 2005, 109, 3851. (12) Fishtik, I.; Datta, R. J. Phys. Chem. A 2004, 108, 5727. (13) Fishtik, I.; Gutman, I.; Nagypal, I. J. Chem. Soc., Faraday Trans. 1996, 92, 3525. (14) Fishtik, I.; Datta, R. J. Chem. Inf. Comput. Sci. 2003, 43, 1259. (15) Luttge, A.; Bolton, E. W.; Rye, D. M. Contrib. Mineral. Petrol. 2004, 146, 546. (16) Holland, T. J. B.; Powell, R. J. Metamorph. Geol. 1998, 16, 509.

Phase Stability Relations in Invariant Systems - American Chemical

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