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Solid-Solution Solubilities and Thermodynamics: Sulfates, Carbonates and Halides Pierre Glynn U.S. Geological Survey 432 National Center Reston, Virginia 20192

INTRODUCTION This review updates and expands an earlier study (Glynn 1990), that is presently out of print. The principal objectives of this chapter are (1) to review the thermodynamic theory of solid-solution aqueous-solution interactions, particularly as it pertains to lowtemperature systems (between 0 and 100°C), and (2) to summarize available data on the effects of ionic substitutions on the thermodynamic properties of binary sulfate solidsolutions. Selected carbonate and halide solid-solutions are also considered. Studies of solid-solution aqueous-solution (SSAS) systems commonly focus on measuring the partitioning of trace components between solid and aqueous phases. The effect of solid-solution formation on mineral solubilities in aqueous media is rarely investigated. Several studies, however, have examined the thermodynamics of SSAS systems, describing theoretical and experimental aspects of solid-solution dissolution and component-distribution reactions (Lippmann 1977, 1980; Thorstenson and Plummer 1977, Plummer and Busenberg 1987, Glynn and Reardon 1990, 1992; Glynn et al. 1990, 1992; Glynn 1991, 1992; Glynn and Parkhurst 1992, Königsberger and Gamsjäger 1991, 1992; Gamjäger et al. 2000). The present chapter describes and compares some of the concepts presented by the above authors, and presents methods that can be used to estimate the effect of SSAS reactions on the chemical evolution of natural waters. Sulfate minerals are particularly well suited for the investigation of the thermodynamics of SSAS systems, because their generally high solubilities facilitate the attainment of thermodynamic equilibrium states and their commonly large crystal structures tend to allow considerable ionic substitution. Field or laboratory observations of miscibility gaps, spinodal gaps, critical mixing points, or distribution coefficients can be used to estimate solid-solution excess-freeenergies, which are needed for the calculation of solid-solution solubilities and of potential component-partitioning behavior. Experimental measurements of the solubility and thermodynamic properties of solid solutions are generally not available, particularly in low-temperature aqueous solutions. A database of excess-free-energy parameters is presented here for sulfate solid solutions and also for selected carbonate and halide solid solutions, based on reported compositional ranges observed in natural environments or through experimental synthesis. When available, excess-free-energy parameters obtained from laboratory equilibration experiments are also provided. It is hoped that this chapter will stimulate interest in obtaining thermodynamic data based on well-controlled laboratory experiments, and will encourage further research on the thermodynamics of SSAS systems.

1529-6466/00/0040-0010$05.00

DOI: 10.2138/rmg.2000.40.10

482

Glynn DEFINITIONS AND REPRESENTATION OF THERMODYNAMIC STATES

Several thermodynamic states are of interest in the study of SSAS systems. The following sections discuss the concepts of thermodynamic equilibrium, primary saturation, and stoichiometric saturation. Thermodynamic equilibrium states Thermodynamic equilibrium in a system with a binary solid solution B1-xCxA can be defined by the law-of-mass-action equations: [B+][A-] = KBAaBA = KBAχBAγBA

(1)

[C+][A-] = KCAaCA = KCAχCAγCA -

+

+

(2) -

+

+

where [A ], [B ] and [C ] are the activities of A , B and C in the aqueous solution; aBA and aCA, χBA and χCA, and γBA and γCA are the activities, mole fractions, and activity coefficients, respectively, of components BA and CA in the equilibrium solid solution. KBA and KCA are the solubility products of pure BA and pure CA solids. Equations (1) and (2) can be used to construct phase diagrams that display the series of possible equilibrium states for any given binary SSAS system. By analogy to the pressure versus mole-fraction diagrams used for binary liquid-vapor systems, Lippmann (1977, 1980, 1982) defined a variable ΣΠ = ([A-], [B+]+[C+]) such that adding together Equations (1) and (2) yields the following relation, known as the "solidus" equation: ΣΠeq = KBAχBAγBA + KCAχCAγCA

(3)

where ΣΠeq is the value of the ΣΠ variable at thermodynamic equilibrium. For a complete description of thermodynamic equilibrium, a second relation must be derived from Equations (1) and (2) (Lippmann 1980, Glynn and Reardon 1990, 1992). The "solutus" equation expresses ΣΠeq as a function of aqueous solution composition: ⎛ χ B,aq χ C,aq ⎞ ⎟⎟ + ΣΠeq = 1 / ⎜⎜ ⎝ K BA γ BA K CA γ CA ⎠

(4)

where the aqueous activity fractions χB,aq and χ C,aq are defined as

χB,aq = [B+]/([C+]+[B+]) and, χ C,aq = [C+]/([B+]+[C+]). Solid-solution activity coefficients can be fitted using the following equations: 1n γCA = χ2BA[a0 – a1(3χCA – χBA)+a2(χCA – χBA)(5χCA – χBA)+ … ]

(5)

1n γBA = χ2CA [a0 + a1(3χBA – χCA)+a2(χBA – χCA)(5χBA – χCA)+ … ]

(6)

Equations (5) and (6) are derived from Guggenheim’s expansion series for the excess Gibbs free energy of mixing, GE (Guggenheim 1937, Redlich and Kister 1948): GE = χBAχCART(a0 + a1(χBA – χCA) + a2(χBA – χCA)2 + ...)

(7)

where R is the gas constant, T is the absolute temperature and the ai parameters are dimensionless fitting parameters, occasionally referred to as excess free-energy parameters. Uppercase Ai parameters expressed in Joules/mol are also occasionally used in this paper and refer to the product: Ai = RTai. Equation (6) differs from Equation (5) not only because of a switch between the BA and CA end members, but also because of a sign change in the second term (and other

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even terms) of the series. The convention adopted here follows that adopted in the MBSSAS code (Glynn 1991) for subregular solutions: a negative value for the a1 parameter skews a possible miscibility gap towards the CA end member, that is, a negative value preferentially increases the excess free energy of solid solutions closer to the CA end member. The first two terms of Equations (5) and (6) are generally sufficient to represent accurately the dependence of γBA and γCA composition (Glynn and Reardon 1990, 1992). Indeed, in the case of a solid solution with a small difference in the size of the substituting ions (relative to the size of the non-substituting ion), the first parameter, a0, is usually sufficient to describe the solid solution (Urusov 1974); Equations (5) and (6) then become identical to those of the “regular” solid-solution model of Hildebrand (1936). For the case where both a0 and a1 parameters are needed, Equations (5) and (6) become equivalent to those of the “subregular” solid-solution model of Thompson and Waldbaum (1969), a model used extensively to describe solid-solution thermodynamics at high temperatures (Saxena 1973, Ganguly and Saxena 1987). Equations (5) and (6) can also be shown to be equivalent to the Margules activity-coefficient series (Margules 1895, Prigogine and Defay 1954). Glynn (1991) provided the relations between the “subregular” excess-free-energy parameters used in the Guggenheim series, the Thompson and Waldbaum model (Thompson 1967, Thompson and Waldbaum 1969) and the original Margules series. The Guggenheim parameters A0 and A1 (in dimensional form), the dimensionless Guggenheim parameters a0 and a1, and the Thompson and Waldbaum parameters W12 and W21 for a subregular binary solid solution (or WBC and WCB for components BA and CA) are related by the following equations: WBC = A0 – Al = RT(a0 – al)

(8)

WCB = A0 + Al = RT(a0 + al))

(9)

According to the Thompson and Waldbaum model, the excess free energy of mixing of a binary solid solution can be expressed by the following relation: GE = χBAχCA(WBC χCA + WCB χBA)

(10)

Lippmann’s solidus and solutus curves can be used to predict the solubility of any binary solid solution at thermodynamic equilibrium, as well as the distribution of components between solid and aqueous phases, when the solid-phase and the aqueousphase activity coefficients are known. Figure 1 shows an example of a Lippmann phase diagram for the (Sr,Ba)SO4–H2O system at 25°C, modeled assuming a dimensionless a0 value of 2.0. This value of a0 corresponds to the maximum value that still allows complete miscibility in a regular solid-solution series. Aqueous solutions that plot below the solutus curve are undersaturated with respect to all solid phases, including the pure end-member solids, whereas solutions plotting above the solutus are supersaturated with respect to one or more solid-solution compositions. Primary saturation states Primary saturation is the first state reached during the congruent dissolution of a solid solution, at which the aqueous solution is saturated with respect to a secondary solid phase, (Garrels and Wollast 1978, Denis and Michard 1983, Glynn and Reardon 1990, 1992). This secondary solid will typically have a composition different from that of the dissolving solid. At primary saturation, the aqueous phase is at thermodynamic equilibrium with respect to this secondary solid but remains undersaturated with respect to the primary dissolving solid. The series of possible primary-saturation states for a given SSAS system is represented by the solutus curve on a Lippmann diagram (Fig. 1).

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Figure 1. Lippmann diagram (with stoichiometric and pure-phase saturation curves) for the (Sr,Ba)SO4 - H2O system at 25°C. An a0 value of 2.0 is assumed. End-member solubility products from PHREEQC (Parkhurst and Appelo 2000): 10-6.63 for celestine (SrSO4), 10-9.97 for barite (BaSO4). Points T1 and T2 give the aqueous and solid phase compositions, respectively, of a system at thermodynamic equilibrium with respect to a Sr.5Ba.5SO4 solid. Points P1 and P2 describe the state of a system at primary saturation with respect to the same solid. Point MS1 gives the composition of an aqueous phase at congruent stoichiometric saturation with respect to that solid. The pure BaSO4 saturation curve cannot be distinguished from the solutus curve on this graph.

In the specific case of a “strictly congruent” dissolution process occurring in an aqueous phase with a [B+]/[C+] activity ratio equal to the B+/C+ ratio in the solid, primary saturation can be approximately found by drawing a straight vertical line on the Lippmann diagram from the solid-phase composition to the solutus (Fig. l). For an exact calculation, the following relations may be used to determine the primary saturation state: XB,aq = ƒBχBA

(11)

XC,aq = ƒCχCA

(12)

where ƒB and ƒC are factors correcting for a possible difference in the aqueous speciation and activity coefficients of B+ and C+. The equation used to calculate the value of ΣΠ at primary saturation as a function of solid composition, for a “strictly congruent” dissolution process, may be found by combining Equation (4), the Lippmann solutus equation, with Equations (11) and (12): ⎛ χ BA ƒ B χ CA ƒ C ⎞ ⎟⎟ ΣΠps = 1 / ⎜⎜ + ⎝ K BA γ BA, y K CA γ CA, y ⎠

(13)

where γBA,y and γCA,y refer to the activity coefficients of BA and CA in the secondary

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solid B1-yCyA with respect to which the aqueous solution (at primary saturation) is in temporary thermodynamic equilibrium. ΣΠps refers to the value of the ΣΠ variable as specifically defined at primary saturation. The composition of the B1-yCyA phase will generally not be known. By equating ΣΠps(x) (Eqn. 13) to ΣΠeq(y) (Eqn. 3), the relation between the initial solid composition B1-xCxA and the secondary solid B1-yCyA is obtained: ⎛ χ BA ƒ B χ CA ƒ C ⎞ ⎟⎟ = χCA,yγCA,yKCA + χBA,yγBA,yKBA + 1 / ⎜⎜ ⎝ K BA γ BA, y K CA γ CA, y ⎠

(14)

In the case of a non-ideal solid-solution series, Equation (14) must be solved graphically or by an iterative technique, because γBA,y and γCA,y are typically exponential functions of χCA,y. Stoichiometric saturation states Stoichiometric saturation was formally defined by Thorstenson and Plummer (1977). These authors argued that solid-solution compositions typically remain invariant during solid-aqueous phase reactions in low-temperature geological environments, thereby preventing attainment of thermodynamic equilibrium. Thorstenson and Plummer defined stoichiometric saturation as the pseudo-equilibrium state that may occur between an aqueous phase and a multi-component solid solution, “in situations where the composition of the solid phase remains invariant, owing to kinetic restrictions, even though the solid phase may be a part of a continuous compositional series”. The stoichiometric saturation concept assumes that a solid solution can, under certain circumstances, behave as if it were a pure one-component phase. In such a situation, the dissolution of a solid solution B1-xCxA can be expressed as: B1-xCxA → (1 - x)B+ + xC+ + A-

(15)

Applying the law of mass action then gives the defining condition for stoichiometric saturation states: [C + ]x [ B + ]1− x [ A− ] ⎛ − ΔGr0 ⎞ IAPss = Kss = = exp ⎜⎜ ⎟⎟ 1 ⎝ Rt ⎠

(16)

where x and (1-x) are equal to χCA and χBA respectively and where -ΔG0r is the standard Gibbs free energy change of Reaction (15). The solid has unit activity because it is assumed to behave as if it were a pure one-component phase. According to Thorstenson and Plummer’s (1977) definition of stoichiometric saturation, an aqueous solution at thermodynamic equilibrium with respect to a solid B1xCxA will always be at stoichiometric saturation with respect to that same solid. The converse statement, however, is not necessarily true: stoichiometric saturation does not necessarily imply thermodynamic equilibrium. Stoichiometric saturation states can be represented on Lippmann phase diagrams (e.g. Fig. 1) by relating the total solubility-product variable ΣΠss (defined specifically at stoichiometric saturation with respect to a solid B1-xCxA) to the constant Kss (Eqn. 16) and to the aqueous activity fractions χB,aq and χC,aq: ΣΠss =

K ss

χ

1− x B ,aq

χ Cx ,aq

(17)

In contrast to thermodynamic equilibrium, for which a single (χB,aq, ΣΠeq) point satisfies Equations (1) and (2), stoichiometric saturation with respect to a given solid

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composition is represented by a series of (χB,aq, ΣΠss), points, all defined by Equation (17). The minimum in each stoichiometric saturation curve for a given solid will occur at the point where χC,aq = x and χB,aq = 1-x. This condition can also be generally expressed as the point at which χB,aq/χC,aq = (l-x)/x. This minimum in the stoichiometric saturation curve may be reached through a “strictly congruent” dissolution process occurring in an aqueous phase with a [B+]/[C+]activity ratio equal to the B+/C+ ratio in the solid. The series of minimum stoichiometric points for a series of solid-phase compositions ranging from x = 0 to x = 1 defines the minimum stoichiometric saturation curve (Glynn and Reardon 1990), also called the equal-G curve (Königsberger and Gamsjäger 1992). The minimum saturation curve essentially defines the series of stoichiometric saturation curves that can be constructed in the system and is a function of the end-member solubility products and of the excess-free-energy of mixing, GE, of the solid-solution series: 1− x x ΣΠms = K BA K CA exp(G E / RT )

(18)

In the case of an ideal solid-solution series (GE = 0), the minimum stoichiometric saturation curve will be a linear function of the end-member solubility products when plotted on a log scale. As shown in Figure 1, stoichiometric saturation states never plot below the solutus curve. Indeed, stoichiometric saturation can never be reached before primary saturation in a solid-solution dissolution experiment. The unique point at which a stoichiometric saturation curve (for a given solid B1-xCxA) joins the Lippmann solutus represents the composition of an aqueous solution at thermodynamic equilibrium with respect to the Bl-xCxA solid. Saturation curves for the pure BA and CA end-member solids can also be drawn on Lippmann diagrams (Lippmann 1980, 1982): ΣΠBA =

K BA

χ B ,aq

(19)

ΣΠCA =

K CA

χ C ,aq

(20)

These equations define the families of (χBA, ΣΠBA) and (χCA, ΣΠCA) conditions for which a solution containing A-, B+ and C+ ions will be saturated with respect to pure BA and pure CA solids. Thermodynamic equilibrium with respect to a mechanical mixture of the two pure BA and CA solids, in contrast to a solid solution of BA and CA, is represented on a Lippmann diagram by a single point, namely the intersection of the pure BA and pure CA saturation curves. The coordinates of this intersection are:

χ B,intaq =

K BA K BA + K CA

(21)

ΣΠint = KCA + KBA

(22)

COMPARISON OF SOLID-SOLUTION AND PURE-PHASE SOLLTBILITIES In predicting solid-solution solubilities, one of two possible hypotheses must be chosen. In the first, the solid solution is treated as a one-component or pure-phase solid, requiring a sufficiently short equilibration time, a sufficiently high solid-to-aqueoussolution ratio, and relatively low solubility ofthe solid. These requirements are needed to ensure that no significant recrystallization of the initial solid or precipitation of a secondary solid-phase occurs. For such situations, the stoichiometric saturation concept may apply. The second hypothesis considers the solid as a multi-component solid solution,

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capable of adjusting its composition in response to the aqueous solution composition. This compositional adjustment requires a long equilibration period, a relatively high solubility of the solid, and a relatively low solid-to-aqueous-solution ratio. In this case, the assumption of thermodynamic equilibrium may apply. If the equilibration period is too short for thermodynamic equilibrium to have been achieved, but if an outer surface layer of the solid has been able to recrystallize (because of the high solubility of the solid), the concept of primary saturation may apply. Currently, there are insufficient data to determine the exact conditions for which each of these assumptions may apply, especially in field situations. In many instances, neither one of these hypotheses will explain the observed solubility of a solid solution, which may lie between the “maximum” stoichiometric saturation solubility and the “minimum” primary saturation solubility. Nonetheless, these solubility limits can often be estimated. Stoichiometric saturation solubilities The case of stoichiometric saturation states attained after “strictly congruent” dissolution is examined here. Two hypothetical B1-xCxA solid solutions are considered: (1) calcite (pKsp = 8.48; Plummer and Busenberg 1982) with a more soluble trace NiCO3 component (pKsp = 6.87; Smith and Martell 1976) and (2) calcite with a less soluble trace CdCO3 component (pKsp = 11.31; Davis et al. 1987). Contour plots of saturation indices (SI = log[IAP/Ksp]), with respect to major and trace end-member components, are shown in Figure 2 as a function of the a0 value (assuming a regular solid-solution model) and of the logarithm of the mole fraction of the trace component (where 10-6 ≤ χCA ≤ 0.5). SI values are calculated for major (BA) and trace (CA) components using the relations: ⎡ ⎛ xB ,aq ⎞ x ⎤ SIBA = – log(KBA) + log ⎢ K ss ⎜ ⎟ ⎥ ⎣⎢ ⎝ xC ,aq ⎠ ⎦⎥

⎡ ⎛x SICA = – log(KCA) + log ⎢ K ss ⎜⎜ C , aq ⎢ ⎝ xB ,aq ⎣

⎞ ⎟⎟ ⎠

(1- x )

(23)

⎤ ⎥ ⎥ ⎦

(24)

Kss values are evaluated from Thorstenson and Plummer’s (1977) Equation (22), modified assuming a regular solid-solution model: (1− x ) x (1 − x ) Kss = K BA K CA

(1− x )

x x exp[a0 x(1 − x)]

(25)

Assuming that the dissolution to stoichiometric saturation takes place in initially pure water and that the aqueous activity ratio of the major and minor ions is equal to their concentration ratio, the relation χB,aq/χC,aq = (l-x)/x will apply. Using this relation, applicable only at “minimum stoichiometric saturation” (Glynn and Reardon 1990, 1992), the following equations may be derived from Equations (23), (24) and (25):

a0 x(1- x) ⎛ K CA ⎞ SIBA = x log ⎜ ⎟ + log(1 – x) + 1n(10) ⎝ K BA ⎠

(26)

a0 x(1- x) ⎛ K BA ⎞ SICA = (1 –x)log ⎜ ⎟ + log x + 1n(10) ⎝ K CA ⎠

(27)

The SI contour plots drawn using Equations (26) and (27) show the miscibility gap and spinodal gap lines separating intrinsically stable, metastable, and unstable solid solutions (Prigogine and Defay 1954, Swalin 1972). In natural environments, metastable

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Figure 2. Values of major- and trace-component saturation indices (SI) for solids at congruent stoichiometric saturation (in solid lines): (a) calcite SI at saturation with (Ca,Cd)CO3; (b) otavite SI at saturation with (Ca,Cd)CO3; (c) calcite SI at saturation with (Ca,Ni)CO3; (d) NiCO3 SI at saturation with (Ca,Ni)CO3. Miscibility-gap lines (short dashes) and spinodal gap lines (long dashes) are also shown.

solid-solution compositions may in some cases persist on a geological time scale (depending on the solubility of the solid), whereas unstable solid solutions formed in lowtemperature environments are not likely to do so. Busenberg and Plummer (1989), in their study of magnesian calcite solubilities, observed that the highest known Mg contents in modem natural biogenic calcites correspond to the predicted spinodal composition. Metastable compositions will probably not persist, however, in solid solutions that have higher solubilities or which have been reacted for a longer period of time (or at higher temperatures) than the biogenic magnesian calcites. Although intrinsically unstable solid solutions, formed at low temperatures, may not be found in geological environments, they commonly can be synthesized in the laboratory (e.g.

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strontian aragonites: Plummer and Busenberg 1987; barian strontianites: Glynn et al. 1988) Figures 2a and 2b show the case of a solid-solution series, (Ca,Cd)CO3, with a much less soluble trace end member. If the mole fraction of the trace component is sufficiently high (χCdCO3 > 10-2.8), the aqueous phase at stoichiometric saturation will be supersaturated with respect to the trace end member (except at unrealistic negative a0 values not shown on the plot). The lower solubility of the trace component will generally cause negative SI values (undersaturation) for the major component, except at high a0 values [higher than 7.5 in the case of (Ca,Cd)CO3] for which the solid solutions will generally be metastable or unstable. Calcite SI values in Figure 2a show that the mole fraction of the trace component must be sufficiently high ((χCdCO3 > 10-2.5) for this effect to be measurable in field environments (typical uncertainty on calcite SI values ≈ 0.01 minimum) or in laboratory experiments. A laboratory example of the above principle is given by the “strictly-congruent” dissolution experiment of Denis and Michard (1983) on an anhydrite containing 3.5% Sr (on a mole basis). Analysis of their results shows that maximum SIcelestine and SIgypsum values of 0.37 and -0.04 respectively were attained after four days. Their last sample (after six days) gave SIcelestine and SIgypsum values of 0.35 and -0.04, respectively. Although these results show that stoichiometric saturation was not obtained with respect to the original anhydrite phase (probably because of back-precipitation of gypsum), supersaturation did occur with respect to the less soluble celestine1 component. In the case of solid solutions with a more soluble trace component, aqueous solutions at “minimum stoichiometric saturation” will generally be supersaturated with respect to the major component, and will be undersaturated with respect to the more soluble trace end member (Figs. 2c and 2d). Aqueous solutions at minimum stoichiometric saturation with respect to (Ca,Ni)CO3 solid solutions will be supersaturated with respect to pure calcite at a0 values greater than -2.7. SIcalcite values greater than +0.01, however, will only be found at mole fractions of NiCO3 greater than approximately 10-2.5. In contrast, SI NiCO3 values will exhibit significant undersaturation even at high χ NiCO3 mole fractions (2 < SI NiCO3 < -1 at χ NiCO3 = 0.5). Primary saturation and thermodynamic equilibrium solubilities A detailed discussion of solid-solution solubilities at primary saturation states and at thermodynamic equilibrium states was given by Glynn and Reardon (1990, 1992). Some fundamental principles governing these thermodynamic states are discussed below. The composition of a SSAS system at primary saturation or at stoichiometric saturation will be generally independent of the initial ratio of solid to aqueous solution, but will depend on the initial aqueous-solution composition (prior to the dissolution of the solid). In contrast, the final thermodynamic equilibrium state of a SSAS system attained after a dissolution or recrystallization process will generally depend not only on the initial composition of the system but also on the initial ratio of solid to aqueous solution (Glynn et al. 1990, 1992). An aqueous solution at primary saturation or at thermodynamic equilibrium with respect to a solid solution will be undersaturated with respect to all end-member component phases of the solid solution (Figs. 1, 3, 4 and 5). A positive excess-freeenergy of mixing (that is, a positive a0 value in the case of a regular solid solution) will

1

The mineral celestine is also commonly known as celestite.

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raise the position of the solutus curve relative to that of an ideal solid-solution system. A positive excess-free-energy of mixing will therefore increase the solubility of a solid solution at primary saturation or at thermodynamic equilibrium. The pure end-member saturation curves on a Lippmann diagram offer an upward limit on the position of the solutus. Conversely, a negative excess-free-energy of mixing will lower the position of the solutus relative to that of an ideal solid-solution series. Solid solutions with negative excess-free-energies of mixing may therefore show a greater degree of undersaturation relative to the pure end-member phases, although no examples of this have been found so far in ionic solids.

Figure 3. Lippmann phase diagram for the (Na,K)Fe3(SO4)2(OH)6–H2O system at 25°C. The excess-free-energy-of-mixing parameter a0 was extrapolated from Stoffregen’s (1993) experimental determination of a0 at 200°C. End-member solubility products from PHREEQC (Parkhurst and Appelo 2000): 10-5.28 for natrojarosite [NaFe3(SO4)2(OH)6], 10-9.21 for jarosite [KFe3(SO4)2(OH)6]. The pure jarosite saturation curve cannot be distinguished from the solutus curve on this graph.

The solutus curve, in binary SSAS systems with ideal or positive solid-solution freeenergies of mixing and with large differences (more than an order of magnitude) in endmember solubility products, will closely follow the pure-phase saturation curve of the less soluble end member (except at high aqueous activity fractions of the more soluble component, e.g. Figs. 1 and 3). In contrast, ideal solid solutions with very close endmember solubility products (less than an order of magnitude apart) will have a solutus curve up to 2 times lower in than the pure end-member saturation curves. The factor of 2 is obtained for the case where the two end-member solubility products are equal and can be derived from Equations (4), (21), and (22) (e.g. Fig. 4). Figures 3, 4, and 5 show examples of pure end-member saturation curves plotted on Lippmann phase diagrams. Figure 3, for the natrojarosite-jarosite solid-solution series, is similar to the diagram for the celestine-barite series (Fig. 1) in that there is a large difference between the end-member solubility products. As a result of this large

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Figure 4. Lippmann phase diagram for the (Na,H3O)Fe3(SO4)2(OH)6–H2O system at 25°C. Ideal mixing was assumed for the solid-solution series (a0 = 0). End-member solubility products from PHREEQC (Parkhurst and Appelo 2000): 10-5.28 for natrojarosite [NaFe3(SO4)2(OH)6], 10-5.39 for hydroniumjarosite [H3OFe3(SO4)2(OH)6].

difference, the saturation curve for pure jarosite, the least soluble end member, is very close to the solutus curve, only slightly above it, except in the region where the K/(K+Na) ratio in the aqueous solution (and consequently the χK+ value) goes to 0. In that region, the jarosite saturation curve climbs sharply towards an infinite ΣΠ value, whereas the solutus curve converges to the ΣΠ value of the natrojarosite end-member solubility product. In contrast, the saturation curve for the more soluble end member, natrojarosite, plots well above the solutus curve. This behavior is typical for SSAS systems with large differences in end-member solubility products. Such systems are typically controlled by the less soluble end-member component. Increasing the excess free energy of mixing in these systems, away from the 0 value of an ideal system, typically causes very little effect on the position of the solutus curve; only the solidus curve is affected. Figures 1 and 3 show the difference in the solidus curve for a nearly ideal system (Fig. 3) and for a system with a maximum a0 value that nevertheless allows complete miscibility (Fig. 1). If the excess-free-energy-of-mixing is increased further, a miscibility gap develops as can be seen in the example for the (Pb,Ba)SO4 series (Fig. 6). Figures 4 and 5 provide an example of a SSAS system with close end-member solubility products: the solid-solution series natrojarosite‒hydronium jarosite. Both pure end-member saturation curves in such a system plot well above the solutus curve, in the case of an assumed ideal system (Fig. 4). If the excess free energy of mixing is increased, however, the solutus curve will plot progressively closer to the end-member saturation curves (Fig. 5) and the solidus curve will also be significantly affected. ESTIMATION OF THERMODYNAMIC MIXING PARAMETERS There are two main applications for SSAS theory in the chemical modeling of aqueous systems: (1) the prediction of solid-solution solubilities, (2) the prediction of the

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distribution of trace components between solid and aqueous phases. Currently, a big problem with both types of predictions is the lack of low-temperature data on solidsolution excess-free-energy functions, and therefore on solid-phase activity coefficients. The two-parameter Guggenheim expansion series (the “subregular” model) has been successfully used to fit laboratory solubility data for the systems (Sr,Ca)CO3-H2O (Plummer and Busenberg 1987), (Ba,Sr) CO3-H2O (Glynn et al. 1988), (Ca,Mg) CO3H2O (Busenberg and Plummer 1989), K(CI,Br)-H2O, and Na(CI,Br)-H2O (Glynn et al. 1990, 1992). The one-parameter Guggenheim series (the “regular” model) also has been frequently used (Lippmann 1980, Kirgintsev and Truslulikova 1966). More laboratory determinations of thermodynamic mixing parameters are needed, not only to acquire data on binary and multicomponent solid solutions, but also to test the applicability of the regular and subregular models, and to compare them with other excess-free-energy models. In the mean time, as a better approximation than the commonly used assumption of “ideal” solid solutions, the a0 and a1 dimensionless parameters can be estimated for many binary systems. The MBSSAS computer code (Glynn 1991) uses either observed or estimated (1) miscibility-gap data, (2) spinodal-gap data, (3) critical temperature and

Figure 5. Lippmann phase diagram for the (Na,H3O)Fe3(SO4)2(OH)6–H2O system at 25°C. An a0 value of 2.6 was used for comparison with This a0 value is probably a maximum one, based on Brophy and Shendan’s (1993) experimental observations and assumption of complete mixing for the (Na,H3O,K)-jarosite series at 114°C. The maximum excess-free-energy-ofmixing that would still allow complete miscibility was assumed for that temperature and the corresponding value of a0 was extrapolated down to 25°C. At that temperature, the system exhibits a miscibility gap as can be seen from the eutectic point (A) in the solutus curve. At that point the aqueous solution is in equilibrium with two solid solutions, of compositions M1 and M2 given by the intersections between the solidus curve and a horizontal line passing through the eutectic point (A).

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Figure 6. Lippmann diagram (with stoichiometric and pure-phase saturation curves) for the (Pb,Ba)SO4–H2O system at 25°C. An estimated a0 value of 2.7 is used, based on observed compositions in natural minerals (Palache et al 1951). End-member solubility products from PHREEQC (Parkhurst and Appelo 2000): 10-7.79 for anglesite (PbSO4), 10-9.97 for barite (BaSO4). The miscibility gap in the system is revealed by the presence of a peritectic point (A) which defines the composition of the aqueous solution at equilibrium with two miscibility-gap solids of composition M1 and M2. The spinodal gap in the system is given by the local minimum and maximum in the solidus curve (solid compositions SP1 and SP2).

critical mole-fraction of mixing data, (4) distribution-coefficient data, (5) alyotropic point data (Lippmann 1980) or (6) activity-coefficient data to calculate a0 and a1 parameters for binary solid solutions. Depending on whether one or two datum points are given, the program computes either a regular or subregular model. Miscibility-gap data Miscibility gaps, as determined from mineral compositions observed in the field, can be used to estimate thermodynamic mixing parameters in the absence of more accurate laboratory data. This approach suffers from several problems. The maximum mole fraction of trace component found may not correspond to the miscibility-gap fraction. In this case, if the solid-solution composition is stable, then the calculated excess free energy will be overestimated. On the other hand, if the analyzed “limiting” composition really represents a mechanical mixture of two end members, rather than a solid-solution mineral, the calculated excess free energy may be underestimated. If the mineral was formed at much higher temperatures than the temperature of interest and if the mineral is fairly unreactive at the lower temperature, the maximum solid-solution mole fraction observed may well be metastable or even unstable. The temperature of formation (and of equilibration) of the solid solution may not be known. If a lower temperature is assumed, the excess free energy will be underestimated.

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Additionally, the extrapolation to 25°C of excess free energies estimated at higher temperatures will introduce an additional error; typically, excess enthalpy and excess entropy data are not available for solid-solution series. A partial solid-solution series may not be isomorphous (i.e. the end members may not have the same structure). In that case, the excess-free-energy parameters should be calculated only on a single side of the miscibility gap. On the other side of the miscibility gap, a different model will apply. Despite the above problems, mixing parameters estimated from miscibility-gap information are still an improvement on the assumption of an ideal solid-solution model. The a0 parameters estimated from potential miscibility-gap data provided by Palache et al. (1951), Jambor et al. (this volume), and other investigators, and also from various experimental determinations, are presented in Tables 1, 2, and 3 for several lowtemperature mineral groups. Because of the large uncertainties in the data and in the estimation procedure, a subregular model is usually unwarranted. As a result, the estimated a0 values that are presented should be used only for solid-solution compositions on a single side of the miscibility gap, i.e. only up to the given miscibility fraction. As can be seen in the tables, binary solid-solution series that have a miscibility gap but nevertheless show significant amounts of substitutional variation will have an a0 excessfree-energy parameter slightly above a value of 2 (dimensionless). This assumes that a regular solid-solution model can be applied. This is also illustrated in Figure 7, which provides an easy way of determining the applicable a0 value for a solid-solution series, given its miscibility-gap fraction. Spinodal-gap data Spinodal gaps can be used to estimate low-temperature solid-solution mixing parameters. This method is considerably less accurate than the miscibility-gap estimation technique. The estimation assumes that intrinsically unstable compositions are not likely to form during a precipitation process, although metastable compositions may do so. According to thermodynamic theory, solid-solution compositions inside a spinodal gap are intrinsically unstable, whereas compositions inside the miscibility gap, but outside the spinodal gap are metastable (Prigogine and Defay 1954). The presence of a miscibility gap is necessarily always accompanied by that of a spinodal gap. The (Pb,Ba)SO4 solidsolution series provides an example of a system with hypothesized miscibility and spinodal gaps (Fig. 6). One of the problems with the theory of spinodal gaps, at least for low-temperature SSAS systems, is that supposedly intrinsically unstable solid solutions can commonly be precipitated from aqueous solutions in the laboratory. An inability of the regular and subregular excess-free-energy models to describe adequately the relation between miscibility and spinodal gaps could offer one possible explanation. Another possible explanation might be that the thermodynamic properties of solids initially precipitated from aqueous solution are typically quite different from those of well crystallized phases. The initial precipitates are generally much finer grained, have a higher proportion of crystal defects, and may contain water, occluded solution, or other impurities. Defects and impurities in these initial solids, high surficial energy of the solids, and strong hydration bonds of the substituting ions may favor the formation of extensive solidsolution series. Plummer and Busenberg (1987) were able to synthesize a complete suite of compositions in their investigation of the strontianite-aragonite solid-solution series, despite the presence of extensive spinodal and miscibility gaps, that were later determined on the basis of measurements of stoichiometric saturation states. Solids with compositions

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inside the spinodal gap, however, were much more difficult to synthesize (Busenberg, pers. commun.). Similarly, Glynn et al. (1988, and unpublished data) were able to synthesize and conduct stoichiometric saturation experiments on the entire witheritestrontianite series, despite the supposed presence of extensive spinodal and miscibility gaps (Table 2b). Investigations of the barite-anglesite and barite-celestine series also indicate the ability to synthesize solids over the entire compositional range, despite the possible Table la. Estimated mixing parameters for binary sulfate solid solutions at 25°C. References correspond to author initials and year of publication (JNA = Jambor et al., this vol.) syn. = synthetic solids; nat. = natural solids; thermo equil. = thermodynamic equilibrium experiment

System (main, trace)

Misc. Frac. (Xtrace)

Temp. (°C)

Stuctur

e

Estimated a0 at 25°C

Reference

PBF 1951 PBF 1951 W 1933 and present paper INA 2000 PBF 1951

Melanterite group (monoclinic heptahydrates) (Fe,Co)SO4⋅7H2O (Fe,Mn)SO4⋅7H2O (Fe,Mn)SO4·7H2O

25 25 0

mono. mono. mono.

Solid-Solution Solubilities and Thermodynamics: Sulfates, Carbonates and Halides Pierre Glynn U.S. Geological Survey 432 National Center Reston, Virginia 20192

INTRODUCTION This review updates and expands an earlier study (Glynn 1990), that is presently out of print. The principal objectives of this chapter are (1) to review the thermodynamic theory of solid-solution aqueous-solution interactions, particularly as it pertains to lowtemperature systems (between 0 and 100°C), and (2) to summarize available data on the effects of ionic substitutions on the thermodynamic properties of binary sulfate solidsolutions. Selected carbonate and halide solid-solutions are also considered. Studies of solid-solution aqueous-solution (SSAS) systems commonly focus on measuring the partitioning of trace components between solid and aqueous phases. The effect of solid-solution formation on mineral solubilities in aqueous media is rarely investigated. Several studies, however, have examined the thermodynamics of SSAS systems, describing theoretical and experimental aspects of solid-solution dissolution and component-distribution reactions (Lippmann 1977, 1980; Thorstenson and Plummer 1977, Plummer and Busenberg 1987, Glynn and Reardon 1990, 1992; Glynn et al. 1990, 1992; Glynn 1991, 1992; Glynn and Parkhurst 1992, Königsberger and Gamsjäger 1991, 1992; Gamjäger et al. 2000). The present chapter describes and compares some of the concepts presented by the above authors, and presents methods that can be used to estimate the effect of SSAS reactions on the chemical evolution of natural waters. Sulfate minerals are particularly well suited for the investigation of the thermodynamics of SSAS systems, because their generally high solubilities facilitate the attainment of thermodynamic equilibrium states and their commonly large crystal structures tend to allow considerable ionic substitution. Field or laboratory observations of miscibility gaps, spinodal gaps, critical mixing points, or distribution coefficients can be used to estimate solid-solution excess-freeenergies, which are needed for the calculation of solid-solution solubilities and of potential component-partitioning behavior. Experimental measurements of the solubility and thermodynamic properties of solid solutions are generally not available, particularly in low-temperature aqueous solutions. A database of excess-free-energy parameters is presented here for sulfate solid solutions and also for selected carbonate and halide solid solutions, based on reported compositional ranges observed in natural environments or through experimental synthesis. When available, excess-free-energy parameters obtained from laboratory equilibration experiments are also provided. It is hoped that this chapter will stimulate interest in obtaining thermodynamic data based on well-controlled laboratory experiments, and will encourage further research on the thermodynamics of SSAS systems.

1529-6466/00/0040-0010$05.00

DOI: 10.2138/rmg.2000.40.10

482

Glynn DEFINITIONS AND REPRESENTATION OF THERMODYNAMIC STATES

Several thermodynamic states are of interest in the study of SSAS systems. The following sections discuss the concepts of thermodynamic equilibrium, primary saturation, and stoichiometric saturation. Thermodynamic equilibrium states Thermodynamic equilibrium in a system with a binary solid solution B1-xCxA can be defined by the law-of-mass-action equations: [B+][A-] = KBAaBA = KBAχBAγBA

(1)

[C+][A-] = KCAaCA = KCAχCAγCA -

+

+

(2) -

+

+

where [A ], [B ] and [C ] are the activities of A , B and C in the aqueous solution; aBA and aCA, χBA and χCA, and γBA and γCA are the activities, mole fractions, and activity coefficients, respectively, of components BA and CA in the equilibrium solid solution. KBA and KCA are the solubility products of pure BA and pure CA solids. Equations (1) and (2) can be used to construct phase diagrams that display the series of possible equilibrium states for any given binary SSAS system. By analogy to the pressure versus mole-fraction diagrams used for binary liquid-vapor systems, Lippmann (1977, 1980, 1982) defined a variable ΣΠ = ([A-], [B+]+[C+]) such that adding together Equations (1) and (2) yields the following relation, known as the "solidus" equation: ΣΠeq = KBAχBAγBA + KCAχCAγCA

(3)

where ΣΠeq is the value of the ΣΠ variable at thermodynamic equilibrium. For a complete description of thermodynamic equilibrium, a second relation must be derived from Equations (1) and (2) (Lippmann 1980, Glynn and Reardon 1990, 1992). The "solutus" equation expresses ΣΠeq as a function of aqueous solution composition: ⎛ χ B,aq χ C,aq ⎞ ⎟⎟ + ΣΠeq = 1 / ⎜⎜ ⎝ K BA γ BA K CA γ CA ⎠

(4)

where the aqueous activity fractions χB,aq and χ C,aq are defined as

χB,aq = [B+]/([C+]+[B+]) and, χ C,aq = [C+]/([B+]+[C+]). Solid-solution activity coefficients can be fitted using the following equations: 1n γCA = χ2BA[a0 – a1(3χCA – χBA)+a2(χCA – χBA)(5χCA – χBA)+ … ]

(5)

1n γBA = χ2CA [a0 + a1(3χBA – χCA)+a2(χBA – χCA)(5χBA – χCA)+ … ]

(6)

Equations (5) and (6) are derived from Guggenheim’s expansion series for the excess Gibbs free energy of mixing, GE (Guggenheim 1937, Redlich and Kister 1948): GE = χBAχCART(a0 + a1(χBA – χCA) + a2(χBA – χCA)2 + ...)

(7)

where R is the gas constant, T is the absolute temperature and the ai parameters are dimensionless fitting parameters, occasionally referred to as excess free-energy parameters. Uppercase Ai parameters expressed in Joules/mol are also occasionally used in this paper and refer to the product: Ai = RTai. Equation (6) differs from Equation (5) not only because of a switch between the BA and CA end members, but also because of a sign change in the second term (and other

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even terms) of the series. The convention adopted here follows that adopted in the MBSSAS code (Glynn 1991) for subregular solutions: a negative value for the a1 parameter skews a possible miscibility gap towards the CA end member, that is, a negative value preferentially increases the excess free energy of solid solutions closer to the CA end member. The first two terms of Equations (5) and (6) are generally sufficient to represent accurately the dependence of γBA and γCA composition (Glynn and Reardon 1990, 1992). Indeed, in the case of a solid solution with a small difference in the size of the substituting ions (relative to the size of the non-substituting ion), the first parameter, a0, is usually sufficient to describe the solid solution (Urusov 1974); Equations (5) and (6) then become identical to those of the “regular” solid-solution model of Hildebrand (1936). For the case where both a0 and a1 parameters are needed, Equations (5) and (6) become equivalent to those of the “subregular” solid-solution model of Thompson and Waldbaum (1969), a model used extensively to describe solid-solution thermodynamics at high temperatures (Saxena 1973, Ganguly and Saxena 1987). Equations (5) and (6) can also be shown to be equivalent to the Margules activity-coefficient series (Margules 1895, Prigogine and Defay 1954). Glynn (1991) provided the relations between the “subregular” excess-free-energy parameters used in the Guggenheim series, the Thompson and Waldbaum model (Thompson 1967, Thompson and Waldbaum 1969) and the original Margules series. The Guggenheim parameters A0 and A1 (in dimensional form), the dimensionless Guggenheim parameters a0 and a1, and the Thompson and Waldbaum parameters W12 and W21 for a subregular binary solid solution (or WBC and WCB for components BA and CA) are related by the following equations: WBC = A0 – Al = RT(a0 – al)

(8)

WCB = A0 + Al = RT(a0 + al))

(9)

According to the Thompson and Waldbaum model, the excess free energy of mixing of a binary solid solution can be expressed by the following relation: GE = χBAχCA(WBC χCA + WCB χBA)

(10)

Lippmann’s solidus and solutus curves can be used to predict the solubility of any binary solid solution at thermodynamic equilibrium, as well as the distribution of components between solid and aqueous phases, when the solid-phase and the aqueousphase activity coefficients are known. Figure 1 shows an example of a Lippmann phase diagram for the (Sr,Ba)SO4–H2O system at 25°C, modeled assuming a dimensionless a0 value of 2.0. This value of a0 corresponds to the maximum value that still allows complete miscibility in a regular solid-solution series. Aqueous solutions that plot below the solutus curve are undersaturated with respect to all solid phases, including the pure end-member solids, whereas solutions plotting above the solutus are supersaturated with respect to one or more solid-solution compositions. Primary saturation states Primary saturation is the first state reached during the congruent dissolution of a solid solution, at which the aqueous solution is saturated with respect to a secondary solid phase, (Garrels and Wollast 1978, Denis and Michard 1983, Glynn and Reardon 1990, 1992). This secondary solid will typically have a composition different from that of the dissolving solid. At primary saturation, the aqueous phase is at thermodynamic equilibrium with respect to this secondary solid but remains undersaturated with respect to the primary dissolving solid. The series of possible primary-saturation states for a given SSAS system is represented by the solutus curve on a Lippmann diagram (Fig. 1).

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Figure 1. Lippmann diagram (with stoichiometric and pure-phase saturation curves) for the (Sr,Ba)SO4 - H2O system at 25°C. An a0 value of 2.0 is assumed. End-member solubility products from PHREEQC (Parkhurst and Appelo 2000): 10-6.63 for celestine (SrSO4), 10-9.97 for barite (BaSO4). Points T1 and T2 give the aqueous and solid phase compositions, respectively, of a system at thermodynamic equilibrium with respect to a Sr.5Ba.5SO4 solid. Points P1 and P2 describe the state of a system at primary saturation with respect to the same solid. Point MS1 gives the composition of an aqueous phase at congruent stoichiometric saturation with respect to that solid. The pure BaSO4 saturation curve cannot be distinguished from the solutus curve on this graph.

In the specific case of a “strictly congruent” dissolution process occurring in an aqueous phase with a [B+]/[C+] activity ratio equal to the B+/C+ ratio in the solid, primary saturation can be approximately found by drawing a straight vertical line on the Lippmann diagram from the solid-phase composition to the solutus (Fig. l). For an exact calculation, the following relations may be used to determine the primary saturation state: XB,aq = ƒBχBA

(11)

XC,aq = ƒCχCA

(12)

where ƒB and ƒC are factors correcting for a possible difference in the aqueous speciation and activity coefficients of B+ and C+. The equation used to calculate the value of ΣΠ at primary saturation as a function of solid composition, for a “strictly congruent” dissolution process, may be found by combining Equation (4), the Lippmann solutus equation, with Equations (11) and (12): ⎛ χ BA ƒ B χ CA ƒ C ⎞ ⎟⎟ ΣΠps = 1 / ⎜⎜ + ⎝ K BA γ BA, y K CA γ CA, y ⎠

(13)

where γBA,y and γCA,y refer to the activity coefficients of BA and CA in the secondary

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solid B1-yCyA with respect to which the aqueous solution (at primary saturation) is in temporary thermodynamic equilibrium. ΣΠps refers to the value of the ΣΠ variable as specifically defined at primary saturation. The composition of the B1-yCyA phase will generally not be known. By equating ΣΠps(x) (Eqn. 13) to ΣΠeq(y) (Eqn. 3), the relation between the initial solid composition B1-xCxA and the secondary solid B1-yCyA is obtained: ⎛ χ BA ƒ B χ CA ƒ C ⎞ ⎟⎟ = χCA,yγCA,yKCA + χBA,yγBA,yKBA + 1 / ⎜⎜ ⎝ K BA γ BA, y K CA γ CA, y ⎠

(14)

In the case of a non-ideal solid-solution series, Equation (14) must be solved graphically or by an iterative technique, because γBA,y and γCA,y are typically exponential functions of χCA,y. Stoichiometric saturation states Stoichiometric saturation was formally defined by Thorstenson and Plummer (1977). These authors argued that solid-solution compositions typically remain invariant during solid-aqueous phase reactions in low-temperature geological environments, thereby preventing attainment of thermodynamic equilibrium. Thorstenson and Plummer defined stoichiometric saturation as the pseudo-equilibrium state that may occur between an aqueous phase and a multi-component solid solution, “in situations where the composition of the solid phase remains invariant, owing to kinetic restrictions, even though the solid phase may be a part of a continuous compositional series”. The stoichiometric saturation concept assumes that a solid solution can, under certain circumstances, behave as if it were a pure one-component phase. In such a situation, the dissolution of a solid solution B1-xCxA can be expressed as: B1-xCxA → (1 - x)B+ + xC+ + A-

(15)

Applying the law of mass action then gives the defining condition for stoichiometric saturation states: [C + ]x [ B + ]1− x [ A− ] ⎛ − ΔGr0 ⎞ IAPss = Kss = = exp ⎜⎜ ⎟⎟ 1 ⎝ Rt ⎠

(16)

where x and (1-x) are equal to χCA and χBA respectively and where -ΔG0r is the standard Gibbs free energy change of Reaction (15). The solid has unit activity because it is assumed to behave as if it were a pure one-component phase. According to Thorstenson and Plummer’s (1977) definition of stoichiometric saturation, an aqueous solution at thermodynamic equilibrium with respect to a solid B1xCxA will always be at stoichiometric saturation with respect to that same solid. The converse statement, however, is not necessarily true: stoichiometric saturation does not necessarily imply thermodynamic equilibrium. Stoichiometric saturation states can be represented on Lippmann phase diagrams (e.g. Fig. 1) by relating the total solubility-product variable ΣΠss (defined specifically at stoichiometric saturation with respect to a solid B1-xCxA) to the constant Kss (Eqn. 16) and to the aqueous activity fractions χB,aq and χC,aq: ΣΠss =

K ss

χ

1− x B ,aq

χ Cx ,aq

(17)

In contrast to thermodynamic equilibrium, for which a single (χB,aq, ΣΠeq) point satisfies Equations (1) and (2), stoichiometric saturation with respect to a given solid

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composition is represented by a series of (χB,aq, ΣΠss), points, all defined by Equation (17). The minimum in each stoichiometric saturation curve for a given solid will occur at the point where χC,aq = x and χB,aq = 1-x. This condition can also be generally expressed as the point at which χB,aq/χC,aq = (l-x)/x. This minimum in the stoichiometric saturation curve may be reached through a “strictly congruent” dissolution process occurring in an aqueous phase with a [B+]/[C+]activity ratio equal to the B+/C+ ratio in the solid. The series of minimum stoichiometric points for a series of solid-phase compositions ranging from x = 0 to x = 1 defines the minimum stoichiometric saturation curve (Glynn and Reardon 1990), also called the equal-G curve (Königsberger and Gamsjäger 1992). The minimum saturation curve essentially defines the series of stoichiometric saturation curves that can be constructed in the system and is a function of the end-member solubility products and of the excess-free-energy of mixing, GE, of the solid-solution series: 1− x x ΣΠms = K BA K CA exp(G E / RT )

(18)

In the case of an ideal solid-solution series (GE = 0), the minimum stoichiometric saturation curve will be a linear function of the end-member solubility products when plotted on a log scale. As shown in Figure 1, stoichiometric saturation states never plot below the solutus curve. Indeed, stoichiometric saturation can never be reached before primary saturation in a solid-solution dissolution experiment. The unique point at which a stoichiometric saturation curve (for a given solid B1-xCxA) joins the Lippmann solutus represents the composition of an aqueous solution at thermodynamic equilibrium with respect to the Bl-xCxA solid. Saturation curves for the pure BA and CA end-member solids can also be drawn on Lippmann diagrams (Lippmann 1980, 1982): ΣΠBA =

K BA

χ B ,aq

(19)

ΣΠCA =

K CA

χ C ,aq

(20)

These equations define the families of (χBA, ΣΠBA) and (χCA, ΣΠCA) conditions for which a solution containing A-, B+ and C+ ions will be saturated with respect to pure BA and pure CA solids. Thermodynamic equilibrium with respect to a mechanical mixture of the two pure BA and CA solids, in contrast to a solid solution of BA and CA, is represented on a Lippmann diagram by a single point, namely the intersection of the pure BA and pure CA saturation curves. The coordinates of this intersection are:

χ B,intaq =

K BA K BA + K CA

(21)

ΣΠint = KCA + KBA

(22)

COMPARISON OF SOLID-SOLUTION AND PURE-PHASE SOLLTBILITIES In predicting solid-solution solubilities, one of two possible hypotheses must be chosen. In the first, the solid solution is treated as a one-component or pure-phase solid, requiring a sufficiently short equilibration time, a sufficiently high solid-to-aqueoussolution ratio, and relatively low solubility ofthe solid. These requirements are needed to ensure that no significant recrystallization of the initial solid or precipitation of a secondary solid-phase occurs. For such situations, the stoichiometric saturation concept may apply. The second hypothesis considers the solid as a multi-component solid solution,

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capable of adjusting its composition in response to the aqueous solution composition. This compositional adjustment requires a long equilibration period, a relatively high solubility of the solid, and a relatively low solid-to-aqueous-solution ratio. In this case, the assumption of thermodynamic equilibrium may apply. If the equilibration period is too short for thermodynamic equilibrium to have been achieved, but if an outer surface layer of the solid has been able to recrystallize (because of the high solubility of the solid), the concept of primary saturation may apply. Currently, there are insufficient data to determine the exact conditions for which each of these assumptions may apply, especially in field situations. In many instances, neither one of these hypotheses will explain the observed solubility of a solid solution, which may lie between the “maximum” stoichiometric saturation solubility and the “minimum” primary saturation solubility. Nonetheless, these solubility limits can often be estimated. Stoichiometric saturation solubilities The case of stoichiometric saturation states attained after “strictly congruent” dissolution is examined here. Two hypothetical B1-xCxA solid solutions are considered: (1) calcite (pKsp = 8.48; Plummer and Busenberg 1982) with a more soluble trace NiCO3 component (pKsp = 6.87; Smith and Martell 1976) and (2) calcite with a less soluble trace CdCO3 component (pKsp = 11.31; Davis et al. 1987). Contour plots of saturation indices (SI = log[IAP/Ksp]), with respect to major and trace end-member components, are shown in Figure 2 as a function of the a0 value (assuming a regular solid-solution model) and of the logarithm of the mole fraction of the trace component (where 10-6 ≤ χCA ≤ 0.5). SI values are calculated for major (BA) and trace (CA) components using the relations: ⎡ ⎛ xB ,aq ⎞ x ⎤ SIBA = – log(KBA) + log ⎢ K ss ⎜ ⎟ ⎥ ⎣⎢ ⎝ xC ,aq ⎠ ⎦⎥

⎡ ⎛x SICA = – log(KCA) + log ⎢ K ss ⎜⎜ C , aq ⎢ ⎝ xB ,aq ⎣

⎞ ⎟⎟ ⎠

(1- x )

(23)

⎤ ⎥ ⎥ ⎦

(24)

Kss values are evaluated from Thorstenson and Plummer’s (1977) Equation (22), modified assuming a regular solid-solution model: (1− x ) x (1 − x ) Kss = K BA K CA

(1− x )

x x exp[a0 x(1 − x)]

(25)

Assuming that the dissolution to stoichiometric saturation takes place in initially pure water and that the aqueous activity ratio of the major and minor ions is equal to their concentration ratio, the relation χB,aq/χC,aq = (l-x)/x will apply. Using this relation, applicable only at “minimum stoichiometric saturation” (Glynn and Reardon 1990, 1992), the following equations may be derived from Equations (23), (24) and (25):

a0 x(1- x) ⎛ K CA ⎞ SIBA = x log ⎜ ⎟ + log(1 – x) + 1n(10) ⎝ K BA ⎠

(26)

a0 x(1- x) ⎛ K BA ⎞ SICA = (1 –x)log ⎜ ⎟ + log x + 1n(10) ⎝ K CA ⎠

(27)

The SI contour plots drawn using Equations (26) and (27) show the miscibility gap and spinodal gap lines separating intrinsically stable, metastable, and unstable solid solutions (Prigogine and Defay 1954, Swalin 1972). In natural environments, metastable

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Figure 2. Values of major- and trace-component saturation indices (SI) for solids at congruent stoichiometric saturation (in solid lines): (a) calcite SI at saturation with (Ca,Cd)CO3; (b) otavite SI at saturation with (Ca,Cd)CO3; (c) calcite SI at saturation with (Ca,Ni)CO3; (d) NiCO3 SI at saturation with (Ca,Ni)CO3. Miscibility-gap lines (short dashes) and spinodal gap lines (long dashes) are also shown.

solid-solution compositions may in some cases persist on a geological time scale (depending on the solubility of the solid), whereas unstable solid solutions formed in lowtemperature environments are not likely to do so. Busenberg and Plummer (1989), in their study of magnesian calcite solubilities, observed that the highest known Mg contents in modem natural biogenic calcites correspond to the predicted spinodal composition. Metastable compositions will probably not persist, however, in solid solutions that have higher solubilities or which have been reacted for a longer period of time (or at higher temperatures) than the biogenic magnesian calcites. Although intrinsically unstable solid solutions, formed at low temperatures, may not be found in geological environments, they commonly can be synthesized in the laboratory (e.g.

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strontian aragonites: Plummer and Busenberg 1987; barian strontianites: Glynn et al. 1988) Figures 2a and 2b show the case of a solid-solution series, (Ca,Cd)CO3, with a much less soluble trace end member. If the mole fraction of the trace component is sufficiently high (χCdCO3 > 10-2.8), the aqueous phase at stoichiometric saturation will be supersaturated with respect to the trace end member (except at unrealistic negative a0 values not shown on the plot). The lower solubility of the trace component will generally cause negative SI values (undersaturation) for the major component, except at high a0 values [higher than 7.5 in the case of (Ca,Cd)CO3] for which the solid solutions will generally be metastable or unstable. Calcite SI values in Figure 2a show that the mole fraction of the trace component must be sufficiently high ((χCdCO3 > 10-2.5) for this effect to be measurable in field environments (typical uncertainty on calcite SI values ≈ 0.01 minimum) or in laboratory experiments. A laboratory example of the above principle is given by the “strictly-congruent” dissolution experiment of Denis and Michard (1983) on an anhydrite containing 3.5% Sr (on a mole basis). Analysis of their results shows that maximum SIcelestine and SIgypsum values of 0.37 and -0.04 respectively were attained after four days. Their last sample (after six days) gave SIcelestine and SIgypsum values of 0.35 and -0.04, respectively. Although these results show that stoichiometric saturation was not obtained with respect to the original anhydrite phase (probably because of back-precipitation of gypsum), supersaturation did occur with respect to the less soluble celestine1 component. In the case of solid solutions with a more soluble trace component, aqueous solutions at “minimum stoichiometric saturation” will generally be supersaturated with respect to the major component, and will be undersaturated with respect to the more soluble trace end member (Figs. 2c and 2d). Aqueous solutions at minimum stoichiometric saturation with respect to (Ca,Ni)CO3 solid solutions will be supersaturated with respect to pure calcite at a0 values greater than -2.7. SIcalcite values greater than +0.01, however, will only be found at mole fractions of NiCO3 greater than approximately 10-2.5. In contrast, SI NiCO3 values will exhibit significant undersaturation even at high χ NiCO3 mole fractions (2 < SI NiCO3 < -1 at χ NiCO3 = 0.5). Primary saturation and thermodynamic equilibrium solubilities A detailed discussion of solid-solution solubilities at primary saturation states and at thermodynamic equilibrium states was given by Glynn and Reardon (1990, 1992). Some fundamental principles governing these thermodynamic states are discussed below. The composition of a SSAS system at primary saturation or at stoichiometric saturation will be generally independent of the initial ratio of solid to aqueous solution, but will depend on the initial aqueous-solution composition (prior to the dissolution of the solid). In contrast, the final thermodynamic equilibrium state of a SSAS system attained after a dissolution or recrystallization process will generally depend not only on the initial composition of the system but also on the initial ratio of solid to aqueous solution (Glynn et al. 1990, 1992). An aqueous solution at primary saturation or at thermodynamic equilibrium with respect to a solid solution will be undersaturated with respect to all end-member component phases of the solid solution (Figs. 1, 3, 4 and 5). A positive excess-freeenergy of mixing (that is, a positive a0 value in the case of a regular solid solution) will

1

The mineral celestine is also commonly known as celestite.

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raise the position of the solutus curve relative to that of an ideal solid-solution system. A positive excess-free-energy of mixing will therefore increase the solubility of a solid solution at primary saturation or at thermodynamic equilibrium. The pure end-member saturation curves on a Lippmann diagram offer an upward limit on the position of the solutus. Conversely, a negative excess-free-energy of mixing will lower the position of the solutus relative to that of an ideal solid-solution series. Solid solutions with negative excess-free-energies of mixing may therefore show a greater degree of undersaturation relative to the pure end-member phases, although no examples of this have been found so far in ionic solids.

Figure 3. Lippmann phase diagram for the (Na,K)Fe3(SO4)2(OH)6–H2O system at 25°C. The excess-free-energy-of-mixing parameter a0 was extrapolated from Stoffregen’s (1993) experimental determination of a0 at 200°C. End-member solubility products from PHREEQC (Parkhurst and Appelo 2000): 10-5.28 for natrojarosite [NaFe3(SO4)2(OH)6], 10-9.21 for jarosite [KFe3(SO4)2(OH)6]. The pure jarosite saturation curve cannot be distinguished from the solutus curve on this graph.

The solutus curve, in binary SSAS systems with ideal or positive solid-solution freeenergies of mixing and with large differences (more than an order of magnitude) in endmember solubility products, will closely follow the pure-phase saturation curve of the less soluble end member (except at high aqueous activity fractions of the more soluble component, e.g. Figs. 1 and 3). In contrast, ideal solid solutions with very close endmember solubility products (less than an order of magnitude apart) will have a solutus curve up to 2 times lower in than the pure end-member saturation curves. The factor of 2 is obtained for the case where the two end-member solubility products are equal and can be derived from Equations (4), (21), and (22) (e.g. Fig. 4). Figures 3, 4, and 5 show examples of pure end-member saturation curves plotted on Lippmann phase diagrams. Figure 3, for the natrojarosite-jarosite solid-solution series, is similar to the diagram for the celestine-barite series (Fig. 1) in that there is a large difference between the end-member solubility products. As a result of this large

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Figure 4. Lippmann phase diagram for the (Na,H3O)Fe3(SO4)2(OH)6–H2O system at 25°C. Ideal mixing was assumed for the solid-solution series (a0 = 0). End-member solubility products from PHREEQC (Parkhurst and Appelo 2000): 10-5.28 for natrojarosite [NaFe3(SO4)2(OH)6], 10-5.39 for hydroniumjarosite [H3OFe3(SO4)2(OH)6].

difference, the saturation curve for pure jarosite, the least soluble end member, is very close to the solutus curve, only slightly above it, except in the region where the K/(K+Na) ratio in the aqueous solution (and consequently the χK+ value) goes to 0. In that region, the jarosite saturation curve climbs sharply towards an infinite ΣΠ value, whereas the solutus curve converges to the ΣΠ value of the natrojarosite end-member solubility product. In contrast, the saturation curve for the more soluble end member, natrojarosite, plots well above the solutus curve. This behavior is typical for SSAS systems with large differences in end-member solubility products. Such systems are typically controlled by the less soluble end-member component. Increasing the excess free energy of mixing in these systems, away from the 0 value of an ideal system, typically causes very little effect on the position of the solutus curve; only the solidus curve is affected. Figures 1 and 3 show the difference in the solidus curve for a nearly ideal system (Fig. 3) and for a system with a maximum a0 value that nevertheless allows complete miscibility (Fig. 1). If the excess-free-energy-of-mixing is increased further, a miscibility gap develops as can be seen in the example for the (Pb,Ba)SO4 series (Fig. 6). Figures 4 and 5 provide an example of a SSAS system with close end-member solubility products: the solid-solution series natrojarosite‒hydronium jarosite. Both pure end-member saturation curves in such a system plot well above the solutus curve, in the case of an assumed ideal system (Fig. 4). If the excess free energy of mixing is increased, however, the solutus curve will plot progressively closer to the end-member saturation curves (Fig. 5) and the solidus curve will also be significantly affected. ESTIMATION OF THERMODYNAMIC MIXING PARAMETERS There are two main applications for SSAS theory in the chemical modeling of aqueous systems: (1) the prediction of solid-solution solubilities, (2) the prediction of the

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distribution of trace components between solid and aqueous phases. Currently, a big problem with both types of predictions is the lack of low-temperature data on solidsolution excess-free-energy functions, and therefore on solid-phase activity coefficients. The two-parameter Guggenheim expansion series (the “subregular” model) has been successfully used to fit laboratory solubility data for the systems (Sr,Ca)CO3-H2O (Plummer and Busenberg 1987), (Ba,Sr) CO3-H2O (Glynn et al. 1988), (Ca,Mg) CO3H2O (Busenberg and Plummer 1989), K(CI,Br)-H2O, and Na(CI,Br)-H2O (Glynn et al. 1990, 1992). The one-parameter Guggenheim series (the “regular” model) also has been frequently used (Lippmann 1980, Kirgintsev and Truslulikova 1966). More laboratory determinations of thermodynamic mixing parameters are needed, not only to acquire data on binary and multicomponent solid solutions, but also to test the applicability of the regular and subregular models, and to compare them with other excess-free-energy models. In the mean time, as a better approximation than the commonly used assumption of “ideal” solid solutions, the a0 and a1 dimensionless parameters can be estimated for many binary systems. The MBSSAS computer code (Glynn 1991) uses either observed or estimated (1) miscibility-gap data, (2) spinodal-gap data, (3) critical temperature and

Figure 5. Lippmann phase diagram for the (Na,H3O)Fe3(SO4)2(OH)6–H2O system at 25°C. An a0 value of 2.6 was used for comparison with This a0 value is probably a maximum one, based on Brophy and Shendan’s (1993) experimental observations and assumption of complete mixing for the (Na,H3O,K)-jarosite series at 114°C. The maximum excess-free-energy-ofmixing that would still allow complete miscibility was assumed for that temperature and the corresponding value of a0 was extrapolated down to 25°C. At that temperature, the system exhibits a miscibility gap as can be seen from the eutectic point (A) in the solutus curve. At that point the aqueous solution is in equilibrium with two solid solutions, of compositions M1 and M2 given by the intersections between the solidus curve and a horizontal line passing through the eutectic point (A).

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Figure 6. Lippmann diagram (with stoichiometric and pure-phase saturation curves) for the (Pb,Ba)SO4–H2O system at 25°C. An estimated a0 value of 2.7 is used, based on observed compositions in natural minerals (Palache et al 1951). End-member solubility products from PHREEQC (Parkhurst and Appelo 2000): 10-7.79 for anglesite (PbSO4), 10-9.97 for barite (BaSO4). The miscibility gap in the system is revealed by the presence of a peritectic point (A) which defines the composition of the aqueous solution at equilibrium with two miscibility-gap solids of composition M1 and M2. The spinodal gap in the system is given by the local minimum and maximum in the solidus curve (solid compositions SP1 and SP2).

critical mole-fraction of mixing data, (4) distribution-coefficient data, (5) alyotropic point data (Lippmann 1980) or (6) activity-coefficient data to calculate a0 and a1 parameters for binary solid solutions. Depending on whether one or two datum points are given, the program computes either a regular or subregular model. Miscibility-gap data Miscibility gaps, as determined from mineral compositions observed in the field, can be used to estimate thermodynamic mixing parameters in the absence of more accurate laboratory data. This approach suffers from several problems. The maximum mole fraction of trace component found may not correspond to the miscibility-gap fraction. In this case, if the solid-solution composition is stable, then the calculated excess free energy will be overestimated. On the other hand, if the analyzed “limiting” composition really represents a mechanical mixture of two end members, rather than a solid-solution mineral, the calculated excess free energy may be underestimated. If the mineral was formed at much higher temperatures than the temperature of interest and if the mineral is fairly unreactive at the lower temperature, the maximum solid-solution mole fraction observed may well be metastable or even unstable. The temperature of formation (and of equilibration) of the solid solution may not be known. If a lower temperature is assumed, the excess free energy will be underestimated.

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Additionally, the extrapolation to 25°C of excess free energies estimated at higher temperatures will introduce an additional error; typically, excess enthalpy and excess entropy data are not available for solid-solution series. A partial solid-solution series may not be isomorphous (i.e. the end members may not have the same structure). In that case, the excess-free-energy parameters should be calculated only on a single side of the miscibility gap. On the other side of the miscibility gap, a different model will apply. Despite the above problems, mixing parameters estimated from miscibility-gap information are still an improvement on the assumption of an ideal solid-solution model. The a0 parameters estimated from potential miscibility-gap data provided by Palache et al. (1951), Jambor et al. (this volume), and other investigators, and also from various experimental determinations, are presented in Tables 1, 2, and 3 for several lowtemperature mineral groups. Because of the large uncertainties in the data and in the estimation procedure, a subregular model is usually unwarranted. As a result, the estimated a0 values that are presented should be used only for solid-solution compositions on a single side of the miscibility gap, i.e. only up to the given miscibility fraction. As can be seen in the tables, binary solid-solution series that have a miscibility gap but nevertheless show significant amounts of substitutional variation will have an a0 excessfree-energy parameter slightly above a value of 2 (dimensionless). This assumes that a regular solid-solution model can be applied. This is also illustrated in Figure 7, which provides an easy way of determining the applicable a0 value for a solid-solution series, given its miscibility-gap fraction. Spinodal-gap data Spinodal gaps can be used to estimate low-temperature solid-solution mixing parameters. This method is considerably less accurate than the miscibility-gap estimation technique. The estimation assumes that intrinsically unstable compositions are not likely to form during a precipitation process, although metastable compositions may do so. According to thermodynamic theory, solid-solution compositions inside a spinodal gap are intrinsically unstable, whereas compositions inside the miscibility gap, but outside the spinodal gap are metastable (Prigogine and Defay 1954). The presence of a miscibility gap is necessarily always accompanied by that of a spinodal gap. The (Pb,Ba)SO4 solidsolution series provides an example of a system with hypothesized miscibility and spinodal gaps (Fig. 6). One of the problems with the theory of spinodal gaps, at least for low-temperature SSAS systems, is that supposedly intrinsically unstable solid solutions can commonly be precipitated from aqueous solutions in the laboratory. An inability of the regular and subregular excess-free-energy models to describe adequately the relation between miscibility and spinodal gaps could offer one possible explanation. Another possible explanation might be that the thermodynamic properties of solids initially precipitated from aqueous solution are typically quite different from those of well crystallized phases. The initial precipitates are generally much finer grained, have a higher proportion of crystal defects, and may contain water, occluded solution, or other impurities. Defects and impurities in these initial solids, high surficial energy of the solids, and strong hydration bonds of the substituting ions may favor the formation of extensive solidsolution series. Plummer and Busenberg (1987) were able to synthesize a complete suite of compositions in their investigation of the strontianite-aragonite solid-solution series, despite the presence of extensive spinodal and miscibility gaps, that were later determined on the basis of measurements of stoichiometric saturation states. Solids with compositions

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inside the spinodal gap, however, were much more difficult to synthesize (Busenberg, pers. commun.). Similarly, Glynn et al. (1988, and unpublished data) were able to synthesize and conduct stoichiometric saturation experiments on the entire witheritestrontianite series, despite the supposed presence of extensive spinodal and miscibility gaps (Table 2b). Investigations of the barite-anglesite and barite-celestine series also indicate the ability to synthesize solids over the entire compositional range, despite the possible Table la. Estimated mixing parameters for binary sulfate solid solutions at 25°C. References correspond to author initials and year of publication (JNA = Jambor et al., this vol.) syn. = synthetic solids; nat. = natural solids; thermo equil. = thermodynamic equilibrium experiment

System (main, trace)

Misc. Frac. (Xtrace)

Temp. (°C)

Stuctur

e

Estimated a0 at 25°C

Reference

PBF 1951 PBF 1951 W 1933 and present paper INA 2000 PBF 1951

Melanterite group (monoclinic heptahydrates) (Fe,Co)SO4⋅7H2O (Fe,Mn)SO4⋅7H2O (Fe,Mn)SO4·7H2O

25 25 0

mono. mono. mono.