Internally consistent thermodynamic data for rock-forming minerals in the system Si0 2 -Ti02-Al203-Fe203-CaO-MgO-FeO-K20-Na20-H20-C0 2

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1 Eur.. Mineral. 997,9, nternally consistent thermodynamic data for rock-forming minerals in the system Si0 2 -Ti02-Al203-Fe203-CaO-MgO-FeO-K20-Na20-H20-C0 2 MATTHAS GOTTSCHALK GeoForschungsZentrum Potsdam, Telegrafenberg A7, D-4473 Potsdam gottschalk@gfz-potsdam.de Abstract: An internally consistent thermodynamic data set has been derived utilizing results from all available experimental phase equilibria in a simultaneous iterative regression approach using /T vs. n K rec { plots. Simultaneous evaluation of all reactions allows the extraction of an internally consistent data set of ΔH and S values with V, cp, a and ß otherwise constrained. An obective function has been defined which requires maximum consistency with available experimental results and minimum deviation from available calorimetric data. One advantage of this procedure is that all available experimental results, compatible or not, are always visible during the calculations. Over 5300 experimental results from 244 contributions for 253 reactions involving 94 phase components have been used. The extracted data set reproduces 92% of the available phase equilibria experiments reported in the literature. t can be shown that for some phases the configurational entropy must be considered (e.g. CaAl-pyroxene, clinochlore, mullite, phlogopite). With the addition of a configurational entropy for these phases the reproduction of the available experimental results is significantly improved. The configurational entropy for these phases is chiefly responsible for the observed differences between this and other available internally consistent data sets. n these other data sets non-addition of configurational entropy for these phases is compensated for by adustments in the enthalpy and/or in other refined thermophysical parameters such as heat capacities. The configurational entropies used here are quite straightforward and involve mostly tetrahedral Si/Al and octahedral Mg/Al disorder. Key-words: internally consistent set of thermodynamic data, enthalpy of formation, third law entropy. phases are not considered to be pure, activitycomposition relations are required. n the case of fluids an equation of state is needed for the calcu- lation of the fugacities. These particular thermo- dynamic properties and functions are available to varying extents from calorimetry, spectroscopy, X-ray crystallography and the study of experi- mental phase equilibria. The available properties and functions for vari- ous mineral and fluid components have been col- lected in a series of thermodynamic data bases/ sets. These have been used for geo-thermobaro- metry, calculations of specific phase equilibria in P-T-x space and the kinetic treatment of mineral reactions. The scope, character and extent of these available data bases, however, are quite different. ntroduction Petrological phase equilibria can be calculated utilizing basic thermodynamic principles, if the required thermodynamic properties and functions for the phases are available. For each phase component these properties are the standard enthalpy of formation Δ f H, the standard entropy S, the heat capacities over the relevant range of temperatures (c P ), and the molar volumes as a function of temperature and pressure (V, α, ß). f a phase transition occurs, the appropriate changes of these properties must be known. Order-disorder transitions and variable compositions of a phase as a function of pressure, temperature and bulk composition must be also taken into account. f the /97/ $ E. Schweizerbart'sche Verlagsbuchhandlung, D-7076 Stuttgart

2 76 M. Gottschalk Some data collections like Woods & Garrels (987) consist only of values for reference enthalpies and entropies. The data set by Robie et al. (979) also includes heat capacities and molar volumes. Enthalpies, entropies and heat capacities in either data set are, to a large extent, based on calorimetric measurements. Other data bases consider the constraints under which the derived extensive and intensive thermodynamic functions for each phase component must correctly reproduce experimentally determined phase equilibria. n these so-called internally consistent data sets, experimental results for a large body of phase equilibria are used in conunction with calorimetric observations in order to determine enthalpies and entropies for each phase component considered. Such internally consistent data sets have been published by Zen (972), Helgeson et al. (978), Hemingway et al (982), Halbach & Chatteree (984), Berman et al. (986), Berman (988), Holland & Powell (985, 990) and more recently by Chatteree et al. (994). The two internally consistent data sets most widely used by petrologists today are those of Berman (988) and Holland & Powell (990). The internally consistent data sets of Berman (988) and Holland & Powell (990) differ greatly in the technique used in their derivation as well as in the extent of the derived properties. n the case of Holland & Powell (990), their data set was calculated using a least square regression technique (REG) which resulted in refined values for A f H. Berman (988) derived his data set using a linear (mathematical) programming algorithm (MAP) with a quadratic obective function. This involved adusting the values for AfH, S and V for each phase component and in some cases also the coefficients for α, ß and c P as well. Besides the technical and mathematical differences, the two approaches reflect some differences in philosophy. n refining only values for AfH, Holland & Powell (990) assume that all other parameters and functions are well known and fixed. Thus, with respect to the input parameters and functions, the derived values for AfH complement an already existent data set. On the other hand, Berman (988) considered that the parameters and functions for each phase component have some experimental error and are therefore targets for optimization. As a consequence, the data set derived by Berman (988), using the MAP method, results in a homogenous solution for all experimental observations (thermophysical and phase equilibria) considered. n addition, the formalism of this approach is capable of incorporating experimental results for activity-composition relations, order-disorder phenomena as well as fluid phase properties. Difficulties with the MAP technique arise, however, because all experimental observations of mineral equilibria and thermophysical measurements used in the algorithm have to be consistent within their constraints. Therefore contradictory experimental observations, which are quite often encountered in experimental petrology, are not allowed and must be eliminated. As a consequence, decisions must be made prior to the derivation as to which experimental observations are to be included and which will not be considered further. These decisions are not unequivocal and certainly influence the final result. The derived values are then a closed solution for a selected set of consistent constraints. Obections against the REG technique are that only A f H values are refined and that due to the algorithm used, the highest probability of equilibrium is thought to be at the center of each experimentally determined bracket. Direct measurements of reference entropies S are not available for many phases and so must be. Because of the unknown state of order at reference conditions, the S values may have a high uncertainty influencing the derived AfH values. n the present study an approach is presented which builds upon the method used by Powell & Holland (985) and Holland & Powell (985, 990). n addition to optimized values for A f H as in Holland & Powell (985, 990), values for S are also derived for a large collection of phase components which complement existing thermophysical parameters and functions. While Holland & Powell used a single fitting process, the approach presented here iterates toward a solution. The iterative part eliminates the constraint that the equilibrium is assumed to be at the center of each experimentally determined bracket. This iterative least square method (REG) is based on plotting /T vs. n K red. The REG approach utilizes a large set of experimental results (> 5300) for phase equilibria from the literature in the form of /T vs. n K red plots with one /T vs. n K red plot constructed for each mineral reaction. All known experimental results for the reaction are plotted on the diagram. With the exception of H 2 O-CO 2 fluids, experimental results involving phases with extensive solid solutions are mostly avoided. This keeps the influence of activity-composition rela-

3 nternally consistent thermodynamic data 77 tions on the derived values for AfH and S to a minimum. n the second step mineral reactions in the form of /T vs. n K red plots are simultaneously evaluated, resulting in an extraction of the internally consistent data set of A f H and S values for the phase components considered. During this process all collected experimental results are always visible. The final refinement of the data set then indicates which experiments either are or are not consistent with the derived data set. Notations standard state property G Gibbs free energy H enthalpy ΔfH enthalpy of formation from the elements K equilibrium constant K red equilibrium constant reduced to standard conditions P pressure MPa] R gas constant (8.344 /K) S entropy S third law entropy at standard conditions T temperature K] V volume a activity c P heat capacity n, number of moles of component / / fugacity x mol fraction α coefficient of thermal expansion ß coefficient of compressibility Δ change of a property for a reaction µ chemical potential v stoichiometric coefficient A R vector of n K red values B matrix of the stoichiometric coefficients D vector of third law entropies and enthalpies of formation E identity matrix W matrix holding the weighting factors fob] obective function Thermodynamic relations n general the change of Gibbs free energy for a phase equilibrium (ΔG (PT) ) at pressure P and temperature T is given by ΔG (P>T) = 2^ v t - G /(P/Γ) () where v, is the stoichiometric coefficient of the / th phase component, k is the total number of phase components involved in the equilibrium and G, is the Gibbs free energy of the / th phase component at P and T. Further consideration requires the definition of standard states and reference conditions. Solids are considered to be in standard state if they are of pure end-member composition (pure phase component) and reference conditions are 0. MPa and K. Fluids are treated as pure hypothetical ideal gases at a reference pressure of 0. MPa ( bar) and temperature T. ΔG (P?T ) is then given by ΔG (PiT) = AH - TAS - \ -^r dtdt T + ΔV s o (PiT) dp + RT n K {?: (P,T) (2) where Δ designates an equivalent relationship for each parameter or function as in eq. () except for ΔV Ö S(PT). n this case Δ designates only the change in volume of the pure solid phase components. The pressure dependence of the Gibbs free energy of the fluid phase components is incorporated in the equilibrium constant AΓ (PT) \nk, P,T) Σ v > ln flk ) f(0. 7(0.,T) + RT ^ v, n a t (3) where / /( p T) is the fugacity of the / h pure fluid phase component, m is total number of fluid phase components, and a x are the activities of the / th phase component. The molar volumes of solids at elevated pressure and temperature are evaluated with the compressibility coefficient ß (eq. 4) and the thermal expansion coefficient α (eq. 5). ''"'' ~ftv, (4, ap T,n dvd OLfVi (5) The combination and integration of eqs. (4) and (5) lead to eq. (6) with the assumption that α and ß are not a function of pressure and temperature: ~α (T-)-ß'/(P-0.) V^P.T) = V, (0.,) ' (6) Although α and ß are treated as constants, this relationship mimics the real volume change as a

4 78 M. Gottschalk Table l. Heat capacity coefficients with sources. àkermanite ákermanite albite almandine almandine analbite analcite aπdalusite andradite anorthite anorthite anthophyllite antigorite aragonite aragonite boehmite brucite brucite CaAl-pyroxene calcite calcite carpholite chloritoid chrysotile clinochlore clinoenstatite clinozoisite clinozoisite coesite cordierite corundum α-cristobalite ß-cristobalite deerite diaspore diaspore diopside dolomite dolomite edenite enstatite fayalite ferrocordierite ferroglaucophane ferroactinolite ferrosilite forsterite gehlenite gehlenite glaucophane grossularite grossularite grunerite hedenbergite α-hematite ß-hematite hercynite ilmenite α-iron α-iron α-iron α-iron α-iron γ-iron adeite adeite kalsilite kaolinite kaolinite kyanite larnite αmarnite laumontite a \ K mol fc 03 -^ K 2 mol c 06 M K 3 mol d 0-3 e lo" 4 f 0-6 & \ ÜL f K > io- 7 K 2 K l/2 mol mo mol mol used from to K] Robie etat (979) 800 Berman& Brown (985) 400 Hemingway et at (98) 000 Bohlen etat (986) 800 Holland & Powell (990) 400 Hemingway et at (98) 800 Berman & Brown (985) 800 Hemingway et at (99b) 250 Robie etat (987) 250 Hemingway etat (982) 800 Berman & Brown (985) 200 Hemingway (99) 800 Berman & Brown (985) 600 Robie etat (979) 800 Berman & Brown (985) 700 Hemingway etat (99a) 500 Robie etat (979) 800 Berman & Brown (985) 800 Hemingway et at (982) 650 Robie etat (979) 800 Berman & Brown (985) 800 Holland & Powell (990) 800 Holland & Powell (990) 800 Berman & Brown (985) 800 Berman (988) 600 Robie etat (979) 900 Hemingway et at (982) 800 Berman & Brown (985) 800 Berman (988) 700 Robie etat (979) 800 Hemingway etat (982) 523 Robie etat (979) 800 Robie etat (979) 800 Holland & Powell (990) 650 Hemingway et at (982) 800 Berman & Brown (985) 600 Robie etat (979) 900 Robie etat (979) 800 Berman & Brown (985) 800 Holland & Powell (990) 800 Holland & Powell (985) 800 Berman (988) 800 Holland & Powell (990) 800 Holland & Powell (990) 800 Holland & Powell (990) 800 Berman (988) 800 Robie etat (979) 500 Hemingway & Robie (984) 800 Berman & Brown (985) 800 Holland (988) 850 Hemingway etat (982) 800 Berman (988) 800 Holland & Powell (990) 800 Haselton^αZ. (987) 950 Robie etat (979) 800 Robie etat (979) 800 Holland & Powell (990) 650 Anovitz <?f α/. (985) 800 Barin et at (977) 000 Basin etat (977) 042 Barin etat (977) 060 Barin etat (977) 84 Barin etat (977) 665 Barin etat (977) 000 Robie etat (979) 800 Berman & Brown (985) 800 Holland & Powell (990) 550 Hemingway et at (982) 800 Berman & Brown (985) 800 Hemingway et at (99b) 970 Hemingway et at (982) 800 Hemingway etat (982) 750 Helgeson et at (978)

5 nternally consistent thermodynamic data 79 Table. (Cont.) lawsonite leucite (tetra.) leucite (cubic) lime lime magnesite magnesite α-magnetite ß-magnetite margarite meionite merwinite microcline monticellite mullite muscovite muscovite Na-phlogopite α-nepheline ß-nepheline γ-nepheline nickel nickel nickel nickel α-nio ß-NiO γ-nio γ-nio γ-nio paragonite pargasite periclase phlogopite phlogopite prehnite prehnite pumpellyite pyrope pyrope pyrophyllite pyrophyllite α-quartz ß-quartz rankinite rutile sanidine sanidine sapphirine sillimanite sphene sphene spinel spurrite staurolite talc tilleyite tremolite α-tridymite ß-tridymite VesuVianite wairakite wollastonite wollastonite c-wollastonite wuestite zoisite zoisite carbondioxide water oxygen a K mol b 0 3 K 2 mol c 0 6 K 3 mol <f 0-3 e 0-4 /xlo-6 g ' 0-7 \ f_l K K /2 mol mo- f ^ l " mol mol used from to K] Berman & Brown (985) 955 Robie etal (979) 800 Robie etal. (979) 000 Hemingway etal. (982) 800 Berman & Brown (985) 800 Robie etal. (979) 800 Berman & Brown (985) 848 Robie etal. (979) 800 Robie etal. (979) 200 Hemingway et al. (982) 200 Holland & Powell (985) 800 Berman & Brown (985) 00 Hemingway etal. (98) 500 Sharp etal (986) 800 Robie etal. (979) 000 Robie etal. (979) 800 Berman & Brown (985) 800 Holland & Powell (990) 467 Robie etal. (979) 80 Robie etal. (979) 525 Robie etal. (979) 600 Holmes etal (986) 63 Holmes etal. (986) 700 Holmes etal. (986) 728 Holmes etal. (986) 525 Holmes etal. (986) 565 Holmes etal. (986) 800 Holmes etal. (986) 200 Holmes etal. (986) 2000 Holmes etal. (986) 800 Berman & Brown (985) 800 Holland & Powell (990) 800 Robie etal. (979) 900 Robie & Hemingway (984b) 800 Berman (988) 800 Hemingway etal. (982) 800 Berman & Brown (985) 800 Holland & Powell (990) 800 Robie etal (979) 800 Berman & Brown (985) 750 Hemingway etal. (982) 800 Berman & Brown (985) 844 Hemingway (987) 800 Hemingway (987) 200 Hemingway etal. (982) 800 Berman & Brown (985) 250 Hemingway et al (98) 800 Berman & Brown (985) 300 Kiseleva(976) 2000 Hemingway etal (99b) 250 Robie etal. (979) 800 Berman & Brown (985) Hemingway & Robie (984) 300 Berman & Brown (985) Robie etal. (979) 432 Berman (988) 800 Robie etal. (979) 600 Valley etal (985) 000 Helgeson etal (978) 995 Richtt etal. (99) 398 Richtletal. (99) 800 Richetetal. (99) 652 Robie etal (979) 900 Hemingway etal (982) 800 Berman & Brown (985) Robie etal (979) Robie etal. (979) Robie etal (979)

6 80 M. Gottschalk Table 2. Molar volumes, thermal expansions and compressibilities with sources. äkermanite albite almandine analbite analcite andalusite andradite anorthite anthophyllite antigorite aragonite boehmite brucite CaAl-pyroxene calcite carpholite chloritoid chrysotile clinochlore clinoenstatite clinozoisite coesite cordierite corundum α-cristobalite ß-cristobalite deerite diaspore diopside dolomite edenite enstatite fayalite ferrocordierite ferroglaucophaπe ferroactinolite ferrosilite forsterite gehlenite glaucophane grossularite grunerite hedenbergite hematite hercynite ilmenite iron adeite kalsilite kaolinite kyanite larnite of-larnite laumontite lawsonite leucite lime magnesite magnetite margarite meionite merwinite microcline monticellite mullite muscovite Na-phlogopite nepheline nickel NiO paragonite pargasite periclase phlogopite V \ MPa mol Robie etal (979) Robie etal (979) Berman (988) Robie etal (979) Helgeson etal. (978) Robie & Hemingway (984a) Robie et al (987) Hemingway etal (982) Holland & Powell (985) Berman etal. (986) Robie etal. (979) Hemingway et al. (982) Robie etal (979) Hemingway et al (982) Robie etal (979) Chopin & Schreyer (983) Rao & ohannes (979) Robie etal. (978) Holland & Powell (985) Stephenson et al. (966) Seki(959) Robie etal. (979) Robie etal. (979) Hemingway etal. (982) Robie etal (979) Robie etal (979) Holland & Powell (990) Hemingway et al. (982) Robie etal. (979) Robie etal. (979) Colvilleétfα/. (966) Chernosky et al (985) Robie etal. (979) Holdaway& Lee (977) Holland (988) Ernst (966) Sueno etal. (976) Robie etal. (979) Hemingway et al. (982) Holland (988) Hemingway et al (982) Newton & Wood (980) Haselton et al. (987) Robie etal (979) Robie etal (979) Robie etal (979) Robie etal. (979) Robie etal. (979) Robie etal. (967) Hemingway etal (982) Robie & Hemingway (984a) Hemingway etal. (982) Hemingway et al (982) Helgeson et al (978) Holland & Powell (985) Robie etal (979) Hemingway etal (982) Robie etal. (979) Hemingway (987) Hemingway et al (982) Holland & Powell (985) Robie etal (979) Robie etal (979) Helgeson etal. (978) Robie etal (979) Robie etal. (979) Carman (974) Robie etal (979) Robie etal (979) Robie etal (979) Helgeson et al. (978) Westrich & Holloway (98) Robie etal (979) Robie etal (979) a 0 6 M-l K Skinner (966) Skinner (966) Skinner (966) Skinner (966) Winter & Ghose (979) Holland & Powell (990) 6.00 Skinner (966) Skinner (966) Skinner (966) Karpinskaya & Ostrovskiy (982) Markgraf & Reeder (985) 32. Holland & Powell (990) Holland & Powell (990) Holland & Powell (985) Holland & Powell (985) Holland & Powell (985) Holland & Powell (985).80 Skinner (966) Skinner (966) Skinner (966) Skinner (966) 4.80 Skinner (966) Holland & Powell (990) Finger & Ohashi (976) Reeder & Markgraf (986) 3.00 Holland & Powell (990) Holland & Powell (985) Skinner (966) 5.48 Holland & Powell (990) Holland & Powell (990) 3.2 Holland & Powell (990) Sueno etal. (976) 38.0 Hazen(976) Skinner (966) Holland (988) Skinner (966) Holland & Powell (990) Holland & Powell (990) 3.30 Skinner (966) Skinner (966) Wechsler & Prewitt (984) 4.20 Skinner (966) Cameron etal. (973) Holland & Powell (990) Winter & Ghose (979) Holland & Powell (985) Holland & Powell (990) Markgraf & Reeder (985) Skinner (966) Holland & Powell (985) 4.0 Holland & Powell (985) Skinner (966) 9.00 Holland & Powell (985) Skinner (966) Holland & Powell (985) Holland & Powell (990) Skinner (966) Holland & Powell (985) Holland & Powell (990) Skinner (966) ß 0 6 MPT Birch (966) Birch (966) Birch (966) Brace etal (969) Holland & Powell (990) Birch (966) Martens etal (982) Martens etal (982) Holland & Powell (990) Holland & Powell (990) Hazen& Finger (976a) 0inger(977) Levien& Prewitt (98) Mirwald etal (984) Birch (966) Berman (988) Berman (988) Holland & Powell (990) Le\ien etal. (979) Martens et al. (982) Holland & Powell (990) Olinger (977) Birch (966) Holland & Powell (990) Holland & Powell (990) Holland & Powell (990) Bass &Weidner (984) Olinger (977) Holland (988) Hazen& Finger (976b) Zhang etal (992) Holland & Powell (990) Birch (966) Holland & Powell (990) Wechsler & Prewitt (984) Birch (966) Birch (966) Holland & Powell (990) Brace etal. (969) Holland & Powell (990) Martens etal (982) Birch (966) Birch (966) Sharp etal. (987) Birch (966) Holland & Powell (990) Birch (966) Holland & Powell (990) Birch (966) Hazen& Finger (976a)

7 nternally consistent thermodynamic data 8 Table 2. (Cont.) prehnite pumpellyite pyrope pyrophyllite α-quartz ß-quartz rankinite rutile sanidine sapphirine sillimanite sphene spinel spurrite staurolite talc tilleyite tremolite α-tridymite ß-tridymite vesuvianite wairakite wollastonite c-wollastonite wuestite zoisite V MPa mol Hemingway etal. (982) Schiffman&Liou(980) Robie etal. (979) Hemingway etal. (982) Hemingway (987) Hemingway (987) Hemingway et al. (982) Robie etal. (979) Robieetal. (979) PDS-File Robie & Hemingway (984a) Robie etal. (979) Robie etal. (979) PDS-File Holdaway etal. (993) Robie etal. (979) PDS-File Robie etal. (979) Robie etal. (979) Robie etal. (979) Valley etal. (985) Helgeson et al. (978) Hemingway etal. (982) Hemingway etal. (982) Robie etal. (979) Hemingway etal. (982) a 0 6 *] Holland & Powell (990) Skinner (966) Taylor & Bell (97) Ghiorsoefα/. (979) Ghiorso etal. (979) Skinner (966) Holland & Powell (985) Winter & Ghose (979) Holland & Powell (985) Skinner (966) Holland & Powell (990) Holland & Powell (985) Suenoetal. (973) Skinner (966) Skinner (966) Holland & Powell (985) Holland & Powell (985) Skinner (966) Holland & Powell (985) ß 0 6 MPä Holland & Powell (990) Hazen& Finger (976b) Birch (966) Birch (966) Birch (966) Birch (966) Brace etal. (969) Birch (966) Vaidyaefα/. (973) Comodi etal. (99) Berman (988) Berman(988) function of pressure and temperature. With eq. (6) the second integral of eq. (2) is calculated for n solid phase components involved in the phase equilibrium: ΔV S (BT) - Σ vr e<cr-) (i_,-ß;'(p-o,) } (7) where k = n + m. The last three terms of eq. (2) can be combined to form the reduced equilibrium constant n K-red RTlnff^ TpdTdT P which leads to eq. (9): + Δ Vs(p,T)dP + RT n K (P ΔG (P,T) AH -TAS + KT\nK re (8) (9) Evaluation of the right hand side of eq. (9) leads to the following conditions: ΔG > 0 => reactants stable ΔG = 0 => equilibrium ΔG < 0 => products stable (0) Setting eq. (9) to zero allows the calculation of any phase equilibrium under the condition that every parameter or function on the right hand side of eq. (9) is known. f phase equilibria are studied experimentally, usually the results can be categorized in the same way as in eq. (0), either reactants or products are stable. Usually, in a single experiment equilibrium can not be observed, but the results of several experiments allow the interpolation of the equilibrium conditions. Thermodynamic evaluation of the right hand side of eq. (9) should be in agreement with experimental results if all relevant thermodynamic parameters and functions are correct. n this case these parameters and functions would be consistent with the experimental results. f this is not the case, then some or even all of these parameters and functions can be altered in such a way that they will become consistent with the assumption that the experimental results are correct. This is fundamentally the basic philosophy behind an internally consistent data set. n addition to experimental results for phase equilibria, additional constraints (e.g. thermophysical measurements) can also be considered. The methodology involved in the construction of an internally consistent data set can take a number of different approaches. Here experimental results for mineral equilibria from the literature were col-

8 82 M. Gottschalk lected and for each experiment an n K red value was calculated. For each mineral reaction a /T vs. n K red plot was then generated and from these plots and with the use of eqs. (9) and (0), A f H and S values were extracted. nput data for the calculation of n K re d n evaluating the n K red term in eq. (9) for each available experimental result, the heat capacities for all components, the molar volumes and the coefficients of the thermal expansion and compressibility of the solids are required. f any fluids are involved, their functions of state are also required. f mixed phases are considered, activity models must be taken into account. Lastly, some phases show phase transitions as a function of P and T, while others reveal a continuous change in the degree of order, all of which must be considered thermodynamically. Values for most of those parameters are available and their P and T dependencies have been measured. When these are not available, they must be. Table l lists the coefficients for the heat capacities and Table 2 the molar volumes, the coefficients of thermal expansion and compressibility used for the calculations below, along with sources. Heat capacities n most cases heat capacities are given in the form proposed by Haas & Fisher (976): c P = a + bt + ct n !) r γl/2 The extrapolation of heat capacities presented in this form to high temperatures is not possible, or at the very least, problematic (Berman & Brown, 985). Therefore Berman & Brown (985) and Berman (988) did a refit of the original calorimetric data using a modified form which can be extrapolated to high temperatures: n a conservative approach, the original c P functions were used here in their validity range up to T max > with extr apolations only done u sing the for m and theparametersproposedbyber man & Brown (985). n the case of extrapolation, the necessary integrationswereperformedinintervals: πr2 v ' G(p,T) _ G( P) ) - - S(P,) C^max ~ ) (3) T max max 0 i Q - Y dtdt - 5 (P, Tm u) (T - T max ) - Y dtdt Tmax One advantage of this approach is that the lambda transition does not have to be treated separately. The algebraic form used by Berman & Brown (985) is unsuitable for the rapid increase in heat capacity observed for phases which undergo lambda transitions. n contrast the Haas & Fisher (976) form is able to reproduce the typical lambda shape for c P. At T max, however, c P values calculated for a mineral phase using Berman & Brown (985) (eq. 2) and the fundamental c P data are usually not identical. n the case of anorthite for example, at a T max of 250 K c P goes from to /K when switching from experimental c P data in Hemingway et al. (982) to the calculated value in Berman & Brown (985). Adding. /K to the c P function of Berman & Brown (985) and applying eq. (3) up to 800 K results in a maximum energy difference of 7 in Δ f H in contrast to A f H when the original c P function is used. This difference falls well within the error range in A f H for anorthite, which is approximately 500. For the two phases spurrite and tilleyite measured heat capacity data were not available. Therefore the heat capacities for these particular phases had to be using an approach similar to that of Helgeson et al (978), i.e. as an approximation, the heat capacities for these two phases were calculated as the summation of other relevant phases or components using the following equation: k c P = ^v icpi (4) = The heat capacity for spinel was taken from Robie et al. (979) and corrected for order-disorder. The heat capacity for spinel in Table is valid for a totally ordered spinel. An energy term considering equilibrium disorder is considered separately. Molar volume Molar volumes for solid phase components (endmember compositions) are well known from X-ray diffraction studies. T

9 nternally consistent thermodynamic data 83 Thermal expansion The coefficients for thermal expansion are known for many phases. Unknown thermal expansion coefficients were using the observation that for phases with similar structures (e.g. phyllosilicate, amphiboles etc.) the thermal expansion coefficients are similar. Compressibility For some of the phases considered, compressibility coefficients were also unknown. For the estimation of these values an approach was chosen very similar to that used by Powell & Holland (985). Anderson & Nafe (965) and Wang (978) observed the following general relationship between the molar volume V and the compressibility coefficient ß: n V -k'$ for M = constant (5) where M is the coefficient M/n of the molar mass per formula unit and the number of atoms n. This leads to: n ß = 4 n + k with k = -n k ' (6) Compressibility values from the literature (Table 3) for various phases were plotted as n ß vs. n V /n (Fig. ) and the following fitted expression was derived: k = M M M 2 (7) Using eqs. (6) and (7), the known compressibility coefficients can then be reproduced to within a maximum error of± 25% (Table 3). The unknown compressibility coefficients were then using this relationship. Treatment of phase transitions Some phase transitions, e.g. the α-quartz/ß-quartz transition, are spontaneous, and under reference conditions ß-quartz is not preserved. The thermophysical parameters and functions of ß-quartz are not accessible at temperatures below 855 K. To avoid the difficulties associated with their estimation, modifications of phases with spontaneous phase transitions were not treated as separate phase components. Any changes in the thermodynamic parameters and functions were considered at transition conditions and treated as first order transitions. n the case of additional lambda behavior, as in quartz, the lambda shape of the transition in Table 3. Values used for the evaluation of unknown compressibility coefficients. albite andalusite anorthite aragonite calcite clinochlore corundum diopside dolomite enstatite forsterite grossularite adeite kyanite magnesite muscovite periclase phlogopite pyrope quartz rutile sanidine sillimanite spinel talc tremolite wollastonite V (o.,) MPal L mol M formula weight MM n number of atoms atoms] n \ MPa Lmol atom-l M n \ g Lmol atom tabulated Γ * LMPa ß 0 6 calc. with eq. (6) LMPa

10 84 M. Gottschalk CO n V Fig.. Plot of ln ß values vs. n V /n. Lines are valid for fixed M/n values. c P is given by the original c P function (cf. Hemingway 987, for quartz). Using tabulated values for the transition temperature T t, and the entropy and volume change at T t, Δ5 t and AV t at 0. MPa, the transition boundary in P-T space can be calculated using the relationship: fáp_ dt V AS t AV t (8) Phase transitions for cristobalite, hematite, iron, larnite, leucite, magnetite, nepheline, quartz, tridymite and wollastonite were treated in this way. n Table 4 the transformation temperatures, the change in entropies, and change in molar volumes for each of these phase transitions are listed. All other polymorphs, e.g. andalusite, kyanite and sillimanite, which are preserved at reference conditions ( K, 0. MPa), were treated as separate phases. Table 4. Values used for the calculation of modification boundaries. phase transition α-cristobalite ^ ß-cristoba ite α-hematite ^ ß-hematite α-iron ^ α-iron (Currie point) α-iron ^± γ-iron larnite ^ α'-larnite tetragonal leucite ^ cubic leucite α-magnetite ^ ß-magnetite α-nepheline ^ ß-nepheline ß-nepheline ^ γ-nepheline α-quartz ^ ß-quartz α-tridymite ^ ß-tridymite low wollastonite ^ high wollastonite high wollastonite ^± c-wollastonite Tt(0. MPa) K] ASt at T t UK mod Fugacities for pure H2O and CO2 and activities on the H2O-CO2 binary ΔV t at T t LMPa mol n most mineral reactions studied experimentally, at least one fluid component is directly involved. Usually these reactions involve either dehydration or decarbonation or both. For the thermodynamic treatment of fluid components a function of state

11 nternally consistent thermodynamic data 85 is needed. For pure H 2 O and pure CO 2, a number of functions of state exist (e.g. Halbach & Chatteree, 982; Shmulovich & Shmonov, 975). For mixtures of H 2 O and CO 2, different functions of state have been proposed (Holloway, 976; Flowers, 979; Bowers & Helgeson, 983; Kerrick & acobs 98). Functions of state for either H 2 O or CO 2 are characterized in most cases by high precision. Functions valid for mixed H 2 O/CO 2 fluids sacrifice some of their precision for the sake of flexibility. To preserve consistency in all calculations, the fugacities of pure H 2 O and CO 2 and activities of H 2 O-CO 2 mixtures were calculated using the equation of state by Kerrick & acobs (98). This function of state reproduces the available experimental volume data reasonably well in the temperature and pressure range of q C and MPa (Ferry & Baumgartner, 987). Above 000 MPa, the function of state by Belonoshko & Saxena (992) was used. For pure H 2 O and CO 2 fluids at temperatures between 200 and 350 -C, the function of state for CO 2 by Shmulovich & Shmonov (975) and the function of state for H 2 O by Halbach & Chatteree (982) were used. Solid solution Most experiments are conducted in relatively simple chemical systems in which only very limited formation of solid solutions takes place. Sometimes, however, natural phases are used as starting materials. n such cases the activities for any deviation from pure end-member composition were calculated assuming ideal mixing. n some experimentally investigated mineral systems, two phases of a solid solution series coexist, forming a miscibility gap. n such a case appropriate activities had to be calculated. For the system calcite-dolomite activities were calculated using the mixing model formulated by Gottschalk & Metz (992). n the system diopside-enstatite activities were calculated using the mixing model by Lindsley et al. (98). Tschermaks substitution (MgSi T ^ AA T ) was considered only for the pyroxenes, since only very few experimental results exist for other phases such as amphiboles and phyllosilicates. The thermodynamics of the tschermaks substitution in these phases is also poorly understood, and tschermaks substitutions are not only a function of pressure and temperature but also of the phase assemblage. Activities due to tschermaks substitution in pyroxenes were calculated following the approach by Gasparik (986). n cordierite the H 2 O-content is at least a function of pressure and temperature (Mirwald et al., 979). Following the approach of Mirwald et al. (979) and Newton & Wood (979) the equilibrium of the reaction cordierite + n H 2 O ^= H 2 O- cordierite is described by the following thermodynamic equation: 0 = AH-TAS + ΔV solid dp + RT n O-cordierite / n H,0 (9) Because many investigated reactions deal with cordierite and water, parameters for eq. (9) were determined so that this equation is able to reproduce the experimentally determined H 2 O contents. t is not clear how much water can be incorporated in the cordierite structure as a maximum. Experimental data from Mirwald et al. (979) suggest that at high pressures more than mol H 2 O can be built into the structure. From a crystallographic point of view 2 moles of H 2 O is the theoretical maximum. Therefore n was set to 2. With the assumption of ideal mixing in eq. (9), it was not possible to reproduce the experimental data base with the required precision (McPhail et al, 990). Therefore in addition to the temperature dependence of ΔG a regular solution model was added. This leads to eq. (20) 0 = AH-TAS - \-γ- dtdt + \AV s áv + RT n T 0-*) 2 /H 2 C + (l-2x)(ff e -TSex) (20) where x is the mol fraction of hydrous cordierite. The heat capacity of hydrous cordierite is not well known. Only few data exist at low temperatures (Carey, 990). Therefore a constant d' was added to the d parameter for dry cordierite and used as an additional fit parameter. Using the experimental results of Mirwald et al. (979), the following parameters were derived: ΔH = k, ΔS = /K, ΔV = /MPa, H e = k, S ex = /K, d' = n Fig. 2 the experimental results from Mirwald et al. (979), Schreyer & Yoder (964), Armbruster & Bloss (982), Bobersky & Schreyer (990), Mukhopa-

12 86 M. Gottschalk CO α..cuu i ' ' i i ' i H i i i# i i i i i i i i i i if i i i i i i i i i.32,»,, * O' /, 0.92,' 0.83 '' / -07/ / -78 i / O / 0.9, 0.93 / ,.24' / * *.' to ' / 0-90/ ' 74 O.70, * '.07/ ' *' y.* * Or * O.88,0.82 * 0.63, * H ' * ' * * * \ * fo- 95 ' *.89 GO.83.' O.68 ** / / / *' - 86 &?''.**' 0.59 \ i - 2o#/ /, α ; 8 o -s *%o^ * * * i.io # * ' ', * * * * - * * * * ".00' * * 9'*.75*)** H " * Λ - - * *.65,--*.80 ^ ^Q *,66<0.6 ^ - O " ^ &54- "\&.4?.45g.6 dßi * " " " (ΓQ..5 6ZOD.59.70«Sa37._ D O-Dc Ä3 - drgd.5 A n ^&>- 35.3<2D.30 ] ^.6, - l, T C] O Schreyer & Yoder (964) O Mirwald et al. (979) Δ Armbruster & Bloss (982) V Bobersky & Schreyer (990) * Mukhopadyay & Holdaway (994) O Carey (995) D Skippen & Gunter (996) Fig. 2. Water content of cordierite as a function of pressure P and temperature T (numbers are «H 2 O in cordierite). Symbols are experimental results by various authors. Curves are calculated using eq. (20). dyay & Holdaway (994), Carey (995), Skippen & Gunter (996) are shown in conunction with the calculated equilibrium H 2 O-content using eq. (20). While only the experimental results of Mirwald et αl (979) were used for the derivation of the parameters in eq. (20), most of the other experimental results are also reproduced quite well. The activities of cordierite and H 2 O-cordierite can be calculated by the following expressions: (\-x)-(h eλ-ts n _ γ2 e "H^O-cordierite ~~ Λ c RT and ^cordierite ~" \ * X) Q Order-disorder x-(h-7s ex) (2) Some phases are substantially more ordered at low temperature than at high temperature. The degree of order has a significant effect on their Gibbs free energy. However for most phases the state of order is not known as a function of temperature or even at reference conditions. n the case of gehlenite, K-feldspar, Na-feldspar and spinel there is sufficient experimental data available that these energy effects could be taken into account. Thus at the temperature and pressure conditions of each experiment for which a K red value had to be determined involving one of these phases the degree of order and its associated energy was calculated. The degree of order and its associated change in the Gibbs free energy for gehlenite was calculated by the approach taken by Waldbaum (973), Waldbaum & Woodhead (975) and Charlu et αl (98). For the K-feldspars the approach by Thompson et αl (974) and Ho vis (974) was used. Order-disorder in Na-feldspar was calculated using the model by Sale et αl (985). A totally disordered phase was chosen as the standard state for the alkali-feldspars. The values for A f H and S for the ordered state of each of these phases were evaluated using the approaches given above.

13 nternally consistent thermodynamic data 87 n the case of spinel above C and 0. MPa the kinetics of the order-disorder process is very fast. At lower temperatures the spinel is not in equilibrium and the disordered state at approximately C is frozen in (Wood et al, 986). Therefore heat capacity measurements at high temperatures should include the energetics of the order-disorder process, whereas at low temperatures, the heat capacity is that of spinel in a metastable state of disorder because of slow solid state diffusion rates. For spinel formed in hydrothermal experiments ( C), however, it can be assumed that it is already closer to its state of equilibrium order. Therefore the heat capacity of spinel by Robie et al. (979) was corrected with the order-disorder model by Sack & Ghiorso (99). The heat capacity for spinel in Table is the heat capacity of totally ordered spinel at all temperatures. The experimental data base For the extraction of the internally consistent data set a total of over 5300 experimental results (half brackets) for 253 different mineral reactions involving 94 rock-forming minerals and the fluid components H 2 O, CO 2 and O 2 were collected. Virtually all experimental results were accepted with only minor restrictions. The following restrictions did apply. For many solid solution series activitycomposition relationships are not well known {e.g. Ganguly & Saxena, 987); also considerable deviation from ideal mixing can be expected. Therefore experiments conducted with phases showing extensive solid solutions could not be used and were not considered any further. Equilibria involving a melt phase were neglected. Some restrictions were imposed on the P-T conditions. Because of the limited P-T validity of the functions of state for H 2 O-CO 2 mixtures, experimental results involving H 2 O and CO 2 were restricted to the range of q C and MPa. Up to 000 MPa the function of state by Kerrick & acobs (98) and above 000 MPa the function of state by Belonoshko & Saxena (992) was used. f only pure fluids were involved in a reaction the results of experiments were considered down to C using functions of state valid for these fluids (Shmulovich & Shmonov, 975; Halbach & Chatteree, 982). n the case that no fluid at all was involved, experiments up to C and 3000 MPa were included in the study. Experiments at higher pressures could not be considered because of insufficient knowledge concerning the pressure and temperature dependence of V for solids. Errors and error propagation Both the experimental results and all other input parameters and functions used for the calculation of n K red values have uncertainties. n addition to the experimental errors reported in the literature, an uncertainty of 5% was assigned to the fugacities and activity coefficients of the fluid components and also to all activities of the solid phase components. n most cases where mixed fluid phases were involved an error of x C o, ± 0.02 was applied to the fluid composition. Extraction procedure The two internally consistent data sets most commonly used in petrology are those of Holland & Powell (990) and Berman (988). The methods used in the derivation of these two data sets are quite different. The regression algorithm (REG) used by Holland & Powell (990) and the linear/ mathematical programming approach (MAP) used by Berman (988) can be summarized as follows. The regression algorithm (REG) used by Holland & Powell (985) and Holland & Powell (990) to optimize AfH is given in Powell & Holland (985). n this procedure AH values were determined using experimentally derived mineral equilibria for each reaction. This was done by setting eq. (9) to zero (equilibrium) and solving for AH. AH = TAS - RT n K red (22) Deducing the equilibrium conditions by looking for experimental brackets and knowing all thermodynamic parameters and functions except A f H allows the calculation of AH for each phase equilibrium in the form of AH brackets. By taking the center of each AH bracket and utilizing a relationship in the form of eq. () for AH, individual A f H values for each phase component could be calculated using a least square approach. The linear/mathematical programming approach (MAP) used by Berman et al. (986) and Berman (988) is based on a set of inequalities. For every available experimental result an inequality of the form 0 H - T AS + RT n K red (23)

14 88 M. Gottschalk is written, where the V and the '<' inequality signs were determined by whether products or reactants, respectively, were observed as being stable (cf. eq. 0). Additional constraints were formulated for the thermophysical parameters and coefficients (à f H ( \ S, V, c P, α, and ß) where needed. These additional inequalities restricted the range of values in which these parameters and coefficients were allowed to vary during the optimization process. The range of values were usually experimental uncertainties to the thermophysical data. The linear programming technique solves such a system of inequalities for values of variables which are consistent with all experimental constraints (i.e. inequalities). The range of possible values obtained is called the region of feasible solution. The best solution in this solution space was derived using an obective function. This obective function minimizes the difference between measured (e.g. calorimetric) and extracted values. Since the obective function is not linear but quadratic, mathematical programming is used to derive an optimized solution. The focus of the present study is to extend the method proposed by Powell & Holland (985) such that AfH and S are optimized over a series of iterations. This allows the equilibrium points between brackets to be chosen as a best fit to the database overall rather than arbitrarily as a halfway point. Since almost all relevant experimental data is included, this particular approach has the additional feature of sorting out which sets of experimental data are in distinct disagreement with the data base as a whole. This allows the internally consistent data base itself to chose which experimental data sets are relevant to its construction. The approach is also free of irreversible decisions, i.e. the size of the region of feasible solution. The iterated least square algorithm (REG), which is used here, also takes advantage of eq. (9). At equilibrium conditions, i.e. ΔG( P)T) = 0, eq. (9) can be rearranged to which has the form of a linear relationship y = mx + b, where /T is the variable. n a /T vs. n K red plot, the equilibrium of a chemical reaction in P-T-x space is reduced to a straight line with a slope of AH /R and an intercept of Δ57R. At any temperature only one n K red equilibrium value exists. Experimental philosophy, in most cases, is to use brackets to determine equilibrium conditions. Experimental results very rarely, if ever, report equilibrium. f n K red values are calculated for these non-equilibrium conditions and plotted on a /T vs. n K red plot, these experimental constraints do not plot themselves on the equilibrium line (eq. 0). f a mineral reaction is formulated in such a way that AH is always positive, then experimental results for mineral reactions where the products are stable plot above and where the reactants are stable plot under the equilibrium line. Therefore all experimental results for a mineral reaction can be treated as half brackets above and below this equilibrium line. Fig. 3a illustrates this point for the reaction calcite + muscovite + 2 quartz ^ anorthite + sanidine + CO 2 + H 2 O (Hewitt, 973). Here experiments with coexisting anorthite + sanidine (empty symbols) plot below and coexisting calcite + muscovite + quartz (filled symbols) plot above the equilibrium line. The calculated equilibrium line for this reaction and therefore AH and AS are well constrained by these experiments. Another advantage of a /T vs. n K red plot is that it checks the consistency of all available experimental data for the reaction in question. However, this is true only if all input data used for the calculation of n K red are valid, i.e. heat capacities, partial molar volumes, coefficients for thermal expansion and compressibility, fugacities of fluid species, activities of components in mixed phases, and treatment of orderdisorder processes. Fig. 3b shows an additional example of a /T vs. n K red plot for experimental results from the reaction diopside + 3 dolomite F 2 forsterite + 4 calcite + 2 CO 2 (Käse & Metz, 980; Richter, 977, 980). n this particular case activities for coexisting calcite and dolomite on their solvus are also taken into account. For each n K red value, an error is also calculated and plotted. The horizontal error bar designates the error in temperature, the vertical bar incorporates all other uncertainties which contribute to the error of n K red. These two error bars create an error rectangle. Not all errors considered are independent of each other (e.g. fugacities are temperature dependent). Therefore an error rectangle is only a first, but reasonable approximation, because temperature uncertainties for hydrothermal experiments are usually less than 0 9 C and less than 5 9 C for piston cylinder runs. n this temperature range fugacities and other parameters such as molar volumes of solids and activities of the phase components vary little. However, plotting

15 nternally consistent thermodynamic data \ L f ΛWHE Λ.HE.HE HET^ÄUβft ' ' Y HE ^P KACS-4 ΔH = k ΔS = /K LE\ ( 5 An + Sa + C02 + H20 " Cal + Ms + 2 Qtz \\ LΛ H \ o \ HE \... \ /T*000 /K] r~\ r K *Ψ \ K RΨΛ. 4 Cal + 2 Fo + 2 C02 ~~'~' ' ' '""' ' '"' ' " CMS- ΔH = k ΔS = /K Di + 3 Dol f* RVfsF ] -] \ \4 \ /T* 000 /K] C] C] Fig. 3. /T vs. n Kred plots for the reactions: a) calcite + muscovite + 2 quartz ^ anorthite + sanidine + CO2 + H2O (HE: Hewitt, 973). b) diopside + 3 dolomite F 2 forsterite + 4 calcite + 2 CO2 (K: Käse & Metz, 980; R: Richter, 977, 980). The following notation applies for all /T vs. n K re d plots: filled symbols represent stable reactants, empty symbols represent stable products, hatched symbols represent experiments with no observed reaction, cross hatched symbols are for reported equilibria. The number of corners of a symbol gives information on the pressure (circles 00 MPa, two circles 200 MPa, triangle 300 MPa, square 400 MPa etc.). the real shape of the error region onto a /T vs. n K red plot would reduce the readability of the plot dramatically. For an experimental result to be in accordance with the equilibrium line, at least one corner of the error rectangle associated with an experimental result, which is considered to represent the 2G region of the calculated n K red value, must be on the correct side of the equilibrium line. n Fig. 3b, the symbol for the experimental half bracket at 000 MPa and C, which shows stable coexistence of diopside + dolomite, plots on the wrong side of the equilibrium line. The actual upper right corner of the associated error rectangle, however, lies on the correct side. So within the error range of the experimental results an equilibrium line can be drawn which separates the stable coexisting phases diopside + dolomite from forsterite + calcite. An equilibrium line can not be chosen arbitrarily, but is defined by AH and ΔS of the reaction. The constant standard values of ΔfH and S for each phase component contribute to its slope and intercept. f a set of such /T vs. n K red plots is evaluated simultaneously, one for each mineral reaction considered, optimized values for AfH and S can be derived. t is, however, important to note that the available experimental results used in a /T vs. n K red plot only bracket the equilibrium line and are not used as data points through which the equilibrium line is fitted. Again, the experiments are used to constrain the calculated equilibrium line. t is of no importance if some of these experimental constraints are far from the equilibrium line, as long as they are on the correct side. The extraction procedure can be described briefly as follows. For every mineral reaction a

16 90 M. Gottschalk l/t vs. ln A^/plot was drawn. As a first step, for each l/t vs. ln K red plot one or two points were chosen which were thought to be very close to a plausible equilibrium line. The experimental data constraining these points had to be well determined. Both reactants and products had to be stable within close proximity. Utilizing a least squares formalism, a set of enthalpies (A f H ) and entropies (S ) was found for which the equilibrium lines were as close as possible to the chosen equilibrium points. n addition, the A f H and S values were constrained to a minimum deviation from their calorimetric values. Because the ln K red equilibrium points in each plot were initially chosen manually their selection was not necessarily obective. Therefore the results after the first extraction run were not final. Successive iterations were used until an internally consistent data set was achieved. After each run, the equilibrium lines for each mineral reaction were plotted in the l/t vs. ln K red plots using extracted AfH and S values from the previous run. f the selected point in the l/t vs. ln K red plot used for the previous iteration cycle was not identical to the current equilibrium line, but the separation of the phase assemblages with this new equilibrium line was still in agreement with the experimental results, a point from this new equilibrium line was chosen as input for the next iteration cycle. For some reactions contradictory sets of experimental results were available. f the new equilibrium line was compatible with another available set of experimental results than the one used for the previous iteration, again, a point from this new equilibrium line was chosen as input for the next iteration cycle, using the other set of experimental results as constraints. The brackets confining the equilibrium line in each l/t vs. ln K red plot were of different widths. The vertical width, which is the width of the bracket with respect to ln K red, can be used as a measure of the validity and quality of the bracket. The reciprocal width was used as a normalizing and weighting factor for this reaction in the least square algorithm. Reactions with wide brackets were less important to the least square algorithm than reactions with tight brackets. Also the reciprocals of the errors of the available calorimetric AfH and S values were used as weighting factors. Some derived A f H and S values showed a tendency to shift away from tabulated calorimetric values. Some third law entropies especially showed a tendency drift to higher values, indicating significant disorder in these phases at elevated temperatures. When this was observed, the constraint which linked the derived S to the calorimetric value was weakened using a higher variance and therefore a lower weighting factor. For some phases calorimetric values with significant differences in AfH and in some cases S (e.g. garnets) were available. At the starting point of the extraction process it was not obvious which values were more reliable. As a consequence, if the result of an iteration cycle preferred a different calorimetric value for AfH and/or S than the one used for the previous iteration, the calorimetric value which came closest to the preferred value was used as the input parameter for the next iteration. Each iteration cycle generated a new input data set. This process was repeated several times until a stable solution was obtained. The formalism behind each extraction cycle is itself very similar to the approach (REG-technique) used by Powell & Holland (985). n this special case, their approach was adapted to the l/t vs. ln K red plots. For each mineral reaction the reduced equilibrium constant ln K red can be expressed by eq. (24) at a fixed temperature T. A more general form of this expression is eq. (25). For the / th reaction constraint, here or 2 for each reaction, the reduced equilibrium constant ln K red at temperature T, is given by: \r\k n T; (25) For a total of n reaction constraints this can be written as a matrix product: A R = B D (26) The vector A R includes the reduced equilibrium constant ln K red for each reaction at a given temperature T/. These ln K red values are the selected points in each l/t vs. ln K red \Aot. The elements of the rows of matrix B are the stoichiometric coefficients for each phase component in each reaction. The A f H and S values of each phase component are listed in vector D, or: Kred,T l A R = (27) ln K redtn l n Vn Vi Vπ Vi^ R R RT, RT, \ \ V^ _ V^ _ V^ Vnk_ R R RT n RT n (28)

17 nternally consistent thermodynamic data 9 D u ST s: A f H, c A f H * (29) n the next step an obective function for the optimization process must be defined. As discussed above, the following criteria were used for the extraction: () assemblages in each /T vs. n K red plot should be perfectly separated by the equilibrium line, i.e. as close as possible to the observed (obs) n K red values; (2) all third law entropies and enthalpies of formation should be close to the values determined by calorimetry. These criteria lead to the following obective function f ob : /.=ΣiΣ^;- ) n K" where the parameters F are the associated variances of n K Kd, AfH, S and therefore \F are the weighting factors. The solution to f oh is obtained by using eq. (33). D = (B T WA B+EWf,,-) (: BT WA A R +EWD-^D (33) The weighting matrix W AR for the manually selected points in the /T vs. n K red plots and the tabulated calorimetric values W D»«"> are defined as follows: (34) Δ/"y -VΓ ') (30) F o tab l =l The superscript tab stands for a calorimetric, tabulated value. Using a least squares approach, the minimum for the obective function can be determined with eq. (3). D = (B T B + E)- (B T A R + E Y> otab ) (3) D tab is a vector which includes the tabulated, i.e. calorimetric, reference entropies and enthalpies. E is the identity matrix. D is the result and includes the optimized AfH and S values of each phase component. The following modified obective function (cf. Powell & Holland, 985) for the weighting scheme was used for the extraction of the internally consistent data set, ob] Σ (v* s"- (AfHi AHL FΛ,H;,o tab \ S}-ST, S'",Δ / H: T, L ) ' ln?red,t. (32) W n F AfH <; 0 %wr* (35) Again vector D contains the optimized AfH and S values. Calorimetric enthalpies and entropies of certain simple phases, especially elements and oxides, are considered to be well known experimentally. With this in mind, A f H and S values of corundum, hematite, iron, periclase, quartz, rutile, CO 2, and O 2 were used as anchors, along with the enthalpies and entropies for andalusite, kyanite and sillimanite from Hemingway et al. (99b). For application in geothermobarometry it is essential to estimate the errors associated with the derived enthalpies and entropies. The least squares logic itself provides an estimate of the uncertainties for the values of enthalpy and entropy through the covariance matrix. This approach was used by

18 92 M. Gottschalk 700 *C0 2 Fig. 4. Phase diagram for the CMS subsystem which includes the phases calcite, dolomite, quartz, talc, tremolite, diopside, forsterite and wollastonite. Phase equilibria are calculated using the derived data set. Error ranges of 2o~ for each of the calculated equilibria are also shown as gray shaded areas. Powell & Holland (985) and Holland & Powell (985, 990). However this approach does not seem to be appropriate for the procedure used here. The resulting errors which the covariance matrix delivers are dependent on the input parameters. The closer the input parameters are to a final result, the smaller will be the calculated errors using the covariance matrix. n the iterative approach used here, the input data were already relatively close to the final result, making errors calculated from the covariance matrix meaningless. To get at least some idea about the confidence of the extracted A f H and S values, errors were using a Monte Carlo simulation, with a variance assigned to each input parameter of the final iteration. Values within this selected range had to be still feasible with respect to the experimental results for each mineral reaction and the calorimetric results. Values within this range were then selected randomly using a Gaussian distribution. With these randomly chosen input parameters the regression was calculated again. This procedure was repeated 000 times, every time using a new randomly chosen set of input parameters. Results were then averaged and the standard deviation (α) calculated. t is clear that while the resulting errors reflect the assigned variance of the input parameters, it at least gives some level of confidence for the extracted A f H and S values. The consequence of these errors on a phase plot is illustrated in Fig. 4 for the system CaO-MgO-SiO 2 -H 2 O-CO2 and the relevant phases calcite, dolomite, quartz, talc, tremolite, diopside,

19 nternally consistent thermodynamic data 93 Table 5. Derived S and comparison with those of other sources. Standard entropies >S àkermanite albite almandihe analbite analcite andalusite andradite anorthite anthophyllite antigorite aragonite boehmite brucite CaAl-pyroxene calcite carpholite chloritoid chrysotile clinochlore clinoenstatite clinozoisite coesite cordierite corundum cristobalite deerite diaspore diopside dolomite edenite enstatite fayalite ferroactinolite ferrocordierite ferroglaucophane ferrosilite forsterite gehlenite glaucophane grossularite grunerite hedenbergite hematite hercynite ilmenite iron adeite kalsilite kaolinite kyanite larnite laumontite lawsonite leucite lime magnesite magnetite margarite meionite merwinite microcline monticellite mullite muscovite Na-phlogopite nepheline nickel NiO paragonite pargasite periclase phlogopite prehnite pumpellyite Kmol S ±2( ( ( ( ( ( ( < ( ( < < Berman (988) S Holland & Powell (990) S Helgeson etal (978) S Halbach & Chatteree Chatteree etal (984) (994) 5 S G Robie etal. (979) S Hemingway etal (982) ±2α S ±2(

20 94 M. Gottschalk Table 5. (Cont.) Standard entropies S" pyrope pyrophyllite quartz rankinite rutile sanidine sapphirine sillimanite sphene spinel spurrite staurolite talc tilleyite tremolite tridymite vesuvianite wairakite wollastonite wuestite zoisite carbondioxide water oxygen Kmol S ±2( Berman (988) S Holland & Powell (990) S Helgeson etal. (978) S Halbach & Chatteree Chatteree (984) S etal. (994) S ±2( Robie etal. (979) Hemingway ±2o etal. (982) S ±2α Table 6. Derived AfH values and comparison with those of other sources. Standard enthalpies ΔfΓ àkermanite albite almandine analbite analcite andalusite andγadite anorthite anthophyllite antigorite aragonite boehmite brucite CaAl-pyroxene calcite carpholite chloritoid chrysotile clinochlore clinoenstatite clinozoisite coesite cordierite corundum cristobalite deerite diaspore diopside dolomite edenite enstatite fayalite ferroactinolite ferrocordierite ferroglaucophane ferrosilite forsterite k mol ΔfH ±2o Berman (988) Δß Holland & Powell (990) ΔfH ±2o Helgeson etal. (978) Δfl Halbach & Chatteree (984) ΔfH Chatteree etal. (994) ΔfH ±2α Robie etal. (979) Δß Hemingway ±2( etal. (982) ΔfH ±2 O.fr. 0.7:.9.0C o.3e

21 nternally consistent thermodynamic data 95 Table 6. (Cont.) Standard enthalpies ΔH ΔfH ±2CT Berman (988) ΔH Holland & Powell (990) ΔfH Helgeson Halbach & etal. Chatteree (978) (984) ±2α Δß ΔH Chatteree etal. (994) ΔH ±2α Robie Hemingway etal. etal. (979) (982) Δfl ±2o ΔH ±2 a

22 96 M. Gottschalk Table 7. Calorimetric S values from various sources. calorimetric values from various sources ákermanite almandine andalusite andradite anthophyllite boehmite CaAl-pyroxene clinochlore diaspore diopside enstatite fayalite ferrosilite forsterite gehlenite grossularite grossularite grossularite hedenbergite ilmenite kaolinite kyanite lawsonite magnesite magnetite margarite meionite monticellite paragonite phlogopite prehnite pyrope pyrope sillimanite staurolite wollastonite zoisite S ±2o Hemingway etal (986) Metz etal. (983) Robie & Hemingway (984a) Robie etal. (987) Krupkaetal. (985) Hemingway etal. (99a) Haselton et al. (984) Henderson etal. (983) Perkins etal. (979) Krupka etal. (985) Krupka etal (985) Robie etal. (982a) Bohlenétfa/. (983) Robie etal. (982b) Hemingway & Robie (984) Haselton & Newton (980) Westrum(979) Kolesnik et al. (977) Haselton etal (987) Anovitzetal. (985) Hemingway & Kittrik (978) Robie & Hemingway (984a) Perkins etal (980) Hemingway et al. (977) Westrum & Gronvold (969) Perkins etal (980) Moecheretal (985) Sharp etal. (986) Robie & Hemingway (984b) Robie & Hemingway (984b) Perkins etal. (980) Haselton & Newton (980) Kolesnike?α/. (977) Robie & Hemingway (984a) Hemingway & Robie (984) Krupka etal (985) Perkins etal. (980) forsterite, wollastonite, CO 2 and H 2 O at 00 Mpa, with an error of 2o for the calculated equilibria using the data listed in Tables 5 and 6. Results and evaluation Table 8. Calorimetric AfH values from various sources. n Tables 5 and 6 the extracted Δ f H and S values are listed together with values from other internally consistent data sets from Berman (988), Holland & Powell (990), Helgeson et al (978), Halbach & Chatteree (984), Chatteree et al (994), Hemingway et al. (982) and the data set from Robie et al (979) which is mainly a collection of calorimetric measurements. n addition single calorimetric determinations from various sources are listed in Tables 7 and 8. The final set of /T vs. n K red plots provide information on how well the calculated phase equilibria match the available experimental results. A selection of these /T vs. n K red plots is presented in Fig The complete set of /T vs. n K red plots can be used as a catalogue of existing experimental results. When looking through this catacalorimetric values from various sources Δ f H ±2Q ákermanite ákermanite ákermanite almandine Hemingway etal (986) Charlu etal (98) Brousseetal (984) Chatillion-Colinetefα/. (983) anorthite Charlu etal (978) anorthite Newton etal (980) CaAl-pyroxene Charlu etal. (978) cordierite Charlu etal. (975) diopside Charlu etal. (978) diopside enstatite Kiselewa etal (980) Brousse etal (984) fayalite forsterite Thierry etal (98).883 Brousse etal (984) gehlenite grossularite grossularite merwinite Charlu etal. (98) Charlu etal. (978) Krupka etal (979a) Brousse etal. (984) monticellite Brousse etal (984) muscovite paragonite phlogopite pyrophyllite Krupka etal. (979b) Robie & Hemingway (984b) Clemens etal (987) Krupka etal. (979b) wollastonite Charlu etal. (978) wollastonite Kiselevaetal (980) logue it becomes obvious which reactions are experimentally well determined and for which reactions only a small amount of data exists. Moreover, one may easily separate compatible from incompatible experimental results, see how well the internally consistent data set reproduces these experimental results, or deduce from systematic departures of experimental results from the calculated equilibrium line errors in either fugacities, activities or thermophysical constants or functions. For some of the phases considered here, values for AfH and S were previously scarce or not even available. Refined values for analcite, carpholite, chloritoid, deerite, edenite, ferroactinolite, ferrocordierite, ferroclaucophane, grunerite, larnite, laumontite, leucite, mullite, rankinite, sapphirine, spurrite, tilleyite, wairakite and vesuvianite are now available. While values for some of these phases are also available in the data set of Holland & Powell (990), the differences between their values for these phases and the values presented here are considerable. From Tables 5 and 6 it can be seen that for many phases the derived A f H and S values are similar to previous determinations. For others sig- The complete set of final /T vs. n Kred diagrams together with a complete list of references for the 253 reactions considered can be either obtained in printed form from the author or through the E.M. Editorial Office - Paris, or by WWW under the address

23 nternally consistent thermodynamic data 97 nificant differences arise. Most experimental results are in agreement with the derived data set. From a total of over 5300 experimental halfbrackets considered, only 405 are not compatible within the 2α errors. There are several possible explanations for the discrepancies between the experimental data and the internally consistent data set derived here. These can be summarized briefly as follows. However, due to their complexity some of these arguments are speculative. The parameters and functions needed for the calculation of an n K red value for each experimental half-bracket may not provide the required data for the pertinent physical conditions (P and/or T). Published heat capacities above 00 -C are sometimes of limited value. Experimental results from mineral reactions at high temperatures where no fluid components were involved especially tend to be poorly reproduced. These particular reactions have very low AS values, so any errors in heat capacity cannot be neglected. Also, at high temperatures solid solutions between end-members are more pronounced. t seems reasonable to assume that in some cases this fact was not accounted for correctly here. Other phases like clinochlore and sapphirine are known to change composition as a function of pressure, temperature and coexisting phases. Up to now these effects have not been very well understood quantitatively. Therefore only one fixed composition for these phases was chosen as a first approximation. t is possible that some portion of the observed configurational entropy is due to this compositional effect. t is also possible that in some phases polymorphic transitions or order-disorder processes were not accounted for well enough. Experimental brackets for natural phases with large amounts of other components also tend to fall off the equilibrium line. This is mainly due to an insufficient knowledge of the mixing behavior and its associated Gibbs free energy in such systems. Also, especially in some older publications, the composition of product phases are documented insufficiently. For some mineral reaction sets, the experimental results are contradictory. For example, in one case it is obvious that the experimental results are not for the mineral reaction stated. Walter (963) published results for the reaction forsterite + calcite F X monticellite + periclase + CO 2 but monticellite was never observed (Fig. 2a). Clearly these results are contradictory to those from Zharikow et al. (977), which are reproduced perfectly by the derived equilibrium line. Since none of the experiments reported in the literature was excluded, these erroneous results (4 for this reaction) must contribute to the 405 experimental results not reproduced by the data set. As discussed by Berman et al (986), experimental results derived by the "weight loss method" are particularly problematic. They are usually published as equilibrium data. n many cases these results are not compatible with others and often fall off the derived equilibrium line. Lastly, the uncertainty stated for these experiments seems to be too low (Berman et al, 986). n this study the uncertainties published with the equilibria data are used. The following systematics apply to the /T vs. n K red plots (Fig. 5-7). The reactions are arranged in chemical subsystems. Elements are sorted in the following order: K, Na, Al, Ca, Fe, Mg, Si. Hydrogen, oxygen and carbon are not explicitly mentioned. For example, the designation ACMS refers to the subsystem Al 2 O 3 -CaO-MgO- SiO 2 -H 2 O-CO 2. The same symbols are used as in Fig. 3a,b. Abbreviations and formulae for phase components are summarized in Table 9. The /T vs. n K red plots for the reactions ACMS- and ACMS-2 (Fig. 5a,b) are examples of how equilibrium lines are constrained by the experimental results. With regard to the error rectangle, which is defined by the horizontal and vertical error bars, all experimental results by Chernosky & Berman (986a, 988) are in agreement with the equilibrium line. n contrast, experimental results involving vesuvianite and pumpellyite are scarce (ACMS-7 and ACMS-8 in Fig. 5c,d), and only a few brackets exist. Calorimetric data are not available for these phases, so the derived values have to be interpreted as being preliminary. The experimental results by Hochella et al. (982) for reaction ACMS-7 are nicely reproduced, those from Olesch (978) are not. For reaction ACMS-8 two sets of experiments by Hinrichsen & Schiirmann (969) and Schiffman & Liou (980) exist. Both sets are contradictory and only the experimental results of Schiffman & Liou (980) agree with the calculated equilibrium line. All experiments for reaction ACS-2 (Fig. 6a) plot above the equilibrium line. These experiments were conducted at high temperatures above C. Because the same behaviour was not observed for other reactions involving the same phases at lower equilibrium temperatures, it seems likely that the available heat capacity functions, at least for some of the phases, are not valid

24 98 M. Gottschalk Table 9. Abbreviation and formula for each phase component. äkermanite albite almandine analbite analcite andalusite andradite anorthite anthophyllite antigorite aragonite boehmite brucite CaAl-pyroxene calcite carpholite chloritoid chrysotile clinochlore clinoenstatite clinozoisite coesite cordierite corundum cristobalite deerite diaspore diopside dolomite edenite enstatite fayalite ferrocordierite ferroglaucophane ferroactinolite ferrosilite forsterite gehlenite glaucophane grossularite grunerite hedenbergite hematite hercynite ilmenite iron adeite kalsilite kaolinite kyanite Ak Ab Aim Aab Anl And Adr An Ath Ant Arg Ca 2 MgSi ] NaAlSi 3 Og] Fe 3 Al 2 (Si04)3] NaAlSi ] NaAlSi ] H 2 0 Al 2 Si0 5 Ca 3 Fe 2 (Si0 4 ) 3 ] CaAl 2 Si 2 Og] Mg 7 Si /(OH) 2 ] Mg 4 8Si /(OH) 62 ] CaC0 3 Bhm AlOOH Brc Cats Cal Crp Cld Ctl Cchl Cen Czo Coe Crd Crn Crs Dee Dsp Di Dol Ed En Fa Fcrd \ Fgln Fac Fs Fo Gh Gin Grs Gru Hd Hem He Dm Fe d Kls Kin Ky Mg(OH) 2 CaAlAlSi0 6 ] CaC0 3 MgAl 2 Si 2 06/(OH) 4 ] FeAl 2 Si04/0/(OH) 2 ] Mg6Si 4 Oio/(OH)8] Mg 5 AlAlSi 3 O 0 /(OH) 8 ] Mg 2 Si 2 O β ] Ca^tSiCVShOyO/OH] Si0 2 Mg 2 Al 3 AlSi 5 0i 8 ] A Si0 2 Fei 2 Fe 6 Sii o/(oh)io] AlOOH CaMgSi ] CaMg(C0 3 ) 2 larnite laumontite lawsonite leucite lime magnesite magnetite maγgarite meionite merwinite microcline monticellite mullite muscovite Na-phlogopit nepheline nickel NiO paragonite pargasite periclase phlogopite prehnite pumpellyite pyrope pyrophyllite quartz rankinite rutile NaCa 2 Mg5AlSi /(OH) 2 ] sanidine Mg 2 Si ] Fe 2 Si0 4 ] Fe 2 Al 3 AlSi 5 0i 8 ] Na 2 Fe 3 Al 2 Si /(OH) 2 ] Ca 2 Fe 5 Si /(OH) 2 ] Fe 2 Si ] Mg 2 Si0 4 ] Ca 2 Al 2 SiO? ] Na 2 Mg 3 Al 2 Si /(OH) 2 ] Ca 3 Al 2 (Si0 4 ) 3 ] Fe 7 Si /(OH) 2 ] CaFeSi ] Fe FeAl FeTi0 3 Fe NaAlSi ] NaAlSi0 4 ] water Al4Si40io/(OH)8] oxygen Al 2 Si0 5 sapphirine sillimanite sphene spinel spurrite staurolite talc tilleyite tremolite tridymite vesuvianite wairakite wollastonite wuestite zoisite Lrn Lmt Lws Lc Lim Ca 2 Si0 4 ] CaAl 2 Si 4 Oi 2 ] 4 H 2 0 CaAl 2 Si /(OH) 2 ]-H 2 0 NaAlSi ] CaO Mgs MgC0 3 Mag Fe Mrg CaAl 2 Al 2 Si 2 Oio/(OH) 2 ] Me Ca 3 (Al 2 Si ) 3 ] CaC0 3 Mrw - Ca 3 Mg(Si0 4 ) 2 ] Mk KAlSi ] Mtc CaMgSi0 4 ] Mul Al 6 Si 2 Oi 3 Ms KAl 2 AlSi 3 O 0 /(OH) 2 ] e Nphl NaMg 3 AlSi 3 Oio/(OH) 2 ] Ne NaAlSi0 4 ] Ni Ni NiO NiO Pg NaAl 2 AlSi 3 Oio/(OH) 2 ] Prg NaCa 2 M g4 AlAl 2 Si /(OH) 2 ] Per MgO Phi KMg 3 AlSi 3 O 0 /(OH) 2 ] Prh CaAlAlSi 3 Oio/(OH) 2 ] Pmp Ca 4 Al 4 (MgAl)Si i/(oh) 7 ] Prp Prl Qtz Rnk Rt Sa Spr Sil Spn Spl Spu St He Tly Tr Trd Ves carbondioxide C0 2 C0 2 H 2 0 H 2 0 O2 c^ Mg 3 Al 2 (Si0 4 ) 3 ] Al 2 Si 4 Oio/(OH) 2 ] Si0 2 Ca 3 Si ] Ti0 2 KAlSi ] Mg 7 Al Al 9 Si ] Al 2 Si0 5 CaTitSiOyO] MgAl Ca 2 Si0 4 ] CaC0 3 Fe 4 Ali 8 Si (OH) 2 Mg 3 Si 4 Oio/(OH) 2 ] Ca 3 Si ] 2 CaC0 3 Ca 2 M g5 Si /(OH) 2 ] Si0 2 Ca 9 Mg(MgAl 7 )Al 4 Sii 8 069/(OH) 9 ] Wrk CaAl 2 Si 4 Oi 2 ]»2H 2 0 Wo Ca 2 Si ] Wus FeO Zo Ca 2 Al 3 Si0 4 /Si /0/OH] in this extreme high-temperature range. n the reactions ACS- and ACS-4 (Fig. 6b,c) clinozoisite and zoisite are involved. All brackets except one by Best & Graham (978) for reaction ACS- 4 are in agreement with the equilibrium line. This plot illustrates how experiments by different groups are consistent (Best & Graham, 978; ohannes & Ziegenbein, 980; Goldsmith, 98; Chatteree et al, 984; enkins et al, 985). The plot for reaction ACS-7 (Fig. 6d) shows two sets of experimental results which are obviously not consistent. The experiments by Storre & Nitsch (974) were conducted using natural margarite, whereas enkins (984) used synthetic starting materials. No activity data for margarite in solid solution exists, and the assumption of ideal mixing does not reconcile the contradictory experimental results. The derived entropy value for margarite is approximately 6 /K higher than those from other internally consistent data sets

25 nternally consistent thermodynamic data y \ C B \ _,,_._. CB^W"! ACMS- ΔH = k ΔS = /K 2 Dol + Cchl ~ ] λ \ c 8 CB^ ^ΓCB k * CB 2 Cal + 3 Fo + Spl + 2 C H20 \ F (a) /T*000 /K] C] C] ΔH = k ΔS = /K Ves /T*000 /K] E o\ SLXQV SL*Qä VèSL SL S SL-? S V*SL SLT* T T ".,,, Γ " -πr. -, ACMS-8 π ΔH = k Γ ΔS = /K L v «H 25 Pmp SL sπwr\»e-/λ--». *sl V^ >,H 3 sly *Hf- \ HS (2^ siy 29 Czo + 4 Grs + 5 Cchl + 6 Qtz + 53 H HS ^ * /T*000 /K] \sl C] C] Fig. 5. a)acms-l. CB: Chernosky & Beπnan (986a, 988). - b) ACMS-2. CB: Chernosky & Berman (986a, 988). - c) ACMS-7. O: Olesch (978), H: Hochella et ai (982). - d) ACMS-8. HS: Hinrichsen & Schiirmann (969), SL: Schiffman & Liou (980) (see text and Fig. 3 for further explanation).

26 200 M. Gottschalk 7.0. i i i 7.0 " ' ' ' ' ' ' f Γ 3 An + Crn G8 GT ACS-2 ΔH = k ] ΔS = /KM 3 Ky + Grs i $ GT ] An + H 20 ACS- ΔH = k ΔS = /K ^ Ky + 2 Czo + Qtz ] V \ * r\ \ /T*000 /K] /T*000 /K] C] L ' ' ' ' F X 4 ~^ "" Γ " r " G ÉÊ G f 4 An + H20 A AG ff Cλ G Ψ X r l-t-γ cy>.... "π ACS-4 ΔH = k Δ S = /K Ky + Qtz + 2 Zo \ ΨC C* \ 4 \ T H <N /T*000 /T*000 /K] C] C] Fig. 6. a) ACS-2. G: Gasparik (984). - b) ACS-. : enkins et al (985). - c) ACS-4. BG: Best & Graham (978), : ohannes & Ziegenbein (980), G: Goldsmith (98), C: Chatteree et al (984), : enkins et al (985). - d) ACS-7. SN: Storre & Nitsch (974), : enkins (984) (see text and Fig. 3 for further explanation).

27 nternally consistent thermodynamic data 20 and from calorimetric measurement (Perkins et al, 980). This could indicate some configurational entropy due to order-disorder effects at experimental conditions. More /T vs. n K red plots in the subsystem ACS are shown in the Fig. 7. The plot for the reaction ACS-30 demonstrates that the bracket by Boettcher (970) is not consistent with those by Hays (967), Huckenholz et al. (975) and Shmulovich (974). The derived entropy value for CaAl-pyroxene is about 0.5 /K higher than that of the calorimetric value. Total disorder of Al and Si on their tetrahedral sites would add a configurational entropy of.53 /K. Experiments with CaAl-pyroxene were conducted at temperatures above 00 g C. The observed higher entropy indicates that at these temperatures the CaAl-pyroxene seems to show significant tetrahedral disorder. The entropy value derived for prehnite is /K, which is nearly identical to the value of proposed by Helgeson et al. (978). The calorimetrically derived entropy is /K (Perkins et al, 980). Chatteree et al (994) argued that the use of natural prehnite could be the cause of the observed discrepancies. Third law entropies for laumontite and wairakite are considerably lower than predicted by Helgeson et al (978). Reactions in the subsystem AMS involve such important phases as clinochlore, cordierite, pyrope, sapphirine and others. Variable compositions, especially tschermaks substitutions, variable H 2 O- contents, and order-disorder processes adding configurational entropy contributions, make this subsystem difficult to quantify in general. The calorimetrically determined third law entropy for clinochlore by Henderson et al (983) is /K. All available internally consistent data sets report an additional configurational contribution between 24 and 38 /K. The data set derived here adds 3.6 /K. Total octahedral disorder would contribute 22.5 /K and total tetrahedral disorder 8.7 /K. The order-disorder behavior of clinochlore at elevated pressures and temperatures, however, is totally unknown. The same is true for the composition of clinochlore as a function of pressure, temperature and bulk composition. As to what extent the additional entropy contribution is an artifact from variable phase composition cannot be answered with our present knowledge. Examples of equilibria with clinochlore are shown in Fig. 8a, 8d and 9a. For pyrope, different calorimetric determinations for third law entropies do exist. A value of /K was measured by Haselton & Westrum (980) and /K by Kolesnik et al (977). The derived value of /K is close to that of Kolesnik et al (977), while the data sets of Berman (988) and Holland & Powell (990) favor the value from Haselton & Westrum (980). However, experimental results with pyrope, even at high temperatures, are nicely reproduced with the value presented here. Some reaction equilibria are very sensitive to the degree of disorder in spinel (e.g. AMS-9, Fig. 8b). The degree of disorder in the spinel used for these experiments is mostly unknown. n this study equilibrium disorder was assumed, though this assumption may not be correct in every case. Future experiments must consider the degree of disorder in spinel before and after each experimental run. A number of reactions in the AMS subsystem considered here involve sapphirine, the composition of which is known to be a function of the phase assemblage. No quantitative determinations or activity models exist. Here sapphirine was treated as a phase with a fixed composition (Mg 7 Al9O4Al 9 Si3O3 6 ]). Despite this simplification, the existing experimental results are nicely reproduced. This may indicate that compositional variations in sapphirine are minor. As mentioned above, the tschermaks substitution was considered only for enstatite. Many reactions in the AMS subsystem involve talc. t is also known that the Al-content of talc is a function of pressure, temperature and phase assemblage, but no reliable data exist to quantify this. Also no Al-contents are given for the existing experimental results. n the calculations talc was assumed to be of a pure end-member composition. Despite this assumption, experimental results involving talc reproduced well. The existing experimental results are apparently not sensitive enough to ustify a correction for tschermaks substitution in talc at pressures below 500 MPa. Experimental results involving kaolinite (e.g. AS-4, Fig. 9b) are not well reproduced. Reported equilibria at low temperatures do not lie on the derived equilibrium line. A much higher entropy would be required (around 460 /K) to make the equilibrium line compatible with an internally consistent data set. One may speculate that the reason for this behavior is that at low temperatures and under hydrothermal conditions additional

28 202 M. Gottschalk F F F F F \^k - \ ^NK - \im* NK V^NK Nl- ^\ NK^ ' " " " ' ' " " " O N ACS-26 H ΔH = k M ΔS = /K π 4 Lws #\ \ H Ky + Qtz + 2 Zo + 7 H20 2 An + 2 Gh + 3 Wo vw N N \N i s "tuw H HHΛ ,,,,.., i Γ\? V H]\ ^ HV t A ; 'H \é Y^H ' ' ' ' ' ' ' ' Δpq on M ΔH = k 4 Grs (30 π /T*00Q /K] /T*000 /K] C] Γ F 33.0 b c^.,,-.,. v. i... ACS-45 ΔH = k H Δ S = /K ' ' ' Γ ' V N' ^ "" T "".,,... i... ACS-62 ΔH = k H ΔS = /K 3.0 L h 25.0 h 23.0 BT N^f^\ A+ N N ki B 3) >Λ N µ N^Vc 6Zo ~ymλfc N Lws + 2 Qtz + 2 H20 L Lmt LX\+L iw L An + 2 Grs + Crn + 3 H20 \ \ 9.0 L...Λ /T*000 /K] 4.0 \ /T*000 /K] C] C] Fig. 7. a) ACS-26. NK: Newton & Kennedy (963), N: Nitsch (968), C: Chatteree et al (984). - b) ACS-30. H: Hays (967), B: Boettcher (970), H: Huckenholz et al (975), S: Shmulovich (974). - c) ACS-45. N: Newton (966), B: Boettcher (970), C: Chatteree et al. (984). - d) ACS-62. N: Nitsch (968), L: Liou (97b) (see text and Fig. 3 for further explanation).

29 nternally consistent thermodynamic data L ' T ""' y^ E sstx s f... 5 Crd + 8 H20 Ci f s ss*, "- # Γ VssA s r,, r.. i. Γ AMS-5 ΔH = k ΔS = /K 8And + 2Cchl + Qtz T\ SS ΛSS ssqi ^ S S \ss ~ _ /T*000 /K] ] Crd + 5 Fo 5Γ "^ i r " r ' ' i ^ " ' ' AMS-9 ΔH = k ΔS = /K 5 En + 2 Spl /T*000 /K] s } tfy C] AMS-22 ΔH = k ΔS = /K 3 En + 2 Sil 0.0 H^ p blf P fh pi P -.0 ; Ψ 2 Prp + 2 Qtz Nv /T*000 /K] /T*000 /K] C] Cl Fig. 8. a) AMS-5. SS: Seifert & Schreyer (970). - b) AMS-9. FY: Fawcett & Yoder (966), S: Seifert (974), H: Herzberg (983). - c) AMS-22. H: Hensen (97), P: Perkins (983). - d) AMS-3. FY: Fawcett & Yoder (966), C: Chernosky (978), M: McPhail et al. (990) (see text and Fig. 3 for further explanation).

30 204 M. Gottschalk b /T*000 /K] /T*000 /K] C] C] K*" Γ M \ K C? r \ M Q ' '.,,,,,,,.,,,,,, "π AS-2 ΔH = k ΔS = /K 2 Dsp \ \ c \ A w ç?\ c ' ' ' ' ' ' ' ' ' ' ' k ΔS = /K CF Prl AS Crn + H20 \ H A H LH FGfeg! FoF X H cr v ' CF Crn + 4 Qtz + H20 \ *l CF /T*0QQ /T*000 /K] C] C] Fig. 9. a) AMS-32. C: enkins & Chernosky (986). - b) AS-4. A: Althaus (966), VK: Velde & Kornprobst (968), T: Thompson (970), H: Hemley et al (980). - c) AS-2. FG: Fyfe & Godwin (962), FH: Fyfe & Hollander (964), M: Matsushima et al (967), H: Haas (972), H: Hemley et al (980). - d) AS-7. CF: Carr & Fyfe (960), C: Chatteree et al (984) (see text and Fig. 3 for further explanation).

31 T, T, nternally consistent thermodynamic data ' "" Γ \ t MP Γ N^A* ^MP m C LÄT GG M K M EKi MRS MPT CMS- ΔH = k ΔS = /K 3 Dol + 4 Qtz + H AP 3 Cal + Tic + 3 C02 f f i oo^ f Y ^AQG TOG Λ λxgg MPY \ MPY 0 GGT /T*000 /K] s i /T*000 /K] C] C] w f Di h2c0 2 EM \ G r -Γ-...,,,,...,.,.., CMS-8 H ΔH = k M Δ S = /K π Dol + 2 Qtz W M V ^ \siλ N SKW i t t E t MM R k M A w M K $\$ M. NrSm..,.,...,... CMS-9 ΔH = k ΔS = /K u 3 Cal + 8 Fo + 9 C02 + H20 M Dol + Tr 'M u ^5 M>i v\ << λ ω i /T*0QQ /K] /T*000 /K] C] C] Fig. 0. a) CMS-. G: Gordon & Greenwood (970), MP: Metz & Puhan (970, 97), EK: Eggert & Kerrick (98), G: Großmann (984), P: Puhan & Metz (987). - b) CMS-4. SKW: Slaughter et al (975), EK: Eggert & Kerrick (98), G: Gottschalk (993). - c) CMS-8. SKW: Slaughter et al (975), K: acobs & Kerrick (98), EK: Eggert & Kerrick (98), W: Wyllie et al (983), G: Gottschalk (990), M: Metz unpublished. - d) CMS-9. M: Metz (967, 976), R: Richter (977, 980) (see text and Fig. 3 for further explanation). MY w π - 3 q

32 206 M. Gottschalk interlayer water is incorporated and the experiments at these conditions involved halloysite rather than kaolinite. The interlayer water would be very loosely bonded and even under saturated conditions it could be easily lost during quenching. The pressure and temperature dependence of the equilibrium water content is unknown, therefore the derived enthalpy and entropy for kaolinite seems to be unreliable. This problem should also be kept in mind when calculating low-temperature equilibria involving kaolinite. Additional /T vs. n K red plots for the AS system are shown in Fig. 9c,d. n the subsystem CMS for reactions involving calcite, dolomite, quartz, talc, tremolite, diopside forsterite and wollastonite, nearly all experiments are reproduced by the derived data set {e.g. Fig. loa-d, lla-c). Some high-pressure ( MPa) experiments {e.g. CMS-8 in Fig. 8c and CMS-3 in Fig. lib) involving calcite and dolomite are on the wrong side of the equilibrium line. This could be due to an incorrect compressibility coefficient for the two carbonates under these conditions. Another reason could be incorrect fugacities of CO 2 at these pressures. The plot for the reaction CMS-5 (Fig. lie) shows at least two different sets of experimental results which are not consistent. Those experiments which agree with the calculated equilibrium line were conducted in the pure subsystem. n experiments by enkins (983), the tremolite includes a significant tschermakite component. Therefore an activity term for tremolite was added assuming ideal mixing. The fact that these experiments, despite this correction, still do not lie on the correct side of the equilibrium line demonstrates that the system tremolite - tschermakite does not behave ideally. The experimental results from Zhou & Hsu (992) are, to a large degree, not in agreement with the derived data set {e.g. Fig. lid). These results were obtained under the false assumption that the composition in the H 2 O-CO 2 fluid did not change significantly during the course of the reaction. The enthalpies and entropies derived for rankinite, spurrite and tilleyite differ significantly from those of Holland & Powell (990). Nevertheless, the plots in Fig. 2b-d for example constrain the involved equilibria reasonably well, so the derived values are believed to be reliable. n Fig. 3 several examples of /T vs. n K red plots for reactions involving iron-bearing phases are shown. The reactions shown in Fig. 4 involve potassium. From the plot for the reaction KAS-6 (Fig. 4d) it can be seen that the experimental results of Evans (965) are not reproduced. These results were obtained using the "weight loss" method. As in other internally consistent data sets, the derived entropy for muscovite indicates that muscovite is substantially ordered, which is in contrast to X-ray studies by Bailey (984). The abundant 2M muscovite should show substantial disorder. The derived enthalpy for magnesite is.663 k higher than observed by Berman (988). n the plots for the reactions MS- and MS-2 (Fig. 5a,b) the experiments conducted at pressures >700 MPa are not in agreement with the derived data set. The reason for this is not clear, but perhaps is due to some kind of pressure effect. The compressibility of one phase could be in error at these pressures. A likely candidate is magnesite. Fig. 5c,d and 6a-d are further examples for /T vs. n K red plots in this subsystem. Examples for the subsystem NAS are shown in Fig. 7. Except for a few brackets {e.g. Gasparik (985) for reaction NAS-2), good agreement can be observed between the experimental data and the derived equilibrium lines. Discussion The advantages and disadvantages of the least squares (REG) and linear programming (MAP) approaches (Berman, 988; Holland & Powell, 985, 990) are the subect of some debate. These arguments are detailed in Berman et al. (986), Holland & Powell (990), Engi (992), Powell & Holland (993) and Olbricht et al (994). Both the recent approach by Olbricht et al. (994) (Bayes estimation, BE), which improves the linear programming technique, and the proposed iterated least squares (REG) approach used here makes it necessary to reconsider some aspects of the discussion. The REG approach is criticized mainly because it favors the center of the enthalpy brackets, which do not necessarily represent the most likely position for the equilibrium. Because of kinetic considerations, i.e. reaction progress is very low close to the equilibrium point, not all points within the bracket have the same probability of being the equilibrium point {e.g. Heinrich et al, 989). The REG technique allows the data base as a whole to select these points. Visually selected equilibrium points from the center of the brackets

33 ,,,, nternally consistent thermodynamic data b ' ' l t *à$*t* R yr9bi ' Di + 2Fo + 5C H 20 \.SLEL sn^ R'?^ *." CMS-0 ΔH = k ΔS = /K ^R 5 Cal + 3 Tr RFi^Ä } CB S C \ VfB \ <Y \ ACB CBΨ /T*000 /K] ] \ \r ,,,, f\ H t HSN^ E t N... HSN \ K Wo + 2 C02 Γ^TH,,, "π CMS-3 ΔH = k ΔS = /K >fll'ht i Γ 2 Cal + 2 Qtz ] G ( o G Q ( /T*000 /K] G ] ] G X. ] X G ] C] C] ! ' ^Ä NÉ Γ FV AM vn ' i c MS-5 A U _L. π Δπ k ΔS = /K ATΓ\ * Γ! 2 Fo + 2 Tr Di + 5 En + 2 H 20. \f i \ /T*000 /K] /T*000 /K] C] C] Fig.. a) CMS-0. R: Richter (977, 980), CB: Chernosky & Berman (986b, 988). - b) CMS-3. HT: Harker & Tuttle (956), G: Greenwood (967b), HSN: Haselton et al (978), K: acobs & Kerrick (98) (the results of Ziegenbein & ohannes (974, 982) are also in agreement but are not shown here for the sake of simplicity). - c) CMS-5. : enkins (98, 983), SM: Skippen & McKinstry (985). - d) CMS-23: ZH: Zhou & Hsu (992) (see text and Fig. 3 for further explanation).

34 208 M. Gottschalk ' ' T Γ \ L zs^^ zs^ γ π,,, r- CMS-36 ΔH = k ΔS = /K \ zs.zs.zs zs^tλs zs2á\ 2 Spu + Wo ] 6 Lrn + 2 C0 2 \xzs zs\ ^ZS záz s\ /T*000 /K] /T*000 /K] C] C] CMS-39 ΔH = k ΔS = /K /T*000 /K] /T*000 /K] C] C] Fig. 2. a) CMS-35. W: Walter (963), ZSB Zharikow et al (977). - b) CMS-36. ZS: Zharikow & Shmulovich (969). - c) CMS-38. ZS: Zharikow & Shmulovich (969). - d) CMS-39. ZS: Zharikow & Shmulovich (969) (see text and Fig. 3 for further explanation).

35 nternally consistent thermodynamic data 209 in each l/t vs. n K red p\ot serve only as input for the first iteration. Successive iterative runs no longer rely on these midpoints, since new positions are chosen with respect to the results of the previous iteration. The mathematical programming approach (MAP) of Berman (988) optimizes A f H, S and V and, in some cases, the coefficients involved in heat capacities, compressibilities, and thermal expansion. This flexibility makes the MAP technique very attractive. On the other hand, because of the highly interrelated thermodynamic network, these parameters are not totally independent of each other. The criticism of this technique lies in the fact that it attempts to optimize too many parameters. For example heat capacities c Py molar volumes V and other coefficients are experimentally determined with relatively high precision. This raises the question whether or not it is really necessary to refine these values, or if this is a required tool for MAP to expand the feasible solution region artificially. Lastly, solutions obtained by the MAP technique are in many cases artificially close to the boundaries of the inequalities (Olbricht et al., 994). The 'bouncing' against these borders is however avoided by the BE and REG approach. The BE and REG approaches concentrate on the values for Δ f H and S. n contrast the REG method of Holland & Powell (990) does not optimize S values. As previously noted, however, configurational entropies for some phases cannot be neglected, and for many phases no direct measurements but only estimations are available. This makes it necessary to optimize S values. n contrast to the REG and REG approach, the MAP and the BE methods do ensure consistency with all experimental results used. This is a disadvantage in the respect that in order to find a region of feasible solution all non-consistent inequalities must be canceled. Every such a cancellation of an inequality has an influence on the final solution and therefore introduces subectivity. This is illustrated in Fig. 8. Fig. 8a shows 4 experimentally consistent constraints for A f H and S of a reaction. n Fig. 8b two additional but contradicting constraints,. and., are introduced. No region for a feasible solution exists. To reestablish a region of feasible solution, one of the two contradicting constraints has to be canceled. n Fig. 8c the constraint. and in Fig. 8d constraint. is canceled. The selection of which experimental result is canceled has a severe influence on the region of feasible solution as well as on the solution itself. The solution depends on which constraint is canceled. The two possible regions of feasible solution do not overlap. However, the two experimental results in this hypothetical case lie very close together, and it is very difficult if not impossible to decide which of the constraints is acceptable and which is not. Both the REG and REG approaches always provide a solution. But with the help of the final l/t vs. n K red plots at the end of the REG optimization process, it is easily visualized which experimental results are not in accordance with the derived data set. Differences also exist in the experimental data bases used. n Holland & Powell (990), 208 reaction enthalpy brackets were formed using an unknown number of experimental results. These 208 enthalpy brackets and 25 additional A f H constraints where then used as input parameters in the REG approach, and the enthalpies of 23 phase components were refined. Berman (988) used 200 half-brackets, i.e. experimental results, involving 80 different equilibria and additional 0 A f H and 67 V constraints to optimize A f H, S and V of 67 phases. The data set presented here consists of optimized A f H and S values for 94 phase components using over 5300 half-brackets involving 253 mineral reactions. n addition, 60 calorimetric or tabulated constraints for A f H and S were considered. While a number of these 5300 experimental results are redundant, they emphasize experimentally well-established stability regions for certain phase assemblages. Therefore it is very reasonable to propagate all these experimental results through the whole process. The MAP and BE approaches are elegant but highly subective, i.e. one must choose or reect which experimental results are used as constraints. The biggest advantage of the REG approach is its capability to evaluate the consistency of experimental results with the help of l/t vs. n K red plots and to derive an internally consistent data set from these. The ideal procedure would be to start with an REG analysis to check for internal consistency between experimental results in order to obtain a good primary data set for A f H and S values. Such a data set could then be fine-tuned using either the MAP or BE process. Lastly, in addition to order-disorder effects, the composition of many phases is a function of pressure, temperature and phase assemblage. For example white micas, biotites, talc, chlorite, amphiboles, pyroxenes and sapphirine are known to

36 20 M. Gottschalk ,, -T- T,,,, AFS-5 ΔH = k Δ S = /K 2 Sil + Aim ] \ r \^ WΛ i^ ' ' ' ' ' ' ' AFS-22 ΔH = k N Δ S = /K 3 Cld ].0 CΓ B 3 He + 5 Qtz B B &?, ] L Aim + 2 Crn + 3 H ^ /T*000 /K] /T*000 /K] C ] C] ME i o CFS-3 ΔH = k ΔS = /K 2 Fac MEVO ^o \p H FS-2 ΔH = k ΔS = /K H 3 Qtz + 2 Mag ] Fa + 5 Qtz + 4 Hd + 2 H /T*000 /K] Fa + 02 H "*2ßME «6ME TAP /T*00Q /K] H C ä\ C] C] Fig. 3. a) AFS-5. B: Bohlen et al (986). - b) AFS-22. G: Ganguly (969). - c) CFS-3. E: Ernst (966). - d) FS-2. C: Chou (978), H: Hewitt (978), ME: Myers & Eugster (983), O: O'Neill (987) (see text and Fig. 3 for further explanation).

37 nternally consistent thermodynamic data f r, i, HEÄ "Wr.ti ôc^h L 5 Sa + 3 Tr + 6 C H20 ' -Γ r T, '.--T--r-, KACMS-4 L ΔH = k ΔS = /K l 4 6 Cal + 5 Phi + 24 Qtz HAM m H Ä H M HΓiyβpcrfKVL H i E< s HE Γ A HE\ ] HET \ (a) /T*000 /K] ^λ s^% s^m ' ' L 3 Crd + 8 Sa + 8 H20 Γ \ ΛS\ s^ _<ák-\ ^KA,.. i.. i.. r KAMS-0 ΔH = k ΔS = /K *π 6 Ms+ 2 Phi+ 5 Qtz i s A. T i (B) /T*000 /K] C] SA\ s VXLABF Sfef ^ L. i , KAMS- ΔH = k H ΔS = /K Cchl + Ms + 2 Qtz Crd + Phi + 4 H20 t s 26.0 s W /T*000 /K] (d) /T*000 /K] C] C] Fig. 4. a) KACMS-4. HE: Hewitt (973). b) KAMS-0. S: Seifert (976). - c) KAMS-. S: Seifert (970), BF: Bird & Fawcett (973). - d) KAS-6. E: Evans (965), AH: Althaus et αl. (970), D: Day (973), C: Chatteree & ohannes (974) (see text and Fig. 3 for further explanation).

38 22 M. Gottschalk ' ^ V F r F.,. r r \ Oi G^ Tlc + 3CQ2 \ 4 Ψ A. G ^ (ok G (ok vq/ \ (c <3 el ' l ' " '! " "! " ^ MS- H ΔH k N ΔS' ' = /K π, L Mgs + 4Qtz + H GrV Λ y t. # \ *\ A 9) /T*000 /K] 3 i ' ' V '"" Γ r η - γy\ yγ F Γ Γ ^ Gr fcsfrvλ ^* 0p ' ' ' ' ' ' ' ' ' ' ' MS-2 L r \ 4 Fo + 5 C02 + H20 \,,,,,,,, V,,,,,,,, i» M ΔH = k N ΔS = /K λ 5 Mgs + Tic /T*000 /K] r i ] E o K K 4Fc > + 6H2 ' ' ' < V o * < O i \ cγ] p <fev w,...,...,,.,..,,.,.^ M MS-6 H ΔH ' = k M ΔS C p Σ w^ Ψ\ F, ' = /K Γl Brc + ctγ V as Λ /T*000 /K] C ] E cl K%Γ) K v p\ te p A \ c á Ψ \ \c 2Fo + 2Tlc+8H20 c... Γ..,,..... MS-7 U ΔH = k ΔS = /K W C lfsc 5 Ctl 4 s i & C*W^Ä /T*000 /K] i i C] C] Fig. 5. a) MS-. G: Greenwood (967a), : ohannes (969), Gr: Großmann & Metz (984). - b) MS-2. Gr: Greenwood (967a), : ohannes (969). - c) MS-6. P: Pistorius (963), K: Kitahara et al. (966), : ohannes (968). - d) MS-7. P: Pistorius (963), K: Kitahara et al (966), SC: Scarfe & Wyllie (967), C: Cheraosky (973, 978) (see text and Fig. 3 for further explanation).

39 nternally consistent thermodynamic data "m G^ A - 3 En + 2 Qtz + 2 H ,,,, C k, r^-r-t-p, MS -2 ΔH = k M ΔS = /K H s r # 2 Tlc l * c r vyc /T*000 /K] Γ Λ\ ov\ Γ A c T 3 Ath + 4 Qtz + 4 H !,,,,!,, t ic <ê> %** * GV G VΛAC CYTΓ Ale \ i 4 ' Γ MS-3 M ΔH = k Γ Δ S = /K 7 Tic \ ' c ] * * Ψ /T*000 /K] C] ,, \ MS -5 ΔH = k ΔS = /K ] r 2 Ath 7.0 \ A G Ä n \Γ 4k G 6.0 c AC A c i 7 En + 2 Qtz + 2 H /T*000 /K] /T*000 /K] C] C] Fig. 6. a) MS-2. G: Greenwood (963), P: Pistorius (963), K: Kitahara et al (966), S: Skippen (97), C: Chernosky et al. (985). - b) MS-3. G: Greenwood (963), C: Chernosky et al (985). - c) MS-5. G: Greenwood (963), C: Chernosky et al (985). - d) MS-23: HT: Harker & Tuttle (955), GH: Goldsmith & Heard (96), M: ohannes & Metz (968), P: Phillip (988) (see text and Fig. 3 for further explanation).

40 24 M. Gottschalk ^ H^S TG Aab LBfH ^ B4 ' " " ' NAS-2 ' ' ' ΔH = k ΔS = /K i d + Qtz LB LB 0 NS NsuHB V M N S NS NS 4 NfflNS LB N NS hns "π \ \ H >NS ^s^w^ f f _ \^ ^H ^BSΓH NAS-9 ΔH = k ΔS = /K Pg H Λ. ] NST H; \. Ky + d + HzO /T*000 /K] /T*000 /K] C ] C] h c \ è ΛK ' ' *, ? Aab + And + H20 C v9\ c C \A 9 NAS-6 ΔH = k ΔS = /K c Pg + Qtz \ 5.0 ' ' ',... L i, i, -.Q.-,,, ,,, r- \ LSé Aab + Ne + 2 H20 ' ' ' ' ' ' ' NAS >-23 ΔH = k ΔS = /K ΛL ^k 2 Anl ^\ * <& L \Γ ftkl- \ ] /T*000 /K] /T*000 /K] C] Cl Fig. 7. a) NAS-2: BL: Birch & LeCompte (960), NS: Newton & Smith (967), E: Essene et al (972), : ohannes et al (97), HB: Hays & Bell (972), HW: Huang & Wyllie (975), H: Holland (980), G: Gasparik (985). - b) NAS-9. H: Holland (979). - c) NAS-6. C: Chatteree (972). - d) NAS-23. L: Liou (97a) (see text and Fig. 3 for further explanation).

41 nternally consistent thermodynamic data 25 Fig. 8. The plots a)-d) demonstrate that for MAP and BE the final region of a feasible solution depends on which of the contradicting experimental results is canceled. n a) 4 experimental consistent constraints are shown (gray area is the region of feasible solution). n b) two additional but contradicting constraints,. and., are introduced. No region for a feasible solution exists. To reestablish a region for a feasible solution one of the two contradicting constraints must be canceled. n c) the constraint. and in d) constraint. is canceled. As can be seen, the regions of feasible solution in c) and d) do not overlap. show such behavior. Up to the present time this has not been well understood quantitatively and experimental data is sparse. Certainly this is one area in which more experimental data is definitely needed. Acknowledgements: Parts of this study are from the author's Ph.D. thesis research conducted at the Eberhard-Karls-University, Tubingen (Gottschalk, 990). The author would like to thank K. Bucher, N.D. Chatteree, W. Heinrich, H.-. Massonne and P. Metz who read an earlier version of this manuscript. Also thanks to.a.d. Connolly, D.E. Harlov and two anonymous reviewers. Their critical statements and the resulting discussions were of great help in improving the manuscript. References Althaus, E. (966): Die Bildung von Pyrophyllit und Andalusit zwischen 2000 und 7000 bar H 2 O-Druck. Naturwissenschaften, 53, Althaus, E., Karotke, E., Nitsch, K.H., Winkler, H.G.F. (970): An experimental reexamination of the upper stability limit of muscovite plus quartz. N. b. Mineral. Mh., 970, Anderson, O.L. & Nafe,.E. (965): The bulk modulusvolume relationship for oxide compounds and related geophysical problems.. Geophys. Res., F, Anovitz, L.M., Treiman, A.H., Essene, E.., Hemingway, B.S., Westrum, E.F., Wall, V.., Burriel, R., Bohlen, S.R. (985): The heat capacity of ilmenite and phase equilibria in the system Fe-Ti-O. Geochim. Cosmochim. Ada, 49, Armbruster, T. & Bloss, F.D. (982): Orientation and effects of H 2 O and CO 2 in cordierite. Am. Mineral., 67, Bailey, S.W. (984): Review of cation ordering in micas. Clays Clay Minerals, 32, Barin,., Knacke, O., Kubaschewski,. (977): Thermochemical properties of inorganic substances. Springer Verlag, Berlin. Bass,.D. & Weidner, D.. (984): Elasticity of singlecrystal orthoferrosilite. /. Geophys. Res., 89, Belonoshko, A.B. & Saxena, S.K. (992): A unified equation of state for fluids of C-H-O-N-S-Ar composition and their mixtures up to high temperatures and pressures. Geochim. Cosmochim. Ada, 56, Berman, R.G. (988): nternally-consistent thermodynamic data for minerals in the system Na 2 O-K 2 0- CaO-MgO-FeO-Fe 2 O 3 -Al 2 O3-SiO 2 -TiO 2 -H 2 O-C0 2.. Petrol, 29, Berman, R.G. & Brown, T.H. (985): Heat capacity of minerals in the system Na 2 O-K 2 O-CaO-Mg0-FeO-

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44 28 M. Gottschalk Reaktionen in metamorphen kieseligen Karbonatgesteinen. Fortschr. Miner., 62, Haas, H. (972): Diaspore-corundum equilibria determined by epitaxis of diaspore on corundum. Am. Mineral, 57, Haas,.L. & Fisher,.R. (976): Simultaneous evaluation and correlation of thermodynamic data. Amer.. Set, 276, Halbach, H. & Chatteree, N.D. (982): An empirical Redlich-Kwong-type equation of state for water to C and 200 kbar. Contrib. Mineral. Petrol, 79, (984): An internally consistent set of thermodynamic data for twentyone CaO-Al 2 O3-SiO2-H 2 O phases by linear programming. Contrib. Mineral Petrol, 88, Harker, R.. & Tuttle, 0.E (955): Studies in the system CaO-MgO-CO 2 : The thermal dissociation of calcite, dolomite and magnesite. Amer.. Set, 253, (956): Experimental data on the P co -T curve for the reaction: calcite + quartz F N wollastonite + carbon dioxide. Amer.. Scl, 254, Haselton, H.T., r., Hemingway, B.S., Robie, R.A. (984): Low-temperature heat capacities of CaAl 2 SiO 6 glass and pyroxene and thermal expansion of CaAl 2 SiO 6 pyroxene. Am. Mineral, 69, Haselton, H.T. & Newton, R.C. (980): Thermodynamics of pyrope-grossular garnets and their stabilities at high temperatures and high pressures.. Geophys. Res., 85, Haselton, H.T, Robie, R.A., Hemingway, B.S. (987): Heat capacities of synthetic hedenbergite, ferrobustamite, and CaFeSi 2 Oo glass. Geochim. Cosmochim. Acta, 5, Haselton, H.T, Sharp, W.E., Newton, R.C. (978): CO 2 fugacity at high temperatures and pressures from experimental decarbonation reactions. Geophys. Res. Lett., 5, Haselton, H.T & Westrum, E.E (980): Low-temperature heat capacities of synthetic pyrope, grossular, and pyrope 6 o-grossular 4 o. Geochim. Cosmochim. Acta, 44, Hays,.F. (967): Lime-alumina-silica. Geophys. Lab. Ann. Rep., 65, Hays,.F. & Bell, P.M. (972): Albite-adeite-quartz equilibrium: A hydrostatic determination. Geophys. Lab. Ann. Rep., 72, Hazen, R.M. (976): Effects of temperature and pressure on the crystal structure of forsterite. Am. Mineral, 6, Hazen, R.M. & Finger, L.W. (976a): The crystal structures and compressibilities of layer minerals at high pressure.. Phlogopite and chlorite. Am. Mineral, 63, (976b): Crystal structures and compressibilities of pyrope and grossular to 60 kbar. Am. Mineral, 63, Heinrich, W., Metz, P., Gottschalk, M. (989): Experimental investigation of the kinetics of the reaction tremolite + dolomite F 8 forsterite + 3 calcite + 9 CO 2 + H 2 O. Contrib. Mineral. Petrol, 02, Helgeson, H.C., Delany,.M., Nesbitt, H.W., Bird, D.K. (978): Summary and critique of the thermodynamic properties of rock-forming minerals. Amer.. Scl, 278A, pp Hemingway, B.S. (987): Quartz: heat capacities from 340 to 000 K and revised values for the thermodynamic properties. Am. Mineral, 72, (99): Thermodynamic properties of anthophyllite and talc: Corrections and discussion of calorimetric data. Am. Mineral, 76, Hemingway, B.S., Evans, H.T., Nord, G.L., Haselton, H.T., Robie, R.A., McGee,.. (986): Åkermanite: phase relations in the heat capacity and thermal expansion, and revised thermodynamic data. Can. Mineral, 24, Hemingway, B.S., Haas,.L., Robinson, G.R. (982): Thermodynamical properties of selected minerals in the system Al 2 O 3 -CaO-SiO 2 -H 2 O at.5 K and bar (0 5 pascal) pressure and at higher temperatures. Geol. Surv. Bull, 544, pp. 70. Hemingway, B.S. & Kittrik,.A. (978): Revised values for the Gibbs free energy of formation of Al(0H) 4aq ]~, diaspore, boehmite, and bayerite at.5 K and bar, the thermodynamic properties of kaolinite to 800 K and bar, and the heats of solution of several gibbsite samples. Geochim. Cosmochim. Acta, 42, Hemingway, B.S., Krupka, K.M., Robie, R.A. (98): Heat capacities of the alkali feldspars between K from differential scanning calorimetry, the thermodynamic functions of the alkali feldspars from.5 to 400 K, and the reaction quartz + adeite # analbite. Am. Mineral, 66, Hemingway, B.S. & Robie, R.A. (984): Heat capacity and thermodynamic functions for gehlenite and staurolite: with comments on the Schottky anomaly in the heat capacity of staurolite. Am. Mineral, 69, Hemingway, B.S., Robie, R.A., Apps,.A. (99a): Revised values for the thermodynamic properties of boehmite, AO(OH), and related species and phases in the system Al-H-O. Am. Mineral, 76, Hemingway, B.S., Robie, R.A., Evans, H.T., Kerrick, D.M. (99b): Heat capacities and entropies of sillimanite, fibrolite, andalusite, kyanite, and quartz and the Al 2 SiO 5 phase diagram. Am. Mineral, 76, Hemingway, B.S., Robie, R.A., Fisher,.R., Wilson, W.H. (977): Heat capacities of gibbsite, Al(OH) 3, between 3 and 480 K and magnesite, MgCO 3, between 3 and 380 K and their standard entropies at.5 K, and the heat capacities of calorimetry conference benzoic acid between 2 and 36 K.. Res. US. Geol. Surv., 5,

45 nternally consistent thermodynamic data 29 Hemley,.., Montoya,.W., Marinenkon,.W., Luce, R.W. (980): Equilibria in the system Al 2 O 3 -SiO 2 - H 2 O and some general implications for alteration/ mineralization processes. Econ. Geol, 75, Henderson, C.E., Essene, E.., Anovitz, L.M., Westrum, E.F., Hemingway, B.S., Bowman,.R. (983): Thermodynamics and phase equilibria of clinochlor, (Mg 5 Al)Si 3 A0 ( /(0H) 8 ]. Trans. Amer. Geophys. Union, 64, 466. Hensen, B.. (97): Cordierite-garnet equilibrium as a function of pressure, temperature, and iron-magnesium ratio. Geophys. Lab. Ann. Rep., 7, Herzberg, C.T. (983): The reaction forsterite + cordierite ^ aluminous orthopyroxene + spinel in the system MgO-Al 2 O 3 -SiO2. Contrib. Mineral. Petrol, 84, Hewitt, D.A. (973): Stability of the assemblage muscovite-calcite-quartz. Am. Mineral, 58, (978): A redetermination of the fayalite-magnetitequartz equilibrium between 650 and C. Amer.. Scl, 278, Hinrichsen, T. & Schurmann, K. (969): Untersuchungen zur Stabilität von Pumpellyit. N. b. Mineral. Mh., 969, Hochella, M.F., Liou,.G., Keskinen, M.., Kim, H.S. (982): Synthesis and stability relations of magnesium idocras. Econ. Geol,, Holdaway, M.., Gunst, R.F., Mukhopadhyay, B., Dyar, M.D. (993): Staurolite end-member molar volumes determined from unit-cell measurements of natural specimens. Am. Mineral, 78, Holdaway, M.. & Lee, S.M. (977): Fe-Mg cordierite stability in high-grade pelitic rocks based on experimental, theoretical, and natural observations. Contrib. Mineral. Petrol, 63, Holland, T.B. (979): The experimental determination of the reaction paragonite ^ adeite + kyanite + H 2 O and internally consistent thermodynamic data for part of the system Na 2 O-Al 2 O 3 -SiO 2 -H 2 O, with application to eclogites and blueshists. Contrib. Mineral. Petrol, 68, (980): The reaction albite # adeite + quartz determined experimentally in the range C Am. Mineral, 65, (988): Prelimary phase relations involving glaucophane and applications to high pressure petrology: new heat capacity and thermodynamic data. Contrib. Mineral. Petrol, 99, Holland, T.B. & Powell, R. (985): An internally consistent thermodynamic dataset with uncertainties and correlations: 2. Data and results.. metamorphic Geol, 3, (990): An enlarged and updated internally consistent thermodynamic dataset with uncertainties and correlations: the system K 2 O-Na 2 O-CaO-MgO- Mn0-FeO-Fe 2 O 3 -Al Ti0 2 -Si0 2 -C-H-O 2.. metamorphic Geol, 8, Holloway,.R. (976): Fugacity and activity of molecular species in supercritical fluids, in "Thermodynamics in Geology", D.G. Fraser ed., Dordrecht- Holland/Boston, 6-8. Holmes, R.D., O'Neill, H.S.C, Arculus, R.. (986): Standard Gibbs free energy of formation for Cu 2 O, NiO, CoO, and Fe 2 O 3 : High resolution electrochemical measurements using zirconia solid electrolytes from K. Geochim. Cosmochim. Acta, 50, Hovis, G.L. (974): A solution calorimetric and X-ray investigation of Al-Si distribution in monoclinic potassium feldspars, in "The feldspars", W.S. MacKenzie &. Zussman ed., Manchester University Press, Huang, W.-L. & Wyllie, P. (975): Melting reactions in the system NaAlSi 3 O 8 -KAlSi SiO 2 to 35 kilobars, dry and with excess water.. Geol, 83, Huckenholz, H.G., Holzel, E., Lindgruber, W. (975): Grossularite, its solidus and liquidus relations in CaO-Al Si0 2 -H 2 0 system up to 0 kbar. N. b. Mineral. Abh., 24, -46. Huckenholz, H.G. & Yoder, H.S. (97): Andradite stability relations in the CaSiO 3 -Fe 2 O 3 oin up to 30 kb. N. b. Mineral. Abh., 4, acobs, G.K. & Kerrick, D.M. (98): Devolatilization equilibria in H 2 O-CO 2 and H 2 0-C0 2 -NaCl fluids: an experimental and thermodynamic evaluation at elevated pressures an temperatures. Am. Mineral, 66, enkins, D.M. (98): Experimental phase relations of hydrous peridotites modelled in the system H 2 O- CaO-MgO-Al 2 O 3 -SiO 2. Contrib. Mineral. Petrol,, (983): Stability and composition relations of calcic amphiboles in ultramafic rocks. Contrib. Mineral Petrol, 83, (984): Upper-pressure stability of synthetic margarite plus quartz. Contrib. Mineral Petrol, 88, enkins, D.M. & Chernosky,.V. (986): Phase equilibria and properties ofmg-chlorite. Am. Mineral, 7, enkins, D.M., Newton, R.C., Goldsmith,.R. (985): Relative stability of Fe-free zoisite and clinozoisite.. Geol, 93, ohannes, W. (968): Experimental investigation of the reaction forsterite + H 2 O ;= serpentinite + brucite. Contrib. Mineral. Petrol, 9, (969): An experimental investigation of the system MgO-SiO2-H 2 O-CO2. Amer.. Scl, 267, ohannes, W., Bell, P.M., Mao, H.K., Boettcher, A.L., Chipman, D.W., Hays,.F, Newton, R.C., Seifert, F. (97): An interlaboratory comparison of pistoncylinder pressure calibration using the albite-breakdown reaction. Contrib. Mineral. Petrol, 32,

46 220 M. Gottschalk ohannes, W. & Metz, P. (968): Experimentelle Bestimmung von Gleichgewichtsbeziehungen im System MgO-CO 2 -H 2 0. N. 3b. Mineral Mh., 968, ohannes, W. & Ziegenbein, D. (980): Stabilität von Zoisit in H 2 O-CO 2 -Gasphasen. Fortschr. Miner., 58, Karpinskaya, T.B. & Ostrovskiy, N.A. (982): Compressibility of brucite and the reaction of its formation from oxide as related to the possible existence of hydrothermal fluids in the mantle. nt. Geol. Rev., 24, Käse, H.R. & Metz, P. (980): Experimental investigation of the metamorphism of siliceous dolomites. Contrib. Mineral. Petrol, 73, Kerrick, D.M. & acobs, G.K. (98): A modified Redlich-Kwong equation for H 2 O, CO 2, and H 2 O-CO 2 mixtures at elevated pressures and temperatures. Amer. 3. Set, 28, Kiseleva, LA. (976): Thermodynamic parameters of natural ordered sapphirine and synthetic disordered specimens. Geochem. nt., 976, Kiseleva, LA., Ogorodova, L.P., Topor, N.D., Chigareva, O.G. (980): A thermochemical study of the CaO- MgO-SiO 2 system. Geochem. nt., 980, Kitahara, S., Takenouchi, S., Kennedy, G.C. (966): Phase relations in the system MgO-Si0 2 -H 2 0 at high temperatures and pressures. Amer.. Sci., 264, Kolesnik, Y.N., Nogteva, V.V., Paukov, LY. (977): The specific heat of pyrope at 3 to 300 K and the thermodynamic parameters of some natural varieties of garnet. Geochem. nt., 977, Krupka, K.M., Hemingway, B.S., Robie, R.A., Kerrick, D.M., to,. (985): Low-temperature heat capacities and derived thermodynamic properties of anthophyllite, diopside, dolomite, enstatite, bronzite and wollastonite. Am. Mineral., 70, Krupka, K.M., Robie, R.A., Hemingway, B.S. (979a): The heat capacities of corundum, periclas, anorthite, CaAl 2 Si 2 O 8 glass, muscovite, pyrophyllite, KAlSi 3 O 8 glass, grossular, and NaAlSi 3 O 8 glass between 350 and 000 K. Trans. Amer. Geophys. Union, 58, 523. (979b): High-temperature heat capacities of corundum, periclas, anorthite, CaAl 2 S 2 O 8 glass, muscovite, pyrophyllite, KAlSi 3 O 8 glass, grossular and NaAlSi 3 O 8 glass. Am. Mineral, 64, Levien, L. & Prewitt, C.T. (98): High-pressure crystal structure and compressibility of coesite. Am. Mineral, 66, Levien, L., Prewitt, C.T., Weidner, D.. (979): Compression of pyrope. Am. Mineral, 64, Lindsley, D.H., Grover,.E., Davidson, P.M. (98): The thermodynamics of the Mg 2 Si 2 O 6 -CaMgSi 2 O 6 oin: A review and an improved model, in "Advances in Geochemistry",, R.C. Newton, A. Navrotsky & B. Wood ed., Springer Verlag, Liou,.G. (97a): Analcime equilibria. Lithos, 4, (97b): P-T stabilities of laumontite, wairakite, lawsonite, and related minerals in the system CaAl 2 Si2O8-SiO 2 -H2O.. Petrol, 2, Markgraf, S.A. & Reeder, R.. (985): High-temperature structure refinements of calcite and magnesite. Am. Mineral, 70, Martens, R., Rosenhauer, M., Gehlen, K.v. (982): Compressibilities of carbonates, in "High-pressure researches in geoscience", W. Schreyer ed., Matsushima, S., Kennedy, G.C, Akella,., Haygrath,. (967): A study of equilibrium relations in the system A 2 O 3 -Si0 2 -H 2 0 and A H 2 O. Amer. 3. Scl, 265, McPhail, D.C., Berman, R.G, Greenwood, H.. (990): Experimental and theoretical constraints on aluminium substitution in gagnesium chlorite, and a thermodynamic model for H 2 0 in magnesium cordierite. Can. Min., 28, Metz, G.W., Anovitz, L.M., Essene, E.., Bohlen, S.R., Westrum, E.F., Wall, V. (983): The heat capacity and phase equilibria of almandine. Trans. Amer. Geophys. Union, 64, 346. Metz, P. (967): Experimentelle Bildung von Forsterit und Calcit aus Tremolit und Dolomit. Geochim. Cosmochim. Acta, 3, (976): Experimental investigation of the metamorphism of siliceous dolomites.. Equilibrium data for the reaction: tremolite + dolomite F 8 forsterite + 3 calcite + 9 CO2 + H2O for the total pressure of 3000 and 5000 bars. Contrib. Mineral. Petrol, 58, Metz, P. & Puhan, D. (970): Experimentelle Untersuchung der Metamorphose von kieselig dolomitischen Sedimenten. Die Gleichgewichtsdaten der Reaktion 3 Dolomit + 4 Quarz + H 2 O F Talk + 3 Calcit + 3 CO 2. Contrib. Mineral. Petrol, 26, (97): Korrektur zur Arbeit: Experimentelle Untersuchung der Metamorphose von kieselig dolomitischen Sedimenten. Contrib. Mineral Petrol, 3, Mirwald, P.W., Malinowski, M., Schulz, H. (984): sothermal compression of low-cordierite to 30 kbar (25 q C). Phys. Chem. Minerals,, Mirwald, P.W., Maresch, W.V., Schreyer, W. (979): Der Wassergehalt von Mg-Cordierit zwischen C und C sowie 0.5 und kbar. Fortschr. Miner., 58, Moecher, D.P., Essene, E.., Westrum, E.F. (985): S of an intermediate scapolite and phase equilibrium constraints on Al-Si disorder. Trans. Amer. Geophys. Union, 66, 390. Mukhopadhyay, B. & Holdaway, M.. (994): Cordierite-garnet-sillimanite-quartz:. New experimental calibration in the system FeO-Al 2 O 3 -SiO 2 -H 2 O and certain P-T-x H o relations. Contrib. Mineral Petrol, 6,

47 nternally consistent thermodynamic data 22 Myers,. & Eugster, H.R (983): The system Fe-Si-O: oxygen buffer calibrations to 500 K. Contrib. Mineral Petrol, 82, Newton, R.C. (966): Some calc-silicate equilibrium relations. Λmer.. Set, 264, Newton, R.C, Charlu, T.V., Kleppa, O. (980): Thermochemistry of the high structural state plagioclase. Geochim. Cosmochim. Acta, 44, Newton, R.C. & Kennedy, G.C. (963): Some equilibrium reactions in the oin CaAl 2 Si 2 O8-H 2 O.. Geophys. Res., 68, Newton, R.C. & Smith,.V. (967): nvestigation concerning the breakdown of albite at depth in the earth.. Geol, 75, Newton, R.C. & Wood, B.. (979): Thermodynamics of water in cordierite and some petrologic consequences of cordierite as a hydrous phase. Contrib. Mineral. Petrol, 68, (980): Volume behavior of silicate solid solutions. Am. Mineral, 65, Nitsch, K.-H. (968): Die Stabilität von Lawsonit. Naturwissenschaften, 55, 368. O'Neill, H.S. (987): Quartz-fayalite-iron and quartzfayalite-magnetite equilibria and the free energy of formation of fayalite (Fe 2 SiO 4 ) and magnetite (Fe 3 O 4 ). Am. Mineral, 72, Olbricht, W., Chatteree, N.D., Miller, K. (994): Bayes estimation: A novel approach to derivation of internally consistent thermodynamic data for minerals, their uncertainties, and correlations. Part : Theory. Phys. Chem. Minerals, 2, Olesch, M. (978): Obere thermische Stabilität von Vesuvian (docras) bis 2 kbar und Vesuvian + Quarz bis 5 kbar im System CaO-MgO-Al 2 O 3 -SiO 2 -H 2 O. Fortschr. Miner., 56, 99. Olinger, B. (977): Compression studies of forsterite (Mg 2 SiO 4 ) and enstatite (MgSiO 3 ). in "High pressure research applications in Geophysics", M.H. Manghmani & S.. Akimoto ed., Perkins, D. (983): The stability ofmg-rich garnet in the system CaO-MgO-Al 2 O 3 -SiO 2 at C and high pressure. Am. Mineral, 68, Perkins, D., Essene, E.., Westrum, E.F., Wall, V.. (979): New thermodynamic data for diaspore and their application to the system Al 2 O 3 -SiO 2 -H 2 O. Am. Mineral, 64, Perkins, D., Westrum, E.F., Essene, E. (980): The thermodynamic properties and phase relations of some minerals in the system CaO-Al 2 O 3 -SiO 2 - H 2 O. Geochim. Cosmochim. Acta, 44, Phillip, R.W. (988): Phasenbeziehungen im System MgO-H 2 O-CO 2 -NaCl. Dissertation, ETH Zurich. Pistorius, C.W.F.T. (963): Some phase relations in the system MgO-SiO 2 -H 2 O to high pressures and temperatures. N. b. Mineral Mh., 963, Powell, R. & Holland, T. (993): The applicability of least squares in the extraction of thermodynamic data from experimentally bracketed mineral equilibria. Am. Mineral, 78, Powell, R. & Holland, T..B. (985): An internally consistent thermodynamic dataset with uncertainties and correlations:. Methods and a worked example.. metamorphic Geol, 3, Puhan, D. & Metz, P. (987): Experimental equilibrium data for the reactions 3 dolomite + 4 quartz + H 2 O # talc + 3 calcite + 3 CO 2 and 5 talc + 6 calcite + 4 quartz F 3 tremolite + 6 CO H 2 O at a total gas pressure of 5000 bars. N. b. Mineral Mh., 987, Rao, B.B. & ohannes, W. (979): Further data on the stability of staurolite + quartz and related assemblages. N. b. Mineral. Mh., 979, Reeder, R.. & Markgraf, S.A. (986): High-temperature crystal chemistry of dolomite. Am. Mineral, 7, Richet, P., Robie, R.A., Hemingway, B.S. (99): Thermodynamic properties of wollastonite, pseudowollastonite and CaSiO 3 glass and liquid. Eur.. Mineral, 3, Richter, R. (977): Experimentelle Bestimmung der Gleichgewichtsdaten der Reaktion 3 Tremolit + 5 Calcit = Diopsid + 2 Forsterit + 3 H 2 O + 5 CO 2. Master Thesis, Gottingen. (980): Bildungsbedingungen von Diopsid- und Forsterit-fiihrenden Paragenesen bei der Metamorphose kieseligen Calcit-Dolomit-Sedimenten. Dissertation, Gottingen. 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