Calculations of fluid ternary solid solution equilibria: An application of the Wilson equation to fluid (Fe,Mn,Mg)TiO 3 equilibria at 600 C and 1 kbar

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1 Amercan eralogst, Volume 84, pages 7 84, 999 Calculatons of flud ternary sold soluton equlbra: An applcaton of the Wlson equaton to flud (Fe,,MgTO equlbra at 6 C and kbar. SHIUE Geoscence Insttute, Hyogo Unversty of Teacher Educaton, Hyogo , Japan ASTRACT The Wlson equaton (Wlson 964 s appled to (Fe,,MgTO sold solutons for obtanng the mxng propertes of the ternary sold soluton at 6 C and kbar. The present study utlzes data on caton exchange between (Fe,Cl (aq and (Fe,TO, between (,MgCl (aq and (,MgTO, and between (Fe,MgCl (aq and (Fe,MgTO (Kubo et al. 99. The molar excess Gbbs energy (G ex s the followng: G ex (kj/mol 7.6[ FeTO ln( FeTO +.4 TO +.96 MgTO + TO ln(.8 FeTO + TO +.9 MgTO + MgTO ln(.46 FeTO +.7 TO + MgTO ], where stands for the mole fracton of the subscrpted component. The predcted compostons of (Fe,,MgCl (aq fluds n equlbrum wth the ternary sold solutons are n good agreement wth the expermental values. INTRODUCTION The Margules equaton has been used by many nvestgators to descrbe the mxng propertes of mneral sold solutons (e.g., Ganguly and Saxena 987. y means of the Margules equaton, the molar excess Gbbs energy (G ex of a ternary sold soluton s wrtten as follows (e.g., Mukhopadhyay et al. 99: G ex ( W + W + ( W + W + ( W + W + C ( where and W stand for the mole fracton of the subscrpted component and the Margules parameter for the subscrpted par, respectvely. The term C s the ternary nteracton parameter, whch s ndependent of the composton and the bnary Margules parameters. If the ternary system conssts of only symmetrc regular bnares, the ternary nteracton parameter s equal to zero (Mukhopadyay et al. 99. Otherwse, there are no reasons for assumng C wthout expermental confrmaton. As an alternatve method for descrbng the mxng propertes of multcomponent solutons, the Wlson equaton (Wlson 964 has been used for real soluton mxtures. The molar excess Gbbs energy of a ternary soluton s wrtten as follows: ex G ln jj ( j where R s the unversal gas constant, T s the absolute temperature, and j stands for the Wlson parameter for the par of components and j. y defnton, s equal to n Equaton. y an approprate dfferentaton of Equaton, the actvty coeffcent of component (γ s expressed as follows: *E-mal:yshbue@sc.hyogo-u.ac.jp ln γ ln j j j k k k jkj j. ( The actvty coeffcents of the components n the ternary soluton can be computed by combnng the Wlson parameters for the three bnary solutons whose components make up the ternary system. Thus, the man advantage of the Wlson equaton s ts potental applcablty to the ternary soluton wthout the ternary nteracton parameter. When the number of the components s n, the equatons for G ex and actvty coeffcents can be obtaned by substtutng n for the lmt sum n Equatons and. Therefore, we can compute actvty coeffcents for solutons of arbtrary number of components by combnng the Wlson parameters for bnary mxtures. The major drawback of the Wlson equaton s ts napplcablty to partally mscble solutons. Renon and Prausntz (968 modfed the Wlson equaton by ntroducng addtonal parameters to make t applcable to those mxtures. Applcatons of ther equaton (the NL equaton to partally mscble multcomponent mxtures mght requre more complcated calculatons than those of the Wlson equaton. Ths study focuses on an applcaton of the Wlson equaton to mneral sold solutons. As an llustratve example, ths study analyzes the expermental results on flud (Fe,,MgTO equlbra (Kubo et al. 99. Although there are many studes on caton exchange reactons between mnerals and aqueous solutons, only a few studes deal wth both bnary and ternary sold solutons. The reason for choosng flud (Fe,,MgTO equlbra s the avalablty of such expermental data. ased on the results of the three bnary caton exchange experments, the present study obtans the mxng propertes of the ternary sold soluton. The present study computes the compostons of (Fe,,MgCl (aq n equlbra -4/99/9 7$. 7

2 76 SHIUE: WILSON EQUATION AND FLUID-(Fe,,MgTO EQUILIRIA wth (Fe,,MgTO sold solutons and compares these computed results wth the expermental data. Ths study also compares the Wlson equaton wth the Margules equaton n regard to the predctablty of the flud-phase compostons. A subsdary purpose of the present study was to use the nonlnear programmng technque of rtt and Luecke (97 for obtanng the Wlson parameters and Gbbs energy of reactons. Although the computatonal technque s more complcated than ordnary lnear least-square regresson, the results of the calculaton obey the law of mass acton. THE WILSON EQUATION The Wlson equaton s based on the Flory-Huggns equaton (Flory 94; Huggns 94 and the concept of local volume fracton (Wlson 964, both of whch have been only rarely referred to n the lterature of geology, mneralogy, and geochemstry. The dervaton of the Flory Huggns equaton was gven n detal by Flory (944 and s not repeated here. Only the resultant Flory Huggns equaton for a ternary mxture s shown below. Thereafter, ths paper derves the Wlson equaton by ntroducng the concept of local volume fracton followng the method of Prausntz et al. (986. Accordng to the Flory Huggns equaton, the molar Gbbs energy of mxng (G mxng s expressed as follows: G mxng ( lnφ + lnφ + lnφ (4 where,, and stand for the mole fractons of the solvent, solute, and solute, respectvely, and φ, φ, and φ are the volume fractons of the solvent, solute, and solute, respectvely. Concept of local volume fracton In a ternary soluton beng consdered here, the focus s on a central molecule of type. The probablty of fndng a molecule of type j, relatve to fndng a molecule of type, adjacent to the central molecule s expressed n terms of the mole fracton and two oltzmann factors: j λ j j exp λ exp where j and stand for the mole fractons of components j and around the central molecule of type. The terms λ j and λ are related to the potental energes of a j and an par, respectvely. The ternary nteracton parameter does not appear n Equaton. It should be noted that Equaton consders only the nearest-neghbor molecules around the central molecule n a sphercal lqud-lke envronment. The equaton then assumes that the lattce stes on whch the molecules are located could be dsplaced by the nteracton energes between molecules. Wlson (964 defned the local volume fracton of component (φ as follows: V φ V + V + V ( (6 where V, V, and V stand for the molar volumes of the subscrpted components n ther pure state. It should be noted that φ + φ + φ s not always equal to (Prausntz et al As a result, the local volume fracton s not equvalent to the volume fracton and should be consdered as a conceptual quantty n the mcroscale envronment. The Wlson parameter ( j s defned as follows: j V V j j exp λ λ. (7 y the above defnton, s equal to. y combnng Equatons to 7, φ s expressed as follows: φ. (8 + + Wlson (964 substtuted the local volume fractons of the components, nstead of ther volume fractons, nto the Flory Huggns equaton (Eq. 4. After rearrangement, G mxng can be expressed as follows: G mxng {[ ln( + + ] + [ ln( + + ] + [ ln( + + ] ( ln + ln + ln }. (9 Therefore, the molar excess Gbbs energy (G ex can be derved as follows: ( Gex Gmxng ln + ln + ln ( whch s equvalent to Equaton. Actvty coeffcents of the ndvdual components can be obtaned from the partal dervatves of the excess Gbbs energy wth respect to the number of moles of that component as follows: ln γ ng n ex TPn,, j ( where n and n stand for the total number of moles of all the components and the number of moles of component, respectvely. Usng Equatons 9 and, the actvty coeffcent of component can be expressed as Equaton. APPLICATIONS TO FLUID-(FE,MNTIO, FLUID- (MN,MGTIO, AND FLUID-(FE,MGTIO EQUILIRIA Kubo et al. (99 carred out caton exchange experments on the followng systems at 6 C and kbar by usng the method of Uchda et al. (989: (A FeTO + Cl (aq TO + FeCl (aq ( TO + MgCl (aq MgTO + Cl (aq (C MgTO + FeCl (aq FeTO + MgCl (aq Total molarty of metal chlorde n the aqueous phase was. M n all ther experments. Only (Fe,,MgTO was observed as the sold product. y usng the Wlson equaton for descrbng the actvty coeffcents for the components of the sold phase, the Gbbs energy of reacton A can be wrtten as follows:

3 SHIUE: WILSON EQUATION AND FLUID-(Fe,,MgTO EQUILIRIA 77 G A TO γ TO Fe ln K ln ln FeTO γ FeTO FeFeTO + TO + ln + FeTO + FeTO TO Fe TO + + Fe FeTO TO TO FeTO Fe + Fe FeTO FeTO FeTO Fe TO ( where K stands for the equlbrum constant and Fe and desgnate the ratos of the molaltes of the subscrpted speces to the total molalty of metal chlorde n the flud phase. The values are expressed as follows: Fe mfecl (aq m + m + m FeCl (aq Cl (aq MgCl (aq mcl (aq m + m + m FeCl (aq Cl (aq MgCl (aq ( (4 G for Reacton or C can be obtaned by changng the subscrpts approprately. For example, Mg denotes the followng rato. Mg mmgcl (aq m + m + m FeCl (aq Cl (aq MgCl (aq ( The present study gnores the onc speces and assumes that the ratos of actvty coeffcents of the neutral aqueous speces are equal to. Kubo et al. (99 reported the concentratons of the aqueous speces n unts of molarty. ecause the present study calculates the ratos of concentratons, the converson factor of molar scale nto molal scale does not appear n Equatons to. Values of G for reactons A C and the Wlson parameters were obtaned by usng the followng constrants: G G + G C A MgFe Mg Fe FeMg Mg Fe (6 (7 The dervaton of Equaton 7 was gven n Hala (97. y mposng the above constrants on the calculatons, the calculated G values as well as the Wlson parameters are nternally consstent. For the calculaton of all the values, the nonlnear programmng technque derved by rtt and Luecke (97 was used n the present study. Sold and flud compostons were also computed for each expermental data pont. Ths study consders that the standard errors of measurements of both compostons are. n terms of mole fracton or the rato of molalty. Substtutons of the calculated compostons of the sold and flud phases and the Wlson parameters nto Equaton make the resultant lnk values become constant. The same s true for reactons and C. Detaled calculaton procedures are shown n the Appendx. The results of the calculatons of G / and the Wlson parameters are lsted n Table. The expermental and computed compostons are summarzed n Tables 4. From Equaton and the equvalent expressons for reactons and C, we get the followng equatons: Fe GA TO TO exp γ ln (8 FeTO γ FeTO G MgTO MgTO Mg exp γ ln (9 TO γ TO Mg GC FeTO FeTO Fe exp γ ln ( MgTO γ MgTO If we specfy the sold compostons, actvty coeffcents of the components n the sold phase are determned by Equaton. y usng the actvty coeffcents at specfc sold compostons and the G value for the relevant reacton, the fludphase compostons can be obtaned from the left-hand sdes of Equatons 8. Then caton exchange sotherms for reactons A,, and C are computed from Equatons 8, 9, and, respectvely (Fgs.. The computed sotherms show good agreement wth the expermental results of Kubo et al. (99. Actvtes of end-member components are plotted aganst the mole fractons for the three bnary sold solutons (Fg. 4. The (Fe,TO and (Fe,MgTO sold solutons show almost deal behavor whereas (,MgTO shows dstnct nondealty. It s possble to compute the G value and the Wlson parameters for each reacton and to compare the results wth the nternally consstent values n Table. Calculatons on each reacton gave nternally nconsstent results. Furthermore, the teratve computatons often showed dvergences due to the nonlnearty of the Wlson equaton. Therefore, ths study dd not attempt to extract thermodynamc quanttes from the ndvdual exchange reactons. Calculatons of flud (Fe,,MgTO equlbra y substtutng the Wlson parameters nto Equaton, the molar excess Gbbs energy of (Fe,,MgTO sold soluton at 6 C and kbar s expressed as follows. G ex (kj/mol 7.6[ FeTO ln( FeTO +.4 TO +.96 MgTO + TO ln(.8 FeTO + TO +.9 MgTO + MgTO ln(.46 FeTO +.7 TO + MgTO ] ( Table. G for reactons A C and Wlson parameters for (Fe,TO, (,MgTO, and (Fe,MgTO sold solutons Reacton A: FeTO + Cl (aq TO + FeCl (aq Reacton : TO + MgCl (aq TO + Cl (aq Reacton C: MgTO + FeCl (aq FeTO + MgCl (aq GA G GC. 46(. 4,. 97(. 6,. 487(. Fe.8(., Fe.4(.7, Mg.7(.87 Mg.9(.7, FeMg.96(., MgFe.46(.8 Notes: Values n parentheses ndcate the standard errors. s 7.6 (kj/mol at 6 C.

4 78 SHIUE: WILSON EQUATION AND FLUID-(Fe,,MgTO EQUILIRIA ln γ FeTO ln FeTO + TO Fe + MgTO FeMg FeTO + + TO Fe + + MgTO MgFe + + FeTO TO Fe MgTO FeMg FeTO Fe TO MgTO Mg FeTO MgFe TO Mg MgTO ( ln ln ln + + FeTO Fe + + TO + + MgTO MgFe + + γ TO FeTO Fe TO MgTO Mg FeTO TO Fe MgTO FeMg FeTO Fe TO MgTO Mg FeTO MgFe TO Mg MgTO ln + + FeTO FeMg + + TO Mg + + MgTO + + γ MgTO FeTO MgFe TO Mg MgTO FeTO TO Fe MgTO FeMg FeTO Fe TO MgTO Mg FeTO MgFe TO Mg MgTO ( (4 y usng Equaton, actvty coeffcents of FeTO, TO, and MgTO n the ternary sold soluton are expressed as follows: On the bass of the above expressons of actvty coeffcents, the compostons of (Fe,,MgCl (aq n equlbra wth (Fe,,MgTO were computed. The sold compostons were taken from the expermental results of Kubo et al. (99. The method of Shallcross et al. (988 was used for the present calculaton. Let us defne Z values as follows. Z Fe, Z Mg, Z Mg Fe From the sold compostons, actvtes of the end-member components can be computed. Thus, Z, Z, and Z values are obtaned from Equatons 8. If we use Z and Z values, values are calculated as follows: Fe Mg ZZ ZZ + Z + Z ZZ + Z + ZZ + Z + ( (6 (7 FIGURE. The caton exchange sotherm computed n ths study (sold lne and the expermental data ponts (Kubo et al. 99 for the reacton FeTO +Cl (aq TO +FeCl (aq. Fe and Fe stand for the mole fracton of FeTO n the sold phase and the molalty rato defned by Equaton. TALE. Expermental and calculated results on the caton exchange reacton FeTO + Cl (aq TO + FeCl (aq Expermental results* Calculated results FeTO Fe FeTO Fe * Kubo et al. (99. Calculated results of ths study. Other solutons of values are possble when we use the set of Z and Z or the set of Z and Z. Therefore, three sets of values are possble. The weghted average values of flud-phase compostons can be obtaned from the followng equatons: Fe ZZ ZZ ZZ Z Z Z + Z + + ZZ + Z + + ZZ + Z + TO FeTO MgTO (8 Z Z Z + Z + + ZZ + Z + + ZZ + Z + (9 TO FeTO MgTO Z Z Z + Z + + ZZ + Z + + ZZ + Z + ( TO FeTO MgTO

5 SHIUE: WILSON EQUATION AND FLUID-(Fe,,MgTO EQUILIRIA 79 TALE. Expermental and calculated results on the caton exchange reacton TO + MgCl (aq MgTO + Cl (aq Expermental results* Calculated results TO TO * Kubo et al. (99. Calculated results of ths study. TALE 4. Expermental and calculated results on the caton exchange reacton MgTO + FeCl (aq FeTO + MgCl (aq Expermental results* Calculated results MgTO Mg MgTO Mg * Kubo et al. (99. Calculated results of ths study. FIGURE. The caton exchange sotherm computed n ths study (sold lne and the expermental data ponts (Kubo et al. 99 for the reacton TO +MgCl (aq MgTO +Cl (aq. and stand for the mole fracton of TO n the sold phase and the molalty rato defned by Equaton 4. FIGURE. The caton exchange sotherm computed n ths study (sold lne and the expermental data ponts (Kubo et al. 99 for the reacton MgTO +FeCl (aq FeTO +MgCl (aq. Mg and Mg stand for the mole fracton of MgTO n the sold phase and the molalty rato defned by Equaton. The weghts correspond to the mole fractons of the components common to Z and Z j. Computed flud compostons and the expermental results of Kubo et al. (99 are compared n Fgure. The agreement s generally good. Although the computed compostons devate from the expermental compostons n the regon of low Fe, the devatons are wthn ±.8 n terms of the value for any component (Table. It can be concluded that flud compostons n equlbra wth the ternary sold solutons may be predcted from the bnary caton exchange experments. COMPARISON WITH THE MARGULES EQUATION In the ntroducton, t was argued that the Margules equaton wthout the ternary nteracton parameter s not accurate for descrbng the mxng property of the ternary sold soluton consstng of asymmetrc bnares. The accuracy of the compo-

6 8 SHIUE: WILSON EQUATION AND FLUID-(Fe,,MgTO EQUILIRIA FIGURE 4. Actvty (a vs. composton ( relatons of (Fe,TO, (,MgTO, and (Fe,MgTO sold solutons at 6 C and kbar. stons of the flud phase predcted by the Margules equaton and by the Wlson equaton can now be compared. y an approprate dfferentaton of Equaton, actvty coeffcents of components and n a bnary sold soluton are wrtten as follows (e.g., Mukhopadhyay et al. 99. lnγ [W + (W W ] ( lnγ [W + (W W ] ( FIGURE. Flud compostons n equlbra wth (Fe,,Mg TO sold solutons. Flud compostons ted by lnes correspond to the same sold compostons. The expermental compostons of sold and flud phases are taken from Kubo et al. (99. The te lnes are omtted when the computed compostons closely agree wth the expermental compostons. From the expermental results lsted n Tables 4, G values for reactons A C and the Margules parameters were computed for the three bnary sold solutons. The constrants (Eqs. 6 and 7 were not used n the calculaton. The computaton of three unknowns for each reacton was carred out wth the nonlnear programmng technque, followng the procedure shown n Appendx. The results of the calculaton are lsted n Table 6. y settng C, actvty coeffcents of the components n the ternary sold soluton of (Fe,,MgTO were computed on the bass of the expermental compostons of the sold phase (Table. Actvty coeffcents of the three components can be wrtten as follows (Mukhopadhyay et al. 99:

7 SHIUE: WILSON EQUATION AND FLUID-(Fe,,MgTO EQUILIRIA 8 TALE. Expermental and calculated results on the caton exchange reactons between (Fe,,MgTO and (Fe,,MgCl (aq Expermental results* Calculated results FeTO TO Fe Fe * Kubo et al. (99. Calculated results of ths study. TALE 6. G for reactons A C and Margules parameters* for (Fe,TO, (,MgTO, and (Fe,MgTO sold solutons GA G GC. (. 8,. 98(. 4,. 47(. 7 W Fe.7(.77, W Fe.4(.66, W Mg.49(., W Mg.9(.866, W MgFe.(.96, W FeMg 6.7(.77 Notes: Values n parentheses ndcate the standard errors. s 7.6 (kj/mol at 6 C. * Calculated from the results on each bnary reacton. Reactons A,, and C are ndcated n Table and n the text. before (see Equatons 8 to. After the computaton of the flud phase compostons, the sum of the dstances were obtaned between the expermental compostons and the calculated compostons n the ternary system. The overall ft to the expermental data usng the Wlson equaton s better than wth the Margules equaton havng no ternary nteracton parameter (Σdstances.98 vs..7. Thus, the predcton of the ternary sold soluton flud equlbra by the Wlson equaton s more accurate than that by the Margules equaton on the bass of data on bnary caton exchange reactons. REFERENCES CITED rtt, H.I. and Luecke, R.H. (97 The estmaton of parameters n nonlnear, mplct models. Technometrcs,, 47. Flory, P.J. (94 Thermodynamcs of hgh polymer solutons. Journal of Chemcal Physcs, 9, (944 Thermodynamcs of heterogeneous polymers and ther solutons. Journal of Chemcal Physcs,, Ganguly, J. and Saxena, S.K. (987 Mxtures and eral Reactons, 9 p. Sprnger Verlag, erln. Hala, E. (97 Note to run Prausntz one parameter local composton equaton. Industral and Engneerng Chemstry Process Desgn and Development,, 68. Huggns, M.L. (94 Solutons of long chan compounds. Journal of Chemcal Physcs, 9, 44. Kubo, T., Uchda, E., Furukawa,., and Ima, N. (99 Expermental study on on exchange equlbra between sold soluton (Fe +, +,Mg + TO and aqueous (Fe +, +,Mg + Cl soluton. Kobutsugaku Zassh (Journal of eralogcal Socety of Japan,, 9-67 (n Japanese wth Englsh abstract. Mukhopadhyay,., asu, S., and Holdway, M.J. (99 A dscusson of Margules type formulaton for multcomponent solutons wth a generalzed approach. Geochmca et Cosmochmca Acta, 7, Prausntz, J.M., Lchtenthaler, R.N., and Azevedo, E.G. (986 Molecular Thermodynamcs of Flud-phase Equlbra. nd. ed., 6 p. Prentce Hall, New ork. Renon, H. and Prausntz, J.M. (968 Local compostons n thermodynamc excess functons for lqud mxtures. Amercan Insttute of Chemcal Engneers Journal, 4, 44. Shallcross, D.C., Herrmann, C.C., and McCoy,.J. (988 An mproved model for the predcton of multcomponent on exchange equlbra. Chemcal Engneerng Scence, 4, Uchda, E., Gma, M., and Ima, N. (989 Expermental studes on on exchange equlbra between mnerals and aqueous chlorde soluton n the system CaWO 4 - FeWO 4 -WO 4 under supercrtcal condton. Geochemcal Journal,, Wlson, G.M. (964 Vapor lqud equlbrum. I. A new expresson for the excess free energy mxng. Journal of Amercan Chemcal Socety, 86, 7. lnγ FeTO W Fe TO ( FeTO + W Fe FeTO TO ( FeTO W Mg TO MgTO W Mg TO MgTO + W MgFe FeTO MgTO ( FeTO + W FeMg MgTO ( FeTO ( lnγ TO W Fe FeTO TO ( TO + W Fe FeTO ( TO + W Mg MgTO ( TO +W Mg TO MgTO ( TO W MgFe FeTO MgTO W FeMg FeTO MgTO (4 lnγ MgTO W Fe FeTO TO W Fe FeTO TO + W Mg TO MgTO ( MgTO + W Mg TO ( MgTO + W MgFe FeTO ( MgTO + W FeMg FeTO MgTO ( MgTO ( Compostons of (Fe,,MgCl (aq n equlbra wth (Fe,,MgTO were computed by the same method descrbed MANUSCRIPT RECEIVED MA 8, 998 MANUSCRIPT ACCEPTED MA, 999 PAPER HANDLED J. WILLIAM CARE APPENDI rtt and Luecke (97 developed a nonlnear programmng technque for obtanng the parameters n varous types of regresson equatons. Ther method was used n the present study to obtan the Gbbs energy of reactons, the Wlson parameters, and the compostons of sold and flud phases. To avod lengthy notatons and equatons, the followng abbrevatons are used hereafter: G, A R T Fe, Fe, G 4, MgFe, 6 Mg, 7 FeMg. Usng the constrant equatons shown n the text (Eqs. 6 and 7, the other two unknowns ( G / for reacton C and MgFe were obtaned from the above seven parameters. In ths study, the parameter F was defned as follows for all the data ponts relatng to reacton A:

8 8 SHIUE: WILSON EQUATION AND FLUID-(Fe,,MgTO EQUILIRIA + F + ln ln ( + ( ( ( ( + + (A where and stand for the calculated values of FeTO and Fe, respectvely, for the -th expermental data pont. Smlarly, by denotng and as the calculated values of TO and for the -th expermental data pont, F for reacton s defned as follows. + F 4 + ln ln ( + 6 ( ( 6 ( ( (A For reacton C, the followng expresson can be derved through the constrant equatons. ( F ( 4 ln ln ( + 7 ( ( ( ( (A 6 where and stand for the calculated values of MgTO and Mg for the -th expermental data pont. In the present study, calculatons were carred out under the condton that all of the F values should be, whch s equvalent to the condton that lnk for each reacton becomes constant. In other words, the computed results satsfy the law of mass acton. Standard errors of the measurements of both and values are assumed to be. ( σ. Then ths study mnmzes the value of Q, whch s defned as follows: [ ] meas meas Q σ ( x + ( y (A4 where the superscrpt, meas, stands for the measured value. The calculated values for sold and flud compostons are desgnated as x and y, respectvely. We defne L, whch corresponds to the weght of the -th expermental datum, as follows. F L + σ (A The superscrpt desgnates the value at the j-th teraton. Also a matrx C(u,v and a functon E defned below are obtaned. E C(u,v F F + u F v L F meas ( + ( L meas (A6 (A7 Then true parameters for values, b ( 7, are obtaned from the followng smultaneous lnear equaton. C(, C(, C(, C(, 4 C(, C(, C(, C( 4, C(, C(, C(, C(, 4 C( 4, C( 4, C( 4, C( 44, C(, C(, C(, C(, 4 C(, 6 C(, 6 C(, 6 C(, 64 C(, 7 C(, 7 C (, 7 C( 74, C(, C(, 6 C(, 7 C(, C(, 6 C(, 7 C(, C(, 6 C(, 7 C( 4, C( 46, C( 47, C(, C(, 6 C(, 7 C(, 6 C(, 66 C(, 67 C(, 7 C(, 76 C(, b b b b 4 b b 6 b 7 E E E E (A8 4 E E 6 7 E

9 SHIUE: WILSON EQUATION AND FLUID-(Fe,,MgTO EQUILIRIA 8 After solvng Equaton A8, the sold and flud compostons are computed by the followng equatons. meas x + λ σ meas y + λ σ (A9 (A After b, x, and y values are obtaned, those are substtuted nto Equatons A A and A A7 as,, and. Then the newly computed F, E, and C(u,v are substtuted nto Equatons A8 A. Successve teratons are repeated untl all of the unknowns and the compostons of sold and flud phases are converged. The followng convergence crtera were adopted as follows: where λ meas F + ( + L F meas meas ( + F b b F F meas ( (j+ (j+ u u. (u 7, u., (j+.. + F ( + ( + ( + ( + meas F meas F meas F meas F meas b b b b b L L (A ( 7