- Email: [email protected]
EQMIN, a Microsoft® Excel© spreadsheet to perform thermodynamic calculations: A didactic approach
Computers & Geosciences
Pergamon PII: SOO98-3004(96)00006-4
Vol. 22, No. 6, pp. 639450, 1996
Copyright Q 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0098.3004/96 $15.00 + 0.00
EQMIN, A MICROSOFT@ EXCEL0 SPREADSHEET PERFORM THERMODYNAMIC CALCULATIONS: DIDACTIC APPROACH
TO A
JORDI DELGADO MARTiNt Departament de Cristal.lografia, Mineralogia i Dip&its Minerals, Universitat de Barcelona, Marti i Franquis s/n, 08028 Barcelona, Spain (e-mail: [email protected]) (Received
27 September
1995; revised
23 November
1995)
Abstract-Knowledge of the basic laws of thermodynamics is of prime importance in understanding processes occurring in the Earth at any place and at any scale. Calculations related to the construction of phase diagrams, petrogenetic grids, geothermobarometry, etc. rely heavily on the proper manipulation of thermodynamic parameters. EQMIN is a Microsoft e Excel” worksheet constructed to help students learn the basic thermodynamic relationships and calculus routines. It is a flexible system with which it is possible to evaluate from the basic thermodynamic parameters almost all the thermodynamic functions in frequent use, as well as creating phase diagrams, retrieval of thermodynamic data from phase equilibria experiments, etc. Copyright Q 1996 Elsevier Science Ltd Ke.y
Words: Thermodynamics, Phase equilibria, Spreadsheet, Mineralogy, Petrology.
INTRODUCTION
the past two decadesan increasingnumberof programsto perform geochemicallyrelevant thermodynamic calculations have become available, for example SUPCRT (Helgeson and others, 1978), PHREEQE (Parkhurst, Thorstenson,and Plummer, 1980), EQ3/6 (Wolery and others, 1983), PTX (Perkins, Brown, and Berman, 1986), THERM0 (Perkins, Essene,and Wall, 1987), PTA (Brown, Berman, and Perkins, 1989),CHILLER (Reedand Spycher,1988),THERMOCALC (Holland andPowell, 1990), VERTEX (Connolly, 1990), TWQ (Berman, 1991),SUPCRT92(Johnson,Oelkers,and Helgeson,1992),amongmany others. They resulted from the needto avoid cumbersomerepetitive operations and to assess self-consistentthermodynamic databases usefulfor calculationsrelatingto problems ranging from the Earth surfaceto the lower crust. Although there are differencesin the philosophy behind each program and their P-T-x applicability range,all usethe samebasicthermodynamicrelationships becauseall are basedstrictly on the laws of thermodynamics.All theseprogramswerewritten in a computer language (FORTRAN, PASCAL, BASIC, etc.) and then compiled to offer a compact presentation,making their useeasier.This approach proved to be a successas shown by the latest programs, which are more user friendly and powerful
During
tNew address: E.T.S. Ingenieros de Caminos, Universidad de La Co&a, Campus de Elvitia, s/n, 15192 La Coruda, Spain.
than ever before. However, from a didactic point of view, computer programspresentseveralhandicaps that prevent themfrom beingthe ideal teachingtool. First, if the final purposeis to learn how a program performs its calculations, studentsunfamiliar with computer languageswill have difficulties reading a medium-sizesource code, a problem even for a trained eye. Moreover, if the source code is not available, the program becomesa “black box” and the output data has to be believed dogmatically. Second,the thermodynamicdata for eachphaseto be consideredin a reaction are not available freely becauseusuallythey are held in a parallel file. So, if it is necessaryto evaluate the effect of taking into account someeffect or to modify one or several reaction parameters,it is necessaryto exit the program,call an editor to changethe data and then rerun the application. Finally, output data usually are constrainedin a way that to obtain the required output it is necessaryto perform off-line calculations with the raw data given by the program. Most of these limitations arise becausethe programs have beenconceivedprimarily for researchand only incidentally for teaching. We have tried to harmonizeteachingand research needsby a new approachto automatedthermodynamiccalculations.This hasbeendone with the aid of a commercial spreadsheet.This type of software offers several advantages over research-oriented thermodynamicprograms:(1) for peopleunfamiliar with programming languages they constitute a smooth transition from hand calculatorsand, with minimum training it is possibleto start to work
639
J. D. Martin
640
almost immediately; (2) spreadsheets are popular and are available in many departments, or are available readily; (3) portability problems are almost non-existent because the commercial firms ensure compatibility among the most successful spreadsheets; (4) thermodynamic data, results, and any intermediate operations are shown on the screen together with graphic, printing, and other utilities; (5) as the thermodynamic parameters are available to the user at the same time the problem is solved, it is possible to modify them in order to maximize or minimize effects induced by conventional assumptions or approximations.
In this expression a is the activity of a mineral component (a = 1 if its composition is equal to tha: of the ideal end-member composition at any P, T) The Gibbs free energy of any balanced reaction ma] be calculated from the stoichiometric addition of the individual properties of participating phases A,G(P,
T) = c v,ArGo(i,
PO, To)
A,$
+RTln WORKSHEET
DESCRIPTION
The system we describe in this paper has been conceived to develop the necessary student skills to perform any of basic thermodynamic calculations. EQMIN uses similar expressions for the heat capacity, expansivity, and compressibility functions as the program THERM0 (Perkins, Essene, and Wall, 1987), although intermediate calculations are more extensive. The primary purpose of EQMIN is to evaluate the Gibbs free energy of substances at any pressure (P) and temperature (T) from conventional reference values (P”, To) using an integrated form of the Gibbs-Duhem equation A,G(P,
T) = A,G”(Po,
+ j;AfVdP
To)
-ITI
ArSdT+RTlna.
(1)
1173 15 Starting
iteration temperature= Temperature increment= Working pressure= Starting iteration pressure= Pressure increment= X(&O)= X(CO2)= A$
tolerance
limiti
800 10 5000 1
100
Figure 1. Aspect of page “Input”
flu; Cl
(21 >
where i represents the ith mineral phase in thl: reaction, v is the stoichiometric coefficient of the ith phase in the reaction, a, is the activity of the ith phase in the reaction, and A,G(P, T) is the Gibbs free energy value of the reaction at the P, T of interest. For a solid-fluid reaction the volume integral con. siders just the solid phases, and we have to add tb: term v,RT Inf, (where j is a fluid species, and v and f the corresponding stoichiometric coefficient and fugacity) to account for the volumetric change from P” to P of fluid j. EQMIN is an EXCEL” workbook arranged ill several interrelated pages. All the pages have a table format. Page 1 (termed “constants”) states the refer, ence temperature (To) and pressure (PO) and som’. other basic constants used by the different equation (gas constant, R; conversion between calories ancl joules; etc.). Page 2, named “input” (Fig. 1), contain<
K
lOOO/Tm=
0 852
“C “C
‘bar bar 100 bar 0.5 05
dT
050MPa
Jmo?
of workbook EQMIN. It is used to state operating conditions for calculations.
EQMIN DlO
641
-3*harc+3*siMm-S*cm
ti.i
1AIDIE(GiIIJIKILlMIP1Q
-lR eactions 1
ArWae, ( J mor)
A@ww
&%a)
CPl
cP2
cP3
cw
Expansivity EXPI EXP2
3.36 162.63 4.01 12.60 8.59
3.36 162.63 4.01 12.60 8.59 43.88
28.01 47 02 2.89 076 -2.13
-1089 -65.67 -0.31 0.22 0.54
-5.70 -5.69 -1.29 -1.05 0.24
9.62 0.49 5.38 6 80 1.42
0.10 0.07 -1.88 -2 13 -0.26
457.81
19944
-225.89
-0.99
cm3mop
(J ml
-1895.02 -2896.80 90823.48 3805.58
s%Y
42335.35 2610.00
7676.69 3920 00 3871.11 1310.00 127989021 14905.60
43.88
-1037.51
0.12
-1151 0 58 2 53
1 95 13.07
Figure 2. Page “Reactions,” which is used by EQMIN to introduce formulas which represent chemical reactions. Note, in upper part of figure, formula “ = 3*herc + 3*sil-alm-S*crn” in cell DlO (A#(p’, TO)) which represents reaction of interest. Here, sil, alm, and cm are the symbols associated with the thermodynamic data ranks of hercinite, sillimanite, almandine, and corundum, respectively. In this example, EQMIN works simultaneously with six different reactions.
the input parameters for the calculations (P, T, and X,,,) together with five additional entries (starting iteration temperature and pressure,temperatureand
pressureincrement,and a cut-off limit for the calculation) usefulif an automatedcalculationwith macro programmingispreferred.Page3 (“reactions”) isthe areaselectedto makeexplicit the reactionsof interest (Fig. 2). Each reaction occupies a single line which follows a column arrangementidenticalto that of the thermodynamic data of pages 5-7 (“elements,” “minerals,” and “gases”), EQMIN is not constrained to the calculation of one reaction at a time. An unlimited number of them can be evaluated simultaneously, taking care to add the samenumber of output areasasreactions.In page 4 (“output”), we provide an output areafor the calculationswhich are the Gibbs free energyof the reaction[A,G(P, T)] and its translation in terms of the equilibrium constant: K(P, T), logK(P, T), and In K(P, T) (Fig. 3). Pages5-7 (“elements,” “minerals,” and “gases”) contain the basic thermodynamic properties of elements,minerals, and gasesand liquids, respectively. Figure 4 showsan example of the aspectof the “minerals” page. The thermodynamic parameters compiled are: A,H”(Po, To), AfGo(Po, TO), S”(Po, To), ArSo(Po, To), coefficientsfor the constant pressureheat capacity (A, B, C, D), isobaric thermal expansion(E, F) and isothermiccompressibility (G, H) functions and molar volume [V’(P”, To)] at the standard pressureand tempera-
ture. All these values are referred to their corresponding sources(p. 10 and 11)with their applicableP-T range, when appropriate. Together with the basic data, each page contains also the result of someintermediatecalculationsat the T, P selected in the input area, for example,the volume increase of the phasefrom PO, To to the P and T of interest, heat capacity at T together with the enthalpy, entropy, and Gibbs free energy increasefrom To to T, and other thermodynamic functions [(HtSpecific equations are f%)/T; - (G! - H&,)/T]. given in Appendix 1. Page 8 (“eos”) is used for the calculation of H,O and CO, end-member fugacities,aswell astheir nonidealmixing, according to the polynomial expressionsgiven by Powell and Holland (1985),updatedin Holland and Powell (1990). On page 9 (“ac-models”), activity models for solid phasesare entered.Pages10 and 11 (“a . . z” and “1 . . n”) are used to keep a record of the sourcesfrom which the thermodynamic data have been collected or a brief description of their method of estimation. Finally, page 12 (“info”) provides a list of the substancefull name, formula, and symbol currently presentin EQMIN. More detailed cell descriptions are given in Appendix 2. The referencestatesusedare the elementsin their stable statesat 1 bar and 298.15K. The standard statefor fluids is pure fluid at 1 bar and T, and for solidphasesthe correspondingstandardstateis that
642
1
J. D. Martin
output
2 3
4
A.Gm
5 6 7 8 9
A&p) ArG,n
110 11
dp,dT
(Jmol-I)= (JmoT’)= (JmoT’)= KF logQG)=
R. 1 973.47
R. 3 -1716.37
2115.35
-6677.07
-90408.47
-5703 60 1.79E+OO 0.25
-9585.74 2.67E+OO 043
-547.72 -226408
-6723.06 -4607.71
1.26E+OO 0.10 .“x’5:
(k:$$%-/
dT/dP (Y? kbaf’)=
R. 4
R. 2 -10604 67
-::;* 41.90
-15 54
R. 5 3831.72 -6175 34
R. 6 14512.17
R. 7 0 00
R. 8 0.00
-22685.29
-2343.62
-8173.12
0.00 0.00
0.00 0.00
1.6OE+OO 0.21
127E+OO 0 10
2.31E+OO 0.36
l.OOE+OO 0 00
l.OOE+OO 0.00
:;;o
;:;1
:;;7
#iDN/O! O.OO
37.17
95 11
56.90
#iDN/O!
#iDN/O! O O0 #iDN/O!
-117.50
12 13 14
15 16
17 18 19
Figure 3. Summary screen, in the page termed “Output,” where selected output for each of reactions processed are shown. There are output data for six different reactions (see Fig. 2). Warning message #i DIVjO! reflects absence of more than six reactions in example considered.
of the pure substance at the pressure and temperature of interest (Robie, Hemingway, and Fisher, 1979; Cox, 1979; Chase and others, 1985). The thermodynamic data given in the worksheet were collected from a variety of sources (Robie,
1.0 10 10
-5261237.0 -2589899.1 -5774860.6
Hemingway, and Fisher, 1979; Robinson and other;. 1982; Halbach and Chatterjee, 1984; Chase and others, 1985; Moecher, Essene, and Anovitz, 198X: Anovitz and others, 1993; etc.). They do not constitute a self-consistent thermodynamic database and
-4938260
0
-2441720 -54320000
0
10 -5161365.2 -47978369 1.0 -4237774 1 -4012000.0 10 -12069957 1 -113433000 1.0 -1207430.4 -1127793.0 0.9 -120359.7 -124900.6 10 -256115.0 -255960.0 10 -4344733.7 -4077481.0 10 -135186.3 -132658 0
(z, @zz) (31) (89 c,~s,
342.60 91.39 316.82 373.60 199.30 (49) 534.50 0 87.99 (KC’) 110.00 (6s) 245.00 (40) 213 68 (66) 20040
(9 (22) (~2) (63) (1) (49) 0 (106) (60 (so) cr,
CPl
cF2
-1083 27 -497 00 -114996 -1219.28 -757.25 -2437.22 -267 11 15.23
800.44 277.30 981.27 58495 8747 1259.82 81.62 -292.69
-57.01 -6.59 -134.68 74.76 114.92 18.93
-896.37
280.27
Figure 4. Example of Part of page “Minerals” where thermodynamic data of solid substances elements are compiled. Tables with same format have been constructed for chemical elements and gases. See text and Appendix 2 for more details.
190 51
other than and liquids
.’
EQMIN so, depending on the purpose of the calculations, care should be taken to consult the original works for limitations on data use. The use of self-consistent data-bases (Holland and Powell, 1990; Berman, 1988; for example), was not considered a critical point in setting up EQMIN. In spite of the obvious benefits accessible through their use (see detailed discussions in Powell and Holland, 1985; Berman, 1988; Engi, 1992; and references therein), we preferred an eclectic approximation for the following reasons: (1) it is possible to modify any of the data records without disturbing the rest of the dataset and to enlarge the database to an indefinite size at any moment; (2) EQMIN is not constrained to a limited number of system components and/or phases, similar to many of the self-consistent databases, so, EQMIN is applicable to a much more wider geological context; (3) everyday experience tells us that we need to retrieve, estimate, refine, and compare thermodynamic data for a great variety of substances and this can be done efficiently with the aid of EQMIN through the incorporation of new phases, trial and error procedures, etc. It is possible, however, to incorporate into EQMIN any of the self-consistent databases. This task is left to the discretion of every instructor. The use of Powell and Holland’s (1985) polynomials to calculate fluid fugacities in the system H,O-CO, obeys its simple mathematical formulation. It is compatible fully with the experimental data currently available for H20, CO, and their mixing according to the equation of state of Redlich-Kwong, modified by Kerrick and Jacobs (1981). Limitations on the pressure-temperature ranges for which these equations apply are quoted in Powell and Holland (1985) Holland and Powell (1990) and Kerrick and Jacobs (1981). At present no other fugacity coefficients can be calculated with EQMIN, although it would be a relatively simple task to add some other equations of state to handle in a more realistic way with the volumetric properties of other gases and their mixtures. EQMIN can perform calculations with component activities different from 1. This can be achieved by modifying the default value of 1 given in every row of column 2 of pages 5-7. Fugacity values for H,O and CO2 are set automatically by the worksheet as far as a fluid composition has been provided in the input area. For other gases, fugacity values must be fixed or supplied through additional equations of state. EQMIN does not implement any kind of general activity model for minerals. Nevertheless, particular activity models (ideal or nonideal) may be merged to EQMIN for substances of interest, at the instructor’s convenience. We provide a couple of examples of two-constant (asymmetric) solid solution models with EQMIN: one for the FeS-ZnS system (Delgado, 1995) and the other for the Fe vacants in pyrrhotite (Froese and Gunter, 1976).
643 WORKSHEET
OPERATION
Operation of the worksheet is related to the final purpose of the calculations. If it is desired to evaluate the individual properties of a single substance, by fixing a T and P, we only need to read the results in the corresponding cells of the pages that contain the thermodynamic data (5-7). However, if a reaction is to be evaluated, a formula must be constructed. According to Equation (2) reaction properties are additive from those of single substance properties on a stoichiometric basis. Because of the general arrangement of spreadsheets (such as grids where columns are represented by letters and rows by numbers), such operations can be carried out efficiently, fast and cleanly. In EQMIN, each single substance occupies a particular row with the thermodynamic data spread within its different cells. We name that the rank of the thermodynamic data of a given substance. Reactions involve always substances in different rows, so their ranks are added or subtracted following the stoichiometry of the reaction. Let us see an example. The polymorphic transition between andalusite and sillimanite involves these two phases, whose thermodynamic data are located in rows 10 (rank DlO to ARlO) and 110 (rank DllO to AR1 10). Column D contains ArH”(To, P”). So the formula = Dl l&DlO, located in column D of the page “reactions,” represents the standard enthalpy transformation andalusite = change of the sillimanite. at To, PO; column E accommodates ArGo(To, P”) and the formula = El l&El0 actually indicates A,G(T”, PO); and so on up to column AR. It is necessary to type the formula just once, because copying it to the adjacent cells makes the worksheet automatically update the column designator (i.e. the letter). To construct a reaction it is necessary to know the row number for each of the participating substances It is useful to associate to a particular row the name of the substance whose data it contains, to avoid having to remember row numbers. So every rank of thermodynamic data is linked to a name which is the full name of the corresponding substance and a symbol. Symbols are the acronyms given by Kretz (1983) although the list has been enlarged to substances not considered by this author. Full names of substances and their corresponding symbols are given in Appendix 3. The expected result of the calculations depends on the final purpose of the calculation itself. As a general procedure we provide A,G, equilibrium constants, and Clapeyron’s slopes of reactions. However, additional outputs may be generated by typing new formulae in the “output” page. EQMIN does not generate specific plots, for the same reason. It is left to the user to generate custom plots directly using the graphic tools offered by the spreadsheet. Figure 5 (A, B, C, D) shows an example of a selected graphic output for a problem related to the stability of Al,SiO, polymorphs. Any of the usual
644
J. D. Martin 4000
A
T= 1073.15K 2000
-6000 ' 0
2500
5000
I 7500
I loo00
Pressure(bar)
Pressure (bar) -2215000
10000 /
/
P= 7000 bar
-2225000 i 460
465
470
41s
480
Temperature (“C)
0
0
250
500
750
mo
Temnerature (‘0
Figure 5. Example of custom plots using results generated by EQMIN. A, Isothermal plot to look for stable-metastable equilibrium in three-phase, one-component system Al,SiO,. B, Same plot (A) under isobaric conditions. C, Standard Gibbs free energy of andalusite, sillimanite, and kyanite at 7 kbar and T. D, Polythermic-polybaric phase diagram of Al,Si05 system calculated with EQMIN.
geochemicalplots may be generatedwith amMicrosofte EXCELC spreadsheetand their construction also may be automated with macro programming (Christie and Langmuir, 1994; Sidder, 1994). The selectedresultsin the output area or in the zone of intermediateoperations allow the user to generate plots which range from any of the thermochemical parametersconsidered,aswell as their responses to changesof the statevariables,up to completephase diagrams.
([HI: - H&,]/Tand -[Gt - #&]/T, respectively)a+e sensitive to the thermodynamic properties of the elementsfrom which they arecalculated:if oneof the elementsfrom which the formation magnitudeof a substance(e.g. entropy of formation, free energy (,f formation, etc.) iscalculatedundergoesa phasetran\formation (melting, vaporization, symmetry changu, etc.) which is not reflected in the proper thermctchemicalparameter,it will propagatean error to the formation parametersof the substance.This is true also for substancesother than the elementswhit h undergo phasetransformations (magnetite,quartr. LIMITATIONS OF USE AND EXAMPLES hematite,etc.). EQMIN doesnot provide any type c,f From a thermodynamicpoint of view, the most model to calculate energy contributions due 1’~ important limitation to using EQMIN is related to i-type or any other type of phase transitio...l. the thermodynamicdata usedand their applicable Instead, we have fitted the heat capacity function 1,~ range. Accuracy of the enthalpy and Gibbs free the correspondingexperimental measurementsLo energyof formation valuesof substances at T, aswell to and from a temperatureas close as possible1’3 as their enthalpy and Gibbs free energy functions the transition point, which is satisfactory for rn0.t
EQMIN
645
Table 1. Standard state properties of reactions HM (6 Fe,O, = 4 Fe,O, + 0,), QFM (3 SO, + 3 Fe,O, = 3Fe,SiO, + 0,) and its difference with respect to first reaction (A(HM)), at 400°C and 2000 A,G”(F-,T”) (Jmol-‘1
V*(pST”)
S”(P”,T”) CPl
(JmW’K”)
CPZ
CP3
CP4
EXPI
EXPZ
COMPl
COMPZ
J bar-’
IogK
HM
415374
265.31
-5247.4
2729.4
960.3
-1016.4
-6.5
60.3
-0.6
-2. I
-0.35
-24.4
QFM
462696
244.88
3555.8
-1975.2
-644.3
684.1
2.1
-44.8
0.1
5.2
-1.79
-28.9
AWM)
-47322
20.43
-8803.2
4704.6
1604.6
-1700.5
-8.6
105.1
-0.7
-7.3
1.44
4.5
calculation purposes, according to Helgeson and others (1978), Anovitz and others (1993) and Saxena and others (1993). Many examples of the use of EQMIN could be given. A good starting point could be to change the standard temperature and pressure and observe the behavior of the thermodynamic parameters. Another typical application would be the location of an equilibrium point for a reaction in the P-T space: after setting up the reaction in the corresponding zone of the worksheet, temperature and pressure are fixed. By iteration of Tat constant P (or vice versa), we observe the change in the value of ArG(P, T). A sign switch in this value indicates that we have surpassed the equilibrium (perhaps because the increment in P or T is too big). When ArG(P, T) = 0, or close to that value, the equilibrium condition (A,G’(P, T) = 0 or AGleacfantS= AGpraducts) is satisfied. With more than one reaction it is possible to draw a complete phase diagram. Similar calculations may be
I 0
I
1
900
loo0
1 1100
1200
L
I
1300
I
1400
Temperature (“C) Figure 6. Plot showing procedure followed to constrain standard Gibbs free energy of grossular from experiments by Goldsmith (1980) and Koziol and Newton (1988) for reaction 3 anorthite = grossular + 2 kyanite + quartz, assuming that the rest of the thermodynamic parameters are known. Solid circles indicate grossular stable; empty circles indicate the stability of anorthite; crosses indicate no reaction. Equilibrium point we chose was at approximately midpoint of experimental bracket (1200 C and 26,250 bar), although it could have been selected at any other reasonable point within any of the reversals. With the Gibbs free energy of grossular estimated, we calculated locus of reaction to compare agreement with rest of experimental reversals.
performed under isobaric, isothermic, and isobaricisothermic conditions, giving rise to temperature activity, temperature-composition, pressure-activity, pressure composition, and activity-activity diagrams. Specific calculations such as the normalizedto-a-reaction diagrams (for example, A log fO,(HM); HM is the hematite-magnetite oxygen buffer) also may be done easily by writing the reaction from which the rest will be normalized in the proper area of the worksheet (page “reactions”), like any other reaction. The rest of the reactions only have to be subtracted from the first (Table 1). Retrieval of thermodynamic data from phase equilibria experiments may be of particular interest as a preliminary exercise to develop a self-consistent thermodynamic dataset. The complexity of the problem is related directly to the amount of information we have about the phase of interest and how restrictive are the criteria used to evaluate the quality of the data retrieved. In the most simple situation, just one reversal which would allow us to compare the Gibbs free energy or enthalpy of an unknown phase can be considered, provided that all the other thermodynamic parameters of the problem phase and the rest of the participating substances are known. This is done by following the steps: (1) Arbitrarily enter the value 0 to the cell that holds the ArGo(Po, To) of the problem substance. (2) Construct the reaction in the page “reactions.” It is convenient to assign a stoichiometric coefficient of 1 to the problem substance. (3) Use the reversely bracketed equilibrium point to fix T and P within it. At this point A,G must be 0 because the Gibbs free energy of reactants and products cancel out. (4) Under those conditions, the direct reading of the A,G in the output area (which is nonzero of course) will give us the A,G”(Po, To) of our problem substance. (5) Copy this value to the cell we assigned previously a zero value. A,G(P, T) then, obviously, becomes 0, which fulfils the desired and established equilibrium condition. The uncertainty region defined by the experimental bracket also is a measure of the uncertainty of the thermodynamic quantity evaluated. So the thermodynamic magnitude may be estimated accurately. If more than one reversal is available, it is possible to have a better constraint of the thermochemical parameters to estimate. Moreover, if we have a set of experiments for the same reaction which define its Clapeyron’s slope, we can evaluate also the entropy and heat capacity of a problem substance. If instead of a single reaction we
646
J. D. Martin
have a set of reactions experimentally studied, we will he one step ahead on the way to making a self-consistent thermodynamic dataset. Figure 6 shows an example of the aforementioned procedures. Manipulation, selection, evaluation, and judgement of experimental quality are not simple tasks and instructor advice is of major importance at this point. EQMIN, although primarily designed to he used in a Lotus 1230 2.3 spreadsheet in DOS environments, evolved with time to the Microsoft’ Excel0 5.0 spreadsheet, under Microsoft@ Windows” 3.1. EQMIN has heen exported successfully to other PC spreadsheets (Borland% Quattro Pro0 2.0) and Apple Macintosh@ computers. EQMIN was developed in a 486-DX-2, 33 MHz PC computer with 4 Mbyte RAM. Real system requirements for PC computers are less restrictive, in particular for the Lotus 123O version. The current size of the EQMIN worksheet for EXCEL0 5.0 is about 800 kbytes (420 kbytes the Lotus 123c version ) with 122 mineral phases, 39 chemical elements, and 20 fluid species (see Appendix 3).
AVAILABILITY
EQMIN is available through FTP either in Lotus 123c 2.1 or Excel0 5.0 formats by request to [email protected], or from the server at IAMG.ORG. Alternatively, the author will supply a copy of the worksheet after receiving a nonformatted 3.5 in DD or HD floppy disk together with a selfaddressed, stamped envelope at: Departament de Cristal.lografia, Mineralogia i Dip&its Minerals, Universitat de Barcelona; C/Marti i Franquirs s/n, 08028 Barcelona, Spain. Acknowledgments-The original manuscript has been improved by the advice and comments of Dr C. Ayora and Dr S. Gali. Thanks are also given to Dr Helen M. Lang and two anonymous reviewers for their kind suggestions. This work has been financed through the project AMB93-0326 of the Spanish Comisibn Interministerial de Ciencia y Tecnologia (CICYT).
REFERENCES
Anovitz, L. M., and Essene, E. J., 1987, Compatibility of geobarometers in the system CaO-FeO-Al,@Si02-TiO, (CFAST): implications for garnet mixing models: Jour. Geology, v. 95, no. 5, p. 633645. Anovitz, L. M., Essene, E. J., Metz, G. W., Boblen, S. R., Westrum, E. F. Jr, and Hemingway, B. S., 1993, Heat capacity and phase equilibria of almandine, Fe,AlzSi,O,,: Geochim. Cosmochim. Acta, v. 57, no. 17, p. 4191-4204. Berman, R. G., 1988, Internally-consistent thermodynamic data for minerals in the system Na,O-K,O-CaO-MgOFeO-Fe*O,-A&O,-SiO,-TiO,-H,O-CO,: Jour. Petrology, v. 29, no. 2, p. 445-522. Berman, R. G., 1991, Thermobarometry using multiequilibrium calculations: a new technique, with petrological^__applications: Can. Mineralogist, v. 29, no. 4, ^.. p. n33455.
Birch, F., 1966, Compressibility: elastic constants, in Clark., S. P. Jr, ed., Handbook of physical constants: Geol. Sot:. Am. Mem. 97, p. 97-173. Brace, W. F., Scholz, C. H., and La Mori, P. N., 196(,. Isothermal compressibilities of kyanite, andalusite, and sillimanite from synthetic aggregates: Jour. Geophyi. Res., v. 74, p. 2089-2098. Brown, T. H., Berman, R. G., and Perkins, E. H., 198’4, PTA-SYSTEM: A GeO-Calc software package for the calculation and display of activity-temperature-pressure phase diagrams: Am. Mineralogist, v. 74, no. 3-1, p. 485487. Chase, M. W. Jr, Davies, C. A., Downey, J. R. Jr, Frurip, D. J., McDonald, R. A., and Syverud, A. N., 198.5, JANAF thermochemical tables (3rd ed.): Jour. Phyi. Chem. Ref. Data, v. 14-1, 1856 p. Christie, D. M., and Langmuir, C. H., 1994, Automated X‘\ plots from Microsoft Excel: Computers & Geoscience,. v. 20, no. 1, p. 47-52. Connolly, J. A. D., 1990, Multivariate phase diagrams: an algorithm based on generalized thermodynamics: Am. Jour. Sci., v. 290, no. 6, p. 666-718. Cox, J. D., 1979, Manual of symbols and terminology for physicochemical quantities and units. Appendix 17;: Notation for states and processes, significance of the word standard in chemical thermodynamics, and rl:marks on functions used in thermodynamic tables: Pure and Applied Chemistry, v. 51, p. 393403. Delgado, J., 1995, An integrated thermodynamic model for sphalerite geobarometry from 300 to 800°C and up 113 10,000 bar (abst.): Boletin Sot. Espaiiola Mineral., v. 1k, no. 2, p. 4142. Engi, M., 1992, Thermodynamic data for minerals: a critical assessment, in Price, G. D., and Ross, N. L., eds., the stability of minerals: Mineralogical Society Series, Chal’man & Hall, London, p. 267-328. Froese, E., and Gunter, A. E., 1976, A note on the pyrrhotite-sulfur vapor equilibrium: Econ. Geology, )I. 71, no. 8, p. 1589-1594. Goldsmith, J. R., 1980, The melting reactions of anorthire at high pressures and temperatures: Am. Mineralogist, I/, 65, no. 34, p. 272-284. Halbach, H., and Chatterjee, N. D., 1984, An interna1.y consistent set of thermodynamic data for twentyone CaO-A120,-SiO,-H,O phases by linear parametric pr’ Igramming: Contrib. Mineral. Petrol., v. 88, no. l-2, p. 14-23. Helgeson, H. C., Delany, J. M., Nesbitt, H. W., and Biril, D. K., 1978, Summary and critique of the thermodynamic properties of rock-forming minerals: Am. Josr. Sci., v. 278-A, 229 p. Hemingway, B. S., Robie, R. A., Evans, H. T. Jr, and Kerrick, D. M., 1991, Heat capacities and entropies If sillimanite, fibrolite, andalusite, kyanite, and quartz and the AI,SiO, phase diagram: Am. Mineralogist, v. 76, no. 9-10, p. 1597-1613. Holland, T. J. B., and Powell, R., 1990, An enlarged arid updated internally consistent thermodynamic dataset with uncertainties and correlations: The system K,ONa,0-Ca0-Mg0-Mn0-Fe0-Fe,O,-Al,O,-TiO,-SiCJ,~C-HZ-O,: Jour. Metamorphic Geology, v. 8, no. j, p. 89-124. Johnson, J. W., Oelkers, E. H., and Helgeson, H. C., 1992, SUPCRT92: A software package for calculating the standard molal thermodynamic properties of minera.s, gases, aqueous species, and reactions from 1 to 5000 bdr and 0 to 1000°C: Computers & Geosciences, v. 18, nr). 7, p. 899-947. Kerrick, D. M., and Jacobs, G. K., 1981, A modified Redlich-Kwong equation for H,O, CO,, and H,O-I?,(:), mixtures at elevated pressures and temperatures: Am. Jour. Sci., v. 281, no. 6, p. 735-767.
EQMIN Koziol, A. M., and Newton, R. C., 1988, Redetermination of the anorthite breakdown reaction and improvement of the plagioclase-garnet-AI,SiO,-quartz geobarometer: Am. Mineralogist, v. 73, no. 34, p. 216-223. Kretz, R., 1983, Symbols for rock-forming minerals: Am. Mineralogist, v. 68, no. 2, p. 277-279. Metz, G. W., Anovitz, L. M., Essene, E. J., Bohlen, S. R., Westrum, E. F. Jr, and Wall, V. J., 1983, The heat capacity and phase equilibria of almandine (abst.): EOS. Trans. Am. Geophys. Union, v. 64, p. 347. Moecher. D. P.. Essene. E. J.. and Anovitz. L. M.. 1988, Calculation ‘and application of clinopyroxene-garnet: plagioclase-quartz geobarometers: Contrib. Mineral. Petrol., v. 100, no. I, p. 92-106. Parkhurst, D. L., Thorstenson, D. C., and Plummer, L. N., 1980, PHREEQE: a computer program for geochemical calculations: U.S. Geol. Survey Water Resources Invest. Rept., no. 80-96, 210 p. Perkins, D., Essene, E. J., and Wall, V. J., 1987, THERMO: a computer program for calculation of mlxedvolatile equilibria: Am. Mineralogist, v. 72, no. 34, p. 446447. Perkins, E. H., Brown, T. H., and Berman, R. G., 1986, PT-SYSTEM, TX-SYSTEM, PX-SYSTEM: three programs which calculate pressure-temperaturecomposition phase diagrams: Computers & Geosciences, v. 12, no. 6, p. 749-755. Powell, R., and Holland, T. J. B., 1985, An internally consistent thermodynamic dataset with uncertainties and correlations: 1. Methods and a worked example: Jour. Metamorphic Geology, v. 3, no. 4, p. 327-342. Reed, M. H., and Spycher, N. F., 1988, Chemical modeling of boiling, condensation, fluid-fluid mixing and water-rock reaction using programs CHILLER and SOLVEQ (abst.): Am. Chemical Sot. National Meeting, 1976th (Los Angeles, California) Sept. 25-30, abstract no. 26. Robie, R. A., and Hemingway, B. S., 1984, Entropies of kyanite, andalusite, and sillimanite: additional constraints on the pressure and temperature of the Al,SiO, triple point: Am. Mineralogist, v. 69, no. 34, p. 298-306. Robie, R. A., Hemingway, B. S., and Fisher, J. R., 1979, Thermodynamic properties of minerals and related substances at 298.15 K and 1 bar (lo5 pascals) and at higher temperatures: U.S. Geol. Survey Bull. 1452, 456 p. Robinson, G. R. Jr, Haas, J. L. Jr, Schafer, C. M., and Haselton, H. T., 1982, Thermodynamic and thermophysical properties of selected phases in the MgO-Si02-H, O-CO,, CaO-Al, O,-SiO,-H, 0-CO2 , and Fe-FeO-Fe,O,-SiO, systems, with special emphasis on the properties of basalts and their mineral components: U.S. Geol. Survey Open-File Rept. 83-79, 429 p. Saxena, S. K., Chatterjee, N., Fei, Y., and Shen, G., 1993, Thermodynamic data on oxides and silicates: an assessed data set based on thermochemistry and high pressure phase equilibrium: Springer-Verlag, Berlin, 428 p. Sidder, G. B., 1994, PETROCALCPLOT, Microsoft EXCEL macros to aid petrologic interpretation: Computers & Geosciences, v. 20, no. 6, p. 1~41-1061. Skinner, B. J.. 1966. Thermal exoansion. in Clark. S. P. Jr. ed., Handbook of Physical Constants: Geol. Sot. Am: Mem. 97, p. 75-96. Taylor, D., 1987, Thermal expansion data: XI. Complex oxides AB,O, and the garnets: Br. Ceram. Trans. Jour., v. 86, no. -1, p. 148-153 Wang, H., and Simmons, G., 1972, Elasticity of some mantle crystal structures: 1. Pleonaste and hercynite spine]: Jour. Geophys. Res., v. 77, p. 43794392. Wolery, T. J., Jackson, K. J., Bourcier, W. L., Bruton, C. J., Viani, B. E., Knauss, K. G., and Delany, J. M., 1990, Current status of the EQ3j6 software package for geochemical modeling, in Melchior, D. C., and Basset, R. CAGE0
22,G-c
647
L., eds., Chemical Modeling of Aqueous Systems: Am. Chem. Sot. Symposium Series, v. 2, no. 416, p. 104-l 16.
APPENDIX
1
Main calculations performed by the EQMIN
worksheet
(1) Heat capacity function of phase i-@(T) C lo2 + D lo5 CpO(i,PO,T)=A +B.lO-‘T+--p. JF T2 (2) Isobaric thermal expansion function AI”&
PO, T) = E(T - To). lO-4 +F(T*
- To*). lo-*.
(3) Isothermic compressibility function AV’(i,
P, T”) = G(P
-PO)
10m5+ H(P’-
P@)
lo-“.
(4) Molar volume at P, T of phase i V(i, P, T) = p(i,
PO, To) +
AVO(i, PO, T) - AV’(i,
P, To).
(5) Standard molar enthalpy of formation at PO, To of phase i ArHo@, PO, To) = A,GO(i, PO, To) + To. ArSO(i, PO, To). (6) Standard molar enthalpy change from To to T of phase i AH’(i, PO, T) =
’ Q(T)
s TO =A(T-To)+B~10-3(T2-To’)
d’f
2
(7) Standard molar entropy change from To to T of phase i ASO(i, PO, T) =
= Gw ~ s ro
dT T
(8) Standard molar Gibbs free energy change from To to of phase i
T
AG’(i, PO, T) = S’(i, PO, T’)(T A ,ln(T’)+
10-3BTo-p
- To)
102C IO’D
2
f-
p
2T”’
1
.(T-To)
.4(TlnT-T)+~-4.102,/‘?+~]‘. TO
(9) Standard molar Gibbs free energy change from P” to of phase i
P
AGO(i, P, To) = V”(i, PO, T’)(P + [10-4E(T - To) -
+ IO-*F(T*
10-5G(P2 - PO’) 2
- PO) - To2)] .(P -PO)
IO-“H(P3 -
- P@) 3
J. D. Martin
648 (10)
Apparent
A,G(i,
molar
Gibbs
P, T) = A,GO(i, + AG”(i,
(I 1) Equilibrium
Standard
A,S’(i, (13)
WV’, -RT.
entropy
(21)
of formation
Fluid
Standard
molar
entropy
lnf
PO, TO).
(22)
(23)
T)
Mixing
Mixing
(24)
free energy
- @-
function
HTo) = _ 1 T
(16)
Standard
ArHo&
A,G’(i,
Hm)
’1
molar
enthalpy
molar
Gibbs
PO, T) = A,H’(i,
of formation
at T of phase i
Go2
free energy
of formation
k%,crco, coefficient
X&o
(%OtH*O
H,O
activity
(27)
CO,
activity
(28)
Molar
n S’(i, PO, T) - c v,S’(elementj, ,=I [ (18) Polynomial “a ” to calculate H,OCO, (Holland and Powell, 1989) a=a,+a,P+a,P2+a.,Pm’+a,P-* to calculate 1989)
PO, T)
1
(%Or”*O
- %*ckco2))
2&0,~ hLLco2
- %OrH20))
of CO, +
fraction
a co2 - xcoz
Yco?
of CO,
Gibbs
= 1 - X”,O’
free energy
T) = 1 v,A,G’(i,
of reaction
PO, To)
A,S,dT+RTln (30)
Clapeyron’s
slope
fugacities
cell description
~&cl
YH*O
fugacities
APPENDIX Selected
+
.
(P=kbar). Hz&CO,
A,G(P,
of H,O
a H*O - X&O
xc,, Molar
2
of the EQMIN
(P = kbar).
RT
(26)
(29)
+ 0.38P
RT
at T of
PO, T)
(P = kbar).
w(CO,-H,O)
coefficient
Activity
In ko2 =
(P = kbar).
1000
+ 0.26P
= 17.8 - 0.014T
PO, To)
- T.
(19) Polynomial “b” (Holland and Powell,
=
_ SC+, PO T)
T
PO, T) = A,HO(i,
(17) Standard phase i
(25)
(HT-
(Hol-
w(H,O-CO,)
parameter
Activity
PO, T). In ho
Gibbs
fugacities
+ c,P-’
= (a + bT + cT2). RT
parameter
c+OrH~O
at T of phase i
So& PO, T) = S’(i, PO, To) + ASo& (15)
(P = kbar:.
fugacity
of phase i
function HTo) = AH”(Po, T
H,Q-CO,
+ c,P-’
T)
0 uzocO~ = 8.3 - 0.007T
T
b6P-’
c=r,+c2P+c,P~2+CqP~“2
PO, To) = S”(i, PO, To) - i SO(elementj, ,=I
Enthalpy
(20) Polynomial “c” to calculate land and Powell, 1989)
of a reaction
T)=
W(14)
+b,P-‘12+
PO, To)
constant
molar
b=b,+b,P+b,P-‘+b,P-2
at T of phase i
PO, T) + AGO(i, P, To) + RT In CI,.
InK,(P, (12)
free energy
workbook
at P, T
649
650
J. D. Martin
Pergamon PII: SOO98-3004(96)00006-4
Vol. 22, No. 6, pp. 639450, 1996
Copyright Q 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0098.3004/96 $15.00 + 0.00
EQMIN, A MICROSOFT@ EXCEL0 SPREADSHEET PERFORM THERMODYNAMIC CALCULATIONS: DIDACTIC APPROACH
TO A
JORDI DELGADO MARTiNt Departament de Cristal.lografia, Mineralogia i Dip&its Minerals, Universitat de Barcelona, Marti i Franquis s/n, 08028 Barcelona, Spain (e-mail: [email protected]) (Received
27 September
1995; revised
23 November
1995)
Abstract-Knowledge of the basic laws of thermodynamics is of prime importance in understanding processes occurring in the Earth at any place and at any scale. Calculations related to the construction of phase diagrams, petrogenetic grids, geothermobarometry, etc. rely heavily on the proper manipulation of thermodynamic parameters. EQMIN is a Microsoft e Excel” worksheet constructed to help students learn the basic thermodynamic relationships and calculus routines. It is a flexible system with which it is possible to evaluate from the basic thermodynamic parameters almost all the thermodynamic functions in frequent use, as well as creating phase diagrams, retrieval of thermodynamic data from phase equilibria experiments, etc. Copyright Q 1996 Elsevier Science Ltd Ke.y
Words: Thermodynamics, Phase equilibria, Spreadsheet, Mineralogy, Petrology.
INTRODUCTION
the past two decadesan increasingnumberof programsto perform geochemicallyrelevant thermodynamic calculations have become available, for example SUPCRT (Helgeson and others, 1978), PHREEQE (Parkhurst, Thorstenson,and Plummer, 1980), EQ3/6 (Wolery and others, 1983), PTX (Perkins, Brown, and Berman, 1986), THERM0 (Perkins, Essene,and Wall, 1987), PTA (Brown, Berman, and Perkins, 1989),CHILLER (Reedand Spycher,1988),THERMOCALC (Holland andPowell, 1990), VERTEX (Connolly, 1990), TWQ (Berman, 1991),SUPCRT92(Johnson,Oelkers,and Helgeson,1992),amongmany others. They resulted from the needto avoid cumbersomerepetitive operations and to assess self-consistentthermodynamic databases usefulfor calculationsrelatingto problems ranging from the Earth surfaceto the lower crust. Although there are differencesin the philosophy behind each program and their P-T-x applicability range,all usethe samebasicthermodynamicrelationships becauseall are basedstrictly on the laws of thermodynamics.All theseprogramswerewritten in a computer language (FORTRAN, PASCAL, BASIC, etc.) and then compiled to offer a compact presentation,making their useeasier.This approach proved to be a successas shown by the latest programs, which are more user friendly and powerful
During
tNew address: E.T.S. Ingenieros de Caminos, Universidad de La Co&a, Campus de Elvitia, s/n, 15192 La Coruda, Spain.
than ever before. However, from a didactic point of view, computer programspresentseveralhandicaps that prevent themfrom beingthe ideal teachingtool. First, if the final purposeis to learn how a program performs its calculations, studentsunfamiliar with computer languageswill have difficulties reading a medium-sizesource code, a problem even for a trained eye. Moreover, if the source code is not available, the program becomesa “black box” and the output data has to be believed dogmatically. Second,the thermodynamicdata for eachphaseto be consideredin a reaction are not available freely becauseusuallythey are held in a parallel file. So, if it is necessaryto evaluate the effect of taking into account someeffect or to modify one or several reaction parameters,it is necessaryto exit the program,call an editor to changethe data and then rerun the application. Finally, output data usually are constrainedin a way that to obtain the required output it is necessaryto perform off-line calculations with the raw data given by the program. Most of these limitations arise becausethe programs have beenconceivedprimarily for researchand only incidentally for teaching. We have tried to harmonizeteachingand research needsby a new approachto automatedthermodynamiccalculations.This hasbeendone with the aid of a commercial spreadsheet.This type of software offers several advantages over research-oriented thermodynamicprograms:(1) for peopleunfamiliar with programming languages they constitute a smooth transition from hand calculatorsand, with minimum training it is possibleto start to work
639
J. D. Martin
640
almost immediately; (2) spreadsheets are popular and are available in many departments, or are available readily; (3) portability problems are almost non-existent because the commercial firms ensure compatibility among the most successful spreadsheets; (4) thermodynamic data, results, and any intermediate operations are shown on the screen together with graphic, printing, and other utilities; (5) as the thermodynamic parameters are available to the user at the same time the problem is solved, it is possible to modify them in order to maximize or minimize effects induced by conventional assumptions or approximations.
In this expression a is the activity of a mineral component (a = 1 if its composition is equal to tha: of the ideal end-member composition at any P, T) The Gibbs free energy of any balanced reaction ma] be calculated from the stoichiometric addition of the individual properties of participating phases A,G(P,
T) = c v,ArGo(i,
PO, To)
A,$
+RTln WORKSHEET
DESCRIPTION
The system we describe in this paper has been conceived to develop the necessary student skills to perform any of basic thermodynamic calculations. EQMIN uses similar expressions for the heat capacity, expansivity, and compressibility functions as the program THERM0 (Perkins, Essene, and Wall, 1987), although intermediate calculations are more extensive. The primary purpose of EQMIN is to evaluate the Gibbs free energy of substances at any pressure (P) and temperature (T) from conventional reference values (P”, To) using an integrated form of the Gibbs-Duhem equation A,G(P,
T) = A,G”(Po,
+ j;AfVdP
To)
-ITI
ArSdT+RTlna.
(1)
1173 15 Starting
iteration temperature= Temperature increment= Working pressure= Starting iteration pressure= Pressure increment= X(&O)= X(CO2)= A$
tolerance
limiti
800 10 5000 1
100
Figure 1. Aspect of page “Input”
flu; Cl
(21 >
where i represents the ith mineral phase in thl: reaction, v is the stoichiometric coefficient of the ith phase in the reaction, a, is the activity of the ith phase in the reaction, and A,G(P, T) is the Gibbs free energy value of the reaction at the P, T of interest. For a solid-fluid reaction the volume integral con. siders just the solid phases, and we have to add tb: term v,RT Inf, (where j is a fluid species, and v and f the corresponding stoichiometric coefficient and fugacity) to account for the volumetric change from P” to P of fluid j. EQMIN is an EXCEL” workbook arranged ill several interrelated pages. All the pages have a table format. Page 1 (termed “constants”) states the refer, ence temperature (To) and pressure (PO) and som’. other basic constants used by the different equation (gas constant, R; conversion between calories ancl joules; etc.). Page 2, named “input” (Fig. 1), contain<
K
lOOO/Tm=
0 852
“C “C
‘bar bar 100 bar 0.5 05
dT
050MPa
Jmo?
of workbook EQMIN. It is used to state operating conditions for calculations.
EQMIN DlO
641
-3*harc+3*siMm-S*cm
ti.i
1AIDIE(GiIIJIKILlMIP1Q
-lR eactions 1
ArWae, ( J mor)
A@ww
&%a)
CPl
cP2
cP3
cw
Expansivity EXPI EXP2
3.36 162.63 4.01 12.60 8.59
3.36 162.63 4.01 12.60 8.59 43.88
28.01 47 02 2.89 076 -2.13
-1089 -65.67 -0.31 0.22 0.54
-5.70 -5.69 -1.29 -1.05 0.24
9.62 0.49 5.38 6 80 1.42
0.10 0.07 -1.88 -2 13 -0.26
457.81
19944
-225.89
-0.99
cm3mop
(J ml
-1895.02 -2896.80 90823.48 3805.58
s%Y
42335.35 2610.00
7676.69 3920 00 3871.11 1310.00 127989021 14905.60
43.88
-1037.51
0.12
-1151 0 58 2 53
1 95 13.07
Figure 2. Page “Reactions,” which is used by EQMIN to introduce formulas which represent chemical reactions. Note, in upper part of figure, formula “ = 3*herc + 3*sil-alm-S*crn” in cell DlO (A#(p’, TO)) which represents reaction of interest. Here, sil, alm, and cm are the symbols associated with the thermodynamic data ranks of hercinite, sillimanite, almandine, and corundum, respectively. In this example, EQMIN works simultaneously with six different reactions.
the input parameters for the calculations (P, T, and X,,,) together with five additional entries (starting iteration temperature and pressure,temperatureand
pressureincrement,and a cut-off limit for the calculation) usefulif an automatedcalculationwith macro programmingispreferred.Page3 (“reactions”) isthe areaselectedto makeexplicit the reactionsof interest (Fig. 2). Each reaction occupies a single line which follows a column arrangementidenticalto that of the thermodynamic data of pages 5-7 (“elements,” “minerals,” and “gases”), EQMIN is not constrained to the calculation of one reaction at a time. An unlimited number of them can be evaluated simultaneously, taking care to add the samenumber of output areasasreactions.In page 4 (“output”), we provide an output areafor the calculationswhich are the Gibbs free energyof the reaction[A,G(P, T)] and its translation in terms of the equilibrium constant: K(P, T), logK(P, T), and In K(P, T) (Fig. 3). Pages5-7 (“elements,” “minerals,” and “gases”) contain the basic thermodynamic properties of elements,minerals, and gasesand liquids, respectively. Figure 4 showsan example of the aspectof the “minerals” page. The thermodynamic parameters compiled are: A,H”(Po, To), AfGo(Po, TO), S”(Po, To), ArSo(Po, To), coefficientsfor the constant pressureheat capacity (A, B, C, D), isobaric thermal expansion(E, F) and isothermiccompressibility (G, H) functions and molar volume [V’(P”, To)] at the standard pressureand tempera-
ture. All these values are referred to their corresponding sources(p. 10 and 11)with their applicableP-T range, when appropriate. Together with the basic data, each page contains also the result of someintermediatecalculationsat the T, P selected in the input area, for example,the volume increase of the phasefrom PO, To to the P and T of interest, heat capacity at T together with the enthalpy, entropy, and Gibbs free energy increasefrom To to T, and other thermodynamic functions [(HtSpecific equations are f%)/T; - (G! - H&,)/T]. given in Appendix 1. Page 8 (“eos”) is used for the calculation of H,O and CO, end-member fugacities,aswell astheir nonidealmixing, according to the polynomial expressionsgiven by Powell and Holland (1985),updatedin Holland and Powell (1990). On page 9 (“ac-models”), activity models for solid phasesare entered.Pages10 and 11 (“a . . z” and “1 . . n”) are used to keep a record of the sourcesfrom which the thermodynamic data have been collected or a brief description of their method of estimation. Finally, page 12 (“info”) provides a list of the substancefull name, formula, and symbol currently presentin EQMIN. More detailed cell descriptions are given in Appendix 2. The referencestatesusedare the elementsin their stable statesat 1 bar and 298.15K. The standard statefor fluids is pure fluid at 1 bar and T, and for solidphasesthe correspondingstandardstateis that
642
1
J. D. Martin
output
2 3
4
A.Gm
5 6 7 8 9
A&p) ArG,n
110 11
dp,dT
(Jmol-I)= (JmoT’)= (JmoT’)= KF logQG)=
R. 1 973.47
R. 3 -1716.37
2115.35
-6677.07
-90408.47
-5703 60 1.79E+OO 0.25
-9585.74 2.67E+OO 043
-547.72 -226408
-6723.06 -4607.71
1.26E+OO 0.10 .“x’5:
(k:$$%-/
dT/dP (Y? kbaf’)=
R. 4
R. 2 -10604 67
-::;* 41.90
-15 54
R. 5 3831.72 -6175 34
R. 6 14512.17
R. 7 0 00
R. 8 0.00
-22685.29
-2343.62
-8173.12
0.00 0.00
0.00 0.00
1.6OE+OO 0.21
127E+OO 0 10
2.31E+OO 0.36
l.OOE+OO 0 00
l.OOE+OO 0.00
:;;o
;:;1
:;;7
#iDN/O! O.OO
37.17
95 11
56.90
#iDN/O!
#iDN/O! O O0 #iDN/O!
-117.50
12 13 14
15 16
17 18 19
Figure 3. Summary screen, in the page termed “Output,” where selected output for each of reactions processed are shown. There are output data for six different reactions (see Fig. 2). Warning message #i DIVjO! reflects absence of more than six reactions in example considered.
of the pure substance at the pressure and temperature of interest (Robie, Hemingway, and Fisher, 1979; Cox, 1979; Chase and others, 1985). The thermodynamic data given in the worksheet were collected from a variety of sources (Robie,
1.0 10 10
-5261237.0 -2589899.1 -5774860.6
Hemingway, and Fisher, 1979; Robinson and other;. 1982; Halbach and Chatterjee, 1984; Chase and others, 1985; Moecher, Essene, and Anovitz, 198X: Anovitz and others, 1993; etc.). They do not constitute a self-consistent thermodynamic database and
-4938260
0
-2441720 -54320000
0
10 -5161365.2 -47978369 1.0 -4237774 1 -4012000.0 10 -12069957 1 -113433000 1.0 -1207430.4 -1127793.0 0.9 -120359.7 -124900.6 10 -256115.0 -255960.0 10 -4344733.7 -4077481.0 10 -135186.3 -132658 0
(z, @zz) (31) (89 c,~s,
342.60 91.39 316.82 373.60 199.30 (49) 534.50 0 87.99 (KC’) 110.00 (6s) 245.00 (40) 213 68 (66) 20040
(9 (22) (~2) (63) (1) (49) 0 (106) (60 (so) cr,
CPl
cF2
-1083 27 -497 00 -114996 -1219.28 -757.25 -2437.22 -267 11 15.23
800.44 277.30 981.27 58495 8747 1259.82 81.62 -292.69
-57.01 -6.59 -134.68 74.76 114.92 18.93
-896.37
280.27
Figure 4. Example of Part of page “Minerals” where thermodynamic data of solid substances elements are compiled. Tables with same format have been constructed for chemical elements and gases. See text and Appendix 2 for more details.
190 51
other than and liquids
.’
EQMIN so, depending on the purpose of the calculations, care should be taken to consult the original works for limitations on data use. The use of self-consistent data-bases (Holland and Powell, 1990; Berman, 1988; for example), was not considered a critical point in setting up EQMIN. In spite of the obvious benefits accessible through their use (see detailed discussions in Powell and Holland, 1985; Berman, 1988; Engi, 1992; and references therein), we preferred an eclectic approximation for the following reasons: (1) it is possible to modify any of the data records without disturbing the rest of the dataset and to enlarge the database to an indefinite size at any moment; (2) EQMIN is not constrained to a limited number of system components and/or phases, similar to many of the self-consistent databases, so, EQMIN is applicable to a much more wider geological context; (3) everyday experience tells us that we need to retrieve, estimate, refine, and compare thermodynamic data for a great variety of substances and this can be done efficiently with the aid of EQMIN through the incorporation of new phases, trial and error procedures, etc. It is possible, however, to incorporate into EQMIN any of the self-consistent databases. This task is left to the discretion of every instructor. The use of Powell and Holland’s (1985) polynomials to calculate fluid fugacities in the system H,O-CO, obeys its simple mathematical formulation. It is compatible fully with the experimental data currently available for H20, CO, and their mixing according to the equation of state of Redlich-Kwong, modified by Kerrick and Jacobs (1981). Limitations on the pressure-temperature ranges for which these equations apply are quoted in Powell and Holland (1985) Holland and Powell (1990) and Kerrick and Jacobs (1981). At present no other fugacity coefficients can be calculated with EQMIN, although it would be a relatively simple task to add some other equations of state to handle in a more realistic way with the volumetric properties of other gases and their mixtures. EQMIN can perform calculations with component activities different from 1. This can be achieved by modifying the default value of 1 given in every row of column 2 of pages 5-7. Fugacity values for H,O and CO2 are set automatically by the worksheet as far as a fluid composition has been provided in the input area. For other gases, fugacity values must be fixed or supplied through additional equations of state. EQMIN does not implement any kind of general activity model for minerals. Nevertheless, particular activity models (ideal or nonideal) may be merged to EQMIN for substances of interest, at the instructor’s convenience. We provide a couple of examples of two-constant (asymmetric) solid solution models with EQMIN: one for the FeS-ZnS system (Delgado, 1995) and the other for the Fe vacants in pyrrhotite (Froese and Gunter, 1976).
643 WORKSHEET
OPERATION
Operation of the worksheet is related to the final purpose of the calculations. If it is desired to evaluate the individual properties of a single substance, by fixing a T and P, we only need to read the results in the corresponding cells of the pages that contain the thermodynamic data (5-7). However, if a reaction is to be evaluated, a formula must be constructed. According to Equation (2) reaction properties are additive from those of single substance properties on a stoichiometric basis. Because of the general arrangement of spreadsheets (such as grids where columns are represented by letters and rows by numbers), such operations can be carried out efficiently, fast and cleanly. In EQMIN, each single substance occupies a particular row with the thermodynamic data spread within its different cells. We name that the rank of the thermodynamic data of a given substance. Reactions involve always substances in different rows, so their ranks are added or subtracted following the stoichiometry of the reaction. Let us see an example. The polymorphic transition between andalusite and sillimanite involves these two phases, whose thermodynamic data are located in rows 10 (rank DlO to ARlO) and 110 (rank DllO to AR1 10). Column D contains ArH”(To, P”). So the formula = Dl l&DlO, located in column D of the page “reactions,” represents the standard enthalpy transformation andalusite = change of the sillimanite. at To, PO; column E accommodates ArGo(To, P”) and the formula = El l&El0 actually indicates A,G(T”, PO); and so on up to column AR. It is necessary to type the formula just once, because copying it to the adjacent cells makes the worksheet automatically update the column designator (i.e. the letter). To construct a reaction it is necessary to know the row number for each of the participating substances It is useful to associate to a particular row the name of the substance whose data it contains, to avoid having to remember row numbers. So every rank of thermodynamic data is linked to a name which is the full name of the corresponding substance and a symbol. Symbols are the acronyms given by Kretz (1983) although the list has been enlarged to substances not considered by this author. Full names of substances and their corresponding symbols are given in Appendix 3. The expected result of the calculations depends on the final purpose of the calculation itself. As a general procedure we provide A,G, equilibrium constants, and Clapeyron’s slopes of reactions. However, additional outputs may be generated by typing new formulae in the “output” page. EQMIN does not generate specific plots, for the same reason. It is left to the user to generate custom plots directly using the graphic tools offered by the spreadsheet. Figure 5 (A, B, C, D) shows an example of a selected graphic output for a problem related to the stability of Al,SiO, polymorphs. Any of the usual
644
J. D. Martin 4000
A
T= 1073.15K 2000
-6000 ' 0
2500
5000
I 7500
I loo00
Pressure(bar)
Pressure (bar) -2215000
10000 /
/
P= 7000 bar
-2225000 i 460
465
470
41s
480
Temperature (“C)
0
0
250
500
750
mo
Temnerature (‘0
Figure 5. Example of custom plots using results generated by EQMIN. A, Isothermal plot to look for stable-metastable equilibrium in three-phase, one-component system Al,SiO,. B, Same plot (A) under isobaric conditions. C, Standard Gibbs free energy of andalusite, sillimanite, and kyanite at 7 kbar and T. D, Polythermic-polybaric phase diagram of Al,Si05 system calculated with EQMIN.
geochemicalplots may be generatedwith amMicrosofte EXCELC spreadsheetand their construction also may be automated with macro programming (Christie and Langmuir, 1994; Sidder, 1994). The selectedresultsin the output area or in the zone of intermediateoperations allow the user to generate plots which range from any of the thermochemical parametersconsidered,aswell as their responses to changesof the statevariables,up to completephase diagrams.
([HI: - H&,]/Tand -[Gt - #&]/T, respectively)a+e sensitive to the thermodynamic properties of the elementsfrom which they arecalculated:if oneof the elementsfrom which the formation magnitudeof a substance(e.g. entropy of formation, free energy (,f formation, etc.) iscalculatedundergoesa phasetran\formation (melting, vaporization, symmetry changu, etc.) which is not reflected in the proper thermctchemicalparameter,it will propagatean error to the formation parametersof the substance.This is true also for substancesother than the elementswhit h undergo phasetransformations (magnetite,quartr. LIMITATIONS OF USE AND EXAMPLES hematite,etc.). EQMIN doesnot provide any type c,f From a thermodynamicpoint of view, the most model to calculate energy contributions due 1’~ important limitation to using EQMIN is related to i-type or any other type of phase transitio...l. the thermodynamicdata usedand their applicable Instead, we have fitted the heat capacity function 1,~ range. Accuracy of the enthalpy and Gibbs free the correspondingexperimental measurementsLo energyof formation valuesof substances at T, aswell to and from a temperatureas close as possible1’3 as their enthalpy and Gibbs free energy functions the transition point, which is satisfactory for rn0.t
EQMIN
645
Table 1. Standard state properties of reactions HM (6 Fe,O, = 4 Fe,O, + 0,), QFM (3 SO, + 3 Fe,O, = 3Fe,SiO, + 0,) and its difference with respect to first reaction (A(HM)), at 400°C and 2000 A,G”(F-,T”) (Jmol-‘1
V*(pST”)
S”(P”,T”) CPl
(JmW’K”)
CPZ
CP3
CP4
EXPI
EXPZ
COMPl
COMPZ
J bar-’
IogK
HM
415374
265.31
-5247.4
2729.4
960.3
-1016.4
-6.5
60.3
-0.6
-2. I
-0.35
-24.4
QFM
462696
244.88
3555.8
-1975.2
-644.3
684.1
2.1
-44.8
0.1
5.2
-1.79
-28.9
AWM)
-47322
20.43
-8803.2
4704.6
1604.6
-1700.5
-8.6
105.1
-0.7
-7.3
1.44
4.5
calculation purposes, according to Helgeson and others (1978), Anovitz and others (1993) and Saxena and others (1993). Many examples of the use of EQMIN could be given. A good starting point could be to change the standard temperature and pressure and observe the behavior of the thermodynamic parameters. Another typical application would be the location of an equilibrium point for a reaction in the P-T space: after setting up the reaction in the corresponding zone of the worksheet, temperature and pressure are fixed. By iteration of Tat constant P (or vice versa), we observe the change in the value of ArG(P, T). A sign switch in this value indicates that we have surpassed the equilibrium (perhaps because the increment in P or T is too big). When ArG(P, T) = 0, or close to that value, the equilibrium condition (A,G’(P, T) = 0 or AGleacfantS= AGpraducts) is satisfied. With more than one reaction it is possible to draw a complete phase diagram. Similar calculations may be
I 0
I
1
900
loo0
1 1100
1200
L
I
1300
I
1400
Temperature (“C) Figure 6. Plot showing procedure followed to constrain standard Gibbs free energy of grossular from experiments by Goldsmith (1980) and Koziol and Newton (1988) for reaction 3 anorthite = grossular + 2 kyanite + quartz, assuming that the rest of the thermodynamic parameters are known. Solid circles indicate grossular stable; empty circles indicate the stability of anorthite; crosses indicate no reaction. Equilibrium point we chose was at approximately midpoint of experimental bracket (1200 C and 26,250 bar), although it could have been selected at any other reasonable point within any of the reversals. With the Gibbs free energy of grossular estimated, we calculated locus of reaction to compare agreement with rest of experimental reversals.
performed under isobaric, isothermic, and isobaricisothermic conditions, giving rise to temperature activity, temperature-composition, pressure-activity, pressure composition, and activity-activity diagrams. Specific calculations such as the normalizedto-a-reaction diagrams (for example, A log fO,(HM); HM is the hematite-magnetite oxygen buffer) also may be done easily by writing the reaction from which the rest will be normalized in the proper area of the worksheet (page “reactions”), like any other reaction. The rest of the reactions only have to be subtracted from the first (Table 1). Retrieval of thermodynamic data from phase equilibria experiments may be of particular interest as a preliminary exercise to develop a self-consistent thermodynamic dataset. The complexity of the problem is related directly to the amount of information we have about the phase of interest and how restrictive are the criteria used to evaluate the quality of the data retrieved. In the most simple situation, just one reversal which would allow us to compare the Gibbs free energy or enthalpy of an unknown phase can be considered, provided that all the other thermodynamic parameters of the problem phase and the rest of the participating substances are known. This is done by following the steps: (1) Arbitrarily enter the value 0 to the cell that holds the ArGo(Po, To) of the problem substance. (2) Construct the reaction in the page “reactions.” It is convenient to assign a stoichiometric coefficient of 1 to the problem substance. (3) Use the reversely bracketed equilibrium point to fix T and P within it. At this point A,G must be 0 because the Gibbs free energy of reactants and products cancel out. (4) Under those conditions, the direct reading of the A,G in the output area (which is nonzero of course) will give us the A,G”(Po, To) of our problem substance. (5) Copy this value to the cell we assigned previously a zero value. A,G(P, T) then, obviously, becomes 0, which fulfils the desired and established equilibrium condition. The uncertainty region defined by the experimental bracket also is a measure of the uncertainty of the thermodynamic quantity evaluated. So the thermodynamic magnitude may be estimated accurately. If more than one reversal is available, it is possible to have a better constraint of the thermochemical parameters to estimate. Moreover, if we have a set of experiments for the same reaction which define its Clapeyron’s slope, we can evaluate also the entropy and heat capacity of a problem substance. If instead of a single reaction we
646
J. D. Martin
have a set of reactions experimentally studied, we will he one step ahead on the way to making a self-consistent thermodynamic dataset. Figure 6 shows an example of the aforementioned procedures. Manipulation, selection, evaluation, and judgement of experimental quality are not simple tasks and instructor advice is of major importance at this point. EQMIN, although primarily designed to he used in a Lotus 1230 2.3 spreadsheet in DOS environments, evolved with time to the Microsoft’ Excel0 5.0 spreadsheet, under Microsoft@ Windows” 3.1. EQMIN has heen exported successfully to other PC spreadsheets (Borland% Quattro Pro0 2.0) and Apple Macintosh@ computers. EQMIN was developed in a 486-DX-2, 33 MHz PC computer with 4 Mbyte RAM. Real system requirements for PC computers are less restrictive, in particular for the Lotus 123O version. The current size of the EQMIN worksheet for EXCEL0 5.0 is about 800 kbytes (420 kbytes the Lotus 123c version ) with 122 mineral phases, 39 chemical elements, and 20 fluid species (see Appendix 3).
AVAILABILITY
EQMIN is available through FTP either in Lotus 123c 2.1 or Excel0 5.0 formats by request to [email protected], or from the server at IAMG.ORG. Alternatively, the author will supply a copy of the worksheet after receiving a nonformatted 3.5 in DD or HD floppy disk together with a selfaddressed, stamped envelope at: Departament de Cristal.lografia, Mineralogia i Dip&its Minerals, Universitat de Barcelona; C/Marti i Franquirs s/n, 08028 Barcelona, Spain. Acknowledgments-The original manuscript has been improved by the advice and comments of Dr C. Ayora and Dr S. Gali. Thanks are also given to Dr Helen M. Lang and two anonymous reviewers for their kind suggestions. This work has been financed through the project AMB93-0326 of the Spanish Comisibn Interministerial de Ciencia y Tecnologia (CICYT).
REFERENCES
Anovitz, L. M., and Essene, E. J., 1987, Compatibility of geobarometers in the system CaO-FeO-Al,@Si02-TiO, (CFAST): implications for garnet mixing models: Jour. Geology, v. 95, no. 5, p. 633645. Anovitz, L. M., Essene, E. J., Metz, G. W., Boblen, S. R., Westrum, E. F. Jr, and Hemingway, B. S., 1993, Heat capacity and phase equilibria of almandine, Fe,AlzSi,O,,: Geochim. Cosmochim. Acta, v. 57, no. 17, p. 4191-4204. Berman, R. G., 1988, Internally-consistent thermodynamic data for minerals in the system Na,O-K,O-CaO-MgOFeO-Fe*O,-A&O,-SiO,-TiO,-H,O-CO,: Jour. Petrology, v. 29, no. 2, p. 445-522. Berman, R. G., 1991, Thermobarometry using multiequilibrium calculations: a new technique, with petrological^__applications: Can. Mineralogist, v. 29, no. 4, ^.. p. n33455.
Birch, F., 1966, Compressibility: elastic constants, in Clark., S. P. Jr, ed., Handbook of physical constants: Geol. Sot:. Am. Mem. 97, p. 97-173. Brace, W. F., Scholz, C. H., and La Mori, P. N., 196(,. Isothermal compressibilities of kyanite, andalusite, and sillimanite from synthetic aggregates: Jour. Geophyi. Res., v. 74, p. 2089-2098. Brown, T. H., Berman, R. G., and Perkins, E. H., 198’4, PTA-SYSTEM: A GeO-Calc software package for the calculation and display of activity-temperature-pressure phase diagrams: Am. Mineralogist, v. 74, no. 3-1, p. 485487. Chase, M. W. Jr, Davies, C. A., Downey, J. R. Jr, Frurip, D. J., McDonald, R. A., and Syverud, A. N., 198.5, JANAF thermochemical tables (3rd ed.): Jour. Phyi. Chem. Ref. Data, v. 14-1, 1856 p. Christie, D. M., and Langmuir, C. H., 1994, Automated X‘\ plots from Microsoft Excel: Computers & Geoscience,. v. 20, no. 1, p. 47-52. Connolly, J. A. D., 1990, Multivariate phase diagrams: an algorithm based on generalized thermodynamics: Am. Jour. Sci., v. 290, no. 6, p. 666-718. Cox, J. D., 1979, Manual of symbols and terminology for physicochemical quantities and units. Appendix 17;: Notation for states and processes, significance of the word standard in chemical thermodynamics, and rl:marks on functions used in thermodynamic tables: Pure and Applied Chemistry, v. 51, p. 393403. Delgado, J., 1995, An integrated thermodynamic model for sphalerite geobarometry from 300 to 800°C and up 113 10,000 bar (abst.): Boletin Sot. Espaiiola Mineral., v. 1k, no. 2, p. 4142. Engi, M., 1992, Thermodynamic data for minerals: a critical assessment, in Price, G. D., and Ross, N. L., eds., the stability of minerals: Mineralogical Society Series, Chal’man & Hall, London, p. 267-328. Froese, E., and Gunter, A. E., 1976, A note on the pyrrhotite-sulfur vapor equilibrium: Econ. Geology, )I. 71, no. 8, p. 1589-1594. Goldsmith, J. R., 1980, The melting reactions of anorthire at high pressures and temperatures: Am. Mineralogist, I/, 65, no. 34, p. 272-284. Halbach, H., and Chatterjee, N. D., 1984, An interna1.y consistent set of thermodynamic data for twentyone CaO-A120,-SiO,-H,O phases by linear parametric pr’ Igramming: Contrib. Mineral. Petrol., v. 88, no. l-2, p. 14-23. Helgeson, H. C., Delany, J. M., Nesbitt, H. W., and Biril, D. K., 1978, Summary and critique of the thermodynamic properties of rock-forming minerals: Am. Josr. Sci., v. 278-A, 229 p. Hemingway, B. S., Robie, R. A., Evans, H. T. Jr, and Kerrick, D. M., 1991, Heat capacities and entropies If sillimanite, fibrolite, andalusite, kyanite, and quartz and the AI,SiO, phase diagram: Am. Mineralogist, v. 76, no. 9-10, p. 1597-1613. Holland, T. J. B., and Powell, R., 1990, An enlarged arid updated internally consistent thermodynamic dataset with uncertainties and correlations: The system K,ONa,0-Ca0-Mg0-Mn0-Fe0-Fe,O,-Al,O,-TiO,-SiCJ,~C-HZ-O,: Jour. Metamorphic Geology, v. 8, no. j, p. 89-124. Johnson, J. W., Oelkers, E. H., and Helgeson, H. C., 1992, SUPCRT92: A software package for calculating the standard molal thermodynamic properties of minera.s, gases, aqueous species, and reactions from 1 to 5000 bdr and 0 to 1000°C: Computers & Geosciences, v. 18, nr). 7, p. 899-947. Kerrick, D. M., and Jacobs, G. K., 1981, A modified Redlich-Kwong equation for H,O, CO,, and H,O-I?,(:), mixtures at elevated pressures and temperatures: Am. Jour. Sci., v. 281, no. 6, p. 735-767.
EQMIN Koziol, A. M., and Newton, R. C., 1988, Redetermination of the anorthite breakdown reaction and improvement of the plagioclase-garnet-AI,SiO,-quartz geobarometer: Am. Mineralogist, v. 73, no. 34, p. 216-223. Kretz, R., 1983, Symbols for rock-forming minerals: Am. Mineralogist, v. 68, no. 2, p. 277-279. Metz, G. W., Anovitz, L. M., Essene, E. J., Bohlen, S. R., Westrum, E. F. Jr, and Wall, V. J., 1983, The heat capacity and phase equilibria of almandine (abst.): EOS. Trans. Am. Geophys. Union, v. 64, p. 347. Moecher. D. P.. Essene. E. J.. and Anovitz. L. M.. 1988, Calculation ‘and application of clinopyroxene-garnet: plagioclase-quartz geobarometers: Contrib. Mineral. Petrol., v. 100, no. I, p. 92-106. Parkhurst, D. L., Thorstenson, D. C., and Plummer, L. N., 1980, PHREEQE: a computer program for geochemical calculations: U.S. Geol. Survey Water Resources Invest. Rept., no. 80-96, 210 p. Perkins, D., Essene, E. J., and Wall, V. J., 1987, THERMO: a computer program for calculation of mlxedvolatile equilibria: Am. Mineralogist, v. 72, no. 34, p. 446447. Perkins, E. H., Brown, T. H., and Berman, R. G., 1986, PT-SYSTEM, TX-SYSTEM, PX-SYSTEM: three programs which calculate pressure-temperaturecomposition phase diagrams: Computers & Geosciences, v. 12, no. 6, p. 749-755. Powell, R., and Holland, T. J. B., 1985, An internally consistent thermodynamic dataset with uncertainties and correlations: 1. Methods and a worked example: Jour. Metamorphic Geology, v. 3, no. 4, p. 327-342. Reed, M. H., and Spycher, N. F., 1988, Chemical modeling of boiling, condensation, fluid-fluid mixing and water-rock reaction using programs CHILLER and SOLVEQ (abst.): Am. Chemical Sot. National Meeting, 1976th (Los Angeles, California) Sept. 25-30, abstract no. 26. Robie, R. A., and Hemingway, B. S., 1984, Entropies of kyanite, andalusite, and sillimanite: additional constraints on the pressure and temperature of the Al,SiO, triple point: Am. Mineralogist, v. 69, no. 34, p. 298-306. Robie, R. A., Hemingway, B. S., and Fisher, J. R., 1979, Thermodynamic properties of minerals and related substances at 298.15 K and 1 bar (lo5 pascals) and at higher temperatures: U.S. Geol. Survey Bull. 1452, 456 p. Robinson, G. R. Jr, Haas, J. L. Jr, Schafer, C. M., and Haselton, H. T., 1982, Thermodynamic and thermophysical properties of selected phases in the MgO-Si02-H, O-CO,, CaO-Al, O,-SiO,-H, 0-CO2 , and Fe-FeO-Fe,O,-SiO, systems, with special emphasis on the properties of basalts and their mineral components: U.S. Geol. Survey Open-File Rept. 83-79, 429 p. Saxena, S. K., Chatterjee, N., Fei, Y., and Shen, G., 1993, Thermodynamic data on oxides and silicates: an assessed data set based on thermochemistry and high pressure phase equilibrium: Springer-Verlag, Berlin, 428 p. Sidder, G. B., 1994, PETROCALCPLOT, Microsoft EXCEL macros to aid petrologic interpretation: Computers & Geosciences, v. 20, no. 6, p. 1~41-1061. Skinner, B. J.. 1966. Thermal exoansion. in Clark. S. P. Jr. ed., Handbook of Physical Constants: Geol. Sot. Am: Mem. 97, p. 75-96. Taylor, D., 1987, Thermal expansion data: XI. Complex oxides AB,O, and the garnets: Br. Ceram. Trans. Jour., v. 86, no. -1, p. 148-153 Wang, H., and Simmons, G., 1972, Elasticity of some mantle crystal structures: 1. Pleonaste and hercynite spine]: Jour. Geophys. Res., v. 77, p. 43794392. Wolery, T. J., Jackson, K. J., Bourcier, W. L., Bruton, C. J., Viani, B. E., Knauss, K. G., and Delany, J. M., 1990, Current status of the EQ3j6 software package for geochemical modeling, in Melchior, D. C., and Basset, R. CAGE0
22,G-c
647
L., eds., Chemical Modeling of Aqueous Systems: Am. Chem. Sot. Symposium Series, v. 2, no. 416, p. 104-l 16.
APPENDIX
1
Main calculations performed by the EQMIN
worksheet
(1) Heat capacity function of phase i-@(T) C lo2 + D lo5 CpO(i,PO,T)=A +B.lO-‘T+--p. JF T2 (2) Isobaric thermal expansion function AI”&
PO, T) = E(T - To). lO-4 +F(T*
- To*). lo-*.
(3) Isothermic compressibility function AV’(i,
P, T”) = G(P
-PO)
10m5+ H(P’-
P@)
lo-“.
(4) Molar volume at P, T of phase i V(i, P, T) = p(i,
PO, To) +
AVO(i, PO, T) - AV’(i,
P, To).
(5) Standard molar enthalpy of formation at PO, To of phase i ArHo@, PO, To) = A,GO(i, PO, To) + To. ArSO(i, PO, To). (6) Standard molar enthalpy change from To to T of phase i AH’(i, PO, T) =
’ Q(T)
s TO =A(T-To)+B~10-3(T2-To’)
d’f
2
(7) Standard molar entropy change from To to T of phase i ASO(i, PO, T) =
= Gw ~ s ro
dT T
(8) Standard molar Gibbs free energy change from To to of phase i
T
AG’(i, PO, T) = S’(i, PO, T’)(T A ,ln(T’)+
10-3BTo-p
- To)
102C IO’D
2
f-
p
2T”’
1
.(T-To)
.4(TlnT-T)+~-4.102,/‘?+~]‘. TO
(9) Standard molar Gibbs free energy change from P” to of phase i
P
AGO(i, P, To) = V”(i, PO, T’)(P + [10-4E(T - To) -
+ IO-*F(T*
10-5G(P2 - PO’) 2
- PO) - To2)] .(P -PO)
IO-“H(P3 -
- P@) 3
J. D. Martin
648 (10)
Apparent
A,G(i,
molar
Gibbs
P, T) = A,GO(i, + AG”(i,
(I 1) Equilibrium
Standard
A,S’(i, (13)
WV’, -RT.
entropy
(21)
of formation
Fluid
Standard
molar
entropy
lnf
PO, TO).
(22)
(23)
T)
Mixing
Mixing
(24)
free energy
- @-
function
HTo) = _ 1 T
(16)
Standard
ArHo&
A,G’(i,
Hm)
’1
molar
enthalpy
molar
Gibbs
PO, T) = A,H’(i,
of formation
at T of phase i
Go2
free energy
of formation
k%,crco, coefficient
X&o
(%OtH*O
H,O
activity
(27)
CO,
activity
(28)
Molar
n S’(i, PO, T) - c v,S’(elementj, ,=I [ (18) Polynomial “a ” to calculate H,OCO, (Holland and Powell, 1989) a=a,+a,P+a,P2+a.,Pm’+a,P-* to calculate 1989)
PO, T)
1
(%Or”*O
- %*ckco2))
2&0,~ hLLco2
- %OrH20))
of CO, +
fraction
a co2 - xcoz
Yco?
of CO,
Gibbs
= 1 - X”,O’
free energy
T) = 1 v,A,G’(i,
of reaction
PO, To)
A,S,dT+RTln (30)
Clapeyron’s
slope
fugacities
cell description
~&cl
YH*O
fugacities
APPENDIX Selected
+
.
(P=kbar). Hz&CO,
A,G(P,
of H,O
a H*O - X&O
xc,, Molar
2
of the EQMIN
(P = kbar).
RT
(26)
(29)
+ 0.38P
RT
at T of
PO, T)
(P = kbar).
w(CO,-H,O)
coefficient
Activity
In ko2 =
(P = kbar).
1000
+ 0.26P
= 17.8 - 0.014T
PO, To)
- T.
(19) Polynomial “b” (Holland and Powell,
=
_ SC+, PO T)
T
PO, T) = A,HO(i,
(17) Standard phase i
(25)
(HT-
(Hol-
w(H,O-CO,)
parameter
Activity
PO, T). In ho
Gibbs
fugacities
+ c,P-’
= (a + bT + cT2). RT
parameter
c+OrH~O
at T of phase i
So& PO, T) = S’(i, PO, To) + ASo& (15)
(P = kbar:.
fugacity
of phase i
function HTo) = AH”(Po, T
H,Q-CO,
+ c,P-’
T)
0 uzocO~ = 8.3 - 0.007T
T
b6P-’
c=r,+c2P+c,P~2+CqP~“2
PO, To) = S”(i, PO, To) - i SO(elementj, ,=I
Enthalpy
(20) Polynomial “c” to calculate land and Powell, 1989)
of a reaction
T)=
W(14)
+b,P-‘12+
PO, To)
constant
molar
b=b,+b,P+b,P-‘+b,P-2
at T of phase i
PO, T) + AGO(i, P, To) + RT In CI,.
InK,(P, (12)
free energy
workbook
at P, T
649
650
J. D. Martin