American Mineralogist, Volume 77, pages 1038-1049, 1992

Thermodynamic modeling of the C-H-O-S fluid system

hNcr'llc Snr, S. K. S.lxBflA,Planetary Geochemistry Program, Departrnent of Mineralogy and Petrology, Box 555, Uppsala University, S-751 22,

Uppsala, Sweden

Ansrru.cr

The pressure-volume-temperature (P-V-T) relations and fugacities of pure fluid phasesin the C-H-O-S system (HrO, COr, CH4, CO, O2,H2, 52, SO2, COS, HrS) are determinedfrom the equation of corresponding state and other empirical equations from low tem-perature and pressure to moderate temperature and pressure. The intermolecular potentialtheory is used to calculate the pressure-volume-temperature-composition (P-V-T-fi and,pressure-volume-temperature-activity (P-V-T-a) relations in the C-H-O-S fluid mixture.The model is useful in calculations of phase equilibria and fluid proportions over thetemperature-pressure range of the crust of the Earth.

INrnooucrroN Volkov, l97l; Ohmoto and Kerrick,1977;' Bumrss, 1981;Ferry and Baumgartner, 1987; Poulson and Ohmoto,

Knowledge of the composition of a C-H-O-S multi- 1989; Hall et al., l99l).component fluid in equilibrium with carbonate * silicate This study extends the work of Belonoshko and Saxena* oxide + sulfide assemblages is important in under- (l99la, l99lb)andSaxenaandFei(1987a, 1987b, 1988a,standing the origin and evolution of the heterogeneous 1988b)toincludealowerpressure-temperaturerangefromphase equilibria involving fluids in natural geological sys- critical points up to 2000 oC and 20 kbar and the S-bear-tems and industrial processes. The main fluid species in ing species. We determined new equations for HrO andthis system are HrO, CO2, CHo, CO, 02, H2, 52, SO2, COS, the four S-bearing species and used new fits for H, com-and HrS. pleting a C-H-O-S multicomponent fluid model that is

Thermodynamic properties for the C-H-O fluid system applicable over a sufficient range of geologically interest-have been successfully modeled by several versions of ing temperatures and pressures. We used all available ex-modified Redlich-Kwong (MRK) equation (e.g., Hollo- perimental P-V-T data on pure fluid species and P-V-way, 1977,1982; Flowers, 1979; Delany and Helgeson, T-XandP-V-T-adataonfluidmixtures(Kennedy, 1954;1978; Kerrick and Jacobs, l98l; Bottinga and Richet, TakenouchiandKennedy,l964;Jusaet a1.,1965;Tziklisl98l;Halbach and Chatterjee, 1982; Flowers and Helge- and Koulikova, 19651, Babb et al., 1968; Burnham et al.,son, 1983; Connolly and Bodnar, 1983; Rimbach and 1969; Greenwood, 1969, 1973; Presnall,1969; Robert-Chatterjee, 1987), Ursell-Mayers virial equation (e.g., son and Babb, 1970; Ryzhenko and Malinin, 1971;Haar et al., 1984), Levelt Sengers nonclassical equation Mel'nik, 1972, 1978a, 1978b; Shmonov and Shmulo-(e.g., Levelt Sengers et al., 1983a, 1983b; Sengers and vich, 1974; Tziklis etal., l975; Anguset a1., 1976;Tan-Levelt Sengers, 1984, 1986; Hill and White, 1985), Saul- ishita et al.,1976; Tziklis, 1977; Schmidt,1979; Hilbert,Wagnerequation(e.g., SaulandWagner, 1987, 1989; Sato 1979; Shmulovich and Shmonov, 1978; Shmulovich etet al., 1989), Schmidt-Wagner equation (e.g., Friend et al., 1980, 1982; Hanafusa et al., 1983 Z,akirov, 1984;al., 1989),unifiedfundamentalequation(e.g.,Hil l, 1990; Frantz, 1990; SternerandBodnar, l99l).TodemonstrateJohnson and Norton, I 99 I ), and some other polynomials the applicability of this complete model of multicompo-(e.g., Grevel, 1990; Holland and Powell, l99l). The cor- nent C-H-O-S fluids, we also used the experimental dataresponding-state formulation was used by Ross and Ree from several experiments on phase equilibria (e.9., Eug-(1980)andSaxenaandFei(1987a, 1987b, 1988b)tostudy sterandWones, 1962; FrenchandEugster, 1965; French,the P-V-T relationships, and a molecular-potential mix- 19661, Craig and Scott, 1974; Chou, 1978, 1986, 1987;ing model was used to investigate the P-V-T-X and P-V- Chou and Cygan, 1990; Barton and Toulmin,1964;Bur1,?"-a relations in the C-H-O fluid mixtures by Saxena and 197 l, 1972; Gamble, 1978, 1982; Burton, 1978; Hasel-Fei (1988a). The theory of perturbation of liquid (e.g., ton et al., 1978; Burton et al., 1982; Myers and Eugster,Weeks et al., l97l) and the method of molecular dynam- 1983; Whitney, 1984; Jakobsson and Oskarsson, 1990;ics(e.g.,BrodholtandWood, 1990; BelonoshkoandSax- Kishima, 1986, 1989; UlmerandLuth, 199l,amongoth-ena l99la, 199lb; Fraser and Refson, 1992) are also uti- ers).lized in simulating the P-V-T relations of C-H-O fluids Applications of this model of C-H-O-S multicompo-at high temperatures and pressures. However, few studies nent fluid to some geological systems, including calcula-treat the complete C-H-O-S fluid system (Ryzhenko and tions of some important buffers of .fo* fn, fco, fcno, ?nd

0003-004x/92l09 l 0- l 038$02.00 I 038

SHI AND SAXENA: C-H-O-S FLUIDS 1039

f",, the phase equilibria and fluid proportions in the Fe-Si-O-H-S system, the stability of graphite under a num-ber ofbufer conditions, and the stability ofhedenbergiteand andradite, will be presented in other papers.

EqulrroNs oF srATE FoR PURE rHASES

The law ofcorresponding states

z: ll_: r(p.,l:.\ (r)R 7 l

r \ t , ' f ,

where Z is the compressibility factor, P. is the reducedpressure (P/P.,), and I, is the reduced temperature (Z/f,), has been used in modeling the C-H-O fluid phasesat moderate to high pressures and temperatures by Mel'nik(1972,1978a), Ross and Ree (1980), and Saxena and Fei(1987a, 1987b, 1988b). The equation ofSaxena and Fei(1987a,1987b,1988b), in terms of compressibility factoras a function of pressure and temperature, is

ze,r): A(r) + B(T)P, + cQ)P? + DQ)P? (2)

for the corresponding-state gases (COr, CO, CH4, and Or)and Hr. The formulations of temperature-dependent co-efficients A(T), B(T), CQ), DQ) can be generalized asfollows:

Q(T),r-,txta' ' : Qt I QtT, I Q.T;'

+ Q^T?-t Q,Ta, + QuT?

* Q,T; ' + QrlnT, (3a)

Q(T) r r . r t t ^ t : Qr - l QrT ; ' * QrT; "

+ QoT , ' * Q rT ;o (3b)

where Q,-Q, are constants listed in Table l. This gen-eralization is useful in organizing the coefrcients in Equa-tion 2 in tabular form, avoiding the writing of severalequations. This is also computationally convenient.

Since the molar Gibbs free energy of a species requiresthe JV dP term, we use the following relations for ex-pressing the molar volume as a function of P and ?":

ve,D -RrzlP',r) ()' P

i.e., for CO2, CO, CH4, o2, and Hr,

Ve,r) : RTIA(T)P-' + B(T)P ;1 + C(T)P;,P+ D(T)P;3P'z1. (5)

Table I gives the values for the temperature-dependentcoeftcients in Z(P,T) and V(P,T) (Eqs. 2, 3, 5) used ordetermined in this study. Many of these values have beenadopted from previous studies. For individual pure HrOphase, many equations are available in literature. Aspointed out by Saxena and Fei (1987a), the virial-typeformulations (Saxena and Fei, 1987a, 1987b; Holland andPowell, l99l) are not useful at or near critical range be-

cause, first, the specific partial derivatives of the statesurface diverge to +o at critical point and V(P,T) changesdiscontinuously across the saturation curve and, second,there are substantial discrepancies between the theoreti-cal and experimental P-V-T data. We therefore adoptedthe Saul-Wagner equation (Saul and Wagner, 1987, 1989),which covers the range from the melting line to 1000 'C

and 25 kbar as well as the critical range. For other C-H-Ophases (COr, CO, CH4, 02, and Hr), we chose the resultsfrom Saxena and Fei (1987a, 1987b, 1988b), shown inTable 1, parts a and b. Our calculated P-V-T relationsfor HrO, CO2, CHo, and H, ars not significantly differentfrom those obtained from some of the other existingmodels (e.g., Holloway, 1977, 1982; Halbach and Chat-terjee, 1982; Haar et al., 1984; Rimbach and Chatterjee,1987; Friend et al., 1989; Hill, 1990; Holland and Powell,l99l; Johnson and Norton, l99l).

In this study, we determined the equation of state usingthe corresponding-state equation for H, and pure S-bear-ing species (S2, SO2, COS, and HrS). In order to use thetheory of corresponding state, the critical points (P.,,2".)for each species must be known. These data are knownfor Hr, SOr, COS, and HrS, shown in Table 2, but had tobe estimated for S, (see discussion below).

The values determined for the temperature-dependentcoefficients for H, are given in Table l, part b, which aresame as those in Saxena and Fei (1987a) at pressuresbelow I kbar. A reoptimization led to an improved fit ofthe data above I kbar; therefore the data are slightly dif-ferent from those shown in Saxena and Fei (1988b) atpressures above I kbar. Figures la-lc show the P-V-Trelations for Hr. There is excellent agreement betweenour theoretical curve and the data of Presnall (1969). Theprecision of the fit with the experimental data for H fromthis set of parameters, in terms of 6Z(P,T), is quite good:the maximum of 6Z(P,T) is +0.0186, which is muchsmaller than +0.0253 in Saxena and Fei (1988b). Theerror in volume is generally smaller than +2.50/0. For adiscussion of fit of the experimerLtal P-V-T data on spe-cies COr, CO, CHo, and Or, refer to Saxena and Fei (1987a,1987b).

The P-V-T relationships of four S-bearing species inthe C-H-O-S system (S2, SO2, COS, and HrS) were as-sessed by using the volume-explicit equation (Eqs. 5, 3a)in this study. Some experimental P-V-T data are avail-able for SO, (Mel'nik, 1978b) and HrS (Reamer et al.,1950; Mel'nik, 1978b; Rau and Mathia, 1982). Becausethe misfit of P-V-T data calculated directly from the cor-responding-state equation derived in Saxena and Fei(1987a) is substantial for both SO, and HrS, the param-eters in the equation for them should be adjusted. Theoptimized parameters for the equation of state are pre-sented in Table l, parts c and d. Figure 2 shows the P-V-Trelationships for SO, at pressures from 5 bars to 10 kbar,and Figure 3 shows the relationships for HrS from 100bars to l0 kbar. The theoretical curves fit the experimen-tal points very well. The precision of the fit with the ex-perimental data for these two species from this set of

1040 SHI AND SAXENA: C-H.O-S FLUIDS

TABLE 1. Modeling of P-y-f properties of C-H-O-S supercritical fluids

(1a) Corresponding-states fluid species (O,, CO", CO, CH., S, COS)P (bar) A (r) B (r) c (r) D (I)

< 1 000'

1 000-5000.

>5000--

P (bar)

1.0000001.000E + 00000

-5 .917E - 010002.0614E + 0000

-2.235E + 0000

-3 .941E - 01

A (r)

00.9827E - 010

-0.2709E + 000009j22E - 0200000005.513E - 0203.934E - 02000

(1b) H,B (r)

00

-0.1030E - 0200.1427E - 010000

-1 .416E - 0400

-2.835E - 0600

-1.894E - 060

-1 .109E - 050

-2.189E - 050

c (r)

005.053E - 1100

-6.303E - 2100

D (r)

Qo

Q2

o"QO

o.o"

o.

< 1 000.

>1000

n

o.n

Q"

o.a,o"

1.0000002.2615E + 000

-6.8712E + 010

-1.0573E + 0400

-1.6936E - 01

A (r)

00.9827E - 010

-0.2709E + 000

-2.6707E - 0402.0173E - 0104.5759E + 00003.1452E - 05

(1c) SO,B (r)

00

-0.1030E - 0200.1427E - 01

-2.3376E - 0903.4091E - 070

-1 .4188E - 03003.01 17E - 10

c (r)

-3.2606E - 1502.4402E - 120

-2.4027E - 09000

P (bar)

1 -1 0000

P (bar)

q

o"Q4

o,o"a,

0.92854E + 000.43269E - 01

-0.24671E + 0000.24999E + 000

-0.53182E + 00-0 .16461E - 01

A (r)

0.84866E - 03-0.18379E - 02

0.66787E - 010

-o.29427E - 0100.29003E - 010.54808E - 02

(1d) H,SB (r)

-0.35456E - 030.23316E - 040.94159E - 030

-0.81653E - 0300.23154E - 030.55542E - 04

c (r)1-500

500-1 0000

0.14721E + 010 . 1 1 1 7 7 E + 0 10.39657E + 010

-0.10028E + 0200.45484E + 01

-0.38200E + 010.59941E + 00

-0.15570E - 020.45250E - 0100.36687E + 000

-0.79248E + 000.26058E + 00

0.16066E + 000.10887E + 000.29014E + 000

-0.99593E + 000

-0.18627E + 00-0.45515E + 00

0.22545E - 010.17473E - 020.48253E - 010

-0.19890E - 0100.32794E - 01

-0.10985E - 01

-0.28933E + 00-0.70522E - 01

0.39828E + 000

-0.50533E - 0100.11760E + 000.33972E + 000.57375E - 03

-0.20944E - 05-0.11894E - 02

00.14661E - 020

-0.75605E - 03-0.27985E - 03

o.

a,o.

o"Q4

o"o"a,o,

. Data from Saxena and Fei (1987a)."* Data from Saxena and Fei (1987b).

Trar-e 2. The critical data of C-H-O-S fulid ohases

SHI AND SAXENA: C-H-O-S FLUIDS 1041

473 673 873 LO73 t273 1473 t6?3 7A73 ZO73

T (K)

473 673 873 1073 1273 t4?3 1673 18?3 e0?3

T (K)

P", (bar)

20000

1 ?500

1 5000

12500

1 0000

7500

5000

2500

Phase i", (K)

HrOCo.CHocoo2H2D2

So.cosH,S

647.25304.15191 . 051 33.15154.7533.25

208.15430.95377.55373.55

221.192573.865946.406934.957150.763812.969672.95478.729565.861290.0779

-F

md

! aoo-!

o- 3ooF

- zoo

90

F 8 0\ z o

t s 6 0

19 to

Note.'The critical data for H,O, CO., CH4, CO, 02, H,, SOr, COS, andH2S are from Mills (1974), JANAF (1985), Prausnitz et at. (1986), Reid(1986), Weast et al. (1988). For S,, the critical data are assumed in thisstudy.

parameters is 6Z(P,T): +0.080 and 6V(P,T) : +4.0o/o.There are no experimental data on S, and COS.

In a recent study by Poulson and Ohmoto (1989), thecritical point of S, was assumed to be the same as that ofS (4. : I 3 l4 K, P., : 209 .7 4 bars, taken from Mathews,1972). Based on such values, a number of calculationsfor geochemical equilibria were perfbrmed. But this choiceresulted in unrealistic P-V-T relationship at moderatetemperature range (500-1200 K) and unreasonable R?"ln /values over a large temperature and pressure range.They also set the fugacity coefrcient for pure S, as 1.0and calculated the nonideal mixing parameters a and b(in Holloway-Flowers MRK equation) by using the crit-ical point of S. Obviously, these assumptions would in-troduce significant errors in the calculations. To expresslhe P-V-T relationship of S, as precisely as possible, weestimated its critical point by searching for the relation-ship between the molecular entropies and critical points.Figure 4 shows such a relationship for nonpolar (Or, Hr,CHo, 52, F2, Cl2,Ir, Nr, etc.) and polar species (H2O, HrS,CO2, CO, SO2, SO, COS, CSr, HF, HCl, HBr, HI, etc.),from which one notes the relation of physical and ther-modynamic variables of certain molecules to their elec-tronic structures. The S atom has an electronic structurens2np4, and S, has a d-overlapping covalence bondingvery similar to that of Or, CO, and SO, as well as of COr,COS, and SOr. The statistical and thermodynamic basisfor this kind of choice may be supported by more precisequantum mechanical calculation. Although the relation-ships are vague, we demonstrate in the following sectionsand further studies in heterogeneous phase equilibria (tobe published in succeeding papers) that the selected val-ues of ?"". and P". are reasonable and useful in the calcu-lations of various thermodynamic equil ibria in theS-bearing systems over geological temperature and pres-sure conditions. Our choice for S, is f. : 208.15 K, P-: 72.95 bars (see Fig. a).

Because experimental P-V-T data for both S, and COSare not available, we assumed that S, and COS behavesimilarly to Or, CO,, CO, and CHo and applied the prin-ciple of corresponding state for S, and COS species, usingEquations 2, 3a, 3b, and 5. For S, and COS we simply

?73

100

0 l -273

120

1 1 0

100

30

20

-2?B 6?3 to?3 1473 1873 2??3 2673 3073

T (K)Fig. l. P-V-T relation of H,: (a) l-100 bars; @) 200 bars-l

kbar; (c) l-10 kbar. The theoretical curves are compared withexperimental data from Presnall (1969).

H2

(.)

"r;^ \e,/

sevjl.

bo ball

100 bars

1042 SHI AND SAXENA: C-H-O-S FLUIDS

E soom-

O 2OO

l-.<

d-'!

-i 3oo

m.1

O 2OO

100

7 400-

-.. 3oo

O- zoo

^ 140

H 120

-.. 100a'!

3 8 0

20000

17500

15000

I 2500

1 0000

7500

5000

2500

,"9ro to'l

too bal:

0 L273 473 673 1073 tZ73 t473 1673 1873 20?3

I r / \r ( I \ ]

Soz

(b )

so"/^S '/

Y -s"a?

Kq

273 473 873 1073 IZ73 L473 1673 1873 2073

T (K)

HzS

(")

bol9

2?3 373 573 6?3 773

T (K)9?3

1 8 0

40

1 6 0

2?3 673 10?3 1473 LA73 2273 2673 30?3

T (K)Fig. 3. P-V-T relation of H,S: (a) 100-500 bars, compared

with data from Rau and Mathia (1982); (b) 600 bars-l0 kbar,compared with data from Mel'nik (1978b).

TTTNNTVTOOYNAMIC PROPERTIES OF PURE PHASES

The equation of state ofa substance defines the func-tional relations among its intensive variables and physi-cal surroundings. The thermodynamic status of the sub-stance is constrained by the Gibbs-Duhem equation, fromwhich all thermodynamic properties, including the Gibbsfree energy, Helmholtz free energy, internal energy, fu-gacity, enthalpy, entropy, isobaric heat capacity, isocho-ric heat capacity, isothermal compressibility, and isobaricexpansivity, can be calculated.

The standard thermodynamic properties, includingHl1,.zs,, Sf,,rnr,, and Cp(l,T), for all ten pure fluid phasesare listed in Table 3. Saul and Wagner (1989) calculatedthe thermodynamic properties of HrO as functions oftemperature and pressure. For corresponding-state C-H-Ogases and S-bearing species, the Gibbs free energy andfugacity, as functions of both temperature and pressure,are calculated as follows.

The Gibbs free energy of formation of a pure phase ata given temperature and pressure, referring to a standardstate of I bar and 298.15 K. can be calculated from

20L273 6?3 1073 r4?3 1873 227X 2673 3073

T (K)Fig.2. P-V-T relation of SO,: (a) 5-100 bars; (b) and (c) 200

bars-10 kbar, compared with data from Mel'nik (1978b).

adopted the same temperature-dependent coefficients asthose of Or, CO2, CO, and CHo (Table l, part a).

Tabulated data on volumes of C-H-O-S fluid speciesfrom low temperatures and pressures to high tempera-tures 4nd pressures may be obtained from the authors.

Gr(P,r): I:"(#),dr + [, (#).* (6a)

G,(P,D -- G,(t,D * f ' 'g,r1o,

where

G{l,T): l lPr,,rnrr - ISf,.,sar

* f'"r,{r,r) o, - , ['"Ef o, (7)and for fluid phase

[," ,1r,rrdP: RZln fe,D.

The fluid fugacity /and fugacity coefrcient ^y for purephase are given by

tn f(P,T'): I:t#dP + rn f(P",r) (e)

and

tn v@,7) : I:"t#dP - ln P + ln {Po,T) (ro)

with an assumption as follows: l(Po: I bar) : l; f(po:I bar) : 1 .

The integral in the above equations (Eqs. 6b, 8) can berewritten for convenience as shown below. Using Equa-tion 2, we obtain

J^'-?*:A(n-a,+ B(T)(P, - Po)

*fs7 - 4,1

*ofs _ 4swhere P. : P/P., and Po. : Ps/P*.

Taeu 3. Thermodynamic properties AGft,,n of pure C-H-O-S fluid phases

SHI AND SAXENA: C-H-O-S FLUIDS 1043

(6b)

0 Lt20 140 160 180 200 220

S (J/ rnoI ,K)

120 140 160 180 200 220 240 260 2AO

s (J /moI .K)Fig. 4. The relations between molecular entropies and criti-

cal data. (a) 7". plotted against molecular entropy. (b) P". plottedagainst molecular entropy. Data points except for those of S,and SO are from Mills (1974), JANAF (1985), Prausnitz et al.(1986), Reid (1986), and Weast et al. (1988). Group symbols:

/r r \ open circle for the group Or, Nr, CO, SO, Sr, and Hr; solid circle\ r U for the group COr, CHo, SOr, COS, and CSr; plus sign for the

group Fr, Clr, and Ir; diamond for the group HF, HCl, HBr, andHI; star for HrO and HrS.

- 500

400

F 3oo

(8)

? 1 6 0

- 120Lo

9 . 8 0

( ' )

rHzO

OHFrsr Clz. .+ So"

o *H:S tcos

HclO HErtco,

CH.. - -o^ oS,

" a . O 5 U -

o P^-o"

o

( b )

HBr 'HzS""i 0 iHr cl" sozuss. - cb". i o5" .

HFO-+ so cos

r q a l " ?

O O.- N" coo

l s ,

C"(I) : a + bT + cT 2 + dT2 + eT 3 + fT-ou + glT

LHlrp""t S10,,r..r aPhase (J/mol) (J/mol.K) (101 (10*)

c c l(10") (10 9 (109

s(1 0')

HrOCo.CHo

H,S,SO,cosH.S

-241818.46-393 509.38

-74809.92-110524.s4

0.000.00

1 28 365.1 2-296 829.70-142088.64

-20627.12

188.715 0.46461213.635 0.70728186.155 1 .2901197.564 0.38578205.029 0.39450130.574 0.23948228.070 0.38408248.11't 0.71828231.459 0.42441205.685 0.65459

6.320 0.03.464 0.0

22.21 0.01.781 0.00.9067 0.005279 0.0

-0.4635 0.02.354 0.00.70732 4.60649.326 0.0

-7.957*2.492

-22.91-0.9971

0.006 039-1.525

0.6930-2.083

- 52.16610.54

0.0 - 1 .6630.0 - 1 .8760.0 -7.6560.0 -0.77250.0 -0.61010.0 0.095360.0 -0.099360.0 -1.6420.40721 -1 74320.0 -2.884

5.966-0.8822-0.4849

0.65130.56094.962

-0.0463-2.672

-27 5730.9168

Note; The AHl1.*.p Sff,*.y for all phases are selected from Weast et al. (1988); C" (D coetficients for C-H-O fluid phases are same as in Saxena andFei (1987b); Cp (D coefficients for S-bearing species are assessed in this study, based on the data given in JANAF (1985); fin K. The real Cp (f)coefficients equal the listed values multiplied by the relative factors. For example, a: 0.46461 x 10, for HrO.

LO44

TABLE 4, Interaction energy parameters of C-H-O-S fluid spe-cies.

Species

SHI AND SAXENA: C.H-O.S FLUIDS

TABLE 5. The binary excess free energies of mixing for equimolarcompositions of binary mixtures at P: 1 kbar and I: 500'C

1. HrO2. CO" 0.6153. CHo 0.2834. CO5. O. 0.3086 . H ,7. S, 0.38. SO, 0.39. COS 0.3

10. H,S 0.3

1 84 1.59z.b52.60

0.10 1 .951.600.79

0.0 2.01.61 3.720.0 2.00.0 2.0

2.6503.7603.5653 7603.1092.9603.03.03.03.0

380.00424.16227.13100.20194.3036.70

200.0200.0200.0200.0

Binarymixture

q''(J/mol) No

Binary q'mixture (J/mol)

- The physicochemical meaning ol a', o, elx, tL, d rcler to Saxena andFei (1 988a), Prausnitz et al. (1 986). The values for C-H-O fluid species aretaken from Saxena and Fei (1988a); the values for S-bearing species areselected from Prausnitz et al. (1986) or assumed in this study (italic num-bers), all of which will be taken as zero if ideal mixing is considered.

Tabulated data and plots on RZln /of C-H-O-S fluidspecies from low temperatures and pressures to high tem-peratures and pressures may also be obtained from theauthors.

INrnnLcrroN rN FLUrD MrxruRES

For ideal mixing, the activity coefrcient of each com-ponent in the fluid mixture is unity, i.e.,

(fr),.,: fPX,: ̂ r?PX, (r2)where ff1)P, T and X, are the fugacity and mole fractionof component i in the fluid mixture afi f? and 7P are,respectively, the fugacity and fugacity coefficient of purenonideal component i. For nonideal mixing, the individ-ual fluid fugacity U!)"., is calculated by

(fy),., : Jla,: {lyyPX, (13)

where a, and 7i are, respectively, the activity and activitycoefficient of component I in the fluid mixture.

At low temperatures and high pressures, the multicom-ponent fluid may be nonideal in such a way that the in-teractions among the components are the functions oftemperature, pressure, and fluid bulk composition. Sax-ena and Fei (1988a) proposed a fluid mixture model forhigh-temperature and high-pressure C-H-O fluid, basedon the intermolecular potential theory (Prausnitz et al.,1986; Reid, 1986). The main ideas can be briefly outlinedas follows.

The regular solution energy parameter W,, for the in-termolecular mixing reaction between the ith and 7thmolecules can be defined as

W,,: zllu - Ydf,j + I))l (14)

where f,r, lr, and I, represent the intermolecular poten-tials and z denotes the coordination number. The inter-molecular potential I consists of a Lennard-Jones poten-tial, as given by Stockmayer (Tien and Lienhard, l97l):

where e, o, p, and r are characteristic energy, equilibriumintermolecular distance, dipole moment, and lenglh scale,respectively. Since the length scale r relates the volumet-ric properties of interacting molecules, I and thereforeW,, are functions of temperature and pressure.

The excess free energy G". for binary intermolecularmixing reaction is given by the van Laar equation

1 .

3.4.5.

Lq

1 0 .1 1 .12.1 31 4 .1 5 .16 .1 7 .1 8 .19 .20.21.22.23.

H,O-CO, 1087.98H,O-CH. 1094.94H,O-CO 1080.25H,O-O, 1169.74H,O-H, 1238.47H,O-S, 1 189.31H,O-SO, 429.46H,O-COS 1123.84H,O-H,S 1198.42co,-cHo 0.95co,-co -22.31cor-o, 22.21co2-H' 40 89co,-s, 24.81co,-so, 359.16co,-cos 33.79co,-H,s 23.34cHo-co -7.68cH4-o' 7.77cHo-H, 23.64cH4-s' 9.08cH4-so' 34297cH.-cos 15.33

24. CH4-H'S 8.1625. CO-O, 2.7226. CO-H, 32.3027. CO-S, 3.9728. CO-SO, 337.9629. CO-COS 13.0630. co-H,s 2.5931. O,-H" 2.4332. O.-S, 0.0533. O,-SO, 335-3034. O,-COS 1.2235. O,-H,S -0.01

36. H,-S, 1.2337. H,-SO, 336.6838 H,-COS -0.55

39. H2-H'S 1.6940. s,-so, 335.3941. S,-COS 0.7342. S2-H'S O.O243. SO,-COS 334.9444. SO,-H,S 337.0645. COS-H,S 0.99

Cr*: w,rX,x,Q,ei/(X,Q, + Xjqj).

Then the activity coemcients 7, can be expressed as

R?"ln "y, : W,S/Q + q,X,/q/,)'

( l 6)

( l 7)

( l 8)

( le)

where, in turn, 4, represents the effective volumes. Theactivity coefficients in ternary or higher order solution arecalculated from the Kohler formulation.

The excess molar volume Z"*, excess entropy S"., andexcess enthalpy I1'. for binary mixtures are defined as

u,, = (ac'.\\ dP l .

: X,X,q,q,/(X,s, * X,d(9hAp),

^ /ac*\r . - = \ d r / "

: X,X,q,q;/(X,n, * *,D(#),

Hex: Gex + ?",S* - PW

: X,X,q,q,/(X,Q, -t Xiei)

[ / - \ " /_\6- l. :o. l( :) -(:) l-qL\r/ \r/ I 13

(15) ,1,,, .,(#), -,(*;) (20)

SHI AND SAXENA: C-H-O-S FLUIDS 1045

The formulations of derivativ es (6 W,, / 6 P) r. and (6 W,, / 6 7-1,are, however, very complicated. One can calculate theexcess functions for each fluid mixture by using the freeenergy minimization program Solgasmix (Eriksson, 1975;Saxena and Eriksson, 1985). The mixing model is alsoused for the C-H-O-S fluid system here, by assuming thatsome of the interaction parameters for S-bearing speciesmay be comparable to those of similar species in theC-H-O fluid. Table 3 lists the interaction energy param-eters chosen for the nonideal C-H-O-S fluid in this study.Since the uncertainty in W,, comes mainly from volu-metric property, the applicable temperature-pressure rangeof intermolecular potential formulation is the same as ofthe pure fluid equation.

This study demonstrates the following:l. Within the geological temperature-pressure envi-

ronments, the most significant interaction energies arethose in binary mixtures involving HrO, and the nextimportant are those in binary mixtures involving SOr.This is because both HrO and SO, have large dipole mo-ments (1.84 and 1.61, respectively; see Table 4). Thepolynomial coefficients for excess energies and activitiesas functions of temperature, pressure, and compositionregressed from intermolecular potential calculations forthese nonideal (polar-polar, polar-nonpolar) binary sub-systems may be obtained from the authors. As an ex-ample of the binary-excess free energy of mixing, we listsuch energies for equimolar compositions at a pressureof I kbar and temperature of 500 "C in Table 5. Table 5also shows that the interaction energies in the binary sys-tem involving two low-polarizability molecules are neg-ligible.

2. The interaction parameter I4/,; decreases swiftly withincreasing temperature or decreasing pressure (see Fig. 5).

3. The independently calculated P-V-T-X ar;.d P-V-T-arelations for HrO-COr, H2O-H2, H2O-CH,, and CH.-CObinary fluid mixtures are consistent with available exper-imental data for HrO-CO, (Takenouchi and Kennedy,1964; Greenwood, 1969, 1973; Eggler and Burnham,1978; Egeler and Kadik, 1979; Chou and Williams, 1979;Shmulovich et al., I 980, I 982; Kerrick and Jacobs, 1 98 1 ;Zak\rov, 1984; Sterner and Bodnar, l99l), for HrO-H,(Shaw, 1963; Seward and Franck, 198 l; Smith et al., 1983;Wormald and Colling, 1985; Rimbach and Chatterjee,1987), and for HrO-CHo (Welsch, 1973; Jacobs and Ker-rick, 198la, l98lb; Smith et al., 1983). No experimentaldata are available for comparison for the system involv-ing S-bearing species. The only way to check the S-bear-ing species is to calculate the phase equilibria in the sys-tem involving these species and compare the results withexperiments.

As an example of the applicability of our mixing mod-el, we show the comparison of calculated P-V-T-X rela-tions of an HrO-CO, binary mixture with the most recentexperimental data from Sterner and Bodnar (1991) inFigure 6. Note that the experimental errors of Zrro-"o, or2". in experiments are about 0.5 to 1.0 cm3lmol; thesecome mainly from uncertainties in pressure (ono), tem-

6000

i +ooo

\ sooot-

pi zooo

400 500 600

T1 000

7000

6000

^ 5000

fi +ooo

3000

2000

1 000

5oo 4oo 5oo 600 7oo Boo 9oo looo

T ( 'c)Fig. 5. The temperature and pressure dependence ofthe bi-

nary interaction parameter Wufor the (a) HrO-CHo and (b) HrO-HrS binary mixtures.

perature (orr), and fluid composition (o). Therefore, thefit is very good, especially at high pressures and high tem-peratures. The mixing data from Shmulovich et al' (1982)

were discussed before by Saxena and Fei (1988a).

CoNcr,usroNs

The corresponding-state equation for COr, CHo, CO,Or, Hr, 32, SO2, COS, and HrS, the Saul-Wagner equationfor HrO, and the intermolecular-potential formulation forthe fluid mixtures (in the calculation of low to mediumtemperature-pressure fluid fugacities and activities of thespecies in the C-H-O-S system) are recommended as apractical solution to a study of the multicomponent be-havior of magmatic, metamorphic, and hydrothermal

( a ) H 2 O - C H a

700 800 900

( 'c)

(b) H,o-HrS

700 800 900

( 'c )

(a ) +oo"c

Sterner & Bodnar (1991)o l * Z . 3 E 4 * 5 k b a r

r046 SHI AND SAXENA: C-H-O-S FLUIDS

0 . B0 1 . 0 01 0 L0 .00 o.2a 0 .40

90

80

E70.1

\ 6 0m

a-t

o 5 0

3+oI

#30

80

70

60

50

40

30

20

I

m

H

- 8 0

E?o

g60

50

J 4 0

1 0 L0 .00 0 .20 0 .40 0 .60

Xo,o

0 .40 0 .60

X""o

0 .60

Xn,o

0 . 8 0 1 . 0 0

1 0 0

90

1 1 0

1 0 0

90

BO

70

60

50

40

30

20

fluids. All the available experimental data on P-V-T, P-V-T-X, and P-Y-Z-a relations for C-H-O-S pure species andmixtures are reproduced well over the temperature-pres-sure range ofgeological interest.

As discussed here and in Saxena and Fei (1987 a, I 987b,1988b), this set ofcorresponding-state equations may beeffectively and reliably applied with precision in the crustaltemperature-pressure range from critical points up to 2500K and 20 kbar. To calculate equilibria involving carbon-ate * silicate * oxide + sulfide + fluid. a valid C-H-O-Sfluid model must be applied. This multicomponent C-H-O-S fluid model will be used in calculating bufers of f,,.f.r, .f*r, fcuo, a\dfs, and phase equilibria involving car-bonates (including graphite), silicates, oxides, sulfides, and

20

1 . 0 01 0 L0 .00 o .20 0 .40 0 .60 0 .80 1 .00

Xn.o

fluids. The pressure-temperature range from moderate tohigh for fluids is covered by the molecular-dynamics sim-ulated data of Belonoshko and Saxena rl991a. l99lb).With this work, we have extended the range to low pres-sures and temperatures. A computer program to calculatethe compound properties of the pure fluid (P-Z-?" andfugacities) in the pressure range from I bar to I Mbarand in the temperature range from 400 to 2500 K is avail-able from the authors.

AcxNowr,nocMENTS

The authors thank the Swedish Natural Science Research Council (NFR)for financial support. Thoughtful reviews by I-Ming Chou, Jackie Dixon,and M.S. Ghiorso helped improve the presentation.

10 r-0 . 0 0 0 .20 0 .80

Fig. 6. P-V-T-X relations of HrO-CO, binary mixture, compared with experimental data from Sterner and Bodnar (1991); (a)400'C, (b) 500'C, (c) 600'C, (d) 700 "C.

(b ) 5oo 'c

Steruer & Bodnar (1991)O 1 ) r Z o 3 0 4 * 5 k b a r

(c ) ooo 'c

Sterner & Bodnar (199f)* Z o g E 4 * 5 0 g f U a .

(d) ?oo"c

Steraer & Bodnar (1991). 3 0 4 * 5 0 6 k b a r

SHI AND SAXENA: C-H-O-S FLUIDS 1047

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M*ruscnrpr RECEIvED OcrossR. 30, 1991MrNuscnrpr AccEPTED M;v 4, 1992