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Methane: An equation of state with application to the ternary system H2O-CO2-CH4
Ceorhimics et Cosmm-himira Acra Vol. 45. pp. 607 to 614. 1981 Printed in Great Britain. All righIs reserved
00167037/81,~swo7-oBI2.~~ Copyright Q 1981. Pcrgsmon Press Ltd
Methane: an equation of state with application to the ternary system H@-C02-CH4 GARY K. JACOBS*and DERRILLM. KERRICK Department of Geosciences, The Pennsylvania State University, University Park, PA 16802, U.S.A. (Received 4 September 1980; accepted in revisedjmn
25 November
1980)
Abstract-The following hard-sphere modified Redlich-Kwong (HSMRK) equation of state was obtained by least squares fitting to available P-V-T data for methane (P in bars; T in Kelvins; v in cm’mol-‘; b = 60.00cm3mol-1; R = 83.14cm3barmol-‘K-r):
p _ RW + Y + Y' - Y? _ c(T) + d(T)/u + e(T)/u* u(v + b)T’
vu - Y?
y = b,4v c(T) = 13.403 x lo6 + (9.28 x l@)T + 2.7 T2 d(T) = 5.216 x 10’ - (6.8 x 10’)T + (3.28 x 103)T2
e(T) = (-2.3322
x 10”) + (6.738 x 10s)T
+ 13.179 x los)T* For the P-T range of experimental data used in the fit (50 to 8600 bars and from 320 to 670 K), calculated volumes and fupeity coefficients for CH. relative to experimentally determined volumes and fugacity coefficients have average percent deviations of 0.279 and 1.373, respectively. The HSMRK equation, which predicts linear isochores over a wide P-T range, should yield reasonable estimates of fugacity coefficients for CH* to pressures and temperatures well outside the P-T range of available P-V-T data. Calculations for the system HsG-C02-CHI. using the HSMRK equations for Hz0 and CO1 of KENUCKand JACDEZS (1981) and the HSMRK equation for CHI of this study, indicate that compared to the binary HzG-CO2 system, small amounts of CHI in the ternary system H2G-C02-CH4 slightly increases the activity of HsO, and significantly decreases the activity of COz.
HOWAY (1977) demonstrated that the modified Redlich-Kwong (MRK) equation of DE SANT~Set al. IN ADDITK~Nto HZ0 and CO1, methane (CH,) can be (1974) could be used for the estimation of fugacity an important species in geologic fluids (ROEDDER, coefficients for pure HZ0 and CO2 and for H20-C01 1972; HOLL~TER and BURRU~~, 1976; BEWR and mixtures at elevated pressures and temperatures. PINEAU, 1979; CRAWFORDet al., 1979a, b; HOUIS’IER HOLLOWAY(1977) combined his MRK equations for et al., 1979; KONI&RUP-MADSMet. al., 1979; MULLIS, HZ0 and CO1 with original Redlich-Kwong (RK) 1979; HENDELand HOLLLTITR,1981). The presence of equations (REDLICHand KWONG, 1949) for several small amounts of CH* in H+CO&H, fluids may other gases (CO, CH, and HZ) to calculate fluid comsignificantly affect the stabilities of mineral assempositions in the C-O-H system.t RYZHENKOand blages relative to assemblages in equilibrium with bi- VOLKOV (1971), MEL’NIK (1978) and O~TR~~~KY and nary H@-C02 fluids. Therefore, it is important to be RYZHENKO(1978) have developed equations of state able to calculate the thermodynamic properties of for several gases, including CH*, but none are readily CH* at elevated temperatures and pressures, both as adapted to the estimation of fugacities for species in a pure species and in H+COs-CH, mixtures. Calmulticomponent mixtures. culations of fluid compositions in the C-O-H-(* S) KERRICKand JAMBS (1981) have developed a hardsystem at elevated pressures and temperatures (for sphere modified Redlich-Kwong equation for H20, example FRENCH, 1966; EUIXITX and SKIPPEN,1967; CO1, and HIO-CO1 mixtures. It recently has been OHMOID and KERRICK, 1977; FROST 1979), where shown (TARAKADet al., 1979) that such equations of CHI can be an abundant species under certain constate, utilizing a hard-sphere repulsive term (CARNAditions, will also be improved if accurate fugacity HAN and STARLING, 1969), may be superior to other coefficients for CH* can be calculated. equations predicting thermodynamic properties of Resent address: Department of Geology, Grand Val- supercritical, aqueous fluid mixtures. FLOWERS (personal communication) has suggested that, in general, ley State Colleges, Allendale, MI 49401, U.S.A. t HOLLOWAY'S (1977) analysis for mixtures is in error molecular dynamics models of repulsive terms pro(see FLOWERS1979). vide better agreement for supercritical fluids, although
INTRODUCTION
l
ci.c.*. 45:5--*
607
608
GARYK. JACOBS and DERRILLM.KERRICK
functions of a(v). which could be fit to the equation: such treatment is not necessarily limited to a hardsphere model. By fitting the hard-sphere modified a(v) = c(T) + Redlich-Kwong (HSMRK) equation of KERRICKand 1’ V2 JACOBS (1981) to available P-V-T data for CHI (TSIKLISet al., 1971; ANGUSet al., 1978; FRANCESCONI, Substitution into eqn (1) yields: 1978), a HSMRK equation for CHI has been develp _ RT(1 + y + y’ - y3) oped which accurately predicts molar volumes and v( 1 - y)3 fugacity coefficients within the P-T limits of the experimental P-V-T determinations, and should yield c(T) + d(T)jv + e(T)/d reasonable estimates of molar volumes and fugacity (3) T*v(v + b) coefficients for CH* at pressures and temperatures above the P-T limits of the experimental P-V-T de- Using a value of b = 60cm3 mol- ‘, least squares terminations. In addition, because the equation is analysis yields the following temperature functions for compatible with the HSMRK equation of KERRICK c, d and e (stated errors give 95”/, confidence intervals) : and JACOBS(1981) for Hz0 and CO*, activities in the c(T) = (13.403 It: 6.9) x lo6 system H#-C02-CH4 may be calculated for a wide
do + e(T)
range of P-T conditions.
+ (9.28 k 2.9) x lo4 T + (2.7 f 28.6)T2
NOMENCLATURE ai
activity of species i
4T.v)
attractive term function of HSMRK equation (bar cm6 K* molwL) anorthite covolume (cm3 mol- I) parameter in attractive term function (bar cm6 K* mot- *) calcite parameter in attractive term function (bar cm9 K’ molm3) parameter in attractive term function (barcm’2K*moI-4) fugacity grossular chemical components subscript denoting a mixture number of moles pressure (bars, unless specified otherwise) quartz gasconstant (83.14cm3barK-‘mol-‘) temperature (Kelvins, unless specified as Celsius) total volume (cm’) molar volume (cm’ mol- ‘) wollastonite mole fraction
an b c cc d r f gr ij M ; (32 R T V v wo X
reduced density (b/4u) fugacity coefficient (f/P) of pure species i fugacity coefficient of species i in a mixture
EQUATION OF STATE The following HSMRK equation (KERRICK and JACOBS,1981) was fit to available P-V-T data for CH4: P=
RT(1 + y + yL - y3) ~(1 - JJ)~
a(T, 4 - T+o(o + b)
(1)
Sources for the molar volumes (293 observations) used in the least squares fitting, performed with ‘Minitab 2’ (RYAN et al., 1976). were: (1) ANGUS ef al. (1978); 320-470K and 50-3000 bars; 470-670K and 50-400 bars, (2) FRANCESCONI (1978); 47&670K and 300-2200 bars, and (3) TSIKLISet al. (1971); 320-670K and 2025-8600 bars. The approach, detailed by KERRICK and JACOBS(1981), consisted of using P-V-T data to find a value for h which resulted in isothermal
(4)
d(T) = (5.216 + 0.88) x lo9 - (6.8 + 3.7) x 106T + (3.28 & 3.7) x
103T2
(5)
e(T) = (- 2.3322 + 0.32) x 10” + (6.738 f 1.3) x lOaT + (3.179 + 1.3) x 105TL
(6)
The hard-sphere repulsive term [the first term on the right side of eqn (I)] has been derived by CARNAHANand STARLING(1969). It is interesting to note that when y - 0 (as v becomes very large), this term reduces to the same form as the van der Waals repulsive term [RT/(u-b)]. The attractive term [the second term on the right side of eqn (l)] combines the denominator of the original Redlich-Kwong equation (REDLICHand KWONG, 1949) with a numerator which is a function of temperature and volume, rather than a constant value as proposed by RJZDLICHand KWONG(1949). The numerator, which consists of temperature-dependent coefficients in an inverse volume relationship [see eqn (2)], closely resembles a virial expansion. This form of the attractive term lends support to the choice of mixing rules [eqns (10) and (1 l)], since these mixing rules can be derived directly from a virial equation for gas mixtures (PluusNrrz, 1969, pp. 98-100). The b value for CH4, which represents the covolume of the gas. is approximately twice the value predicted by the original RK equation (REDLICHand KWONG, 1949). The b values for H20 and CO2 in KERRICK and JACOBS(1981) are similarly approximately twice the value predicted by the original RK equation. Because the value of the covolume (the volume of 1 mol of close-packed gas molecules) is difficult to evaluate precisely (i.e. measure), and because the HSMRK is a modified form of the original RK equation, the physical significance of the difference in b values for the two equations is questionable. Additionally, because all versions of the MRK equation. including that of the present study, are seni-empiri-
Methane: an equation of state with application to the ternary system H20-C02-CH4 cal, parameters such as b have limited physical significance. The key requirement for b values in our HSMRK equation is that a(r) functions fit the form of eqn (2), so that reasonable estimates of fugacity coefficients can be calculated at pressures exceeding the maximum pressure of P-V-T measurements (see KERRICKand JACOBS,1981). THERMODYNAMIC PROPERTIES OF THE HIO-COI-CH, SYSTEM After substitution of eqn (3) for P, fugacity coefficients for a pure species can be calculated by evaluating (PRAUSNITZ,1969, p. 41):
- RT In Z + RT(Z - l),
(7)
609
calculated with eqns (4-6) for CH*, and eqns (8-13) of KERRICKand JACOBS(1981) for Hz0 and C02. Equations (IO) and (11) are theoretically valid only for the virial equation of state. However, it was shown earlier that the attractive term chosen for this study [eqn (2)] closely resembles a virial expansion, thereby supporting the choice of eqn (10) and (11) for the mixing rules of the attractive term parameters. PRAUSNITZ (1969, p. 44) suggests that substitution of a term representing molecular radii (b113)rather than molecular volume (b) in a mixing rule such as eqn (9) does not significantly affect the resulting fugacity coefficients at moderate pressures, although the effect has not been studied at high pressures. The fugacity coefficient of a species i in a mixture is calculated by the equation (F’RAUSNJTZ 1969, p. 41): RTln4?=
6
[(3,.“..,
- ?]dF
which yields:* -RTlnZ IntiP”“=
8y - 9Jf2 + 33’” --_
(1 - Y?
In Z -
RT3’2(v + b)
d e RT3’2u(v + b) - RT3’2u2(o + b)
Integrating and rearranging (12), after substitution of eqn (3) for P with b, c, d,,,, and e,,, substituted for b, c. d and e yields: In 4;“”
d - RT”‘bv -
(12)
C
4Ym - 3Y; + 5 4Yln- 2Yi = (1-Ym)2 ( bm’ (1 - Y,)’ )
e RT3122bv2
e
+-RT312b2v
(8)
Evaluation of eqn (8) is accomplished by first solving eqn (3) for v at the pressure and temperature of interest by an iterative technique. Fugacity coeficients for CH,, CO2 and H20 in mixtures of H20-C02-CH. can be calculated after first choosing a set of mixing rules for b, c, d and e. KERRICK and JACOBS(1981) demonstrate that despite the controversy concerning mixing rules, those originally proposed by van der Waals appear adequate for H20-C02 fluids. These same mixing rules were therefore chosen for an analysis of the system H20-C02-CH,:
&em + RT3’22b,v;(v,
b, = F bixi + Cm =
C 1 CijXiXj i i
[&.*t(‘+)]
(10) -
3hem RT3’2b&,
where C[j =
(CfCj)+
3he, + b,) - RT3’22b;u,(v,,, + b,)
(11)
Equations (10) and (1 l), which also apply for d,,, and are evaluated after the values for c, d and e are
e,,
* KERRICKand JACOBS(1981) present a detailed derivation of eqn (8).
+ b,)
- InZ
KERRICK and JACXXS (1981) provide a detailed derivation of the expression In 41’” for binary H20-C02 fluids, which is easily extended to the ternary system H20-C02-CH9 [eqn (13)J. Note that it is necessary to solve eqn (3) for v, at the pressure and temperature of interest after substitution of b,, c,,,, d,,, and e, for b, c, d and e. The required expres-
610
GARY
K. JACOES and DERRILLM. KERRICK
sions for P,, Pd and P, are partial derivatives of the mixing rules, [eqns ( 10) and (1 1)] : P, = 2Cf C CfXj
I -
P* = 2d) 1 df Xj + d,
(15) I
C t?jXj+2e,
P,=2e!
,
The activity of species i is then calculated from: a, =
(rpf”‘” ‘> xi
(17)
4p”’
It is important to note that eqns (13-16) only when eqn (12) is evaluated using eqn bined with mixing rules (9-11). Any change the mixing rules or the form of the HSMRK invalidates eqns (13-16).
are valid (3) comin either equation
400 COMPARISON OF CALCULATED VOLUMES AND FUGACITY COEFFICIENTS WITH PREVIOUS WORK
In order to evaluate the accuracy and usefulness of our HSMRK equation for CH,, two analyses are important: (1) a comparison. of calculated molar volumes and fugacity coefficients with experimentally determined volumes and fugacity coefficients and (2) an evaluation of the accuracy of molar volumes and fugacity coethcients calculated at pressures and temperatures above the P-T limits of experimental P-V-T determinations. Figure 1 is a comparison between volumes calculated with eqn (3) and the
-2
0 2 4 Percent Dtfference
Fig. 1. Histogram showing the frequency of deviation of CH* molar volumes calculated with eqn (3) from experimental P-V-T measurements (TSIKLISet al., 1971; ANGUS et
al.. 1978; FRANCESCONI. 1978). Percent difference = (v,.,~ - c._,) x ~OO/U,,~.
800
1200
1600
T(K) Fig. 2. Isochores (cm’ mol- ‘) of CH, calculated with eqn (3) as a function of pressure and temperature. Circles represent P-T coordinates calculated with eqn (3).
experimentally determined volumes used in the least squares regression of eqn (1). The average percent deviation of the calculated volumes compared to the experimental volumes is 0.28%, with few values exceeding l.oo/,. The largest differences (3-4%) occur only at the highest temperatures (T > 550K) and at approx. 8ooObars. Fugacity coefficients derived from experimental data at T > 320K are relatively scarce. ANGUS et al. (1978) supply the most comprehensive set of fugacity coefficients. The agreement between fugacity coefficients calculated with eqn (8) and those at P I 1000 bars and T = 320-67OK from ANGUS et al. (1978) is always better than 1.0%. At pressures above looobars (up to P= 10 kbar), the differences range from approx. 2-18x, with most being approx. 2-S%. Interpretation of these differences at P > loo0 bars is difficult, however, because ANGUSet al. (1978) used no experimental data in their analysis above 1000 bars for temperatures greater than 310K. Figure 2 may be used to attempt to evaluate how well our HSMRK equation for CH, extrapolates to pressures and temperatures above the P-T limits of the experimental data used in the regression of eqn (1). Each of the four isochores in Fig. 2, though somewhat curved, can be represented by a straight line. The calculated isochores remain essentially linear to ZOOOK (P=4kbar for c = 75; P = 15kbar for c = 45). Though not shown in Fig. 2, a calculated isochore for I: = 35 is also a linear function to 2000K (P = 35 kbar). The linear isochores imply that volume and fugacity coefficients may be reasonably estimated at P-T conditions well beyond those oi existing
Methane: an equation of state with application to the ternary system H20-C02-CH4
611
study involve extrapolations exceeding the P-T limits of available P-V-T data, the significance of the large differences at the higher pressures and temperatures is tions cannot be precisely evaluated. difficult to evaluate. One of the main advantages of OSTF~OVSKY and RYZHENKO(1978) calculated the fugacity of CH4 at T 2 lOOOK and P L 20 kbar by the HSMRK equation over the corresponding state evaluating a volume function obtained through equations of RYZHENKOand VOLKOV(1971) and graphical extrapolation of lower pressure-temperaMEL’NIK(1978), however, is that the HSMRK quature compressibility factors. However, since their distion is easily adapted to the calculation of activities in cussion of the method and results is brief, a detailed multicomponent mixtures. comparison is precluded. RYZHENKO and VOLKOV KERRICKand JACOBS(1981) have shown that their HSMRK equation predicts volumetric properties of (1971), using a corresponding states approximation, Hz0-C02 fluids that are in reasonable agreement calculated fugacity coefficients for CH4 from 373 to 1273K and from 100 to 1OOOObars.Comparison of with the available experimental data (FRANCK and T~~DHEIDE,1959; T~~DHEIDEand FRANCK, 1963; their fugacity coefficients with those calculated with GREENWOOD, 1969, 1973). Available experimental data our HSMRK equation [eqn (8)] yields minimum and maximum differences of -94.1 and +9.70/, respectfor COz-CH4 mixtures (REAMER et al., 1944) is at temperatures (T s 510K) too low for calculations ively, with an average difference of approx. -5.37;. involving mixtures with the HSMRK equation for The poorest agreement is at T -c 650K and P > 6000 bars. These large differences may be attriCO2 (KERRICKand JACOBS,1981; JACOSSand KERbuted to the fact that the data of FRANCESCONI (1978) RICK, 1981); thus, a comparison with the data of REAMERet al. (1944) is precluded. However, experiand TSIKLISet al. (1971) were not incorporated into of volumes in the system the equation of state for CH4 of RYZHENKOand VOL- mental determinations H+CH4 (WELSCH,1973) at 673K and from 400 to KOV(1971). MEL’NIK(1978), utilizing another version 1000 bars allows an evaluation of u, calculated with of the corresponding states principle, calculated fugaour HSMRK equation for this binary system. Figure 3 city coefficients for CH4 from 400 to 15000 bars. For most of the P-T conditions tabulated by MEL’NIK shows that calculated values of c, for HrO-CH4 (19783, his fugacity coefficients are systematically low mixtures at 673K and from 400 to loo0 bars are in close agreement with the experimental values of when compared to those calculated with our WELXH (1973). Unfortunately, there is presently no HSMRK equation [qn (8)]. The minimum and maxidata for HtO-C02-CH4 mixtures mum differences of -36.5 and +3.9% occur at experimental which allows an evaluation of the accuracy of activi14OOK-7 kbar and 6OOK-5 kbar, respectively, and ties for H20, CO* and CH4 in this ternary system the average deviation is approx. - 5.8%. Because both calculated with our HSMRK equation. However, the the method of MEL’NIK(1978) and that of the present agreement of the calculated properties with the experimentally determined properties for the binary systems discussed above strengthens the application of the HSMRK equation to the H@-C02-CH4 system. P-V-T data. It must be noted that until more data is available, however, the accuracy of such extrapola-
IMPLlCATJONS Calculations for the system H#-C02CH4 indicate that compared to activities in binary H20-C02 fluids, small amounts of CH4 in H20-C02-CH4 fluids may significantly affect the activity of HZ0 and CO*. Figure 4 illustrates this effect for pressures and temperatures corresponding to typical contact and regional metamorphic conditions (Figs 4a and b, respectively). For both P-T regimes, the presence of up to 20mol% CH4 slightly increases the activity of HZ0 relative to its value in binary H20--CO2 fluids, whereas the activity of CO2 is significantly decreased. Similar relations were suggested by HOLLOWAY (1977), although the increase in the activity of HZ0 was more 0.6 0.8 1.0 0 0.2 0.4 pronounced in HOLLOWAY’S (1977) analysis. However, XH,O FLOWERS(1979) has pointed out a problem with HOLLOWAY’S (1977) analysis, which when accounted for, Fig. 3. Comparison of calculated and experimental values of u,,,~~for the H,O-CH, system at 673K. The solid lines may lower the activity of HZ0 somewhat in Fig, 2 of were calculated with the HSMRK equation for CH4 of this HOLLOWAY(1977). Since CH4 and CO2 are quite study and the HSMRK equation for H,O of KERRICKand JACOBS(198 1). The circles are experimental data of WELSCH similar, in being non-polar molecules (although CO2 does have a quadrupole moment) and having average (1973).
GARY ILJACOBS and DERRIU M.
612
02
0.0
0.4
(a)
0.6
0.8
1.0 0.0
xco*
KERRICK
0.4
0.2
0.6
0.8
1.0
xco,
lb)
Fig. 4. Activity~om~s~tion diagram for Hz0 and CO2 in H#-CUZ-CH~ mixtures. (a): loDo bars 723K; (b): 6000 bars, 873K. Calculated with the HSMRK equation for CHr of this study and the HSMRK equations for Hz0 and CO* of KERRICKand JACOBS(1981) for three constant CHI contents (XC”, = 0.0, 0.05 and 0.20). Dashed lines represent ideal mixing. polarizabilities
of 26 x IOz5 and 26.3 x 10” cm3 p. 60), respectively, the relations in
(PRAUSNITZ, 1969,
Figs 4a and b are not entirely unexpected. In particular, the attractive forces between HZ0 and CHI molecules and between HZ0 and CO2 molecules should be quite similar, since the induction forces (a function of polarizability) between H#-CH4 and HIO-COL molecules, set up by the electric field from nearby polar HZ0 molecules, should be similar in magnitude. In addition, dispersion forces (a function of polarizability and first ionization potential) resulting from H&kCO* and H#-CHI interactions should be similar, since the first ionization potentials of CH.+ and COZ are similar (PRAUSNITZ, 1969, p. 63). This interpretation of the molecular interactions explains the relativefy small increase in the activity of HzO. The reduced activity of CO1 can be attributed lo simple dilution by a similar molecule (CH,). The lowered activity of CO1 in the H*O-rich portion of the H20-C02-CH, system due to the small amounts of CH,, may shift the stability fields of mineral assemblages in equilibrium with the fluid relative to mineral assemblages in equilibrium with binary H&3-CO2 fluids at the same P-T conditions. Figure 5 illustrates this effect for a portion of the system CaO-Al~O~-SiOt-H#-COz-CH~. The Xco, of the isobaric invariant point resulting from the inter~ion of equilibrium (1) and equilib~um (2) is increased by approx. 5 mol% for a fluid with X CH. = 0.20 relative to a fluid with XCHI = 0.0. The expansion of the stability fields for grossular and woliastonite is of ,particular im~rtan~. since grossular and wollastonite are generally considered excellent indicators of H20-rich fluids. However, the presence of
CHo permits assemblages containing grossular or wollastonite to coexist with fluids having considerably higher CO2 contents. In addition, temperatures estimated from mineral assemblages may be significantly in error if CH, is present as an additional species, especially if decarbonation equilibria occurring in
I
0.0
0:
I
L
02
03
04
05
%o,
Fig. 5. A portion of the system CaO-A1203SiOx-H20-C02-CHc at BOO bars calculated with XC% = 0.0 (solid curves) and XCH*= 0.2 (dashed curves). Calculations were done with the computer programs of SLAUGHTERrt ul. (1976) modified to incorporate the HSMRK equations for Hz0 and COz (KERRICK and JACOBS, 1981; JACOBS and KERRICK. 1981) and the HSMRK equation for CH, of this study. Sources for the experimental data used are: reaction f 1); NEWTON (1966) and HUCKENHOLZet al. (1975) and reaction (2); HARKER and TUTTLE(1956) and GREENWWDt 1967).
Methane: an equation of state with application to the ternary system HzO-CO&H,
613
SUMMARY
Fig 6. Activity-composition diagram for HsO and COs in HsO-C02-CH, mixtures at 2.5kbar and 673K. Calculated with the HSMRK equation for CH. of this study and the HSMRK equations for Ha0 and CO1 of KERRICKand JACC~BS (1981) for three constant CH, contents (Xnt, = 0.0, 0.05 and 0.20). Dotted portions of curws imply a region of immiscibihty. Dashed fines represent ideal mixing.
compositions [e.g. equilibria (3) and (4) in Fig S] are used in conjunction with other equilibria. Therefore, when fluid ~rn~iti~s and temperatures are estimated from mineral assemblages one must be careful to consider all possible species which may have been present in the fluid phase and their possible effect on the stability of mineral assemblages. The effect of small amounts of CH., on the activities of Hz0 and CO1 in HsOCOs-CH, &rids is even more pronounced at pressures and temperatures ap proximating those of blueschist facies metamorphism (see Fig. 6). KERRICKand JACOBS(1981) suggest that unmixing in binary H&CO2 fluids may occur at T = 673K and P > 20 kbar. As seen in Fig. 6, for a constant Xcu, = 0.05 and XCH, = 0.20 at 673K, 25 kbar, the size of the implied miscibility gap (note o-X curve for HrO) is not markedly affected and the activity of Hz0 remains constant across the miscibility gap. The activity of COz, however, is significantly lowered and changes as X,, increases from 0.15 to 0.45 (the region of unmixing). Examples of systems which exhibit similar behavior can be found in KING (1969, pp. 156-159) and SAXENA(1973, pp. 32-36). For the H#-C02-CH4 ternary system, this behavior may be rationalized by considering, at constant Xc,,, the re~tionship between isoactivity contours and the miscibility gap. The loci of mixed-volatile equilibria which intersect the ternary solvus will he complicated by the corresponding variation in h, within the miscibility gap. Considering mixed-volatile equilibria on isobaric T-X,, sections, the geometry and interpretation of buffering paths will therefore differ for regions of immiscibility in H@-C02-CH, fhrids compared to corresponding buffering in binary H,O-CO, fluids (see KERRICK, 1974, p. 750). H&)-rich
A hard-sphere modified Redlich-Kwong (HSMRK) equation of state for CH, has been developed which is compatible with the HSMRK equations for Hz0 and COz of KERRICKand JAC~IBS(1981). Calculated molar volumes and fugacity coefficients for pure CH,, are in close agreement with those from experimental P-V-T determinations. The HSMRK equation, which predicts linear isochores for CH*, should yield reasonable estimates of fugacity coe&ients for CH, to pressures and temperatures exceeding the maximum P-T conditions of experimental P-V-T data. Calculations in the H@-CO,-CH, system at elevated pressures and temperatures indicate that small amounts of CH4 in H+COs-CH, fluids slightly increases the activity of HrO, while si~i~ntly decreasing the activity of CO,, relative to activities in binary H@-COs fluids. This effect may significantly shift the stability fields of mineral assemblages in equilibrium with HrO-C0s-CH4 fluids relative to those in equilibrium with H@-CO2 fluids. Acknowledgements-This study represents part of G. K. JACOBS’Ph.D. thesis research at The Pennsylvania State University. Comments by A. C. LA~AGA,D. H. EGCZERand P. C. JLJRS on an early draft of the manuscript were very helpful. G. C. Fro~~ns provided a critical review of the manuscript which greatly improved the final version. This work was supported in part by NSF Grants EAR 76-84199 and EAR 79-08244 to D. M. KERRICKand by an OwensComing Fiberglas Graduate Fellowship to G. K. JACOLQ.
REFERENCES ANGUSS., ARM~T~ONO B. and DE REUCK K. M. (1978) International thermodynamic tables of the fluid state-5/ Methane. international Union ofPureand Applied Chemistry, Chemical Data Series 16, 251 pp. Pergamon Press. BEHARF. and PINEN F. (1979) Analyse de COs. Hz0 hydrocarbures des inclusions fluides par chromatographie en phase gazeuze: application aux fentes aipines et aux roches rn~~rno~hiqu~. Buti. Miner& 102. 61 l-621. CARNAHANN. F. and STARLINGK. E. (1969) Equation of state for non-attracting rigid spheres. J. Chem. Phys. 51. 635-636. CRAWFORD M. L., FILERJ. and WOODC. (1979) Saline fluid inclusions associated with retrograde meta~~hism. Bull. Mineral. 102, 562-568. CRAWFORDM. L., KRAUS D. W. and HOLL~VER L. S. (1979) Petroloaic and fluid inclusion studv of talc-silicate rocks; Prince &pert, British Columbia. &r. J. Sci. 279, 1135-1159. DE SANTISR., BREEDVEL_D G. J. F. and PRAUSNITZJ. M. (1974) Th~~ynamic properties of aqueous gas mixtures at advanced pressures. Ind. Eng. Chem. Process Des. Dev. 13.374-377.
EUGSTERH. P. and SK~PPENG. B. (1967) Igneous and metamorphic reactions involving 8as equilibria, In Researches in Geochemistry II (ed. P. II. Abelson), pp. 492-520. Wiley. FLOVVERS G. C. (1979) Correction of Holloway’s (1977) adaptation of the modified Redlich-Kwong equation of state for calculation of the fugacities of molecular species in supercritical fluids of geologic interest. Contrih. Mineral. Perrol. 69. 315-318.
614
GARY K. JACO~~S and DERRILLM. KERRICK
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Folge 22.232-245. FRENCH 8. M. (1966) Some geological implications of equi-
librium between graphite and a C-H-O gas phase at high temperatures and pressures. Rev. Geophys. 4. 223-253. FROST B. R. (1979) Mineral equilibria involving mixedvolatiles in a C-O-H fluid phase: The stabilities of graphite and siderite. Am J. Sci. 279, 10334059. GREENWOODH. J. (1967) Woll~tonite: stability in H2C&C02 mixtures and occurrence in a contact-metamorphic aureole near Salmo, British Columbia, Canada. Am. Mineral. 56, 16741688. GREENWOOD H. J. (1969) The compressibility of gaseous mixtures of carbon dioxide and water between 0 and 500 bars pressure and 450° and 800°C. Am. J. Sci. 267A. 191-208. GREENWOODH. J. (1973) Thermodynamic properties of gaseous mixtures of Hz0 and CO2 between 450” and 800°C and 0 to 500 bars. Am. .I. Sci. 273, 561-571. HARKER R. 1. and TUTTLE 0. F. (1956) Experimental data on the Pm,-T curve for the reaction: calcite + quartz = woliastonite + carbon dioxide. Am. J. Sci. ud 239-256. HENDEJ_ E. and HOLL~STER L. S. (1981) An empirical solvus for CO*-H20-2.6 wt% salt. Geochim. Cosmochim. Acta. 45,225-228.
HOLLISIERL. S. and BURRUS R. C. (1976) Phase equilibria in fluid inclusions from the Khtada Lake metamorphic complex. Geochim. Cosnwchim. Acfa 40, 163-175. HOLL~~TER L. S., BCR~LJ~~ R. C., HENRYD. L. and HENDEL E. (1979) Physical conditions during uplift of metamorphic terranes, as recorded by fluid inclusions. Bull. Mineral. 102 555-561. HOLLOWAY J. R. (1977) Fugacity and activity of molecular species in supercritical fluids. In Thermod&amics in Geotoav led. D. G. Fraser). D. 161-181. Reidel. HU&E&OLZ H. G., H&L E. and ~I~DHU~R W. (1975) Gross&rite, its sotidus and liquidus rehtions in the CaO-A120s-Si02-Hz0 system up to 10 kbar. Neues Jakrb. Mineral. Abh. 124, l-46. JACOSSG. K. and KERRICKD. M. (1981) APL and FORTRAN programs for a new equation of state for HIO. COz, and their mixtures at supercritical conditions. Cornput. Geosci. (in press). KERRICKD. M. (1974) Review of metamornhic mixed-volatile (H,O-CO;) eqbilibria. Am. Mineral: 59. 729-762. KERRICKD. M. and JACOBSG. K. (1981) A modified Redlich-Kwong equation for HtO. COz. and H&-CO, mixtures at elevated pressures and temperatures. Am. J. Sci. 281. in press. KING M. B. (1969) Phase Equjfibrium in Mixrures. New York, Pergamon Press. 584 pp. KONNERUPMADSENJ., LARSEN E. and ROSE-HANSENJ.
(1979) Hydrocarbon-rich fluid inclusions in minerals from the- alkaline Dimaussaq intrusion. South Greenland. Bull. Mineral. 102, 642-653. MEL’NIKY. P. (1978) Thermodynamic properties of carbon monoxide and methane at high temperatures and pressures-a new correlation based on the principle of corresponding states. Geokhimiya 11, 1677-1691. (trawl.) Geo&em. Inr. 6.49-62, 1978. MULLISJ. (1979) The system methane-water as a geologic thermometer and barometer from the external part of the Central Alps. Bull. Mineral. 102, 526536. NEWTON R. C. (1966) Some talc-silicate equilibrium relations. Am. J. Sci. 264, 204-222. OHMOTOH. and KERRICKD. M. (1977) Devolatilization equilibria in graphitic systems. Am. J. Sci. 277. 1013-1044. @~TROVSKY I.A. and RYZHMKO B. N. (1978) Fugacities of gases and some mineralogical reactions at very high pressures and temperatures (an extrapolation). Phys. Chem. Mineral. 2. 297-303.
PRAUSNITZJ. M. (1969) Molecular Thermodynamics of Fluid-Phase Equilibria, 523 pp. Prentice-Hail. Ruha~ H. H., Ou>s R. H.. SAGEB. H. and LACEYW. N. (1944) Phase equilibria in hydrocarbon systems. CH,-C02 system in the gaseous region. Ind. Eng. Chm. 36, 88-90. REDLICH0. and KWONGJ. N. S. (1949) An equation of state. Fugacities of gaseous sohtions. Chem. Reu. 44. 233-244.
ROEDDERE. (1972) The composition of fluid inciusions. In Data ofGeochemistry (ed. M. FL&her), 6th edn, 164 pp. U.S. Geol. Survey Prof. Pap. 440 JJ. RYANI’. A. JR, JOINERB. L. and RYANB. F. (1976) Minicab Student Handbook, 341 pp. Duxbury Press. RYZHENKO B. N. and VOLKOVV. P. (197f) Fugacity coefficients of some gases in a broad range of temperatures and pressures. ~kh~iya 7,760-773. @ran&) Geochem. Int. & 468481 (1971). SAXENAS. K. (1973) Thermodynamics of Rock-Forming Crystailine Solutions, 188 pp. Springer-Verlag. SLAUGHTER J., WALLV. J. and KERRICKD. M. (1976) APL computer programs for thermodynamic calculations of equilibria in P-T-&, space. Contrib. Mineral. Petrol. 54, 1.57-17f. TARAKADR. R., SPENCERC. F. and ALDERS. B. (1979) A comparison of eight equations of state to predict gasphase density and fupacity. fnd. Eng. Chem. Process Des. Dev. lg. 726-739. T~~DHEIDE K. and FRANCKE. U. (1963) Das Zweiphasengebiet und die kritische Kurve im System KohlendioxydWasser bis zu Drucken von 3500 bar. Z. Phys. Chem. 37. 387-W.
TSIKLISD. S., LINSHIT~L. R. and TSSIMMERMAN S. S. Molar volumes and thermodynamic properties of ane at high pressures and temperatures. Dokl. Nauk SSSR I!#& 384-386. (transl.) Dokl. Akad. SSSR I%, 423-425. WECSCH H. (1973) Die Systeme Xenon-Wasser
(1971) methAkad. Nauk
und Methan-Wasser bei hohen D&ken und Temperaturen. Ph.D. Thesis, Institute for Physical Chemistry, Karlsruhe.
00167037/81,~swo7-oBI2.~~ Copyright Q 1981. Pcrgsmon Press Ltd
Methane: an equation of state with application to the ternary system H@-C02-CH4 GARY K. JACOBS*and DERRILLM. KERRICK Department of Geosciences, The Pennsylvania State University, University Park, PA 16802, U.S.A. (Received 4 September 1980; accepted in revisedjmn
25 November
1980)
Abstract-The following hard-sphere modified Redlich-Kwong (HSMRK) equation of state was obtained by least squares fitting to available P-V-T data for methane (P in bars; T in Kelvins; v in cm’mol-‘; b = 60.00cm3mol-1; R = 83.14cm3barmol-‘K-r):
p _ RW + Y + Y' - Y? _ c(T) + d(T)/u + e(T)/u* u(v + b)T’
vu - Y?
y = b,4v c(T) = 13.403 x lo6 + (9.28 x l@)T + 2.7 T2 d(T) = 5.216 x 10’ - (6.8 x 10’)T + (3.28 x 103)T2
e(T) = (-2.3322
x 10”) + (6.738 x 10s)T
+ 13.179 x los)T* For the P-T range of experimental data used in the fit (50 to 8600 bars and from 320 to 670 K), calculated volumes and fupeity coefficients for CH. relative to experimentally determined volumes and fugacity coefficients have average percent deviations of 0.279 and 1.373, respectively. The HSMRK equation, which predicts linear isochores over a wide P-T range, should yield reasonable estimates of fugacity coefficients for CH* to pressures and temperatures well outside the P-T range of available P-V-T data. Calculations for the system HsG-C02-CHI. using the HSMRK equations for Hz0 and CO1 of KENUCKand JACDEZS (1981) and the HSMRK equation for CHI of this study, indicate that compared to the binary HzG-CO2 system, small amounts of CHI in the ternary system H2G-C02-CH4 slightly increases the activity of HsO, and significantly decreases the activity of COz.
HOWAY (1977) demonstrated that the modified Redlich-Kwong (MRK) equation of DE SANT~Set al. IN ADDITK~Nto HZ0 and CO1, methane (CH,) can be (1974) could be used for the estimation of fugacity an important species in geologic fluids (ROEDDER, coefficients for pure HZ0 and CO2 and for H20-C01 1972; HOLL~TER and BURRU~~, 1976; BEWR and mixtures at elevated pressures and temperatures. PINEAU, 1979; CRAWFORDet al., 1979a, b; HOUIS’IER HOLLOWAY(1977) combined his MRK equations for et al., 1979; KONI&RUP-MADSMet. al., 1979; MULLIS, HZ0 and CO1 with original Redlich-Kwong (RK) 1979; HENDELand HOLLLTITR,1981). The presence of equations (REDLICHand KWONG, 1949) for several small amounts of CH* in H+CO&H, fluids may other gases (CO, CH, and HZ) to calculate fluid comsignificantly affect the stabilities of mineral assempositions in the C-O-H system.t RYZHENKOand blages relative to assemblages in equilibrium with bi- VOLKOV (1971), MEL’NIK (1978) and O~TR~~~KY and nary H@-C02 fluids. Therefore, it is important to be RYZHENKO(1978) have developed equations of state able to calculate the thermodynamic properties of for several gases, including CH*, but none are readily CH* at elevated temperatures and pressures, both as adapted to the estimation of fugacities for species in a pure species and in H+COs-CH, mixtures. Calmulticomponent mixtures. culations of fluid compositions in the C-O-H-(* S) KERRICKand JAMBS (1981) have developed a hardsystem at elevated pressures and temperatures (for sphere modified Redlich-Kwong equation for H20, example FRENCH, 1966; EUIXITX and SKIPPEN,1967; CO1, and HIO-CO1 mixtures. It recently has been OHMOID and KERRICK, 1977; FROST 1979), where shown (TARAKADet al., 1979) that such equations of CHI can be an abundant species under certain constate, utilizing a hard-sphere repulsive term (CARNAditions, will also be improved if accurate fugacity HAN and STARLING, 1969), may be superior to other coefficients for CH* can be calculated. equations predicting thermodynamic properties of Resent address: Department of Geology, Grand Val- supercritical, aqueous fluid mixtures. FLOWERS (personal communication) has suggested that, in general, ley State Colleges, Allendale, MI 49401, U.S.A. t HOLLOWAY'S (1977) analysis for mixtures is in error molecular dynamics models of repulsive terms pro(see FLOWERS1979). vide better agreement for supercritical fluids, although
INTRODUCTION
l
ci.c.*. 45:5--*
607
608
GARYK. JACOBS and DERRILLM.KERRICK
functions of a(v). which could be fit to the equation: such treatment is not necessarily limited to a hardsphere model. By fitting the hard-sphere modified a(v) = c(T) + Redlich-Kwong (HSMRK) equation of KERRICKand 1’ V2 JACOBS (1981) to available P-V-T data for CHI (TSIKLISet al., 1971; ANGUSet al., 1978; FRANCESCONI, Substitution into eqn (1) yields: 1978), a HSMRK equation for CHI has been develp _ RT(1 + y + y’ - y3) oped which accurately predicts molar volumes and v( 1 - y)3 fugacity coefficients within the P-T limits of the experimental P-V-T determinations, and should yield c(T) + d(T)jv + e(T)/d reasonable estimates of molar volumes and fugacity (3) T*v(v + b) coefficients for CH* at pressures and temperatures above the P-T limits of the experimental P-V-T de- Using a value of b = 60cm3 mol- ‘, least squares terminations. In addition, because the equation is analysis yields the following temperature functions for compatible with the HSMRK equation of KERRICK c, d and e (stated errors give 95”/, confidence intervals) : and JACOBS(1981) for Hz0 and CO*, activities in the c(T) = (13.403 It: 6.9) x lo6 system H#-C02-CH4 may be calculated for a wide
do + e(T)
range of P-T conditions.
+ (9.28 k 2.9) x lo4 T + (2.7 f 28.6)T2
NOMENCLATURE ai
activity of species i
4T.v)
attractive term function of HSMRK equation (bar cm6 K* molwL) anorthite covolume (cm3 mol- I) parameter in attractive term function (bar cm6 K* mot- *) calcite parameter in attractive term function (bar cm9 K’ molm3) parameter in attractive term function (barcm’2K*moI-4) fugacity grossular chemical components subscript denoting a mixture number of moles pressure (bars, unless specified otherwise) quartz gasconstant (83.14cm3barK-‘mol-‘) temperature (Kelvins, unless specified as Celsius) total volume (cm’) molar volume (cm’ mol- ‘) wollastonite mole fraction
an b c cc d r f gr ij M ; (32 R T V v wo X
reduced density (b/4u) fugacity coefficient (f/P) of pure species i fugacity coefficient of species i in a mixture
EQUATION OF STATE The following HSMRK equation (KERRICK and JACOBS,1981) was fit to available P-V-T data for CH4: P=
RT(1 + y + yL - y3) ~(1 - JJ)~
a(T, 4 - T+o(o + b)
(1)
Sources for the molar volumes (293 observations) used in the least squares fitting, performed with ‘Minitab 2’ (RYAN et al., 1976). were: (1) ANGUS ef al. (1978); 320-470K and 50-3000 bars; 470-670K and 50-400 bars, (2) FRANCESCONI (1978); 47&670K and 300-2200 bars, and (3) TSIKLISet al. (1971); 320-670K and 2025-8600 bars. The approach, detailed by KERRICK and JACOBS(1981), consisted of using P-V-T data to find a value for h which resulted in isothermal
(4)
d(T) = (5.216 + 0.88) x lo9 - (6.8 + 3.7) x 106T + (3.28 & 3.7) x
103T2
(5)
e(T) = (- 2.3322 + 0.32) x 10” + (6.738 f 1.3) x lOaT + (3.179 + 1.3) x 105TL
(6)
The hard-sphere repulsive term [the first term on the right side of eqn (I)] has been derived by CARNAHANand STARLING(1969). It is interesting to note that when y - 0 (as v becomes very large), this term reduces to the same form as the van der Waals repulsive term [RT/(u-b)]. The attractive term [the second term on the right side of eqn (l)] combines the denominator of the original Redlich-Kwong equation (REDLICHand KWONG, 1949) with a numerator which is a function of temperature and volume, rather than a constant value as proposed by RJZDLICHand KWONG(1949). The numerator, which consists of temperature-dependent coefficients in an inverse volume relationship [see eqn (2)], closely resembles a virial expansion. This form of the attractive term lends support to the choice of mixing rules [eqns (10) and (1 l)], since these mixing rules can be derived directly from a virial equation for gas mixtures (PluusNrrz, 1969, pp. 98-100). The b value for CH4, which represents the covolume of the gas. is approximately twice the value predicted by the original RK equation (REDLICHand KWONG, 1949). The b values for H20 and CO2 in KERRICK and JACOBS(1981) are similarly approximately twice the value predicted by the original RK equation. Because the value of the covolume (the volume of 1 mol of close-packed gas molecules) is difficult to evaluate precisely (i.e. measure), and because the HSMRK is a modified form of the original RK equation, the physical significance of the difference in b values for the two equations is questionable. Additionally, because all versions of the MRK equation. including that of the present study, are seni-empiri-
Methane: an equation of state with application to the ternary system H20-C02-CH4 cal, parameters such as b have limited physical significance. The key requirement for b values in our HSMRK equation is that a(r) functions fit the form of eqn (2), so that reasonable estimates of fugacity coefficients can be calculated at pressures exceeding the maximum pressure of P-V-T measurements (see KERRICKand JACOBS,1981). THERMODYNAMIC PROPERTIES OF THE HIO-COI-CH, SYSTEM After substitution of eqn (3) for P, fugacity coefficients for a pure species can be calculated by evaluating (PRAUSNITZ,1969, p. 41):
- RT In Z + RT(Z - l),
(7)
609
calculated with eqns (4-6) for CH*, and eqns (8-13) of KERRICKand JACOBS(1981) for Hz0 and C02. Equations (IO) and (11) are theoretically valid only for the virial equation of state. However, it was shown earlier that the attractive term chosen for this study [eqn (2)] closely resembles a virial expansion, thereby supporting the choice of eqn (10) and (11) for the mixing rules of the attractive term parameters. PRAUSNITZ (1969, p. 44) suggests that substitution of a term representing molecular radii (b113)rather than molecular volume (b) in a mixing rule such as eqn (9) does not significantly affect the resulting fugacity coefficients at moderate pressures, although the effect has not been studied at high pressures. The fugacity coefficient of a species i in a mixture is calculated by the equation (F’RAUSNJTZ 1969, p. 41): RTln4?=
6
[(3,.“..,
- ?]dF
which yields:* -RTlnZ IntiP”“=
8y - 9Jf2 + 33’” --_
(1 - Y?
In Z -
RT3’2(v + b)
d e RT3’2u(v + b) - RT3’2u2(o + b)
Integrating and rearranging (12), after substitution of eqn (3) for P with b, c, d,,,, and e,,, substituted for b, c. d and e yields: In 4;“”
d - RT”‘bv -
(12)
C
4Ym - 3Y; + 5 4Yln- 2Yi = (1-Ym)2 ( bm’ (1 - Y,)’ )
e RT3122bv2
e
+-RT312b2v
(8)
Evaluation of eqn (8) is accomplished by first solving eqn (3) for v at the pressure and temperature of interest by an iterative technique. Fugacity coeficients for CH,, CO2 and H20 in mixtures of H20-C02-CH. can be calculated after first choosing a set of mixing rules for b, c, d and e. KERRICK and JACOBS(1981) demonstrate that despite the controversy concerning mixing rules, those originally proposed by van der Waals appear adequate for H20-C02 fluids. These same mixing rules were therefore chosen for an analysis of the system H20-C02-CH,:
&em + RT3’22b,v;(v,
b, = F bixi + Cm =
C 1 CijXiXj i i
[&.*t(‘+)]
(10) -
3hem RT3’2b&,
where C[j =
(CfCj)+
3he, + b,) - RT3’22b;u,(v,,, + b,)
(11)
Equations (10) and (1 l), which also apply for d,,, and are evaluated after the values for c, d and e are
e,,
* KERRICKand JACOBS(1981) present a detailed derivation of eqn (8).
+ b,)
- InZ
KERRICK and JACXXS (1981) provide a detailed derivation of the expression In 41’” for binary H20-C02 fluids, which is easily extended to the ternary system H20-C02-CH9 [eqn (13)J. Note that it is necessary to solve eqn (3) for v, at the pressure and temperature of interest after substitution of b,, c,,,, d,,, and e, for b, c, d and e. The required expres-
610
GARY
K. JACOES and DERRILLM. KERRICK
sions for P,, Pd and P, are partial derivatives of the mixing rules, [eqns ( 10) and (1 1)] : P, = 2Cf C CfXj
I -
P* = 2d) 1 df Xj + d,
(15) I
C t?jXj+2e,
P,=2e!
,
The activity of species i is then calculated from: a, =
(rpf”‘” ‘> xi
(17)
4p”’
It is important to note that eqns (13-16) only when eqn (12) is evaluated using eqn bined with mixing rules (9-11). Any change the mixing rules or the form of the HSMRK invalidates eqns (13-16).
are valid (3) comin either equation
400 COMPARISON OF CALCULATED VOLUMES AND FUGACITY COEFFICIENTS WITH PREVIOUS WORK
In order to evaluate the accuracy and usefulness of our HSMRK equation for CH,, two analyses are important: (1) a comparison. of calculated molar volumes and fugacity coefficients with experimentally determined volumes and fugacity coefficients and (2) an evaluation of the accuracy of molar volumes and fugacity coethcients calculated at pressures and temperatures above the P-T limits of experimental P-V-T determinations. Figure 1 is a comparison between volumes calculated with eqn (3) and the
-2
0 2 4 Percent Dtfference
Fig. 1. Histogram showing the frequency of deviation of CH* molar volumes calculated with eqn (3) from experimental P-V-T measurements (TSIKLISet al., 1971; ANGUS et
al.. 1978; FRANCESCONI. 1978). Percent difference = (v,.,~ - c._,) x ~OO/U,,~.
800
1200
1600
T(K) Fig. 2. Isochores (cm’ mol- ‘) of CH, calculated with eqn (3) as a function of pressure and temperature. Circles represent P-T coordinates calculated with eqn (3).
experimentally determined volumes used in the least squares regression of eqn (1). The average percent deviation of the calculated volumes compared to the experimental volumes is 0.28%, with few values exceeding l.oo/,. The largest differences (3-4%) occur only at the highest temperatures (T > 550K) and at approx. 8ooObars. Fugacity coefficients derived from experimental data at T > 320K are relatively scarce. ANGUS et al. (1978) supply the most comprehensive set of fugacity coefficients. The agreement between fugacity coefficients calculated with eqn (8) and those at P I 1000 bars and T = 320-67OK from ANGUS et al. (1978) is always better than 1.0%. At pressures above looobars (up to P= 10 kbar), the differences range from approx. 2-18x, with most being approx. 2-S%. Interpretation of these differences at P > loo0 bars is difficult, however, because ANGUSet al. (1978) used no experimental data in their analysis above 1000 bars for temperatures greater than 310K. Figure 2 may be used to attempt to evaluate how well our HSMRK equation for CH, extrapolates to pressures and temperatures above the P-T limits of the experimental data used in the regression of eqn (1). Each of the four isochores in Fig. 2, though somewhat curved, can be represented by a straight line. The calculated isochores remain essentially linear to ZOOOK (P=4kbar for c = 75; P = 15kbar for c = 45). Though not shown in Fig. 2, a calculated isochore for I: = 35 is also a linear function to 2000K (P = 35 kbar). The linear isochores imply that volume and fugacity coefficients may be reasonably estimated at P-T conditions well beyond those oi existing
Methane: an equation of state with application to the ternary system H20-C02-CH4
611
study involve extrapolations exceeding the P-T limits of available P-V-T data, the significance of the large differences at the higher pressures and temperatures is tions cannot be precisely evaluated. difficult to evaluate. One of the main advantages of OSTF~OVSKY and RYZHENKO(1978) calculated the fugacity of CH4 at T 2 lOOOK and P L 20 kbar by the HSMRK equation over the corresponding state evaluating a volume function obtained through equations of RYZHENKOand VOLKOV(1971) and graphical extrapolation of lower pressure-temperaMEL’NIK(1978), however, is that the HSMRK quature compressibility factors. However, since their distion is easily adapted to the calculation of activities in cussion of the method and results is brief, a detailed multicomponent mixtures. comparison is precluded. RYZHENKO and VOLKOV KERRICKand JACOBS(1981) have shown that their HSMRK equation predicts volumetric properties of (1971), using a corresponding states approximation, Hz0-C02 fluids that are in reasonable agreement calculated fugacity coefficients for CH4 from 373 to 1273K and from 100 to 1OOOObars.Comparison of with the available experimental data (FRANCK and T~~DHEIDE,1959; T~~DHEIDEand FRANCK, 1963; their fugacity coefficients with those calculated with GREENWOOD, 1969, 1973). Available experimental data our HSMRK equation [eqn (8)] yields minimum and maximum differences of -94.1 and +9.70/, respectfor COz-CH4 mixtures (REAMER et al., 1944) is at temperatures (T s 510K) too low for calculations ively, with an average difference of approx. -5.37;. involving mixtures with the HSMRK equation for The poorest agreement is at T -c 650K and P > 6000 bars. These large differences may be attriCO2 (KERRICKand JACOBS,1981; JACOSSand KERbuted to the fact that the data of FRANCESCONI (1978) RICK, 1981); thus, a comparison with the data of REAMERet al. (1944) is precluded. However, experiand TSIKLISet al. (1971) were not incorporated into of volumes in the system the equation of state for CH4 of RYZHENKOand VOL- mental determinations H+CH4 (WELSCH,1973) at 673K and from 400 to KOV(1971). MEL’NIK(1978), utilizing another version 1000 bars allows an evaluation of u, calculated with of the corresponding states principle, calculated fugaour HSMRK equation for this binary system. Figure 3 city coefficients for CH4 from 400 to 15000 bars. For most of the P-T conditions tabulated by MEL’NIK shows that calculated values of c, for HrO-CH4 (19783, his fugacity coefficients are systematically low mixtures at 673K and from 400 to loo0 bars are in close agreement with the experimental values of when compared to those calculated with our WELXH (1973). Unfortunately, there is presently no HSMRK equation [qn (8)]. The minimum and maxidata for HtO-C02-CH4 mixtures mum differences of -36.5 and +3.9% occur at experimental which allows an evaluation of the accuracy of activi14OOK-7 kbar and 6OOK-5 kbar, respectively, and ties for H20, CO* and CH4 in this ternary system the average deviation is approx. - 5.8%. Because both calculated with our HSMRK equation. However, the the method of MEL’NIK(1978) and that of the present agreement of the calculated properties with the experimentally determined properties for the binary systems discussed above strengthens the application of the HSMRK equation to the H@-C02-CH4 system. P-V-T data. It must be noted that until more data is available, however, the accuracy of such extrapola-
IMPLlCATJONS Calculations for the system H#-C02CH4 indicate that compared to activities in binary H20-C02 fluids, small amounts of CH4 in H20-C02-CH4 fluids may significantly affect the activity of HZ0 and CO*. Figure 4 illustrates this effect for pressures and temperatures corresponding to typical contact and regional metamorphic conditions (Figs 4a and b, respectively). For both P-T regimes, the presence of up to 20mol% CH4 slightly increases the activity of HZ0 relative to its value in binary H20--CO2 fluids, whereas the activity of CO2 is significantly decreased. Similar relations were suggested by HOLLOWAY (1977), although the increase in the activity of HZ0 was more 0.6 0.8 1.0 0 0.2 0.4 pronounced in HOLLOWAY’S (1977) analysis. However, XH,O FLOWERS(1979) has pointed out a problem with HOLLOWAY’S (1977) analysis, which when accounted for, Fig. 3. Comparison of calculated and experimental values of u,,,~~for the H,O-CH, system at 673K. The solid lines may lower the activity of HZ0 somewhat in Fig, 2 of were calculated with the HSMRK equation for CH4 of this HOLLOWAY(1977). Since CH4 and CO2 are quite study and the HSMRK equation for H,O of KERRICKand JACOBS(198 1). The circles are experimental data of WELSCH similar, in being non-polar molecules (although CO2 does have a quadrupole moment) and having average (1973).
GARY ILJACOBS and DERRIU M.
612
02
0.0
0.4
(a)
0.6
0.8
1.0 0.0
xco*
KERRICK
0.4
0.2
0.6
0.8
1.0
xco,
lb)
Fig. 4. Activity~om~s~tion diagram for Hz0 and CO2 in H#-CUZ-CH~ mixtures. (a): loDo bars 723K; (b): 6000 bars, 873K. Calculated with the HSMRK equation for CHr of this study and the HSMRK equations for Hz0 and CO* of KERRICKand JACOBS(1981) for three constant CHI contents (XC”, = 0.0, 0.05 and 0.20). Dashed lines represent ideal mixing. polarizabilities
of 26 x IOz5 and 26.3 x 10” cm3 p. 60), respectively, the relations in
(PRAUSNITZ, 1969,
Figs 4a and b are not entirely unexpected. In particular, the attractive forces between HZ0 and CHI molecules and between HZ0 and CO2 molecules should be quite similar, since the induction forces (a function of polarizability) between H#-CH4 and HIO-COL molecules, set up by the electric field from nearby polar HZ0 molecules, should be similar in magnitude. In addition, dispersion forces (a function of polarizability and first ionization potential) resulting from H&kCO* and H#-CHI interactions should be similar, since the first ionization potentials of CH.+ and COZ are similar (PRAUSNITZ, 1969, p. 63). This interpretation of the molecular interactions explains the relativefy small increase in the activity of HzO. The reduced activity of CO1 can be attributed lo simple dilution by a similar molecule (CH,). The lowered activity of CO1 in the H*O-rich portion of the H20-C02-CH, system due to the small amounts of CH,, may shift the stability fields of mineral assemblages in equilibrium with the fluid relative to mineral assemblages in equilibrium with binary H&3-CO2 fluids at the same P-T conditions. Figure 5 illustrates this effect for a portion of the system CaO-Al~O~-SiOt-H#-COz-CH~. The Xco, of the isobaric invariant point resulting from the inter~ion of equilibrium (1) and equilib~um (2) is increased by approx. 5 mol% for a fluid with X CH. = 0.20 relative to a fluid with XCHI = 0.0. The expansion of the stability fields for grossular and woliastonite is of ,particular im~rtan~. since grossular and wollastonite are generally considered excellent indicators of H20-rich fluids. However, the presence of
CHo permits assemblages containing grossular or wollastonite to coexist with fluids having considerably higher CO2 contents. In addition, temperatures estimated from mineral assemblages may be significantly in error if CH, is present as an additional species, especially if decarbonation equilibria occurring in
I
0.0
0:
I
L
02
03
04
05
%o,
Fig. 5. A portion of the system CaO-A1203SiOx-H20-C02-CHc at BOO bars calculated with XC% = 0.0 (solid curves) and XCH*= 0.2 (dashed curves). Calculations were done with the computer programs of SLAUGHTERrt ul. (1976) modified to incorporate the HSMRK equations for Hz0 and COz (KERRICK and JACOBS, 1981; JACOBS and KERRICK. 1981) and the HSMRK equation for CH, of this study. Sources for the experimental data used are: reaction f 1); NEWTON (1966) and HUCKENHOLZet al. (1975) and reaction (2); HARKER and TUTTLE(1956) and GREENWWDt 1967).
Methane: an equation of state with application to the ternary system HzO-CO&H,
613
SUMMARY
Fig 6. Activity-composition diagram for HsO and COs in HsO-C02-CH, mixtures at 2.5kbar and 673K. Calculated with the HSMRK equation for CH. of this study and the HSMRK equations for Ha0 and CO1 of KERRICKand JACC~BS (1981) for three constant CH, contents (Xnt, = 0.0, 0.05 and 0.20). Dotted portions of curws imply a region of immiscibihty. Dashed fines represent ideal mixing.
compositions [e.g. equilibria (3) and (4) in Fig S] are used in conjunction with other equilibria. Therefore, when fluid ~rn~iti~s and temperatures are estimated from mineral assemblages one must be careful to consider all possible species which may have been present in the fluid phase and their possible effect on the stability of mineral assemblages. The effect of small amounts of CH., on the activities of Hz0 and CO1 in HsOCOs-CH, &rids is even more pronounced at pressures and temperatures ap proximating those of blueschist facies metamorphism (see Fig. 6). KERRICKand JACOBS(1981) suggest that unmixing in binary H&CO2 fluids may occur at T = 673K and P > 20 kbar. As seen in Fig. 6, for a constant Xcu, = 0.05 and XCH, = 0.20 at 673K, 25 kbar, the size of the implied miscibility gap (note o-X curve for HrO) is not markedly affected and the activity of Hz0 remains constant across the miscibility gap. The activity of COz, however, is significantly lowered and changes as X,, increases from 0.15 to 0.45 (the region of unmixing). Examples of systems which exhibit similar behavior can be found in KING (1969, pp. 156-159) and SAXENA(1973, pp. 32-36). For the H#-C02-CH4 ternary system, this behavior may be rationalized by considering, at constant Xc,,, the re~tionship between isoactivity contours and the miscibility gap. The loci of mixed-volatile equilibria which intersect the ternary solvus will he complicated by the corresponding variation in h, within the miscibility gap. Considering mixed-volatile equilibria on isobaric T-X,, sections, the geometry and interpretation of buffering paths will therefore differ for regions of immiscibility in H@-C02-CH, fhrids compared to corresponding buffering in binary H,O-CO, fluids (see KERRICK, 1974, p. 750). H&)-rich
A hard-sphere modified Redlich-Kwong (HSMRK) equation of state for CH, has been developed which is compatible with the HSMRK equations for Hz0 and COz of KERRICKand JAC~IBS(1981). Calculated molar volumes and fugacity coefficients for pure CH,, are in close agreement with those from experimental P-V-T determinations. The HSMRK equation, which predicts linear isochores for CH*, should yield reasonable estimates of fugacity coe&ients for CH, to pressures and temperatures exceeding the maximum P-T conditions of experimental P-V-T data. Calculations in the H@-CO,-CH, system at elevated pressures and temperatures indicate that small amounts of CH4 in H+COs-CH, fluids slightly increases the activity of HrO, while si~i~ntly decreasing the activity of CO,, relative to activities in binary H@-COs fluids. This effect may significantly shift the stability fields of mineral assemblages in equilibrium with HrO-C0s-CH4 fluids relative to those in equilibrium with H@-CO2 fluids. Acknowledgements-This study represents part of G. K. JACOBS’Ph.D. thesis research at The Pennsylvania State University. Comments by A. C. LA~AGA,D. H. EGCZERand P. C. JLJRS on an early draft of the manuscript were very helpful. G. C. Fro~~ns provided a critical review of the manuscript which greatly improved the final version. This work was supported in part by NSF Grants EAR 76-84199 and EAR 79-08244 to D. M. KERRICKand by an OwensComing Fiberglas Graduate Fellowship to G. K. JACOLQ.
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