Contrib Mineral Petrol (1994) 117: 362-374
9 Springer-Verlag 1994
S. Michael Sterner 9 Kenneth S. Pitzer
An equation of state for carbon dioxide valid from zero
to extreme pressures
Received: 24 November 1993 / Accepted: 17 March 1994
Abstract A new form of equation of state is described
with application to carbon dioxide from 215 K to
T > 2000 K and from zero pressure to more than 105 bar
(10 GPa). The equation was calibrated using properties
predicted by existing formulations at low to moderate
PTconditions, original experimental PVTdata at higher
pressures, corresponding states comparisons at higher
temperatures and using shock compression data at still
higher PTs. Extensive comparisons illustrating the correlation of our new EOS with available phase equilibria
and volumetric data are provided. Fugacities of carbon
dioxide at high pressures and temperatures predicted
using our EOS are in agreement with mineral equilibria
calculated from internally consistent thermodynamic
data for minerals.
Introduction
Thermophysical data bases for water and carbon dioxide pure-component fluids are now available over extensive ranges of both temperature and pressure. While a
number of equations of state (EOS) accurately describe
the physical and thermodynamic properties of water
over much of the PTrange of geological interest (Haar
et al. 1984; Saul and Wagner 1989; Hill 1990), correlations of the thermophysical properties of carbon dioxide
over the PT range of available information generally
S. M. Sterncr 1 (~])
Bayerisches Geoinstitut, Universitfit Bayreuth,
Postfach 1012 51,
D-95440 Bayreuth, Germany
K. S. Pitzer
Department of Chemistry and Lawrence Berkeley Laboratory,
University of California,
Berkeley, CA 94720 USA
1 Present address:
Battelle Pacific Northwest Laboratory,
Riehland, WA 99352, USA
Editorial responsibility: J. Hoefs
take the form of separate equations valid over restricted
PTranges or single analytical expressions with different
numerical values for adjustable parameters corresponding to different regions of P, T and p. Although such
equations have been successful in describing properties
of interest over limited ranges of physical conditions,
they frequently extrapolate poorly outside the limits of
the primary data base from which they were derived.
Those formulations having general validity over extended PTranges tend to reproduce the available therlnophysical data for carbon dioxide with considerably lower accuracy.
The accuracy with which a given algebraic formulation can be made to reproduce the available data is
largely dependent on the number of adjustable parameters (assuming a suitable choice for the general form of
the equation). While increasingly complex formulations
may allow improved correlation with available data,
they also yield additional composition-dependent adjustable parameters that must be evaluated upon extension to binary or higher order fluid mixtures. Thus, extension of any of the above EOS for water to mixed
fluids would result in an expression of enormous complexity - one for which it is doubtful that the adjustable
parameters could be adequately constrained using the
information presently available for fluid mixtures.
An immediate attraction to the simpler EOS formulations is their potential for extension to higher-order
systems. Thus, many of the extended P T range equations, in combination with appropriate mixing rules and
limited physical data, can provide reasonable predictions of mixed-fluid properties that are useful in some
geochemical applications. The accuracy of predictions
by these EOS is limited by the accuracy with which they
describe the pure-component endmembers and the
availability of data for calibrating the adjustable
parameters in composition-dependent terms. For most
geologically important fluid mixtures, the latter consideration entails the calibration of mixing terms using experimental data at low pressures and temperatures. Provided the pressure and temperature dependence of the
363
mixing terms is simple, that the compositional dependence itself is not unduely complex and that magnitudes
of properties predicted for mixtures at high P and Tare
dominated by terms describing the pure component
endmembers, reasonable predictions at extreme conditions may be forthcoming (subject to our ability to evaluate them).
Here we introduce a new form of equation of state for
fluids with application to the CO2 pure-component system that is continuously valid from zero to extreme
pressures ( > 10 GPa) and temperatures from the CO2
triple point ( ~ 215 K) to T > 2000 K. The EOS has been
calibrated over a broad range of temperatures and pressures using available thermophysical data for carbon
dioxide augmented by corresponding states comparisons with other substances. Extensive comparisons illustrating the correlation of our EOS with the available
data for CO2 are provided. Potential application of the
new form of EOS to mixed fluids (specifically, CO2-H20)
is discussed in a companion paper (Pitzer and Sterner
1994).
Data base
At low-to-moderate pressures and temperatures there is
an extensive array of thermodynamically interrelated
data on the properties of pure CO2. Many of these data
were used or considered by Ely et al. (1989) during the
development of their 32-term EOS which describes the
thermophysical properties of CO2 between 215 and
330 K and below 300 bar (including the region of liquidvapor separation) with high accuracy. The correlation
equation of Altunin and Gadetskii (1971)predicts CO2
properties over the larger range of temperature and
pressure to ~ 1273 K and 3 kbar. This equation was
adopted in the IUPAC study of Angus et al. (1976) after
evaluation of the available experimental data and these
authors provide extensive discussions of the agreement
between available data bases and EOS-predicted properties. These two equations are in good agreement in the
region of common validity.
At higher pressures there are primary volumetric
data: (a) for 2_<P_<7kbar and T=323, 373, 473, 573
and 673 K (Tsiklis et al. 1969), (b) for 1 _<P_<8 kbar and
T-681.35, 789.35, 883.45 and 980.65 K (Shmonov and
Shmulovich 1974), and (c) for 323-750 K and 0.5
4.1 kbar (Juza et al. 1965). Information at very high
pressures and temperatures is available from shockcompression experiments (Nellis etal. 1991; Schott
1991). Theoretical intermolecular potentials derived
from these data have been used to predict volumetric
properties of high-density liquid CO2 at pressures and
temperatures away from the primary Hugoniot. At high
pressures and low temperatures (296 K) there are data
from diamond-anvil pressure cell measurements on
solid CO2 (Olinger 1982; Liu 1984). Finally, experimental data are available on the equilibrium boundaries in
PTspace of five mineralogical decarbonation reactions
occurring between 800 and 1800 K and at pressures to
~ 4 5 kbar (most information in the range 1200-1800 K
and 10-45 kbar). These data may be used in conjunction
with thermodynamic data for the mineral phases to constrain the fugacity of CO2 over the corresponding PT
range.
The primary data base for parameter evaluation of
our CO2 EOS was constructed as follows. Values of
Pc=73.748 bar, Tc=304.127 K and pc=0.01063 mol/
cm 3 were accepted for the critical parameters based on
discussions in Ely et al. (1989) and Altunin and Gadetskii (1971). At low pressures, we adopted values predicted by the Ely et al. (1989) EOS for 215 _<T< 330 K and
P < 300 bar. This includes values for liquid and vapor
densities and saturation pressures along the two-phase
curve. For pressures and temperatures outside this region but below 1200 K and 2 kbar, we used values calculated with the Altunin and Gadetskii (1971) EOS;
there seemed no need to re-evaluate the extensive array
of experimental data that was used in the construction
of these two equations. Over the range 215 < T_< 1000 K
and 2 < P < 8 kbar we used the primary experimental
data from Ysiklis etal. (1969) and Shmonov and
Shmulovich (1974) as well as values predicted using Eq.
5 of Shmonov and Shmulovich (1974) for intermediate
temperatures. This equation describes their data and
also those of Tsiklis et al. (1969) and Juza et al. (1965)
very well at pressures above 2 kbar. Since the temperature dependency is very simple, this equation was assumed to be valid for temperatures below that of the
lowest measurement (i.e., down to those of the supercooled liquid, ~ 100 K below).
At still higher pressures and temperatures, we
interpolated from the curves of Nellis et al. (1991) which
include the results of Schott (1991) for a series of
values along the shock-compression Hugoniot for
1000 < T< 4000 K. The data base was further augmented in this region by some values calculated for us by
Dr. F.H. Ree from his model potential which fits both
the shock-compression data and the static measurements of Shmonov and Shmulovich (1974). Specifically,
this was an "exponential-six" potential (Ree 1983)
with parameters of "Set A" of Nellis et al. (1991). The
values we used were for the range 800<T_<2000 K
and densities of 1.1 <Pco2<2.0 g/cm 3. At pressures of
80 _ P < 500 kbar and low temperatures we included
some data for solid, ultra-compressed CO2 from the diamond-anvil study of Liu (1984). We assume that the
pressure for the liquid exceeds that of the solid at the
same density, but only by a small amount in the range of
very high total pressure.
Finally, values for the second virial coefficient included in the data base were generated using the Ely et al.
(1989) EOS for 215 < T< 330 K, and the Altunin and
Gadetskii (1971) EOS for 330< T< 1200 K. For higher
temperatures, BT values were estimated using corresponding states comparisons with data from Dymond
and Smith (1980) for N2 to 1600 K, Ne to 3000 K and
He to 40,000 K.
364
Table 1 Coefficients c~,i of EOS for CO2
Ci,1
Ci,2
1
***
2
***
3
***
4
***
5
***
6
***
7--0.39344644E+12
8
***
9
***
10
***
Ci,3
***
***
***
***
***
***
+0.90918237E+8
***
+ 0.22995650E + 8
***
Ci,4
+0.18261340E+7
***
***
--0.13270279E + 1
+0.12456776E+0
***
+0.42776716E+6
+0.40282608E+ 3
-0.78971817E+5
+0.95029765E+ 5
+0.79224365E+ 2
+ 0.66560660E-- 4
+0.59957845E-- 2
-0.15210731E+0
+0.49045367E+ 1
+0.75522299E+0
-0.22347856E+2
+0.11971627E+3
-0.63376456E+ 2
+0.18038071E+2
Ci,5
***
+0.57152798E-- 5
+0.71669631E--4
+ 0.53654244E- 3
+ 0.98220560E- 2
***
***
***
***
***
Ci,6
+ 0.30222363E- 9
+0.62416103E- 8
--0.71115142E--7
+ 0.55962121E-- 5
Equation of state
Fitting procedure
The new equation of state for CO2 written as an explicit
equality for the residual Helmholtz energy is:
The coefficients of the CO2 EOS were evaluated from
the primary data base using a semi-global, non-linear,
least-squares optimization procedure. In the first phase,
the coefficients ai of Eqs. 1 and 3 were optimized along
the ciritical isotherm (T~= 304.127 K) in an isothermal
regression using primary volumetric data from the
sources given above (including the diamond-anvil data
for ultra-compressed solid CO2 at T~ To), as well as
residual Helmholtz energy data up to ,,~2 kbar calculated with the Ely et al. (1989) and Altunin and Gadetskii
(1971) equations, a value for the second virial coefficient
o f B r = -116.1 cm3/mol from the Ely et al. (1989) EOS
and the requirements of OP/Sp= 0 and 02p/sp 2 = 0 at the
critical density.
The critical isotherm parameters were then held constant while the temperature dependence of the coefficients al of Eqs. 1 and 3 (i.e., coefficients c;j of Eq. 2) were
evaluated in a polythermal regression using the remainder of the data described above, with the additional requirement of the equality of chemical potentials between liquid and vapor at Tand Psat (as calculated from
Eqs. 2, 3 and 4 in Ely et al. 19891). Fugacity data for CO2
from mineral equilibria calculations were not used in
the evaluation of our EOS parameters because of the
uncertainties in the various available mineral data bases
as well as in the experimental location of the reaction
boundaries.
(
Ar~S/RT = alp +
1
/)
a2 + a3p + a4p 2 + asp 3 + a6p 4
-
-
--(a~jo) (ex p aIoP- 1)
(l)
where p is the density in mol/cm 3 and a~ through alo have simple
temperature dependences represented by combinations of various
terms in the polynomial:
ai=c~, ~ T-4-}-c,,2 Z-2-}-ci,3 r - l §
rq-c~, 6 T 2 ( T i n K). (2)
Values of the coefficients cij are given in Table 1. A total
of 28 non-zero parameters were used to describe the
data base satisfactorily.
The form of Eq. 1 affords an excellent quantitative
description of the properties of CO2 over a broad range
of densities from ideal gas behavior in the limit of zero
pressure (zero density) to approximately seven times
critical density. At high densities, the first term in Eq. 1,
al p, is dominant. The second term exerts major influence at intermediate densities whereas the exponential
terms guide the behavior in the region from low to nearcritical densities.
The EOS expressed in terms of pressure has the following form:
p ,--e
/RT=p+a,p--p
2 [ a3 + 2a4p + 3aspZ + 4aop 3 \
~(a2+a3P+a4pZ+a,p3+a6p4)2)
Comparisons
+ aTp2exp-a~p
+ agpaexp ,,,op
Volumetric data
(3)
where the temperature dependences of the coefficients a~ are given
by Eq. 2 above.
In the limit of zero pressure, the second virial coefficient,
B T = O ( A ~ / R T ) / O P Ip_o is given by:
B r = al -- a3/a~ + a7 + a9
(4)
Compressibilities (Z =PV/RT} predicted by our new
EOS are compared with available PVTdata in Figs. 1-3.
For the PTrange P_<2 kbar and 215 ___T_< 1200 K, the
average absolute percent deviation in compressibility
between values predicted by our EOS and those in our
primary data base generated using the Ely et al. (1989)
where ai are given by Eq. 2.
1 Amended values for coefficients of Eqs. 2, 3 and 4 in Ely et al.
1989 provided by J.F. Ely are tabulated by Sterner and Bodnar
1991
365
Fig. l a - d Comparison between experimental data
of Tsiklis et al. (1969; TL & T
at T= 373, 573 K and
2 < P < 7 kbar), Shmonov and
Shmulovich (1974; S & S at
' T=789.35, 980.65 K and
1 <P_<8 kbar) and values calculated using the Altunin and
Gadetskii (1971; A & G at
P < 2 kbars) EOS with compressibility (PV/RT) isotherms
predicted by our new EOS.
Reduced density =p/p,, where
Pc = 0.01063 mol/cm 3
/
7
373.00K
6
9
T
L & T
i
i
i
I
i
t
i
i
I
i
i
i
i
789.35K
6
9
(1969)
/
S & S (1974)
[
5
5
b-"
C
>
IX.
i
7
3
I-rr4
>
o- 3
2
2
4
0
i
i
0
p
i
I
i
i
i
i
I
i
I
I
I
0
I
1
2
3
Reduced Density
a
i
i
~
i
I
i
i
i
i
I
i
i
i
i
_
I
~
I
I
I
0
I
I
i
t
1
I
i
i
l
I
I
2
3
Reduced Density
I
'
~
'
'
i
'
'
~
'
i
I
I
I
I
I
I
I
t
I
i
.'
'
'
I
I
'
I
7
7
6
5
Irr 4
>
t-w4
>
13_
13_
3
2
2
0
~
0
b
3
I
1
~
t
r
~
I
~
~
~
2
I
0
3
Reduced Density
and Altunin and Gadetskii (1971) equations was 1.34%.
Volumetric data were incorporated into the calibration
of our EOS by simultaneously minimizing pressure deviations for p<p~, and density deviations for P>Pc.
Within the above range, maximum deviations occurred
at ~ 350 K and ~ 300 bar where the errors in predicted
pressure and density simultaneously exceeded 1% (i.e.,
2.8 and 1.2%, respectively). Compressibilities predicted
using the Altunin and Gadetskii (1971) EOS are indicated with open circles and those from Ely et al. (1989) are
shown by open triangles in Figs. 1 and 2.
Compressibilities predicted by our EOS are compared with experimental data of Tsiklis et al. (1969;
P _<7 kbar) in Figs. 1a and b, with data of Shmonov and
Shmulovich (1974; P_<8 kbar) in Figs. lc and d, and
with values generated using Eq. 5 of Shmonov and
~
d
I
i
I
I
2
3
Reduced Density
Shmulovich (1974) at 304.127 and 1000 K in Figs. 2a
and b. The average absolute deviation in density between values predicted by our EOS and those reported
by Tsiklis et al. (1969) was 0.50% with a maximum of
2.2% at 673 K and 7 kbar. The average deviation between our EOS and data from Shmonov and
Shmulovich (1974) was 0.79 % with a maximum of 2.1%
at 980.65 K and 8 kbar.
Agreement between our EOS and the data of Tsiklis
et al. (1969) is excellent (i.e., deviations in p < 1.2%) at
all pressures for T<473 K, and pressures below 6 kbar
for the 573 and 673 K data. For pressures less than
3 kbar our EOS agrees well with data fi'om Shmonov
and Shmulovich (1974) (see Figs. lc and d). However, at
higher pressures - particularly at high temperatures,
larger deviations are noted (i.e., + ~ 2% in density). The
366
200
__+-
175
Critical
Isotherm
150
I--- 125
iT"
+
9
o
100
f
[
. EH&B(1989)
1001
1000 K
9 N~t
N et al (1991)
Ree (1992)
+ SR~
& S (1974)
s~
o A&
75
5
9
20
III
I
I
I
I
I
I
I
~
~
[
I
I
I
I
I
I
I
~
I
I
I
I
.2000 K
A & G (1971)
I
r
Net al (1991)
15
I
/
l
/
/
(3_
, ,
/
/
Liu(1984)
,1~,,i
/
/
S & S (1974)
A & G (1971)
0
10
0
5
25
0
,,,i .... ~
0
a
1
2
0
3
4
5
6
Reduced Density
7
.... I*''
0
b
i
L i
,
i
i
I
r
i
,
i
I
,
i
i
i
1
2
3
4
Reduced Density
Fig. 2a-e Comparison between experimental data of Liu (1984)
for solid CO~, shock-compression data for liquid CO~ from Nellis
et al. (1991), values calculated using equations of Shmonov and
Shmulovich (1974; S & S for 2 < P < 8 kbar), Altunin and Gadetskii (1971; A & G at 0.3<P<2kbar at T~ and at P < 2 k b a r at
T= 1000 K), Ely et al. (1989; EH &B at P<0.3 kbar at T~) and
values provided by Dr. F.H. Ree with compressibility (PV/RT)
isotherms predicted by our new EOS. Critical temperature
for COz is T~=304.127K. Reduced density=p/p~ where p~=
0.01063 mol/cm 3
Z - p ~ isotherms predicted by our EOS have considerably greater curvature at high pressures and temperatures than implied by the data of S h m o n o v and
Shmulovich (1974). In fact, the 980.65 K experimental
data actually show an inflection in the Z - p ~ isotherm
and become slightly concave d o w n w a r d for P > 4 kbar
(Fig. ld). Equation 5 of Shmonov and Shmulovich
(1974) mimics this behavior and upon extrapolation to
higher temperatures, the feature becomes increasingly
pronounced.
The curvature of the Z - p ~ isotherms at high pressures and temperatures implied by the Shmonov and
Shmuiovich (1974) data is both anomalous by corresponding states comparisons with other fluids (e.g.,
HzO: see Pitzer and Sterner 1994) and unsupported by
the shock-compression data at higher pressures and
temperatures from Nellis et al. (1991) and Schott (1991).
Furthermore, because the Shmonov and Shmulovich
(1974) data represented the most extreme conditions of
P and T o f all the P V T d a t a available prior to the shockcompression measurements, it seems likely that this feature is responsible for much of the disparate behavior
found between the various "semi-empirical" EOS available for CO2 when they are extrapolated above 1000 K
and 10 kbar. The cause of the apparent aberrant behavior is unknown. It is plausible that it resulted from
the smoothing procedure used by Shmonov and
Shmulovich (1974) in the preparation of their Table 1,
3
'
5
Ill
0
I l l l l l l l l l l l l l l l l l l
1
2
3
....
, ....
o t200K
[]
L
o/U/
800 K
000K
o,oo
0
t
./J/
///,',
350 K
Iii
ml I
II1
rl I !
l
o
--43-- "-~- --v--
1
- -r
i
0
5
;//; '//t
I ///
-
4
Reduced Density
I
2
Ill
0
i
i
1
i
/
304.127K~ ~
/
290K
\",.
270K
~
//
250K
~-~
220 K 9 saturation
i
i
i
i
i
i
i
i
i
1
2
Reduced Density
i
3
Fig. 3 Comparison between values calculated using equations
of Shmonov and Shmulovich (1974; S &S for 2_<P_< 8 kbar);
Altunin and Gadetskii (1971; A & G at 0.3_<P<2kbar for
T_<330K and at P < 2 k b a r for T>330K) and Ely etal. (1989;
EH & B at P < 0.3 kbar and T_<330 K) with compressibility (PV/
RT) isotherms predicted b y our new EOS. Critical isotherm at
T~= 304.127 K is shown with heavy solid line. Reduced density =p/
p,, where p~:= 0.01063 mol/cm3. Filled circles represent intersection
of isotherms with saturation surface
or it may have resulted from a systematic experimental
error.
Compressibilities predicted by our EOS are compared with diamond-anvil pressure cell measurements
of Liu (1984) for solid COe in the range of 24 to 358 kbar
367
Table 2 Compressibility factor ( Z = PV/RT)
T(K)
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
1050
1100
1150
1200
1250
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
1950
2000
Pressure (kbar)
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
20.0
30.0
40.0
50.0
1.71
1.56
1.46
1.41
1.38
1.36
1.36
1.36
1.36
1.36
1.35
1.35
1.34
1.33
1.32
1.32
1.31
1.30
1.30
1.29
1.29
1.28
1.27
1.27
1.26
1.26
1.25
1.25
1.24
1.24
1.24
1.23
1.23
1.22
1.22
1.22
3.21
2.85
2.59
2.41
2.26
2.16
2.07
2.00
1.95
1.90
1.86
1.82
1.79
1.76
1.74
1.71
1.69
1.67
1.65
1.63
1.61
1.60
1.28
1.57
1.55
1.54
1.53
1.52
1.51
1.50
1.49
1.48
1.47
1.46
1.45
1.45
4.60
4.04
3.63
3.33
3.09
2.90
2.75
2.62
2.52
2.43
2.35
2.28
2.22
2.17
2.12
2.08
2.04
2.01
1.97
1.94
1.92
1.89
1.87
1.84
1.82
1.80
1.78
1.77
1.75
1.73
1.72
1.70
1.69
1.68
1.67
1.65
5,93
5,17
4.62
4.21
3.88
3.61
3.40
3.22
3.07
2.94
2.82
2.73
2.64
2.56
2,49
2.43
2.38
2,33
2.28
2.24
2.20
2.16
2.13
2.10
2.07
2,04
2,02
1.99
1,97
1.95
1.93
1.91
1.89
1.88
1.86
1.85
7.21
6,26
5.57
5.04
4.63
4.30
4.02
3.79
3.60
3.43
3.28
3.16
3.04
2.94
2.85
2.77
2.70
2.64
2.58
2.52
2.47
2.43
2.38
2.34
2.31
2.27
2.24
2.21
2,18
2.16
2.13
2.11
2.08
2.06
2.04
2.02
8.46
7.31
6.49
5.85
5.36
4.95
4.62
4.34
4.11
3.90
3.73
3.57
3.43
3.31
3.20
3.11
3.02
2.94
2.86
2.80
2.74
2.68
2.63
2.58
2.53
2.49
2.45
2.42
2.38
2.35
2.32
2.29
2.26
2.24
2.21
2.19
9.67
8.34
7.38
6.64
6.06
5.59
5.21
4.88
4.60
4.37
4.16
3.98
3.82
3.67
3.54
3.43
3.32
3.23
3.14
3.06
2.99
2.93
2.86
2.81
2.75
2.70
2.66
2.62
2.58
2.54
2.50
2.47
2.44
2.41
2.38
2.35
10.86
9.34
8.25
7.41
6.75
6.22
5.77
5.40
5.09
4.82
4.58
4.37
4.19
4.02
3.88
3.74
3.62
3.52
3.42
3.33
3.24
3.17
3.10
3.03
2.97
2.91
2.86
2.81
2.76
2.72
2.68
2.64
2.60
2.57
2.54
2.51
12.03
10.33
9.10
8.16
7.42
6.82
6.33
5.92
5.56
5.26
4.99
4.76
4.55
4.37
4.20
4.05
3.92
3.80
3.69
3.58
3.49
3.40
3.32
3.25
3.18
3.12
3.06
3.00
2.95
2.90
2.85
2.81
2.77
2.73
2.69
2.66
13.17
11.29
9.93
8.89
8.08
7.42
6.87
6.42
6.03
5.69
5.39
5.14
4.91
4.70
4.52
4.36
4.21
4.07
3.95
3.83
3.73
3.63
3.55
3.46
3.39
3.32
3.25
3.19
3.13
3.07
3.02
2.97
2.93
2.89
2.85
2.81
23.80
20.21
17.63
15.67
14.14
12.90
11.87
11.02
10.28
9.65
9.10
8.62
8.19
7.81
7.46
7.15
6.87
6.62
6.38
6.17
5.97
5.79
5.62
5.46
5.31
5.18
5.05
4.93
4.81
4.71
4.61
4.51
4.43
4.34
4.26
4.19
33.48
28.31
24.60
21.79
19.60
17.83
16.37
15.15
14.11
13.21
12.43
11.74
11.14
10.59
10.11
9.67
9.27
8.91
8.58
8.27
7.99
7.73
7.49
7.27
7.06
6.86
6.68
6.51
6.35
6.20
6.06
5.92
5.79
5.67
5.56
5.45
42.56
35.88
31.10
27.50
24.68
22.42
20.55
18.99
17.66
16.51
15.52
14.65
13.87
13.18
12.56
12.00
11.50
11.04
10.61
10.23
9.87
9.54
9.23
8.95
8.68
8.44
8.20
7.99
7.78
7.59
7.41
7.24
7.07
6.92
6.77
6.64
51.18
43.07
37.27
32.91
29.50
26.76
24.50
22.62
21.02
19.64
18.44
17.39
16.46
15.63
14.88
14.21
13.60
13.05
12.54
12.07
11.65
11.25
10.88
10.54
10.22
9.92
9.65
9.38
9.14
8.91
8.69
8.48
8.29
8.10
7.93
7.76
in Fig. 2a. We assumed that the pressure for the liquid
exceeds that of the solid at the same density, but only by
a small amount at very high densities. Thus, these data
were used to guide the critical isotherm at pressures
above ~ 80 kbar (Z > 65).
Predictions by our EOS are compared with compressibility values interpolated from shock-compression
data along the Hugoniot from Nellis et al. (1991) and
values calculated using their model potential with "Set
A" parameters at 1000 and 2000 K in Figs. 2b and c,
respectively. Absolute deviations between the Hugoniot
data (10 values for 1000 < T_<4000 K) and densities predicted by our EOS are < 1.1% over this temperature
interval.
A general comparison between compressibilities predicted with our new EOS and values generated using the
equations of Ely et al. (1989), Altunin and Gadetskii
(1971) and Shmonov and Shmulovich (1974) for Z < 3 at
several temperatures between 220 and 1200 K is presented in Fig. 3. Maximum pressures along isotherms
shown in Fig. 3 range from ~1.5 kbar at 2 2 0 K to
6.0 kbar at 1200 K. Note that data shown along the
'
'
'
I
'
'
'
r
'
'
'
~E 0 ~~~'~'~i
'
'
i
I
'
~~~
%O,~ -loo
rf'l
I
rial Coefficient
~
z~
o
-200
t
200
N2 D&S (1980)
Ne D&S (1980)
c02 A&G(1971)
v
t
i
I
600
i
i
~
I
i
i
i
I
i
1000 1400
Temperature (K)
i
i
i
1800
Fig. 4 Comparison between values of second virial coefficient B
predicted with new EOS and those given for CO2 at T_< 1273 K
(Altunin and Gadetskii 1971) and for N2 and Ne at T_> 1000 K
(Dymond and Smith 1980)
368
I 'i
r-
i i
i i i
i i
i [
t i
~
i i
i i
i i
i [
i i
i i i
i i
i i
i i i
i i
i i
i i
i [ i
Saturation Properties
.O
i i
77
i
,
-ff
100x(obs-calc)/obs
PlJq
"~
13)
. . . . pvop
- - - - Psa,~.
"-~c9~
0
/
/
..13
'.3;50;''/
[] 304.50
9 304.127
z~ 303.70
9 303.25
76
75
/
/o
J
~
/
/
v
74
.........
09
73
0
13_
(3_
72
t Deviation from Eqs 2, 3 and 4 of Ely et al., (1989)
- 5
I I I I I I I I t l t l l t t l t
215
t t l l l l l t t t r
235
t l l l l l l l l l l l l l l l
255
275
295
71
0.5
Temperature (K)
Fig. 5 Deviations between saturation properties calculated using
Eqs. 2, 3 and 4 of Ely et al. (1989) and those predicted with our
EOS. p~q, p~v and Psat are the saturated liquid and vapor densities, and saturation pressure, respectively
1200 K isotherm for Z > 1.5 are values calculated with
Eq. 5 of Shmonov and Shmulovich (1974) and represent
extrapolations some 220 K above their highest reported
experimental measurements. Values calculated with the
Shmonov and Shmulovich (1974) equation above
1000 K were not used in our primary data base and are
included here only for comparative purposes. A brief
compilation of Z values calculated with our EOS over
the range 1 < P < 50 kbar and 250 _<T_< 2000 K is given
in Table 2.
Second virial coefficient (B~) values for T_<2000 K,
taken from the sources described above in the data
base section, are compared with those calculated using
our EOS (Eq. 4) in Fig. 4. Similar agreement is found
at higher temperatures when our calculated values
are compared on a corresponding-states basis with
Ne up to 3000 K and with He up to 40,000 K. Inclusion of corresponding-states-based values for Br had a
major influence on the behavior of our EOS for
1200 _<T_< 2000 K and pressures below those of the
Fig. 7a-b Isochores (constant
density contours) for CO2 predicted with our EOS (Eq. 3).
p,,, = 0.01063 mol/cm 3
10
1.0
1.5
Density (mole/crn3xl00)
Fig. 6 Comparison between compressibilities calculated using
CO2 EOS of Ely et al. (1989; symbols) with PV/RTisotherms predicted by our new EOS in the near-critical region (solid curves).
Critical isotherm at ~ = 304.127 K. Liquid-vapor curve predicted
by Ely et al. (1989) EOS is given by the solid curve. P-p coordinates
for the liquid-vapor curve predicted by our EOS are given by the
intersections of the isotherms and the horizontal dashed lines
Nellis et al. (1991) model calculations (i.e., P < 10 kbar).
The importance of this was readily apparent during later mineral equilibria calculations.
Near-critical and saturation behavior
Low PTphase equilibria for carbon dioxide predicted
by our new EOS are compared with data from Ely et al.
(1989) in Fig. 5. Considering the extensive PTrange of
validity of our equation, the agreement with experimentally derived phase relations is excellent. Critical
parameters predicted with our new EOS are:
Pc=73.7713 bar, Tc=304.143 K and p~=0.01054 mol/
cm 3 (see "data base" section for comparison with accepted values). Sub-critical, liquid-vapor phase separation is predicted by our EOS with maximum deviations
,
250
9
-g 8
-Q
.~
n
~-" 200
7
6
5
4
3
2
1
0
u 15o
co
100
t3_
5O
273 473 673 873 10731273
Temperature (K)
0
500
1500 2500 3500
Temperature (K)
369
5~
1 : MgCO~ + MgSIO~ = Mg=SiO4 + CO~
----
7/
AI
2: MgOOa § SIO2 = MgSIOa + CO2
40
....
v9
-.-
o,
,,,,;
//
20
"Y
lo
///" +',"
////
0
/
AJ
I
~"
i~,1 /
I1
.Jl
ill
/~/
9
/ _.
"~ /
#//
A
/
,,."
/
A~
,',&
13_
~
,~
,' /
#
;,;
A
, ,,
~" /
30
I
l 9 /
3:MgCO~.M~O+CO~
v
,.,.
//
I.~
^
.,/
0 / , 7. ,.
**
I
0
i
/
...i
~-~-~'~'
Isochores for CO2
''
'
''
773 973 1173 1373 1573 1773
'
30
I
'
I
'
I
'
I f
4: ?.~CO~..+?Od.M~io3+~
. /
0:CaOO.+S,~;o.~,o~,oo,
7 /
I
/
.+
/'
./../
,//.
(II
.13
@20
u/
o ./..//"/
//,
/?;
/ / o//
co
/ 5,:
i'//'', i , , , ,
0
b
773 973 1173 1373 15731773
.I i
1: MgCOa + MgSiO~ = Mg~SiO4 + CO2
---
A',
.11""
3: MgOOa = MgO + CO 2
5: CaCO~
..(3
....
,..~ 3
,_(~
0..
//o/ f ~
/)
.~,
-//
1 {I
.~
~
~""
('"
o
~
~"-+-"~
773
.~1""
oI /
/
" ~ ~
//.
ol
"
~.~-"
,
973
Temperature
i
1073
Reaction 2:
MgCO3 + SiO2 = MgSiO 3 + C O 2
(magnesite + coesite = o-enstatite + CO2)
Reaction 3:
M g C Q = MgO +CO2
(magnesite -- periclase + CO2)
Reaction 4:
MgCO3 + TiO2 = MgTiO3 +CO2
(magnesite + rutile = geikielite + CO2)
i
@./i
i
Thermophysical predictions of the new EOS may be
further evaluated by comparison with fugacities of carbon dioxide 0Coo) derived from experimental data on
high P Tmetamorphic decarbonation reactions. The calculations require complete thermodynamic data for
each mineral phase as well as a reference state Gibbs
energy and heat capacity expression for CO2. This information is available for many minerals over a broad P T
range from a variety of sources (e.g. Robie et al. 1979;
Chase et al. 1985; Berman 1988; Holland and Powell
1990). Five reactions have been selected from the literature for which there exist both experimental data on the
location of the equilibrium boundary in PTspace and
ample thermodynamic data to calculate the Gibbs energies of each mineral at the appropriate PTconditions:
Reaction 1: MgCO3 + MgSiO3 =Mg2SiO 4 + C O a
(magnesite + o-enstatite = forsterite + C02)
.1,
~ i
I "1(>
i
873
.11"
.11""
?/I
ii
2
0
.
oe
/~/ /
-
e
. .
- - . - -+Si02<>i= CaSlO~ + C02
I...
lsochores (constant volume projections) for carbon
dioxide predicted by our equation are shown in Figs. 7a
and b. Based on the earlier discussions of the agreement
between our equation and available data, isochores calculated using Eq. 3 are considered to be valid from zero
pressure and T~215 K to pressures of at least 10 GPa
(105 bar) and T~2000 K. Furthermore, the new EOS
provides reasonable extrapolation of the properties of
CO2 to still higher pressures and temperatures.
Mineral equilibria
//'n /::/
lo
in Psat, Pliq a n d Pvap of: _<0.7%, ~0.1% and _<1.0%,
respectively, for all but near-critical temperatures (i.e.,
(T~-5) < T< T~) where deviations in liquid and vapor
densities reach 3-4% (Fig. 5).
The near-critical behavior of our equation is shown
in Fig. 6. Data from Ely et al. (1989) within the one-fluid
phase field and along the solvus are compared with our
results along several isotherms. The extent of the twophase field predicted using our EOS is outlined by the
ends of the dashed horizontal lines.
,
i
1173
(K)
Fig. 8a--e Comparison between five experimentally studied decarbonation reactions and mineral equilibrium boundaries calculated using our new EOS and the internally-consistent data bases
of "Berman" (heavy lines) and "Holland and Powell" (light lines).
Details of the two primary mineral data bases are given in the text.
Experimental data for each reaction were taken from the following sources. Reaction 1 : Goldsmith and Heard (1961), Harker and
Tuttle (1955), Irving and Wyllie (1975), Milder and Berman (1991);
Reaction 2: Haselton etal. (1978); Reaction 3: Haselton et al.
Reaction 5: CaCO~+ SiO a = CaSiO3 +CO2
(calcite +/Y-quartz = wollastonite + CO2)
(1978), Johannes (1969), Newton and Sharp (1975); Reaction 4:
Haselton et al. (1978); Reaction 5: Haselton et al. (1978), Harker
and Turtle (1956). The stabilities of the carbonate assemblages a r e
indicated with open symbols while those of the decarbonated assemblages are given byfilled symbols
370
Table 3 Fugacity of CO2 (lnfco~)
T(K)
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
1050
1100
1150
1200
1250
1300
1350
1400
1450
1500
1550
1600
1650
1700
1750
1800
1850
1900
1950
2000
Pressure (kbar)
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
20.0
30.0
40.0
50.0
4.51
5.45
6.03
6.40
6.65
6.81
6.93
7.00
7.05
7.09
7.11
7.13
7.14
7.15
7.15
7.16
7.16
7.16
7.16
7.16
7.15
7.15
7.15
7.15
7.14
7.14
7.14
7.13
7.13
7.13
7.13
7.12
7.12
7.12
7.11
7.11
6.16
6.93
7.40
7.69
7.88
8.00
8.09
8.14
8.17
8.19
8.20
8.21
8.21
8.20
8.20
8.19
8.18
8.17
8.17
8.16
8.15
8.14
8.13
8.12
8.11
8.10
8.09
8.08
8.08
8.07
8.06
8.05
8.05
8.04
8.03
8.02
7.73
8.32
8.65
8.84
8.96
9.02
9.06
9.07
9.07
9.06
9.05
9.03
9.02
9.00
8.98
8.96
8.93
8.92
8.90
8.88
8.86
8.84
8.82
8.81
8.79
8.78
8.76
8.75
8.73
8.72
8.71
8.70
8.68
8.67
8.66
8.65
9.23
9.63
9.83
9.92
9.95
9.96
9.94
9.91
9.87
9.83
10.69
10.90
10.96
10.95
10.90
10.84
10.76
10.69
10.61
10.54
10.47
10.41
10.35
10.29
10.23
10.18
10.13
10.09
10.05
10.01
9.97
9.93
9.90
9.87
9.84
9.81
9.78
9.76
9.73
9.71
9.68
9.66
9.64
9.62
9.60
9.58
12.12
12.14
12.06
11.94
11.81
11.68
11.55
11.43
11.31
11.21
ll.ll
11.02
10.94
10.86
10.78
10.72
10.65
10.60
10.54
10.49
10.44
10.40
10.36
10.32
10.28
10.24
10.21
10.18
10.15
10.12
10.09
10.06
10.04
10.01
9.99
9.97
13.52
13.34
13.13
12.90
12.69
12.49
12.31
12.14
11.98
11.84
11.72
11.60
11.49
11.39
11.30
11.22
11.14
11.07
11.00
10.94
10.88
10.83
10.78
10.73
10.69
10.64
10.60
10.56
10.53
10.49
10.46
10.43
10.40
10.37
10.34
10.32
14.89
14.52
14.17
13.84
13.54
13.28
13.04
12.82
12.63
12.46
12.30
12.16
12.03
11.91
11.80
11.70
11.61
11.52
11.44
11.37
11.30
11.24
11.18
11.12
11.07
11.02
10.97
10.93
10.88
10.84
10.81
10.77
10.74
10.70
10.67
10.64
16.23
15.68
15.19
14.76
14.38
14.04
13.75
13.49
13.26
13.05
12.86
12.69
12.54
12.40
12.27
12.16
12.05
11.95
11.86
11.78
11.70
11.62
11.55
11.49
11.43
11.37
11.32
11.27
11.22
11.18
11.13
11.09
11.05
11.02
10.98
10.95
17.56
16.82
16.19
15.65
15.19
14.79
14.45
14.14
13.87
13.63
13.41
13.22
13.04
12.88
12.73
12.60
12.48
12.37
12.26
12.17
12.08
11.99
11.92
11.84
11.78
11.71
11.65
11.59
11.54
11.49
11.44
11.40
11.35
11.31
11.27
11.23
30.02
27.45
25.49
23.95
22.70
21.66
20.78
20.03
19.38
18.82
18.32
17.87
17.47
17.12
16.79
16.50
16.23
15.99
15.76
15.56
15.37
15.19
15.02
14.87
14.73
14.59
14.47
14.35
14.24
14.13
14.03
13.94
13.85
13.77
13.69
13.61
41.53
37.19
33.98
31.48
29.48
27.84
26.46
25.29
24.29
23.41
22.65
21.97
21.36
20.82
20.33
19.88
19.48
19.11
18.77
18.46
18.17
17.91
17.66
17.43
17.22
17.01
16.83
16.65
16.48
16.33
16.18
16.04
15.91
15.78
15.66
15.55
52.42
46.39
41.95
38.54
35.82
33.60
31.75
30.18
28.84
27.67
26.65
25.75
24.94
24.22
23.58
22.99
22.46
21.97
21.52
21.11
20.73
20.38
20.06
19.75
19.47
19.21
18.96
18.73
18.51
18.30
18.11
17.93
17.75
17.59
17.43
17.28
62.85
55.17
49.56
45.26
41.85
39.07
36.76
34.81
33.14
31.69
30.43
29.31
28.32
27.43
26.63
25.91
25.25
24.65
24.10
23.59
23.13
22.70
22.30
21.92
21.58
21.25
20.95
20.66
20.39
20.14
19.90
19.67
19.46
19.26
19.07
18.88
9.79
9.75
9.71
9.67
9.64
9.60
9.57
9.54
9.51
9.48
9.45
9.42
9.40
9.37
9.35
9.33
9.31
9.29
9.27
9.25
9.23
9.21
9.20
9.18
9.17
9.15
Experimental phase equilibria data for the above reactions are summarized in Figs. 8a-c. The data were taken
from the sources indicated in the figure caption. Discussions of the uncertainties in pressure and temperature
associated with the various experiments can be found in
the original citations and elsewhere (Berman 1988;
Chernosky and Berman 1989; Milder and Berman
1991; Holland and Powell 1990). Some of the data
shown in Figs. 8a c represent true experimental reversals while others are pseudo-reversals, representing the
stability of the indicated assemblage (products or reactants) in the absence of the other. For the purposes of
the present analysis, we have taken raw experimental
data from the original citations, then adopting the presentation style used by others (e.g., Berman 1988), expanded the implied equilibrium bracket by the addition
(or subtraction) of Crp and ~rr - the estimated uncertainties in the experimental pressure and temperature. Thus,
in each case, ap and aT were subtracted from the data
points representing the stability of the carbonate-bearmg assemblage, and added to the points corresponding
to the d e c a r b o n a t e d side of the reaction. The following
estimated uncertainties were assigned to the experimental data shown in Fig. 8: to the data from cold-seal or
internally-heated reaction vessels at T < 1 3 0 0 K and
P_< 7 kbar, experimental uncertainties were estimated
at a / , = 5 0 b a r and o-r=10~ to the data from pistoncylinder experiments at higher pressures a n d / o r temperatures, uncertainties were estimated at o-t,-- 1 kbar and
o-r= 15 ~ F r o m a review of the literature cited above, is
seems that for m u c h of the data, these are conservative
estimates. Also, as suggested by Milder and Berman
(1991), a - 3 kbar correction was applied to the data of
Irving and Wyllie (1975) for Reaction 3 based on calibrations by H u a n g and Wyllie (1975).
Fugacities of carbon dioxide predicted by our EOS
m a y be calculated using Eq. 1 and the relationship:
lnf= [lnp + A"*~/R T+ P/pR 7]p ,t p + In (R 7}- 1
(5)
E q u a t i o n 5 yields CO2 fugacities relative to the ideal gas
at 1 bar and T ( i . e . , f / P = 1 as P-+ 0). A brief compilation
of In (fco2) values calculated with our EOS over the
range 1 _<P < 50 kbar and 250 _< T__<2000 K is given in
Table 3.
371
Equilibrium boundaries for Reactions 1 5 shown in surement (750K). The extrapolations assumed by
Figs. 8a c were calculated using two separate sets of Berman and by Holland and Powell are reasonable and
thermodynamic data. The heavy curves were calculated nearly the same, but a large uncertainty in Cp at high T
using CO2 fugacities predicted by our new EOS and the remains.
The discrepancies at low pressure between experiinternally-consistent thermodynamic data base of
Berman (1988) with revisions to M g C O 3 and C a C O 3 mentally derived mineral equilibria for Reaction 5 and
from Milder and Berman (1991). The lighter curves in reaction boundaries calculated using our EOS most
Fig. 8 were calculated using our EOS together with the likely result from uncertainties in the mineral data
internally-consistent data set of Holland and Powell bases regarding the standard state properties or heat
(1990). Data for Geikielite (MgTiO3) used in both calcu- capacity of one of the phases. Virtually all EOS valid
lations were taken from the "TWEEQU" mineral data- at these temperatures (i.e., 773_< T<973 K), including
base of Berman (1991, June 1992 version). In the follow- our own, predict essentially ideal behavior for CO2 (i.e.
ing discussions, these two principal data compilations lnfco2~lnP) at pressures below 1 kbar a prediction
with the noted revisions will be refered to as the that is corroborated by accurate volumetric data over
"Berman" and "Holland and Powell" data bases. In the corresponding PTrange.
each set of calculations, the equilibrium boundary for
Reaction 2 was derived using thermodynamic properties for coesite - a high pressure polymorph of SiO2. Comparison with other equations of state
Because ~-quartz is the stable SiO2 polymorph below
30 kbar and g 1273 K the curves drawn for Reaction Comparisons like those in Fig. 8, between experimental2 represent metastable extension of this equilibrium be- ly derived mineral equilibria and calculations using varlow these PTconditions. Also, equilibrium boundaries ious sources of thermodynamic data, commonly accomfor Reactions 1 and 2 were calculated using thermody- pany the introduction of new equations of state for geonamic properties for the ortho-enstatite polymorph of logic fluids (e.g., Milder and Berman 1991 ; Holland and
MgSiQ.
Powell 1991; Kerrick and Jacobs 1981). Typically, those
Reaction boundaries predicted for the equilibria EOS which appear the most consistent with mineral
shown in Fig. 8 calculated with our new EOS and the equilibria data tend to correlate the available volumetBerman data base (heavy curves) are in excellent agree- ric data less accurately and vice versa. Possible explanament with observed mineral stabilities. Of the five reac- tions for this behavior may be explored by examining
tions considered, calculated reaction boundaries lie out- the inter-relationships between the mineral equilibria
side the experimental constraints only over the tempera- data and the fugacities and volumetric properties of
tures interval 1373_< T_< 1573 K for Reaction 1 and at CO2 as illustrated in Figs. 9 and ! 0.
very low pressures for Reaction 5. Agreement between
Volumetric properties and fugacities of carbon dioxexperimental data and equilibrium boundaries calculat- ide at 1000 and 1600 K predicted by several CO2 EOS
ed using our EOS and the Holland and Powell data base are compared in Figs. 9 and 10 (as PV/RTand lnJco2).
(light curves) is somewhat poorer for the reactions we Also shown are CO2 fugacities derived from experimenconsidered. Equilibria for three of the four magnesite tal mineral equilibria data. Approximate, and in some
(MgCO3) bearing reactions (Reactions 1-3) lie at lower cases, extrapolated locations of equilibria boundaries
temperatures at high pressures, when calculated with for Reactions 1-5 at 1000 and 1600 K were infcrred
the Berman data base, than in corresponding calcula- from the experimental data shown in Fig. 8. Fugacities
tions using the Holland and Powell data. These dis- of CO2 corresponding to these PTconditions were then
crepancies arise primarily from differences in the specific calculated using the Berman data base.
volume of magnesite \tV['MgCO3]
r ~ predicted by the different
At temperatures as high as 1000 K, and pressures
expressions used in the respective data bases. When the below ~ 8 kbar, compressibilities and fugacities predictsame expression for V~gco3
•. r is substituted into both data ed by the various equations are comparable at the scales
bases, the equilibrium boundaries predicted for Reac- shown in Figs. 9a and b. For the pressure interval
tions 1,2 and 3 become nearly coincident. Differences in 10_<P< 50 kbar (Fig. 10), however, compressibilities
the two V~i[co~expressions arise from the limited exper- predicted by the different equations diverge rapidly
imental data for the temperature dependence of the yielding a correspondingly large range of values for
thermal expansion coefficient. One-atmosphere cell ln(Jco~). By contrast, at 1600 K there already exist subparameters for magnesite were determined by Markgraf stantial discrepancies between predicted compressibiliand Reeder (1985) up to 773 K using X-ray techniques. ties at low pressures (Fig. 9a) and an ensuingly wide
Extrapolation of these data to the temperature range of array of derived fugacities (Fig. 9b). At 1600 K and
interest in the present application gives rise to large un- 10 <__P <__50 kbar (Fig. 10), compressibilities and fugaccertainties, although the experimental mineral equi- ities continue to diverge with increasing pressure, and
libria imply that Vf;
r
calculated using the Berman changes in the relative ordering of the curves of both
MgCO3
expression is more suitable at these conditions. For PV/RT and ln(/co2) above and below ~10 kbar are
magnesite there is also a considerable uncertainty in the readily apparent. At low pressures, our EOS predicts
heat capacity above the highest temperature of mea- larger compressibilities than most other equations,
372
4
_
'
I
~
I
'
I
'
I
.//'~/
-
1000.00K / f
I-- 3
9
...
.
~
. ."
lOOOOO
15
~
F"~,
//
-
///
..'"
,/z'.>..'t,'
10
13.
13_
2
E.Z//
.4'
~~ ' I
0
2
i
i
4
I
6
i
I
~160,
/
1600.00K
I
-
, ,,~, ~/ ~ / 2,,-~- f; J- '/5 ~ 1
,'Xz'~" ..".,~"'..~J-"
5
r7
/
."
Y
0,0K
i
i
8
10
10
20
30
40
50
Pressure (kbar)
12
25
1000.00K
1000.00K
23
/
10
/
/
/
.,
/
//
21
c3
o
i-
619
o
t'-
1600.00K
8
- - --. . . .
- - - ........
r (1)
14)
1600.00K
--17
This Work
Kerr]ck & Jacobs (1981)
Bottlnga & Richel (1981 }
Holloway (1983)
Belonoshkc and Saxena {1991 l
Mader and Berman (1991)
15
(3)
(5) __ Th,sWo,k
13
6
0
b
2
4
6
8
Pressure (kbar)
10
11
b
I
10
i
I
20
i
- --.-. . . .
Kerriek & Jacobs (1981)
Bottinga & Rlchet (1981)
Holloway (1983)
- - - -
BelonoshkoandSaxena(1991)
........
Mader and Berman (1991)
t
30
i
I
40
i
I
50
Comparisonofcompressibilities(PV/RT;a)andfugacities
Fig. 9 Comparison of compressibilities (PV/RT; a) and fugacities
(as lnfco2; b) for CO2 at 1000 and 1600K and 0_<P_<10kbar
predicted by several available EOS. Compressibility data at
P < 8 kbar (filled circles in a) were calculated using equation in
Shmonov and Shmulovich (1974), open circle at ~ 9 kbar is from
Ree calculation (see text). Approximate values for hffco2 at
1000 K derived from experimental data for Reactions 3 and 5 in
Fig. 8 are as indicated in b
Fig.
whereas in the higher pressure range (Fig. 10a), our EOS
predicts lower PV/RT values than all EOS except Belonoshko and Saxena (1991).
Accurate correlation of the volumetric properties for
CO2 shown in Figs. 9a and 10a, requires more flexibility
(curvature of the compressibility isotherms) at high pressures and temperatures than allowed by the Modified
Redlich-Kwong-type EOS. The magnitudes and temperature dependences of parameters established for these
equations lead to an almost negligible contribution of
the "attractive" term by 1600 K. The result is a nearly
linear pressure dependence of the compressibility at high
temperatures (e.g., note linearity of 1600 K isotherms
predicted by equations of Holloway 1981; Kerrick and
Jacobs 1981; Milder and Berman 1991; shown in
Figs. 9a and 10a). The use of different numerical values
for parameters corresponding to different density ranges
in the Bottinga and Richet (1981) EOS partially over-
10
(as lnfco,; b) for CO2 at 1000 and 1600 K and 10_<P_<50 kbar
predicted by several available EOS. Compressibility data (open
circles in a) are from Ree calculation (see text). Approximate values for lnfco2 at 1600 K derived from experimental data for Reactions 1-5 in Fig. 8 are as indicated in b
373
comes this difficulty but only at the cost of introducing
discontinuities. Similarly, the ability of the comparatively simple expression used by Belonoshko and Saxena
(1991) to predict a relatively slow increase in PV/RTwith
increased pressure for 10_< P_< 50 kbar results from the
restricted region of validity of this EOS (i.e., P _>5 kbar).
Fugacities of CO2 derived from mineral equilibria
and shown in Figs. 9b and 10b, reflect only the experimental uncertainties in P and T outlined above. Additional error arises from uncertainties in the primary
thermodynamic data base. For example, typical uncertainties associated with the reference state enthalpies of
formation of minerals in Reactions 1 5 (see Appendix in
Berman 1988) are on the order of _+0.1%. Propagation
of these uncertainties through the calculation of lnfc Q
from experimental mineral equilibria data assuming a
worst-case scenario (i.e., errors in products and reactants having opposite sign) yields an additional uncertainty for data shown on Figs. 9b and 10b of a few
tenths of a log unit. Specifically, inclusion of this source
of error in the lnfco2 calculated from experimental data
for Reactions 1 would shift the position of the corresponding line segment on Fig. 10b by _+0.4 at 1600 K.
Similar considerations for Reactions 3 result in additional uncertainties of +0.24 and _+0.16 at 1000 and
1600 K, respectively.
Expansion of the experimental "brackets" by the
above uncertainties yields a rather "forgiving" set of
predicted fco~ that can be readily satisfied by most EOS
for CO2 claiming validity in this range, it is for this
reason that the mineral equilibria data were excluded
from the primary data base used to calibrate our new
EOS. On the other hand, many of the uncertainties inherent in the mineral equilibria calculations are highly
correlated. For example, the standard state properties,
and the P- and T-dependencies of the heat capacity and
specific volume for ortho-enstatite, regardless of their
associated uncertainties, are identical for Reactions !
and 2. Similarly, thermodynamic properties used for
calculations involving magnesite must be the same for
each Reactions 1~4. Furthermore, the statistical procedures used in the construction of internally-consistent
data bases such as those of Berman (1988) and Holland
and Powell (1990), result in refinement of thermodynamic parameters for some minerals, thereby reducing their
associated uncertainties relative to those estimated for
each quantity individually. Unfortunately, the outcome
of such refinement procedures depends on peculiarities
of the primary data bases and also on the statistical
procedures used; thus, estimating uncertainties in the
thermodynamic parameters associated with each mineral remains somewhat subjective.
Summary and conclusions
The new EOS proposed herein correlates phase equilibria, volumetric data and other thermophysical properties for carbon dioxide from the triple point at
216.58 K to T> 2000 K and from zero pressure to more
than 105 bar using a total of 28 adjustable parameters.
The equation predicts reasonable volumetric properties
at extreme pressures- in part due to the limited use of
high-order density terms. Similarly, the EOS provides
reasonable extrapolations to temperatures well above
2000 K owing to a carefully chosen temperature dependence and constraints imposed by corresponding-states
considerations. Of the available EOS for CO2, those that
yield reasonable predictions of Jco~ in the PTrange appropriate for mineral equilibria calculations generally
reproduce the lower PTvolumetric and phase equilibria
data with much lower precision, and vice versa. Thus,
compared with these equations, our new EOS provides
accurate correlation of the available data for CO2 over
a larger range of pressure and temperature. Several of
the other equations have repulsive terms of the van der
Waals or the Carnahan and Starling type which yield
infinite pressures at a finite volume (e.g., Holloway 1981;
Kerrick and Jacobs 1981; M/ider and Berman 1991).
Such equations are clearly inappropriate in the domain
of very high pressure because we now know that the
repulsive interaction between molecules at short distances is not as sudden as had long been thought (Barker 1989).
Although not included in the primary data set used
to calibrate the coefficients of our new EOS, fugacities of
carbon dioxide at high pressures and temperatures predicted using our equation are in excellent agreement
with mineral equilibria calculated with the internally
consistent thermodynamic data for minerals of Berman
(1988), and are in good agreement with those calculated
using data from Holland and Powell (1990). It is worth
noting that it is indeed possible to reconcile the available volumetric and mineral equilibria data using a single continuous analytical expression.
The new EOS has also been applied to water (Pitzer
and Sterner 1994) yielding a correlation of the available
thermophysical properties over the entire range from
the vapor and liquid below the critical temperature to
T_> 2000 K and from zero pressure to more than 105 bar
with good agreement. The potential for extension of the
equation to mixed fluids is also discussed by Pitzer and
Sterner (1994).
Acknowledgements Critical reviews of earlier versions of this
manuscript by Paddy O'Brien and Teresa S. Bowers have significantly improved the final draft. We thank Dr. F.H. Ree for providing the values calculated from his equation for CO2. Discussions
with Paddy O'Brien regarding the implementation of the Berman
and Holland and Powell data bases were most helpful. Calibration
of the coefficients in our equations of state was accomplished
using the nonlinear optimization routine "MINIG6" provided by
Monte Boisen and Lee Johnson of the Department of Mathematics, Virginia Polytechnic Institute & State University. This
research was supported by the Bayerisches Geoinstitut, and by
the Director, Office of Energy Research, Office of Basic Energy
Sciences Division of Engineering and Geosciences, of the United
States Department of Energy under Contract DE-AC0376SF00098.
374
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