Fluid Phase Equilibria 238 (2005) 77–86
An accurate model for calculating C2H6 solubility in pure
water and aqueous NaCl solutions
Shide Mao, Zhigang Zhang, Jiawen Hu, Rui Sun, Zhenhao Duan ∗
State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, P.O. Box 9825, Beijing 100029, China
Received 28 June 2005; received in revised form 14 September 2005; accepted 15 September 2005
Abstract
An accurate model is presented to calculate the solubilities of C2 H6 in pure water (273–444 K and 0–1000 bar) and in aqueous NaCl solutions
(273–348 K, 0–16 bar and 0–6.3 mol kg−1 ). This model is based on a specific particle interaction theory for liquid phase and a new accurate equation
of state developed in this study for vapor phase. Precision of the model is within or close to the uncertainty of experimental solubilities (about 7%).
A FORTRAN code is developed for this model and can be downloaded from the website: www.geochem-model.org/programs.htm.
© 2005 Elsevier B.V. All rights reserved.
Keywords: C2 H6 ; Solubility; Water; Aqueous NaCl solutions; Equation of state
1. Introduction
C2 H6 is one of the most important gases in nature, and has
been found in natural gases [1–3], coalbed gas [4] and fluid inclusions [5–8]. Accurate prediction of C2 H6 solubility in pure water
or in aqueous NaCl solutions over a wide range of temperature,
pressure and ionic strength, especially where data do not exist or
where the data are of poor quality, is important for geochemical
applications [9,10]. There have been quite a few experimental
studies on the solubility of C2 H6 in pure water and aqueous NaCl
solutions. However, these data are very scattered and cover only
a limited temperature–pressure (T–P) space, inconvenient to use.
Therefore, theorists have devoted extensive efforts to the modeling of C2 H6 solubility in water or aqueous NaCl solutions
in order to interpolate between the data points or extrapolate
beyond the data range [11–15]. Although several models on
C2 H6 solubility have been published, they either deviate from
experimental data by a big margin or are limited in a narrow T–P
region. Up to now, no model can predict C2 H6 solubility accurately both in pure water and in aqueous NaCl solutions over a
large T–P region.
Based on a cubic equation of state and a mixing rule of Huron
and Vidal [16], Sorensen et al. [11] predicted gas solubility in
∗
Corresponding author.
E-mail address: [email protected] (Z. Duan).
0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.fluid.2005.09.014
pure water and aqueous salt solutions. They tried to model C2 H6
solubility in pure water (303.15–523.15 K and 0–2000 atm) and
in NaCl solutions (273.15–303.15 K and 1–16 atm). However,
the average deviation of the calculated C2 H6 solubility from
experimental measurements is up to 25.0% in the C2 H6 –H2 O
system and up to 14.3% in the C2 H6 –H2 O–NaCl system.
Mohammadi et al. [12] developed a model based on modified Patel–Teja EOS and non-density-dependent mixing rule
to predict the vapor–liquid equilibria of the C2 H6 –H2 O system. The model is accurate for the solubility in pure water in a
small T–P range (274.26–343.08 K and 3.73–49.52 bar). Carroll
and Mather [13] presented a model (C–M model) for the solubility of light hydrocarbons in water and aqueous solutions
of alkanolamines by using Henry’s law and Peng–Robinson
EOS. The temperature range for the C2 H6 –H2 O system is
310.95–444.25 K and the highest pressure is limited below
300 bar. The average deviation of the calculated C2 H6 solubilities is 6.69%, as can be seen later. This model cannot predict
C2 H6 solubility at higher pressures. For instance, at 344.15 K
and 1000 bar, the deviation is over 20%. Li et al. [14] developed
a model (L–V model) to predict the solubility and gas–liquid
equilibrium for gas–water and light hydrocarbon–water systems
with modified UNIFAC [17] and Soave–Redlich–Kwong EOS
[18], covering a range of 310–444 K and 10–700 bar for the
C2 H6 –H2 O system. This model cannot accurately predict C2 H6
solubility in water below about 420 K. Soreide and Whitson [15]
developed a model (S–W model) to calculate the C2 H6 solu-
S. Mao et al. / Fluid Phase Equilibria 238 (2005) 77–86
78
l(0)
bility in pure water and aqueous NaCl solutions, ranging from
311 to 473 K and from 14 to 690 bar. It is in good agreement
with experimental solubility in pure water at T > 311 K, but the
average deviation from experimental results is over 15% at low
temperatures (273–310 K). In addition, it is not accurate for the
solubility in aqueous NaCl solutions, with the average deviation
from experiments of about 20% in the studied region. Kim et al.
[19] predicted C2 H6 solubility in aqueous NaCl solutions from
311 to 411 K using an EOS with a modified Debye–Huckel electrostatic term. This model has similar accuracy to L–V model,
but is only applicable from 27 to 136 bar. Notice that most of
the above mentioned models are for general applications, not
specifically for ethane. Errington et al. [20] made a Monte Carlo
simulation about the phase equilibria of the C2 H6 –H2 O mixtures (523–573 K and 200–3000 bar), but the simulated C2 H6
solubilities deviate significantly (up to 30% on an average) from
the experimental results of Danneil et al. [21]. Economou [22]
adopted the same Monte Carlo simulation to predict the phase
equilibria of the C2 H6 –H2 O system, and the results are no better
than those of Errington et al. [20]. Voutsas et al. [23] calculated
water–hydrocarbon phase equilibria using EOS and the cubic
plus association and statistical associating fluid theory, where
the calculated C2 H6 solubilities in water deviate from experimental data of Danneil et al. [21] by more than 25%. McCabe
et al. [24] studied the solubility of alkanes in near-critical water.
In this article, we present a more accurate model covering
a larger T–P range in the C2 H6 –H2 O and C2 H6 –H2 O–NaCl
systems using an approach proposed by Duan et al. [25]. In
this approach, the chemical potential of C2 H6 in vapor is calculated with the equation of state developed in this study, and
the chemical potential of C2 H6 in liquid phase is described by
the specific interaction model of Pitzer [26]. The framework
of the model is presented in Section 2. A brief description of
the C2 H6 equation of state is given in Section 3. In order to
settle the controversy over the experimental measurements, the
available data are reviewed in Section 4. The last section shows
that the new model can calculate C2 H6 solubility in pure water
(273–444 K and 0–1000 bar), and in aqueous NaCl solutions
(273–348 K, 0–16 bar and 0–6.3 mol kg−1 ) with accuracy close
to that of experiments (about 7% on an average), and the results
are obviously superior to the literature models currently available.
µlC2 H6 (T, P, m) = µC2 H6 (T, P) + RT ln αC2 H6 (T, P, m)
2. Phenomenological description of gas solubility as a
function of pressure, temperature and composition
where PHs 2 O is the saturated pressure of pure water which can be
s
calculated from an empirical equation (see Appendix A), ϕH
2O
refers to the fugacity coefficient of pure water calculated from
the EOS of Duan et al. [30], xH2 O the mole fraction of water
P
in liquid and P s VH2 O dP is calculated with the equations in
C2 H6 solubility in aqueous solutions depends on the balance between the chemical potential of C2 H6 in the liquid phase
µlC2 H6 and that in the vapor phase µvC2 H6 . The potential can be
written in terms of fugacity in vapor phase and activity in the
liquid phase:
v(0)
µvC2 H6 (T, P, y) = µC2 H6 (T ) + RT ln fC2 H6 (T, P, y)
v(0)
= µC2 H6 (T ) + RT ln yC2 H6 P
+ RT ln ϕC2 H6 (T, P, yC2 H6 )
(1)
l(0)
= µC2 H6 (T, P) + RT ln mC2 H6
+ RT ln γC2 H6 (T, P, m)
(2)
l(0)
where µC2 H6 , the standard chemical potential of C2 H6 in liquid, is defined as the chemical potential in hypothetically ideal
v(0)
solution of unit molality [27] and µC2 H6 , the standard chemical potential in vapor, is the hypothetical ideal gas chemical
potential when the pressure is set to 1 bar.
At phase equilibrium µlC2 H6 = µvC2 H6 , and we obtain
l(0)
v(0)
µC2 H6 (T, P) − µC2 H6 (T )
yC H P
ln 2 6 =
− ln ϕC2 H6 (T, P, y)
m C 2 H6
RT
+ ln γC2 H6 (T, P, m)
(3)
v(0)
In the parameterization, the reference value µC2 H6 can be set
l(0)
to 0 for convenience, because only the difference between µC2 H6
v(0)
and µC2 H6 is necessary. Since there is a small mole fraction of
water in the vapor phase, the fugacity coefficient of C2 H6 in
gaseous mixtures differs very little from that of pure C2 H6 at
273–444 K. Therefore, ln ϕC2 H6 can be approximated from the
EOS for pure C2 H6 (see Section 3), which means that the Lewis
fugacity rule is applied. In our previous studies [25,28,29], the
mole fraction of gas component i (not water), yi , is calculated
from
yi =
P − P H2 O
P
(4)
where the partial pressure of water in vapor, PH2 O , can be
approximated in two approaches. One approach is to approximate it as the saturated pressure of pure water, which will lead
l(0)
to errors (up to 5%) for µC2 H6 /RT and ln γCl 2 H6 . However, these
errors can be cancelled to a large extent in the parameterization.
The second approach is to approximately regard PH2 O as H2 O
fugacity:
 P

PHs O VH2 O dP
s
2

PH2 O = PHs 2 O ϕH
x
exp 
(5)
2 O H2 O
RT
H2 O
Appendix B. At high temperatures and pressures, the second
approach can improve the prediction of the vapor composition
(T > 473 K), as indicated in Table 1. However, in the studied
region (273–444 K and 0–1000 bar), the difference between the
vapor composition predicted from the two approaches is trivial which cause little change for the ethane solubility in water
and aqueous NaCl solutions. In this article, we adopt the first
approach to calculate PH2 O for convenience.
S. Mao et al. / Fluid Phase Equilibria 238 (2005) 77–86
79
Table 1
Comparison of the model with experimental C2 H6 mole fraction in vapor phase of the C2 H6 –H2 O system
exp
T (K)
P (bar)
PHs 2 O (bar)
PH2 O (bar)
yCs 2 H6
yC2 H6
yC2 H6
Dev yCs 2 H6 (%)
Dev yC2 H6 (%)
473.15
473.15
473.15
473.15
473.15
473.15
473.15
473.15
523.15
523.15
523.15
523.15
523.15
523.15
523.15
523.15
573.15
573.15
573.15
573.15
573.15
573.15
573.15
200
500
1000
1500
2000
2500
3000
3500
200
500
1000
1500
2000
2500
3000
3500
500
1000
1500
2000
2500
3000
3500
15.4036
15.4036
15.4036
15.4036
15.4036
15.4036
15.4036
15.4036
39.5735
39.5735
39.5735
39.5735
39.5735
39.5735
39.5735
39.5735
86.5384
86.5384
86.5384
86.5384
86.5384
86.5384
86.5384
20.821
24.4505
31.6514
40.5335
51.4877
64.9153
78.8854
102.457
61.8875
72.6658
93.7585
118.8
149.072
185.6088
229.4072
282.1757
132.1966
173.7296
219.7655
274.0007
338.7432
415.6821
506.6905
0.923
0.9692
0.9846
0.9897
0.9923
0.9938
0.9949
0.9956
0.8021
0.9209
0.9604
0.9736
0.9802
0.9842
0.9868
0.9887
0.8269
0.9135
0.9423
0.9567
0.9654
0.9712
0.9753
0.8959
0.9511
0.9683
0.973
0.9743
0.974
0.9737
0.9707
0.6906
0.8547
0.9062
0.9208
0.9255
0.9258
0.9235
0.9194
0.5902
0.7308
0.7729
0.7877
0.79
0.7853
0.7756
0.88
0.93
0.93
0.93
0.93
0.93
0.93
0.93
0.69
0.781
0.85
0.885
0.902
0.902
0.902
0.902
0.587
0.734
0.828
0.855
0.855
0.855
0.855
4.89
4.22
5.87
6.42
6.7
6.86
6.98
7.05
16.25
17.91
12.99
10.01
8.67
9.11
9.4
9.61
40.87
24.46
13.8
11.89
12.91
13.59
14.07
1.81
2.27
4.12
4.62
4.76
4.73
4.7
4.38
0.09
9.44
6.61
4.05
2.61
2.64
2.38
1.93
0.55
−0.44
−6.65
−7.87
−7.6
−8.15
−9.29
PHs 2 O : saturated pressure of water; PH2 O : partial pressure of water calculated from Eq. (5); yCs 2 H6 : mole fraction of C2 H6 calculated through saturated pressure of
exp
water; yC2 H6 : mole fraction of C2 H6 calculated from Eq. (5) and Dalton’s law; yC2 H6 : experimental mole fraction of C2 H6 [21]; dev yCs 2 H6 : deviation of mole fraction
of C2 H6 calculated from PHs 2 O ; dev yC2 H6 : deviation of mole fraction of C2 H6 calculated from Eq. (5).
ln γC2 H6 is expressed as a virial expansion of excess Gibbs
energy [26].
ln γC2 H6 =
c
+
2λC2 H6 −c mc +
c
In order to calculate ϕC2 H6 (T, P), we developed an equation
of state for pure C2 H6 based on the formula of Duan et al. [30]:
2λC2 H6 −a ma
a
ξC2 H6 −a−c mc ma
3. The equation of state of C2 H6
(6)
Z=
a
where λ and ξ are second- and third-order interaction parameters, respectively; c and a mean cation and anion, respectively.
Substituting Eq. (6) into Eq. (3) yields
µC2 H6
yC2 H6 P
=
2λC2 H6 −c mc
− ln ϕC2 H6 +
mC2 H6
RT
c
+ 2λC2 H6 −a ma +
ξC2 H6 −c−a mc ma
= 1+
a
c
a
(7)
Following Pitzer et al. [31], we choose the following equation
l(0)
for the P–T dependence of λ’s, ξ’s and µC2 H6 /RT :
Par(T, P) = c1 + c2 T +
c3
c6 P
c7
+ c 4 T 2 + c5 P + 2 +
T
T
P
(8)
Eqs. (7) and (8) form the basis of our model parameterization.
a1 + a2 /Tr2 + a3 /Tr3
a4 + a5 /Tr2 + a6 /Tr3
+
Vr
Vr2
a7 + a8 /Tr2 + a9 /Tr3
a10 + a11 /Tr2 + a12 /Tr3
+
Vr4
Vr5
a15
a13
a15
+ 3 2 a14 + 2 exp − 2
Tr V r
Vr
Vr
+
l(0)
ln
Pr Vr
Tr
(9)
Pr =
P
Pc
(10)
Tr =
T
Tc
(11)
Vr =
V
Vc
(12)
Vc =
RTc
Pc
(13)
where Pc and Tc are critical pressure and critical temperature, respectively; R is the universal gas constant
S. Mao et al. / Fluid Phase Equilibria 238 (2005) 77–86
80
Table 3
Experimental data of C2 H6 solubility
Table 2
Parameters of Eq. (9) for pure C2 H6 and water
Parameters
C2 H6
Water
References
System
T (K)
P (bar)
Na
a1
a2
a3
a4
a5
a6
a7
a8
a9
a10
a11
a12
a13
a14
a15
1.17251079D−002
−0.12275519
−0.21790069
3.88523929D−002
−0.18300538
0.14631598
−3.87281285D−004
4.60403075D−003
−3.73899089D−003
4.81844769D−005
−2.89809201D−004
2.55723237D−004
4.06315933D−002
0.68348632
6.55999984D−002
8.64449220D−02
−3.96918955D−01
−5.73334886D−02
−2.93893000D−04
−4.15775512D−03
1.99496791D−02
1.18901426D−04
1.55212063D−04
−1.06855859D−04
−4.93197687D−06
−2.73739155D−06
2.65571238D−06
8.96079018D−03
4.02000000D+00
2.57000000D−02
[3]
[12]
[21]
[46,47]
[48]
[48]
[50]
[51,52]
[53]
[54]
[55]
[56]
[57]
[57]
[58]
[60]
[60]
[61]
[62]
[62]
[65]
[66]
[67]
Water
Water
Water
Water
Water
NaCl (M not clear)
Water
Water
Water
Water
Water
Water
Water
0–2.95 M NaCl
Water
Water
0.25–2.1 M NaCl
0–6.29 M NaCl
Water
0.5–2.1 M NaCl
Water
Water
Water
283.2–303.2
274.26–343.08
473.15–673.15
310.9–444.3
285.5–345.6
285.75–344.85
273.51–353.12
310.9–377.6
344.15
273.15–288.15
278.15–298.15
274.7–312.9
273.2–293.2
273.15–293.15
278.2–318.2
283.15–303.15
283.15–303.15
283.15–348.15
273.15
273.15
278.15–308.15
298.15–363.15
275.44–323.15
5–40
3.73–49.52
200–3700
4.1–685
1.01
1.01
1.01
25.7–260.3
200–1000
6.6
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01
1.01–5.07
1.01–16
1.01
Not clear
0.51–1.11
17
23
77
75
14
4
9
9
4
7
5
6
2
8
3
10
20
168
3
30
4
2
23
(83.14467 bar cm3 K−1 mol−1 ); V is the molar volume. Note that
Vc is not the real critical volume. The parameters of the EOS are
fitted to the experimental PVT measurements of C2 H6 [32–45],
and the results are listed in Table 2. The critical properties of
C2 H6 are: Tc = 305.33 K and Pc = 48.718 bar. The fugacity coefficient of C2 H6 can be derived from Eq. (9):
ln ϕ(T, P) = Z − 1 − ln Z +
a1 + a2 /Tr2 + a3 /Tr3
Vr
a4 + a5 /Tr2 + a6 /Tr3
a7 + a8 /Tr2 + a9 /Tr3
+
+
2Vr2
4Vr4
a10 + a11 /Tr2 + a12 /Tr3
a13
+
5
5Vr
2Tr3 a15
a15
a15
× a14 + 1 − a14 + 1 + 2 exp − 2
Vr
Vr
(14)
+
The total average deviation of predicted volumes from experimental results (273–700 K and 0–1000 bar) is 0.22%, and the
averaged deviation of saturated pressures is 0.99%, with a maximum of 1.93%. The average deviations of saturated vapor and
liquid volumes are 1.34% and 0.27% respectively, while the corresponding maxima are 2.21% and 0.78%, respectively.
a
N: number of measurements.
4. Review of solubility data of C2 H6
The solubilities of C2 H6 in pure water have been measured
over a wide P–T range, but the measurements for aqueous NaCl
solutions are limited in a small P–T range (Table 3). Since the
solubilities of C2 H6 are much lower than those of CH4 , CO2 ,
H2 S, etc., their measurements showed larger uncertainties.
The most extensive measurements of C2 H6 solubility in water
include those reported by Refs. [12,21,46–50]. The experimental data at high pressures are reported by Refs. [21,46,47,51–53].
The other experimental data are for low pressures. We find
that most of the data sets for C2 H6 solubility in pure water
are consistent with each other, except for those [3,54] whose
isobaric solubility data are apparently deviated from others’
data. In addition, some C2 H6 solubilities of Morrison and Billett [48] (1 atm and 285.5–303.7 K) and Mohammadi et al. [12]
(>303 K) are incompatible with other data at the same P–T
range. Only the C2 H6 solubility data of Danneil et al. [21] fall
in the high T–P range (473–673 K and 200–3500 bar). However, these data sets may not be reliable. We find that C2 H6
Table 4
Interaction parameters of Eq. (8)
l(0)
T–P coefficient
µC2 H6 /RT
λC2 H6 −Na
ξC2 H6 −Na−Cl
c1
c2
c3
c4
c5
c6
c7
54.1127956327964
−8.583893829070893E−002
−7736.34284365169
4.167222742396957E−005
1.209399974354395E−003
84.2894899992575
2.65430360892582
−5.556284666975641E−003
−208.023819748501
−0.188409958505841
6.058865412902489E−004
−0.110132531032777
S. Mao et al. / Fluid Phase Equilibria 238 (2005) 77–86
81
Table 5
Calculated C2 H6 solubility in water (mol kg−1 water) at vapor–liquid equilibria
P (bar)
1
5
10
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
T (K)
273.15
293.15
313.15
333.15
353.15
373.15
393.15
413.15
433.15
453.15
473.15
0.00421
0.02008
0.03761
0.00208
0.01019
0.01940
0.00123
0.00634
0.01225
0.04195
0.04693
0.04952
0.05173
0.05371
0.05551
0.05717
0.05872
0.06016
0.06150
0.06276
0.06394
0.06504
0.06607
0.06703
0.06792
0.06876
0.06953
0.07024
0.07090
0.00079
0.00459
0.00904
0.03395
0.04330
0.04655
0.04909
0.05129
0.05327
0.05508
0.05675
0.05831
0.05976
0.06111
0.06238
0.06357
0.06469
0.06573
0.06670
0.06761
0.06845
0.06923
0.06996
0.00044
0.00367
0.00749
0.03038
0.04296
0.04766
0.05090
0.05359
0.05596
0.05810
0.06006
0.06187
0.06356
0.06513
0.06660
0.06797
0.06926
0.07046
0.07158
0.07263
0.07361
0.07452
0.07536
0.00304
0.00669
0.02958
0.04503
0.05203
0.05649
0.06002
0.06305
0.06576
0.06821
0.07046
0.07254
0.07448
0.07628
0.07797
0.07954
0.08100
0.08237
0.08365
0.08484
0.08595
0.08697
0.00236
0.00614
0.03075
0.04954
0.05961
0.06600
0.07089
0.07499
0.07859
0.08183
0.08477
0.08748
0.08998
0.09231
0.09447
0.09648
0.09836
0.10010
0.10173
0.10324
0.10464
0.10594
0.00119
0.00540
0.03349
0.05680
0.07088
0.08014
0.08714
0.09291
0.09791
0.10235
0.10637
0.11003
0.11341
0.11652
0.11941
0.12208
0.12457
0.12688
0.12903
0.13101
0.13285
0.13455
0.00378
0.03746
0.06713
0.08664
0.10009
0.11028
0.11862
0.12577
0.13206
0.13771
0.14284
0.14753
0.15184
0.15581
0.15949
0.16289
0.16604
0.16895
0.17164
0.17412
0.17641
0.00006
0.04210
0.08081
0.10790
0.12739
0.14238
0.15462
0.16504
0.17417
0.18231
0.18965
0.19633
0.20244
0.20806
0.21323
0.21799
0.22239
0.22644
0.23017
0.23361
0.23676
0.04610
0.09775
0.13559
0.16385
0.18596
0.20408
0.21949
0.23292
0.24484
0.25555
0.26525
0.27408
0.28217
0.28958
0.29638
0.30264
0.30839
0.31366
0.31849
0.32292
solubilities in water keep constant from 200 to 3500 bar at
473.15 K, which may be unreasonable. Many experimental data
for other gases (e.g. CH4 and CO2 ) show obvious variations with
pressure, as we reviewed [25,28]. Therefore, in our parameterization, we adopt the following C2 H6 solubility data in pure water
[46–49,51,52,55–60] and a portion of data of Morrison and Bil-
lett [48] (1 atm and 303.7–345.6 K) and Mohammadi et al. [12]
(274.26–303 K). The maximum P and T are high up to 444 K and
1000 bar.
Experimental C2 H6 solubilities in aqueous NaCl solutions
include these [48,57,60–62]. Most of these data are measured at
1 atm, where the NaCl concentration is up to 6.3 mol kg−1 . These
Table 6
Calculated C2 H6 solubility deviations in water and aqueous NaCl solutions
References
System
T (K)
P (bar)
Na
AAD (%)
MAD (%)
Winkler [59]
Culberson et al. [46]
Eucken and Hertzberg [57]
Claussen and Polglase [56]
Morrison and Billett [48]
Czerski and Czaplinski [62]
Wetlaufer et al. [58]
Anthony and Mcketta [51,52]
Ben-Naim et al. [55]
Wen and Hung [65]
Yaacobi and Bennaim [60]
Rettich et al. [49,67]
Dhima et al. [53]
Wang et al. [3]
Mohammadi et al. [12]
Eucken and Hertzberg [57]
Mishnina et al. [61]
Czerski and Czaplinski [62]
Yaacobi and Bennaim [60]
Water
Water
Water
Water
Water
Water
Water
Water
Water
Water
Water
Water
Water
Water
Water
NaCl(aq)
NaCl(aq)
NaCl(aq)
NaCl(aq)
273.51–353.12
310.9–444.3
273.2–293.2
274.7–312.9
303.7–345.6
273.15
278.2–318.2
310.9–377.6
278.15–298.15
278.15–308.15
283.15–303.15
275.44–323.15
344.15
283.2–303.2
274.26–303
273.15–293.15
283.15–348.15
273
283.15–303.15
1.01
4.1–685
1.01
1.01
1.01
1.01–5.07
1.01
25.7–260.3
1.01
1.01
1.01
0.51–1.11
200–1000
5–40
3.73–41.3
1.01
1.01
3–16
1.01
9
75
2
6
8
3
3
9
5
4
10
23
4
17
19
8
168
24
20
1.71
3.73
2.29
1.75
4.14
12.32
1.81
5.21
1.16
1.61
0.74
8.00
4.27
16.02
5.91
13.58
3.57
7.87
9.51
2.79
12.61
2.81
3.59
7.71
18.21
2.62
7.78
2.59
2.42
1.48
11.53
7.1
26.66
14.14
19.72
11.51
18.88
12.53
AAD: average absolute deviations calculated from this model and MAD: maximal absolute deviations calculated from this model.
a N: number of data points.
82
S. Mao et al. / Fluid Phase Equilibria 238 (2005) 77–86
data are in reasonable agreement with each other. The data [62]
(1.01325 bar and 273 K) are very scattered and therefore not
included in the parameterization. Morrison and Billett [48] did
not report the corresponding ionic concentrations, so his data
cannot be used in this work. The experimental C2 H6 solubilities
in aqueous NaCl solutions that are used in the parameterization
include those [57,60,61] and Czerski and Czaplinski [62] (273 K
and 3–16 bar), which only cover a relatively small T–P range
(273–348 K and 0–16 bar).
5. Parameterization and comparison with experimental
data
Since measurements can only be made in electronically neutral solutions, one of the parameters in Eq. (7) must be assigned
arbitrarily [63]. We set λC2 H6 −Cl to zero and then fit remaining parameters to the experimental solubilities selected above,
l(0)
where µC2 H6 /RT is evaluated from the C2 H6 solubility in
pure water with a standard deviation of 3.96%; λC2 H6 −Na and
Fig. 1. The solubility of C2 H6 in pure water (model predictions vs. experimental data).
S. Mao et al. / Fluid Phase Equilibria 238 (2005) 77–86
83
Fig. 2. The solubility of C2 H6 in aqueous NaCl solutions (model predictions vs. experimental data).
ξC2 H6 −Na−Cl are then evaluated simultaneously to the solubilities
in aqueous NaCl solutions with a standard deviation of 5.25%.
Table 4 gives the optimized parameters.
Using these parameters, the C2 H6 solubility in pure water
(273–444 K and 0–1000 bar) (see Table 5) and aqueous NaCl
solutions (273–348 K, 0–16 bar and 0–6.3 M) can be calculated.
Due to lack of reliable experimental solubility data (T > 444 K),
this model maybe fail in the near-critical region. Similar solubility calculations are also made from other models (C–M, L–V
and S–W models). Table 6 shows the C2 H6 solubility deviations
in water and aqueous NaCl solutions calculated from our model.
Figs. 1 and 2 show the comparisons between the experimental
results and our model prediction. As can be seen from the figures, most experimental data are described by this model within
or close to experimental uncertainty. Our model not only covers
a wider range, but also is more accurate. The average deviations
of C2 H6 solubility in water calculated from C–M, L–V, S–W
and this model are 6.69%, 15.56%, 7.06% and 3.40% compared
with extensive experimental data [12,46,47,50,53,55]. The average deviations of C2 H6 solubility in aqueous NaCl solutions
calculated from S–W and this model are 20.13% and 4.74%
compared with experimental measurements [60–62].
As demonstrated by Fig. 1a–h, our model and S–W model are
apparently better than other two models, but our model is better
than S–W model at low temperatures (Fig. 1f–h). Our model
is also much better than S–W model at high temperature for
salt-containing systems (see Fig. 2c–d). Fig. 2e shows that solubility of C2 H6 in aqueous NaCl solutions varies almost linearly
with pressure below 10 bar, which indicates a good Henry’s law
behavior. According to Sloan [64], C2 H6 in aqueous NaCl solutions (2.05 M) will form clathrate hydrate at 273.15 K when P is
over 10 bar. Fig. 3 shows that the predicted C2 H6 solubilities
as a temperature function at given pressure exhibit a minimum. The minima vary from about 370 K at 50 bar to 326 K at
1000 bar.
The heats of solution, partial molar volumes and Henry’s
constants of C2 H6 in water can also be derived from the above
solubility model:
l(0)
v(0)
∂
Hm − Hm
µl(0) − µv(0)
=
−
RT 2
∂T
RT
= c2 −
c3
2c6 P
+ 2c4 T −
2
T
T3
(15)
S. Mao et al. / Fluid Phase Equilibria 238 (2005) 77–86
84
(16) and P is total pressure. At given temperature, we can set P
slightly above PHs 2 O for the calculation of Henry’s constant.
The predicted molar heats of solution, partial molar volumes
and Henry’s constants are compared with some experimental
results (Tables 7 and 8), which exhibits a good agreement. These
offer extra information on the good reliability of the model.
6. Conclusions
Fig. 3. The isobaric solubility of C2 H6 in pure water predicted from this model.
Table 7
Molar heat of solution and partial molar volume of C2 H6 in water
T (K)
−Hms (kJ mol−1 )
P (bar)
288.15
298.15
308.15
1
1
1
a
b
c
22.31
19.52
16.65
22.56
19.30
16.21
21.64
19.24
16.82
V C2 H6 (l)
298.15
310.93
344.26
377.59
410.93
444.26
1
358
358
358
358
358
(cm3
mol−1 )
c
d
53.49
53.81
54.97
56.53
58.38
60.45
53.27
e
53.00
53.56
57.25
63.74
63.99
a: Dec and Gill [68,69]; b: Olofsson et al. [70]; c: this study; d: Tiepel and
Gubbins [71]; e: Kobayashi and Katz [72].
µl(0) − µv(0)
c6 V C2 H6 (l)
= RT c5 + 2
RT
T
−V C2 H6 (l) (P − PHs 2 O )
yC2 H6 ϕC2 H6 P
exp
kH (T ) =
xC2 H6
RT
∂
= RT
∂P
l(0)
(16)
(17)
v(0)
where Hm − Hm is the molar heat of solution of C2 H6 ,
V C2 H6 (l) the partial molar volume of C2 H6 calculated from Eq.
Table 8
Henry’s constants (kH ) of C2 H6 in water
T (K)
kH1 (bar)
kH2 (bar)
273.15
293.15
310.93
344.26
377.59
410.93
444.26
12962
25780
39651
62711
71186
64388
49977
12797
25534
39430
62062
68949
60357
44991
kH3 (bar)
Based on a new highly accurate equation of state for C2 H6
developed in this study and the theory of Pitzer [26], an accurate model for solubility of C2 H6 in pure water and aqueous
NaCl solutions has been developed. This model gives results
within or close to experimental uncertainty (about 7%) in pure
water (273–444 K and 0–1000 bar) and aqueous NaCl solutions
(273–348 K, 0–16 bar and 0–6.3 M). Comparison with experimental measurements and other models indicates that our model
can predict C2 H6 solubility both in pure water and aqueous
NaCl solutions with higher accuracy and wider P–T region
than previous models. A FORTRAN code is developed for this
model and can be downloaded from the website: www.geochemmodel.org/programs.htm.
List of symbols
m
molality of C2 H6 or salts in liquid phase
P
total pressure, that is PC2 H6 + PH2 O in bar
Par
parameter
R
universal gas constant, which is
83.14467 bar cm3 mol−1 K−1
T
absolute temperature in Kelvin
y
mole fraction of C2 H6 in vapor phase
Greek letters
α
activity
ϕ
fugacity coefficient
γ
activity coefficient
λC2 H6 –ion interaction parameter
µ
chemical potential
ξC2 H6 –cation–anion interaction parameter
Subscripts
a
anion
c
cation
Superscripts
l
liquid
v
vapor
(0)
standard state
Acknowledgements
41558
64436
68658
59683
47811
kH1 : calculated from this model; kH2 : from Prini and Crovetto [73]; kH3 : from
Kobayashi and Katz [72].
We thank the two anonymous reviewers and Dr. Peter Cummings for their constructive suggestions. This work is supported
by Zhenhao Duan’s “One Hundred Scientist Project” funds
awarded by the Chinese Academy of Sciences and his outstanding young scientist funds (#40225008) awarded by National
Natural Science Foundation of China.
S. Mao et al. / Fluid Phase Equilibria 238 (2005) 77–86
Appendix A. The empirical model for pure water
pressure
References
The empirical model to calculate pure water pressure has the
following form:
P=
Pc T
Tc
[1 + c1 (−t)1.9 + c2 t + c3 t 2 + c4 t 3 + c5 t 4 ]
(A1)
where c1 = −38.640844; c2 = 5.8948420; c3 = 59.876516;
c4 = 26.654627; c5 = 10.637097; T is the temperature (K),
t = (T − Tc )/Tc ; Tc and Pc are critical temperature and
critical pressure of water, respectively (Tc = 647.29 K and
Pc = 220.85 bar).
Appendix B.
P
PHs
P
s
PH
s
H2 O VH2 O
= PV
Vr
Vrs
V dP of pure water
V dp = PV |PV
Ps
2O
2O
−
− PHs 2 O VHs 2 O
V
VHs
P dV
2O
− P c Vc
Vr
Vrs
Pr dVr
(B1)
Pr dVr
=
Vr
Vrs
+
85
Tr
a1 Tr + a2 /Tr + a3 /Tr2
+
Vr
Vr2
a4 Tr + a5 /Tr + a6 /Tr2
a7 Tr + a8 /Tr + a9 /Tr2
+
Vr3
Vr5
a10 Tr + a11 /Tr + a12 /Tr2
Vr6
a13 a15
a13 a14
a15
a15
dVr
+ 2 3 exp − 2 + 2 5 exp − 2
Tr V r
Vr
Tr V r
Vr
a1 Tr + a2 /Tr + a3 /Tr2
= Tr ln Vr −
Vr
+
a4 Tr + a5 /Tr + a6 /Tr2
a7 Tr + a8 /Tr + a9 /Tr2
−
2Vr2
4Vr4
a10 Tr + a11 /Tr + a12 /Tr2
a13 a14
a15
−
+
exp
−
5Vr5
2a15 Tr2
Vr2
a13 (1 + a15 /Vr2 )
a15
Vr
−
exp − 2
(B2)
s
Vr
2a15 Tr2
Vr
−
where Pr , Tr and Vr of pure water are defined in the same way
as those of C2 H6 in Section 3. The parameters (a1 –a15 ) [30] of
pure water are listed in Table 2. The critical properties of water
used are: Tc = 647.25 K and Pc = 221.19 bar.
[1] V. Dieckmann, M. Fowler, B. Horsfield, Org. Geochem. 35 (7) (2004)
845–862.
[2] R. Sassen, S.T. Sweet, D.A. DeFreitas, J.A. Morelos, A.V. Milkov, Org.
Geochem. 32 (8) (2001) 999–1008.
[3] L.-K. Wang, G.-J. Chen, G.-H. Han, X.-Q. Guo, T.-M. Guo, Fluid Phase
Equilib. 207 (2003) 143–154.
[4] A.R. Scott, In Situ 18 (2) (1994) 185–208.
[5] S. Makhoukhi, C. Marignac, J. Pironon, J.M. Schmitt, C. Marrakchi,
M. Bouabdelli, A. Bastoul, J. Geochem. Exploration 78 (9) (2003) 545–
551.
[6] K.D. Newell, R.H. Goldstein, Chem. Geol. 154 (1999) 97–111.
[7] T.A.S. Saka, Geochim. Cosmochim. Acta 55 (5) (1991) 1395–1405.
[8] E.F.F. Sturkell, C. Broman, P. Forsberg, P. Torssander, Eur. J. Miner. 10
(3) (1998) 589–606.
[9] N. Gnanendran, R. Amin, Fluid Phase Equilib. 221 (1–2) (2004)
175–187.
[10] G. Raabe, J. Janisch, J. Koehler, Fluid Phase Equilib. 185 (1–2) (2001)
199–208.
[11] H. Sorensen, K.S. Pedersen, P.L. Christensen, Org. Geochem. 33 (2002)
635–642.
[12] A.H. Mohammadi, A. Chapoy, B. Tohidi, D. Richon, Ind. Eng. Chem.
Res. 43 (2004) 5418–5424.
[13] J.J. Carroll, A.E. Mather, Chem. Eng. Sci. 52 (4) (1997) 545–552.
[14] J. Li, I. Vanderbeken, S. Ye, H. Carrier, P. Xans, Fluid Phase Equilib.
131 (1997) 107–118.
[15] I. Soreide, C.H. Whitson, Fluid Phase Equilib. 77 (1992) 217–240.
[16] M.J. Huron, J. Vidal, Fluid Phase Equilib. 3 (1979) 255–271.
[17] U. Weidlich, J. Gmehling, Ind. Eng. Chem. Res. 26 (1987) 1372–1381.
[18] G. Soave, Chem. Eng. Sci. 27 (6) (1972) 1197.
[19] A.-P. Kim, E. Stenby, A. Fredenslund, Ind. Eng. Chem. Res. 30 (1991)
2180–2185.
[20] J.R. Errington, G.C. Boulougouris, I.G. Economou, J. Phys. Chem. B
102 (1998) 8865–8873.
[21] A. Danneil, K. Todheide, E.U. Franck, Chemie Ingenieur Technik 39
(13) (1967) 816.
[22] I.G. Economou, Fluid Phase Equilib. 183–184 (2001) 259–269.
[23] E.C. Voutsas, G.C. Boulougouris, I.G. Economou, D.P. Tassios, Ind.
Eng. Chem. Res. 39 (2000) 797–804.
[24] C. McCabe, A. Galindo, P.T. Cummings, J. Phys. Chem. B 107 (44)
(2003) 12307–12314.
[25] Z. Duan, N. Moller, J.H. Weare, Geochim. Cosmochim. Acta 56 (1992)
1451–1460.
[26] K.S. Pitzer, J. Phys. Chem. 77 (1973) 268–277.
[27] K. Denbigh, The principles of chemical equilibrium, Cambridge University Press, Cambridge, 1971.
[28] Z. Duan, R. Sun, Chem. Geol. 193 (2003) 257–271.
[29] R. Sun, W. Hu, Z. Duan, J. Sol. Chem. 30 (6) (2001) 561–573.
[30] Z. Duan, N. Moller, J.H. Weare, Geochim. Cosmochim. Acta 56 (1992)
2605–2617.
[31] K.S. Pitzer, J.C. Peiper, R.H. Busey, J. Phys. Chem. Ref. Data 13 (1984)
1–102.
[32] J.A. Beattie, N. Hadlock, N. Poffenberger, J. Chem. Phys. 3 (2) (1935)
93–96.
[33] H.H. Reamer, R.H. Olds, B.H. Sage, W.N. Lacey, Ind. Eng. Chem. 36
(1944) 956–958.
[34] A. Michels, W.V. Straaten, J. Dawson, Physica 20 (1954) 17–23.
[35] D.R. Douslin, R.H. Harrison, J. Chem. Thermodyn. 5 (1973) 491–512.
[36] G.C. Straty, R. Tsumura, J. Res. Nat. Bur. Stand. A. Phys. Chem. 80
(1976) 35–39.
[37] A.K. Pal, G.A. Pope, Y. Arai, N.F. Carnahan, R. Kobayashi, J. Chem.
Eng. Data 21 (1976) 394–397.
[38] M. Jaeschke, A.E. Humphreys, P.V. Canegham, M. Fauveau, R.J.-V.
Rosmalen, Q. Pellei, GERG Technical Monograph 4, Vdl-Verlag, Dusseldorf, 1990.
[39] D.G. Friend, H. Ingham, J.F. Ely, J. Phys. Chem. Ref. Data 20 (2) (1991)
275–347.
86
S. Mao et al. / Fluid Phase Equilibria 238 (2005) 77–86
[40] A.F. Alexanders-Estrada, J.P.M. Trusler, J. Chem. Thermodyn. 29 (9)
(1997) 991–1015.
[41] C.M.M. Duarte, H.J.R. Guedes, M.N.D. Ponte, J. Chem. Thermodyn.
00 (2000) 877–889.
[42] M. Funke, R. Kleinrahm, W. Wagner, J. Chem. Thermodyn. 34 (2002)
2001–2015.
[43] M. Funke, R. Kleinrahm, W. Wagner, J. Chem. Thermodyn. 34 (2002)
2017–2039.
[44] P. Claus, R. Kleinrahm, W. Wagner, J. Chem. Thermodyn. 35 (2003)
159–175.
[45] M.H. Lagache, P. Ungerer, A. Boutin, Fluid Phase Equilib. 220 (2004)
211–223.
[46] O.L. Culberson, A.B. Horn, J.J. Mcketta, Trans. Am. Inst. Mining Metallurgical Eng. 189 (1950) 1–6.
[47] O.L. Culberson, J.J. Mcketta, Trans. Am. Inst. Mining Metallurgical
Eng. 189 (1950) 319–322.
[48] T.J. Morrison, F. Billett, J. Chem. Soc. (October) (1952) 3819–3822.
[49] T.R. Rettich, Y.P. Handa, R. Battino, E. Wilhelm, J. Phys. Chem. 85
(1981) 3230–3237.
[50] W. Hayduk, Solubility Data Series, vol. 9, Pergamon Press, Oxford,
1982, p. 9.
[51] R.G. Anthony, J.J. Mcketta, J. Chem. Eng. Data 12 (1) (1967) 17–20.
[52] R.G. Anthony, J.J. Mcketta, J. Chem. Eng. Data 12 (1) (1967) 21–28.
[53] A. Dhima, J.C.D. Hemptinne, G. Moracchini, Fluid Phase Equilib. 145
(1998) 129–150.
[54] K.Y. Song, G. Feneyrou, F. Fleyfel, R. Martin, J. Lievois, R. Kobayashi,
Fluid Phase Equilib. 128 (1997) 249–260.
[55] A. Ben-Naim, J. Wilf, M. Yaacobi, J. Phys. Chem. 77 (1) (1973) 95–102.
[56] W.F. Claussen, M.F. Polglase, J. Am. Chem. Soc. 74 (19) (1952)
4817–4819.
[57] A. Eucken, G.Z. Hertzberg, Phys. Chem. 195 (1950) 1–23.
[58] D.B. Wetlaufer, S.K. Malik, L. Stoller, R.L. Coffin, J. Am. Chem. Soc.
86 (3) (1964) 508–514.
[59] L.W. Winkler, Chem. Ber. 34 (1901) 1408–1422.
[60] M. Yaacobi, A. Bennaim, J. Phys. Chem. 78 (2) (1974) 175–178.
[61] T.A. Mishnina, O.I. Avdeeva, T.K. Bozhovakaya, Materialy Vses.
Nauchn. Issled. 46 (1961) 93–110.
[62] L. Czerski, A. Czaplinski, Ann. Soc. Chim. Polonorum (Poland) 36
(1962) 1827–1834.
[63] C.E. Harvie, J.H. Wear, Geochim. Cosmochim. Acta 48 (1984) 723–
751.
[64] E.K. Sloan, Clathrate Hydrates of Natural Gases, Marcel Decker, 1998.
[65] W.Y. Wen, J.H. Hung, J. Phys. Chem. 74 (1) (1974) 170–175.
[66] E.S. Rudakov, A.I. Lutsyk, Russ. J. Phys. Chem. 53 (1979) 731–733.
[67] T.R. Rettich, Y.P. Handa, R. Battino, E. Wihelm, J. Phys. Chem. 85 (22)
(1981) 3230–3237.
[68] S.F. Dec, S.J. Gill, J. Sol. Chem. 13 (1) (1984) 27–41.
[69] S.F. Dec, S.J. Gill, J. Sol. Chem. 14 (12) (1985) 827–836.
[70] G. Olofsson, A.A. Oshodj, E. Qvarnstrom, I. Wadso, J. Chem. Thermodyn. 16 (1984) 1041–1052.
[71] E.W. Tiepel, K.E. Gubbins, J. Phys. Chem. 76 (21) (1972) 3044–3049.
[72] R. Kobayashi, D.L. Katz, Ind. Eng. Chem. 45 (2) (1953) 446–451.
[73] R.F. Prini, R. Crovetto, J. Phys. Chem. Ref. Data 18 (3) (1989)
1231–1243.