Ind. Eng. Chem. Res. 2009, 48, 4553–4564
4553
Solubility of CO2 in (Water + Acetone): Correlation of Experimental Data and Predictions from Molecular Simulation ´ lvaro Pe´rez-Salado Kamps, and Gerd Maurer* Ilina Urukova, A Thermodynamics, Department of Mechanical and Process Engineering, UniVersity of Kaiserslautern, P.O. Box 30 49, D-67653 Kaiserslautern, Germany
The correlation and prediction of the solubility of gases in (nonelectrolyte as well as in electrolyte) aqueous/ organic solvent mixtures is an important topic in many areas of chemical engineering. Despite the importance of that field, comparatively little attention has been given to that area in both classical thermodynamics (i.e., correlation methods that are based on semiempirical expressions for the excess Gibbs energy) and molecular simulation (where gas solubility can be predicted from intermolecular pair potentials). In the work presented here, recently published experimental data for the solubility of carbon dioxide in aqueous solutions of acetone (covering temperatures from 313 to 395 K at pressures ranging up to about 9 MPa) are used for testing a semiempirical, classical method (for the correlation of such gas solubility phenomena) as well as the Gibbs ensemble Monte Carlo method (GEMC) (for predicting the solubility of carbon dioxide in aqueous solutions of acetone from published intermolecular pair potentials without using any adjustable binary interaction parameter). The correlation method reproduces the experimental data (nearly) within experimental uncertainty, as was expected. The predictions by the GEMC method also agree well with the experimental data. 1. Introduction The solubility of single gases (and of gas mixtures) in aqueous/organic solvent mixtures without as well as with strong electrolytes is of interest in many areas in chemical engineering. Therefore, our research group at the University of Kaiserslautern has been engaged for many years in experimental work as well as modeling work in that area. The experimental work is primarily aiming to provide reliable data for the testing of thermodynamic models. Previously the modeling work was restricted to the development (and testing) of semiempirical expressions for the excess Gibbs energy of such solutions. Such models are to be able to correlate gas solubility data and, when the model has a profound base, to extrapolate to regions of interest that were not covered by experiments. However, such extrapolations should not go too far beyond the experimentally investigated regions, as otherwise the quality of the calculation results might suffer dramatically. On the other side, a lot of progress has been achieved in the field of molecular simulation of thermodynamic properties in recent years. Nowadays, an increasing number of chemical engineers are considering applying these techniques for the prediction of thermodynamic properties from information on (effective) intermolecular pair potentials. The present work is aiming to contribute to both areas. The modeling work starts from recently published experimental results for the solubility of CO2 in acetone and in aqueous solutions of acetone (Jo¨decke et al.1). In the first part of the present publication, these data are used to parametrize a model for the excess Gibbs energy that was particularly developed to describe the solubility of gases in aqueous solutions of mixed solvents, without as well as with strong electrolytes (Pe´rez-Salado Kamps2). That semiempirical model is then also used to compare calculation results with literature data for the solubility of CO2 in acetone as well as in aqueous solutions of acetone. In the second part, the solubility of CO2 in such solutions is predicted by GEMC simulations and compared to the experimental data by Jo¨decke et al.1 * To whom correspondence should be addressed. Tel.: +49 631 205 2410. Fax: +49 631 205 3835. E-mail:
[email protected].
The ternary system (CO2 + water + acetone) reveals a highpressure liquid-liquid-vapor equilibrium (so-called “saltingout by a nearcritical gas” phenomenon) that was investigated by Wendland et al.3 The model by Pe´rez-Salado Kamps has not yet been extended to describe such high-pressure phenomenon. Vice versa, although some efforts have been made to develop a software package to handle such multiphase phenomena (cf., Kristo´f et al.4), there is currently no approved method available for predicting such phenomena by molecular simulation. Therefore, an extension of the present publication that deals with the correlation (using the model by Pe´rez-Salado Kamps) as well as the prediction (by molecular simulation) of the liquid-liquid phase split that might be observed when a homogeneous liquid mixture of (water + acetone) is pressurized by CO2 has to be postponed to future work. 2. Correlation by the Model of Pe´rez-Salado Kamps for the Excess Gibbs Energy 2.1. Model Description. Only an outline of the model is given here. For further details, the reader is recommended to the previous publications (Pe´rez-Salado Kamps,2 Scha¨fer et al.5). The framework for the vapor-liquid equilibrium in the ternary system (CO2 + water + acetone) starts from the extended Raoult’s law for the solvent components (acetone and water, respectively):
(
pskφsk exp
)
Vsk(p - psk) ak ) pykφ(V) k RT
(1)
(for k ) water (W) and acetone (A)), and the extended Henry’s law (on the molality scale) for the gaseous solute CO2: (m) kH,CO (T, x˜A) 2
(
exp
)
∞ VCO (T, x˜A)(p - psmix(T, x˜A) mCO2 2
RT
γ*CO2 ) mo (V) pyCO2φCO (2) 2
pks, φks, Vks, and ak are the pressure, the fugacity coefficient, the molar liquid volume (all at saturation), and the liquid phase
10.1021/ie801015u CCC: $40.75 2009 American Chemical Society Published on Web 04/01/2009
4554 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009
activity, respectively, of solvent component k. The universal gas constant, pressure, and temperature are designated by R, T, and p, respectively, while yk and φ(V) k stand for the mole fraction and the fugacity coefficient of component k in the vapor phase. (m) (∞) (T, x˜A) and VCO (T, x˜A) are Henry’s constant of CO2 (on kH,CO 2 2 the molality scale) in the gas-free liquid solvent mixture at the s (T, x˜A) and the partial saturation pressure of that mixture pmix molar volume of CO2 at infinite dilution in that liquid solvent (m) (∞) s (T,x˜A), VCO (T,x˜A), and pmix mixture. kH,CO (T,x˜A) depend on 2 2 temperature and the composition of the gas-free solvent mixture. This composition is here expressed by the mole fraction of acetone x˜A in the (water + acetone) mixture. mCO2 is the molality of CO2 in the solvent mixture, mo ) 1 mol/(kg solvent mixture), and γ*CO2 is the activity coefficient of CO2 in the liquid phase. When the solubility of CO2 in an aqueous solution of acetone is treated with the model of Pe´rez-Salado Kamps, the following expression results for the activity coefficient of CO2: ln γ*CO2 ) 2
mCO2
(0) βCO o 2,CO2
m
( )
+3
mCO2
2
mo
µCO2,CO2,CO2
(3)
where β(0) CO2,CO2 and µCO2,CO2,CO2 are binary and ternary parameters, respectively, that depend on temperature T and composition of the gas-free solvent mixture x˜A. The corresponding expression for the activities of the solvent components water and acetone (aW and aA, respectively) follows from:
( )(
)(
)
( )(
)(
)
(x) M* mCO2 ∂∆tGCO2 + RT mo ∂x˜A T,p ln γW,UNIQUAC + ln γW,Pitzer + ln γW,conv (4)
ln aW ) ln xW - x˜A
(x) M* mCO2 ∂∆tGCO2 + ln aA ) ln xA + x˜W RT mo ∂x˜A T,p ln γA,UNIQUAC + ln γA,Pitzer + ln γA,conv (5)
xi (i ) W, A) is the mole fraction of a solvent component in the liquid mixture of (CO2 + water + acetone). M* is the relative molar mass of the gas-free solvent mixture divided by 1000. For the binary solvent mixture under consideration, it is M* ) M*W + x˜A(M*A - M*W)
(6)
where M*W ) 0.01801528 (for water) and M*A ) 0.05808004 (x) is the mole fraction scale-based molar (for acetone). ∆tGCO 2 Gibbs energy of transfer of carbon dioxide from pure water to the solvent mixture of water and acetone. That Gibbs energy of transfer depends on temperature, pressure, and the composition of the (gas-free) solvent mixture. It can be calculated from the molality scale-based Henry’s constant of CO2 in water (kH,CO2(T,p,x˜A ) 0) ) kH,CO2,W(T,p)) and in liquid mixtures of (water + acetone) (kH,CO2(T,p,x˜A)) as follows:
(
(x) ∆tGCO (T, p, x˜A) ) RT ln 2
kH,CO2(T, p, x˜A) kH,CO2,W(T, p)
)
( )
- RT ln
M* M*W
(7)
In eqs 4 and 5, ln γi,UNIQUAC (i ) W, A) designates a contribution to the activity coefficient of the solvent component i, which results from the UNIQUAC equation. That contribution does only depend on T and x˜A; it does not depend on the dissolved amount of gas. The equations for ln γi,UNIQUAC were taken from Pe´rez-Salado Kamps.2
Because the equation for the excess Gibbs energy is based on the molality scale, whereas the UNIQUAC equation is based on the mole fraction scale, a so-called conversion term ln γi,conv (i ) W, A) is included in eqs 4 and 5. For the case under consideration (cf., e.g., ref 2): ln γi,conv ) -
( ) mCO2 o
m
[ ( ) ] mCO2
M* + ln 1 +
mo
M* (i ) W, A) (8)
Only the contribution of Pitzer’s equation6,7 to the activity coefficient of the solvent components ln γi,Pitzer (i ) W, A) contains interaction parameters that were adjusted in the present work (see also eq 3). The mathematical expressions for those terms are therefore given here (cf., also ref 2):
[( [( [( [(
ln γW,Pitzer ) -M*W M*x˜A
ln γA,Pitzer ) -M*A M*(1 - x˜A)
mCO2 mo
mCO2 mo
) )
m
mCO2 mo
)
( ) ( ) ( ) ( )
(0) βCO +2 2,CO2
2 ∂β(0) CO2,CO2
mCO2 o
)
2
∂x˜A
(0) βCO +2 2,CO2
∂x˜A
]
3
µCO2,CO2,CO2 mo mCO2 3 ∂µCO2,CO2,CO2 + (9) ∂x˜A mo
2
2 ∂β(0) CO2,CO2
mCO2
+
mCO2 o
m
]
3
µCO2,CO2,CO2 +
mCO2 3 ∂µCO2,CO2,CO2 mo
]
∂x˜A
]
(10)
2.2. Model Parameters and Comparison of Correlation Results for the Gas Solubility with Experimental Data. The model requires information for the Henry’s constant k(0) ˜ A) H,CO2(T,x of CO2 in liquid mixtures of water and acetone at the saturation s pressure of that solvent mixture pmix , the partial molar volume (∞) (T,x˜A), of CO2 at infinite dilution in that solvent mixture VCO 2 (0) and the parameters βCO2,CO2(T,x˜A) and µCO2,CO2,CO2(T,x˜A) for interactions between CO2 molecules in that solvent mixture. The corresponding properties in the pure solvents water and acetone have also to be known: Henry’s constants of CO2 (in water (m) (m) (T) and in acetone kH,CO (T)), partial molar volumes at kH,CO 2,W 2,A (∞) (∞) and VCO ), and infinite dilution in the pure solvents (VCO 2,W 2,A binary and ternary interaction parameters for CO2 in pure water and pure acetone. In addition, some properties of the pure solvents water and acetone are required: the vapor pressure (psW, psA), the molar volume of the saturated liquid (VsW, VsA), and vapor phase fugacity coefficients at saturation (φsW, φAs). Pure component vapor pressures and molar volumes are calculated from correlation equations reported before,1 and fugacity coefficients are calculated from a truncated virial equation of state.8 Details for the calculation of the pure component and mixed second virial coefficients were given previously.1 The vapor pressure of the binary solvent mixture (water + acetone) was calculated applying the UNIQUAC equation to calculate the activities in the liquid. The corresponding parameters (pure component UNIQUAC size and surface parameters (rW, rA, qW, qA) and binary UNIQUAC interaction parameters (ΨWA, ΨAW)) were reported previously.1 The correlation equation for Henry’s constant of CO2 in pure water at the vapor pressure of water (and on a molality scale (m) basis, kH,CO (T)) was adopted from Rumpf and Maurer.9 That 2,W correlation covers temperatures from 273 to 473 K. Henry’s (m) (T) was determined from literature data for the constant kH,CO 2,A solubility of CO2 in acetone. Much of such experimental data is available in the open literature.1,10-31 Whenever possible,16,18,19,22,23,25,26,28,29,31 the usual isothermal extrapolation procedure (cf., e.g., refs 1, 2) was applied to these experimental
Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4555 (∞) VCO /(cm3/mol) 2,A
) -293.9 + 0.3045·(T/K)
(12)
(0) ) 0.02267 - 11.25/(T/K) βCO 2,CO2,A
(13)
µCO2,CO2,CO2,A ) -2.467 × 10-4 + 0.09259/(T/K) (14) Table 1 gives a comparison between the aforementioned experimental data1,26 and the correlation results. Obviously, the quality of the correlation is very satisfactory (see also Figure 1 of ref 1). It must be pointed out that there are many combinations of these three parameters, which may result in a good correlation of the experimental solubility data. Also, because we could not account for any volumetric data in the correlation, the values resulting from eqs 12-14 may not be regarded as realistic. (For (∞) ≈ -200 cm3/mol.) However, example, at T ) 313 K, VCO 2,A for the purpose of the present work, this correlation proves to be sufficient. Table 1 additionally contains a comparison between other literature data and calculation results. This comparison is restricted to literature data at temperatures above about 291 K. Moreover, because we are dealing with gas solubility, the composition of the mixture is restricted to a certain maximum amount of the gas. Literature sources were discarded in that comparison, when they contained information only at pressures of about 0.1 MPa at a maximum. As can be seen from Table 1, only the data by Kir’yakov et al.18 and by Han et al.30 severely disagree with the data by Jo¨decke at al.1 and Adrian and Maurer.26 All other data listed in Table 1 show a good to very good agreement with the data from refs 1 and 26. This is illustrated by way of example in Figure 2, where the solubility pressures reported in refs 1, 19, 21, 25-27, and 31 and the calculation results from the model are plotted versus the molality of CO2 in the liquid at T ≈ 313.15 K (up to the maximum molality of the gas as given in Table 1). The extension of the model for the excess Gibbs energy of the liquid mixture for a binary system (CO2 + (either water or acetone)) to the ternary mixture (CO2 + water + acetone) requires additional information. Henry’s constants (on the (m) (T,x˜A) for the solubility of CO2 in molality scale) kH,CO 2 aqueous solutions of acetone at the vapor pressure of the solvent mixture were determined from the experimental data for the solubility of CO2 in solvent mixtures of (water + acetone) at T ) (313.75, 354.35, and 395.0) K, x˜A ≈ (0.05, 0.1, 0.25, 0.5, 0.75, 0.9, and 0.95), and pressures up to about 10 MPa reported by Jo¨decke et al.1 in the usual way (i.e., in
Figure 1. Henry’s constant of CO2 in acetone (on the molality scale): (b) extrapolated experimental results, Jo¨decke et al.;1 (O) extrapolated experimental results, literature;16,18,19,22,23,25,26,28,29,31 (-) correlation, this work.
data. As can be seen from Figure 1, values resulting from the aforementioned literature data scatter widely, and it is impossible to determine which of them are best. The following four(m) parameter equation for kH,CO (T) was adjusted to these values 2,A exactly matching the three values determined by Jo¨decke et al.1 at T ) 313.75, 354.35, and 395.0 K. The equation holds for temperatures from about 198 to 395 K.
(
ln
(m) kH,CO (T) 2,A
MPa
)
) 152.664 -
6575.78 - 24.5996 ln(T/K) + (T/K) 0.0276539(T/K) (11)
The partial molar volume of CO2 infinitely diluted in water, (∞) VCO , was calculated as recommended by Brelvi and 2,W O’Connell32 (see also ref 2). Pitzer’s6,7 parameters for interactions between carbon dioxide molecules in pure water were (0) neglected (βCO ) µCO2,CO2,CO2,W ) 0) as recommended by 2,CO2,W Rumpf and Maurer.9 The partial molar volume of CO2 infinitely (0) diluted in acetone, V(∞) CO2,A, and the interaction parameters βCO2,CO2,A and µCO2,CO2,CO2,A were fit to the experimental pressures above solutions of (CO2 + acetone) from Jo¨decke et al.1 (at T ) 313.75, 354.35, and 395.0 K) as well as to those data reported by Adrian and Maurer26 (at T ) 313.15 and 333.15 K), which do not exceed CO2 molalities of about 61 mol/(kg of acetone) (or CO2 mole fractions of about 0.78):
Table 1. Comparison of Experimental Data with Correlation/Prediction Results from the Vapor-Liquid Equilibrium (VLE) Model for the Solubility of CO2 in Acetonea
Correlation Jo¨decke et al.1 Adrian and Maurer26 Prediction from VLE Model Kir’yakov et al.18 Katayama et al.19 Traub and Stephan21 Kato et al.22 Giacobbe23 Day et al.25 and Chang et al.27 Bamberger and Maurer28 Lazzaroni et al.29 Han et al.30 Stievanno and Elvassore31
Na
T/K
i
29 16
∼313-395 ∼313-333
0 0
9 16 2 4 82 53 15 12 26 16
∼293 ∼298-313 ∼313 ∼298 ∼293-303 ∼291-313 ∼303-333 323 ∼333-393 ∼291-323
0 0 0 0 1 0 0 0 0 0
(i) pmax /MPa
8.25 8.0 2.5 5.7 4.8 3.4 1.0 5.3 8.1 7.1 13.2 4.0
mCO2,max/(mol/kg)
xCO2,max
∆pj(i)/MPa
∆pj(i)/p(i)/%
44.4 60.6
0.72 0.78
0.037 0.029
0.9 0.7
26.9 52.0 28.6 23.1 4.72 65.2 69.8 63.6 87.0 19.4
0.61 0.75 0.62 0.57 0.22 0.79 0.80 0.79 0.83 0.53
0.401 0.037 0.096 0.172 0.008 0.071 0.167 0.051 1.581 0.152
28.6 1.5 2.6 6.2 2.0 3.6 5.4 1.1 27.6 8.3
N ) number of experimental data points within the given ranges. For i ) 0, p(i) ) p (total pressure); for i ) 1, p(i) ) pCO2 (partial pressure of CO2). N (i) (i) N (i) (i) (i) xCO2 is the mole fraction of CO2 in the liquid mixture of (CO2 + acetone). ∆pj(i) ) 1/N · ∑j)1 |pj,exp - pj,calc |; ∆pj(i)/p(i) ) 1/N · ∑j)1 |(pj,exp - pj,calc )/pj,exp |. a
4556 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009
an isothermal extrapolation procedure).2 The evaluation results were correlated by means of the following equation:
[
(0) (T) + (1 - x˜A)x˜A 0.45 x˜AβCO 2,CO2,A
+ 1750 - (1 - x˜A)x˜A 7.579 + T/K 3024 1672 9.388 (1 - 2x˜A)+ 4.865 (1 - 2x˜A)2 (15) T/K T/K
(m) ln[kH,CO (T, x˜A)/MPa] 2
) (1 -
(0) (0) βCO (T, x˜A) ) (1 - x˜A)βCO (T) + 2,CO2 2,CO2,W
(m) x˜A) ln[kH,CO (T)/MPa] 2,W
{[
(m) (T)/MPa] x˜A ln[kH,CO 2,A
[
]
[
[
x˜AµCO2,CO2,CO2,A(T) + (1 - x˜A)x˜A -0.00725 +
The remaining model parameters (the partial molar volume (∞) (T,x˜A), of CO2 infinitely diluted in (water + acetone), VCO 2 and parameters for interactions between carbon dioxide (0) (T,x˜A), molecules in (water + acetone) (βCO 2,CO 2 µCO2,CO2,CO2(T,x˜A)) were simultaneously fit to the experimental pressures above solutions of (CO2 + water + acetone) from Jo¨decke et al.:1
+ x˜A 3 + cm3/mol cm /mol 1118 302.6 (1 - x˜A)x˜A·103· 1.605 + -0.3126 + · T/K T/K 1629 (1 - 2x˜A) + -3.942 + (1 - 2x˜A)2 (16) T/K
cm3/mol
) (1 - x˜A)
[(
) (
(
)
]
)
]
Only very limited experimental data for the pressure required to dissolve CO2 in liquid mixtures of (water + acetone) can be found in the open literature.3,21,33,34 The data by Kir’yakov et al.34 cover temperatures from 223 to 253 K. They are well outside the correlation range of our model. We therefore refrain from a comparison with these data here. Table 2 shows a comparison of all other literature data with the correlation results. 2.3. Thermodynamic Properties of Solution of CO2 in Mixtures of (Water + Acetone). Following KrichevskyKasarnovsky, one may account for the influence of pressure on the (molality scale based) Henry’s constant of CO2 in solvent mixtures of water and acetone as follows:
(∞) VCO (T) 2,A
(∞) (T) VCO 2,W
2 (18) T/K
Table 2 gives a comparison between the experimental data from ref 1 and the correlation results. Because of the excellent correlation quality (see also Figure 1 of ref 1), the procedure proves to be adequate to reliably describe gas solubility data. However, again, it must be emphasized that there are many combinations of these three parameters, which may result in good correlations of the gas solubility data. Obviously, one (0) can not reliably determine V(∞) CO2, βCO2,CO2, and µCO2,CO2,CO2 from (∞) gas solubility data alone. Instead, VCO should first be 2 determined from experimental information on the influence of the gas on the density of the liquid mixture.
A comparison between the extrapolated data and the correlation results from eq 15 is shown in Figure 3 of ref 1.
(∞) (T, x˜A) VCO 2
]
µCO2,CO2,CO2(T, x˜A) ) (1 - x˜A)µCO2,CO2,CO2,W(T) +
] }
]
119 (17) T/K
(m) (m) kH,CO (T, p, x˜A) ) kH,CO (T, x˜A) × 2 2
[
exp
(∞) VCO (T, x˜A)·{p - psmix(T, x˜A)} 2
RT
]
(19)
s (m) (∞) (T,x˜A) (as well as VCO (T,x˜A) and pmix (T,x˜A)) Knowing kH,CO 2 2 allows one to calculate some thermodynamic properties of solution of CO2 in mixtures of (water + acetone), that is, the change in some thermodynamic properties (such as the molar Gibbs energy G, the molar enthalpy H, and the molar entropy S) when 1 mole of CO2 is transferred from the ideal gas (at temperature T and pressure p° ) 0.1 MPa), to the liquid state (at mCO2/mo ) 1, temperature T, and pressure p) (see, for example, ref 2):
Figure 2. Solubility pressure versus gas molality in a liquid mixture of (CO2 + acetone). Symbols denote experimental results: (0) Jo¨decke et al.1 (T ) 313.75 K), (O) Adrian and Maurer26 (T ) 313.15 K), (2) Katayama et al.19 (T ) 313.15 K), (×) Traub and Stephan21 (T ) 313.15 K), (b) Day et al.25 and Chang et al.27 (T ) 313.15 K), (9) Stievanno and Elvassore31 (T ) 313.15 K), (-) calculated results (T ) 313.15 K), this work.
(m) (m) (T, p, x˜A) ) RT ln[kH,CO (T, p, x˜A)/po] ∆solGCO 2 2
(20)
Table 2. Comparison of Experimental Data with Correlation/Prediction Results from the VLE Model for the Solubility of CO2 in (Water + Acetone) Nb Correlation Jo¨decke et al.1 Prediction from VLE Model de Kiss et al.33 Traub and Stephan21 Wendland et al.3
T/K
x˜A
202
∼313-395
0.05-0.95
18 24 (19)a 40
∼273-298 ∼313 ∼293-333
0.04-0.66 0.007-0.89 0.041-0.88
pmax/MPa 9.7 ∼0.1 10 7.9
mCO2,max/(mol/kg)
∆pj/MPa
48.5
0.075
0.31 18.1 57.3
0.011 1.3 (0.55) 0.3
∆pj/p/% 2.3 10.8 15 (7.4) 7.8
When five data points are discarded. b N ) number of experimental data points within the given ranges. x˜A ) mole fraction of acetone on a gas-free N N basis. ∆pj ) 1/N · ∑j)1 |pj,exp - pj,calc|; ∆pj/p ) 1/N · ∑j)1 |(pj,exp - pj,calc)/pj,exp|. a
(
(m) (T, p, x˜A) ) R ∆solHCO 2
∂
(m) ln[kH,CO (T, p, x˜A)/po] 2
∂(1/T)
)
Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 4557
3. Prediction from Molecular Simulation p,x˜A
(m) (m) (m) ) (∆solHCO - ∆solGCO )/T ∆solSCO 2 2 2
(21) (22)
At standard temperature and pressure (To ) 298.15 K, po ) 0.1 MPa): (m,o) (m) (m) ) ∆solGCO (To, po, x˜A) ) RTo ln[kH,CO (To, po, x˜A)/po] ∆solGCO 2 2 2 (23) (m,o) (m) ∆solHCO ) ∆solHCO (To, po, x˜A) ) 2 2
(
R
(m) ∂ ln[kH,CO (T, po, x˜A)/po] 2
∂(1/T)
)|
x˜A at T)To
(24)
(m,o) (m) (m,o) (m,o) ∆solSCO ) ∆solSCO (To, po, x˜A) ) (∆solHCO - ∆solGCO )/To 2 2 2 2 (25) (m,o) (m,o) Numerical values for ∆solG(m,o) CO2 , ∆solHCO2 , and ∆solSCO2 are given in Table 3 for several solvent mixture compositions (x˜A). 2.4. Thermodynamic Properties of Transfer of CO2 from Pure Water to Mixtures of (Water + Acetone). The molality scale-based Gibbs free energy of transfer of CO2 from pure water to a mixture of (water + acetone) is related to the molality scale-based Henry’s constant of that gas in pure water and in mixtures of (water + acetone) through:2
[
(m) ∆tGCO (T, p, x˜A) ) RT ln 2
Furthermore:
([ ∂ ln
(m) ∆tHCO (T, p, x˜A) 2
)R
(m) kH,CO (T, p, x˜A) 2 (m) kH,CO (T, p) 2,W
(m) kH,CO (T, p, x˜A) 2 (m) kH,CO (T, p) 2,W
∂(1/T)
]
])
p,x˜A
(m) (m) (m) ) (∆tHCO - ∆tGCO )/T ∆tSCO 2 2 2
(26)
(27) (28)
At standard temperature and pressure, and at several solvent (m,o) (m,o) mixture compositions, numerical values for ∆tGCO , ∆tHCO , 2 2 (the Gibbs free energy, the enthalpy, and the entropy and ∆tS(m,o) CO2 of transfer, based on the molality scale) resulting from those equations (or directly from the data given in Table 3) are given in Table 4. Table 3. Standard State (To ) 298.15 K, po ) 0.1 MPa) Thermodynamic Properties of Solution of CO2 in Mixtures of (Water + Acetone) (on the Molality Scale)a x˜A
∆solG(m,o)/(kJ/mol)
∆solH(m,o)/(kJ/mol)
∆solS(m,o)/(J/(mol · K))
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
8.42 7.68 6.82 5.93 5.09 4.34 3.71 3.23 2.88 2.63 2.46
-19.40 -14.97 -12.90 -12.38 -12.74 -13.47 -14.17 -14.61 -14.72 -14.52 -14.23
-93.3 -76.0 -66.1 -61.4 -59.8 -59.7 -60.0 -59.8 -59.0 -57.5 -56.0
a
Based on experimental data from Jo¨decke et al.1
Computer simulation, on a basis of detailed models for the intermolecular interactions, has proven to be a powerful tool to predict thermodynamic properties of fluid systems. The aim of such simulations is to extend the range of experimental investigations, because experimentation is often difficult, tedious, time-consuming, and expensive. It is, for example, highly desirable to develop predictive methods for estimating the solubility of gases in mixed solvents over a broad range of conditions. The main purpose of the present study is to test the reliability of the Gibbs ensemble Monte Carlo (GEMC)35,36 simulation technique (in combination with intermolecular pair potentials for the pure components) for predicting the solubility of carbon dioxide in liquid mixtures of (water + acetone) (see also ref 37). Several simulation studies have been reported in the literature,38-50 for pure acetone, aqueous acetone solutions, and binary mixtures of carbon dioxide and acetone, which give reasonable structural and thermodynamic predictions. This work extends these studies to ternary mixtures of (carbon dioxide + water + acetone).Calculationsarecarriedoutusingtheisothermal-isobaric (NpT) ensemble, where the number of molecules N, pressure p, and temperature T are constant. The intermolecular potentials for the pure components are taken from the literature: for carbon dioxide, the EPM2 potential developed by Harris and Yung;51 for water, the point charge (SPC) model of Berendsen et al.;52 for acetone, the OPLS model of Jorgensen et al.39 Common mixing rules without any adjustable constants are employed for the cross interactions between unlike species. 3.1. Simulation Details and Potential Models. Standard MonteCarlosimulationswereconductedintheisothermal-isobaric Gibbs ensemble35,36 (NpT-GEMC simulations) to study the solubility of carbon dioxide in pure acetone and in aqueous solutions of acetone. Calculations are performed in two separate microscopic regions (subsystems) within the bulk phases, away from the interface. Both subsystems are internally in thermodynamic equilibrium as well as in equilibrium with each other. The general Monte Carlo moves, which were implemented to obtain equilibrium (i.e., equality of temperature, pressure, and chemical potentials in the two regions), were displacements of particles within the subsystems, volume fluctuations, and particle transfers between the two phases. The probability of the transfer moves was varied between 5% and 25%. Each attempt to transfer a molecule from one box to another comprised between 5 and 20 trial insertions (depending on the system’s density). The probability of volume changes was kept at 1%. The probability of particle displacements (translations or rotations) within the subsystems ranged from 74% to 94%. The same Table 4. Standard State (To ) 298.15 K, po ) 0.1 MPa) Thermodynamic Properties of Transfer of CO2 from Pure Water to Mixtures of (Water + Acetone) (on the Molality Scale)a x˜A
∆tG(m,o)/(kJ/mol)
∆tH(m,o)/(kJ/mol)
∆tS(m,o)/(J/(mol · K))
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.74 -1.60 -2.49 -3.33 -4.08 -4.71 -5.19 -5.54 -5.79 -5.96
4.43 6.50 7.02 6.65 5.93 5.23 4.78 4.68 4.87 5.17
17.4 27.2 31.9 33.5 33.6 33.3 33.5 34.3 35.8 37.3
a
Based on experimental data from Jo¨decke et al.1
4558 Ind. Eng. Chem. Res., Vol. 48, No. 9, 2009 Table 5. Geometry Data and Potential Parameters for the Components Studied in This Work carbon dioxide (EPM2, ref 51) rC-O (Å) σO (Å) εO/k (K) σC (Å) εC/k (K) qO (e) qC (e)
a
acetonea (OPLS, ref 39)
water (SPC, ref 52) RHOH (deg) rO-H (Å) σO (Å) εO/k (K) qO (e) qH (e)
1.149 3.033 80.507 2.757 28.129 -0.3256 0.6512
109.47 1.00 3.166 78.197 -0.82 0.41
R(CH3)C(CH3) (deg) R(CH3)CO (deg) r(CH3)-C (Å) rC-O (Å) σO (Å) εO/k (K) σCH3 (Å) εCH3/k (K) σC (Å) εC/k (K) qO (e) qCH3 (e) qC (e)
117.2 121.4 1.507 1.222 2.960 105.750 3.910 80.570 3.750 52.870 -0.424 0.062 0.300
The geometry of the acetone molecule is taken from ref 40.
number of attempts for translational and rotational moves was applied. All simulations were carried out using a total number of 600 molecules. The number of gas molecules varied between 150 and 350, depending on the pressure. The runs started either from an arbitrary initial configuration or from an output configuration of a previous run. After an equilibration period (typically in about 10 to 70 million cycles), where the system evolves toward equilibrium, the so-called “production period” began. Thermodynamic properties were calculated by ensemble averaging during a production period of 10 to 35 million cycles. The statistical uncertainties of the simulation results were estimated by the block averaging technique. Typically, a block consists of about one to two million cycles, depending on the length of the production period. All interactions were described by a combination of LennardJones-12-6 (superscript LJ) and Coulomb (superscript Coul) potentials of the form: Coul uij ) uLJ ij + uij m
)
n
∑∑ a
b
{ [( ) ( ) ] 4εab ij
σab ij rab ij
12
-
σab ij rab ij
6
+
a b 1 qi qj 4πε0 rab ij
}
(29)
where uij is the total interaction energy between two molecules i and j with m and n interaction sites, respectively. ab ab rab ij is the site-site separation distance, and ε ij and σ ij are the energy and the size parameters of the Lennard-Jones potential between sites a and b located at molecules i and j, respectively. qai is a point charge located at site a of molecule i, and qbj is a point charge located at site b of molecule j. ε0 () 8.8542 × 10-12 C2 N-1 m-2) is the permittivity of vacuum. Cross interaction parameters were estimated by applying the conventional Lorentz-Berthelot mixing rules (size parameters, arithmetic mean; energy parameters, geometric mean) except for carbon dioxide and acetone where the geometric mean rule for size as well as energy parameters were used. No adjustable interaction constants were introduced. For all simulations, an Ewald summation technique within vacuum boundary conditions53 was applied for the calculation of longrange electrostatic interactions. Lennard-Jones nonbonded interactions were calculated with a method proposed by Theodorou and Suter.54 Carbon Dioxide. The EPM2 model of Harris and Yung51 is employed for carbon dioxide. Three interaction sites are located on the carbon atom and both oxygen atoms, respectively. Each site is the center of a Lennard-Jones potential with an embedded central point charge. Parameter values were optimized for the reproduction of the vapor-liquid equilibrium of pure carbon
dioxide.51 The potential parameters and charge distributions are given in Table 5. Water. The SPC potential of Berendsen et al.52 models water as a Lennard-Jones sphere located on the oxygen atom and a single negative point charge combined with two positive point charges, which represent the hydrogen atoms. The potential parameters (see Table 5) were adjusted to reproduce the density, the internal energy, and the structure of liquid water at 300 K and pressures close to zero. Acetone. The united-atom OPLS force field developed by Jorgensen et al.39 is used to represent acetone. The methyl groups are treated as single interaction sites centered on the carbon atoms (hydrogen atoms are bonded to carbon atoms). The acetone molecule is modeled by a rigid set of four Lennard-Jones sites (two for the methyl groups, one for the carbon atom, and one for the oxygen atom) and four partial point charges located on the site positions. That OPLS force field was parametrized to reproduce thermodynamic quantities at or near ambient conditions. Model parameters are given in Table 5. 3.2. Thermodynamic Properties. The thermodynamic properties of interest were calculated as block averages. The statistical uncertainty of a simulation result is approximated by the standard deviation from the corresponding ensemble average. For example, the residual internal energy U is calculated as: U)
〈∑
〉
uij
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ReceiVed for reView June 30, 2008 ReVised manuscript receiVed February 16, 2009 Accepted February 25, 2009 IE801015U
