Ind. Eng. Chem. Res. 2002, 41, 1689-1694
1689
Henry’s Constant in Mixed Solvents from Binary Data I. Shulgin† and E. Ruckenstein* Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260
The Kirkwood-Buff formalism was used to derive an expression for the composition dependence of the Henry’s constant in a binary solvent. A binary mixed solvent can be considered as composed of two solvents, or one solvent and a solute, such as a salt, polymer, or protein. The following simple expression for the Henry’s constant in a binary solvent (H2t) was obtained when the binary solvent was assumed ideal: ln H2t ) [ln H2,1(ln V - ln V03) + ln H2,3(ln V01 - ln V)]/ (ln V01 - ln V03). In this expression, H2,1 and H2,3 are the Henry’s constants for the pure single solvents 1 and 3, respectively; V is the molar volume of the ideal binary solvent 1-3; and V10 and V30 are the molar volumes of the pure individual solvents 1 and 3. The comparison with experimental data for aqueous binary solvents demonstrated that the derived expression provides the best predictions among the known equations. Even though the aqueous solvents are nonideal, their degree of nonideality is much smaller than those of the solute gas in each of the constituents. For this reason, the ideality assumption for the binary solvent constitutes a most reasonable approximation even for nonideal mixtures. 1. Introduction The solubility of a gas in a mixture of solvents is a problem of interest in many industrial applications. One example is the removal of acidic compounds from industrial and natural gases.1 The solubility of a gas in a binary mixture containing water has particular importance because it is connected with the solubility of gases in blood, seawater, rainwater, and many other aqueous solutions of biological and environmental significance.2 Therefore, it is important to be able to predict the gas solubility in a mixture in terms of the solvent composition and the solubilities in the individual constituents of the solvent or in one pure component and a selected composition of the mixed solvent. The oldest and simplest relationship between the Henry’s constant in a binary solvent and those in the individual solvents [throughout this paper, only binary mixtures of solvents will be considered, and the following subscripts for the components will be used: 1, first solvent; 2, solute (gas); 3, second solvent] is that proposed by Krichevsky3
ln H2,1 + xb,1-3 ln H2,3 ln H2,t ) xb,1-3 1 3
(1)
where H2,t, H2,1, and H2,3 are the Henry’s constants in the binary solvent 1-3 and in the individual solvents 1 and xb,1-3 are the mole and 3, respectively, and xb,1-3 1 3 fractions of components 1 and 3, respectively, in the binary solvent 1-3. The rigorous thermodynamic equation4 for the excess Henry’s constant in a binary solvent has the form E ∞ ∞ ∞ ) ln γ2,t - xb,1-3 ln γ2,1 - xb,1-3 ln γ2,3 (2) ln H2,t 1 3 ∞ ∞ ∞ ∞ ∞ where γ2,t , γ2,1 , γ2,3 (γ2,t ) limx2f0 γ2,t, γ2,1 ) limx2f0 b,1-2 ∞ b,2-3 γ2 , and γ2,3 ) limx2f0 γ2 ) are the activity coefficients of the solute in the ternary (γ2,t) and binary (γ2,1
* Correspondence author. E-mail:
[email protected]. Fax: (716) 645-3822. Phone: (716) 645-2911, ext. 2214. † E-mail:
[email protected].
and γ2,3) mixtures at infinite dilution. It shows that Krichevsky’s relationship (eq 1) is valid when ∞ ∞ ∞ ln γ2,t ) xb,1-3 ln γ2,1 + xb,1-3 ln γ2,3 1 3
(3)
Although, in principle, eq 3 can be satisfied by some nonideal systems, it is surely valid when the ternary mixture and the binary mixtures (1-2 and 2-3) are ideal. Equation 2 was used as the starting point for the prediction of the Henry’s constant in a binary solvent mixture in terms of binary data, by expressing the ∞ ∞ , γ2,1 , and activity coefficients at infinite dilution (γ2,t ∞ γ2,3) through the Wilson5,6 or the van Laar equations.7,8 The Kirkwood-Buff theory of solution9 was used by O’Connell to develop a semiempirical expression for the prediction of the Henry’s constant in binary solvents from binary data.4 He found that his expression provided better results than that of Krichevsky and that obtained5 by combining the Wilson equation with eq 2. Using a corresponding state method, Campanella et al.10 calculated the Henry’s constant for mixed solvents and obtained good agreement when the excess volume of the nonideal solvent was taken into account. The present authors employed the Kirkwood-Buff theory of solution to obtain expressions for the derivatives of the activity coefficients in a ternary mixture with respect to the mole fractions and applied them to ternary mixtures when the composition(s) of one (or two) component(s) was (were) small.11,12 That approach will be used here to derive new expressions that can predict the Henry’s constant in a binary solvent mixture in terms of binary data. 2. Theory 2.1. Expressions for the Derivatives of the Activity Coefficients in a Ternary Mixture with Respect to Mole Fractions through the Kirkwood-Buff Theory of Solution. For the present purpose, the following two derivatives, obtained in a previous paper,11 are useful
10.1021/ie010911x CCC: $22.00 © 2002 American Chemical Society Published on Web 02/21/2002
1690
Ind. Eng. Chem. Res., Vol. 41, No. 6, 2002
(
)
∂ ln γ2,t ∂xt3
-
)
lim
T,P,xt2
t
(c1 + c2 + c3)(c1[G11 + G23 - G12 - G13] + c3[-G12 - G33 + G13 + G23]) c1 + c2 + c3 + c1c2∆12 + c1c3∆13 + c2c3∆23 + c1c2c3∆123
(4)
(
∂xt3
x2f0
-
)
∂ ln γ2,t
)
T,P,xt2
(c01 + c03)[(c01 + c03)(∆12 - ∆23)xt2)0 + (c01 - c03)(∆13)xt2)0] 2[c01 + c03 + c01 c03(∆13)xt2)0]
and
(
)
(12)
∂ ln γ1,t ∂xt2
and
)
T,P,x3t
(c1 + c2 + c3)(c3[G11 + G23 - G12 - G13] + c2∆12 + c2c3∆123) c1 + c2 + c3 + c1c2∆12 + c1c3∆13 + c2c3∆23 + c1c2c3∆123
lim t
x2f0
(5) xt2
where is the mole fraction of component 2 in the ternary mixture, ck is the bulk molecular concentration of component k in the ternary mixture 1-2-3, and GRβ is the Kirkwood-Buff integral given by
GRβ )
∫0∞(gRβ - 1)4πr2 dr
(6)
In the above expression, gRβ is the radial distribution function between species R and β, r is the distance between the centers of molecules R and β, and ∆Rβ and ∆123 are defined as
∆Rβ ) GRR + Gββ - 2GRβ, R * β
∆123 ) G11G22 + G11G33 + G22G33 + 2G12G13 + 2G12G23 + 2G13G23 - G122 - G132 - G232 2G11G23 - 2G22G13 - 2G33G12 (8) One can verify that the factors in the square brackets in the numerators of eqs 4 and 5 and ∆123 can be expressed in terms of ∆Rβ as follows
G12 + G33 - G13 - G23 )
∆13 + ∆23 - ∆12 2
G11 + G23 - G12 - G13 )
∆12 + ∆13 - ∆23 (10) 2
)
∂xt2
c03(c01 + c03)(∆12 + ∆13 - ∆23)xt2)0
)
2[c01 + c03 + c01 c03(∆13)xt2)0]
T,P,xt3
(13) In eqs 12 and 13, c01 and c03 represent the bulk molecular concentrations of components 1 and 3, respectively, in the gas-free binary solvent 1-3. In addition to eqs 12 and 13, the following expression9 for the derivative of the activity coefficient in a binary mixture with respect to the mole fractions will be used in the next section to derive the basic equation for the Henry’s constant for mixed solvents
(
)
∂ ln γb,1-3 1
(7)
and
∂xb,1-3 3
)
P,T
c03∆13 1 + c01 xb,1-3 ∆13 3
(14)
where xb,1-3 and γb,1-3 are the mole fraction of compo3 1 nent 3 and the activity coefficient of component 1, respectively, in the gas-free binary solvent 1-3. 2.2. Composition Dependence of the Henry’s Constant for a Binary Solvent. To obtain the composition dependence of the Henry’s constant in a binary solvent, one should consider either the derivative (∂ ln H2t/∂xt3)P,T,xt2)0 or the derivative (∂ ln H2t/∂xt1)P,T,xt2)0. To obtain the above derivatives, one can start from the following expression for the Henry’s constant in a binary solvent4
ln H2,t ) lim ln γ2,t + ln f 02(P,T)
(9)
t
(15)
x2f0
and
where f 0i (P,T) is the fugacity of component i.15 The combination of eqs 12 and 15 leads to the result
(
∂xt3
)
-
(c01 + c03)[(c01 + c03)(∆12 - ∆23)xt2)0 + (c01 - c03)(∆13)xt2)0]
∂ ln H2t
∆123 ) -
(
∂ ln γ1,t
(∆12)2 + (∆13)2 + (∆23)2 - 2∆12∆13 - 2∆12∆23 - 2∆13∆23) 4
)
P,T,xt2)0
2[c01 + c03 + c01 c03(∆13)xt2)0]
(11) The insertion of eqs 9-11 into eqs 4 and 5 provides rigorous expressions for the derivatives (∂ ln γ2,t/∂xt3)T,P,xt2 and (∂ ln γ1,t/∂xt2)T,P,xt3 in terms of ∆Rβ and concentrations. It should be noted that ∆ij is a measure of the nonideality13 of the binary mixture R-β because, for an ideal mixture, ∆Rβ ) 0. For the ternary mixture 1-23, ∆123 also constitutes a measure of nonideality. Indeed, id inserting GRβ for an ideal mixture14 into the expression ∆123, one obtains that for, an ideal ternary mixture, ∆123 ) 0. At infinite dilution of component 2, eqs 4 and 5 become
(16) Integration of eq 16 provides the following relation for the composition dependence of the Henry’s constant in a binary solvent mixture at constant temperature and pressure
(c01 + c03)(∆12 - ∆23)xt2)0
∫2[1 + c0 xb,1-3(∆
ln H2t ) -
1
3
13)xt2)0]
dxb,1-3 3
(c01 - c03)(∆13)xt2)0
∫2[1 + c0 xb,1-3(∆ 1
3
13)xt2)0]
dxb,1-3 + A (17) 3
Ind. Eng. Chem. Res., Vol. 41, No. 6, 2002 1691
where A(P,T) is a composition-independent constant of integration. The last equation can be also derived starting from the Gibbs-Duhem equation for a ternary mixture (see Appendix 1). By eliminating (∆13)xt2)0 with the help of eq 14, eq 17 becomes
(∆12 - ∆23)xt2)0
∫
ln H2t ) - (c01 + c03)
[
1 2
(
)] ( ) 2
×
∂ ln γb,1-3 3 b,1-3 1 + x3 dxb,1-3 + 3 b,1-3 ∂x3 P,T (xb,1-3 - xb,1-3 ) ∂ ln γb,1-3 1 3 3 dxb,1-3 3 b,1-3 b,1-3 x1 ∂x3 P,T
∫
(
xb,1-3 1
∂xb,1-3 3
)
(
) (
∂ ln γb,1-3 1 ∂xb,1-3 3
+
P,T
ln H2t ) A(P,T) -
∫
ln H2t ) -
[
+
(∆12 - ∆23)xt2)0
c03)
1 + xb,1-3 3
(
∂
2
)]
ln γb,1-3 3 b,1-3 ∂x3 P,T
(21)
3
B(P,T) ln(xb,1-3 V01 + xb,1-3 V03) 1 3 V03 - V01 (22)
)
P,T
dxb,1-3 + 3
1 + ln γb,1-3 ) + A (19) (ln γb,1-3 1 3 2
It should be noted that this equation can be applied to the solubility of a gas in various kinds of binary mixtures: a mixture of two solvents, or a mixture of a solvent and a solute (salt, polymer, or protein). As already mentioned, the Krichevsky equation (eq 1) is valid when the binary mixtures 1-2 and 2-3 (gas solute/pure solvents) and the ternary mixture 1-2-3 are ideal. However, these conditions are often far from reality. Let us consider, for example, the solubility of a hydrocarbon in a water-alcohol solvent (for instance, water-methanol, water-ethanol, etc.). The activity coefficient16 of propane in water at infinite dilution is ∼4 × 103, whereas the activity coefficients of alcohols and water in aqueous solutions of simple alcohols seldom exceed 10. It is therefore clear that the main contribution to the nonideality of the ternary gasbinary solvent mixture comes from the nonidealities of the gas solute in the individual solvents, which are neglected in the Krichevsky equation.
(23)
(ln H2t)xb,1-3 )0 ) ln H2,1 3
(24)
Combining eqs 22-24 yields the final result
ln H2t ) ×
(ln H2t)xb,1-3 )0 ) ln H2,3 1 and
which, introduced into eq 18, leads to
(c01
3
The main single approximation of this paper is the assumption that (∆12)xt2)0 ) (G11 + G22 - 2G12)xt2)0 and (∆23)xt2)0 ) (G22 + G33 - 2G23)xt2)0 are independent of the composition of the solvent mixture. Consequently, eq 21 becomes
∂ ln γb,1-3 3 ∂xb,1-3 3
1
where B(P,T) ) (∆12 - ∆23)xt2)0/2. The constants A(P,T) and B(P,T) can be obtained using the following extreme expressions
)
P,T
(∆12 - ∆23)xt2)0
1
+ A (18)
(20)
where V is the molar volume of the binary mixture 1-3 and V01 and V03 are the molar volumes of the individual solvents 1 and 3, respectively. Under these conditions, eq 19 becomes
+A ∫2(xb,1-3 V0 + xb,1-3 V0) dxb,1-3 3
d ln γb,1-3 ) -xb,1-3 d ln γb,1-3 xb,1-3 1 1 3 3
(xb,1-3 - xb,1-3 ) ∂ ln γb,1-3 1 3 3
V ) xb,1-3 V01 + xb,1-3 V03 1 3
ln H2t ) -
The first term on the right-hand side of eq 18 involves the ternary mixture through the limiting value (∆12 ∆23)xt2)0, whereas the second involves the gas-free binary solvent. Equation 18 can be transformed using the Gibbs-Duhem equation for a binary system in the second integral on the right-hand side of eq 18. The Gibbs-Duhem equation for a binary system at constant temperature and pressure has the form
Therefore, one can write
For this reason, in a first step, it will be assumed that only the binary solvent (1-3) behaves as an ideal ) 1 and mixture. One can therefore write that γb,1-3 3
ln H2,1(ln V - ln V03) + ln H2,3(ln V01 - ln V) ln V01 - ln V03 (25)
Equation 25 provides the Henry’s constant for a binary solvent in terms of those for the individual solvents and the molar volumes of the pure solvents. This simple equation was obtained using less restrictive approximations than those involved in the Krichevsky equation by assuming that only the binary solvent 1-3 is an ideal mixture. This assumption is reasonable because, as already noted, the nonideality of the binary solvent is much lower than the nonidealities of the solute gas and each of the constituents of the solvent. Equation 19 can, however, be integrated using any of the analytical expressions available for the activity , such as the van Laar, Margules, coefficient ln γb,1-3 3 Wilson, NRTL, etc. To take into account the nonideality of the molar volume, one can use the expression
V01 + xb,1-3 V03 + VE V ) xb,1-3 1 3
(26)
where VE is the excess molar volume. When the integration in eq 19 cannot be performed analytically, one can first perform the integration < 1 to obtain the numerically between 0 < xb,1-3 3 expression
1692
Ind. Eng. Chem. Res., Vol. 41, No. 6, 2002
ln H2,3 - ln H2,1 )
[
∫01V1 1 + xb,1-3 3
-B
(
∂
)]
ln γb,1-3 3 ∂xb,1-3 3 P,T
dxb,1-3 + 3
ln H2,1 - B
∫0 V x1
[
1 + xb,1-3 3
deviations (%) of experimental data from calculated resultsa
1 ∞ ∞ (ln γ1,3 - ln γ3,1 ) (27) 2
∞ ∞ where γ1,3 ) limx1f0 γb,1-3 and γ3,1 ) limx3f0 γb,1-3 . 1 3 Equation 27 allows for the determination of the constant B. Further, eq 19 can be integrated between < x to obtain the Henry’s constant for the 0 < xb,1-3 3 mole fraction x in the binary solvent
ln H2,t )
Table 1. Comparison between Experimental and Calculated Henry’s Constants for a Binary Solvent Mixture
(
)]
∂ ln γb,1-3 3 ∂xb,1-3 3
dxb,1-3 + 3
P,T
1 ∞ [ln γb,1-3 (x) + ln γb,1-3 (x) - ln γ3,1 ] (28) 1 3 2 This procedure allows one to account for the nonideality (activity coefficients and molar volume) of the binary solvent. The analytical expressions obtained using the twosuffix Margules equations15 for the activity coefficients and eq 20 for the molar volume are given in Appendix 2. The extension of eq 25 to a multicomponent solvent is not straightforward and requires additional investigation. 3. Calculations and Comparison with Experimental Data For comparison, we selected the solubilities of gases in aqueous binary solvents because, as noted in ref 6, the prediction of the Henry’s constant for such mixtures is the most difficult and the available methods are not reliable. The results of the calculations are presented and compared in Table 1 and Figure 1 with the Krichevsky equation and an empirical correlation for aqueous mixtures that provided the best results6 among the existing expressions. All of the necessary experimental data [V0i , H2,1, H2,3, and E (Margules parameter)] were taken from the original publications (indicated as footnotes to Table 1) or calculated using the data from Gmehling’s vaporliquid equilibrium data compilation.21 Figure 1 and Table 1 show that the present eq 25 is in much better agreement with experiment than Krichevsky’s eq 1 and equations A2-3-5 from Appendix 2, which involve the Margules expression for the activity coefficient. The new eq 25 provides predictions that are comparable to those of an empirical correlation for aqueous mixtures of solvents,6 which involves three adjustable parameters. However, none of the expressions available, including eq 25, can represent the extremum in the mixed Henry’s constant found in some experiments at low alcohol concentrations.17 Perhaps only very accurate representations of the activity coefficients and excess molar volume of the mixed solvent in the dilute region can explain this anomaly. Table 1 also shows that equations A2-3-5, based on the two-suffix Margules equation, provide results that are comparable to those of Krichevsky’s eq 1 but much less accurate than those of the new eq 25. Numerical
system argon (2)acetone (1)water (3)c helium (2)methanol (1)water (3)d helium (2)ethanol (1)water (3)d helium (2)1-propanol (1)water (3)d helium (2)2-propanol (1)water (3)d oxygen (2)methanol (1)water (3)e oxygen (2)ethanol (1)water (3)e oxygen (2)1-propanol (1)water (3)e oxygen (2)2-propanol (1)water (3)e nitrogen (2)methanol (1)water (3)e nitrogen (2)ethanol (1)water (3)e nitrogen (2)1-propanol (1)water (3)e nitrogen (2)2-propanol (1)water (3)e carbon dioxide (2)1-propanol (1)water (3)e carbon dioxide (2)2-propanol (1)water (3)e methane (2)methanol (1)water (3)f methane (2)ethanol (1)water (3)f average (%) a
predictions of this work Krichevsky’s empirical T (K) eq 1 correlationb eq 25 eq A2-3 288.15 298.15 308.15 298.15
17.0 28.3 29.6 5.7
-
15.3 7.1 8.4 10.8
37.9 39.4 39.3 7.3
298.15
5.6
-
12.1
7.8
298.15
18.2
-
5.6
24.1
298.15
13.5
-
7.9
8.9
273.15 293.15 313.15 273.15 293.15 313.15 273.15 293.15 313.15 273.15 293.15 313.15 293.15 313.15
13.9 7.0 10.4 12.4 16.7 31.5 20.4 45.5 53.5 26.6 41.4 67.7 7.8 7.7
7.6 6.2 10.8 12.2 6.2 6.0 7.9 8.3 13.0 14.3 12.6 9.2 9.9 10.0
22.4 13.3 8.9 21.9 10.3 8.1 13.8 12.5 13.9 19.2 16.2 12.6 17.5 9.7
19.8 9.8 7.4 20.9 16.4 20.5 32.7 35.7 36.1 31.0 27.5 30.6 15.2 8.5
293.15 313.15
17.3 22.7
8.8 14.2
7.8 6.5
15.0 19.5
293.15 313.15
43.9 73.5
9.5 14.5
9.1 20.3
29.9 40.0
293.15 313.15
44.6 61.2
14.5 14.0
8.9 11.8
28.1 28.3
283.15 293.15 303.15 313.15 283.15 293.15 303.15 313.15 293.15
14.1 18.3 23.4 33.7 14.0 21.7 23.4 30.5 5.3
25.5
12.1 6.1 3.7 6.9 16.1 10.2 8.5 6.4 15.0
26.4 25.2 24.1 28.8 21.6 22.5 23.4 22.5 11.7
293.15
33.7
30.3
16.3
24.5
25.8
12.1
11.6
23.5
(i) (100/m)∑i|H2,t(exp)
(i) (i) - H2,t(calc) |/H2,t(exp) , where m is points. b Taken directly from ref 6.
Defined as the number of experimental Experimental data from ref 18. d Experimental data from ref 19. e Experimental data from ref 17. f Experimental data from ref 20. c
calculations based on eqs 27-28 showed that the use of the Wilson equation for the activity coefficients in binary solvent mixtures improved the results obtained via eqs A2-3-5 only slightly. It seems that eqs 27-28, which contain the derivatives of the activity coefficient with respect to the mole fraction, require a much more accurate representation of the vapor-liquid equilibrium than that provided by the two-suffix Margules or Wilson equations. As noted in the literature,4 the above equa-
Ind. Eng. Chem. Res., Vol. 41, No. 6, 2002 1693
Figure 1. Henry’s constants of gases in binary solvent mixtures at 760 mmHg partial pressure (9, experimental;17 A, calculated with the new eq 25; B, calculated with Krichevsky eq 1; in Figure 1a, C represents eq A2-3): (a) oxygen (2) in 1-propanol (1)-water (3) at 40 °C, (b) nitrogen (2) in 2-propanol (1)-water (3) at 40 °C, (c) oxygen (2) in ethanol (1)-water (3) at 40 °C, and (d) carbon dioxide (2) in 1-propanol (1)-water (3) at 40 °C. xalc is the mole fraction of the alcohol in the gas-free mixture of solvents.
tions applied to complex mixtures, such as aqueous mixtures, can lead to results in the wrong direction for the Henry’s constant. Conclusion In this paper, the Kirkwood-Buff formalism was used to relate the Henry’s constant for a binary solvent mixture to the binary data and the composition of the solvent. A general equation describing the above dependence was obtained, which can be solved (analytically or numerically) if the composition dependence of the molar volume and the activity coefficients in the gasfree mixed solvent are known. A simple expression was obtained when the mixture of solvents was considered to be ideal. In this case, the Henry’s constant for a binary solvent mixture could be expressed in terms of the Henry’s constants for the individual solvents and the molar volumes of the individual solvents. The agreement with experiment for aqueous solvents is better than that provided by any other expression available, including an empirical one involving three adjustable parameters. Even though the aqueous solvents considered are nonideal, their degrees of nonideality are much lower than those of the solute gas in each of the constituent solvents. For this reason, the assumption that the binary solvent behaves as an ideal mixture constitutes a reasonable approximation.
Appendix 1 On the basis of the Gibbs-Duhem equations for a ternary mixture, Krichevsky and Sorina22 derived the following equation
( ) ( ∂ ln H2t ∂xt3
)
P,T,xt2f0
)
∂ ln γb,1-3 1 ∂xb,1-3 3
( )
∂ ln γt1 - b,1-3 lim x2f0 ∂x2 x 1
P,T
3
(A1-1) P,T,xt3
Using eq 14 for the first term of the right-hand side of eq A1-1 and eq 13 for the second term, eq A1-1 becomes
(
)
∂ ln H2t ∂xt3
1
)
P,T,xt2f0
c03∆13 + c01 x03∆13
-
1 xb,1-3 3
c03(c01 + c03)
(
)
∆12 + ∆13 - ∆23 2
xt2)0
c01 + c03 + c01 c03∆13 (A1-2)
which after some algebraic transformation acquires the form
1694
(
Ind. Eng. Chem. Res., Vol. 41, No. 6, 2002
)
∂ ln H2t ∂xt3
)
P,T,xt2f0
-(c01 + c03)(∆12 - ∆23)xt2)0 - (c01 - c03)∆13 2(1 + c01 xb,1-3 ∆13) 3
(A1-3)
which coincides with eq 16 in the text. Appendix 2 The activity coefficients of the binary mixture 1-3 can be expressed through the two-suffix Margules equations15 as
) Exb,1-3 ln γb,1-3 1 3
(A2-1)
) Exb,1-3 ln γb,1-3 3 1
(A2-2)
where E is a temperature-dependent constant. Insertion of eq A2-2 into eq 19 provides the following result for the Henry’s constant in the mixed solvent (the molar volume of the mixed solvent being expressed through eq 20)
[
ln H2,t ) A - B
2V01 V03 E ln V ln V + + V03 - V01 (V03 - V01)3
Exb,1-3 3 (xb,1-3 0 0 2 3 (V3 - V1)
]
V03 - xb,1-3 V03 - 2V03) 1 - (xb,1-3 )2] (A2-3) 2E[xb,1-3 3 1
The composition-independent constants A and B can be obtained by combining eq A2-3 with eqs 23 and 24. One thus obtains the following expressions for A and B
A)
ln V03 ln H2,1 - ln V01 ln H2,3 ln V03 - ln V01
(A2-4)
and
B)
(ln H2,1 - ln H2,3)(V03 - V01)3 (ln V03 - ln V01)[(V03 - V01)2 + 2V01 V03E]
(A2-5)
Using eqs A2-3-5, one can calculate the Henry’s constant for any composition of the solvent. Note Added after ASAP Posting This article was released ASAP on 2/21/02 with errors in eqs A1-1 and A1-2 and in footnote a of Table 1. The correct version was posted on 2/25/02. Literature Cited (1) Li, Y. G.; Mather, A. E. Correlation and prediction of the solubility of N2O in mixed solvents. Fluid Phase Equilib. 1994, 96, 119-142.
(2) Wilhelm, E.; Battino, R.; Wilcock, R. J. Low-pressure solubility of gases in liquid water. Chem. Rev. 1977, 77, 219-262. (3) Krichevsky, I. R. Thermodynamics of an infinitely dilute solution in mixed solvents. I. The Henry’s coefficient in a mixed solvent behaving as an ideal solvent. Zh. Fiz. Khim. 1937, 9, 41-47. (4) O’Connell, J. P. Molecular thermodynamics of gases in mixed solvents. AIChE J. 1971, 17, 658-663. (5) Prausnitz, J. M.; Eckert, C. A.; Orye, R. V.; O’Connell, J. P. Computer Calculations for Multicomponent Vapor-Liquid Equilibria; Prentice Hall: Englewood Cliffs, NJ, 1967. (6) Kung, J. K.; Nazario, F. N.; Joffe, J.; Tassios, D. Prediction of Henry’s constants in mixed solvents from binary data. Ind. Eng. Chem. Res. 1984, 23, 170-175. (7) Prausnitz, J. M.; Chueh, P. L. Computer Calculations for High-Pressure Vapor-Liquid Equilibria; Prentice Hall: Englewood Cliffs, NJ, 1968. (8) Boublik, T.; Hala, E. Solubility of gases in mixed solvents. Collect. Czech. Chem. Commun. 1966, 31, 1628-1635. (9) Kirkwood, J. G.; Buff, F. P. Statistical mechanical theory of solutions. I. J. Chem. Phys. 1951, 19, 774-782. (10) Campanella, E. A.; Mathias, P. M.; O’Connell, J. P. Equilibrium properties of liquids containing supercritical substances. AIChE J. 1987, 33, 2057-2066. (11) Ruckenstein, E.; Shulgin, I. Entrainer effect in supercritical mixtures. Fluid Phase Equilib. 2001, 180, 345-359. (12) Ruckenstein, E.; Shulgin, I. The solubility of solids in mixtures containing a supercritical fluid and an entrainer. Fluid Phase Equilib. 2002, in press. (13) Ben-Naim, A. Inversion of the Kirkwood-Buff theory of solutions: Application to the water-ethanol system. J. Chem. Phys. 1977, 67, 4884-4890. (14) Ruckenstein, E.; Shulgin, I. Effect of a third component on the interactions in a binary mixture determined from the fluctuation theory of solutions. Fluid Phase Equilib. 2001, 180, 281-297. (15) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1986. (16) Kojima, K.; Zhang, S. J.; Hiaki, T. Measuring methods of infinite dilution activity coefficients and a database for systems including water. Fluid Phase Equilib. 1997, 131, 145-179. (17) Tokunaga, J. Solubilities of oxygen, nitrogen, and carbon dioxide in aqueous alcohol solutions. J. Chem. Eng. Data 1975, 20, 41-46. (18) Yamamoto, H.; Tokunaga, J.; Koike, K. Solubility of argon in acetone plus water mixed solvent at 288.15, 298.15 and 308.15 K. Can. J. Chem. Eng. 1994, 72, 541-545. (19) Yamamoto, H.; Ichikawa, K.; Tokunaga, J. Solubility of helium in methanol plus water, ethanol plus water, 1-propanol plus water, and 2-propanol plus water solutions at 25 °C. J. Chem. Eng. Data 1994, 39, 155-157. (20) Tokunaga, J.; Kawai, M. Solubilities of methane in methanol-water and ethanol-water solutions. J. Chem. Eng. Jpn. 1975, 8, 326-327. (21) Gmehling, J. et al. Vapor-Liquid Equilibrium Data Collection; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, Germany, 1977-1996; Vol. I. (22) Krichevsky, I. R.; Sorina, G. A. What thermodynamics can state about Henry’s coefficient in a mixed solvent. Dokl. Akad. Nauk 1992, 325, 325-328.
Received for review November 12, 2001 Revised manuscript received January 11, 2002 Accepted January 15, 2002 IE010911X
