4674
Ind. Eng. Chem. Res. 2002, 41, 4674-4680
Salting-Out or -In by Fluctuation Theory E. Ruckenstein* and I. Shulgin† Department of Chemical Engineering, State University of New York at Buffalo, Amherst, New York 14260
In this paper, the Kirkwood-Buff formalism was used to examine the effect of the addition of a salt on the gas solubility. A general expression for the derivative of the Henry constant with respect to the salt concentration was thus derived. The obtained equation was used to correlate the experimental solubilities as a function of the salt molality. The correlation involves one parameter, which has to be determined from the experimental data. In addition, it requires information about the molar volume of the salt solution and the mean activity coefficient of the salt. It has been shown that the experimental solubilities can be well correlated when an accurate expression for the mean activity coefficient of the salt is used. It was also shown that the wellknown Sechenov equation constitutes a particular case of the obtained expression. The general expression allowed one to find a criterion for the prediction of the kind of salting (salting-in or salting-out) for dilute salt solutions. According to this criterion, the kind of salting depends mainly on the molar volume of the salt at infinite dilution. This explains the literature observations that the salts with large molar volumes at infinite dilution usually increase the gas solubility compared to that in pure water. 1. Introduction The gas solubilities in aqueous solutions of salts or organic substances constitute most useful information in many areas of chemical and biochemical engineering, hydrometallurgy, geochemistry, etc. There are many processes, such as the biological and organic reactions, the corrosion and oxidation of materials, the aerobic fermentation, the petroleum and natural gas exploitation, the petroleum refining, the coal gasification, the gas antisolvent crystallization, the formation of gas hydrates, etc., in which the salting effect is relevant.1 The solubility of naturally occurring and atmospheric gases in sodium chloride solutions, and its dependence on pressure, temperature, and the salt concentration, is of interest not only to the physical chemist but also to the geochemist because the aqueous salt solutions containing gases can simulate the brine in the earth’s crust. The information about the solubility of atmospheric gases in water and seawater is also relevant to the understanding of the ecological balance between the freshwater and seawater systems.1 The addition of an electrolyte to water decreases in many cases the gas solubility (salting-out). Similarly, the addition of an organic compound to water can decrease or increase the solubility of gases compared to that in pure water. The present paper is devoted to the examination of the effect of the addition of an inorganic substance, mainly a salt, to water on the gas solubility. Usually the effect of the salt addition on the solubility has been attributed to the greater attraction between the ions and the water molecules than between the nonpolar or slightly polar gas molecules and water.2 Therefore, the interactions between the ions and the water molecules should decrease the number of “free” water molecules available to dissolve the gas.3 This explanation is, * Corresponding author. E-mail:
[email protected]. Fax: (716) 645-3822. Phone: (716) 645-2911/ext. 2214. † E-mail:
[email protected].
however, oversimplified because it ignores the effect of the ions on the water structure.3 Indeed, some electrolytes with large ions, which disorganize the structure of water, increase the solubility (salting-in);4-7 in contrast, electrolytes with small ions, which organize the structure of water, decrease the solubility (salting-out). An interesting observation regarding the effect of an ion on the water structure was made in ref 8, where it was found that the “salting-out” and “salting-in” salts affect differently the partial molar volume of the gas in an electrolyte solution, compared to that in pure water. While the “salting-out” salts decrease this volume, the “salting-in” salts increase it. However, so far no criterion for the prediction of whether a salt generates “saltingout” or “salting-in” was suggested. The oldest but, nevertheless, the most popular equation for the representation of the gas solubility in the presence of a salt is the Sechenov equation9
ln(H2,t/H21) ) kSm
(1)
where H2,t and H21 are the Henry constants in the salt solution and pure water, respectively, kS is the Sechenov coefficient, and m is the molality. The Sechenov coefficient depends on the nature of the salt and solute and the temperature. The following subscripts will be used throughout this paper: component 1 is the solvent, 2 is the gas, and 3 is the salt. Of course, the Sechenov equation (1) can represent both the salting-out and the salting-in; for salting-out the Sechenov coefficient is positive and for salting-in negative. Because of its simplicity, the Sechenov equation has become a very popular tool for the correlation of the gas solubility in salt solutions. It has attracted the attention of theoreticians,5,7,10-11 and several modifications have been suggested.3,12,13 However, it has failed to correlate the gas solubility at high salt concentrations and also was not satisfactory in correlating the solubility of carbon dioxide in a number of salt solutions.14 In addition, recent investigations15,16 indicated that the salting effect is more complex. For
10.1021/ie020348y CCC: $22.00 © 2002 American Chemical Society Published on Web 08/03/2002
Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002 4675
example, the kind of salting effect can be inverted with a change in the gas partial pressure, and this inversion cannot be described by the Sechenov equation. Last but not least, because of its empirical character, the Sechenov equation cannot predict whether a salt will increase the solubility (salting-in) or will decrease it (salting-out). The aim of the present paper is to develop a theoretical approach for the description of the gas solubility in a solvent containing a salt. To achieve this goal, the Kirkwood-Buff formalism17 for ternary mixtures will be used. Recently, such a formalism has been used to predict the gas solubility in mixed solvents18 (mixture of two nonelectrolytes) in terms of the solubilities in the individual solvents. A similar approach will be employed here. The paper is organized as follows: first, the Kirkwood-Buff formalism will be used to derive general expressions for the derivatives of the activity coefficients in ternary mixtures with respect to the mole fractions. Then, the obtained expressions will be applied to the gas solubility in dilute and concentrated salt solutions. Numerical calculation will be carried out for several mixtures, particularly for those for which the Sechenov equation failed to provide an accurate correlation. Finally, a criterion will be proposed for the a priori prediction of the kind of salting (salting-in or saltingout).
Kirkwood-Buff integral given by
GRβ )
ln H2,t ) lim ln γ2,t + ln f 02(P,T) t
(2)
x2f0
where xti and γi,t are the mole fraction and the activity coefficient of component i in the ternary mixture, respectively, f 0i (P,T) is the fugacity of the pure component i,3 P is the pressure, and T is the temperature in K. For the derivative of the Henry constant in a binary solvent with respect to the mole fraction of the electrolyte, one can write
(
)
∂ ln H2,t ∂xt3
P,T,xt2)0
) lim t
x2f0
(
)
∂ ln γ2,t ∂xt3
(3)
T,P,xt2
In a previous paper,20 the Kirkwood-Buff formalism was applied to ternary mixtures and explicit expressions for the partial molar volumes, isothermal compressibility, and the derivatives of the activity coefficients with respect to the mole fractions derived. In particular, the following expression for the derivative (∂ ln γ2,t/∂xt3)T,P,xt2 was obtained:
(
)
∂ ln γ2,t ∂xt3
) T,P,xt2
-{(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)
where ck is the bulk molecular concentration of component k in the ternary mixture 1-2-3 and GRβ is the
(5)
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 the following combinations of the KirkwoodBuff integrals
∆Rβ ) GRR + Gββ - 2GRβ, R * β
(6)
and
∆123 ) G11G22 + G11G33 + G22G33 + 2G12G13 + 2G12G23 + 2G13G23 - G122 - G132 - G232 2G11G23 - 2G22G13 - 2G33G12 (7) One can show18 that the factors in the square brackets in eq 4 and ∆123 can be expressed in terms of ∆Rβ as follows:
2. The Henry Constant in a Salt Solution The Henry constant in a binary solvent is given by the following expression:19
∫0∞(gRβ - 1)4πr2 dr
G12 + G33 - G13 - G23 )
∆13 + ∆23 - ∆12 2
(8)
G11 + G23 - G12 - G13 )
∆12 + ∆13 - ∆23 2
(9)
and
∆123 ) -(∆122 + ∆132 + ∆232 - 2∆12∆13 - 2∆12∆23 2∆13∆23)/4 (10) The substitution of eqs 8-10 into eq 4, and considering infinite dilution of the solute (component 2), yields the following rigorous expressions for the derivative (∂ ln γ2,t/∂xt3)T,P,xt2:
lim
t x2f0
(
)
∂ ln γ2,t ∂xt3
)
T,P,xt2
-[(c01 + c03)((c01 + c03)(∆12 - ∆23)xt2)0 + (c01 - c03)(∆13)xt2)0)]/2(c01 + c03 + c01 c03(∆13)xt2)0) (11) In eq 11 c01 and c03 represent the bulk molecular concentrations of components 1 and 3 in the gas-free binary solvent 1-3. In addition to eq 11, the following expression17 for the derivative of the activity coefficient in a binary mixture 1-3 with respect to the mole fraction of the electrolyte can be written:
(
)
∂ ln γb,1-3 1 ∂xb,1-3 3
P,T
)
c03∆13 1 + c01 xb,1-3 ∆13 3
(12)
and γb,1-3 are the mole fraction of compowhere xb,1-3 3 1 nent 3 and the activity coefficient of component 1 in the gas-free binary solvent 1-3, respectively. The combination of eqs 3, 11, and 12 provides the following expression for the derivative of the Henry constant in a binary solvent:18
4676
(
Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002
)
∂ ln H2,t ∂xt3
-(c01
+
)
P,T,xt2)0
[
(∆12 - ∆23)xt2)0
c03)
2
1+
b,1-3
xb,1-3 3 b,1-3
- x3 1 x1 b,1-3 2 x1
(
(
)]
∂ ln γb,1-3 3 ∂xb,1-3 3
)
∂ ln γb,1-3 3 ∂xb,1-3 3
+
P,T
(13)
P,T
The first term on the right-hand side of eq 13 involves the ternary mixture through the limiting value (∆12 ∆23)xt2)0, while the second involves the gas-free binary solvent. It is important to emphasize that ∆Rβ constitutes a measure of the nonideality21 of the binary mixtures R-β because for an ideal mixture ∆Rβ ) 0. Two cases will be examined in what follows: (a) the case of dilute salt solutions and (b) the case of concentrated ones: (a) Dilute Salt Solutions. In the dilute range, it will be assumed that ∆13 ) 0 (in other words, the mixed solvent 1-3 behaves like an ideal mixture). From eq 12 one can find that in this case
(
)
∂ ln γb,1-3 i ∂xb,1-3 i
)0
i ) 1, 3
(14)
P,T
In a previous paper18 regarding the gas solubility in mixtures of two nonelectrolytes, the ideality approximation for the binary solvent was employed to obtain an expression for the gas solubility. The ideality of the mixed solvents constituted a good approximation because usually the nonideality of the mixture of two nonelectrolytes is much lower than those between each of them and the gas. A similar assumption can be made for dilute aqueous salt solutions. Indeed, the data regarding the activity coefficient of water (γw) in dilute aqueous solutions of sodium chloride22 indicate that |(∂ ln γw/∂xw)P,T| < 0.01 for a molality of sodium chloride smaller than 0.8. Considering, in addition, that (∆12 - ∆23)xt2)0 is independent of composition, eq 13 becomes
Figure 1. The Henry constant of nitrogen in an aqueous solution of sodium sulfate at 25 °C: (O) experimental data;12 (dashed line) the Henry constant calculated with eq 15; (solid line) the Henry constant calculated with the Sechenov equation.
ln
H2,t FwB(P,T) B(P,T) m ) C(P,T) m ≈x3 ≈ 0 H21 1000 V1 (18)
where Fw is the water density and C(P,T) is a composition-independent constant
C(P,T) ) -
Equation 15 can be used to correlate the gas solubility in the presence of a salt if the composition dependence of the molar volume of the binary electrolyte-water mixture is known. Such data are available in the literature for numerous aqueous salt solutions.24 Like that of Sechenov, eq 15 is a one-parameter equation whose parameter B has to be determined from the solubility data. The two equations provide almost the same results (see Figure 1). (b) Concentrated Solutions of Electrolytes. Because the mean activity coefficient of a salt (Kν+Aν-, where ν+ and ν- are respectively the number of cations and anions in the salt molecule) is usually expressed in terms of the salt molality, eq 13 will be converted to the molality scale. First, the Gibbs-Duhem equation for the binary mixture water (1)-electrolyte (3) allows one to rewrite eq 13 as follows:
(
)
∂ ln H2,t
ln H2,t - ln H21 ) -B
b,1-3 x3dx3
∫0
V
(15)
where B ) (∆12 - ∆23)xt2)0/2 and V is the molar volume of the binary gas-free mixture. However, for very dilute electrolyte solutions, one can write23
V ∞3 + (1 - xb,1-3 )V 01 V ) xb,1-3 3 3
(16)
where V 01 and V ∞3 are the molar volume of the pure solvent and the partial molar volume of the electrolyte at infinite dilution, respectively. Using eq 16, eq 15 becomes
ln
B(P,T) ln[1 + (V ∞3 /V 01 - 1)x3] H2,t )(17) H21 V∞-V0 3
1
Because, for very small values of y, ln(1 + y) ≈ y, eq 17 can be rewritten in the form of the Sechenov equation:
Fw(∆12 - ∆23)xt2)0 FwB(P,T) )(19) 1000 2000
∂xt3
)
P,T,xt2)0
[
(∆12 - ∆23)xt2)0
-(c01 + c03)
2
1 + xb,1-3 1
b,1-3
b,1-3
- x3 1 x1 b,1-3 2 x3
(
(
)]
∂ ln γb,1-3 1 ∂xb,1-3 1
)
∂ ln γb,1-3 1 ∂xb,1-3 1
+
P,T
(20)
P,T
Second, for the water activity coefficient in the binary mixture water (1)-electrolyte (3), one can use the relation2
νm d ln(mγ() ) 1000 1000 d ln xb,1-3 d ln γb,1-3 (21) 1 1 M1 M1 where γ( is the mean activity coefficient of the salt, ν ) ν+ + ν-, M1 is the molecular weight of water, and m is the molality of the salt. By combining eqs 20 and 21 with the obvious relations
Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002 4677 Table 1. Information about the Mixtures Used in the Calculations gas and salt solution gas
salt solution
temp, K
composition range, m
ref
O2 O2 N2 CO2 CO2 O2
Na2SO4 + H2O Na2SO4 + H2O Na2SO4 + H2O NaCl + H2O Na2SO4 + H2O NaCl + H2O
298.15 308.15 298.15 298.15 298.15 298.15
0-1.517 0-1.656 0-1.070 0-5.096 0-2.205 0-5.4
a a a b b c
a Yasunishi, A. J. Chem. Eng. Jpn. 1977, 10, 89. b Yasunishi, A.; Yoshida, F. J. Chem. Eng. Data 1979, 24, 11. c Mishina, T. A.; Avdeeva, O. I.; Bozhovskaya, T. K. Mater. Vses. Nauchno-Issled. Geol. Inst. 1961, 46, 93 (as given in Solubility Data Series; Pergamon: New York, 1981; Vol. 7).
xb,1-3 ) 1 )dxb,1-3 1
1000 1000 + M1m 1000M1
(1000 + M1m)2
(22)
dm
Figure 2. The Henry constant of oxygen in aqueous solutions of sodium sulfate at 25 °C: (O) experimental data; (a) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Debye-Hu¨ckel equation; (b) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the extended Debye-Hu¨ckel equation; (c) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Bromley equation; (d) the Henry constant calculated with eq 15.
(23)
and integrating, one obtains an equation for the Henry constant at a given molality m (again B was considered independent of composition):
[
(
) ]
∂ ln γ( H2,t m ∂m P,T ) -B 0 ln dm + H21 1000(1 + 0.001M1m)V 1 m1 - 0.001M1m ν(1 + 0.001M1m) 1 + 2 0 1 + 0.001M1m ∂ ln γ( - 1 dm (24) m ∂m P,T
( )
∫
M1ν 1 + m
∫
(
(
[
) ] )
This equation contains a parameter B that must be calculated from the experimental data. In addition, information about the molar volume of the mixture and the mean activity coefficient of the salt on the molality in the binary mixture water (1)-salt (3) is necessary. 3. One-Parameter Gas Solubility in Aqueous Salt Mixtures: Comparison with Experiment Several aqueous salt mixtures were selected to verify eq 24. They are listed in Table 1. Some mixtures (oxygen or carbon dioxide with water + sodium sulfate or sodium chloride) have been selected because the Sechenov equation did not provide an accurate correlation of the solubility in these mixtures at high molalities. Other mixtures, listed in Table 1, have also been considered. Accurate density data for many aqueous salt mixtures are available in the literature.24 The following expression was used for the analytical representation of the molar volume of aqueous salt solutions:25
V ) x3φ + x1V1 ) 1000V 01 M1m1.5 M1mφ0 +β + (25) 1000 + M1m 1000 + M1m 1000 + M1m where φ is the apparent molar volume of the electrolyte, φ0 ) V b,∞ 3 , V1 is the partial molar volume of water in the binary electrolyte solution, M1 and V 01 are the molecular weight and molar volume of the pure water,
Figure 3. The Henry constant of carbon dioxide in aqueous solutions of sodium sulfate at 25 °C: (O) experimental data; (a) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Debye-Hu¨ckel equation; (b) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the extended Debye-Hu¨ckel equation; (c) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Bromley equation; (d) the Henry constant calculated with eq 15. Table 2. Constant β and O0 in Equation 25 for the Mixtures Investigated mixture
temp, K
NaCl + H2O 298.15 Na2SO4 + H2O 298.15
β, composition φ 0, range, m cm3/mol (cm3 kg0.5)/mol1.5 0-6.1 0-2
16.04 10.43
2.09 12.66
and β is a constant; β and φ0 were evaluated from experimental density data,24 and their values are listed in Table 2. For the mean activity coefficient of the salt, several expressions have been used, such as the Debye-Hu¨ckel equation,2 the extended Debye-Hu¨ckel equation,2 and the Bromley equation.2 The Bromley equation was selected because of its simplicity and its accuracy; of course, other accurate equations2 are also available. The values of the parameter B for all cases examined are listed in Table 3. A comparison between the experimental solubilities and those calculated with eq 24 is presented in Figures 2-5. They show that the values calculated using eq 24 are highly dependent on the activity coefficient employed. Indeed, the correlation based on the DebyeHu¨ckel equation is very poor, particularly at high molalities (see curves a in Figures 2-5). The extended Debye-Hu¨ckel equation (see curves b in Figures 2-5) provides a better agreement but is not yet satisfactory at high molalities. Equation 15 [based on the ideal 1
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Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002
Table 3. Values of B Obtained from Experimental Solubility Data (for the Sources of Experimental Solubility Data, See Table 1) parameter B, cm3/mol mixture
eq 15
eq 24 with the Debye-Hu¨ckel equation
eq 24 with the extended Debye-Hu¨ckel equation
eq 24 with the Bromley equation
oxygen in Na2SO4 + H2O (T ) 298.15 K) oxygen in Na2SO4 + H2O (T ) 308.15 K) nitrogen in Na2SO4 + H2O carbon dioxide in Na2SO4 + H2O oxygen in NaCl + H2O carbon dioxide in NaCl + H2O
-838 -822 -822 -708 -3.8 -6.42
-136 -337 -337 -98 150 -7.53
-221 -209 -209 -39 184 36.9
-149 -141 -141 -151 175 166.4
mation about the other systems is contained in Table 3. For NaCl, the results are not too accurate at high molalities. However, it is well-known3 that most models fail to represent accurately the mean activity coefficient for the NaCl + H2O mixtures at high molalities. One can, therefore, conclude that the gas solubility in aqueous salt solutions can be well described by eq 24 when accurate expressions for the mean activity coefficient of the salt in the binary water + salt mixtures are used. Figure 4. The Henry constant of oxygen in aqueous solutions of sodium chloride at 25 °C: (O) experimental data; (a) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Debye-Hu¨ckel equation; (b) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the extended Debye-Hu¨ckel equation; (c) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Bromley equation; (d) the Henry constant calculated with eq 15.
4. Salting-In or Salting-Out? It is worth mentioning that no criterion is presently available to answer the question of whether the addition of a salt will increase or decrease the solubility of a gas in a solvent. Let us consider eq 15, which is valid for sufficiently small salt concentrations. Of course, the sign of B accounts for the type of salting, because the integral always has positive values: consequently, B should be negative for salting-out and positive for salting-in. Using eq 6, one obtains the following relation for B:
B)
(G11 - G33 - 2G12 + 2G23)xt2)0 2
(26)
The partial molar volume of the solute (component 2) at infinite dilution in the ternary mixture 1-2-3 is given by20
lim V t2 ) V t,∞ 2 ) t
Figure 5. The Henry constant of carbon dioxide in aqueous solutions of sodium chloride at 25 °C: (O) experimental data; (a) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Debye-Hu¨ckel equation; (b) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the extended Debye-Hu¨ckel equation; (c) the Henry constant calculated with eq 24 using for the mean activity coefficient of dissolved salt the Bromley equation; (d) the Henry constant calculated with eq 15.
(water)-3 (electrolyte) mixture approximation] provides a better agreement for not too high molalities [the Sechenov equation and eq 15 provide in all of the cases very similar results (see Figure 1)]. Only when the more accurate Bromley equation has been employed for the mean activity coefficient of the salt was the agreement between experiment and calculation very good. Additional systems, also listed in Table 1, have been examined, and very good agreement was obtained when the Bromley equation was used for the activity coefficient. We have also employed other accurate equations for the mean activity coefficient of the salt, such as the Pitzer equation,2,3 and the agreement was as good as that provided by the Bromley equation. Figures 2-5 examine only some of the systems investigated. Infor-
x2f0
lim {[1 + c1(G11 - G12) + c3(G33 - G23) + t
x2f0
c1c3(-G12G33 + G12G13 - G132 + G13G23 + G11G33 - G11G23)]/(c1 + c3 + c1c3∆13)} (27) Assuming that the mixture 1-3 behaves like an ideal one and using eq 6, one can write
∆13 ) 0
and
2G13 ) G11 + G33
(28)
The combination of eqs 26-28 yields the following expression for B in the dilute region (c3 , c1 and c1 ≈ c01) (for details, see the appendix):
B)-
b,∞ b,∞ b,∞ 0 V t,∞ 2 - V 2 - c3V 3 (V 3 - V 1)
c3V b,∞ 3
(29)
where V b,∞ is the partial molar volume of the solute 2 gas (2) in water at infinite dilution, V b,∞ 3 is the partial molar volume of the electrolyte (3) in water at infinite dilution, and V 01 is the molar volume of pure water.
Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002 4679 Table 4. Aqueous Mixtures with Salting-Ina gas c
CH4 CH4c CH4c CH4c
salt
b V b,∞ 3 , cm3/mol
gas
salt
b V b,∞ 3 , cm3/mol
(Me)4NBr (Et)4NBr (Pr)4NBr (Bu)4NBr
∼114 ∼174 ∼240 ∼301
O2, He, Kr He Ar Ar
(Et)4NBr (Bu)4NBr (Et)4NI (Pr)4NI
∼174 ∼301 ∼185 ∼250
a As given in ref 7. b Data regarding Vb,∞ (T ) 298.15 K) 3 obtained from: Millero, F. J. In Water and Aqueous Solutions: Structure, Thermodynamics and Transport Processes; Horne, R. A., Ed.; Wiley: London, 1972; Chapter 13. c Salting-in was also observed for C2H6, C3H8, and C4H10.
For small values of c3, one can, however, write b,∞ V t,∞ 2 ) V 2 +
( ) ∂V t,∞ 2 ∂c3
c3
(30)
P,T
and consequently eq 29 becomes
B)
( ) ∂V t,∞ 2 ∂c3
b,∞ 0 + V b,∞ 3 (V 3 - V 1)
P,T
V b,∞ 3
(31)
The available information8 indicates that R ) (∂V t,∞ 2 / ∂c3)P,T is usually small;8 e.g., for Ar in CaCl2, KCl, KI, (Me)4NBr, and (Bu)4NBr, R is approximately equal to -1, -0.5, -0.3, +0.4, and +1.5 (cm3/mol)2, respectively. The values of R for methane, oxygen, and hydrogen are 0 also small.8 However, when the difference V b,∞ 3 - V 1 is small, the value of R can affect the sign of B. When R is small and can be neglected, one obtains a very simple criterion for salting-out and salting-in. Namely, salting-out will occur when 0 V b,∞ 3 < V 1
(32)
0 V b,∞ 3 > V 1
(33)
and salting-in when
The above criterion indicates that salting-in occurs for salts with large values of V b,∞ 3 and salting-out for salts with relatively small or negative values of V b,∞ 3 . The literature data are in agreement with this conclusion (see Table 4). Salting-in was also observed16 (not for all conditions) for aqueous solutions of CH3COONH4, CH33 26 and CH COOH (V b,∞ = COONa (Vb,∞ 3 3 = 40 cm /mol), 3 52 cm3/mol).27 To our knowledge, salting-in was not observed for salts with relatively small (or negative) V b,∞ 3 , for example, for aqueous solutions of NaOH 3 26 or KOH (V b,∞ = 3-4 cm3/mol).26 (V b,∞ 3 = -5 cm /mol) 3 The above criterion can also be extended to nonelectrolytes. Indeed, the addition of an alcohol (for alcohols28 0 b,∞ 3 V b,∞ 3 > V 1: for instance, for methanol V 3 ) 38.2 cm / b,∞ 3 mol, for ethanol V 3 ) 55.1 cm /mol, and for 1-pro3 panol V b,∞ 3 ) 70.7 cm /mol) increases the solubilities of oxygen, nitrogen, and carbon dioxide in water.29 However, it should be noted that the above criteria (32) and (33) pertain only to very low salt concentrations and involve the approximation of ideal behavior for dilute solutions of salt in water. 5. Discussion and Conclusion The Kirkwood-Buff formalism was used to derive a general expression for the derivative of the activity
coefficient γ2,t of the gas in a ternary mixture with respect to the mole fraction xt3 of the salt. The derived expression was used to obtain the composition dependence of the Henry constant for a gas dissolved in a mixed solvent. It should be pointed out that the mixed solvent can be composed of two nonelectrolytes or a solvent and a solute, such as a salt. In this paper the emphasis is on the latter case. A general expression for the Henry constant in a salt solution was also obtained, which contains as a particular case the Sechenov equation and which, like the Sechenov equation, is a one-parameter equation. This equation requires information about the molar volume and the mean activity coefficient of the salt in the binary water + salt mixture. The Sechenov equation and the new ones (eqs 15 and 24) have been compared with experimental data. The results obtained with the Sechenov equation underestimate, in agreement with literature observations,13 the gas solubility at high salt concentrations. In contrast, the expressions based on the Debye-Hu¨ckel equation or the extended DebyeHu¨ckel equation for the mean activity coefficient of the salt overestimate the gas solubility. When the derived eq 24 has been combined with an accurate equation for the mean activity coefficient of the dissolved salt, such as the Bromley equation, an accurate correlation for the oxygen solubility in an aqueous solution of sodium sulfate could be obtained. However, even the Bromley equation is not accurate enough to represent the mean activity coefficient of the salt for the NaCl + H2O mixture at high molalities, and this explains the less good prediction obtained for the gas solubilities in NaCl solutions at high molalities. The main advantage of the new equations in comparison with that of Sechenov and its modifications is their clear physical meaning, which allowed one to derive a criterion for predicting the kind of salting. The obtained criterion predicted salting-in for “large” ions and salting-out for relatively small ions, in agreement with the available experimental information. Appendix: Expression for Coefficient B in Equation 15 Equation 27 for the partial molar volume of component 2 at infinite dilution in a binary mixture water (1) + electrolyte (3)20 for ∆13 ) 0, c3 , c1, and c1 ≈ c01 can be recast in the form
V t,∞ 2 ) [1 + c1(G11 - G12)xt2)0 + c3(G33 - G23)xt2)0 + c1c3B(G33 - G11)xt2)0]/(c1 + c3) (A1-1) From eq 26, one obtains
(G23)xt2)0 )
B (G11 - G33 - 2G12)xt2)0 (A1-2) 2 2
which, introduced in eq A1-1, provides the following expression for B: 0 B ) -[2c01 V t,∞ 2 - 2 + 2c1(G12 - G11)xt2)0 +
c3(2G12 - G33 - G11)xt2)0]/ [c3(1 - c01G33 + c01G11)xt2)0] (A1-3)
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Ind. Eng. Chem. Res., Vol. 41, No. 18, 2002
For small values of c3, (G12)xt2)0 can be replaced by30 0 lim G12 ) lim G12 ) G∞12 ) RTkT,1 - V b,∞ 2 t
x2f0
t
x2f0 t
x3f0
(A1-4)
The Kirkwood-Buff integrals for the mixture waterelectrolyte, when one assumes ideal behavior in the dilute region, can be expressed as follows:30 0 0 b,∞ 0 G13 ) G id 13 ) RkT,1 - V 1 - φ1(V 3 - V 1)
(A1-5)
id b,∞ 0 G11 ) G id 11 ) G 13 + V 3 - V 1
(A1-6)
id b,∞ 0 G33 ) G id 33 ) G 13 - (V 3 - V 1)
(A1-7)
and
In the above expressions, R is the universal gas con0 stant, kT,1 is the isothermal compressibility of the pure is the partial molar volume of component solvent, V b,∞ i i at infinite dilution in the binary mixture of component i and solvent, and φ1 is the volume fraction of water. It should be noted that expressions (A1-5)(A1-7) involve ∆13 ) 0 and (∂ ln γb,1-3 /∂xb,1-3 )P,T ) 0 3 3 (see eq 14). Combining eqs A1-4-A1-7 with eq A1-3 leads to eq 31 in the text. Literature Cited (1) Wilhelm, E.; Battino, R.; Wilcock, R. J. Low-pressure solubility of gases in liquid water. Chem. Rev. 1977, 77, 219-262. (2) Zemaitis, J. F.; Clark, D. M.; Rafal, M.; Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics; AIChE: New York, 1986. (3) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1986. (4) Desnoyers, J. E.; Pelletier, G. E.; Jolicoeur, C. Salting-in by quaternary ammonium salts. Can. J. Chem. 1965, 43, 32323237. (5) Masterton, W. L.; Bolocofsky, D.; Lee, T. P. Ionic radii from scaled particle theory of the salt effect. J. Phys. Chem. 1971, 75, 2809-2815. (6) Feillolay, A.; Lucas, M. Solubility of helium and methane in aqueous tetrabutylammonium bromide solutions at 25 and 35 °C. J. Phys. Chem. 1972, 76, 3068-3072. (7) Krishnan, C. V.; Friedman, H. L. Model calculations for Setchenow coefficients. J. Solution Chem. 1974, 3, 727-744. (8) Tiepel, E. W.; Gubbins, K. E. Partial molal volumes of gases dissolved in electrolyte solutions. J. Phys. Chem. 1972, 76, 30443049. (9) Sechenov, I. M. U ¨ ber die konstitution der salzlo¨sungen auf grund ihres verhaltens zu kohlensa¨ure. Z. Phys. Chem. 1889, 4, 117-125.
(10) Long, F. A.; McDevit, W. F. Activity coefficients of nonelectrolyte solutes in aqueous salt solutions. Chem. Rev. 1952, 51, 119-169. (11) Tiepel, E. W.; Gubbins, K. E. Thermodynamic properties of gases dissolved in electrolyte solutions. Ind. Eng. Chem. Fundam. 1973, 12, 18-25. (12) Yasunishi, A. Solubilities of sparingly soluble gases in aqueous sodium sulfate and sulfite solutions. J. Chem. Eng. Jpn. 1977, 10, 89-94. (13) Schumpe, A. The estimation of gas solubilities in saltsolutions. Chem. Eng. Sci. 1993, 48, 153-158. (14) Yasunishi, A.; Yoshida, F. Solubility of carbon dioxide in aqueous electrolyte solutions. J. Chem. Eng. Data 1979, 24, 1114. (15) Sing, R.; Rumpf, B.; Maurer, G. Solubility of ammonia in aqueous solutions of single electrolytes sodium chloride, sodium nitrate, sodium acetate, and sodium hydroxide. Ind. Eng. Chem. Res. 1999, 38, 2098-2109. (16) Xia, J. Z.; Kamps, A. P. S.; Rumpf, B.; Maurer, G. Solubility of H2S in (H2O + CH3COONa) and (H2O + CH3COONH4) from 313 to 393 K and at pressures up to 10 MPa. J. Chem. Eng. Data 2000, 45, 194-201. (17) Kirkwood, J. G.; Buff, F. P. Statistical mechanical theory of solutions. I. J. Chem. Phys. 1951, 19, 774-782. (18) Shulgin, I.; Ruckenstein, E. Henry’s constant in mixed solvents from binary data. Ind. Eng. Chem. Res. 2002, 41, 16891694. (19) O’Connell, J. P. Molecular thermodynamics of gases in mixed solvents. AIChE J. 1971, 17, 658-663. (20) Ruckenstein, E.; Shulgin, I. Entrainer effect in supercritical mixtures. Fluid Phase Equilib. 2001, 180, 345-359. (21) Ben-Naim, A. Inversion of the Kirkwood-Buff theory of solutions: application to the water-ethanol system. J. Chem. Phys. 1977, 67, 4884-4890. (22) Perry, R. L.; Massie, J. D.; Cummings, P. T. An analytic model for aqueous-electrolyte solutions based on fluctuation solution theory. Fluid Phase Equilib. 1988, 39, 227-266. (23) Cooney, W. R.; O’Connell, J. P. Correlation of partial molar volumes at infinite dilution of salts in water. Chem. Eng. Commun. 1987, 56, 341-349. (24) CRC Handbook of Chemistry and Physics, 81st ed.; Lide, D. R., Ed.; CRC Press: Boca Raton, FL, 2000-2001. (25) Harned, H. S.; Owen, B. B. The physical chemistry of electrolytic solutions, 2nd ed.; Reinhold Pub. Corp.: New York, 1950. (26) Millero, F. J. In Water and Aqueous Solutions: Structure, Thermodynamics and Transport Processes; Horne, R. A., Ed.; Wiley: London, 1972; Chapter 13; (27) Lang, W. Setchenov coefficients for oxygen in aqueous solutions of various organic compounds. Fluid Phase Equilib. 1996, 114, 123-133. (28) Franks, F.; Desnoyers, J. E. Alcohol-water mixtures revisited. Water Sci. Rev. 1985, 1, 171-232. (29) Tokunaga, J. Solubilities of oxygen, nitrogen, and carbon dioxide in aqueous alcohol solutions. J. Chem. Eng. Data 1975, 20, 41-46. (30) Shulgin, I.; Ruckenstein, E. Kirkwood-Buff integrals in aqueous alcohol systems: comparison between thermodynamic calculations and X-ray scattering experiments. J. Phys. Chem. B 1999, 103, 2496-2503.
Received for review May 8, 2002 Revised manuscript received July 2, 2002 Accepted July 2, 2002 IE020348Y
