Solubility of Gases in Binary Liquid Mixtures: An Experimental and

Ind. Eng. Chem. Res. 2003, 42, 1439-1450

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GENERAL RESEARCH Solubility of Gases in Binary Liquid Mixtures: An Experimental and Theoretical Study of the System Noble Gas + Trifluoroethanol + Water Ana M. Mainar,† Juan I. Pardo,‡ Jesu ´ s Santafe´ ,† and Jose´ S. Urieta*,† Departamento de Quı´mica Orga´ nica-Quı´mica Fı´sica, Facultad de Ciencias, Universidad de Zaragoza, Ciudad Universitaria, Plaza san Francisco, 50009 Zaragoza, Spain, and Centro Polite´ cnico Superior, Universidad de Zaragoza, Marıá de Luna 3, 50018 Zaragoza, Spain

Solubilities of noble gases (He, Ne, Ar, Kr, and Xe) in mixtures of water + 2,2,2-trifluoroethanol (TFE) at 298.15 K and 101.33 kPa partial pressure of gas are reported. Our procedure for the estimation of these solubilities from the experimental data is described in detail. From these data, the Henry’s constants at the vapor pressure of water (the least volatile component), the standard changes in the Gibbs energy for both the solution process and the solvation process, and the so-called excess Henry’s constant are calculated. Finally, three prediction methods are applied, and their results are compared and discussed. Introduction The importance of solutions and mixtures of nonelectrolytes is difficult to overestimate. Hence, the growth in both the experimental and theoretical thermodynamic study of these systems has been extraordinary. Whereas liquid mixtures have been very extensively studied, this is not the case for solutions of gases in liquids. However, such solutions are not less relevant. For example, the solubilities of gases in liquids provide information about the solvophobicity1 of the liquid solvent, information that can be used for practical purposes such as the choice of adequate media for some chemical processes. At a more fundamental level, these solubilities also allow for the estimation of solvent molecular parameters, as in the case of those corresponding to the Lennard-Jones potential.2 It can be said that gases act as probes for certain characteristics of the solvent into which they dissolve. If data for solutions of gases in pure liquids are not as common as those for binary liquid mixtures, the lack of data is even more pronounced in the case of solutions of gases in binary liquid mixtures. Thus, this kind of solution is well worth considering. In fact, this paper is part of a study on the solubilities of a wide set of gases (He, Ne, Ar, Kr, Xe, H2, N2, O2, CH4, C2H6, C2H4, CF4, SF6, and CO2) in mixtures of water and a fluoro alcohol, namely, 2,2,2-trifluoroethanol (TFE) or 1,1,1,3,3,3hexafluoropropanol. These liquid mixtures have been used to modulate some of the properties influencing the kinetics of Diels-Alder-type reactions.3 The aim of this investigation is to measure the solubilities, then to calculate related thermodynamic properties, and finally to interpret all of these data. * To whom correspondence should be addressed. Phone number: 34 976 761298. Fax number: 34 976 761202. E-mail: [email protected]. † Facultad de Ciencias. ‡ Centro Politećnico Superior.

As a preliminary step, the solubilities of the gases in the pure fluoro alcohols were determined.4,5 Now, we report the solubilities of the noble gases helium, neon, argon, krypton, and xenon in the mixture water + 2,2,2trifluoroethanol (TFE) at a temperature of 298.15 K and a partial pressure of gas of 101.33 kPa. Confirmation of the above-mentioned lack of data is given by the fact that, as far as we know, only one other systematic study of the solubilities of noble gases in a binary liquid mixture has been conducted.6 In our studies, the quantity of alcohol in the mixtures covers a range between 10 and 90% by volume. The solubilities are expressed in terms of the mole fraction of gas dissolved and also the Henry’s constant as a function of the liquid-phase composition as given by the mole fraction of TFE. Estimation of the solubilities required the development of a calculation method, which is explained in detail. The solubility values allow for the calculation of the changes in the standard Gibbs energy for both the solution and solvation processes. A simple method for obtaining the Gibbs energies for the solution process and the solubilities in the systems considered is derived from the relationship between the Gibbs energies and the depth of the potential well of the gases. In turn, the Gibbs energies for the solvation process provide useful information about the behavior of the liquid solvent mixture in the water-rich zone. Finally, three prediction methods, namely, the equation of Krichevsky7 (related to the so-called excess Henry’s constant), the equation of Shulgin and Ruckenstein,8 and the scaled particle theory (SPT)2,9-11 are considered to verify their validity in these ternary solutions. Experimental Section Materials. The gases used were helium (99.995%), argon (99.9990%), krypton (99.95%), and xenon

10.1021/ie020329o CCC: $25.00 © 2003 American Chemical Society Published on Web 03/04/2003

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Ind. Eng. Chem. Res., Vol. 42, No. 7, 2003

(99.995%), all from Air Liquide Espanã, along with neon (99.9%) obtained from J. T. Baker. For the liquids, 2,2,2trifluoroethanol with a stated purity of better than 99% was provided by Fluorchem Ltd., and the water was doubly distilled and deionized with a Millipore device (quality MilliQ). The purity of TFE was checked by GLC. Apparatus and Procedures. The apparatus employed for measuring the solubilities of gases in liquids has been described in detail elsewhere.12 It consists of an all-glass setup similar to that of Ben-Naim and Baer13 that uses a saturation method. A system of burets permits the determination of the volume of gas dissolved in a known quantity of liquid solvent. The gas dissolved is a wet gas, i.e., a mixture of the gas and vapor of the solvent. This solvent fills a vessel that is immersed in a thermostated water bath whose temperature is controlled to within (0.05 K. In turn, the whole apparatus is located in an air bath whose temperature, greater than that of the water to prevent condensation of the solvent from the solution vessel, is also maintained constant to within (0.2 K. The difference between the temperatures in the water bath and air bath is the main characteristic that distinguishes our apparatus and that of Ben-Naim and Baer. This particularity simplifies the manipulation but complicates the treatment of experimental data. To verify the accuracy of the apparatus, the solubilities of argon in water were determined at some temperatures near 298.15 K and then compared with the most reliable values in the literature, which are those provided by Krause and Benson.14 The Henry’s constants obtained by applying a method described earlier15 were 3659.3 MPa at 292.85 K, 4030.4 MPa at 298.20 K, and 4160.1 MPa at 300.15 K, whereas the values of Krause and Benson are 3656.3, 4007.3, and 4132.2 MPa, respectively. In all cases, the deviations are below 1%, which is the estimated uncertainty for the reduction method used. The density of each mixture was obtained once the measurements had been completed. The densities were determined with an Anton Paar DMA-58 vibrating tube density meter with an uncertainty of (10-5 g cm-3. Calculation of the Solubilities. Calculation Method. Among the ways available for expressing the solubility of a gas in a liquid (be it a pure liquid or a mixture), the mole fraction seems the most desirable and informative.16 Therefore, we have developed the following method for the calculation of solubilities expressed as mole fractions of gas dissolved in a liquid mixture. This reduction method is an extension of a previous approach15 based on that proposed by Wilhelm and Battino,17-19 and it takes into account the mentioned specific features of our apparatus. The method is a trial-and-error approach comprising two main parts. The first focuses on the calculation of the number of moles of each of the components in each of the phases for the ternary system. For that purpose, the experimental volumes of both dissolved gas and liquid solvent are used. Then, the corresponding mole fractions are obtained directly by applying the definition of a mole fraction. The nonideal behavior of the gas phase is taken into account through the virial equation. The second part of the reduction method is intended to ensure that the mole fractions obtained are thermodynamically coherent, fulfilling the requirements of the equilibrium equations. The two parts are not independent but are

intertwined. The density of the mixture determined after the solubility measurement is used as the final criterion for convergence as will be shown later. A scheme of the calculation method can be found in Figure 1. The procedure begins by selecting an arbitrary value of the density near the experimental value for the final mixture. This arbitrary initial density corresponds to the binary liquid solvent mixture and allows for the calculation of its composition (mole fractions of 1 and 2) from the density-composition curve. Subsequently, the number of moles of each of the liquid components in that mixture, nL1 and nL2 , can be obtained because the total volume of the solvent, VL, is known. In this calculation, the excess volume is taken into account. The composition of the gas phase in equilibrium at the temperature of the bath, yi,bin, is also determined from the composition of the binary liquid mixture using the data for the corresponding VLE. The central equation in the first part of the reduction method considers that the gas phase is a wet gas; consequently, the number of moles of component 3 (gas) dissolved is given by

n3 ) nT - nV1 - nV2

(1)

where n3 is the effective number of moles of pure gas dissolved, nT is the total number of moles of wet gas dissolved, and nV1 and nV2 are the numbers of moles of the solvents in the gas phase. This expression can be rewritten as

n3 )

PV - nV1 - nV2 RT′ + Bm,ter(y1,y3,T′)P

(2)

where P is the total pressure of the measurement, V is the volume of mercury introduced into the burets (equal to the volume of wet gas dissolved), T′ is the temperature of the air bath, yi is the mole fraction of component i in the gas phase, and Bm,ter is the second virial coefficient for the ternary (solute and solvents) mixture. The values of nV1 and nV2 are given by

nVi )

Ps,12(T)yi,binV RT′ + Bm,bin(y1,T′)Ps,12(T)

(3)

where Ps,12(T) is the vapor pressure of the liquid mixture at the temperature of the water bath and Bm,bin is the second virial coefficient for the binary (solvents only) mixture. To determine simultaneously the values of the number of moles of the solute in the gas phase, n3, and the mole fractions in the gas phase, yi, it is necessary to use an iterative method in which the first step considers that

n3,0 )

(P - Ps,12)V RT′

(4)

where the subscript 0 indicates that this is the first approximate value. This calculation ends when there is no change in the value of n3. Thus, knowing the number of moles of each of the components in each phase, the corresponding mole fractions in the two phases can be determined, and the first part of the reduction method is concluded.

Ind. Eng. Chem. Res., Vol. 42, No. 7, 2003 1441

Figure 1. Scheme for the calculation of the solubilities of gases in binary liquid mixtures considering the nonideal behavior of the gas phase. The boxes have the following meanings: trapezoid, manual entrance; parallelogram, experimental data; rectangle, calculation; diamond, decision.

The second part of the reduction method is introduced because the mole fractions obtained in the first part do not, in principle, strictly fulfill the requirements of gasliquid equilibrium, which are expressed by the following equations /V yiφVi (T,P,{y})P ) xiγi(T,P,{x})Ps,iφs,i P (T,P) i ) 1, 2 (5)

y3φV3 (T,P,{y})P ) x3γ′3(T,P,{x}) KH3,12(T,P)

(6)

where φVi represents the fugacity coefficient of solvent /V represents the fugacity coefficient of saturated i, φs,i pure component i, Ps,i represents the vapor pressure of component i, P is the Poynting correction, γi represents the activity coefficient of liquid component i according to the Lewis-Randall rule, γ′3 is the activity coefficient of the gas according to Henry’s law, and KH3,12 is the

Henry’s law constant for the gaseous component in a liquid mixture of 1 and 2. New values of the mole fractions of components 1 and 2 in the ternary gas phase are calculated from eqs 5 by substituting the values of xi determined in the first part of the reduction method. These mole fractions are denoted by y′1 and y′2 in the scheme of Figure 1 to distinguish them from the values determined in the first part. The value of y′3 is obtained from y′1 and y′2. In this calculation, two assumptions are made to simplify eqs 5. First, we assume that the fugacity coefficients at the low pressures used in this work can be obtained from a virial equation truncated after the second term, that is

[

φVi ) exp (2

n

P

yjBij - Bm,ter) ∑ RT j)1

]

(7)

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On the other hand, the activity coefficients for liquid components 1 and 2 were estimated from the excess Gibbs energies corresponding to their mixture through the well-known relation

γi ) exp

( ) µEi

Table 1. Experimental and literature densities, G, of water and TFE at 298.15 K and 101.33 kPa 10-3F (kg m-3)

(8)

RT

a

with

µE1

)

GEm

dGEm + x2 dx1

(9)

dGEm µE2 ) GEm - x1 dx1

(10)

where µEi is the chemical potential of the ith component and GEm is the molar excess Gibbs energy of the binary mixture of 1 and 2. Given the low solubilities of the gases, the influence of component 3 on the activity coefficients of the liquids, γi, is considered negligible. The new y′i values are compared with those estimated in the first part of the reduction method. If they do not agree, then the values of y′i are used to obtain a new value for Bm,ter, and an iterative cycle is performed until convergence is achieved. In this manner, a new set of mole fractions in the gas phase is calculated. As this set is different from that used in the first part, properties of this part of the method are affected. Then, these properties are recalculated, and the process described restarts again until all of the mole fractions (in both parts) are equal in two successive steps. At this point the mole fractions in the liquid phase are introduced into the equation 3

Ffinal )

xi,terMi ∑ i)1 2

∑ i)1

2

xi,terVLi

E

+V

xi,ter + ∑ i)1

(11) x3V∞L 3

where Mi denotes the molar mass of component i, VLi denotes the molar liquid volume of component i, VE is the excess molar volume of the mixture water + TFE, and V∞L 3 is the molar volume of the gas in the liquid phase at infinite dilution. If the value of Ffinal deviates from the measured value for the density of the solution by less than the experimental uncertainty, the calculation process is finished. If not, another arbitrary value of density for the binary liquid mixture has to be proposed, and the whole procedure takes place as many times as necessary until Ffinal fulfills the criterion expressed in eq 11. In this case, the mole fraction of gas dissolved, x3, is known at the working pressure. Now, the Henry’s law constant at the working pressure is obtained from eq 6 assuming that γ′3 approaches unity given the low solubility values involved. From this Henry’s law constant, the value corresponding to the pressure of the least volatile component, Ps,1, can be determined through the relationship

KH3,12(T,Ps,1) ) KH3,12(T,P) exp

[

]

V∞L 3 (Ps,1 - P) (12) RT

liquid

experimental

literature

water TFE

0.997 07 1.382 09

0.997 05a 1.383 29b 1.381 77c

Reference 21. b Reference 22. c Reference 23.

Table 2. Densities and Excess Volumes for the Mixture H2O (1) + CF3CH2OH (2) at 298.15 K, along with the Fitting Redlich-Kister Equationa and the Standard Deviationb x2

10-3F (kg‚m-3)

106VE (m3‚mol-1)

0.0900 0.2001 0.3065 0.3981 0.4846 0.5751 0.6562 0.7995 0.8733

1.125 58 1.215 54 1.267 10 1.297 81 1.318 87 1.336 46 1.348 76 1.365 56 1.372 32

-0.395 -0.615 -0.659 -0.650 -0.592 -0.529 -0.447 -0.278 -0.176

a 106VE(m3‚mol-1) ) x (1 - x )[-2.3414 - 1.4189(1 - 2x ) 2 2 2 1.2210(1 - 2x2)2 - 0.9070(1 - 2x2)3]. b 106σ(VE) ) 0.004 m3‚mol-1.

The pressure of the least volatile component is the reference state recommended20 for gas-liquid systems when the solvent is a mixture. The mole fraction of gas dissolved at 101.33 kPa partial pressure of gas can be determined from the Henry’s constant at the corresponding total pressure by applying the equations of equilibrium. The described method is routinely applied to all gases, but in general, only the most soluble gases deviate from ideality by more than the estimated error. Calculation of the Solubilities. Necessary Data. Referring to the data that must be used in the calculation of the solubility when the method is applied to the systems considered in this work, we measured the densities of both the pure solvents and their mixtures. The experimental densities of water and TFE are listed in Table 1 and compared with literature values.21-23 Good agreement can be observed. The densities of the mixtures along with the corresponding excess volumes are listed in Table 2, and the excess volumes of the mixtures at 298.15 K as a function of the mole fraction of fluoro alcohol are represented in Figure 2. For this binary mixture, the excess volumes had been already measured at 293.15 K24 and 303.15 K.25 These results are also included in Figure 2, and good agreement between our data and those found in the literature is observed. The vapor pressures for mixtures of water + TFE and the mole fractions yi for this binary system at 298.15 K were obtained from the work of Morcom and Cooney.23 The same paper also provides the values of the excess molar Gibbs energy, GEm, for that mixture. The second virial coefficients were obtained from the literature26 for all of the gases and water, whereas that of TFE was determined by the method of Lin and Stiel.27 The cross second virial coefficients for the mixtures gas + TFE, gas + H2O, and H2O + TFE were calculated by the method proposed by Maris and Stiel.28 The results for both TFE and its mixtures with the gases can be found in an earlier paper.29 The cross second virial coefficients calculated for the systems H2O + TFE and H2O + gas

Ind. Eng. Chem. Res., Vol. 42, No. 7, 2003 1443 Table 3. Coefficients for the Fitting to Eq 13 of the Cross Second Virial Coefficients in Systems Containing Water Obtained through the Method of Maris and Stiel28 for the Temperature Range 268.15-308.15 K, along with the Squared Regression Coefficients system

a0 (m3‚mol-1)

a1 (m3‚mol-1‚K-1)

104a2 (m3‚mol-1‚K-2)

water + TFE + He + Ne + Ar + Kr + Xe

-1.2853 -2.2872 -48.588 7099.9 5419.6 8926.0

0.012 61 0.097 99 -0.117 59 -75.816 -59.052 -107.10

-0.4157 -1.069 17.8 2651.6 2074.8 3779.7

106a3 (m3‚mol-1‚K-3) 0.046 02 3.087 -306.75 -239.78 -439.58

r2 0.9989 0.9998 0.9998 0.9998 0.9998 0.9995

behavior of the gas phase and those considering ideal behavior are within the estimated error for the present systems. The mole fractions of gas dissolved, x3, were fitted using the least-squares method to a polynomial of the type

104x3 ) A0 + A1x2 + A2x22 + A3x23

(14)

The coefficients Ai are gathered in Table 5, along with the corresponding standard deviations, σ. The solubilities and their corresponding fitting curves are presented in Figure 3. The solubilities of the gases increase as usual in the sequence He < Ne < Ar < Kr < Xe. It can be seen that, beginning with pure water, the solubility scarcely increases until x2 ) 0.1 is reached. Then, the solubility begins to increase rapidly in such a way that it follows a straight line in the TFE-rich zone. In the cases of Ne and Xe, an almost imperceptible minimum occurs at x2 ) 0.02. From the solubility, the change in the Gibbs energy accompanying the hypothetical solution process

M (gas, T, 101.33 kPa) f M (T, P, hypothetical solution, x3 ) 1) can be calculated. This change16 is the variation in the partial molar Gibbs energy for the solute Figure 2. Excess molar volumes, VE, and the corresponding fitting function for the binary system H2O (1) + TFE (2) at the temperature 298.15 K. The excess molar volumes at 293.15 K (- ‚ - ‚ -) and 303.15 K (- - -) are also included.

were fitted to the temperature by means of the expression

B12 (m3‚mol-1) ) a0 + a1T + a2T2 + a6T3 (13) for the interval 268.15-308.15 K. The coefficients ai and the squared regression coefficients for the fitting can be found in Table 3. Results and Discussion Solubilities, Gibbs Energies for the Solution Process, and Gibbs Energies for the Solvation Process. The mole fraction of gas dissolved, x3, at 101.33 kPa partial pressure of gas and the logarithm of the Henry’s law constant, KH3,12 (expressed in kilopascals), at the vapor pressure of water are listed in Table 4 for different mole fractions of TFE, x2, in the binary solvent mixture. The estimated relative uncertainty of the mole fraction is (1% in the most unfavorable case. It must be pointed out that the differences between solubilities calculated considering the real

0 ) RT ln KH3,12 ∆G h 3,sol

(15)

The values for this property are included in Table 4. It has been verified30 that there is a good linear relationship between the Gibbs energy for the solution 0 , and the energy parameter for the process, ∆G h 3,sol Lennard-Jones potential (i.e., the depth of the potential well), 3/k. This approximate relationship is improved when only the noble gases are considered, and it is especially interesting in the systems studied in the present paper because the slope of the straight lines 0 connecting the ∆G h 3,sol vs 3/k points is almost invariable for the entire composition range of binary solvent mixtures, as can be observed in Figure 4. The values of 3/k for the gases were obtained from the literature.11 The constancy in the slope is an accidental result of the fact that the slopes for the pure liquids are very similar, -0.0296 for water and -0.0364 for TFE. That constancy allows an empirical estimation of 0 (and then of solubility) for a noble gas in a ∆G h 3,sol 0 water-TFE mixture of a given composition if ∆G h 3,sol (or the solubility) of another noble gas at the same composition is known. This can be done by selecting an average value for the slope. Although the relationship between the slope and the composition of the binary liquid phase is not linear, the arithmetic average of the values for the pure components, i.e., -0.0330, allows for

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Table 4. Solubilities of the Gases, expressed as Mole Fractions, at 101.33 kPa Partial Pressure of Gas and Natural Logarithm of the Henry’s Constants (in kPa) at the Vapor Pressure of Water in the Mixtures H2O (1) + CF3CH2OH (2), as well as the Gibbs Energies for the Solution Process, the Excess Henry’s Constants Both Calculated from the Experimental Data and Predicted with the SPT Model, and the Logarithms of the Henry’s Constants Predicted by eq 23 and by the SPT, along with Their Corresponding Average Deviations x2,bin

104x3

ln KH3,12 (kPa)

∆G h 03,m (kJ‚mol-1)

E ln KH3,12 (kPa)

SR ln KH3,12 (kPa)

SPT ln KH3,12 (kPa)

E,SPT ln KH3,12 (kPa)

0.0853 0.0193 -0.1705 -0.3428 -0.3496 -0.5723 -0.6239 -0.5417 -0.3527

23.37 23.31 23.24 23.15 23.05 22.88 22.66 22.37 21.65 2.7%

22.86 22.66 22.44 22.19 21.96 21.60 21.25 20.92 20.40 4.3%

-0.2712 -0.3673 -0.4672 -0.5713 -0.6518 -0.7456 -0.7874 -0.7646 -0.5407

0.117 94 0.0792 -0.1573 -0.3319 -0.5016 -0.6253 -0.7193 -0.6648 -0.4747

23.19 23.12 23.05 22.96 22.84 22.68 22.49 22.12 21.49 3.1%

22.30 22.10 21.84 21.58 21.30 20.97 20.65 20.22 19.76 7.9%

-0.3263 -0.4263 -0.5409 -0.6471 -0.7451 -0.8289 -0.8656 -0.8262 -0.6131

0.0845 -0.0063 -0.1905 -0.4007 -0.5268 -0.7276 -0.7956 -0.7624 -0.5766

22.05 21.99 21.91 21.81 21.70 21.53 21.29 20.97 20.26 3.6%

21.53 21.30 21.03 20.73 20.43 20.06 19.65 19.26 18.72 5.1%

-0.4776 -0.5888 -0.7103 -0.8290 -0.9279 -1.0187 -1.0567 -1.0057 -0.7338

0.0435 0.0314 -0.1828 -0.4237 -0.6103 -0.7468 -0.8338 -0.7854 -0.5996

21.47 21.40 21.32 21.22 21.12 20.93 20.68 20.33 19.60 3.9%

21.07 20.82 20.55 20.23 19.95 19.53 19.11 18.70 18.16 4.0%

-0.5460 -0.6626 -0.7792 -0.9023 -0.9957 -1.0943 -1.1254 -1.0600 -0.7634

0.1208 0.0454 -0.2101 -0.4857 -0.7058 -0.8702 -0.9454 -0.9174 -0.7156

20.91 20.84 20.75 20.64 20.53 20.33 20.07 19.66 18.94 4.6%

20.36 20.14 19.85 19.55 19.27 18.86 18.44 17.98 17.49 4.3%

-0.6141 -0.7087 -0.8291 -0.9441 -1.0331 -1.1267 -1.1541 -1.0705 -0.7907

helium 0.0294 0.0615 0.0986 0.1442 0.1884 0.2678 0.3630 0.4717 0.6967

0.0692 0.0819 0.1112 0.1541 0.1783 0.2901 0.4118 0.5420 0.9344 average deviation

23.41 23.24 22.93 22.61 22.46 21.98 21.62 21.35 20.81

29.45 29.03 28.28 27.47 27.10 25.90 25.03 24.35 23.00

0.0295 0.0600 0.1002 0.1440 0.1983 0.2695 0.3496 0.4865 0.6793


23.26 23.12 22.75 22.43 22.08 21.72 21.36 20.96 20.51

29.09 28.75 27.83 27.02 26.15 25.25 24.37 23.38 22.27

0.0297 0.0607 0.0993 0.1447 0.1941 0.2649 0.3592 0.4709 0.6771


22.09 21.89 21.57 21.20 20.90 20.45 20.05 19.69 19.15

26.18 25.70 24.90 23.98 23.24 22.12 21.13 20.25 18.89

0.0285 0.0610 0.0984 0.1450 0.1900 0.2681 0.3626 0.4806 0.6833


21.47 21.34 20.99 20.58 20.23 19.81 19.38 19.00 18.45

24.66 24.34 23.46 22.45 21.58 20.53 19.47 18.54 17.16

0.0287 0.0584 0.0984 0.1436 0.1902 0.2668 0.3613 0.4903 0.6773


20.98 20.79 20.38 19.93 19.53 19.07 18.63 18.16 17.64

23.45 22.98 21.97 20.84 19.86 18.72 17.61 16.46 15.16

neon

argon

krypton

xenon

0 the determination of both ∆G h 3,sol and the solubility expressed in terms of mole fractions with errors of less (in the worst cases) than 7 and 38%, respectively. Obviously, if the solubilities of a noble gas over the whole composition range are known, they can be used 0 with the average slope to calculate ∆G h 3,sol and the corresponding solubilities for every other noble gas over that range. In this way, using the parameters obtained for eq 14 applied to argon with the average slope, the 0 greatest deviations in the estimations of ∆G h 3,sol and solubility are 5 and 25%, respectively. In the present

state of the gas solubility models, these values would be acceptable if the experimental ones were not available. Closely related to the Gibbs energy for the solution process is the change in Gibbs energy for the solvation process, ∆G h solv. The relation between these two properties is31

( )

∆G h solv ) ∆G h 3,sol0 - RT ln

RT V12

(16)

where V12 is the molar volume of the liquid mixture of


1 and 2. The process of solvation is defined31 as the process of transferring one molecule of solute from a fixed position in the gas to a fixed position in the liquid at a given pressure and temperature. The values of the Gibbs energy for the solvation process can be found in Figure 5 for the five noble gases. A point to be noted is that the general shape of these curves follows that of 0 the corresponding ∆G h 3,sol curves, but now they all clearly show a maximum in the zone near pure water, whereas this maximum appears only for Ne and Xe 0 is considered. when ∆G h 3,sol It would be interesting at this point to compare the shape of ∆G h solv for our systems with that observed for the noble gases6,32-34 when dissolved in mixtures water and ethanol, i.e., in systems where the alkanol is not fluorinated. The behavior of the curves is similar for most of the composition range, but remarkable differences are observed in the zone near pure water. In the case of ethanol, the Gibbs energy presents a minimum and then a maximum. In the case of the fluoro alcohol, only a slight maximum is observed. For the water-ethanol solvent, the minimum has been associated32,34,35 with a more structured form of water promoted by the presence of ethanol (and also of the gas), which acts as a hydrophobic agent. The hydrophobic effect produced by the ethanol leads to a higher solubility. The maximum would correspond to the destruction of the structure of water and the subsequent decrease in solubility. When ethanol is substituted by TFE, there is no minimum. Thus, it can be concluded that TFE does not promote the structure of water. On the contrary, the maximum indicates that TFE causes a destruction of the structure of water. Consequently, from the solubility data, it can be inferred that TFE is not a hydrophobic agent or, at least, that its hydrophobic effect is much less than that of ethanol and is not reflected by the solubility values. These conclusions do not definitively require the hydrophobicity of TFE to be completely discarded because solubility and Gibbs energy are dependent on the temperature.6,32-34 As the temperature rises, the minimum in each noble gas-water-ethanol system vanishes, and an inflection point appears. It is possible that the minimum would be observed for the noble gaswater-TFE systems if the temperature were decreased, although the fact that there is no inflection point in our solutions probably indicates that this possibility is remote. From the values of ∆G h solv, it can be also concluded that the solvation of these gases is not favored, except for xenon when x2 > 0.7, a result that agrees with the higher solubility of this gas. For the remaining gases, the Gibbs energy for the solvation process is positive over the whole composition range. Excess Henry’s Constants and Prediction Methods. Another property of interest in the case of solubilities of gases in liquid mixtures is the so-called excess E Henry’s constant, ln KH3,12 , which is defined7 by 2

E ln KH3,12 ) ln KH3,12 -

xi ln KH3,i ∑ i)1

(17)

for a binary liquid solvent. The values for the solubilities of the noble gases in the pure components can be found in the literature,5,14 and the calculated excess Henry’s constants for noble gases in mixtures H2O + TFE are listed in Table 4. These results were fitted to the mole

Figure 3. Solubilities expressed as mole fractions of dissolved gas, 104x3, versus the mole fraction of TFE, x2, in the solvent liquid mixture H2O (1) + TFE (2) at 298.15 K and 101.33 kPa partial pressure of gas for the gases: (9) He, (b) Ne, (2) Ar, (1) Kr, and ([) Xe. The fitting curves have also been drawn. (a) Full composition range. (b) Detail of the zone rich in water.

1446


0 Figure 4. Gibbs energies for the solution process, ∆G h 3,sol , versus the depths of the potential wells of the noble gases, 3/k, along with the corresponding linear fitting. The data are represented for different compositions of the mixture H2O (1) + TFE (2): x2 ) (4) 0, (2) 0.2, (O) 0.4, (b) 0.6, (0) 0.8, and (9) 1.

Table 5. Fitting Coefficients and Standard Deviations According to Eq 14 for the Solubilities of Noble Gases in H2O (1) + CF3CH2OH (2) Mixtures at 298.15 K and 101.33 kPa Partial Pressure of Gas gas

A0

A1

A2

A3

104σ

He Ne Ar Kr Xe

0.0487 0.0575 0.2093 0.3779 0.7037

0.6745 0.7189 1.555 2.702 2.403

0.5592 1.764 10.38 21.16 60.30

0.4656 -0.2798 -3.563 -7.165 -26.02

0.0162 0.0175 0.0379 0.0762 0.1385

fraction of fluoro alcohol according to the following rational adjusting equation36 p

Ci(1 - 2x2)i ∑ i)0

x2(1 - x2) ln

E KH3,12

) q

1+

Figure 5. Gibbs energies for the solvation process, ∆G h solv, versus the mole fraction of TFE, x2, in the solvent liquid mixture H2O (1) + TFE (2) at 298.15 K. Table 6. Fitting Coefficients and Standard Deviations According to Eq 18 for the Excess Henry’s Constants in H2O (1) + CF3CH2OH (2) Mixtures at 298.15 K and 101.33 kPa Partial Pressure of Gas gas

C0

He Ne Ar Kr Xe

-2.159 -2.633 -3.065 -3.124 -3.685

C1

C2

C3

D1

0.4611 1.029 1.404 -0.9443 3.078 -1.709 3.469 -1.578 5.325 -1.994 -2.093 4.215 -1.643

D2

σ

0.7520 0.6213 1.1626 0.6830

0.039 0.017 0.016 0.014 0.024

Actually, the excess Henry’s constants can be considered as a measure of the deviation from the Henry’s constants predicted by the expression 2

(18)

Di(1 - 2x2)i ∑ i)0

The coefficients Ci and Di and the standard deviations are gathered in Table 6. Plots of the excess Henry’s constants and their fitting curves are shown in Figure 6. The excess Henry’s constants are negative for all of the gases, except in the zone very rich in water (x2 < 0.1), where they are slightly positive. The greater the gas size, the lower the minimum, which occurs at x2 ) 0.4 in all cases. Negative excess Henry’s constants imply that the solubilities are greater than those expected if the behavior of the gas in the mixtures were described by a linear variation of the natural logarithms of the Henry’s constants when passing from a pure compound to the other. The opposite is true for positive values.

xi ln KH3,i ∑ i)1

ln KH3,12 )

(19)

Average deviations for a given system were calculated through the equation

average deviation )

1

exp pred ln KH3,12 - ln KH3,12

∑| n

ni)1

exp ln KH3,12

|

× 100 (20)

where n is the number of experimental points and the superscripts exp and pred refers to the experimental and predicted values, respectively. Values for the solubilities of the noble gases in the water-TFE mixture predicted by Krichevsky’s method7 show the following average percent deviations: 1.6% for He, 1.9% for neon, 2.2% for argon, 2.4% for krypton, and 2.6% for xenon. Thus, very good agreement is obtained.


Moreover, the positive values of the last property remain unexplained. Finally, according to the result of O’Connell and Prausnitz,20 the excess Henry’s constant must have the same value regardless of the gas considered; obviously, however, this is not the case. These disagreements could be attributed to the fact that the oneparameter Margules expansion is too simple, so the relation between the excess Henry’s constant and the molar excess Gibbs energy of the solvent liquid mixture is probably more complex than expressed by eq 22. Moreover, the existence of ternary contributions is a possibility that should be taken into account. It must be pointed out that equations other than the Margules one considered have been tested,37 but in general, no improvement has been achieved. In reference to the effect of the nonideality of the binary liquid solvent on the solubilities of gases, Shulgin and Ruckenstein8 recently proposed the following expression for predicting the Henry’s constant in ternary systems SR ln KH3,12 ) ln KH3,1(ln V12 - ln V2) + ln KH3,2(ln V1 - ln V12) ln V1 - ln V2 (23)

E Figure 6. Excess Henry’s constants, ln KH3,12 , versus the mole fraction of TFE, x2, in the solvent liquid mixture H2O (1) + TFE (2) at 298.15 K, along with the corresponding fitting curves for the noble gases: (9) He, (b) Ne, (2) Ar, (1) Kr, and ([) Xe. The dashed curve shows the values of the excess molar Gibbs energy, GE, with the sign changed for the binary mixture H2O (1) + TFE (2).

On the other hand, eq 19 is equivalent37 to ∞ ln γ3,12 )

2

∞ xi ln γ3,i ∑ i)1

(21)

The substitution20,37 of different expressions for the activity coefficients in eq 21 has permitted several formulas for the excess Henry’s constant to be obtained. For example, O’Connell and Prausnitz20 demonstrated that the following equation is fulfilled E )ln KH3,12

GE12 RT

(22)

when the excess Gibbs energies for all of the binary mixtures (1-2, 1-3, and 2-3) involved in the gasliquid equilibrium are considered to obey a oneE is the molar exparameter Margules expansion. G12 cess Gibbs energy for the binary mixture of the liquids. This equation indicates that the deviation from ideality of the liquid mixed solvent has a relevant effect on the behavior of gas-liquid systems. E for the mixture water-TFE23 is also repre-G12 sented in Figure 6. It can be observed that there is qualitative agreement between this curve and those of E the excess Henry’s constants. -G12 is negative, as E occurs for the main range of ln KH3,12, and its minimum lies at x2 ) 0.35, very near of the minima of the excess Henry’s constants. However, the numerical values are not coincident, as E . the absolute values are quite a bit greater for ln KH3,12

where Vi is the molar volume of pure component i and the superscript SR indicates the authors. The derivation of eq 23 was carried out on the basis of the KirkwoodBuff formalism assuming that the nonideality of the binary liquid mixture is almost negligible when compared with the nonidealities of the gas-pure liquid systems. Thus, the nonidealities of the gas-pure liquid mixtures would play the primary role in the behavior of the ternary mixture. The logarithms of the Henry’s constant calculated through eq 23 can be found in Table 4, along with their corresponding average percent deviations determined through eq 20. According to its authors,8 better results are expected from eq 23 than from eq 19. As has been stated, eq 19 is equivalent to eq 21, which is surely valid when the ternary and binary gas-pure liquid mixtures are ideal. This would imply, according to Shulgin and Ruckenstein,8 that the nonidealities of the gas-pure liquid systems are neglected. Given the important role of such nonidealities, worse predictions should be obtained with eq 19. However, the deviations are greater for eq 23, as was already observed for other systems.8 This is not as strange as it seems. It is true that eq 20 is rigorously fulfilled when all of the mixtures involved are ideal. However, inspection of eq 20 shows that it states that the nonideality of the ternary system ∞ ) comes exclusively from the nonidealities of the (γ3,12 ∞ ). This is confirmed by gas-pure liquid mixtures (γ3,i the result in eq 22 that clearly indicates that the major source of the deviations between experimental and predicted values is the nonideality of the binary liquid solvent, i.e., the nonideality that has not been considered in the prediction. Therefore, the nonidealities of the gas-pure liquid mixtures are not neglected in the Krichevsky’s method7 (rather, the opposite is true), and this and Shulgin and Ruckenstein’s8 approaches are both grounded on very similar assumptions. This would explain why Krichevsky’s method, although not as well founded as that of Shulgin and Ruckenstein, would perform better in certain cases.

1448


To complete our discussion of prediction methods, it would be illustrative to consider a very different approach, namely, the scaled particle theory.2,9-11 Basically, this is a model that assumes that the solution process can be divided into two steps: (a) formation in the liquid solvent of a cavity of a size suitable to accommodate the atom or molecule of the gas solute and (b) insertion of the gas solute into the cavity, followed by its interaction with the surrounding solvent. The Henry’s constant is given by the expression

ln KH3,12 )

hi ∆G h c ∆G RT + + ln RT RT V12

(24)

where ∆G h c and ∆G h i are the changes in the partial molar Gibbs energies for the steps of formation of the cavity and interaction, respectively. ∆G h c corresponds to the creation of a cavity in a liquid binary mixture constituted by hard spheres and is related to the hard-sphere diameter.38 This diameter is identified with the distance parameter in the Lennard-Jones potential, σi. For nonpolar gases, ∆G h i is the sum of two terms,11 each of which accounts for a class of interaction, namely, dispersion-repulsion and dipole-induced dipole. The interaction step is related to the energy parameter of the Lennard-Jones potential, i/k. The values of σi and i/k for the pure components that are necessary for these calculations were obtained from the literature11 for the noble gases and for water. For TFE, they were determined in a previous paper5 from gas solubility data. The polarizabilities of the gases and the dipole moment of TFE can be found in an earlier paper,4 and the dipole moment used for water, µ ) 1.84 Debyes, is that used by Pierotti39 in the original application of SPT for solubilities in water. The values 0 , along with the predicted by the theory for ∆G h 3,sol corresponding average deviations estimated by means of eq 20, are included in Table 4. It can be seen that the deviations are not as small as those determined for the preceding methods, although, for xenon, the deviation is comparable to that of the Shulgin-Ruckenstein method. The highest discrepancy appears for neon in an usual feature of the SPT model. These deviations are, in general, also greater than those corresponding to both TFE (2.9% for He, 8% for neon, 5.9% for Ar, 4.4% for krypton, and 2.7% for xenon) and water (0.8, 2.1, -9 × 10-4, -9 × 10-3, and -6 × 10-3%, respectively). However, it can be observed that the difference with respect to pure TFE is not important. This is a remarkable achievement if we take into account the fact that SPT assumes that the properties of water and TFE do not undergo any modification when passing from the pure state to the mixed state, an assumption that is quite unrealistic. These good results are corroborated by the values of the excess Henry’s constants calculated using the Henry’s constants provided by the SPT for the gases in both mixed solvent and pure liquids. The excess Henry’s constants so determined occupy the last column in Table 4. The experimental and SPT results are plotted in Figure 7 for neon and krypton, which were selected to represent the behavior observed. For all of the gases, the excess Henry’s constants predicted by the SPT are negative and increase with the size of the gas, and the minima coincide well with those obtained experimentally, although the values predicted by the theory are always smaller. The positive

E Figure 7. Excess Henry’s constants, ln KH3,12 , versus the mole fraction of TFE, x2, in the solvent liquid mixture H2O (1) + TFE (2) at 298.15 K for the gases Ne (s, experimental; ‚‚‚, SPT) and Kr (- - -, experimental; - ‚ - ‚ -, SPT).

E values of ln KH3,12 in the zone very rich in water are not reproduced by the SPT. Indeed, in precisely this zone, the SPT-predicted excess Henry’s constants exhibit negative values greater than those expected in such a way that the entire predicted curve seems to be shifted to the left if we take into account that it must include the point (0,0). To summarize, good qualitative agreement is obtained between the results of the SPT and experiment, with the shortcomings of the model for the pure solvents apparently providing the main source of deviation. Actually, the trend of the deviations for the solubilities in the water-TFE mixture seems to follow the pattern marked by the solubilities in TFE. The somewhat greater discrepancies can be attributed to the fact that the SPT considers the effects of the solvents to be additive. Nevertheless, these conclusions are provisional because they are only valid for the system under study. It would be necessary to apply the theory to more ternary systems to arrive at more reliable conclusions about the general performance of the SPT.

Conclusion In this paper, the solubilities of noble gases in the liquid mixture water-trifluoroethanol have been reported, and the Gibbs energies for both the solution and the solvation processes have been calculated. To calculate solubilities expressed in terms of mole fractions, a method has been developed that could be adapted to other experimental procedures based on the measurement of gas- and liquid-phase volumes. This method takes into account the nonideal behavior of the gas and the influence of the vapor-liquid equilibrium of the solvent mixture on the solubility.


The linear relation between the Gibbs energy for the solution process and the depth of the potential well of the gases, 3/k, allows the solubility of any noble gas in water-TFE mixtures to be determined if the solubility of one of the gases in that liquid mixture is known. This is true because the slopes of the straight lines in plots 0 vs 3/k are almost identical for the pure of ∆G h 3,sol liquids water and TFE. Consequently, the slopes are also very similar over the entire composition range of their mixtures. The Gibbs energies for the solvation process in the studied systems have been compared with those corresponding to solutions of the same gases in mixtures of water and ethanol. The comparison leads to the conclusion that the hydrophobic effect of the TFE seems to be much less significant than that of the alkanol. On the other hand, the so-called excess Henry’s constants have been calculated, and the Henry’s constants have been estimated by means of an equation proposed by Shulgin and Ruckenstein.8 A comparison of the two sets of results leads to the conclusion that, despite their different origins, the two prediction methods include very similar assumptions about the nonidealities of the mixtures involved in the solubilities of gases in binary solvents. Finally, the scaled particle theory was also applied, yielding quite acceptable results. Acknowledgment The authors gratefully acknowledge the Direccioń General de Investigacioń Cientı´fica y Tećnica (DGICYT) for its financial support through Project PB98-1624. Nomenclature Latin Letters ai ) coefficients in eq 13 Ai ) coefficients in eq 14 B ) second virial coefficient Ci, Di ) coefficients in eq 18 G ) Gibbs energy KH ) Henry’s constant n ) number of moles P ) total pressure Ps ) vapor pressure P ) Poynting correction R ) gas constant T ) temperature of the water bath T′ ) temperature of the air bath V ) volume x ) mole fraction in the liquid phase y ) mole fraction in the liquid phase Greek Symbols R ) polarizability ∆ ) change of property ) energy parameter in the Lennard-Jones potential φ ) fugacity coefficient γ ) activity coefficient according to ideality in the sense of the Lewis-Randall rule γ′ ) activity coefficient in the sense of Henry’s law µ ) chemical potential and also dipole moment F ) density σ ) distance parameter in the Lennard-Jones potential and also standard deviation Superscripts exp ) experimental value

E ) excess property L ) liquid pred ) predicted value V ) vapor 0 ) standard state ∞ ) infinite dilution * ) pure component Subscripts bin ) binary mixture c ) step of creation of the cavity i ) step of interaction of solute-solvent m ) mixture sol ) solution solv ) solvation ter ) ternary system 1, 2 ) liquid components 3 ) gas component Abbreviations GLC ) gas liquid chromatography SPT ) scaled particle theory TFE ) 2,2,2-trifluoroethanol VLE ) vapor-liquid equilibrium

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Received for review May 3, 2002 Revised manuscript received December 13, 2002 Accepted December 20, 2002 IE020329O

Solubility of Gases in Binary Liquid Mixtures: An Experimental and

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