PREDICTION OF MUTUAL SOLUBILITY AND INTERFACIAL TENSION IN TWO-PHASE LIQUID SYSTEMS

PREDICTION OF MUTUAL SOLUBILITY AND INTERFACIAL TENSION IN TWO-PHASE LIQUID SYSTEMS Wiesław APOSTOLUK Institute of Inorganic Chemistry and Metallurgy of Rare Elements, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland e-mail: apostoluk@ichn.ch.pwr.wroc.pl ABSTRACT The mutual solubility (expressed in mole fractions) in two-phase liquid systems water/organic solvent has been discussed applying the Kamlet-Taft and Abraham models. The model of Abraham has also been found as a convenient tool for the prediction of interfacial tension in other systems involving organic solvents and such bipolar liquids as glycerol, 1,2- etanediol (ethylene glycol) and formamide. INTRODUCTION The mutual solubility in two-phase liquid systems affects the properties of both phases (e.g. their density and surface tension) and the interfacial tension [1-4]. The equilibrium between both mutually saturated phases in the system water (W) and organic solvent (S) can be written as follows: W (w) + S (s) = W (s) + S (w) (1) The corresponding equilibrium constant (K W/S ) can be defined by means of the mutual solubilities expressed in mole fractions x W,s and x S,w respectively: x W,s xs,w KW / S = (2) (1 x W,s ) (1 xs,w ) Both liquid phases in the systems W/S could be treated separately: (i) the water rich phases as solutions of different solutes in water as solvent; (ii) the organic solvent rich phases as solutions of water in different solvents. As a result, the linear solvation energy relationships (LSER) based on the empirical models of solute-solvent interactions could be adequate to describe the solubility of solvents in water and water in solvents, respectively. For

instance, the mutual solubilities and interfacial tension in 12 W/S systems have been correlated with such property parameters of solvents as E T of Dimroth and Reichardt, DP of Shmidt and π* of Kamlet and Taft [5]. From this point of view, however, the model of Abraham of solute effects [ 6] seems to be appropriate in the first case while the application of the Kamlet and Taft model of solvent effects [7] is reasonable in the second. In the previous paper of Apostoluk and Szymanowski [8] it has been shown that the mutual solubilities of water (x W,s ) and organic solvent (x S,w ) in 26 twophase liquid W/S systems could be used as a descriptors which allow to estimate the interfacial tension (γ) in W/S systems with aliphatic and aromatic hydrocarbons, chloro- and nitrohydrocarbons, aliphatic ethers and ketones with better results than those obtained by means of the Donahue and Bartell approach [3]. From the final correlation which is also valid for the systems with higher alcanols and aniline: γ (3) = 7.97 2.10 log xs,w 14.35 log x W,s it follows that the contribution of water solubility in the solvent rich phase predominates significantly over the contribution of solvent solubility in the water rich phase. Starting from such assumption, Apostoluk and Szymanowski [8] have demonstrated that the interfacial tension in W/S systems studied could be correlated with the empirical property parameters of solvents involved in the Kamlet-Taft model of solvent effects. The same assumption appears in the paper of Lee [9] who has correlated the Lewis acid base surface interaction components with the Kamlet and Taft parameters α and β describing the hydrogen-bond donating and accepting abilities of solvents, respectively. Surprisingly, Freitas and coworkers [10] have found that the work of adhesion in 103 systems W/S could be correlated with the solute property parameters involved in the model of Abraham. These authors have argued that always the application of solute property parameters of solvents dissolved in the water rich phase gives a better fitting of the work of adhesion in W/S systems than that obtained with the property parameters of bulk solvents. It should be pointed out that the model of Kamlet and Taft [7,11] involves when it is necessary the property parameters of organic liquid solutes [14-16]. For non-associating liquids these parameters are equal to those of bulk solvents. In the case of selfassociated liquids the parameters of their monomers (m) differ significantly from the corresponding bulk parameters [7,11,14-16]. The few representative examples of such solvents are given in Table 1. The main goal of the present work is to compare the Abraham [6] and Kamlet-Taft [11] models of solute-solvent interactions applied for the prediction of a mutual solubility in the systems W/S and optimize the estimation of interfacial tension in W/S and other two-phase liquid systems.

Table 1. Parameters of the Kamlet and Taft model for self-associated solvents and their monomers (m) [7,16] Solvent Dipolarity/polarizability HBD-strength HBA-strength π* π * m α α m β β m Water 1.09 0.39 1.17 0.32 0.47 0.15 1-butanol 0.47 0.40 0.84 0.33 0.84 0.45 cyklohexanol 0.45 0.45 0.66 0.31 0.84 0.51 1-octanol 0.40 0.40 0.77 0.33 0.81 0.45 heptanoic acid 0.50 0.50 1.20 0.55 0.45 0.45 phenol 0.72 0.72 1.65 0.61 0.30 0.33 aniline 0.73 0.73 0.26 0.16 0.50 0.50 MODELS The model of Kamlet and Taft of solute-solvent interactions [11] applied for the description of a property P changing in the series of solutions is a linear combination of several contributions of the property parameters of solvents and monomeric molecules (m) of solutes, respectively: P = f ( π *, α, β, δ2, * H π m, αm, β m, Vx, m ) (4) where parameter π* describes the polarity/polarizability effects, parameters α and β are the hydrogen-bond donating and accepting abilities of solvents and solutes (m), respectively. The square of Hildebrand solubility parameter (δ H ) stands for the cohesive energy density of solvents [12] and V x,m denotes the molar intrinsic volume of solute monomers [13]. The last two parameters describe the cavity formation effects. It is important that for non-self associated liquids π* = π * m, α = α m, and β = β m. [11]. The model of Kamlet and Taft involving the property parameters of solutes was successfully used to predict the solubility of organic compounds in water [14] and their distribution in the system water/octanol [15] and other W/S systems [16]. The main drawback of this model is the application of a single parameter π* for the polarity/polarizability effects and fact that for a certain processes the introduction of an additional parameter δ is necessary for the proper

description of solvent polarizability effects [11]. In the two-phase liquid systems W/S the property parameters of net solvents could be used as long as the mole fraction of water in the solvent rich phase is lower than 0.13 [17]. However, there are some indications that this limitation could be omitted if the linear combination of π* and π* 2 and product of α and β parameters is used instead of the cohesive energy density [18]. The similar treatment could be related to the corresponding parameters of solutes. As a result, the modified model is as follows: and/or P = f ( π *, π*2, α, β, αβ, Vx, m ) (5) P = ϕ( π *, * 2 m π m, αm, βm, αm β m, Vx, m ) (6) The model of Abraham [6] used for the analysis of a different phenomena and processes which occur in the systems with condensed phases involves the linear combination of five solute property parameters: H P = f (R,, H, H 2 π2 Σα Σβ, Vx, m ) (7) 2 2 where R 2 is an excess molar refraction of solute, π H 2 denotes the solute dipolarity/polarizability, ΣαH 2 and Σ βh 2 stand for the effective hydrogenbond acidity and basicity of solute, respectively. The molar intrinsic volume V x,m of solute has the same meaning as in the model of Kamlet and Taft. However, in water solutions and W/S systems the molar intrinsic volume of solutes could also be interpreted as a primal measure of their hydrophobicity [13]. CHOICE OF THE DATASET AND COMPUTING The mutual solubilities of water and organic solvents have been taken from the Marcus monograph [19] and the paper [4]. The data for n-decane and n-dodecane [19] have been excluded since the mole fractions of water in these hydrocarbons is extremely small in comparison with n-hexadecane. Those for 2-ethylhexanol and 2-ethyl- hexanoic acid have been taken from Ref. [20]. The property parameters of solvents in the Kamlet and Taft model have been taken from the review and monograph of Marcus [7,19]. The solubility parameters of solvents have been taken from the book of Barton [12]. The molar intrinsic volumes of organic compounds can be easily calculated from their structure [13] or found elsewhere [21]. Except for 1- bromonaphtalene, the property parameters of organic compounds involved in the Abraham model have been taken from the papers [6,21-23]. The excess refraction of 1-bromonaphtalene has been calculated in accordance with Ref. [24] while its dipolarity/polarizability and effective hydrogen-bond basicity

have been evaluated applying the same rules of estimation as those used by Kamlet et al. [15]. The values of interfacial tension in the different W/S systems have been taken from Ref. [4,10,25]. The comparison of four systems involving 14 apolar solvents and water (W), etanediol (E), glycerol (G) and formamide (F) as a second polar liquid phase has been made applying the original data of Jańczuk and coworkers [26]. Both models described above have been used. The calculations have been carried out by means of the multiple regression analysis. The assessment of statistical validity of the correlations derived has been done using the values of determination coefficient (R 2 ), standard deviation of a regressed property P (s.d.) and test function F of Snedecor-Fisher (F-statistics). N always denotes the number of experimental points. RESULTS AND DISCUSSION Water/organic solvent systems Application of the Kamlet and Taft (Eqs. (5) and (6) and Abraham models for the analysis of a mutual solubility in the W/S systems leads to the results presented in Table 2. All correlations derived are of relatively low statistical importance. However, one can conclude that in all cases there are no significant statistical differences between the correlations obtained by means of both forms of Kamlet and Taft model as well as of the Abraham model for logarithmic values of x Ws and x Sw, respectively. Taking into account the correlations for log K W/S it is evident that both forms of the Kamlet and Taft model give better results than the model of Abraham. The mole fraction of water in solvent rich phase increases with increasing hydrogen-bond donating and accepting abilities of solvent molecules, however, the contributions of later terms are always of a greater statistical significance and prevail over the corresponding α- or Σ αh 2 -terms. This conclusion could be easily explained since the monomers of water are rather weak hydrogen-bond acceptors in comparison with momeric molecules of alcohols, carboxylic acids, phenols and amines. Therefore, the monomers of water in solvent rich phase exhibit always in a greater extent the function of hydrogen-bond donors than hydrogen-bond acceptors. The same can be said about the interactions of bulk water with monomers of solvents in water rich phase. The contribution of molar intrinsic volumes of solvents is always negative. It reflects the known fact that in W/S systems with the homologous series of solvents, the mutual solubility of water and solvents as well as the equilibrium constant K W/S gradually decrease. The estimated logarithmic values of increments per one methylene group for x Ws, x Sw and K W/S are approximately equal to 0.08, -0.46 and -0.56, respectively.

The correlations presented below are valid for the interfacial tension: γ = (42.79±2.01) (14.44±5.03) π * m. + (10.54±5.43) π * 2 m (46.84±17.15)α m - (72.18±2.98)β m + (58.38±40.25)α m β m + (0.0705±0.0153)V x,m., (8) = 0.9517, s.d. = 4.17, F = 293, N = 90; deviations >3 s.d.: di(2- chloroethyl) ether (+3.21), ethyl formate (-3.39), carbon disulphide (+3.16). R 2 γ = (40.69±2.36) (12.20±5.52)π* + (6.58±5.99)π* 2 (38.57±3.94)α (70.29±3.34)β + (73.97±7.89)αβ + (0.0871±0.0184)V x,m, (9) R 2 = 0.9409, s.d. = 4.60, F = 224, N = 85; deviations >3 s.d.: di(2- chloroethyl) ether (+3.02), nitromethane (-4.39), carbon disulphide (+3.17). γ = (38.50±1.71) + ( 13.88±1.89)R 2 (19.06±2.47) π H 2 - (22.51±2.53)Σ α 2 H - (56.69±2.78)Σβ H 2 + (0.0873±0.0135)V x,m, (10) = 0.9519, s.d. = 4.00, F = 416, N = 106; deviations >3 s.d.: diiodomethane (+3.01), 2-chloro-2-methylpropane (-3.19). R 2 From the statistical point of view the obtained correlations describe fairly well the changes of interfacial tension in different W/S systems. The best fitting has been obtained with the model of Abraham. It should be noted, however, that the populations of W/S systems compared with the considered models differ each other. This fact can affect in a certain degree the observed differences in the fitting of the interfacial tension by these models. Note also, that the interfacial tension in the indicated systems deviates from the corresponding correlation more than 3 s.d. Theses deviations can be partly attributed to the non-correct values of interfacial tension measured in some systems. It is well known that the accuracy and precision of the interfacial tension measurements depend upon the purity of components of a system and the method used [27,28]. From this point of view, the interfacial tensions reported in Ref. [25] for W/S systems with 1-chloro-2-methylbutane (15.4 mn/m.) and with 1-chloro-2-methylpropane (24.4 mn/m) and 2- chloro-2-methylpropane (23.8 mn/m) seem to be too low in comparison with the surface tension in other W/S systems involving more polar and less hydrophobic solvents like dichloromethane (28.3 mn/m) and chloroform (32.8 mn/m). Therefore, the systems with both chloromethylpropanes should be excluded from the dataset and one can expect the improvement of the final correlations for the interfacial tension in W/S systems. As it has been always mentioned, the square of molar intrinsic volume, (V x,m ) 2, could be used as a fully empiric term describing the negative

deviations from the straight line dependences observed for homologous series of solutes [29-31]. Further calculations have been performed as follows: (i) the mentioned above systems with chloropropanes have been excluded; (ii) systems involving 1-hexene, 1-heptene, 1-octene, 1-nonene, 1- decene and 1-dodecene [33] have been added; (iii) the square of molar intrinsic volume of solvents has been used as an additional descriptor in the Abraham model. As it has been expected, the significant improvement of final correlation has been achieved. γ = (26.68±2.78) + (11.60±1.55)R 2 (16.88±2.02) π H 2 (22.43±2.02)Σ α 2 H - (60.06±2.28)Σ β 2 H + (0.316±0.045)V x,m (0.000945±0.000175)V 2 x,m, (11) R 2 = 0.9698, s.d. = 3.20, F = 584, N = 110. Comparison of W/S, E/S, G/S and F/S systems Further calculations have been performed applying the data for the interfacial tension in four systems with water (W), ethanediol (E), glycerol (G) and formamide (F) involving as a second liquid phase the same 14 solvents: i.e. 11 n-alkanes (from n-hexane to n-hexadecane), benzene, 1- bromonaphtalene and diiodomethane [26]. Again, the Abraham model has been found to fit better the experimental results in each system than both forms of the model of Kamlet and Taft. The correlations based on the Abraham model are presented in Table 3. Table 3. Correlations for estimation of the interfacial tension in different two-phase liquid systems [26] System Equation for γ R 2 s.d. F W/S γ = 50.92 + 40.87R 2 51.04π H 2 113.5Σβ H 2 + 0.00120 V x,m. 0.9950 0.37 513 G/S γ = 28.07 + 15.86R 2 22.71π H 2 51.35Σβ H 2 + 0.00160 V x,m. 0.9939 0.30 530 E/S γ = 14.36 + 19.41R 2 24.99π H 2 53.44Σβ H 2 + 0.00184 V x,m. 0.9957 0.21 760 F/S γ = 25.17 + 19.13R 2 23.66π H 2 72.90Σβ H 2 + 0.00149 V x,m 0.9592 0.82 78 Next step of calculations has been performed assuming that the parameters of Abraham model describe the interfacial tension in each considered system while the parameters of the Kamlet and Taft model describe the polar solvents in these systems. As a result, the interfacial tension in these systems could be quantitatively described with a single correlation:

γ = -(3.06±1.03) + (23.8±1.9)R 2 (30.6±2.2)π H 2 (72.4±8.0)Σβ H 2 + (0.0132±0.0040)V x,m + (261.8±3.7)α (462.7±7.4)αβ. (12) R 2 = 0.9915, s.d. = 1.18, F = 1069, N = 56. The quality of correlation (12) is excellent. On the other hand it predicts fairly well the interfacial tension in the systems water/squalane and formamide/squalane. The interfacial tensions in these systems reported by Fowkes and coworkers [34] are equal 52.3, 29.3 mn/m., respectively, while those predicted from correlation (12) are equal to 54.5, 30.8 mn/m. Introducing these two mentioned systems with squalane [34] to the data reported by Jańczuk et al. [26] we obtain the next correlation: γ = -(2.36±0.91) + (24.2±1.9)R 2 (30.9±2.2)π H 2 (76.1±7.5)Σβ H 2 + (0.00891±0.00251)V x,m + (261.1±3.7)α (461.4±7.3)αβ, (13) R 2 = 0.9916, s.d. = 1.18, F = 1127, N = 58. It should be also mentioned about the systems 1,3-propanediol/squalane and diethyleneglycol/squalane [34] which are formally similar to those investigated by Jańczuk et al. [26]. However, both reported values of interfacial tension in these systems [34] (equal to 17.3 and 12.4 mn/m., respectively) lead to the following result: γ = (766.7±48.8) + (24.3±1.8)R 2 (30.9±2.1)π H 2 (76.1±7.3)Σβ H 2 + (0.00883±0.00244)V x,m - (1094±37)π* + (681.1±23.4)π* 2 (419±52)α (634±79)β + (825±105)αβ, (14) R 2 = 0.9922, s.d. = 1.15, F = 834, N = 60. The quality of correlation (14) is even a slightly better than those of correlations (12) and (13). The decreased value of its F-statistics is affected by the lower number of the degrees of freedom than in the previous cases. It is interesting, however, that except for the molar intrinsic volumes, the contributions of other terms connected with the Abraham model parameters are quite the same as in the correlations (12) and (13). Aliphatic hydrocarbon/polar liquids systems It should be added that Fowkes et al. [34] have been also studied several other systems with squalane and polar organic liquids and reported the corresponding values of interfacial tension, work of adhesion and semiquantitative estimation of solubility of studied liquids in squalane. Nevertheless, these very interesting results have not been successfully

correlated with the parameters neither Abraham nor Kamlet and Taft models. Practically the same results have been obtained for the systems with n- tetradecane and water, glycerol, formamide, 1,3-propanediol, 1,2-ethanediol, N-methylformamide, 1,2-propanediol and dimethylformamide for which the interfacial tensions have been compiled by Israelachvili [35]. It means that the problem of application of considered models for the prediction of interfacial tension and work of adhesion in the two-phase liquid systems seems to be more complicated than it has been previously demonstrated for W/S systems by Apostoluk and Szymanowski [8] as well as Freitas et al. [10]. As a result, further studies are necessary to clarify some crucial questions, concerning for instance the systems with squalane [34]. Having in mind, however, that the model of Abraham fits reasonably well the work of adhesion in W/S system and the interfacial tension in W/S, G/S, E/S F/S systems it is probable that a new sets of data should also be organized in the similar way. In other words, the two-phase liquid systems with the same polar phase should be selected for further calculations and tests of different models. CONCLUSIONS 1. It has been demonstrated that there is no any significant difference in the prediction of mutual solubilities in W/S systems applying the appropriate modifications of the Kamlet and Taft model with property parameters of bulk solvents and/or with the property parameters of their monomeric molecules. The same results of prediction of mutual solubilities in W/S systems have been obtained using the model of Kamlet and Taft with the parameters of bulk solvents and applying the Abraham model with property parameters of monomeric molecules of organic solvents. 2. Both forms of the Kamlet and Taft model involving the property parameters of bulk solvent and/or the property parameters of their monomeric molecules are suitable for the prediction of interfacial tension in W/S systems. However, the application of Abraham model is more reasonable since the results of prediction of interfacial tension in W/S systems are better than those obtained with the Kamlet and Taft model. It has been demonstrated that application of both models permits to predict the interfacial tension with good results in the two-phase liquid systems involving such polar solvents as water, glycerol, ethanediol and formamide as a first liquid phase and with alkanes, benzene, 1- bromonaphatalene and diiodomethane as a second one. 3. There is no any satisfactory correlation which predicts fairly well the interfacial tension in the systems aliphatic hydrocarbon/polar organic solvents applying the parameters of the Kamlet and Taft and Abraham models.

REFERENCES 1. H. M., Backes, Jing Jung Ma, E. Bender, G. Maurer, Chem. Eng. Sci., 45, 275-286, 1990. 2. D. Y. Kwok, W. Hui, R. Lin, A. W. Neumann, Langmuir, 11, 1669-2673, 1995. 3. D. J. Donahue, F. E. Bartell, J. Phys. Chem., 56, 480-484, 1952. 4. Fu Jufu, Li Buqiang, Wang Zhihao, Chem. Eng. Sci., 41, 2673-2679, 1986. 5. E. A. Mezhov, N. L. Khananashvili, V. S. Shmidt, Radiokhimya, 30, 278-281, 1988. 6. M. H. Abraham, Chem, Soc. Rev., 22, 73-83, 1993. 7. Y. Marcus, Chem. Soc. Rev., 22, 409-416, 1993. 8. W. Apostoluk, J. Szymanowski, Solvent Extr. Ion Exch., 14, 635-651, 1996. 9. L.-H. Lee, Langmuir, 12, 1681-1687, 1996. 10. A. A. Freitas, F. H. Quina, F. E. Carroli, J. Phys. Chem. B, 101, 7488, 1997. 11. R. W. Taft, J.-L. Abboud, M. J. Kamlet, M. H. Abraham, J. Solution Chem., 14, 153-185, 1985. 12. A. F. M. Barton, Handbook of Solubility Parameters and Other Cohesion Parameters, CRC Press, Boca Raton, FL, 1983. 13. J. C. McGowan, J. Chem. Tech. Biotechnol., 58, 357-361, 1993. 14. M. J. Kamlet, R. Doherty, M. H. Abraham, P. W. Carr, F. Doherty, R. W. Taft, J. Phys. Chem., 91, 1996-2004, 1987. 15. M. J. Kamlet, R. M. Doherty, M. H. Abraham, Y. Marcus, R. W. Taft, J. Phys. Chem., 92, 5244-5255, 1988. 16. Y. Marcus, J. Phys. Chem., 95, 8886-8891, 1991. 17. Y. Marcus, Solvent Extr. Ion Exch., 10, 527-538, 1992. 18. S. C. Rutan, P. W. Carr, R. W. Taft, J. Phys. Chem., 93, 4292-4297, 1989. 19. Y. Marcus, The Properties of Solvents, Wiley, Chichester, 1998. 20. B. P. Whim, P.G. Johnson, Directory of Solvents, Chapman & Hall, 1996. 21. M. H. Abraham, J. Andonian-Haftvan, G. S. Whiting, A. Leo, J. Chem. Soc. Perkin Trans. 2, 1777-1791, 1994. 22. M. H. Abraham, J. Phys. Org. Chem., 6, 660-684, 1993. 23. M. H. Abraham, A. J. Leo, J. Chem. Soc. Perkin Trans., 1839-1842,1995. 24. M. H. Abraham, G. S. Whiting, R. M. Doherty, W. J. Shuely, J. Chem. Soc. Perkin Trans. 2, 1451-1460, 1990. 25. R. J. Good, E. Elbing, Ind. Eng. Chem., 62, 54-78, 1970 26..B. Jańczuk, W. Wójcik, A. Zdziennicka, J. Colloid Interface Sci, 157, 384-393, 1993. 27. J. Szymanowski, K. Prochaska, J. Colloid Interface Sci., 123, 456-, 1988. 28. J. Szymanowski, G. Cote, I. Blondet, C. Bouvier, D. Bauer, J. L. Sabot, Hydrometallurgy, 44, 163-178, 1997. 29. W. Apostoluk, J. Szymanowski, Colloids Surf. A, 135, 227-234, 1998. 30. W. Apostoluk, J. Szymanowski, Anal. Chim. Acta, 374, 137-147, 1998. 31. W. Apostoluk, B. Gajda, J. Szymanowski, M. Mazurkiewicz, Anal. Chim. Acta, 405, 321-333, 2000. 32. C. J. van Oss, Interfacial Forces in Aqueous Media, Marcel Dekker, 1994. 33. S. Nakahara, H. Masamoto, Y. Arai. J. Chem. Eng. Jpn., 23, 94-95, 1990. 34. F. M. Fowkes, F. L. Riddle Jr., W. E. Pastore, A. E. Weber, Colloids & Surfaces, 43, 367-387, 1990. 35. J. N. Israelachvili, Intermolecular and Surface Forces, Second Ed., Academic Press, 1992.

Table 2. Correlations for mutal solubilities in W/S systems obtained in terms of the Kamlet-Taft and Abraham models P Equation for log P R 2 s.d. F N -2.74 + 1.81 π * m 1.34 π * 2 m + 2.05α m + 3.31β m 0.00543V x,m 0.8758 0.39 149 106 x Ws -2.83 + 1.61π* - 0.86π* 2 + 1.68α + 3.27β 2.66αβ 0.00473V x,m 0.8767 0.36 119 101-2.42 0.21R 2 + 0.66 π 2 H + 2.05Σ α 2 H + 3.22Σβ 2 H 0.00694V x,m 0.8748 0.36 149 107-2.12 + 2.46 π * m 2.19 π * 2 m + 2.72α m + 4.19β m 4.98α m β m 0.0330V x,m 0.8952 0.45 180 127 x Sw -2.13 + 2.57π* - 2.18π* 2 + 1.91α + 4.12β 4.46αβ 0.0330V x,m 0.8954 0.46 174 122-1.62 0.37R 2 + 0.74 π 2 H + 1.01Σ απ H 2 + 3.87Σβ H 2 0.0353V x,m 0.8863 0.46 202 130-4.70 + 4.09 π * m 3.56 π * 2 m + 5.66α m + 7.80β m. 6.07α m β m 0.0401V x,m 0.9243 0.65 215 106 K W/S -4.83 +3.96π* - 2.95π* 2 + 3.98α + 7.86β - 7.74αβ 0.0391V x,m 0.9211 0.68 196 101-3.86 0.51R 2 + 1.11 π 2 H + 3.36Σ α 2 H + 7.67Σβ 2 H 0.0441V x,m 0.9072 0.73 208 107