Determination of Infinite-Dilution Partial Molar Excess Entropies and

Reliable values of partial molar excess entropies and partial molar excess enthalpies at infinite dilution of short-carbon-chain alkanes diluted in ...
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Ind. Eng. Chem. Res. 2003, 42, 4927-4938

4927

Determination of Infinite-Dilution Partial Molar Excess Entropies and Enthalpies from the Infinite-Dilution Activity Coefficient Data of Alkane Solutes Diluted in Longer-Chain-Alkane Solvents Satoru Kato,* Daisuke Hoshino, Hidetaka Noritomi, and Kunio Nagahama Department of Applied Chemistry, Graduate School of Engineering, Tokyo Metropolitan University, 1-1 Minamiohsawa, Hachiohji 192-0397, Japan

Reliable values of partial molar excess entropies and partial molar excess enthalpies at infinite dilution of short-carbon-chain alkanes diluted in long-carbon-chain alkanes are required for the critical verification of conventional activity coefficient models. They are scarce in the literature because of the lack of pertinent criteria for the selection of extensive data for infinite-dilution activity coefficients, γi∞. The criteria for the selection of literature data for ln γi∞ have been established, and reliable values of partial molar excess entropies and partial molar excess enthalpies at infinite dilution have been determined. It has been shown that both are positive and that each quantity is correlated as a linear function with a dispersion force parameter. It has been demonstrated that only a quite limited part of each quantity is accurately predicted by conventional activity coefficient models. 1. Introduction Infinite-dilution activity coefficients of a solute i, γi∞, for alkane/alkane binary systems are of practical importance for the rational design of separation devices in chemical plants and effective treatments of oils in petroleum engineering, where, when γi∞ is combined with a suitable activity coefficient model derived from thermodynamics, a relationship describing phase equilibria can be established. They are also important as fundamental properties for the prediction of Henry’s constants and liquid-liquid partition coefficients.1 For the alkane/alkane binary systems, it is a prominent simplicity that the dispersion force is the only attractive molecular interaction acting on alkane molecules. Therefore, the infinite-dilution activity coefficients of alkane solutes dissolved in alkane solvents are of profound theoretical interest in thermodynamics for testing activity coefficient models or solution models. On the basis of these insights, a great number of experimental data for γi∞ in the alkane/alkane systems have been reported in the literature. Because of the difficulty involved in the measurement of γi∞ for an alkane solute having a long carbon chain diluted in alkane solvents having short carbon chains, the amount of data for γi∞ of heavy molecular weight alkanes in light alkanes is limited, whereas much work has been done to elucidate the characteristics of ln γi∞ for short-carbon-chain alkanes infinitely diluted in long-carbon-chain alkanes; a great number of experimental data are found in the literature. When the carbon number of the solute i, Ni, is close to that of the solvent j, Nj, the ideal solution theory and the regular solution theory have been used for the evaluation of the experimental data.2,3 When Ni , Nj or Ni . Nj, the partial molar excess entropy term of the solute is predominant in the partial molar excess energy of the solute, where either the Flory-Huggins combi* To whom correspondence should be addressed. Tel: 81426-77-2824. Fax: 81-426-77-2821. E-mail: kato-satoru@ c.metro-u.ac.jp.

natorial entropy4-9 or the UNIQUAC combinatorial term9-11 has been used for the expression of ln γi∞. In the case where the partial molar excess enthalpy of the solute has a great effect, the parameters accounting for molecular interactions between a solute and a solvent are introduced and determined from regression with experimental data; both the Flory-Huggins and the UNIQUAC models have been used for this purpose.2,4-7,10,11 These activity coefficient models successfully correlate the experimental data for γi∞. However, it has been clearly recognized that the precision in the prediction of ln γi∞ is inadequately low. The main reason for the low reliability in the prediction procedures may arise from the lack of data, which enables us to evaluate the validity of each solution model; the accurate values of the partial molar excess entropy of solute i at infinite dilution, siE,∞, and the partial molar excess enthalpy of solute i at infinite dilution, hiE,∞, have not been established, although a great many of γi∞ have been reported. The lack of the reliable values of siE,∞ brings us these disadvantages that the combinatorial entropies from conventional solution models, such as the Flory-Huggins and the UNIQUAC models, cannot be critically evaluated. On the other hand, the decisive lack of accurate values of hiE,∞ prevents us from estimating the magnitude of solute-solvent molecular interactions and determining the valid interaction parameters involved in the solution models. Therefore, at the moment, we must allow for some amount of prediction error arising not only from the inadequate expression involved in the enthalpy terms but also from that in the combinatorial entropy term. These awkward failures have been persistently involved in the conventional solution models. Except for highly solvated and associated solutions, the values of hiE,∞ and siE,∞ should be constant over moderate temperature ranges. Following this insight, there have been many attempts to determine the values of hiE,∞ from the slopes of the plot of ln γi∞ linear in reciprocal temperature, 1/T, for these solutes from hydrocarbons to polar substances infinitely diluted in such solvents as hydrocarbons, halogenated compounds,

10.1021/ie030129s CCC: $25.00 © 2003 American Chemical Society Published on Web 09/04/2003

4928 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

ketones, sulfides, acetates, and alcohols.12-15 In these analyses, experimental errors included in ln γi∞ have a critical effect on the certainty of hiE,∞. Trampe and Eckert12 demonstrated that the uncertainty in these values is estimated as follows: for hiE,∞ ) 10 000 J/mol or greater, the uncertainty is 10-20%; for hiE,∞ ) 200010 000 J/mol, the uncertainty is 20-40%; for hiE,∞ ) 2000 J/mol or less, the uncertainty is more than 40%. In the short-carbon-chain alkane/long-carbon-chain alkane systems, the values of hiE,∞ are less than 500 J/mol;16 therefore, the uncertainty is much more noticeable than 40%. In fact, negative values of hiE,∞ were reported for C6/C16, C7/C16, and C8/C16 systems,17 while positive values of 359 and 231 J/mol for C6/C16 and C7/ C16 have also been reported.16 On the other hand, hiE,∞ can also be determined from the limiting slope of heat of mixing calorimetric data. For C6/C16 and C7/C16 at 298.15 K, values from 356 to 464 J/mol have been reported.16,18-20 Unfortunately, none of the calorimetric measurements provides accurate values of the limiting slopes of heat of mixing at infinite dilution because experimental errors arising from the calorimetric measurement become noticeable when the solute approaches infinite dilution. Clearly, both the method of slope analysis and most calorimetric methods in existence are not accurate enough for a hiE,∞ determination, although the direct measurements by calorimetric methods have provided positive values of hiE,∞. Meanwhile, Kato et al.21 collected literature data for ln γi∞ of short-carbonchain alkanes infinitely diluted in long-carbon-chain alkanes. They determined siE,∞ from a relationship -siE,∞/R ) ln γi∞ assuming hiE,∞ ) 0 and proposed a linear equation for ln γi∞ related with Qij, which they called the dispersion force parameter. This linear relationship generates almost half of the value of ln γi∞ predicted by the combinatorial terms from the FloryHuggins and UNIQUAC equations. However, judging from the experimental proof obtained from the calorimetric measurements, hiE,∞ is positive and the error involved in siE,∞ cannot be neglected. These awkward situations concerning siE,∞ and hiE,∞ for the light alkanes infinitely diluted in the heavy alkanes imply that a rational procedure for the data selection from ln γi∞ is needed to determine the accurate values of siE,∞ and hiE,∞. The purpose of the present study is to establish a procedure selecting the data for ln γi∞ from almost all of the data reported in the literature in order to determine the accurate values of siE,∞ and hiE,∞ and to find correlation equations for siE,∞ and hiE,∞. These correlations enable us to evaluate conventional solution models and to provide a rational insight on solution structures in the alkane/alkane solutions at infinite dilution. 2. Characteristics of Experimental Data for ln γi∞ 2.1. Data Sources. Kato et al.21 collected from the literature 215 data for ln γi∞ of the short-carbon-chain alkane solutes infinitely diluted in the long-carbon-chain alkane solvents. Table 1 includes these data from 14 references2,4,5,7,8,10,16,22-28 and those recently collected from a reference.17 A total number of 225 points of data have been found, where the solutes are from 4 e Ni e 10 and the solvents are from 7 e Nj e 36, satisfying Ni < Nj. The system temperature ranges from 280.15 to

Figure 1. Comparison between ln γi∞ measured by three different methods: (a) by GLC and GS for C6/C16 and C6/C20; (b) by GLC and SEC for C6/C18 and C7/C18.

393.35 K. Table 2 summarizes the number of data for each solute-solvent combination measured by different methods: (a) 206 points by gas-liquid chromatography (GLC), (b) 17 points by gas stripping (GS), and (c) two by the static equilibrium cell (SEC). The data from GLC are extensive, and the data for the solutes C6 and C7 and the solvents C16 and C18 are most abundant. 2.2. Measuring Methods. To compare the values of ln γi∞ from different measuring methods, in Figure 1, ln γi∞ has been plotted versus 1/T (a) for the solutesolvent combination of C6/C16 and C6/C20 measured by GLC and GS and (b) for C6/C18 and C7/C18 systems measured by GLC and SEC. It is apparent from Figure 1 that the values of ln γi∞ from one of the three methods are identical with those from the other two methods within an experimental error from the GLC measurements. Experimental errors in GS and SEC measurements are not obvious because of the shortage of data. 2.3. Effect of Temperature on ln γi∞. From thermodynamics, ln γi∞ is related with the partial molar excess entropy of a solute i at infinite dilution, siE,∞, and

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4929 Table 1. List of Experimental Data for γ∞i Nia Njb 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

16 16 16 16 18 20 22 24 7 8 12 12 16 16 16 16 16 18 20 20 22 24 24 24 24 24 24 28 28 28 28 30 32 36 7 12 12 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 18 18 18 18 18 18 18

T (K)

γ∞i

303.15 313.15 343.15 363.15 308.15 353.15 323.15 333.15 293.15 293.15 280.15 298.15 298.15 303.15 313.15 343.15 363.15 308.15 313.15 353.15 333.15 324.45 333.15 333.75 343.55 353.45 355.15 343.15 353.35 373.45 393.35 353.15 353.15 353.15 293.15 280.15 298.15 293.15 293.15 293.15 298.15 298.15 303.15 303.15 303.15 303.15 304.85 313.15 313.15 313.15 313.15 313.15 315.35 323.15 323.15 324.45 333.15 333.15 343.15 343.15 353.15 363.15 303.15 303.15 303.15 303.15 308.15 308.15 308.15

0.889 0.85 0.84 0.83 0.877 0.83 0.793 0.767 0.999 0.97 0.97 0.941 0.825 0.916 0.87 0.859 0.85 0.871 0.83 0.835 0.791 0.73 0.76 0.73 0.74 0.73 0.757 0.719 0.693 0.688 0.694 0.67 0.635 0.606 0.999 0.995 0.988 0.899 0.908 0.893 0.903 0.888 0.903 0.894 0.898 0.904 0.91 0.892 0.892 0.883 0.903 0.904 0.92 0.905 0.889 0.87 0.911 0.884 0.879 0.882 0.878 0.876 0.887 0.869 0.868 0.869 0.864 0.865 0.877

measd byc ref Nia Njb GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GS GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GS GLC GLC GLC GS GLC GS GLC GLC GLC GS GLC GLC GLC GLC GLC GLC GS GLC GLC GS GS GLC GS GLC GLC GLC GLC SEC GLC GLC GLC

23 27 27 27 23 25 23 23 28 28 5 5 26 23 27 27 27 23 23 25 23 22 23 22 22 22 8 23 17 17 17 24 24 24 28 5 5 10 16 7 16 26 16 10 7 23 22 10 27 7 16 23 22 16 10 22 16 10 10 27 10 27 24 16 7 29 16 7 24

6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 20 20 20 20 20 20 20 20 20 22 24 24 24 24 24 24 24 24 28 30 30 30 30 32 36 36 36 36 12 12 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 18 18 18 18 18

T (K)

γ∞i

308.15 313.15 313.15 313.15 313.15 318.15 318.15 323.15 323.15 323.15 324.55 333.15 334.55 343.65 353.15 353.65 373.15 313.15 313.15 313.15 323.15 333.15 333.15 343.15 353.15 353.15 333.15 324.45 333.15 333.75 343.55 349.15 353.45 355.15 361.15 343.15 349.15 353.15 357.15 361.15 353.15 349.15 353.15 357.15 361.15 280.15 298.15 293.15 293.15 298.15 298.15 303.15 303.15 303.15 304.85 313.15 313.15 313.15 315.35 323.15 324.45 333.15 343.15 363.15 293.15 303.15 303.15 303.15 303.15

0.877 0.876 0.867 0.859 0.862 0.856 0.858 0.861 0.855 0.87 0.85 0.9 0.86 0.88 0.91 0.86 0.97 0.834 0.844 0.851 0.825 0.837 0.82 0.816 0.814 0.872 0.804 0.78 0.776 0.78 0.77 0.794 0.76 0.781 0.791 0.735 0.694 0.698 0.697 0.693 0.66 0.64 0.633 0.635 0.628 1.078 1.022 0.924 0.914 0.922 0.931 0.921 0.916 0.929 0.93 0.92 0.898 0.911 0.93 0.921 0.83 0.923 0.903 0.899 0.889 0.878 0.909 0.885 0.909

measd byc ref Nia Njb GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GS GLC GLC GS GLC GS GS GS GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GS GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC SEC GLC GLC GLC GLC

23 24 7 16 7 16 7 7 16 24 22 6 22 22 6 22 6 10 16 23 10 16 10 10 10 25 23 22 23 22 22 2 22 8 2 23 2 2 2 2 24 2 2 2 2 5 5 16 7 16 26 16 7 23 22 16 7 27 22 16 22 16 27 27 29 16 24 7 24

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9

18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 20 20 20 20 24 24 24 24 24 24 24 24 28 28 28 30 30 30 30 32 32 36 36 36 36 16 16 16 16 18 20 24 24 24 24 28 28 28 30 30 30 30 32 32 36 36 36 36 16 16 16

T (K)

γ∞i

308.15 308.15 313.15 313.15 313.15 313.15 318.15 318.15 323.15 323.15 323.15 324.55 333.15 334.55 343.65 353.15 353.65 373.15 313.15 313.15 333.15 353.15 324.45 333.15 333.75 343.55 349.15 353.45 355.15 361.15 353.35 373.45 393.35 349.15 353.15 357.15 361.15 345.15 353.15 349.15 353.15 357.15 361.15 298.15 313.15 343.15 363.15 308.15 353.15 333.15 349.15 355.15 361.16 353.35 373.45 393.35 349.15 353.15 357.15 361.15 345.15 353.15 349.15 353.15 357.15 361.15 313.15 343.15 363.15

0.875 0.893 0.897 0.879 0.873 0.883 0.871 0.879 0.881 0.871 0.891 0.86 0.91 0.89 0.89 0.93 0.88 0.999 0.867 0.875 0.862 0.899 0.8 0.813 0.8 0.79 0.818 0.78 0.81 0.8 0.744 0.751 0.757 0.718 0.72 0.701 0.715 0.695 0.691 0.653 0.65 0.65 0.646 0.986 0.929 0.923 0.919 0.914 0.922 0.832 0.85 0.829 0.825 0.774 0.798 0.794 0.737 0.736 0.734 0.746 0.723 0.718 0.68 0.679 0.678 0.675 0.945 0.94 0.937

measd byc ref Nia Njb GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GS GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC

16 23 24 7 16 7 16 7 7 16 24 22 6 22 22 6 22 6 16 23 16 25 22 23 22 22 2 22 8 2 17 17 17 2 2 2 2 23 24 2 2 2 2 26 27 27 27 23 25 23 2 8 2 17 17 17 2 2 2 2 23 24 2 2 2 2 27 27 27

9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10

24 30 30 30 30 36 36 36 36 16 16 20 24 30 32 36

T (K)

γ∞i

361.16 349.15 353.15 357.15 361.15 349.15 353.15 357.15 361.15 343.15 363.15 353.15 353.15 353.15 353.15 353.15

0.848 0.77 0.766 0.762 0.766 0.727 0.716 0.716 0.717 0.956 0.954 0.969 0.88 0.801 0.781 0.724

measd byc ref GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC

2 2 2 2 2 2 2 2 2 27 27 25 25 25 25 25

a N : number of carbons in the molecule of solute alkane i. b N number of carbons in the molecule of solvent alkane j. c GLC: gasi j: liquid chromatography. GS: gas stripping. SEC: static equilibrium cell.

the partial molar excess enthalpy of solute i at infinite dilution, hiE,∞:

ln γi∞ ) (hiE,∞/RT0)T0/T - siE,∞/R

(1)

where R and T0 refer to the gas constant and an

arbitrary reference temperature normalizing hiE,∞; a temperature 298.15 K is chosen throughout this study. To examine the relationship between ln γi∞ and 1/T, experimental data for ln γi∞ of the solutes from butane to decane have been plotted in Figures 2-8 versus 1/T. Figures 2-8 include the numbers of references reporting

4930 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 2. Number of Experimental γi∞ Data Classified by the Combination of the Solutes and Solvents solute solvent

C4

C5

C6

C7

C8

C7 C8 C12 C16 C18 C20 C22 C24 C28 C30 C32 C36

(a) Data from GLCa (206 Points) 1 1 1 2 2 2 4 4 16 16 3 1 1 23 23 1 1 2 4 4 1 1 1 1 1 6 8 8 4 4 1 3 1 4 4 4 1 1 2 2 1 4 4 4

C16 C20

(b) Data from GSb (17 Points) 1 9 1 1 5

C18

(c) Data from SECc (2 Points) 1 1

C9

C10

3

2 1

2

1

4 0 4

1 1 1

Figure 3. ln γi∞ vs 1/T for pentane.

a GLC: gas-liquid chromatography. b GS: gas stripping. c SEC: static equilibrium cell.

Figure 4. ln γi∞ vs 1/T for hexane. Figure 2. ln γi∞ vs 1/T for butane.

the values of ln γi∞ for the given combinations of solute and solvent. It is obvious from Figures 2-8 that, when a solute and a solvent are fixed, data fluctuations for ln γi∞ are as large as the definite values of slopes, hiE,∞/ 298.15R in eq 1, and the intercepts, -siE,∞/R, are hardly determined. It is this uncertainty involved in the ln γi∞ data which hinders the determination of the accurate values of -siE,∞/R and hiE,∞/298.15R, although extensive data for ln γi∞ are found in the literature. It is obvious21 from Figures 2-8 that ln γi∞ < 0; i.e., γi∞ < 1 for the short-carbon-chain alkanes infinitely diluted in the longcarbon-chain alkanes, satisfying Ni < Nj. Kato et al.21 determined the values of siE,∞ assuming -siE,∞/R ) ln γi∞; i.e., hiE,∞ ) 0. However, these values of -siE,∞/R obviously include errors because calorimetric measurements require an absolute requirement: hiE,∞ > 0. The data for ln γi∞ are often correlated by solution models or activity coefficient models such as the Flory-Huggins, the UNIQUAC, the NRTL, and the Wilson equations assuming appropriate expressions for the combinatorial entropy, -siE,∞/R. However, no definite experi-

mental value for -siE,∞/R has been specified so far, and it is still impossible to rationally evaluate the expressions of combinatorial entropies from conventional solution models. These awkward situations must be improved by the establishment of pertinent criteria for the data selection universally applied to all data collected and listed in Table 1. Not only in the alkane/alkane systems where molecular interactions are weak but also in strongly interacting systems such as alcohols/water binaries, the values of -siE,∞/R are hardly determined because of large data fluctuations involved in ln γi∞. To demonstrate these difficulties, in Figure 9, ln γi∞ has been plotted versus 1/T not only for the C6/C16 alkane system but also for those alcohol/water binaries including methanol, ethanol, 1-propanol, and 1-butanol using the data for ln γi∞ compiled by Kojima et al.29 Further, Table 3 includes the values of -siE,∞/R and hiE,∞/298.15R obtained from the first-order regression according to eq 1 and also their average absolute deviations (AADs). It is obvious from Figure 9 that data fluctuations in ln γi∞ are large in alcohol/water systems and that the AAD of the correlation according to eq 1 is large even in the alkane/alkane system though the absolute fluctuations of ln γi∞ are

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4931

Figure 5. ln γi∞ vs 1/T for heptane.

Figure 7. ln γi∞ vs 1/T for nonane.

Figure 6. ln γi∞ vs 1/T for octane.

Figure 8. ln γi∞ vs 1/T for decane.

small. These results strongly support the necessity of pertinent criteria of data selection for ln γi∞ in the alkane/alkane systems. 3. Criteria of Data Selection for ln γi∞ The linear relationship appearing in eq 1 is expected to hold in alkane/alkane systems because molecular interactions are as weak as the temperature dependency of ln γi∞ is small: the intercept, -siE,∞/R, and the slope, hiE,∞/298.15R, are independent of temperature. Then, in the present study, it is assumed that the accurate relationship between ln γi∞ and 1/T is linear. To evaluate different experimental errors arising from different measuring apparatuses, a first-order regression according to eq 1 was made using a data set consisting of the data for ln γi∞ described in the same reference for the same solute-solvent combination. An average absolute deviation, (AAD)γ, from the first-order regression was determined as follows:

(AAD)γ )

1

n

∑

n i)1

|

|

ln γi∞exp - ln γi∞cal ln γi∞cal

(2)

where n denotes the number of data used for the regression and γi∞cal was calculated using the regressed parameters at the temperature specified by an experimental data, γi∞exp. When a value of ln γi∞ alone is reported in a reference for a solute-solvent combination, the value was not used in the following consideration because it is impossible to determine an intercept and a slope from the sole point. A total number of 53 points were eliminated by this limitation. Furthermore, if only two different values of ln γi∞ for a solute-solvent combination are reported in a reference, though they may belong to an accurate data group, they were eliminated from the following consideration because the uncertainty of the data cannot be determined from the value of (AAD)γ as it is always zero for these data. A total number of 20 points consisting of 10 data sets were eliminated by this limitation. The remaining 38 data sets satisfy n > 2 and are listed in Table 4: -siE,∞/R and hiE,∞/298.15R determined by the first-order regression according to eq 1 along with their (AAD)γ. Kato et al.21 demonstrated that the combinatorial entropies of the short-carbon-chain alkanes infinitely diluted in long-carbon-chain alkanes calculated from the

4932 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 4. Values of -siE,∞/R and hiE,∞/298.15R for the 38 Data Sets Satisfying n > 2 no. Ni Nj

Figure 9. ln γi∞ vs 1/T for C6/C16 and alcohol/water systems. Table 3. Values of -siE,∞/R, hiE,∞/298.15R, and AADs for C6/C16 Alkane and Alcohol/Water Systems solute/solvent hexane/hexadecane methanol/water ethanol/water 1-propanol/water 1-butanol/water

hiE,∞/ na 298.15R -siE,∞/R (AAD)γb (ln γi∞)avec 25 0.1386 -0.2423 27 -0.8709 1.456 43 -1.779 3.241 15 -1.816 4.5733 22 0.1705 3.7944

0.082 0.246 0.089 0.071 0.037

-0.111 0.641 1.571 2.821 3.949

∞ a n: number of data. b (AAD) ) (1/n)∑|(ln γ ∞ γ i exp - ln γi cal)/ln γi∞cal|. c (ln γi∞)ave: average of the values of ln γi∞.

UNIQUAC and the Flory-Huggins models are incorporated into the dispersion force parameter Qij defined as follows:

Qij ) (qi - qj )/qi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

7 9 6 7 8 5 6 4 8 7 6 7 5 7 6 6 7 7 5 6 6 6 7 6 6 9 8 9 8 6 7 6 7 7 6 6 7 7

28 16 16 16 16 16 16 16 36 36 20 18 28 16 18 24 18 24 24 36 16 18 18 30 18 30 28 36 30 16 30 18 16 18 18 16 18 16

measd hiE,∞/ bya nb ref (AAD)γ AADTc 298.15R -siE,∞/R GLC GLC GLC GLC GLC GLC GS GLC GLC GLC GS GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC GLC

3 3 3 3 3 3 7 3 4 4 5 5 3 6 5 4 4 4 4 4 6 6 6 4 4 4 3 4 4 3 4 4 3 4 3 3 3 3

17 27 27 27 27 27 10 27 2 2 10 16 17 16 16 22 24 22 22 2 16 7 7 2 24 2 17 2 2 7 2 22 7 22 6 22 6 22

0.0003 0.0007 0.0008 0.0017 0.0021 0.0058 0.0081 0.0081 0.002 0.0025 0.0094 0.0057 0.0096 0.0152 0.0078 0.0133 0.0101 0.0143 0.0152 0.0062 0.0284 0.0141 0.0148 0.0079 0.0164 0.0097 0.032 0.0125 0.0147 0.044 0.023 0.0594 0.053 0.0872 0.1928 0.1508 0.443 0.2841

0.0028 0.0044 0.0051 0.0116 0.0144 0.0394 0.0434 0.055 0.0583 0.0739 0.0779 0.0887 0.09 0.1178 0.1223 0.1548 0.1572 0.1676 0.1781 0.1843 0.2202 0.2204 0.232 0.2349 0.2554 0.2865 0.2983 0.3702 0.4338 0.6673 0.6793 0.6926 0.8026 1.0165 1.7021 2.4223 3.9112 4.5638

-0.202 0.0647 0.1378 0.1017 0.082 0.1747 0.1409 0.1772 0.249 0.3415 0.2226 0.136 -0.0132 0.0079 0.2757 0.3597 0.3382 0.3506 -0.0561 0.567 -0.0334 0.1602 0.0757 0.0582 0.2888 0.223 -0.3051 0.4433 -0.352 0.1695 0.4195 -0.2354 0.2681 -0.2876 -0.7695 0.7197 -0.9618 1.8533

-0.1252 -0.1182 -0.2454 -0.1901 -0.1517 -0.3051 -0.2495 -0.3305 -0.5976 -0.7179 -0.3959 -0.2647 -0.3581 -0.089 -0.4128 -0.5751 -0.4286 -0.5415 -0.2619 -0.9318 -0.0672 -0.2994 -0.1984 -0.412 -0.407 -0.4538 0.0066 -0.7021 -0.0079 -0.2819 -0.6899 0.0592 -0.3589 0.1252 0.5744 -0.7876 0.7578 -1.8664

Qijd -2.58 -0.69 -1.4 -1.106 -0.875 -1.791 -1.4 -2.334 -3.063 -3.562 -1.961 -1.351 -3.745 -1.106 -1.68 -2.521 -1.351 -2.088 -3.094 -4.201 -1.4 -1.68 -1.351 -3.361 -1.68 -2.071 -2.188 -2.663 -2.407 -1.4 -2.825 -1.68 -1.106 -1.351 -1.68 -1.4 -1.351 -1.106

a GLC: gas-liquid chromatography. GS: gas stripping. b n: number of data. c AADT ) (AAD)γ/[(Tmax - Tmin)/Tave]. d Qij ) (qi - qj)/qi; qi ) (2)(0.848) + 0.540(Ni - 2); qj ) (2)(0.848) + 0.540(Nj - 2).

(3)

where qi and qj refer to the measures of molecular surface areas of alkane i and alkane j, respectively;30 therefore, -siE,∞/R determined from the first-order regression is expected to be correlated with Qij. Then Table 4 includes the values of Qij. As shown in footnote d of Table 4, qi and qj are first-order functions of Ni and Nj, respectively; therefore, the dispersion force parameter is given as Qij ) (Ni - Nj)/(1.141 + Ni) for an alkane i/alkane j binary. Further, hiE,∞/298.15R may also be correlated with Qij because the dispersion forces are the common sources of both molecular interactions originating hiE,∞ and the combinatorial order of molecules originating siE,∞ in the alkane solutions at infinite dilution. In Figures 10 and 11, -siE,∞/R and hiE,∞/ 298.15R have been plotted, respectively, versus Qij for the 38 data sets listed in Table 4. It is apparent from Figures 10 and 11 that further criteria for data selection should be set up to establish definite relationships between -siE,∞/R and Qij and also between hiE,∞/298.15R and Qij; therefore, the following two criteria were further introduced. First, taking into account the requirement from calorimetric measurements, those data sets satisfying hiE,∞ > 0 were selected from the 38 data sets shown in Table 4. A number of 37 points consisting of

Figure 10. Relationship between Qij and -siE,∞/R for the 38 data sets in Table 4.

10 data sets in Table 4 were eliminated by this limitation. In the next step, those data sets having narrow temperature ranges were eliminated as follows. The fluctuations appearing in Figures 10 and 11 seem to arise from the facts that the values of (AAD)γ for some

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4933

Figure 11. Relationship between Qij and hiE,∞/298.15R for the 38 data sets in Table 4.

data sets shown in Table 4 are large and also that uncertain data sets are involved in which extrapolation distances from the lowest inverse temperatures to the axis representing 1/T ) 0 are too far to determine accurate intercepts. For example, as shown in Figure 4, the data for ln γi∞ of hexane diluted in hexatriacontane2 have a small value 0.0062 of (AAD)γ as shown in Table 4 and satisfy hiE,∞ > 0; however, the extrapolation distance to obtain siE,∞/R is 29 times beyond the measured range of 1/T. Hence, to provide high reliability to those data having wide temperature ranges, (AAD)γ data selection was modified to the form (AAD)γ/(∆T/ Tave), where Tave denotes an average temperature of the data used in the first-order regression and ∆T is defined as follows:

∆T ) Tmax - Tmin

(4)

In eq 4, Tmax and Tmin refer to the maximum and minimum temperatures of the data used for the regression. The values of (AAD)γ/(∆T/Tave) have been listed in Table 4; data in Table 4 have been arranged in the increasing order of (AAD)γ/(∆T/Tave). In the present study, a critical value of (AAD)γ/(∆T/Tave), denoted as , was arbitrarily chosen from a range of  < 0.5 and was investigated if the values of -siE,∞/R satisfying (AAD)γ/(∆T/Tave) <  in Table 4 are correlated with Qij. As a result of this investigation, -siE,∞/R has been clearly correlated with Qij covering a wide range from -3.5 < Qij < 0 when  is fixed at 0.1. The data sets finally selected are those 11 from nos. 2-12 in Table 4. In Figure 10, -siE,∞/R of the 11 data sets have been marked with closed circles. It is remarkable from Figure 10 that the emerged relationship between -siE,∞/R and Qij is linear originating from the point (-siE,∞/R, Qij) ) (0, 0), which must be satisfied because of a thermodynamic requirement. In particular, the linearity proved by the seven points in the range -1.8 < Qij < -0.7 is excellent. In Figure 11, hiE,∞/298.15R of the 11 data sets from nos. 2-12 in Table 4 have also been marked with closed circles. It is striking from Figure 11 that a clear linear relationship between hiE,∞/298.15R and Qij has appeared. Because the data of -siE,∞/R as well as hiE,∞/ 298.15R are obviously correlated with Qij, it can be said

Figure 12. Relationship between 1/T and ln γi∞ for the data satisfying criteria A-C.

that pertinent criteria for the data selection of ln γi∞ are established in the present study. The criteria are summarized in short as follows: (A) Select three or more data for ln γi∞ obtained at different temperatures and reported in the same reference for the same solute-solvent combination. (B) Select those data providing a positive partial molar excess enthalpy at infinite dilution if it is determined from first-order regression according to eq 1. (C) Select those data satisfying (AAD)γ/(∆T/Tave) < 0.1, following the definitions in eqs 2 and 4. In Figure 12, ln γi∞ is plotted versus 1/T for the definite data from nos. 2-12 in Table 4 satisfying the criteria A-C. Figure 12 demonstrates that each data set shows a converged linear trend and that the data for the hexane/hexadecane system obtained from different measuring techniques of GLC27 and GS10 conform to the same line. Figure 12 clearly demonstrates that siE,∞ and hiE,∞ are constant over the temperature range of 293.15-363.15 K. In the next step, the constants of proportionality for -siE,∞/R and hiE,∞/298.15R with Qij are determined. In Figure 13, average values of (-siE,∞/R)/Qij, [(-siE,∞/R)/ Qij]ave, and (hiE,∞/298.15R)/Qij, [(hiE,∞/298.15R)/Qij]ave, have been plotted versus critical values of (AAD)γ/(∆T/ Tave), [(AAD)γ/(∆T/Tave)]crit. The average values were determined using those limited data sets in Table 4 satisfying (AAD)γ/(∆T/Tave) < [(AAD)γ/(∆T/Tave)]crit. Values of [(AAD)γ/(∆T/Tave)]crit were arbitrarily chosen from 0.01 to 0.5. The fluctuations of (-siE,∞/R)/Qij and (hiE,∞/298.15R)/Qij were evaluated using the same data by the following (AAD)s and (AAD)h, respectively:

(AAD)s ) (AAD)h ) 1

m

∑

m i)1

|

1

m

∑

mi)1

|

|

(-siE,∞/R)/Qij - [(-siE,∞/R)/Qij]ave [(-siE,∞/R)/Qij]ave

(5)

|

(hiE,∞/298.15R)/Qij - [(hiE,∞/298.15R)/Qij]ave [(hiE,∞/298.15R)/Qij]ave

(6)

4934 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

Figure 13. [(-siE,∞/R)/Qij]ave, [(hiE,∞/298.15R)/Qij]ave, (AAD)s, and (AAD)h vs [(AAD)γ/(∆T/Tave)]crit.

Figure 13 also includes the values of (AAD)s and (AAD)h. The number of data sets satisfying (AAD)γ/(∆T/Tave) < [(AAD)γ/(∆T/Tave)]crit, m, decreases with decreasing [(AAD)γ/(∆T/Tave)]crit. It follows from Figure 13 that both (AAD)s and (AAD)h have minimum values near [(AAD)γ/(∆T/Tave)]crit ) 0.04; therefore, the following correlation equations have been established at this critical value.

-siE,∞/R ) 0.173Qij

m ) 4, (AAD)s ) 0.008

(7)

Figure 14. Relationships between Qij and -siE,∞/R and hiE,∞/ 298.15R including data from calorimetric measurement. Table 5. Values of hiE,∞/298.15R from Calorimetric Measurements at 298.15 K Ni

Nj

(qi - qj)/qi

6 6 6 6

16 16 16 16

-1.4 -1.4 -1.4 -1.4

7

16

-1.11

a

hiE,∞a (J/mol)

hiE,∞/298.15Ra

410 0.165 356 0.144 464 0.187 447 0.180 ave ) 0.169 (AAD ) 0.086) 410 0.165

ref 18 19 20 16 16

Data from ref 16.

hiE,∞/298.15R ) -0.095Qij m ) 4, (AAD)h ) 0.021 (8) The total number of data sets used for establishing eqs 7 and 8 is four, and AADs are as small as (AAD)s ) 0.008 and (AAD)h ) 0.021. In Figure 14, eqs 7 and 8 have been drawn by a dashed line and a solid line, respectively, and the experimental values of -siE,∞/R and hiE,∞/298.15R satisfying the criteria A-C have been plotted versus Qij. In Figure 14, the values of hiE,∞/ 298.15R from calorimetric measurements16,18-20 are also plotted. In Table 5, these values from calorimetric measurements are listed. The average value of hiE,∞/ 298.15R from calorimetric measurements for the solute of hexane in the solvent of hexadecane is 0.169, while their AAD is 0.086 as shown in Table 5. The calorimetric enthalpies of mixing are generally expressed as analytical functions of the mixture composition, while hiE,∞ can be calculated using these functions in extrapolation to the infinite-dilution regions. Unfortunately, there are no calorimetric values for highly diluted solutions; the lower hexane mole fractions in the study of its mixtures with hexadecane were from 0.146 to 0.31.16 Thus, the certainty involved in hiE,∞ by calorimetric measurements is not high either. Taking into consideration the uncertainty involved in the calorimetric data, the partly overlapping data obtained from different measuring methods, i.e., GLC, GS, and calorimetric measurements, can be the proof verifying that the criteria A-C are pertinent and that eqs 7 and 8 are valid as the expressions for -siE,∞/R and hiE,∞/298.15R of short-

Figure 15. Relationship between Qij and ln γi∞/[0.173 - (0.095)(298.15)/T] for all 225 data points.

carbon-chain alkanes infinitely diluted in long-carbonchain alkanes. It should be stressed that reliable values of -siE,∞/R and hiE,∞/298.15R have been established covering all experimental ranges of the solutes and solvents.

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4935 Table 6. Relative Deviations of ln γi∞C,UNIQ and ln γi∞C,F-H from Equation 7 for Hexane in Different Solvents Ni

Nj

∆UNIQa

ln γi∞C,UNIQ

∆FHb

ln γi∞C,F-H

0.173Qij

6 6 6 6 6 6 6 6 6 6

7 8 9 10 11 12 13 14 15 16

-0.6449 -0.3949 -0.2145 -0.082 0.0165 0.0903 0.1457 0.1873 0.2183 0.241

-0.0086 -0.0293 -0.0571 -0.089 -0.1231 -0.1585 -0.1943 -0.2301 -0.2656 -0.3007

-0.6156 -0.3489 -0.1588 -0.0209 0.0805 0.1556 0.2113 0.2525 0.2826 0.3043

-0.0093 -0.0316 -0.0611 -0.0949 -0.1309 -0.168 -0.2054 -0.2428 -0.2797 -0.316

-0.0242 -0.0485 -0.0727 -0.0969 -0.1211 -0.1454 -0.1696 -0.1938 -0.218 -0.2423

∞ ∞ a ∆ b UNIQ ) (ln γi C,UNIQ - 0.173Qij)/0.173Qij. ∆FH ) (ln γi C,F-H - 0.173Qij)/0.173Qij.

γi∞C,K, Thomas and Eckert,32 ln γi∞C,T, and Weidlich and Gmehling,33 ln γi∞C,W. These were calculated as follows:

ln γi∞C,K ) 1 - Φi′ + ln Φi′ Figure 16. Relationships between the dispersion force parameter with -siE,∞/R and hiE,∞/298.15R: (- - -) hiE,∞/298.15R from eq 8; (2) ln γi∞Reg; (s) -siE,∞/R from eq 7; (4) ln γi∞C,K; (O) ln γi∞C,W; (+) ln γi∞C,T; (b) ln γi∞C,MV with molar volumes at 298.15 K; (×) ln γi∞C,UNIQ; (0) ln γi∞C,F-H.

From eqs 1, 7, and 8, ln γi∞ of a short-carbon-chain alkane infinitely diluted in a long-carbon-chain alkane is given as follows:

ln γi∞ ) [0.173 - (0.095)(298.15)/T]Qij

(9)

In Figure 15, the values of ln γi∞ normalized by temperature as ln γi∞/[0.173 - (0.095)(298.15)/T] have been plotted versus Qij for all 225 data points. It follows from Figure 15 that eq 9 gives an excellent representation of ln γi∞. 4. Evaluation of Conventional Solution Models In Figure 16, -siE,∞/R and hiE,∞/298.15R calculated from eqs 7 and 8 have been drawn versus Qij by the solid and dashed lines, respectively. Figure 16 also includes the combinatorial entropies calculated by the Flory-Huggins model, ln γi∞C,F-H, and the UNIQUAC model, ln γi∞C,UNIQ, as follows:

ln γi∞C,F-H ) ln(ri /rj) + 1 - ri /rj

(10)

ln γi∞C,UNIQ ) ln Φi + (z/2)qi ln(θi /Φi ) + li - ljri /rj (11) Φi ) ri /rj

(12)

θi ) qi /qj

(13)

li ) (z/2)(ri - qi ) - (ri - 1)

(14)

lj ) (z/2)(rj - qj ) - (rj - 1)

(15)

where ri and rj denote the measures of molecular volumes of alkane i and alkane j, respectively.30 The coordination number z was fixed at 10 as a common approximation. Moreover, Figure 16 includes modified combinatorial entropies proposed by Kikic et al.,31 ln

5qi(1 - Φi /θi + ln Φi /θi) (16) Φi′ ) ri2/3/rj2/3

(17)

ln γi∞C,T ) ln Φi′′ + 1 - Φi′′

(18)

Φi′′ ) ri3/4/rj3/4

(19)

ln γi∞C,W ) 1 - Φi′′ + ln Φi′′ - 5qi(1 - Φi /θi + ln Φi /θi) (20) In the calculation of these combinatorial entropies, the solute alkanes were chosen from C6 and C10 and the solvent alkanes were from C8, C16, C24, and C32 where Ni < Nj is satisfied. It is apparent from Figure 16 that these conventional combinatorial entropies can satisfactorily be correlated with Qij. The fluctuations appearing in these calculated combinatorial entropies are much smaller than the fluctuations for the definite data satisfying criteria A-C, as shown in Figure 14. The values for -siE,∞/R from both experimental data and conventional solution models have been correlated with Qij for any combination between the solute and the solvent; therefore, it is possible to evaluate the validity of each model by comparing the values predicted from a solution model with experimental data or eq 7: the solid line in Figure 16. It is obvious from Figure 16 that the conventional models can represent a part of the experimental data in a region near Qij ) 0, while the Flory-Huggins and UNIQUAC models predict the -siE,∞/R closest to the solid line. Predicted values of -siE,∞/R from the Flory-Huggins and UNIQUAC models cross the solid line near Qij ) -0.7, while the other models always predict larger values of combinatorial entropies than eq 7 does. In Table 6, relative deviations of ln γi∞C,F-H and ln γi∞C,UNIQ from eq 7 have been listed for hexane, as an example of the solute, diluted in different solvents from heptane to hexadecane. It is clear from Table 6 that the values of ln γi∞C,UNIQ of hexane diluted in the limited solvents from decane to dodecane are close to those from eq 7, where |(ln γi∞C,UNIQ 0.173Qij)/0.173Qij| < 0.1 is satisfied. The solvents from the Flory-Huggins model are further limited from decane to undecane if the same condition is applied for ln γi∞C,F-H instead of ln γi∞C,UNIQ. A similar limitation

4936 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

on solvents occurs for the other cases of solute alkanes. When Qij < -0.7, the values of ln γi∞C,F-H and ln γi∞C,UNIQ are smaller than -siE,∞/R from eq 7, as shown in Figure 16. The critical evaluation of the validity of the conventional solution models cannot adequately be performed in the range -0.7 < Qij because highly accurate data for -siE,∞/R are scarce in this region. The combinatorial entropies from eqs 10 to 20 are independent of temperature. On the other hand, the combinatorial entropy from the Flory-Huggins model introducing the temperature-dependent molar volumes has been widely used.4,6 It is defined as follows:

ln γi∞C,MV ) ln(vi0/vj0) + 1 - vi0/vj0

(21)

where vi0 and vj0 are the molar volumes of the solute i and solvent j, respectively. In Figure 16, the values of ln γi∞C,MV at 298.15 K have been plotted versus Qij for the solutes from C4 to C10 and solvents from C6 to C20, where the molar volumes were calculated by the Rackett equation.34 Figure 16 demonstrates that, although the convergence in the relationship between ln γi∞C,MV and Qij is worse in this case, ln γi∞C,MV at 298.15 K can adequately predict the experimental relationship of ln γi∞C, i.e., eq 7, in the range of -3.5 < Qij < -0.7. The temperature dependency of ln γi∞C,MV is relatively strong because the absolute value of ln γi∞C,MV at 373.15 K is almost 60-90% in magnitude of that from eq 7 if compared at the same Qij. It has immense significance not only that eq 7 has been established based on a large number of experimental data points but also that the Flory-Huggins model approximated with molar volumes at 298.15 K, which adequately predicts eq 7, has been identified. Figure 16 includes the values of ln γi∞ predicted by the regular solution theory, ln γi∞Reg, at 298.15 K for those solutes of hexane, heptane, octane, and decane and the solvents of dodecane and hexadecane. These were calculated as follows:

ln γi∞Reg ) vi0(δi - δj )2/RT

(22)

δi ) [(∆Hvi - RT)/vi0]1/2

(23)

where the enthalpy of vaporization of solute i, ∆Hvi, was estimated by the Watson relation35 with the enthalpy of vaporization at normal boiling point. The values of ln γi∞Reg could not be correlated with Qij, as shown in Figure 16. It is apparent from Figure 16 that the regular solution theory, which predicts hiE,∞ assuming siE,∞ ) 0, is in fair agreement with eq 8 when Qij ) 0. In the present study, -siE,∞/R and hiE,∞/298.15R have been successfully determined based on the temperature dependency of the ln γi∞ data. The main reason for this success lies in the use of the dispersion force parameter Qij, which enables us to correlate -siE,∞/R with Qij from both experimental data and conventional solution models. The second reason is obviously the set up of the criterion C for data selection. Without the criterion C, a definite relationship between -siE,∞/R and Qij could not have been found. These examples imply another possibility for establishing successful criteria for the selection of a great amount of data refusing successful analysis for decades in the field of property measurements or, more widely, plant operations.

5. Solution Structures in Light Alkane/Heavy Alkane Solutions at Infinite Dilution Equations 7 and 8 and Figure 14 demonstrate that both siE,∞ and hiE,∞ are positive. Thus, an enthalpyentropy compensation effect holds in eqs 1 and 9. It follows from eq 9 that ln γi∞ is negative when Ni < Nj and that the ratio of the magnitude of entropy term to that of the enthalpy term at 298.15 K is -1.8 ()-0.173/ 0.095), indicating that the magnitude of the entropy effect is almost twice as strong as that of the enthalpy effect. It can be clearly shown from Figure 14 that the entropy-enthalpy compensation effect appears to hold in any combination of the solute and solvent covering a wide range of Qij from -4 to 0 and that the ratio is independent of the combination of alkanes if Ni < Nj is satisfied. It is demonstrated from eq 9 that the entropy effect much more dominates the partial molar excess energy at infinite dilution, RT ln γi∞, at temperatures higher than 298.15 K. The partial molar excess entropies at infinite dilution of the solute of short-carbon-chain alkanes in the solvent of long-carbon-chain alkanes are always positive; therefore, a disorder occurs around a molecule of the shortcarbon-chain alkane if one of the molecules of the longcarbon-chain alkanes is replaced by a molecule of the solute alkane. A rational explanation for the disorder might be that a molecular parallel framework formed in the solvent alkane having a long carbon chain is partly destroyed by the introduction of a molecule of the solute alkane having a short carbon chain. When Qij , 0, the magnitude of siE,∞ is large, resulting in a strong disorder: a noticeable disorder appears when the carbon chain of the solvent alkane is much longer or when the carbon chain of the solute alkane is much shorter. The reason for the first disorder is deduced as follows: the long-carbon-chain solvent forms a large-scale parallel framework as a result of the strong dispersion forces exerted by the large molecular surface areas arising from the long carbon chain; therefore, if the large-scale parallel framework is destroyed by the introduction of a molecule of the short-carbon-chain solute, the disorder is not confined around the solute molecule but reaches far from it, resulting in a remarkable increase in siE,∞, whereas the second disorder may arise as a result of the following: the shorter the molecule of the solute alkane, the weaker is the molecular attractive interaction between the solute and solvent, which results in failure of the restoration of the order formed as the parallel framework by the solvent molecules. On the other hand, an increase in the disorder of the parallel framework formed by the solvent molecules reflects a decrease in the strength of the attractive molecular interactions, which implies that hiE,∞ is positive. Alkanes are highly ordered in the solid state, existing in many crystalline phases which depend on the chain length.36 In the liquid state, they retain much of their order, as was proven by Barbe and Patterson.37 Using a light scattering technique, Tancrede et al.38 showed clearly that globular or spherical solvents destroy the conformation order in liquid long-chain hydrocarbons.9 However, because of the scarcity of accurate experimental data for thermodynamic properties, such as activity coefficients at infinite dilution, the destruction of the conformation order in liquid long-chain hydrocarbons has not been verified from the viewpoint of thermodynamic properties. The present investigation is the first to clearly demonstrate the destruction of the

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4937

conformation order occurring in the liquid long-carbonchain alkanes in terms of thermodynamic properties such as the partial molar excess entropies derived from the infinite-dilution activity coefficients. 6. Conclusions Partial molar excess entropies, siE,∞, and partial molar excess enthalpies, hiE,∞, at infinite dilution of shortcarbon-chain alkanes in long-carbon-chain alkanes have been determined using the extensive data of infinitedilution activity coefficients, γi∞, in the literature covering the alkane solutes from butane to decane, alkane solvents from heptane to hexatriacontane, and temperatures from 280.15 to 373.15 K. Those obtained are summarized as follows. The criteria of the data selection for the extensive data of ln γi∞ have been established. Reliable values of siE,∞ and hiE,∞ have been determined as the slopes and intercepts obtained from the first-order regression of ln γi∞ with reciprocal temperature. Both siE,∞ and hiE,∞ are positive and proportional to a dispersion force parameter Qij; i.e., (qi - qj)/qi where qi and qj denote the measures of molecular surface areas of the solute alkane i and the solvent alkane j, respectively. The correlation equations have been established as -siE,∞/R ) 0.173Qij and hiE,∞/298.15R ) -0.095Qij; entropy-enthalpy compensation holds in a wide range of -4 < Qij < 0, where the ratio of the entropy term to the enthalpy term at 298.15 K, -298.15siE,∞/hiE,∞, is -1.8 and independent of the system. Conventional solution models such as the Flory-Huggins and the UNIQUAC models predict accurate values of -siE,∞/R at limited ranges near Qij ) -0.7 and 0, while the Flory-Huggins model modified with molar volumes at 298.15 K adequately represents the new correlation for -siE,∞/R. The longer the molecules for the heavy alkane solvent or the shorter a molecule for the light alkane solute, the stronger is the disorder of the parallel framework formed by the solvent alkane molecules. Literature Cited (1) Kato, S. Liquid-Liquid Partition Coefficients: Data Sources and Methods for Prediction. Physico-Chemical Properties for Chemical Engineering; Kagaku Kogyosha: Tokyo, 1999; Vol. 21. (2) Tewari, Y. B.; Martire, D. E.; Sheridan, J. P. Gas-Liquid Partition Chromatographic Determination and Theoretical Interpretation of Activity Coefficients for Hydrocarbon Solutes in Alkane Solvents. J. Phys. Chem. 1970, 74, 2345. (3) Roberts, K. L.; Rousseau, R. W.; Teja, A. S. Solubility of Long-Chain n-Alkanes in Heptane between 280 and 350 K. J. Chem. Eng. Data 1994, 39, 793. (4) Vogel, G. L.; Hamzavi-Abedi, M. A.; Martire, D. E. Activity Coefficients of Nine Normal and Branched Alkanes in n-Octadecane at 303.15 K. J. Chem. Thermodyn. 1983, 15, 739. (5) Letcher, T. M.; Moollan, W. C. The Determination of Activity Coefficients at Infinite Dilution Using G.L.C. with a Moderately Volatile Solvent (Dodecane) at the Temperatures 280.15 and 298.15 K. J. Chem. Thermodyn. 1995, 27, 1025. (6) Mengarelli, A. C.; Bottini, S. B.; Brignole, E. A. Infinite Dilution Activity Coefficients and Solubilities of Biphenyl in Octadecane and Mineral. J. Chem. Eng. Data 1995, 40, 746. (7) Chien, C.-F.; Kopecni, M. M.; Laub, R. J.; Smith, C. A. Solute Liquid-Gas Activity and Partition Coefficients with Mixtures of n-Hexadecane and n-Octadecane with N,N-Dibutyl-2-ethylhexylamide Solvents. J. Phys. Chem. 1981, 85, 1864. (8) Meyer, E.; Stec, K. S.; Hotz, R. D. A Thermodynamic Study of Solute-Solvent Interactions Using Gas-Liquid Chromatography. J. Phys. Chem. 1973, 77, 2140.

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Received for review February 11, 2003 Revised manuscript received June 25, 2003 Accepted June 27, 2003 IE030129S