Infinite-Dilution Activity Coefficient Model for Predicting the SLE of

of a long-carbon-chain alkane solute (j) and a short-carbon-chain alkane solvent (i) has been investigated using SLE data satisfying the Gibbs-Duhem e...
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Ind. Eng. Chem. Res. 2005, 44, 3766-3775

Infinite-Dilution Activity Coefficient Model for Predicting the SLE of Alkane/Alkane Binaries Satoru Kato* Department of Applied Chemistry, Graduate School of Engineering, Tokyo Metropolitan University, 1-1 Minamiohsawa, Hachiohji 192-0397, Japan

An infinite-dilution activity coefficient model for predicting the SLE of a binary system consisting of a long-carbon-chain alkane solute (j) and a short-carbon-chain alkane solvent (i) has been investigated using SLE data satisfying the Gibbs-Duhem equation. It has been demonstrated that the temperature dependency of heat capacities of a solute alkane has a strong effect on the infinite-dilution activity coefficient of the solute, ln γj∞. The criteria for the establishment of infinite-dilution for the alkane/alkane SLE have been proposed. Infinite-dilution Wilson parameters introduced in the residual term of ln γj∞ have been determined from the regression using the SLE data. The Wohl equation combined with the present model accurately predicts the solubilities of the solute alkanes, while conventional models less satisfactorily predict the solubility. The ordered structure formed in a short-chain carbon solvent alkane liquid is partly destroyed, leaving the strength of molecular interactions unchanged when a molecule of an alkane having a longer carbon chain is introduced into the solvent alkane. Introduction Infinite-dilution activity coefficients of a long-carbonchain alkane j (γj∞) in a binary system consisting of a long-carbon-chain alkane (j) and a short-carbon-chain alkane (i) are of practical importance for solving wax formation problems in petroleum engineering and improving the cleaning properties of liquid detergents,1 where, when γj∞ is combined with a suitable activity coefficient model derived from thermodynamics, a solubility curve describing SLE (solid-liquid equilibria) can be established. A prominent simplicity for alkane/alkane binaries is that the dispersion force is the only attractive molecular interaction acting on the 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 the size effects originating in the difference between the solute and the solvent molecular surface areas. On the basis of these insights, a large amount of experimental data for the infinite-dilution activity coefficients of a short-carbonchain alkane i (γi∞) in an alkane solute (j)/alkane solvent (i) binary has been reported;2 they were measured by gas-liquid chromatography or equilibrium cells. However, for the long-carbon-chain alkanes, it is difficult to directly measure γj∞ by these methods. Although the activity coefficients of the long-carbon-chain alkane j (γj) in the alkane (j)/alkane (i) solution can be determined from VLE data, unfortunately, VLE data under dilute conditions have not been reported for xj < 0.01, where xj denotes the mole fraction of the alkane j. On the other hand, to investigate γj∞, a considerable amount of SLE data covering dilute conditions has been reported in the literature3-5 as measured melting points and their equilibrium compositions. These data are useful for determining γj∞. However, in the analyses of these SLE data, simplified methods5-10 for calculating solid and * Telephone: +81-426-77-2824. Fax: +81-426-77-2821. Email: [email protected].

liquid heat capacities have been used, which yield significant errors in γj∞ values when xj approaches zero. Because of this fault, no reliable value of γj∞ is found in the literature. To obtain reliable values of γj∞, the thermal quantities must be accurately determined, and the SLE data for the analysis should be limited to those satisfying the thermodynamic requirement (i.e., the Gibbs-Duhem equation). The activity coefficients in alkane/alkane binaries have been analyzed with the regular solution theory in the literature.5,6,9,11 However, Kato et al.2 showed that, based on the experimental proof for γi∞, the regular solution theory is limited in its application to the range where Ni is nearly equal to Nj; Ni and Nj denote the carbon number of the alkanes i and j, respectively. Meanwhile, many investigators5,6,10,12-16 have used the Flory-Huggins combinatorial entropy for representing the combinatorial term in the partial molar excess energy (RT ln γj) where R and T denote the gas constant and the system temperature, respectively. As demonstrated by Kato et al.,2 the Flory-Huggins model provides inaccurate values of the combinatorial entropies when Qij is remote from -0.7, where Qij denotes a dispersion force parameter defined as (qi - qj)/qi with measures of molecular surface areas of the alkanes i (qi) and j (qj). As for the residual terms representing the SLE data, the Wilson, the UNIQUAC, and the NRTL models have been used in the literature.5,6,10,12-16 These analyses have not established accurate models for representing γj∞ because the Flory-Huggins model is used and the heat capacities have been oversimplified. Recently, Kato et al.17 used the Wohl equation for representing ln γi and ln γj, in which ln γj∞was given by a combination of a combinatorial term, represented by an extension of the experimental proof, and the UNIQUAC residual term. Although they successfully correlated the infinite-dilution UNIQUAC interaction parameters with a dispersion force parameter, (qj - qi)/ qj, a practical method that predicts the SLE covering Ni , Nj with high accuracy must be established using

10.1021/ie040219c CCC: $30.25 © 2005 American Chemical Society Published on Web 03/25/2005

Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3767

experimental data at dilute concentrations prevailing in SLE data. The purpose of the present study is 2-fold: (i) to establish a model for predicting infinite-dilution activity coefficients of the long-carbon-chain alkanes in alkane/ alkane binaries based on the SLE data satisfying infinite-dilution conditions and a thermodynamic requirement represented by the Gibbs-Duhem equation and (ii) to combine this model with the Wohl equation for developing a practical activity coefficient model that enables us to predict activity coefficients with high accuracy and the solution structures in the long-carbonchain alkane/short-carbon-chain alkane binaries. Modeling

measure of the molecular surface area of the shortcarbon-chain alkane i. Considering the case in which xi approaches zero in eq 5, it can be shown that A is identical with ln γi∞. Kato et al.2 showed that the relationships between 1/T and ln γi∞ determined from the reliable γi∞ data of alkane/alkane binaries are linear over a temperature range from 280 to 373 K; they established the expression of the nondimensional infinitedilution partial molar excess entropies of the dilute alkane i to be 0.173Qij from the intercepts and also the enthalpies as -28.3Qij/T from the slopes. Therefore, the present study uses these relationships for a combinatorial term of the short-carbon-chain alkane i (ln γi∞C) and a residual term (ln γi∞R), consisting of ln γi∞ as follows:

A ) ln γi∞

Infinite-Dilution Activity Coefficient Model. It is assumed that a pure alkane solid j is in contact with an alkane solvent i at temperature T and that a solidliquid equilibrium is achieved. The equilibrium mole fraction of the long-carbon-chain alkane j (xj) is given as follows:18

ln γi∞C ) 0.173Qij

(8)

ln γjxj ) Am + At + ACp

(1)

ln γi∞R ) -28.3Qij/T

(9)

(2)

where Qij denotes a dispersion force parameter defined as follows:

( (

∆Hm 1 1 R Tm T

Am ) At )

∆Ht 1 1 R Tt T

∆Cp 1 A Cp ) T dT m RT RT

∫

T

)

)

[ ( [ (

∫T ∆Cp dT T

m

(4)

)] )]

Bqi - A θi qj

(5)

Aqj ln γj ) θi B + 2 - B θj qi

(6)

qixi θ ) 1 - θi qixi + qjxj j

(7)

2

θi )

Qij ) (qi - qj)/qi

(3)

where γj denotes the activity coefficient of the longcarbon-chain alkane j in which the hypothetical supersaturated liquid at T is assigned as the standard state of the activity of the alkane j. Tm and ∆Hm denote the melting point and the heat of melting for the solid alkane j, respectively, while Tt and ∆Ht stand for the phase transition temperature and the heat of phase transition of the alkane j, respectively. Furthermore, ∆Cp denotes CpL - CpS: the difference between the heat capacity of the super-saturated liquid alkane j (CpL) and that of the solid alkane j (CpS) at T. If the values of T and the solubility (xj) are given, the activity coefficient of the long-carbon-chain alkane j is calculated from eq 1. Kato et al.17 demonstrated that the activity coefficients for alkane/alkane binaries can be correlated by the Wohl equation as well as by any other equations, such as the Margules and the UNIQUAC equations; therefore, the present study uses the Wohl equation for expressing the activity coefficients as follows:

ln γi ) θj2 A + 2

) ln γi∞C + ln γi∞R

qi ) (2)(0.848) + 0.540(Ni - 2) qj ) (2)(0.848) + 0.540(Nj - 2) where Ni and qi denote the carbon number and a

On the other hand, B in eqs 5 and 6 is identical with ln γj∞ as follows:

B ) ln γj∞ ) ln γj∞C + ln γj∞R The expression of ln γi∞C has now been clarified using Qij as shown by eq 8; therefore, by exchanging the suffixes i and j in eq 8, the present study represents ln γj∞C as follows:

ln γj∞C ) 0.173Qji

(10)

Qji ) (qj - qi)/qj Kato et al.17 used the UNIQUAC model for representing the residual terms of ln γi∞ and ln γj∞. As shown in the latter section, the UNIQUAC model does not represent the SLE data. In the present study, the Wilson model, which has a simple expression, is examined regarding whether it is suitable for the description of the SLE as follows:

[ [

( (

)] )]

∆λji∞ qi ∆λij∞ + 1 - exp RT qj RT

(11)

∆λij∞ qj ∆λji∞ + 1 - exp ln γj R ) RT qi RT

(12)

ln γi∞R ) ∞

where ∆λij∞ and ∆λji∞ are the infinite-dilution interaction parameters defined only under the infinite-dilution conditions of alkane/alkane binaries, in which the ratio of measures of molecular volumes of the alkanes i and j included in the original Wilson model were replaced by qi/qj and qj/qi, because the dispersion forces dominate intermolecular attractive forces acting in the alkane/ alkane solutions. If the values of T and the solubility (xj) under infinite-dilution conditions are given, γj∞ is

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determined from eq 1; the value of ln γj∞R is also determined; therefore, ∆λij∞ and ∆λji∞ can be determined from eqs 11 and 12. Kato et al.17 has shown that a nondimensional UNIQUAC interaction parameter at infinite-dilution is a fourth-order function of Qji; therefore, the present study regresses the SLE data at infinite dilution using a polynomial expression of ∆λij∞/ RT with Qji. Infinite-Dilution Criteria for the SLE Data. To determine the values of ∆λij∞/RT from the SLE data, the criteria for the achievement of the infinite-dilution condition must be established. The data for SLE can be well represented by a linear expression (ln xj ) A + B/T) over a broad solubility range; therefore, dividing each side of eq 1 with ln xj, an equation is obtained as follows:

1+

ln γj ∆s° - ∆h°/T ) ln xj A + B/T

∆h° ) ∆Ht + ∆Hm +

∫T ∆Cp dT T

∆s° ) ∆Ht/Tt + ∆Hm/Tm +

m

∫TT

m

∆Cp dT T

(13) Figure 1. Relationships between 1/T and xj of long-chain carbon alkanes in heptane.

(14)

(15)

where the following equation holds:

ln γjxj ) ∆s°/R - ∆h°/RT

(16)

When xj approaches zero, γj approaches γj∞, a constant; therefore, in this case, the left side of eq 13 approaches unity, while the temperature simultaneously decreases leaving the right side enthalpy-term dominant. At xj ) 0, T ) Tm holds; therefore, the parameter L defined in eq 17 should approach zero:

L ) -T∆s°/(∆h° - T∆s°) - [-Tm,i∆s°m,i/(∆h°m,i Tm,i∆s°m,i)] (17) where the suffix i denotes the solvent i, and m corresponds to the value at T ) Tm. On the other hand, ∆λij∞/ RT is expected to approach a constant value when xj approaches zero because ∆λij∞/RT in eqs 11 and 12 is defined at infinite-dilution for the alkane j. Therefore, in the present study, the differences in the parameter L from zero are used as the criteria for the achievement of infinite-dilution conditions. SLE Data and Thermal Quantities Data Sources of SLE. A total of 34 sets of SLE data on alkane/alkane binaries has been found in the literature from 11 articles. Table 1 includes the solutesolvent combinations of the 34 sets with the variation ranges in temperature and solubility. The SLE data cover the following ranges: 16 e Nj e 36, 5 e Ni e 12, and 276 < T < 343 K. In Figure 1, the solubilities of the long-carbon-chain alkanes are plotted versus the inverse absolute temperature for the solvent heptane. Figure 1 demonstrates that ln xj almost linearly decreases with increasing 1/T over wide temperature ranges. Heat Capacities and Thermal Quantities. On the basis of heat capacity data, Jin and Wunderlich22

Table 1. List of Solubility Data for Alkane/Alkane Binaries Nj

Ni

T (K)

xj

ref

16 18 19 22 22 23 24 24 24 24 24 25 26 28 28 28 28 28 32 32 32 32 32 32 32 36 36 36 36 36 36 36 36 36

6 7 7 6 7 7 6 7 7 7 12 7 7 5 7 7 10 12 5 6 7 7 8 10 12 5 5 6 7 7 7 8 10 12

276.1-287.4 276.9-297.65 278.25-302.3 291.25-312.0 284.29-317.6 274.7-320.0 279.5-299.7 289.47-323.78 283.3-317.1 279.31-303.38 282.0-305.82 300.7-326.2 303.3-329.5 284.6-301.8 298.2-334.0 287.1-311.9 285.9-309.8 287.4-306.3 288.8-300.8 299.3-341.4 303.0-334.0 289.2-301.5 291.9-343.2 299.5-340.2 296.4-343.2 291.0-303.9 284.6-304.0 284.4-305.4 291.2-307.2 284.4-291.5 297.3-344.4 284.8-303.2 285.7-304.2 285.9-306.3

0.31-0.75 0.12-0.73 0.13-0.78 0.089-0.57 0.049-1.00 0.016-1.00 0.014-0.12 0.050-1.00 0.015-0.54 0.010-0.16 0.012-0.18 0.098-1.00 0.10-1.00 0.003-0.035 0.025-1.00 0.0047-0.11 0.0036-0.081 0.0039-0.049 0.0012-0.0077 0.0037-0.89 0.0102-0.45 0.0013-0.0086 0.0026-1.00 0.0061-1.00 0.0039-1.00 0.00027-0.0024 0.000089-0.00026 0.000092-0.0032 0.00033-0.0040 0.000090-0.00032 0.0005-0.60 0.000093-0.0023 0.00011-0.0026 0.00010-0.0033

8 13 13 15 19 7 8 19 5 9 9 7 7 3 7 3 20 20 3 4 21 3 4 4 4 3 20 20 3 20 5 20 20 20

xj,lim

0.004 0.006 0.007 0.02

0.01

0.0002 0.00022 0.0004 0.0004 0.0008 0.0008 0.0008

proposed a group-contribution method for predicting the heat capacities of liquid alkanes (CpL) as follows:

CpL ) (17.33 + 0.04551T)(Nj - 2) + 2(30.41 + 0.01479T) 90 < T < 430 K (18) Equation 18 agrees with experimental data22-25 within 0.15% error. In the present study, the values of CpL of the liquids and the super-saturated liquids are calculated from eq 18. Table 2 lists the equations of heat capacities for solids used in the analysis. As for the data for Tm, ∆Hm, Tt, and ∆Ht, data points from two to seven for one of the quantities are found in

Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3769 Table 2. Heat Capacities of Solid Alkanes Used in the Analysisa Ni

CpS (J/mol K)

T (K)

max error (%)

ref

16 18 24 32 36 19, 23, 25

4.185 (1.9143 + 0.34208T) 4.185 (1.9058 + 0.37856T) 4.185M (43.916-0.44463T + 0.0015T2 - 1.6667 × 10-6 T3) 4.185M (0.17935-0.00032002T + 4.2857 × 10-6 T2) 2751.5-16.853T + 0.035483T2 2(-5.8969 + 1.0367T - 0.0083149T2 + 2.1166 × 10-5 T3)+ (271.65-2.2183T + 0.00484T2)(Nj - 2) 2(9350.5 + 101.24T - 0.36431T2 + 0.00043823T3) + (-5.4513 + 0.096111T)(Nj - 2)

270-231.3 270-301.3 270-300 260-300 280-310 220-240

0.1 0.1 0.1 0.1 0.1 2

25 25 24 24 22 22, 26

270-330

2

22

22, 26, 28 a

M, molecular weight.

Table 3. Thermal Quantities for Alkanes Ni

Tm (K)

∆Hm (kJ/mol)

Tt (K)

∆Ht (kJ/mol)

16 18 19 22 23 24 25 26 28 32 36

291.33 301.33 304.90 317.10 320.52 323.71 326.65 329.50 334.33 342.10 349.02

53.35 61.72 45.82 48.98 53.56 54.89 57.42 59.52 64.67 75.97 88.82

295.95 316.15 314.62 321.25 320.23 326.45 331.15 338.90 346.98

13.79 28.22 21.75 31.30 26.38 32.23 35.45 42.69 30.82

the literature.1,5,7,12,16,21,23,24,27-29 Table 3 lists thermal quantities used in the analysis. The maximum variation involved in ln γj for the data in Table 1 is less than 5% for the experimental errors involved in these thermal quantities. The values of Tt in Table 3 correspond to the transition from a crystalline phase to a rotator phase.1 The abbreviation of the term ACp in eq 1 can generate an error of about 100% in the values of ln γj. The errors in ln γj are much greater than the errors in γj, when γj is close to unity. However, as shown by eqs 5, 6, 11, and 12, precise values of ln γj are required for the establishment of a reliable activity coefficient model. If ∆Cp is approximated to be constant in eq 4, the following equation is obtained:

ACp )

(

)

∆Cp Tm T ln -1+ R Tm T

(19)

The approximation of ACp by eq 19 yields large errors in ln γj exceeding 200% at the low xj shown in Table 1; it is this approximation which prevents us from obtaining accurate values of ln γj∞; therefore, neither the heat capacity nor its temperature dependency should be neglected in the analysis of SLE data under dilute conditions. Exclusion of Unreliable SLE Data. Of the alkanes shown in Table 3, each of triacosane (C23) and pentacosane (C25) has three transition points in the solid phase.1 However, only one of the heats of transitions, the transition from the crystalline phase to the rotator II phase, is reported in the literature. To retain high reliability of the data in the analyses, the SLE data of these two alkanes, C23 and C25, were excluded from the analysis. Although octacosane (C28) has two phase transition points, these points are close each other; therefore, octacosane is approximated to have only one phase transition point. Following the thermodynamic requirement represented by the Gibbs-Duhem equation, ln γj must approach a constant value when xj approaches zero. In

Figure 2. Relationships between xj and ln γj/θi2 for hexatriacontane.

Figure 2, ln γj/θi 2 is plotted versus xj following the manner indicated by eq 6 using the SLE data for the solute hexatriacontane (C36). Although the data convergence is low, Figure 2 shows that ln γj/θi 2 typically increases with decreasing xj, goes through a plateau, and then decreases. Accordingly, the lower limit of each plateau was identified, and the value of xj corresponding to the lower limit (xj,lim) was determined. In the present investigation, these data satisfying xj < xj,lim were excluded from the analysis because they do not meet the requirement of thermodynamics. Table 1 includes the values of xj,lim. By this exclusion, 86 points of solubility data were eliminated. The remaining 329 points were used for the analysis. Determination of Model Parameters and Prediction of SLE Specification of Infinite-Dilution Conditions. In Figure 3, ∆λij∞/RT determined from eqs 11 and 12 is plotted versus L. The denominator of eq 17 (∆h° - T∆s°) can approach zero, because ∆h° and ∆s° have the same sign; therefore, the value of L covers a wide range from -1200 to 1500. In Figure 3, a limited range covering -20 < L < 0 is drawn for hexatriacontane. Figure 3 shows a typical trend that ∆λij∞/RT approaches a constant value when L approaches zero; therefore, those values of ∆λij∞/RT satisfying -10 < L < 0 were regressed according to a linear equation, ∆λij∞/RT ) a + bL. Thus, the data set that provides a small value of |b| and that shows a significant data convergence was identified, that is, those data sets satisfying the following three criteria were selected: |b| < 0.005, (AAD)L < 1% and

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Figure 3. Relationships between L and ∆λij∞/RT for hexatriacontane. Table 4. Wilson Interaction Parameters at Infinite-Dilution for Long-Chain Carbon Alkanesa Figure 4. Relationships between Qji and ∆λij∞/RT.

SLE Nj Ni Ndata 24 28 28 28 36 36 36 36

7 7 10 12 5 7 8 12

12 12 13 7 13 19 9 9

T (K)

∆λij∞/ RT

bb

279/319 289/334 292/310 299/306 290/304 293/344 297/303 298/306

-3.46 -4.16 -2.99 -2.48 -6.37 -4.89 -4.44 -3.01

-0.0005 -0.0001 -0.0022 -0.0004 -0.0017 -0.0022 -0.0037 0.0026

(AAD)Lc 0.92% 0.79% 0.58% 0.29% 0.47% 0.86% 0.10% 0.21%

Qji

ref

0.6762 0.7206 0.6177 0.5491 0.8347 0.7808 0.7539 0.6462

9 3 20 20 20 3, 5, 20 20 20

Constant Temperature VLE Nj Ni Ndata 6 8 10 12

4 6 6 6

60 13 12 12

T (K)

∆λij∞/RT

dd

(AAD)xe

Qji

ref

253/293 328.15 308.15 308.15

-1.22/-1.33 -1.24 -1.53 -2.0

0.3 0.005 -0.02 0.22

15% 14% 10% 56%

0.280 0.219 0.359 0.457

30 31 31 31

(AAD)xe

Qji

ref

Constant Pressure VLE Nj Ni Ndata P (kPa) 7 6 10 6 12 10

63 7 29

∆λij∞/RT

dd

20/101 -0.74/-0.86 0.3 101 -1.44 0.075 20 -0.87 0.18

6% 0.90% 8.40%

0.123 32 0.359 34 0.152 35

a |b| < 0.005 (AAD) < 1 % and N L data > 3 for SLE data and |d| < 1 for VLE data. b Slope in the correlation as ∆λij∞/RT ) a + bL for SLE. c (AAD)L ) (1/Ndata)∑|(∆λij∞/RT)exp - (∆λij∞/RT)cal)/(∆λij∞/ RT)exp| with (∆λij∞/RT)cal ) a + bL. d Slope in the correlation as ln γj ) c + dxj for VLE. e (AAD)x ) (1/Ndata)∑|(∆λij∞/RT)exp - (∆λij∞/ RT)cal)/(∆λij∞/RT)exp| for (ln γj)cal ) c + dxj.

Ndata > 3. From the data sets selected in this manner, the average values of ∆λij∞/RT in the range -10 < L < 0 are listed in Table 4. The definition of an average absolute deviation, (AAD)L, is given in footnote b in Table 4, where Ndata denotes the number of data points in the range -10 < L < 0. From this analysis, it is concluded that infinite dilution conditions are achieved for the eight data sets shown in Table 4, because the values of L are close to zero and ∆λij∞/RT has reached a constant value. If the Flory-Huggins combinatorial entropy is used instead of eq 10, the convergence and constancy of ∆λij∞/RT shown in Figure 3 cannot be met. Correlation of ∆λij∞/RT with Qji. In Figure 4, the average values of ∆λij∞/RT shown in Table 4 for the eight data sets are plotted versus Qji using closed circles. In

Figure 4, the SLE data sets excluded from Table 4 are also plotted by X marks, where the average values of ∆λij∞/RT in the range -10 < L < 0 were used. It follows from Figure 4 that the values of ∆λij∞/RT determined from the SLE data provide a converged relationship with Qji and that the convergence of those data shown by closed circles is much better than those shown by X marks, because the former was determined from the data achieving infinite-dilution conditions. To find a universal relationship between ∆λij∞/RT and Qji, constant-temperature and constant-pressure VLE data were used for determining ∆λij∞/RT; 32 data sets for constant-temperature VLE of alkane/alkane binaries shown in Table 1 by Kato et al.17 were collected. However, only the data sets satisfying the GibbsDuhem equation were selected for the analysis as follows: first, the values of ln γj were regressed according to a linear equation, ln γj ) c + dxj using the data satisfying xj < 0.4. In the next step, the data sets satisfying |d| < 0.3 were selected because they follow the requirement represented by the Gibbs-Duhem equation. Table 4 includes the average values of ∆λij∞/ RT in the range 0 < xj < 0.4 for the selected data sets and the average absolute deviations from the linear correlations, (AAD)x. The constant-temperature VLE data for hexane/butane binaries30 consist of five data sets covering temperatures from 253.15 to 293.15 K, where each includes 12 points. The other three constanttemperature VLE data sets excluding the hexane/ butane system have significantly constant values of ∆λij∞/RT for the data covering 0 < xj < 0.4, and the deviations from average values are smaller than 5%; therefore, their average values are listed in Table 4. As for the hexane/butane system, ∆λij∞/RT cannot be constant in the range 0 < xj < 0.4 but yields a converged linear relationship with xj; therefore, the linear relationship was extrapolated for determining ∆λij∞/RT at xj ) 0, and the value is listed in Table 4. Six data sets for constant-pressure VLE of alkane/ alkane binaries were found in the literature.32-37 Of these data sets, Table 4 includes three sets which satisfy |d| < 0.3 obtained from the linear regression, ln γj ) c

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+ dxj, for the data covering 0 < xj < 0.4. Table 4 also includes average values of ∆λij∞/RT for the data covering 0 < xj < 0.4, where the AADs from the average values are smaller than 10%. Figure 4 includes the values of ∆λij∞/RT obtained from the constant-temperature and the constant-pressure VLE. It is striking as shown in Figure 4 that the SLE and the VLE data fortuitously conform to the same relationship between ∆λij∞/RT and Qji; the following correlation has been established from the SLE data achieving infinite-dilution conditions, as shown by closed circles, and the VLE data listed in Table 4:

∆λij∞/RT ) -11.642Qji + 56.79Qji2 - 163.71Qji3 + 213.5Qji4 - 110.09Qji5 (20) The solid line in Figure 4 stands for eq 20. Equation 20 originates at the point (Qji, ∆λij∞/RT) ) (0,0), which means that eq 20 meets the thermodynamic requirement: the relationship ∆λij∞/RT ) 0 must be satisfied when the solute is identical with the solvent. This fulfillment, in addition to the fact that eq 20 can represent both the SLE and VLE data, establishes the utmost reliability of eq 20. Furthermore, it should be stressed that eq 20 is a significantly practical relationship because, being established using both isothermal and nonisothermal data, it does not depend on the temperature. In Figure 4, the ∆λij∞/RT values calculated by the infinite-dilution activity coefficient model proposed by Kato et al.17 are plotted versus Qji using a dashed line for the solvent pentane. Their model uses a combination of the Flory-Huggins/UNIQUAC models for combinatorial/residual terms. Figure 4 demonstrates that the dashed line deviates from the experimental data represented by the solid line at Qji > 0.5. Furthermore, even positive values of ∆λij∞/RT are obtained for the solvent octane or heavy alkane solvents; therefore, the combination of the Flory-Huggins/UNIQUAC models does not represent the SLE data covering Qji > 0.5. Of the combinations examined, only the combination, eq 10/Wilson, provides the converged relationship as shown in Figure 4 for the SLE and VLE data. Furthermore, the present model predicts the total pressures within an average error of 1.1 kPa for the constanttemperature VLE data of the 32 binary systems listed in Table 1 by Kato et al.17 The value is identical with the average error predicted by the model of Kato et al.17 Therefore, the present model has a wider application region in Qji. Comparison between Predicted and Experimental Solubilities. In Figure 5, the calculation procedures for the prediction of activity coefficients and solubilities are summarized as a flow diagram. To compare the experimental and calculated solubilities, the solubility of a solute j measured with a solvent i at T (xj) is transformed into the solubility in heptane (xj,heptane), and the transformed solubility in heptane is compared with a calculated solubility. From eq 1, xj,heptane is related to xj as follows:

xj,heptane )

γj γj,heptane

xj

(21)

where γj,heptane is the activity coefficient of the solute j in heptane at T. The transformed values (xj,heptane) are plotted versus 1/T in Figure 6. The 86 points satisfying xj < xj,lim and excluded from the previous analysis are

Figure 5. Flow diagram for the calculation procedures of activity coefficients and solubilities.

Figure 6. Relationships between 1/T and solubilities of long-chain alkanes in heptane. (-) predicted.

included in Figure 6. Figure 6 shows that the convergence of xj,heptane is satisfactory. The solid lines in Figure 6 stand for the solubilities of alkanes in heptane predicted from eqs 1-12 and 20. Figure 6 shows that the model proposed in the present study satisfactorily predicts the solubilities of alkanes except for the dense concentration ranges covering xj > 0.2 for C28 and C36 and the dilute concentration ranges for C28 and C32. The disagreements of C28 in the dense concentration ranges may arise from the experimental errors involved in the solubility data7 because the experimental and predicted solubilities show considerable discrepancy when xj,heptane approaches unity and the solutions approach an ideal solution. The case is the same for the data on C36.5 Any disagreement between measured and predicted solubilities cannot be found at the dense concentration range

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Figure 7. Comparison between the solubilities predicted by the present and conventional models for the solvent of heptane at 298.15 K.

for the data on C32.4 On the other hand, the disagreements at the dilute concentration ranges may arise from the lack of thermal quantities of the solid alkanes. The disagreement of octacosane (C28) arose from the negligence of the phase transition from the rotator III phase to the rotator IV phase1 as stated previously. Although it has not been experimentally verified for C32, the same phase transition1 that occurred for C28 and C30 may occur in the case of C32. Figure 6 shows that the Wohl equation combined with the infinite-dilution activity coefficient model proposed in the present study satisfactorily predicts solubilities at dense concentration ranges. Therefore, it has been demonstrated that both the infinite-dilution activity coefficient model proposed in the present study and the Wohl equation are valid for representing the SLE of alkanes. To compare all the experimental data with calculated solubilities, the solubility of a solute alkane j in a solvent i measured at T (xj) is transformed by eq 21 into the solubility in the solvent heptane at T (xj,heptane)T. Furthermore, (xj,heptane)T is transformed into the solubility at 298.15 K ((xj,heptane)298.15) as follows:

(

xj,heptane

xj,heptane

cal

) ( )

T

xj,heptane

xj,heptanecal

)

(22)

298.15

Equation 22 means that the relative difference between the calculated solubility ((xj,heptanecal)T) and the measured solubility at T transformed by eq 21 into that in heptane ((xj,heptane)T) is projected to that at 298.15 K. In the calculation of the activity coefficients of the solute j in the given solvent i, γj, and (γj,heptane)T, the experimental data of ∆λij∞/RT are used (i.e., those values given by the closed circles and the X marks in Figure 4 are used). In Figure 7, (xj,heptane)298.15 determined from eq 22 with (xj,heptane)T plotted versus Nj using open circles, where (xj,heptane)T is determined from eq 21 with the experimental solubility in the solvent i at T (xj). In Figure 7, the solid line stands for the solubility calculated from the present model for the solute j with the solvent heptane at 298.15 K. Figure 7 shows that the error involved in the transformed solubilities is large when Nj is large, but the solubilities predicted from the present model are almost equal to the averages of the

Figure 8. Relationships between the mole fraction of hexane and excess molar enthalphies for the hexadecane(j)/hexane(i) system.

transformed solubilities except for dotriacontane (C32). Figure 7 includes the solubilities calculated from the ideal solution theory, as shown by a dotted line. It is demonstrated from Figure 7 that, if an activity coefficient is estimated to be equal to unity, there is a risk that the solubility at a dilute condition is predicted to be a small value. Figure 7 includes those values predicted from the ASOG and the UNIFAC groupcontribution methods. In calculations with these methods, eqs 8 and 9 were used for ln γi∞ because these represent experimental proofs, and ln γj∞ was calculated by each model. Figure 7 shows that the ASOG and the UNIFAC group-contribution methods predict large values of solubilities because the activity coefficients are underestimated by the Flory-Huggins combinatorial entropy, which is substantially used in these models and dominates the term of ln γj∞. Because of this reason, none of the ASOG and the UNIFAC group-contribution methods are desirable for predicting the solubility of a long-carbon-chain alkane in a short-carbon-chain alkane. Excess Molar Enthalpies. The data for excess molar enthalpy (HME) or the heat of mixing of the hexadecane (j)/hexane (i) system are reported in the literature;38,39 therefore, the values of HME in the literature are plotted versus the mole fraction of hexane (xi) in Figure 8. On the other hand, the heat of mixing is calculated as follows:

HME ) hiExi + hjExj

(23)

The partial molar excess enthalpy of a solvent alkane i (hiE) is calculated as follows:

∂ ln γi ∂(1/T)

|

x

)

hiE R

(24)

Figure 8 includes the values of HME calculated by the present model using eq 23 between 298.15 and 349 K covering the temperature range of the HME data.38,39 The predicted values for the temperature range conform to a single line as shown in Figure 8. On the other hand,

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the values of HME were determined from the constanttemperature VLE data reported by McGlashan and Williamson40 as follows: first, the values of parameter B in the Wohl equation determined with the constantpressure VLE data were found to be well regressed according to a linear equation with T to be B ) -1.0207 + 261.55/T. Kato et al.17 reported the values of B for the constant-temperature VLE data using the prefixed values of A determined from eqs 8 and 9. In the next step, the values of HME of the hexadecane/hexane binary were calculated from the Wohl equation with the parameters A and B thus determined in the temperature range covering the VLE data, 293.15-330.15 K. Figure 8 includes the calculated values from the VLE data drawn by a dashed line. Figure 8 shows that the values of HME predicted from the present model representing SLE data are close to those from calorimetric measurement and that they are temperature-independent as is the case for the HME values from VLE data. The data for HME from calorimetric measurement are temperature-dependent as shown in Figure 8; thus, it has been clearly demonstrated that the temperature dependency of HME from calorimetric measurement and that from phase equilibria, SLE and VLE, contradict each other. Moreover, as shown by Kato et al.,2 the slope of HME at xi ) 0 is independent of temperature for the hexadecane/hexane binary as experimentally proven by the gas-liquid chromatography data41 and the equilibrium cell data,42 which also contradict the results from the calorimetric measurement. There was a dispute regarding the temperature dependency of HME of the hexadecane/hexane binary. More than 50 yeas ago, van der Waals and Hermans reported a relatively weak temperature dependency,43 while Mcglashan and coworkers43 demonstrated a strong dependency. It should be noted that the contradiction must be precisely examined based on experimental data in the future.

Figure 9. Relationship between Nj and infinite-dilution partial molar excess enthalpies.

Characterization of the Present Model It has been shown in the previous section that the model proposed in the present study successfully predicts the activity coefficients of the alkanes in alkane/ alkane binaries. The model has also shown that a hypothetical super-saturated liquid can be conveniently chosen as a standard state defining activities, if reliable heat capacities and other thermal quantities are obtained. The present model may be more useful than an equation of state approach, which is conveniently used in the description of phase equilibria at high pressures. In Figure 9, hiE,∞/RT and hjE,∞/RT calculated from eq 24 and the infinite-dilution activity coefficients model proposed in the present study are plotted versus Nj for the solvent hexane at 298.15 K. Figure 9 shows that hiE,∞/RT linearly increases with increasing Nj, but hjE,∞/ RT slightly decreases approaching zero; therefore, it is demonstrated from Figure 9 that, if a molecule of a longcarbon-chain alkane is introduced into the liquid of a short-carbon-chain alkane, such as hexane, the strength of the molecular interactions remains almost constant. On the other hand, the infinite-dilution activity coefficient of solvent i from the regular-solution theory (ln γi∞Reg) is calculated.2 In Figure 9, the calculated values of ln γi∞Reg and ln γj∞Reg are also plotted because the regular-solution theory predicts the partial molar excess enthalpies of the solutions where the partial molar excess entropies can be neglected. Figure 9 shows that, as demonstrated by Kato et al.,2 ln γi∞Reg is accurately

Figure 10. Relationships between Nj and infinite-dilution partial molar excess entropies.

identical with hiE,∞/RT, which has been experimentally proven, when Ni is close to Nj. However, Figure 9 also demonstrates that, when Ni , Nj, the regular solution theory cannot accurately predict hiE,∞/RT nor can it estimate the constancy of the strength of molecular interactions around a long-carbon-chain alkane molecule because ln γj∞Reg . hjE,∞/RT. The relationship between the activity coefficient of solvent i and its partial molar excess entropy (siE) is given as follows:

hiE siE ln γi ) RT R

(25)

In Figure 10, -siE,∞/R and -sjE,∞/R calculated from eqs 24 and 25 are plotted versus Nj for the solvent hexane at 298.15 K. Figure 10 includes the Flory-Huggins combinatorial term of a solvent i dissolving in an alkane j at infinite-dilution (ln γi∞C,F-H)2 and the FloryHuggins combinatorial term at infinite dilution modified by the molar volumes (ln γi∞C,F-H(MV)).2 As is the case for the Wilson equation expressed by eqs 11 and 12, measures of molecular volumes are replaced with those of molecular surface areas. When hiE,∞ ) 0, ln γi∞C,F-H

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is identical with -siE,∞/R. Furthermore, Figure 10 includes the combinatorial term at infinite dilution calculated from the free volume model44 (ln γi∞C,FV) as follows:

vi0 - vi/

vi0 - vi/

vj0 - vj

vi0 - vi/

ln γi∞C,FV ) ln

+1/

(26)

where vi/ and vj/ denote van der Waals volume of the alkanes i and j, respectively; these were calculated by the Peng-Robinson equation. The UNIQUAC combinatorial terms are not shown in Figure 10 because they are identical with those from the Flory-Huggins equation within a difference of 2% when Ni , Nj. Now, -siE,∞/R is identical with ln γi∞C, because ln γi∞ is a firstorder function of 1/T as shown by experimental data;2 therefore, in Figure 10, the relationship between Nj and -siE,∞/R has been experimentally proven. It has been demonstrated2 that the Flory-Huggins model (ln γi∞C,F-H) predicts -siE,∞/R less accurately than that modified by molar volumes (ln γi∞C,F-H(MV)) at 298.15 K; Figure 10 shows this trend. It should be noted that -siE,∞/R is temperature-independent, while ln γi∞C,F-H(MV) is considerably temperature-dependent. On the other hand, hjE,∞/RT is close to zero as shown in Figure 9; therefore, -sjE/R is almost identical with ln γj∞. Figure 10 shows an important relationship, 0 < sjE,∞ < siE,∞, demonstrating that the liquid short-carbon-chain alkane (i.e., hexane) forms a weakly ordered structure that is destroyed by the introduction of a molecule of a longcarbon-chain alkane, although the ordered structure formed in the short-carbon-chain alkane is not so strong as that formed in a long-carbon-chain alkane liquid2 because sjE,∞ < siE,∞ holds. As for the prediction of the solution structures by the conventional models, the Flory-Huggins model predicts a relationship (ln γj∞C,F-H < ln γi∞C,F-H); therefore, it predicts the existence of a more strongly ordered structure formed in hexane than those formed in long-carbon-chain alkanes, which is absurd. The situation is the same for the Flory-Huggins model modified with the molar volumes. Similarly, it is demonstrated from Figure 10 that the free volume model does not accurately predict the partial molar excess entropies when Ni , Nj. These outcomes for the solution structures and the strength of molecular interactions are expected to be useful information for improving desirable cleaning properties and skin care properties of liquid detergents and the design of drugs solubilizing in a structured fluid. The model proposed in the present study provides the accurate solubilities of solid alkanes; therefore, the model is expected to promote the protection strategy for wax formation in petroleum engineering. It also provides the values of activity coefficients of low-molecularweight alkanes; therefore, the model can contribute to improving the design of separation devices in petrochemical engineering. Conclusions An infinite-dilution activity coefficient model for predicting the SLE of a binary system consisting of a long-carbon-chain alkane solute (j) and a short-carbonchain alkane solvent (i) has been investigated using SLE data covering solutes from hexadecane to hexatriacontane, solvents from pentane to dodecane and temperatures from 276 to 343 K. The SLE data satisfying the

Gibbs-Duhem equation were used in the analysis. Those obtained are summarized as follows. If the temperature dependency of heat capacities of a solute alkane is neglected at dilute conditions, significant errors appear in infinite-dilution activity coefficients of the solute (ln γj∞). The criteria for the establishment of infinite dilution for alkane/alkane SLE have been proposed as -10 < L < 0, where L is defined by eq 17. The infinite-dilution Wilson parameter introduced in the residual term of ln γj∞ has been determined from the regression using the SLE data as a fifth-order function of a dispersion force parameter (Qji) where the combinatorial term of ln γj∞ is expressed as 0.173Qji following an experimental proof. The Wohl equation combined with the present model accurately predicts the solubilities of the solute alkanes. Literature Cited (1) Dirand, M.; Bouroukoba, M.; Chevallier, V.; Petitjean, D.; Behar, E.; Ruffier-Meray, R. Normal alkanes, multialkane synthetic model mixtures and real petroleum waxes: crystallographic structures, thermodynamic properties, and crystallization. J. Chem. Eng. Data 2002, 47, 115. (2) Kato, S.; Hoshino, D.; Noritomi, H.; Nagahama, K. 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. Ind. Eng. Chem. Res. 2003, 42, 4927. (3) Madsen, H. E. L.; Boistelle, R. Solubility of long-chain n-paraffins in pentane and heptane. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1078. (4) Seyer, W. F. Mutual solubilities of hydrocarbons. II. The freezing point curves of dotriacontane (dicetyl) in dodecane, decane, octane, hexane, cyclohexane and benzene. J. Am. Chem. Soc. 1938, 50, 827. (5) 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. (6) Haulait-Pirson, M.-C.; Huys, G.; Vanstraelen, E. New predictive equation for the solubility of solid n-alkanes in organic solvents. Ind. Eng. Chem. Res. 1987, 26, 447. (7) Provost, E.; Chevallier, V.; Bouroukba, M.; Petitjean, D.; Dirand, M. Solubility of some n-alkanes (C23, C25, C26, C28) in heptane, methylcyclohexane and toluene. J. Chem. Eng. Data 1998, 43, 745. (8) Dernini, S.; Desantis, R. Solubility of solid hexadecane and tetracosane in hexane. Can. J. Chem. Eng. 1976, 54, 369. (9) Brecevic, L.; Garside, J. Solubilities of tetracoxane in hydrocarbon solvents. J. Chem. Eng. Data 1993, 38, 598. (10) Coutinho, H. A. P.; Andersen, S. I.; Stenby, E. H. Evaluation of activity coefficient models in prediction of alkane solidliquid equilibria. Fluid Phase Equilib. 1995, 103, 23. (11) Hammami, A.; Mehrotra, A. K. Liquid-solid-solid thermal behaviour of n-C44H90 + n-C50H102 and n-C25H52 + n-C28H58 paraffinic binary mixtures. Fluid Phase Equilib. 1995, 111, 253. (12) Domanska, U. Solubility of n-paraffin hydrocarbons in binary solvent mixtures. Fluid Phase Equilib. 1987, 35, 217. (13) Domanska, U.; Hofman, T.; Rolinska, J. Solubility and vapor pressures in saturated solutions of high-molecular-weight hydrocarbons. Fluid Phase Equilib. 1987, 32, 273. (14) Domanska, U.; Rolinska, J. Correlation of the solubility of even-numbered paraffins C20H42, C24H50, C26H54, C28H58 in pure hydrocarbons. Fluid Phase Equilib. 1989, 45, 25. (15) Kniaz, K. Solubility of n-docosane in n-hexane and cyclohexane. J. Chem. Eng. Data 1991, 36, 471. (16) Kniaz, K. Influence of size and shape effects on the solubility of hydrocarbons: the role of the combinatorial entropy. Fluid Phase Equilib. 1991, 68, 35. (17) Kato, S.; Hoshino, D.; Noritomi, H.; Nagahama, K. Prediction of activity coefficients using UNIQUAC interaction parameters correlated with constant-temperature VLE data for alkane/ alkane binaries. Fluid Phase Equilib. 2004, 219, 41.

Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3775 (18) Prausnitz, J. M. Molecular Thermodynamics of FluidPhase Equilibria; Prentice-Hall: Upper Saddle River, NJ, 1960. (19) Floter, E.; Hollanders, B.; de Loos, T. W.; de S. Arons, J. The ternary system (n-heptane + docosane + tetracosane): the solubility of mixtures of docosane and tetracosane in heptane and data on solid-liquid and solid-solid equilibria in the binary subsystem (docosane + tetracosane). J. Chem. Eng. Data 1997, 42, 562. (20) Madsen, H. E. L.; Boistelle, R. Solubility of octacosane and hexatriacontane in different n-alkane solvents. J. Chem. Soc., Faraday Trans. 1 1979, 75, 1254. (21) Chang, S.-S.; Maurey, J. R.; Pummer, W. J. Solubilities of two n-alkanes in various solvents. J. Chem. Eng. Data 1983, 28, 187. (22) Jin, Y.; Wunderlich, B. Heat capacities of paraffins and polyethylene. J. Phys. Chem. 1991, 95, 9000. (23) Messerly, J. F.; Guthrie, G. B.; Godd, S. S.; Finke, H. L. Low-temperature thermal data for n-pentane, n-heptadecane, and n-octadecane. J. Chem. Eng. Data 1967, 12, 338. (24) Parks, G. S.; Moore, G. E.; Renquist, M. L.; Naylor, B. F.; McClaine, L. A.; Fujii, P. S.; Hatton, J. A. Thermal data on organic compounds. XXV. Some heat capacity, entropy and free energy data for nine hydrocarbons of high molecular weight. J. Am. Chem. Soc. 1949, 71, 3386. (25) Sirota, E. B.; Singer, D. M. Phase transitions among the rotator phases of the normal alkanes. J. Chem. Phys. 1994, 101, 10873. (26) Parks, G. S.; Huffman, H. M.; Thomas, S. B. Thermal data on organic compounds. VI. The heat capacities, entropies and free energies of some saturated, non-benzenoid hydrocarbons. J. Am. Chem. Soc. 1930, 52, 1032. (27) Finke, H. L.; Gross, M. E.; Waddington, G.; Huffman, H. M. Low-temperature thermal data for the nine normal paraffin hydrocarbons from octane to hexadecane. J. Am. Chem. Soc. 1954, 76, 333. (28) Schaerer, A. A.; Busso, C. J.; Smith, A. E.; Skinner, L. B. Properties of pure normal alkanes in the C17 to C36 range. J. Am. Chem. Soc. 1955, 77, 2017. (29) Messerly, J. F.; Guthrie, G. B.; Todd, S. S.; Finke, H. L. Low-temperature thermal data for n-pentane, n-heptadecane, and n-octadecane. J. Chem. Eng. Data 1967, 12, 338. (30) Hoepfner, A.; Kreibich, U. T.; Schaefer, K. Effect of molecular formula on the thermodynamic properties of binary organic nonelectrolyte systems of nonpolar liquids. Ber. BunsenGes. 1970, 74, 1016. (31) Marsh, K. N.; Ott, J. B.; Costigan, M. J. Excess enthalpies, excess volumes, and excess Gibbs free energies for n-hexane + n-decane at 298.15 and 308.15 K. J. Chem. Thermodyn. 1980, 12, 343.

(32) Leslie, B. H.; Carr, A. R. Vapor pressure of organic solutions, and application of duhring’s rule to calculation of equilibrium diagrams. J. Ind. Eng. Chem. 1925, 17, 810. (33) Beatty, H. A.; Calingaert, G. Vapor-liquid equilibrium of hydrocarbon mixtures. J. Ind. Eng. Chem. 1934, 26, 504. (34) Ogorodnikov, S. K.; Kogan, V. B.; Morozova, A. I. Determination of boiling point of mixtures of substances with considerably different vapor pressures. Zh. Prikl. Khim. 1962, 35, 193. (35) Dejoz, A.; Gonzalez-Alfaro, V.; Miguel, P. J.; Vazquez, M. I. Isobaric vapor-liquid equilibria for binary systems composed of octane, decane, and dodecane at 20 kPa. J. Chem. Eng. Data 1996, 41, 93. (36) Wisniak, J.; Magen, E.; Shachar, M.; Zeroni, I.; Reich, R.; Segura, H. Phase equilibria in the systems hexane + heptane and methyl 1,1-dimethylethyl ether + hexane + heptane. J. Chem. Eng. Data 1997, 42, 458. (37) Wisniak, J.; Embon, G.; Shafir, R.; Segura, H.; Reich, R. Isobaric vapor-liquid equilibria in the systems methyl 1,1dimethylethyl ether + octane and heptane + octane. J. Chem. Eng. Data 1997, 42, 1191. (38) Miller, R. C.; Williamson, A. G. Excess molar enthalpies for (n-hexane + n-hexadecane) and for three- and four-component alkane mixtures simulating this binary mixture. J. Chem. Thermodyn. 1984, 16, 793. (39) Holleman, T. Heats of mixing of liquid binary normal alkane mixtures. Physica 1965, 31, 49. (40) McGlashan, M. L.; Williamson, A. G. Thermodynamics of mixtures of n-hexane + n-hexadecane. Trans. Faraday Soc. 1961, 57, 588. (41) Snyder, P. S.; Thomas, J. F. Solute activity coefficients at infinite dilution via gas-liquid chromatography. J. Chem. Eng. Data 1968, 13, 528. (42) Iwai, Y.; Yamashita, M.; Kohashi, K.; Arai, Y. Measurement and prediction of infinite dilution activity coefficients for C6 hydrocarbons in heavy paraffinic hydrocarbons. Kagaku Kogaku Ronbunshu 1998, 14, 706. (43) Friend, J. A.; Larkin, J. A.; Maroudas, A.; McGlashan, M. L. Temperature-dependence of the heat of mixing of two n-alkanes. Nature 1963, May, 683. (44) Elbro, H. S.; Fredenslund, A.; Rasmussen, P. A New equation for the prediction of solvent activities in polymer solutions. Macromolecules 1990, 23, 4707.

Received for review August 12, 2004 Revised manuscript received January 7, 2005 Accepted February 21, 2004 IE040219C