I m ELSEVIER Fluid Phase Equilibria 123 (1996) 29-38

Thermochemical investigations of hydrogen-bonded solutions. Part 10. Development of expression for predicting excess enthalpies of

ternary two alcohol + inert hydrocarbon systems based upon mobile order theory

Siddharth Pandey, Joyce R. Powell, Mary E. R. McHale, Ann-Soft M. Kauppila, William E. Acree Jr *

Department of Chemistry, University of North Texas, Denton, TX 76203-0068, USA

Received 6 November 1995; accepted 6 March 1996

Abstract

The unconventional solution model of mobile order theory, which has previously been shown to provide a very accurate thermodynamic description of anthracene and pyrene solubilities and chemical potentials in binary alcohol + alcohol solvent mixtures, is extended to other thermodynamic excess functions. An expression is derived for predicting excess molar enthalpies of ternary inert hydrocarbon + two alcohol systems from measured binary data. Applications and limitations of the newly derived predictive expression are assessed using published enthalpy data for 11 ternary alkane + 1-alcohol + 1-alcohol systems.

Keywords: Theory; Excess enthalpies; Ternary solution; Self-association; Alcohol cosolvents; Mobile order theory

I. Introduction

This work continues a systematic search for mixing models which will provide reasonable mathematical descriptions of the thermochemical properties of ternary non-electrolyte solutions which contain components capable of self-association. To date, we have examined both the application and limitations of mobile order theory to describe the solubilities of anthracene dissolved in 24 binary alcohol + alkane [1] and 28 binary alcohol + alcohol solvent mixtures [2,3], and of pyrene dissolved in 24 binary alcohol + alcohol solvent mixtures [4]. The basic model [5-10] assumes all molecular groups perpetually move in the liquid, and that neighbors of a given external atom in a molecule constantly change identity. All molecules of a given kind dispose of the same volume, equal to the

* Corresponding author.

0378-3812/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0378-3812(96)03057-9

30 S. Pandey et al./ Fluid Phase Equilibria 123 (1996) 29-38

deviations with respect bonding which requires group of a neighboring order".

total volume V of the liquid divided by the number N A molecules of the same kind, i.e. Dom A = V/N A. The center of this domain perpetually moves. Highest mobile disorder is achieved whenever groups visit all parts of their domain without preference. Preferential contacts lead to

to this " random" visiting. This is especially true in the case of hydrogen that a hydroxylic hydrogen atom follow most of the time the proton acceptor molecule in its walk through the liquid, thus originating a kind of "mobile

The thermodynamics of mobile order expresses the equilibrium condition in terms of time fractions for the time schedule of a given molecule, and not in terms of concentrations of various entities in the ensemble. Thus, in the case of alcohols one considers the time fraction ~ch: and not the concentrations of the various i-mers in the ensemble (this does not mean that these i-mers do not exist, but their concentrations do not govem the thermodynamic probability). ~/ch: is the fraction of the time during which a given molecule of the ensemble is free from hydrogen bonding, i.e. does not possess the energy of the hydrogen bond, but it is by no means the fraction of the time during which the molecule is free at both sides. A molecule bonded at one side is free from hydrogen bonding only half of the time.

For an inert crystalline solute dissolved in a binary alcohol (B) + alcohol (C) solvent mixture the volume fraction saturation solubility (+~t) is given by [2]:

RT{ln(a~A°'ia/~)/~at ) __ 0.511 -- VA//( x~3V B "~ x~V C )] --I- 0 .5 In[ VA/( xgV B + x~V c )]

- - ( V A I V B ) ~ [ (~B( KBIVB) "3v 6~( KBc/Vc)] / [1 "3v ~)~( KBIVB) + (~( gBc/Vc)]

-(VA/Vc)6~[&~3(KcB/VB) + 6~( Kc/Vc)]/[1 + dp~( KcB/VB) + 6~( Kc/V¢)]}

= VA[ &~(/~A-- ~)2 + &~(~A- 6~) 2 - tb~b~(~- ~)2] (1)

whenever the saturation solubility is sufficiently low that 1 - "ea'l'sat = 1.0. The symbols ~k, 8B and ~'c denote the modified solubility parameters of the solute and s e l f - associating alcohols, respectively. The remaining symbols are defined in detail at the end of this paper. Contributions from nonspecific interactions are incorporated into mobile order theory through the V A [t~B ° ( ~ A - ~'8) 2 + ~bc°(~a- ~ ) 2 __ (~Bo(~CO(~rB_ ~ ) 2 ] term. Through suitable mathematical manipulations, the VA(bB°(~ -- ~B) 2 and VA+C°(~A -- ~'C) 2 terms were eliminated from the basic model in favor of measured solubility

sat sat data in both pure solvents, (+A)B and (+A)C" The derived expression

In (j~at = ~ ln( ~/~at)B "~ ~ In( (])~at)c -- 0.5[ln( xgV B + x~Vc) - qb~ In V,

- &~ In Vc] - (VA/VB)6~[ qb~(KB/VB) + ck~(KBc/Vc) ]/[1 + qb~(KB/V~)

"t- 6~( KBc/Vc)] d- (VAKBdP~3/V2)(1 n t- KB/VB) -1 -- (VA/Vc)dp~[ C~3( KcB/VB)

+dp~( Kc/Vc)]/[1 + c~( Kc~/VB) + d~( Kc/Vc) ] + (VAKcdP~/Vg)(1 q'- Kc/Vc)-

+ VA&~&~(6~ - 6~)2(RT)- ' (2)

does not require a prior knowledge of the solute's enthalpy of fusion and melting point temperature, solid which would be needed to calculate the numerical value of a A at the temperature corresponding to

the solubility measurements. Three previous papers in this series [2-4] have shown that Eq. (2)

S. Pandey et aL / Fluid Phase Equilibria 123 (1996) 29-38 31

provides very reasonable predictions of the saturated mole fraction solubilities of both anthracene and pyrene dissolved in binary alcohol + alcohol solvent mixtures using a single equilibrium constant of K R = K c = KBC = KcB = 5000 cm 3 mol - l . The overall average absolute deviations between pre- dicted and observed solubilities were 1.7 and 2.2%, respectively, for anthracene and pyrene.

The success of Eq. (2) in describing the solubility behavior of anthracene and pyrene in various alcohol + alcohol mixtures suggests that the basic model might be applicable to other thermodynamic properties. To pursue this idea further, an expression for predicting excess molar enthalpies of ternary hydrocarbon + two alcohol mixtures based upon mobile order theory has been derived. The predictive applicability of the newly-derived expression is illustrated using published enthalpy data for four ternary n-heptane + ethanol + 1-alkanol, two ternary n-heptane + 1-propanol + 1-alkanol, four ternary n-octane + ethanol + 1-alcohol [11,12] and cyclohexane + ethanol + 1-propanol [13] mixtures. Excess molar enthalpies of ternary alcohol + two inert hydrocarbon systems were considered in an earlier publication [ 14].

2. Development of predictive expressions based upon mobile order theory

The Gibbs free energy of mixing for the ternary solution containing an inert hydrocarbon (component A) and two alcohol cosolvents (components B and C) is separated into three contribu- tions:

AG2~ ~ = (AGABC)conf + (AGABC)chem "1- (AGABC)phy s (3)

The first term describes the configurational entropy based upon the Huyskens and Haulait-Pirson [ 15] definition of solution ideality

(AGABc)conf = (0.5)st[ n A In thA + n~ In ~b B + n c In 6c + nA In x A + n B In x B + n c In Xc] (4)

whereas the latter two terms in Eq. (3) result from formation of hydrogen-bonded complexes and weak nonspecific physical interactions in the ternary solution, respectively.

The chemical contribution depends upon the functional groups present and the characteristics of the self-associating component. Alcohols have one hydrogen "donor" site and the lone pairs on the oxygen atom provide two "acceptor" sites. The maximum possible number of hydrogen bonds is determined by the number of sites that are in minority. Monofunctional alcohols do form self-associ- ated species, and there is no prior experimental evidence or computational reason to preclude formation of heterogeneous B~Cj complexes, particularly if we are using a single equilibrium constant to describe every monofunctional alcohol's self-association characteristics. For each alcohol, the fraction of time that the alcohol is not involved in hydrogen-bond formation, ~Bh: and ~/Ch:, is calculated as:

1/[1 + + K 6c/V ]

and

(5)

YCh: = 1/[1 + KcBdpB/VB +Kccl, c/Vc] (6)

where KBC and Kc~ refer to the two additional equilibrium constants needed to describe the formation of the new heterogeneous alcohol-alcohol complexes not found in either neat solvent. It

32 S. Pandey et aL / Fluid Phase Equilibria 123 (1996) 29-38

can be readily shown that this set of conditions leads to the following expression for the Gibbs free energy for hydrogen-bonding contribution

(AGABC)chem = nBRT ln[(1 + KB/VB)/ (1 + KBC~B/V B + KBcqbc/Vc)]

+ ncRT In[(1 + K c / V c ) / ( 1 + KCB~PB/V B 4. Kcchc/Vc)] (7)

Physical effects are expressed in terms of the nearly ideal binary solvent (NIBS) model developed by Bertrand and co-workers [16-18]:

(AGgeBC)phys = (nAV A + nBV B + ncVc)- l × (nAV AnBVB AAB + nAVAncV c AAC

+nBVBncVcABc ) (8)

where the generalized weighting factors are approximated by molar volumes and Aij is a binary interaction parameter that is assumed to be independent of composition. Few non-electrolyte systems, even those believed to contain only non-specific interactions, actually obey simple mixing models like Eq. (8) in the strictest sense. Many real mixtures exhibit behavior suggesting a slight compositional dependence in the A/;parameter. From a purely mathematical standpoint, some degree of composi- tional dependency can be incorporated into the model, while still retaining the model 's highly desirable computational simplicity, by eliminating as many of the binary interactional parameters as possible from the final derived equation, as was done in the development of the NIBS model. The NIBS model has been shown to provide accurate predictions for naphthalene, iodine, p-dibromoben- zene, benzil, p-benzoquinone, anthracene, pyrene and carbazole solubilities in systems containing only non-specific interactions when all three weighting factors are approximated with molar volumes [19].

Combining Eqs. (3)-(8), the Gibbs free energy of mixing (per stoichiometric mole of solution) is written as

AG~Xc = (0.5)RT[x A In ~A +xB In ~b B + x c In 6c +XA In x A + x B In x B

+ x c In Xc] + x B RT In[(1 + KB/Va) / (1 + KBCkB/VB + KBCchc/Vc) ]

+ x c RT ln[(1 + K c / V c ) / ( 1 + KCBChB/VB + Kccbc/Vc) ]

+ ( XAV A + XBV B + xcVc) [ ~bgq~ B AAB + ~bA~6 c Aac + ~b B ~b c ABC ] (9)

Standard thermodynamic principles relate the excess enthalpy and Gibbs free energy: mix a Gc/R = ( lo)

Assuming that the molar volumes of the pure components are independent of temperature, differentia- tion of Eq. (9) with respect to 1 / T gives the following expression for A HAESC:

AHEBc = AH°(XB + xc)K{[( rB/VB) 4. ( r c / V c ) ] / [ 1 4- K( qbB/V s 4- thc/Vc)] }

- + K/V ) - a H ° X c ( r / V c ) / ( 1 + K / V c )

+(x,,VA + x VB + xcVc)[ + G 'c Bac + Bc] (1 l)

where Bi; = 3( A i J T ) / 3 ( I / T ) and AH ° is the standard enthalpy of formation of a hydrogen bond. The above expression assumes that self-association constants and standard enthalpies of hydrogen-bond formation for all of the various homogeneous and heterogeneous alcoholic complexes are equal. For

S. Pandey et al. / Fluid Phase Equilibria 123 (1996)29-38 3 3

notational simplicity, the subscript has been dropped from the association constants since all constants have the same numerical value.

For model systems obeying mobile order theory, the excess enthalpies of the two contributing inert hydrocarbon + alcohol sub-binaries

AH& = - a l l ° x~[( K / V K ) / ( l + K/V~) - ( ~ K / V ~ ) / ( 1 + 6~ K/VK)]

+ ( x ~ V A +X°KVK)ck2ck~BAK K = B , C (12)

AH~K= - A H ° x~[ ( K I V K ) / ( 1 + K I V K ) - ( dp~ K / V K ) / ( 1 + dp~ K / V K ) ]

E * +(aH&)ph. (13)

and alcohol + alcohol sub-binary

A HEc = AH°K{[(ck~/VB) + ( d?~/Vc)] /[1 + K ( dp~/V B + ck~/Vc)]}

- x~ AH ° (K/VB) / (1 + K/VB) - AH ° x3(K/Vc) / (1 + K /Vc)

+ (x~V~ + x~V~)6~4,~B~c (14)

a H~c = a H o K { [ ( ~ 4 / V ~ ) + (~,~/V~)]/[1 + r ( ~ 4 / V . + ~,~/V~)]}

- x~ A H ° ( K / V B ) / ( 1 + K / V B ) - A H ° x ~ ( K / V c ) / ( 1 + K / V c )

HB~)ph. (15) +(A E *

would be given by the appropriate reduction of Eq. (11), obtained by setting the number of moles of the third component equal to zero. Here, mole and volume fraction compositions in the ij binary

o o o _ xjVj), respectively. system are x i = 1 - x j° _- x i / ( xi + xi ) and 4) i = 1 - thj- - xiVi/( xiV i @ Careful examination of Eqs. (I 1)-(I 5) reveals that it is possible to mathematically manipulate the

ternary expression, so as to eliminate all three binary interaction parameters in favor of measured binary excess enthalpy data. Performing this mathematical manipulation, one obtains the following expression for A HEBc:

A HEBc = A H ° ( x B + x c) K{[( 4)B/VB ) + ( ~bc/V c ) ] / [ I + K ( 4>B/VB + 4)c/Vc )]}

- x B A H ° ( K / V B ) / ( 1 + K / V B ) -- A H o x c ( K / V c ) / ( 1 + K / V c )

+ ( 4)A + (kB)( XA + XB)(A HAEB)phy, + (~A + 4)C)( XA + Xc)(A HAEC) phy,

+ ( 4)B + 4)C)( XB + Xc)(A HEc)phy, (16)

E * where (AHij)phy s refers to an actual numerical value for the physical contribution to the observed excess enthalpy of the ij binary mixture calculated at mole fraction composition (x~ °, x i°), such that x ° = 1 - x~ = x J ( x ~ + x j). The mathematical manipulation used corresponds to our having evaluated each individual Bij-parameter from a single measured binary excess enthalpy, rather than having obtained the so-called "average" Bq value that would best describe the measured enthalpy data for the entire sub-binary system. We feel that our method is preferable in that very few non-electrolyte solutions obey simple mixing models like Eq. (8). Many real mixtures exhibit behavior suggesting a slightly compositionally dependent Bij interaction parameter. In such cases, parameters should be

3 4 S. Pandey et al. / Fluid Phase Equilibria 123 (1996) 29-38

deduced from binary mixtures whose molecular interactions are as similar to the ternary mixture as possible. We feel that this is best accomplished by maintaining the relative mole numbers of the two components in the ternary and sub-binary systems constant. In the absence of self-association (K c = K B = 0) and cross-association (KBc = KcB = 0), Eq. (16) is mathematically equivalent to the BAB model, which has been shown to provide very reasonable estimates of both ternary (and higher-order multicomponent) thermodynamic and physical properties from measured binary data [20-23].

3. Results and discussion

The chemical literature contains excess enthalpy data for a fairly large number of binary saturated hydrocarbon + alcohol systems, with alcohols ranging in size from methanol to 1-decanol (see for example Refs. [11,24,25]), and for a limited number of ternary inert hydrocarbon + two alcohol systems. Published experimental data can be used to test the limitations and applications of mobile order theory, in terms of its ability both to predict ternary values from measured binary data and to

E * mathematically describe the various sub-binaries with reasonable calculated values for the (A Hij )phys contributions. Rather than performing computations on all systems found in the literature, we selected four ternary n-heptane + ethanol + 1-alkanol, two ternary n-heptane + 1-propanol + 1-alkanol, four ternary n-octane + ethanol + 1-alcohol and cyclohexane + ethanol + 1-propanol mixtures, along with the corresponding 12 hydrocarbon-I-alcohol sub-binaries, as a representative set. These particular systems were selected because the experimental values appear to have been determined with a high degree of precision, the components have different molecular sizes, and there are at least 30 data points in each individual ternary system. The cyclohexane + ethanol + l-propanol system has over 90 data points. Moreover, the excess enthalpies were measured at 298.15 K, which corresponds to the temperature at which the mobile order alcohol self-association constants are KB~or c ) = 5000 cm 3 mol-

E * Table 1 summarizes results of the mathematical description and computation of the (AHKc)phy s contributions via Eq. (13) for the cyclohexane + 1-propanol [13] and n-octane + 1-propanol [11] sub-binary systems, using assumed value of K = 5000 cm 3 mo1-1 and AH ° = - 2 5 . 4 kJ mol - l (average value reported for primary alcohols of ethanol to 1-hexanol, see Huyskens [26]). Examina- tion of the second and third columns of the table reveals that mobile order theory correctly predicts the highly skewed excess enthalpy versus binary mole fraction composition curves; however, numerical values calculated for the chemical contribution exceed the measured excess enthalpies at low alcohol mole fraction. This results in an "S-shaped" E • o (AH~c)phy s versus XAlcoho I < 0.15 which become progressively more negative with increasing molecular size of the inert hydrocarbon. Calculations for the remaining alkane + 1-alkanol binary systems show similar trends. Simple mixing models predict positive excess molar enthalpies in systems of non-specific interactions. For example, in the Scatchard-Hildebrand solubility parameter theory (see [27-29]) the excess enthalpy is

A Hi E -~ ( x°Vi + x~Vj)~Pi°~P;( ~ i - 6j) 2 (17)

a product of molar volume times volume fractions times the difference in solubility parameters squared.

S. Pandey et al . / Fluid Phase Equilibria 123 (1996) 29-38 35

Table 1 Physical contribution to the excess enthalpies (in J mol-l) of binary alkane (A)+ alcohol (B) mixtures based upon mobile order theory

A H ° = - 25.4 kJ tool- l

xA ° (AHEB) Cxp (J mol-l) E * (A H,~)ch~m

A H ° ~ - 20.4 kJ mol - t

(a/4~B~;.. Cyclohexane (A)+ 1-propanol (B) 0.1312 209.0 70.1 138.9 56.3 152.7 0.2257 334.3 120.3 214.0 96.6 237.7 0.2710 387.5 144.2 243.3 115.8 271.7 0.3656 482.3 193.6 288.7 155.5 326.8 0.4635 558.6 244.0 314.6 196.0 362.6 0.5182 598.5 271.2 327.3 218.2 380.3 0.5827 614.1 303.6 310.5 243.8 370.3 0.6604 622.9 340.0 282.9 273.1 349.8 0.7504 597.9 378.0 219.9 303.6 294.3 0.8478 516.2 406.2 110.0 326.2 190.0 0.9345 373.4 383.7 - 10.3 308.2 65.2 0.9806 227.9 252.4 - 24.5 202.7 25.2

n-Octane(A)+ethanol(B) 0.1006 274 81 193 65 209 0.1828 408 147 261 118 290 0.2444 480 197 283 158 322

00.2814 515 225 290 181 334 0.3113 541 249 292 200 341 0.3922 594 312 282 250 344 0.5301 644 415 220 333 311 0.6930 637 524 113 421 216 0.8188 565 580 - 15 466 99 0.8997 467 566 -99 455 12 0.9561 347 455 -108 366 - 19

a Molar volumes (V) used in mobile order computations were 108.76 cm 3 mol- J for cyclohexane, 163.46 cm 3 tool- i for n-octane, 58.69 cm 3 mol-~ for ethanol and 75.10 cm 3 mol-~ for 1-propanol.

From a purely mathematical standpoint, contributions from chemical effects can be reduced by

assuming either a larger numerical value for the K self-association constant or a smaller standard

enthalpy of hydrogen-bond formation, A H °. Literature precedent exists for both options. In the

Mecke -Kempte r and Kre t schmer -Wiebe continuous self-association models each alcohol has a

different equilibrium constant. Small primary alcohols like methanol and ethanol have larger K

values, whereas long alkyl chain and branched alcohols self-associate to a much lesser extent, perhaps

because the steric hindrance in these molecules discourages strong hydrogen-bond formation. The so-called " b e s t " , calculated K and A H ° values for any given alcohol depend upon the specific

systems considered, and whether the data reduction was limited to only the properties of the pure

alcohol or included binary phase equilibria a n d / o r excess enthalpy data. The equilibrium constant and

standard enthalpy of hydrogen-bond formation for methanol o f K = 261 (at 323.15 K) and A H ° =

- 2 1 . 8 6 kJ mol-1 , determined from the enthalpy of vaporization and boiling point temperature of the

alcohol, differ significantly from the numerical values o f K = 450 (at 323.15 K) and A H ° = - 2 5 . 1 0

36 S. Pandey et al. / Fluid Phase Equilibria 123 (1996) 29-38

Table 2 Summarized comparison between experimental excess molar enthalpies and predicted values based upon mobile order theory for select ternary inert hydrocarbon + two alcohol mixtures

Ternary system Dev. (J mol-- l ) a.b

n-Heptane + ethanol + l-propanol c n-Heptane + ethanol + l-pentanol c n-Heptane + ethanol + l-octanol c n-Heptane + ethanol + l-decanol c n-Heptane + l-propanol + l-pentanol c n-Heptane + l-propanol + I-octanol c n-Octane + ethanol + l-propanol c n-Octane + ethanol + 1-pentanol c n-Octane + ethanol + 1-octanol c n-Octane + ethanol + 1-decanol c Cyclohexane + ethanol + l-propanol o

7.3 17.5 16.1 19.3 12.9 13.3 9.0

16.5 16.0 21.5 6.8

a Dev.=(1/N)EKAHEBc)eXp_(AHEac)¢~cl. b Molecular volumes (V) used in the mobile order computations were 147.48 cm 3 tool- ~ for n-heptane, 163.46 cm 3 mol- for n-octane, 108.76 cm 3 tool-1 for cyclohexane, 58.69 cm 3 tool-i for ethanol, 75.10 cm 3 tool-1 for 1-propanol, 108.68 cm 3 tool -1 for 1-pentanol, 158.30 cm 3 mol -~ for l-octanol and 191.56 cm 3 tool -1 for 1-decanol. c Experimental data taken from Ramalho and Ruel [11,12]. d Experimental data taken from Nagata and Kazuma [13].

kJ m o l - ~, which are commonly used in the K r e t s c h m e r - W i e b e model for predicting excess enthalpies [30]. Based upon the preceding discussion and unpublished excess enthalpy calculations for over 100 different binary alcohol + saturated hydrocarbon systems, a prel iminary value of A H0 = - 2 0 . 4 kJ m o l - ~ has been assigned to the standard enthalpy of hydrogen-bond formation so as to give the more

E * realistic set o f most ly posit ive (AHKc)phy s values listed in the last column of Table 1. This particular assumption will need to be re -examined in greater detail whenever more computat ions using mobi le order theory become available.

Table 2 gives a summarized compar ison between experimental excess enthalpies for four ternary n-heptane + ethanol + 1-alkanol, two ternary n-heptane + 1-propanol + 1-alkanol, four ternary n-oc- tane + ethanol + 1-alcohol and cyc lohexane + ethanol + 1-propanol mixtures, and predicted values based upon Eq. (16) with K = 5000 cm 3 mol -~ and A H ° = - 2 0 . 4 kJ m o l - i . As noted previously, each ternary system contains a m i n i m um of 30 individual data points measured at alkane mole fraction ratios of x , ~ / X c ° = 0.3, x A°/x c ° - - 1.0 and x ~ / X c ° - . 3.0, so as to cover a significant portion of the entire ternary composi t ion range. Replicate measurements at select composi t ions indicated that the published data were reproducible to about A HAEBc = -J- 10 J mol - t . Examinat ion of Table 2 reveals that Eq. (16) provides very reasonable estimates of the ternary excess enthalpies of the 11 ternary systems considered. The overall average absolute deviation between predicted and observed was AHAEBc = _+ 14.2 J mol -~, which is slightly less than 1.5 t imes the experimental uncertainty. As a point o f reference, we note that had one elected to predict the excess enthalpies of these 11 ternary systems based solely upon the BAB model (Eq. (16) with all self-association and cross-association equilibrium constants set equal to zero), then calculated individual average absolute deviations would be about 5 - 1 5 J mo1-1 larger for the better o f the BAB predict ive expressions

S. Pandey et al. / Fluid Phase Equilibria 123 (1996) 29-38 37

[20,21]. Excess enthalpies of binary alcohol + hydrocarbon mixtures are highly skewed. The better BAB predictive expression involves computation of weighted mole fraction compositions based upon normalized weighting factors calculated at sub-binary compositions of XAlcoho~ ° = 0.333 and XA~¢oho~ ° = 0.666. The BAB method is discussed in greater detail elsewhere [20,21]. The computations presented here, combined with our earlier solubility studies involving anthracene dissolved in hydrocarbon+ alcohol mixtures [1] and alcohol+alcohol mixtures [2,3] and involving pyrene dissolved in alcohol + alcohol mixtures [4], clearly show that mobile order theory provides a reasonable (although by no means perfect) thermodynamic description of binary and ternary mixtures containing inert hydrocarbons and alcohols.

4. List of symbols

a~ olid

A i j

B i j

mix AG Bc An BC

E * (AH u)phy~

AH °

KB, Kc K i j

n i

V i . . 0 0

X i , Xj

X i

Activity of the solid solute, defined as the ratio of the fugacity of the solid to the fugacity of the pure sub-cooled liquid Binary interaction parameter for describing physical, non-specific molecular interactions between components i and j in the NIBS model for Gibbs free energy Binary interaction parameter for describing physical, non-specific molecular interactions between components i and j in the NIBS model for enthalpy of mixing Gibbs free energy of mixing for the ternary system Excess molar enthalpy of mixing of the ternary non-electrolyte solution Portion of the excess molar enthalpy of the ij binary system at mole fraction compositions x ° and x)', which is attributed to non-specific interactions Standard molar enthalpy of formation of a hydrogen-bond for both the self-association and cross-association of alcohols Mobile order self-association constants for alcohols B and C, respectively. Mobile order cross-association constant for describing the hydrogen-bonding of the monomer i to a "polymeric" alcohol chain ending with alcohol j Number of moles of component Molar volume of component i Mole fraction compositions of the ij binary mixture, calculated as if the third component were not present Mole composition of component i in the ternary solution

4.1. Greek symbols

~Ch: Fraction of the time during which a given molecule C of the ensemble is free from hydrogen-bonding Scatchard-Hildebrand solubility parameter of component i Modified solubility parameter of component i Ideal volume fraction compositions of the ij binary mixture, calculated as if the third component were not present Ideal volume fraction solubility of the solute

38 S. Pandey et al. / Fluid Phase Equilibria 123 (1996) 29-38

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