J Solution Chem (2010) 39: 680–691DOI 10.1007/s10953-010-9526-9

Thermodynamic Investigation of Molecular Interactionsin 1,3-Dioxolane or 1,4-Dioxane + Benzene orToluene + Formamide or +N,N -DimethylformamideTernary Mixtures at 308.15 K and Atmospheric Pressure

Satish Kumar · V.K. Sharma · J.S. Yadav · Il Moon

Received: 18 July 2009 / Accepted: 7 December 2009 / Published online: 5 May 2010© Springer Science+Business Media, LLC 2010

Abstract Excess molar volumes, V E123 of 1,3-dioxolane or 1,4-dioxane (1) + benzene or

toluene (2) + formamide or + N,N -dimethylformamide (3) ternary mixtures at 308.15 Kand at atmospheric pressure have been determined dilatometrically over the entire compo-sition range. The excess molar volumes data of these ternary systems were fitted to theRedlich–Kister equation. The data have been analyzed in terms of Graph theory (model)to understand the nature and strength of molecular interactions existing in these mixtures.It has been observed that V E

123 values predicted by Graph theory compare well with theircorresponding experimental values.

Keywords Excess molar volumes · Connectivity parameter of third degree · Graphtheory · Molecular interactions

1 Introduction

Thermodynamic properties of multicomponent liquid mixtures have attracted the attentionof researchers in the past decade as they are required for the design of several types of in-dustrial equipment, and for many industrial applications such as design calculation, heattransfer, fluid flow and so forth. Moreover, these properties can provide an important toolto extract information about the state of aggregation of components in pure as well as inmixed states and also about the nature and extent of interactions between the constituentsof mixtures. Representation of thermodynamic data for binary and ternary mixtures by the

S. Kumar (�) · Il MoonDepartment of Chemical & Bimolecular Engineering, Yonsei University, 134 Shinchon-dong,Seodaemun-gu, Seoul 120-749, Republic of Koreae-mail: vatssatish@yahoo.co.in

Il Moone-mail: ilmoon@yonsei.ac.kr

V.K. Sharma · J.S. YadavDepartment of Chemistry, Maharshi Dayanand University, Rohtak, India

J Solution Chem (2010) 39: 680–691 681

Redlich–Kister [1] equation and prediction of thermodynamic properties such as molar ex-cess volumes, molar excess enthalpies and excess isentropic compressibilities, etc. by usingvarious models as theoretical tools is a task of considerable importance to gain insight intothe molecular interactions among the constituents of a mixture.

Chemical Graph theory [2] is interested in the nature of chemical structures. The struc-tural formula of compounds represents their molecular graphs, where atoms are representedby dots and bonds joining them by lines. The molecular graph of a molecule represents thetotal information contained in that molecule. Topological indices in numerical form expressthe topology of the chemical species. The construction and investigation of topological in-dices which could describe molecular topology is one of the main directions of chemicalGraph theory. Kier [3] has suggested that the connectivity parameter of third order of amolecule, 3ξ (which depends upon the topology of a molecule), could be utilized to extractinformation about the effect of branching in a molecule and Singh et al. [4] have shown that(3ξ)−1 represents a measure of the probability that the surface area of a molecule interactseffectively with the surface area of other molecules. The concept of the connectivity para-meter of third order of the constituents of binary and ternary mixtures has been utilized todevelop the Graph theoretical approach to evaluate excess molar volumes, molar excess en-thalpies and excess isentropic compressibility of binary [5–7] and ternary [8–11] mixtures.

In our present work we report the excess molar volumes of 1,3-dioxolane or 1,4-dioxane(1) + benzene or toluene (2) + formamide or N,N -dimethylformamide (3) ternary mixturesat 308.15 K. Further, it would be of interest to see how well Graph theory (which involvesthe connectivity parameter of third degree) describes the excess molar volumes data of theseternary mixtures.

2 Experimental

1,3-dioxolane (Fluka USA 99%), 1,4-dioxane (Fluka USA 99%), formamide (FD), N,N -dimethylformamide (DMF), benzene and toluene (AR Grade) were purified by standardmethods [12, 13]. The purities of the purified liquids were checked by measuring their den-sities by a glass pycnometer at 298.15 ± 0.01 K and these agreed to ±5 × 10−5 g·cm−3 withtheir literature values [12, 13] and are reported in Table 1.

Excess molar volumes, V E123, for various (1 + 2 + 3) ternary mixtures were measured

using a three-limbed specially designed dilatometer (shown in Fig. 1). It consists of threelimbs (shown as A, B and C in Fig. 1) having vacuum tested B-7 standard joints at all threelimbs (capacity ≈ 4 cm3 each). All the three limbs were interconnected by small bent tub-ing which was further connected to a thin uniform bored capillary having a reference mark

Table 1 Comparison ofdensities, ρ, of pure liquids alongwith their literature values atT = 298.15 K

Liquids T (K) ρ (g·cm−3)

Experimental Literature

1,3-dioxolane 298.15 1.05885 1.05881 [13]

1,4-dioxane 298.15 1.02792 1.02797 [13]

Benzene 298.15 0.87360 0.87362 [12]

Toluene 298.15 0.86219 0.86220 [12]

Formamide 298.15 1.12926 1.12918 [12]

N,N -dimethyl 298.15 0.94445 0.94447 [12]

formamide

682 J Solution Chem (2010) 39: 680–691

Fig. 1 Schematics of thethree-limbed dilatometer

R on it. A column of mercury (3–5 cm3) was placed in the dilatometer, in order to avoidthe contact of pure liquids, and the dilatometer was weighed. One of the liquids was addedto the first limb A with the help of a hypodermic syringe in such a way that no air bubbleswere entrapped in this bulb. The dilatometer was weighed again. Then the second liquid wasadded to the second limb B in similar fashion and the dilatometer was weighed again. Thenthe third liquid was added to the third limb C in similar fashion and the dilatometer wasweighed again. All the three liquids were degassed before filling the limbs. The dilatometerwas then immersed in a water thermostat. The temperature of water thermostat was con-trolled to ±0.01 K by a mercury-toluene regulator. After thermal equilibrium, the referencemark R and the position of the liquid in the capillary were noted with a cathetometer (OSAW,INDIA) that could read to ±0.001 cm. All three liquids in the dilatometer were then mixedby gently tilting it sideways several times. In order to ensure that the liquid in the capillarystem had the same composition as that of the bulk solution on mixing, the dilatometer (aftermixing the components) was placed in a cold bath so that there was a minimum of liquidvolume in the capillary and then it was again immersed in the water thermostat. This processwas repeated thrice. Once thermal equilibrium was achieved, the position of the referencemark R and that of the level of the liquid in the capillary was again recorded. The dilatometerwas then taken out, cleaned and dried. The composition of mixture was changed by varyingthe amount of mercury, liquid components and position of mercury in the dilatometer. Molarexcess volumes of the ternary mixtures were calculated by the following relation:

V E/cm3·mol−1 = (�h/cm) × (capillary coefficient/cm2)/{(nA + nB + nC)/mol} (1)

where �h is the change in height of liquid level in the capillary before and after mixing andnA, nB and nC are the number of moles of liquid A, B and C, respectively. The capillarycoefficient was the cross-sectional area of the capillary. The radius of the capillary tube wasdetermined from the weight of a length of mercury column in the capillary. The length ofthe mercury column at various positions in the capillary was read by a traveling microscope(OSAW, Ambala) that could read up to ±0.001 cm. The density of mercury was taken from

J Solution Chem (2010) 39: 680–691 683

Fig. 2 Excess molar volume of1,3-dioxolane (1) + benzene (2)+ formamide (3) atT = 308.15 K; —, represents theexperimental data in front of theplane; - - -, represents theexperimental data behind theplane; x1, x2 and x3 are the molefraction of component 1, 2 and 3,respectively

the literature [14]. The cross-sectional area of the capillary was calculated from the aver-age length of the mercury column in the capillary and the volume of this mercury column.The reliability of the dilatometer was checked by measuring V E values for benzene + cy-clohexane mixtures at 298.15 K and these values agreed with their corresponding literaturevalues [15]. The uncertainties in our measured V E

123 values are ±0.5%.

3 Results and Discussion

Excess molar volumes, V E123, of 1,3-dioxolane or 1,4-dioxane (1) + benzene or toluene (2) +

FD or + DMF (3) ternary mixtures were determined dilatometrically at 308.15 K (recordedin Table 2 and plotted in Figs. 2–5) and were fitted to the Redlich–Kister [1] equation:

V E123 = x1x2

[2∑

n=0

V(n)

12 (x1 − x2)n

]+ x2x3

[2∑

n=0

V(n)

23 (x2 − x3)n

]

+ x1x3

[2∑

n=0

V(n)

13 (x1 − x3)n

]+ x1x2x3

[2∑

n=0

V(n)

123(x2 − x3)nxn

1

](2)

where x1, x2 are the mole fractions of components (1) and (2) in the (1 + 2 + 3) ternarymixture. V

(n)

12 (n = 0–2), etc. are parameters characteristic of (1 + 2), (2 + 3) and (1 + 3)

binary mixtures and have been taken from literature [16, 17]. V(n)

123 (n = 0–2) are adjustableparameters characteristic of (1 + 2 + 3) ternary mixtures and were determined by fittingexperimental V E

123 data to Eq. 3⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

V E123 − x1x2

2∑n=0

V(n)

12 (x1 − x2)n

− x2x3

2∑n=0

V(n)

23 (x2 − x3)n

− x1x3

2∑n=0

V(n)

13 (x1 − x3)n

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

[x1x2x3

] =2∑

n=0

V(n)

123(x2 − x3)nxn

1 (3)

684 J Solution Chem (2010) 39: 680–691

Table 2 Comparison ofmeasured V E

123 values for thevarious (1 + 2 + 3) ternarymixtures, evaluated from graphtheory, with their correspondingexperimental values atT = 308.15 K along with theirroot mean square deviations,(σ(V E

123)); molar volumeinteraction parameters(χ∗

12, χ∗33), etc. and connectivity

parameters of third degree, 3ξi(i = 1 to k) are also included

x1 x2 V E123 (cm3·mol−1)

Experimental Graph theory

1,3-Dioxolane (1) + Benzene (2) + Formamide (3)

0.0580 0.0595 −0.095 −0.106

0.1080 0.8250 −0.105 −0.105

0.1520 0.5120 −0.292 −0.236

0.1910 0.1125 −0.250 −0.265

0.2269 0.7258 −0.146 −0.222

0.2550 0.1999 −0.204 −0.206

0.2655 0.6500 −0.202 −0.226

0.3502 0.2246 −0.184 −0.184

0.3843 0.4020 −0.246 −0.231

0.4347 0.3120 −0.201 −0.202

0.4612 0.2150 −0.149 −0.140

0.5308 0.3270 −0.203 −0.203

0.5842 0.1820 −0.112 −0.092

0.6478 0.2580 −0.148 −0.171

0.6712 0.1051 −0.061 −0.019

0.7351 0.1501 −0.080 −0.080

0.7910 0.0991 −0.052 −0.028

0.8108 0.0650 −0.030 −0.024

(3ξ1) = 0.601, (3ξ2) = 0.666, (3ξ3) = 0.440

χ ′33 = 1.678, χ∗

12 = −0.893, χ∗23 = −1.0821, χ∗

13 = 0.697,

σ(V(E)123 ) = 0.027

1,3-Dioxolane (1) + Toluene (2) + Formamide (3)

0.0991 0.5520 −0.480 −0.682

0.1050 0.1525 −0.350 −0.395

0.1221 0.7810 −0.326 −0.581

0.1521 0.6210 −0.550 −0.722

0.1928 0.3010 −0.383 −0.391

0.2010 0.2856 −0.369 −0.369

0.2516 0.1025 −0.186 −0.100

0.2716 0.2500 −0.316 −0.322

0.3102 0.2102 −0.261 −0.261

0.3642 0.0460 −0.114 −0.020

0.4125 0.4002 −0.220 −0.200

0.4389 0.1978 −0.212 −0.241

0.4530 0.0503 −0.124 −0.020

0.5371 0.2020 −0.226 −0.245

0.5702 0.1458 −0.115 −0.157

0.6107 0.1823 −0.213 −0.213

0.7310 0.0961 −0.080 −0.086

0.7775 0.1320 −0.150 −0.139

0.8715 0.0467 −0.033 −0.033

(3ξ1) = 0.601, (3ξ2) = 0.840, (3ξ3) = 0.440

χ ′33 = −2.948, χ∗

12 = −1.269, χ∗23 = −0.109, χ∗

13 = 0.140,

σ(V E123) = 0.090

J Solution Chem (2010) 39: 680–691 685

Table 2 (Continued)

χ∗12, χ ′

33, . . . , etc. are

in cm3·mol−1

x1 x2 V E123 (cm3·mol−1)

Experimental Graph theory

1,4-Dioxane (1) + Benzene (2) + N,N -dimethylformamide (3)

0.0215 0.0995 −0.150 −0.328

0.1058 0.3894 −0.190 −0.190

0.1525 0.5210 −0.075 −0.020

0.1925 0.4020 −0.085 −0.081

0.2269 0.7521 −0.057 −0.110

0.2450 0.3332 −0.084 −0.153

0.3531 0.4764 −0.072 −0.133

0.4012 0.1847 −0.186 −0.186

0.4715 0.2250 −0.032 −0.118

0.5080 0.2100 −0.023 −0.109

0.5130 0.1858 −0.057 −0.122

0.5819 0.1254 −0.091 −0.104

0.6202 0.1005 −0.084 −0.084

0.6314 0.1111 −0.066 −0.085

0.6667 0.1020 −0.017 −0.073

0.7424 0.0521 −0.025 −0.025

(3ξ1) = 0.801, (3ξ2) = 0.666, (3ξ3) = 0.589

χ ′33 = 6.417, χ∗

12 = 0.149, χ∗23 = −2.316, χ∗

13 = 0.271, σ(V E123) = 0.065

1,4-Dioxane (1) + Toluene (2) + N,N -dimethylformamide (3)

0.0630 0.8913 −0.084 −0.086

0.1352 0.7535 −0.148 −0.141

0.1269 0.8010 −0.110 −0.115

0.2103 0.6521 −0.135 −0.135

0.2510 0.0510 −0.040 −0.036

0.3123 0.6825 −0.052 −0.018

0.3720 0.5825 −0.065 −0.065

0.4965 0.3571 −0.072 −0.072

0.5315 0.3020 −0.103 −0.064

0.5655 0.2821 −0.092 −0.060

0.5791 0.2250 −0.116 −0.058

0.6328 0.1862 −0.114 −0.059

0.6750 0.1050 −0.080 −0.080

0.7329 0.0990 −0.058 −0.070

0.7882 0.0866 −0.038 −0.059

(3ξ1) = 0.801, (3ξ2) = 0.840, (3ξ3) = 0.589

χ ′33 = −2.322, χ∗

12 = −0.027, χ∗23 = 0.870, χ∗

13 = −0.55,

σ(V E123) = 0.121

by the least squares method. Such parameters along with the standard deviation, σ(V(n)

123)

defined by Eq. 4 are reported in Table 3.

σ(V E123) = [(V E

123)exp t1 − (V E123)calc.Eq. 2]/(m − n)]0.5 (4)

686 J Solution Chem (2010) 39: 680–691

Fig. 3 Excess molar volume of1,3-dioxolane (1) + toluene (2)+ formamide (3) atT = 308.15 K; —, represents theexperimental data in front of theplane; - - -, represents theexperimental data behind theplane; x1, x2 and x3 are the molefraction of component 1, 2 and 3,respectively

Fig. 4 Excess molar volume of1,4-dioxane (1) + benzene (2) +N,N -dimethylformamide (3) atT = 308.15 K; —, represents theexperimental data in front of theplane; - - -, represents theexperimental data behind theplane; x1, x2 and x3 are the molefraction of component 1, 2 and 3,respectively

Fig. 5 Excess molar volume of1,4-dioxane (1) + toluene (2) +N,N -dimethylformamide (3) atT = 308.15 K; —, represents theexperimental data in front of theplane; - - -, represents theexperimental data behind theplane; x1, x2 and x3 are the molefraction of component 1, 2 and 3,respectively

where (V E123)calc.Eq. 2 are the values evaluated by employing Eq. 2; m is the number of data

points and n is the number of adjustable parameters of Eq. 3.We are unaware of any previously published V E

123 data with which to compare our results.The measured V E

123 values for 1,3-dioxolane or 1,4-dioxane (1) + benzene or toluene (2) +

J Solution Chem (2010) 39: 680–691 687

Table 3 Redlich–Kisterparameters, V

(n)123 (n = 0–2), and

standard deviations, σ(V E123),

obtained from Eqs. 2 and 4,respectively, at T = 308.15 K

V(n)123, σ(V E

123), etc. are in

cm3·mol−1

T (K) V(0)123 V

(1)123 V

(2)123 σV E

123

1,3-Dioxolane (1) + Benzene (2) + Formamide (3)

308.15 −1.250 −10.321 −100.000 0.002

1,3-Dioxolane (1) + Toluene (2) + Formamide (3)

308.15 −6.258 −95.066 −536.492 0.003

1,4-Dioxane (1) + Benzene (2) + N,N -dimethylformamide (3)

308.15 4.622 50.824 −175.586 0.001

1,4-Dioxane (1) + Toluene (2) + N,N -dimethylformamide (3)

308.15 −2.527 25.524 150.092 0.001

FD or DMF ternary mixtures are negative over the whole composition range. Negative V E123

values are due to more efficient packing and/or a greater attractive intermolecular interactionin the mixtures than in pure liquids. The negative V E

123 values mean there is a contraction ofvolume which can be attributed to specific interaction between the delocalized π -electroncloud of an aromatic hydrocarbon and a lone pair of electrons on oxygen atom of the cyclicether, FD or DMF. Furthermore, negative values of V E

123 for 1,3-dioxolane (1) + benzene(2) + FD as compared to 1,3-dioxolane (1) + toluene (2) + FD are due to the presenceof the bulky –CH3 group in toluene which causes repulsion during the approach of FD orDMF to the aromatic ring and so to 1,3-dioxolane molecule as compared to benzene. Thesame behavior is also found in the case of 1,4-dioxane (1) + benzene (2) + DMF (3) and1,4-dioxane (1) + toluene (2) + DMF (3) ternary mixtures.

The V E123 values of these mixtures have also been evaluated in terms of Graph theory.

4 Graph Theory

If atoms in the structural formula of a molecule are represented by letters and the bondsjoining them by lines, the resulting graph then provides the total information contained inthat molecule [18–20]. Consequently if δν

m, δνn , δν

o , etc. represent the degree of m,n and o

etc. vertices of the graph of a molecule, then connectivity indices of the first, second andthird order of the molecules are defined by the following relations [21]

1ξ =∑m<n

(δνmδν

n)−0.5 (5)

2ξ =∑

m<n<o

(δνmδν

nδνo )

−0.5 (6)

3ξ =∑

m<n<o<p

(δνmδν

nδνoδ

νp)−0.5 (7)

where δνm, etc. values explicitly reflect the valency of the atoms forming the bonds and

is expressed as [21] δνm = Zm − h, where Zm is the maximum valency of the atom and

h is the number of hydrogen atoms attached to it. Consequently, for carbon in benzene,δν

C = 4 − 1 = 3 and for carbon in 1,3-dioxolane, δνC = 4 − 2 = 2.

The connectivity parameter of the first order of a molecule is assumed to be a measureof oscillation of the bond length (as it depends on the order of its closest vertices takentwo at a time), and as the bond length remains constant with concentration and temperature,

688 J Solution Chem (2010) 39: 680–691

1ξ would be independent of temperature and concentration. The connectivity indices of 2ξ

second order appear first to encode some information about the effect of branching on thestructural formulation of a molecule, but Kier [21] has suggested that information regarding,the effect of branching in molecules can be obtained by evaluation of 3ξ of a molecule. Sincethe addition of 1 to 2 in mixtures leads to change in topology of 1 or 2 in the (1+2) mixture,and as V E is a packing effect, therefore V E data of these mixtures should reflect changes inthe topology of 1 or 2.

Study of the excess molar volumes, V E, and molar excess enthalpies, H E, of 1,3-dioxolane (1) + FD or DMF (2) mixtures by Graph theory [22] have led to the proposalthat 1,3-dioxolane exists as monomers and FD or DMF exist as associated molecular enti-ties and (1 + 2) mixtures formation involve the processes (I) formation of unlike contactsbetween (1) and (2); (II) establishment of unlike contacts then leads to depolymerizationof 2n to yield monomers of (2); and (III) monomers of (1) and (2) then undergo specificinteraction to form 1:2 molecular entities. Consequently, V E and H E data of these mixturesare described [22] by Eq. 8

H E =[

x1x2(3ξ1/

3ξ2)

x1 + x2(3ξ1/3ξ2)

][χ ′

22 + x2χ12

](8)

Further analysis of V E and H E data of binary 1,3-dioxolane (1) + benzene or toluene (2)and 1,4-dioxane or 1,3-dioxolane (1) + water (2) mixtures by Graph theory have revealed[8, 23, 24] that 1,3-dioxolane or 1,4-dioxane, benzene or toluene exist as monomers in thepure state and these mixtures are characterized by interactions between the delocalized π -electron clouds of aromatic hydrocarbons (benzene or toluene) and the π -electron cloudof ethereal oxygen atom of 1,3-dioxolane or 1,4-dioxane. Therefore, 1,3-dioxolane or 1,4-dioxane (1) + benzene or toluene (2) mixture formation involves processes (I) formation ofunlike contacts between 1 and 2, and (II) specific interaction between 1 and 2 to form 1:2molecular entities. The thermodynamic properties, XE (X = V or H ) for these mixtures areexpressed [23] by Eq. 9

XE =[

x1x2(3ξ1/

3ξ2)

x1 + x2(3ξ1/3ξ2)

][χ ′

22 + x2χ12

](9)

where x1, x2 are the mole fractions and χ ′12, χ ′

22 and χ12 are interaction energy parametersfor unlike, like contacts and specific interactions between 1 and 2 components of (1 + 2)mixtures. 3ξ1 and 3ξ2 are connectivity indices of third order of the 1st and 2nd componentsof (1 + 2) mixture and have the same significance as described elsewhere [4]. The variousassumptions in deriving Eqs. 8 and 9 were justified as XE (X = V or H ) data of studiedmixtures were reproduced by employing χ ′

12, χ12, etc. (evaluated from XE data at x1 = 0.4,and 0.5).

If a third component like FD or DMF (3) is added to 1,3-dioxolane or 1,4-dioxane (1) +benzene or toluene (2) binary mixtures, then ternary (1 + 2 + 3) mixtures formation wouldinvolve the processes: (I) establishment of unlike contacts (a) 1–2, (b) 2–3n, and (c) 1–3n

between 1, 2 and 3 components of (1 + 2 + 3) mixtures; (II) the establishment of unlikecontacts cause depolymerization of 3n polymer to yield its respective monomers; and (III)monomers of 1, 2 and 3 then undergo specific interactions to form (a) 1:2, (b) 2:3, and(c) 1:3 molecular entities. Consequently if χ ′

12 χ ′23 and χ ′

13 are the molar volume interactionparameters of unlike (1–2), (2–3) and (1–3) contacts, then changes in V E due to processesI (a)–(c) would be given [25–27] by Eq. 10

�X1(X = V ) = x1S2χ′12 + x2S3χ

′23 + x3S1χ

′13 (10)

J Solution Chem (2010) 39: 680–691 689

S1 etc. are defined [25] by

S1 = x1ν1

/∑x1ν1 (11)

where v1 is the molar volume of component 1. Consequently Eq. 10 reduces to Eq. 12

�X1 =[x1x2ν2

/∑x1ν1

]χ ′

12 +[x2x3ν3

/∑x2ν2

]χ ′

23 +[x3x1ν1

/∑x3ν3

]χ ′

13 (12)

Further, if χ ′33 and χ12, χ ′

12, χ ′′12 are the molar volume interaction parameters for 3–3 contacts

and specific interactions (leading to the formation of 1:2, 2:3, and 1:3 molecular entities),then changes in a thermodynamic property XE (X = V ) due to processes II and III (a)–(c)are given [25–27] by the relations:

�X2 = x22x3ν3χ

′33

/∑x1ν1 (13)

and

�X3 = x1x22ν2χ12

/∑x1ν1 (14)

�X4 = x2x23ν3χ

′12

/∑x2ν2 (15)

�X5 = x3x21ν3χ

′′12

/∑x3ν3 (16)

The overall change in a thermodynamic property, XE (X = V ), due to processes I (a)–(c),and III (a)–(c) then can be expressed by Eq. 17

XE(X = V ) =[x1x2ν2

/∑x1ν1

][χ ′

12 + x2χ12]

+[x2x3ν3

/∑x2ν2

][χ ′

23 + x3χ′12 + x2χ

′33]

+[x1x3ν3

/∑x3ν3

][χ ′

13 + x1χ′′12] (17)

Because ν2/ν1 = (3ξ1/3ξ2) [3] then Eq. 17 reduces to Eq. 18

XE(X = V ) = [x1x2(3ξ1/

3ξ2)/x1 + x2(3ξ1/

3ξ2)][χ ′12 + x2χ12]

+ [x2x3(3ξ2/

3ξ3)/x2 + x3(3ξ1/

3ξ3)][χ ′23 + x3χ

′12 + x2χ

′33]

+ [x1x3(3ξ3/

3ξ1)/x3 + x1(3ξ3/

3ξ1)][χ ′13 + x1χ

′′12] (18)

Further, if it be assumed that χ ′12 ≈ χ12 = χ∗

12; χ ′23

∼= χ ′12

∼= χ∗23; χ ′

13 ≈ χ ′′12 = χ∗

13, then Eq. 18reduces to Eq. 19

XE(X = V ) = [x1x2(3ξ1/

3ξ2)/x1 + x2(3ξ1/

3ξ2)][(1 + x2)χ∗12]

+ [x2x3(3ξ2/

3ξ3)/x2 + x3(3ξ2/

3ξ3)][(1 + x3)χ∗23 + x2χ

′33]

+ [x1x3(3ξ3/

3ξ1)/x3 + x1(3ξ3/

3ξ1)][(1 + x1)χ∗23] (19)

Equation 19 has four unknown parameters (χ∗12, χ

∗23, χ

∗13, χ

′33) and for the present analysis,

we employed V E123 data at four arbitrary compositions to evaluate them. These parameters

690 J Solution Chem (2010) 39: 680–691

were then utilized to predict V E123 values of various ternary mixtures at other values of x1, x2.

Such predicted V E123 values along with χ∗

13, χ∗23, χ

∗13, χ

′33, etc. parameters are recorded in

Table 2 and also compare well with their corresponding experimental values.Examination of data in Table 2 reveals that V E

123 values predicted by Graph theory com-pare well with their corresponding experimental values. Even in those cases where calcu-lated and experimental values are not in agreement, they are of same sign and magnitude.This may be due to the ternary 1–2–3 etc. contacts which have not been taken into consid-eration for the derivation of Eq. 19.

5 Conclusions

Excess molar volumes were determined for the 1,3-dioxolane or 1,4-dioxane (1) + benzeneor toluene (2) + FD or + DMF (3) ternary mixtures at 308.15 K and at atmospheric pressure.The observed data has also been analyzed in terms of Graph theory (model). The valuescalculated by Graph theory (model) are very close to the experimental values. This showsthe validity of this theory (model) for the studied mixtures. Further, the V E

123 values arenegative for all of the studied ternary systems. These negative V E

123 values represent strongermolecular interactions in the systems than in pure liquids.

Acknowledgements This work was supported by the Ministry of Education of Korea by its BK21 programat Yonsei University, GAS plant R&D centre and president of Yonsei University.

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