Topological Investigations of Some Cyclic Ether-Water-Alkanol Ternary Mixtures: Molar Excess Volumes

Journal of Solution Chemistry, Vol. 34, No. 7, July 2005 ( C© 2005)DOI: 10.1007/s10953-005-4990-3

Topological Investigations of Some CyclicEther-Water-Alkanol Ternary Mixtures:Molar Excess Volumes

V. K. Sharma1,∗ and Satish Kumar1

Received June 6, 2004; revised February 22, 2005

Molar excess volumes, V Ei jk , of 1,3-dioxolane or 1,4-dioxane (i) + water ( j) + propan-

1-ol or + propan-2-ol (k) ternary mixtures have been determined dilatometrically overthe entire composition range at 308.15 K. The resulting data have been analyzed interms of (1) the graph theoretical approach (which involves the topology of the mixtureconstituents), (2) the Sanchez and Locombe theory and (3) the Flory theory. It wasobserved that V E

i jk values predicted by the graph theory compare reasonably well withtheir corresponding experimental values. However, although V E

i jk values calculated bythe Sanchez and Lacombe and Flory theories are of same sign and magnitude, thequalitative agreement is poor.

KEY WORDS: Molar volumes; excess volumes; 1,3-dioxolane; 1,4-dioxane; propan-1-ol; propan-2-ol.

1. INTRODUCTION

The physical properties of mixtures depend upon the manner in which thecomponents are associated with each other. Recent studies(1−4) have shown thatthe graph theoretical approach, which is based upon the topology of the con-stituents of the mixture, can be utilized to extract information about, (1) the stateof the components in their pure and mixture states, and (2) the nature of andextent of molecular interactions operating between the components. In our earlierpublications(5,6) we reported molar excess volumes and molar excess enthalpiesfor 1,3-dioxolane or 1,4-dioxane (i) + alkanol ( j) mixtures and the data have beenanalyzed in terms of the graph theoretical approach. It is of interest to see howwell the graph theory describes the molar excess volumes data of ternary mixtures,when a third component such as water is added to 1,3-dioxolane (i) or 1,4-dioxane

1Department of Chemistry, Maharshi Dayanand University, Rohtak 124001, India; e-mail: [email protected].

839

0095-9782/05/0700-0839/0 C© 2005 Springer Science+Business Media, Inc.

840 Sharma and Kumar

Table I. Comparision of Densities, ρ, ofPure Liquids with Their Literature Values at

298.15 K

ρ/(g-cm−3)

Liquids Expt. Lit.

1,3-dioxolane 1.05885 1.05881(9)

1,4-dioxane 1.02792 1.02797(9)

water 0.99708 0.99704(10)

propan-1-ol 0.79980 0.79976(11)

propan-2-ol 0.78091 0.78095(11)

+ alkanol ( j) binary mixtures. This prompted us to measure molar excess vol-ume data for 1,3-dioxolane (i) or 1,4-dioxane (i) +water ( j) + propan-1-ol orpropan-2-ol (k) ternary mixtures.

2. EXPERIMENTAL

1,3-dioxolane (Fluka, 99%, USA), 1,4-dioxane (AR Grade, 99%), propan-1-ol (AR Grade, 95%) and propan-2-ol (AR Grade 95%) were purified by stan-dard methods.(7,8) Deionized water was doubly distilled with a small amount ofKMnO4 containing two or three pellets of NaOH. The middle fraction was col-lected for use in preparing solutions. The purities of the purified liquids werechecked by measuring their densities with a pycknometer at (298.15 ± 0.01) Kand the resulting densities (reported in Table I) agreed to within ±(5 × 10−5)g-cm−3 of their literature values.(9−11) Molar excess volumes, V E

i jk , for ternarymixtures were measured in a three-limbed dilatometer as described elsewhere.(12)

The temperature of the water bath was controlled to ±0.01 K and changes inthe liquid level of the dilatometer capillary were measured with a cathetometer(Osaw, India) that could be read to ±0.001 cm. The uncertainties in our measuredV E

i jk values are about 0.5%. The reliability of the dilatometer measurements waschecked by measuring V E values for benzene (i) + cyclohexane ( j) mixtures at298.15 K and these values agreed with their corresponding literature values(13)

within the reported accuracy.

3. RESULTS

The measured V Ei jk data for ternary 1,3-dioxolane or 1,4-dioxane (i) + water

( j) + propan-1-ol or + propan-2-ol (k) mixtures at 308.15 K are recorded inTable II and are plotted in Figs. 1–4. These results were represented by the

Cyclic Ether-Water-Alkanol Ternary Mixtures 841

Table II. Comparison of Measured V Ei jk Values for the Various (i + j + k) Ternary Mixtures at

308.15 K with the Corresponding Values Evaluated from Graph, Flory, and Sanchez and LacombeTheories; Also Included Are the Various V (n)

i jk (n= 0, 1, 2) Parameters, Along with the Standard

Deviation σ V Ei jk ) and the Interaction Energy Parameters (χ ′, χi j ... )a

V E / (cm3-mol−1)

xi x j Exptl. Graph Sanchez Flory

1,3-dioxolane (i) + water ( j) + propan-1-ol (k)0.0893 0.5506 −0.141 −0.126 −0.706 0.6440.1325 0.5074 −0.198 −0.129 −0.650 0.6650.1550 0.6512 −0.180 −0.180 −0.774 0.3380.2802 0.6015 −0.348 −0.348 −0.768 0.1230.3126 0.6202 −0.485 −0.498 −0.671 −0.0390.3515 0.5090 −0.309 −0.309 −0.877 0.2010.5051 0.2707 0.129 0.082 −0.554 0.5060.5632 0.2185 0.109 0.181 0.496 0.5150.5703 0.3388 −0.322 −0.326 −0.532 0.1190.6072 0.2697 −0.078 −0.119 0.795 0.2400.7009 0.1760 −0.022 −0.070 −0.703 0.280

V (0)i jk = 13.571, V (1)

i jk = 85.525, V (2)i jk = −1103.097, σ (V E

i jk ) = 0.003, (ξi ) = 0.601, (ξ j ) = 1.010,(ξ k ) = 0.701, χ∗= − 3.387, χ∗

i j = − 1.877, χ∗jk= − 0.190, χ∗

ik = 3.029, χ∗∗i j = 418.5, χ∗∗

jk =14.6, χ∗∗

ik = 962.4, χ ′′i j = 28.6, χ ′′

jk = 21.1, and χ ′′ik = 131.4.

1,3-dioxolane (i) + water( j) + propan-2-ol (k)0.0954 0.8530 −0.107 −0.107 −0.894 −0.0840.1559 0.7983 −0.113 −0.097 −0.743 −0.1470.1716 0.7659 −0.078 −0.056 −0.820 −0.1110.2243 0.6900 −0.029 −0.026 −0.820 −0.0660.2340 0.6802 −0.035 −0.045 −0.809 −0.0670.3481 0.5181 0.324 0.083 −0.813 0.0560.4271 0.4673 0.048 0.175 −0.576 0.0110.5465 0.3571 0.192 0.261 −0.520 0.0300.6012 0.1020 −0.324 −0.324 −0.353 0.5110.6750 0.2500 0.246 0.294 −0.529 0.0320.7712 0.1728 0.184 0.262 −0.397 0.0290.8356 0.1111 0.164 0.125 −0.413 0.052

V (0)i jk = 39.122, V (1)

i jk = 382.609, V (2)i jk = −3073.697, σ (V E

i jk ) = 0.003, (ξi ) = 0.601, (ξ j ) = 1.010,(ξ k ) = 0.902, χ∗ = 8.419, χ∗

i j = − 0.923, χ∗jk = 1.722, χ∗

ik = −3.577, χ∗∗i j = 418.5, χ∗∗

jk =−1030.4, χ∗∗

ik = 965.9, χ ′′i j = 28.6, χ ′′

jk = 121.8, χ ′′ik = 134.2.

1,4-dioxane (i) + water ( j) + propan-1-ol (k)0.0465 0.8512 −0.339 −0.256 −0.256 0.0430.0879 0.8861 −0.395 −0.204 −0.370 −0.3190.1637 0.6058 −0.433 −0.433 −0.369 0.3180.2510 0.5080 −0.340 −0.358 −0.274 0.3760.3031 0.4666 −0.295 −0.295 −0.241 0.3670.4750 0.3031 −0.126 −0.257 −0.249 0.4370.5717 0.2720 −0.135 −0.135 −0.178 0.277


Table II. Continued

V E / (cm3-mol−1)

xi x j Exptl. Graph Sanchez Flory

0.6320 0.2010 −0.099 −0.176 −0.025 0.3520.6830 0.1226 −0.206 −0.187 −0.248 0.4790.7332 0.1026 −0.148 −0.148 −0.147 0.4090.7466 0.0962 −0.136 −0.138 −0.197 0.3940.8780 0.0650 −0.066 −0.055 −0.077 0.125

V (0)i jk = 6.325, V (1)

i jk = 150.325, V (2)i jk = −1200.320, σ (V E

i jk ) = 0.004, (ξi ) = 0.801, (ξ j ) = 1.010,(ξ k ) = 0.701, χ∗=1.638, χ∗

i j = 2.214, χ∗jk = − 3.133, χ∗

ik = 0.253, χ∗∗i j = 403.0, χ∗∗

jk = 14.6and χ∗∗

ik = 1051.1, χ ′′i j = 32.3 χ ′′

jk = 21.2, and χ ′′ik = 118.7

1,4-dioxane (i) + water ( j) + propan-2-ol (k)0.1320 0.6501 −0.321 −0.321 −1.844 0.0710.2895 0.5201 −0.196 −0.376 −1.389 0.0190.3012 0.3015 −0.153 −0.153 −0.774 0.4370.3301 0.5921 −0.242 −0.511 −0.925 −0.2320.4305 0.4117 −0.141 −0.343 −0.824 0.0110.4536 0.4107 −0.296 −0.365 −0.636 −0.0280.5465 0.3571 −0.363 −0.363 −0.530 −0.0660.6750 0.2510 −0.170 −0.260 −0.516 −0.0390.7114 0.2566 −0.350 −0.350 −0.301 −0.1170.7868 0.1688 −0.116 −0.188 −0.383 −0.0360.8613 0.1064 −0.058 −0.109 −0.247 −0.0160.8801 0.0844 −0.020 −0.061 −0.270 0.004V (0)

i jk = 40.210, V (1)i jk = 150.320, V (2)

i jk = −500.210, σ (V Ei jk ) = 0.003, (ξi )= 0.801, (ξ j ) = 1.010,

(ξ k ) = 0.902, χ∗= − 0.4508, χ∗i j =−1.825, χ∗

jk= −0.720, χ∗ik = −1.661, χ∗∗

i j = 403.0, χ∗∗jk =

−1105.7, χ∗∗ik = 1119.9, χ ′′

i j = 32.3, χ ′′jk = 540.4, and χ ′′

ik = 137.0.

aThe units of the χ∗, χ∗i j , etc., are cm3-mol−1; the χ∗∗

i j and χ ′′i j , etc., are in J-mol−1; and the V (n)

i jk (n =0, 1, 2) and (σ V E

i jk ) are in cm3-mol−1.

equation

V Ei jk = xi x j

[ 2∑n=0

V (n)i j (xi − x j )

n

]+ x j xk

[ 2∑n=0

V (n)jk (x j − xk)n

]

+ xi xk

[ 2∑n=0

V (n)ik (xk − xi )

n

]+ xi x j xk

[ 2∑n=0

V (n)i jk (x j − xk)n xn

i

](1)

The V (n)i j (n = 0, 1, 2) are parameters characteristic of the (i + j), ( j + k) and (i +

k) binary mixtures and their values have been taken from the literature.(5−6,14−16)

The V (n)i jk (n = 0, 1, 2) are parameters characteristic of the ternary mixtures and

their values were determined by fitting to the experimental V (n)i jk data using matrix


Fig. 1. Molar excess Volumes, V Ei jk of 1,3-Dioxoplane (i) + water ( j) + propan-1-ol at 308.15 K.

Eq. (2)

V Ei jk − xi x j [

2�

n=0V (n)

i j (xi − x j )n]

− x j xk[2�

n=0V (n)

jk (x j − xk)n]

− xk xi [2�

n=0V (n)

jk (xk − xi )n]

[xi x j xk

]−1

= V (0)i jk + V (1)

i jk (x j − xk)xi + V (2)i jk (x j − xk)2x2

i (2)

by the least-squares method. The resulting parameters, along with their standarddeviations, σ (V (n)

i jk ), are recorded in Table II. The standard deviations are given byEq. (3),

σ(V E

i jk

) = [(V E

i jk)exp tl − (V E

i jk

)calc.Eq.(1)

]/(m − p)

]0.5(3)


Fig. 2. Molar excess Volumes, V Ei jk of 1,3-Dioxoplane (i) + water ( j) + propan-2-ol (k) at 308.15 K.

where m is the number of data points and p is the number of adjustable parametersin Eq. (3).

4. DISCUSSION

We are unaware of any V (n)i jk data for the present ternary mixtures with which

to compare our results.

4.1. Conceptual Aspect of the Graph Theoretical Approach and Results

According to the mathematical discipline of graph theory, if atoms in thestructural formula of a molecule are represented by dots and the bonds joiningthem by lines, then resulting graph describes the total amount of information(17−19)

contained in that molecule. Keir(20) suggested the use of a molecular connectivityparameter of the third degree of the molecule to extract information about theeffect of branching in molecules and Singh(21) defined connectivity parameters of


Fig. 3. Molar excess Volumes, V Ei jk of 1,4-dioxane (i) + water ( j) + propan-1-ol (k) at 308.15 K.

the first, second, and third degree of molecule by Eqs. (4)–(6).

1ξ =∑l<m

(δv

l δvm

)−0.5(4)

2ξ =∑

l<m<n

(δv

l δvmδv

n

)−0.5(5)

3ξ =∑

l<m<n<o

(δv

l δvm δv

nδvo

)−0.5(6)

where the δvl , etc., represent explicitly the valence of the atoms forming the

bonds (δv = valence δ) and their values can be evaluated via the relation δv =Zm − hm ,(22) where Zm denotes the maximum valency of an atom and hm is the


Fig. 4. Molar excess Volumes, V Ei jk , of 1,4-Dioxone (i) + water ( j) + propan-2-ol (k) at 308.15 K.

number of hydrogen atoms attached to it. Singh et al.(23) have suggested thatthe 1/3ξ parameter of a molecule represents a measure of the probability thatthe surface area of a molecule interacts effectively with the surface area of othermolecules and this parameter can be utilized to predict molar excess volumes andmolar excess enthalpies of binary(1,2,4) and ternary(3,24−26) mixtures.

Thermodynamic investigations of molar excess volumes, V E, and molar ex-cess enthalpies, H E, of water (i) + propan-1-ol ( j) by the graph theoreticalapproach have revealed(14) that water and propan-1-ol are associated molecularentities and that the (i+ j) mixture formation involves the processes: (1) formationof unlike contacts between i and j that cause their depolymerization to occur toyield their respective monomers, and (2) monomers of i and j that undergo inter-actions to form the i : j molar complex. Consequently, changes in thermodynamicproperties, XE(X = V or H ), can be expressed by Eq. (7).

XE(X = V or H ) = [xi x j (3ξi/ξ j )/xi + x j (

3ξi/ξ j )][(1 + x j )χ′ij + 2xiχ

∗] (7)


where the χ ′i j and χ∗ are molar volume interaction parameters representing unlike

and like contacts between the i and j components of the binary mixtures. Furtheranalysis of V E and H E data for 1,3-dioxolane or 1,4-dioxane (i) + propan-1-ol +propan-2-ol or water ( j) binary systems by the graph theoretical approach have ledto the postulation(5,6) that 1,3-dioxolane and 1,4-dioxane (i) in the pure state existas monomers, whereas propan-1-ol and propan-2-ol exist as associated molecularentities. The thermodynamic properties XE(X = V or H ) for these mixtures wereexpressed by Eq. (8)

XE = [xi x j

(3ξi/ξ j

)/xi + x j

(3ξi

/ξ j

)][(1 + x j )χ

′ij + x jχ12] (8)

where χ12 is the molar volume interaction parameter for specific interactionsbetween the components of the mixture. The basic assumptions needed in derivingEqs. (7) and (8) were justified as the XE data (X = V or H ) of these binary mixtureswere reproduced by employing the χ ′

i j , χ12 parameters (calculated from XE dataat xi = 0.4 and 0.5).

If a third component such as water ( j) is added to 1,3-dioxolane or 1,4-dioxane (i) + propan-1-ol or propan-2-ol (k) binary mixtures, then ternary mixtureformation should involve the processes: (1) establishment of (a) i − jn , (b) jn − kn

and (c) i − kn unlike contacts between the i, j and k components of (i+ j+k)ternary mixtures, (2) formation of unlike contacts between i, j and k leads todepolymerization of (a) in and (b) kn to yield their respective monomers, and (3)monomers of i, j and k then undergo specific interactions to form (a) i : j (b) j :kand (c) i :k molecular entities. Consequently, if χ ′

i j , χ ′jk and χ ′

ik are the molarvolume interactions parameters for unlike (i − j), ( j − k) and (i − k) contacts,then the change in a molar thermodynamic property X (X = V ) due to processes1 (a)–(c) is given(27−29) by Eq. (9)

�Xi (X = V ) = xi S jχ′i j + x j Skχ

′jk + xk Siχ

′ik (9)

where the Si are defined(29) by

Si = xivi/�xivi (10)

where vi is the molar volume of component (i). Consequently, Eq. (9) reduces toEq. (11).

�X1 = xi x jv jχ′i j/�xivi + x j xkvkχ

′jk/�x jv j + xk xiv jχ

′ik/�xkvk (11)

Again, if χ ′j j , χ ′

kk and χ12, χ′12, χ ′′

12 are the molar volume interaction parametersfor j − j and k − k contacts and specific interactions, between the i, j and kcomponents, then changes in the molar property X (X = V ) due to processes2(a)–(b) and 3(a)–(b) are given by(27−29) the relations:

�X2 = x2i x jv jχ

′i i/�xivi + x2

k xivkχ′kk/�xkvk (12)


and

�X3 = xi x2j v jχ12/�xivi + x j x

2k vkχ

′12/�x jv j + xk x2

i χ ′′12/�xkvk (13)

The overall change in a thermodynamic property XE(X=V ), due to processes1 (a)–(c), (2) (a)–(b) and 3 (a) –(c) can then be expressed by Eq. (14)

XE(X = V ) =3∑

i=1

�Xi = xi x jv j/�xivi [χ′i j + xiχ

′i i + x jχ12]

+ x j xkv j/�x jv j [χ′jk + xkχ

′12] + xi xkvi/�xivi [χ

′ik + xkχ

′kk + xiχ

′′12] (14)

Because v j/vi = (3ξi/3ξ j ), (23) Eq. (14) reduces to Eq. (15).

V Ei jk = xi x j (

3ξi/3ξ j )/xi + x j (

3ξi/3ξ j ) [χ ′

i j + xiχ′i i + x jχ12]

+ x j xk(3ξ j/3ξk)/x j + xk(3ξ j/

3ξk) [χ ′jk + xkχ

′12]

+ xi xk(3ξk/3ξi )/xk + xi (

3ξk/3ξi ) [χ ′

ik + xkχ′kk + xiχ

′′12] (15)

Further, if it is assumed that χ ′i j

∼= χ12 = χ∗i j , χ ′

jk∼= χ ′

12∼= χ∗

jk, χ ′ik ≈ χ ′′

12 =χ∗

ik, and χ ′i i = χ ′

ik∼= χ∗, then Eq. (15) can be re-expressed as Eq. (16).

V Ei jk = [xi x j (

3ξi/3ξ j )/xi + x j (

3ξi/3ξ j )] [(1 + x j ) χ∗

i j + xiχ∗]

+ [x j xk(3ξ j/3ξk)/x j + xk(3ξ j/

3ξk)] [(1 + xk)χ∗jk]

+[xi xk(3ξk/3ξi )/xk + xi (

3ξk/3ξi )] [(1 + xi )χ

∗ik + xkχ

∗] (16)

Equation (16) contains four unknown parameters (χ∗i j , χ

∗jk, χ∗

ik, χ∗), and for the

present analysis we used V Ei jk data at four arbitrary compositions to evaluate the

parameter values. These parameters were then utilized to predict V Ei jk values for

various ternary mixtures at other values of xi and x j . Such predicted V Ei jk values,

along with the corresponding χ∗ik, χ∗, andχ∗

i j parameters, are recorded in Table II.The V E

i jk data for the studied ternary mixtures were next analyzed in terms ofthe Sanchez and Locombe theory and the Flory theory.

4.2. Sanchez and Lacombe Theory

According to the Lacombe and Sanchez theory, V Ei jk for a ternary mixture is

given(30,31) by the relations

V Ei jk = rmixvmix

[v̄mix −

k∑i=i

φi v̄i

](17)


φi = mi (ρ∗i )−1

[k∑

i=i

(miρ∗i )−1

](18)

mi = xi mi

[k∑

i=i

(xi mi )−1

](19)

rmix =k∑

i=i

xiri (20)

ri = ri [v∗i (v∗

mix)−1] (21)

v∗mix =

k∑i=i

φ∗i v∗

i (22)

v̄mix = 1/ρ̄mix (23)

φi = mi (ρ∗i v∗

i )−1

[k∑

i=i

(mi/ρiv∗i )−1

](24)

All the terms in these equations have the same significance as describedelsewhere.(30,31)

Evaluation of V Ei jk values using Eqs. (17)–(24) requires a knowledge of the

reduced density of ternary mixture, ρi jk , which in turn can be evaluated from theequation of state of the mixture, Eq. (25),

(ρ̄i jk)2 + P̄ + [RT/ ∈∗mix][ln[1 − ρ̄i jk] + [1 − [rmix]−1][ρ̄i jk] = 0 (25)

where

∈∗mix=

[k∑

i=i

φ2i ∈i i −�φiφ jχ

∗∗i j

](26)

The χ∗∗i j are interaction energy parameters for the various (i + j), ( j + k) and (i

+ k) binary mixtures. These parameters have been evaluated using the single H E

value (xi = 0.5) of the binary submixtures (i + j), ( j + k) and (i + k) availablein the literature,(5,6,14,16) by using Eq. (27)

H Ei j = 2φiφ j ρ̄i jχ

∗∗i j + RT rmix

[�

(φ0

i ρ̄i − ρ̄i jφi)(T̄i )

−1 (27)

The V Ei j value (at xi = 0.5) was used to predict χ∗∗

i j values for 1,3-dioxolane (i) +water ( j) binary mixtures for which H E values are not available in the literature.


Because (i + j), ( j + k) and (i + k) binary submixtures of the (i + j + k)ternary mixtures satisfy Eq. (28),

(ρ̄i j )2 + P̄ + [RT/ ∈∗

i j ][ln[1 − ρ̄i j ] + [1 − [rmix]−1][ρ̄i j ]

= (−0.0221) − (−0.0551) (28)

it follows that the present ternary mixtures will also not satisfy Eq. (25). If theaddition of kth component to the (i + j) mixture does not drastically alter thei − j , j − k and i − k contacts, then ternary mixtures will also satisfy an equationof state like Eq. (28). Because ternary (i + j + k) mixtures are considered tobe composed of (i + j), ( j + k) and (i + k) binary mixtures, then the extentto which the ternary mixture deviates from Eq. (25) would be nearly one-thirdof the sum by which binary (i + j), ( j + k) and (i+k) mixtures deviate fromEq. (28). Consequently, the equation of state for the present ternary mixtures canbe expressed by Eq. (29).

(ρ̄i jk)2 + P̄ + [RT/ ∈∗mix][ln[1 − ρ̄i jk] + [1 − [rmix]−1][ρ̄i jk]

= 1/3[�R.H.S. of Eq. (28) for (i + j), ( j + k) and (i + k) binary mixtures]

(29)

The V Ei jk values were then calculated using the established equation of state, and the

resulting values are recorded along with χ∗∗i j , χ∗∗

jk , and χ∗∗ik parameters in Table II.

4.3. Flory Theory

According to Flory’s theory, V Ei jk values for a ternary mixture are given

by:(32,33)

V Ei jk = V̄ E

cal

[k∑

i=i

xiv∗i

](30)

v̄i = [1 + αi (T/3)/(1 + αi T )3]3 (31)

V̄ Ecal = v̄

7/30 [(4/3) − (v̄0)1/3]−1[T̄ − T̄0] (32)

T̄0 = (v̄

1/30 − 1

)/v̄

4/30 (33)

v̄∗i = vi/v̄i (34)

v̄0 = �φiv∗i (35)

T̄ = [�(φi P∗i T̄i/�φi P∗

i )][1 − (φiθ jχ′′i j )(�φi P∗

i )−1]−1 (36)

T̄i = (v̄

1/3i − 1

)/v̄

4/3i (37)


P∗ = �φi P∗i − �φ∗

i θ jχ′′i j (38)

P∗i = αi T v̄2

i [(KT )i ]−1 (39)

All of the terms have the same significance as described elsewhere.(32,33) Eval-uation of V E

i jk values by the Flory theory requires a knowledge of the reducedtemperature, T̄ , which in turn depends upon the adjustable parameters θ j , χ

′′ij ,

etc., of the (i + j), ( j + k) and (i + k) binary submixtures of the (i + j + k)ternary mixtures. These parameters were determined by fitting their values to theH E values(5,6,14,16) at xi = 0.5 using Eq. (40),

H E = �xi P∗i

(V −1

i − V −1cal ) + xi V

∗i θ jχ

′′i j V

−1cal (40)

Various parameters for pure components were determined using values of theisothermal compressibility (KT ) reported in the literature.(34) The KT values forthose liquids that were not available were calculated by using their �HVvalues(34)

in the manner suggested by Hildebrand.(35) Such values of V Ei jk evaluated using

Eqs. (32) through (40), along with the corresponding χ ′′i j , etc. parameters, are

recorded in Table II, where they are also compared with their correspondingexperimental values.

Examination of the data in Table II reveals that the V Ei jk values predicted by

the graph theory compare well with their corresponding experimental values. Evenin those cases where the calculated and experimental values are not in agreement,they are of the same sign and magnitude. However, V E

i jk values calculated byemploying Sanchez and Lacombe and Flory theories are of same sign, but thequalitative agreement is poor. The failure of the Lacombe and Sanchez theory topredict the correct magnitude of V E

i jk may be due to the assumption that the χ∗∗i j

parameters are independent of composition and that the right hand side of theequation of state for ternary mixtures is unchanged over the entire compositionrange. The failure of the Flory theory to calculate correctly the magnitude ofthe V E

i jk values may be due to the various assumptions made in evaluating theparameters of the pure components, which were not available in the literature, andalso due to the nature of the components of the ternary mixtures.

ACKNOWLEDGMENTS

The authors are grateful to the Head of the Chemistry Department and au-thorities of M. D. University, Rohtak, for providing research facilities.

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