ids 133 (2007) 33–38www.elsevier.com/locate/molliq

Journal of Molecular Liqu

Statistical thermodynamics and viscous flow mechanism ofquaternary liquid systems at 298.15 K

R.K. Shukla ⁎, Sonu Dwivedi, A.N. Dubey

Department of Chemistry, V.S.S.D. College, Nawabganj, Kanpur 208 002, India

Received 1 July 2005; accepted 20 May 2006Available online 20 September 2006

Abstract

Assuming a quaternary liquid mixture to be made up of six binaries, statistical mechanical theory of Flory has been extended to develop theexpression for dynamic viscosity of multi-component systems using the concept of absolute rate and free volume theories of liquid state. Areasonable agreement has been achieved between theory and the experiment for n-hexadecane, carbon tetrachloride, n-benzene and n-hexanequaternary system at 298.15 K. An attempt has also been made to explain the nature of the molecular interactions involved, in the light of excessthermodynamic functions of which sign and magnitude depend upon the chain length of the component liquids.© 2006 Published by Elsevier B.V.

Keywords: Thermodynamics; Viscosity; Quaternary; Interactions; Liquid mixtures

1. Introduction

A better understanding about the viscous flow mechanism ofmulticomponent systems is of considerable physico-chemicalinterest in designing calculations involving separations, heattransfer, mass transfer and fluid flow. A substantial amount ofwork has been done on binary [1–5] and ternary [6] liquidmixtures and it is still in progress. However, correlationmethods for viscosity data of quaternary liquid mixtures arevery rare except for the work of Heric and Brewer [7].Bloomfield and Dewan [3,4] used the statistical theory of Floryin the prediction of viscous behaviour of binary liquid mixtures.To our knowledge, no systematic experimental study has beenmade on the viscosity of quaternary systems and very few haveapplied the statistical mechanical approach of Flory [9,10] toquaternary liquid systems so far. Flory's statistical thermo-dynamic study of viscous flow mechanism lays stressparticularly on the equation of state contribution to explainmolecular size and shape [2,3] in terms of different parametersi.e. lattice distortion and disorder parameters [11,12], condensa-tion effect [2,13], steric hindrance [14,15], coupling of tortionaloscillations [17], and nature and extent of non-ideality [18]

⁎ Corresponding author.

0167-7322/$ - see front matter © 2006 Published by Elsevier B.V.doi:10.1016/j.molliq.2006.05.003

arising from shape factor and molecular interactions of theliquid mixture. The viscous flow mechanism is thus a directconsequence of the liquid mixture to molecular structure havingtheoretical correlation with various macroscopic properties, i.e.ultrasonic velocity, molar volume, free energy and surfacetension. Bertrand et al. [19,20] developed a general equation forestimating the viscosity and Gibbs free energy of activation forviscous flow of multi-component liquid systems from theknowledge of the properties of their contributory binaries. Themain objective of the present paper is to develop theoreticalexpressions for the quaternary liquid system (n-hexadecane,carbon tetrachloride, n-benzene, n-hexane) at 298.15 K whichare already available for binaries and test their validity.

A comparative study and its correlation with molecularinteractions have also been made in the present context based onexcess thermodynamic functions. Another aim is to investigatemore closely the conceptual framework, phenomenologicalbehaviour and mechanism involved in the viscous flow of thequaternary liquid mixture.

2. Experimental/materials and methods

Component liquids hexadecane, carbon tetrachloride, ben-zene, hexane obtained from the BDH Chemicals Ltd, Poole,

34 R.K. Shukla et al. / Journal of Molecular Liquids 133 (2007) 33–38

England of Anal R grade, were purified and dried in accordancewith the usual procedure [34].

Viscosity measurements were carried out by a calibratedmodified three limbed Cannon–Ubbelohde dilution viscometerwhich is a suspended level design of Ubbelohde at 298.15 Kwith an accuracy of ±4×10−3 kg m−1 s−1. For a capillaryviscometer, viscosity is given by the following equation whichaccounts the Hagenbach–Couette correction.

g ¼ hpqgr4t8VL

−KPV8pLt

ð1Þ

where η is the dynamic viscosity, h is the mean hydrostatichead, ρ is the density of liquid, g is acceleration due to gravity,r is the radius of the capillary, L is the length of the capillary, t isthe efflux time of a volume of liquid and K is the kinetic energycoefficient which depends on the geometrical shape of thecapillary ends. Since most of these parameters are constant for aparticular viscometer the above equation can be written as

g ¼ At−Bt

� �ð2Þ

where A and B are calibration constants. Thus from Eq. (l)the Hagenbach–Couette correction can be ignored if thecapillary is sufficiently long. The viscometer was immersed ina thermostated water bath maintained at ±0.01 °C and flowtime was recorded. Care was taken that temperature shouldnot only be constant but also uniform, as a change of±0.01 °C in temperature causes approximately 0.02% changein viscosity.

3. Theory and calculation

Combining the absolute rate [21] and free volume [22]theories of liquid viscosity, one obtains [23,24], the expressionfor the viscosity of liquid mixture.

g ¼ Aexp DG#=RT þ rv⁎=vf� � ð3Þ

whereΔG# is the free energy of activation per mole, R is the gasconstant, v⁎ is the enthalpy volume which must be available fora molecular segment jumping to its new site, vf is the freevolume per segment in the mixture, r is a factor of the orderunity and η is the viscosity of liquid. Free energy of activation[25] for a quaternary liquid system can be expressed as

DG# ¼X4i¼1

xiDG#i −amDG

RM ð4Þ

where the residual free energy of mixing, ΔGMR , is closely

related to the excess free energy of mixing and αm is a constantof the order unity. Substituting the value of ΔG# from Eq. (4)into Eq. (3) we get

lng ¼X4i¼1

xilngi−DGRM=RT þ v⁎ 1=vf−

X4i¼1

xi=vf

!" #ð5Þ

The residual free energy of mixing can, in turn the split intoenthalpy and entropy contributions

DGRM ¼ DHM−TDSRM ð6Þ

where ΔHM is the enthalpy of mixing per mol and ΔSR is theresidual entropy of mixing per mole. Substitution of Eq. (6) in toEq. (5), and changing v⁎ and vf to terms of v, we get

lng ¼�X4

i¼1

xilngi−DHM=RT þ DSRM=Rþ ð1=v−1Þ

−�X4

i¼1

xi=v−1��

ð7Þ

where free volume per segment is defined as the differencebetween the total volume per segment and the hard corevolumes, i.e. vf=v−v⁎.

Eq. (7) displays explicitly the various contributions to themixture viscosity, the ideal mixture viscosity, the enthalpy andresidual entropy of mixing, and the difference in free volumebetween mixture and pure components. Different models havebeen proposed to relate energy to volume, the van der Waalsrelation, u=−1/v, is widely used by Flory.

The residual free energy of mixing may be defined as

DGRM ¼ DGM−DGcomb ð8Þ

where ΔGM is the free energy of mixing and ΔGcomb is thecombinatorial free energy. Obviously, ΔGM

R becomes identicalwith GE, if ΔGcomb is represented by the ideal mixing law. If,however, the component molecules differ in size, at least one ofthem being a chain molecule, ΔGComb and ΔSComb may beexpressed by polymer solution theory

DGcomb ¼ −TDScomb ¼ kT lnZcomb ¼ RTX4i¼1

NilnWi ð9Þ

This equation includes a contact interaction term and equationof state term. If the molecules of the four components are ofcomparable size and shape, ΔGcomb should be given appro-priately by the ideal mixing law and the remaining term may beidentified with the excess free energy, GE.

Following the assumptions made by Flory [8,9], the equationof reduced partition function directly gives the value of residualfree energy for a quaternary liquid mixtures as

DGRM ¼ 3ðriNiv

⁎i Þ1 N 4

X4i¼1

WiP⁎i T iln½ðv1=3i −1Þ=ðv1=3i −1Þ�

þ DHM ð10ÞIgnoring the difference between energy and enthalpy of the

condensed system at low pressure, we have extended theexpression of enthalpy for the quaternary liquid mixture, theresulting equation is

DHM ¼ E0ðmixÞ−E0ð1Þ−E0ð2Þ−E0ð3Þ−E0ð4Þ

¼ ðrNv⁎Þ1 N 4

X4i¼1

ðWiP⁎i Þ=ðvi−PT

i =vÞ ð11Þ

35R.K. Shukla et al. / Journal of Molecular Liquids 133 (2007) 33–38

where E0 represents the energy term for liquid mixtures andpure components. Eq. (11) allows to express the result in analternative form

DHM ¼ ðrNv⁎Þ1 N 4

�X4i¼1

ðWiP⁎i Þ=ð1=vi−1=vÞ þ

X4i¼1

X1j¼4

xijðWihj=vÞ" #

ð12ÞSubstituting the value ofΔHM from Eq. (12) into Eq. (10) we

get the expression for residual free energy of mixing which onsimplification gives

DGRM ¼

�X4i¼1

NiP⁎i v

⁎i ð1=vi−1=vÞ þ 3T ilnðv1=3i −1Þ=ðv1=3−1Þ

þX4i¼1

X1j¼4

ðNiv⁎i hjXij=viÞ

�ð13Þ

The value of residual free energy of mixing is substituted inEq. (7) and N, the number of particles is replaced by x to get thefinal expression for the viscosity of a quaternary liquid mixture,which is expressed as

lng ¼"X4

i¼1

xilngi−

(X4i¼1

xip⁎i v⁎i ð1=vi−1=vÞ

þ 3T ilnðv1=3i −1Þ=ðv1=3−1Þ

þX4i¼1

X1j¼1

ðxiv⁎i hjxij=viÞg=RT

þ 1=ðv−1Þ−X4i¼1

xi=ðvi−1Þ( )#

ð14Þ

It is assumed that random mixing of four components takesplace and that a species i neighbours at any given site is equal toits site fraction θi which is defined here as

hið1 N 4Þ ¼ ðsiriNiÞ1 N 4

rN

¼ wið1 N 4Þ

wið1 N 4Þ þP4i¼1

P1j¼4

Wjðv⁎i =vTj Þ; ipj ð15Þ

We have defined the segment fraction as

Wið1 N 4Þ ¼ðriNiÞ1 N 4

rN¼ xið1 N 4Þ

xið1 N 4Þ þP4i¼1

P1j¼4

xjðv⁎j =v⁎i Þ; ipj ð16Þ

The characteristic pressure for the pure components is givenby the equation P⁎= sn/2v⁎2. By analogy we define assumingxij−xji

xij ¼ sDn=2v⁎2 ð17Þ

Substituting the values of characteristic pressure for the purecomponents and on simplification, we have

−E0= rN ¼X4i¼1

WiP⁎i −X4i¼1

X1j¼4

ðWihjxijÞ" #

v⁎=v; ipj

ð18ÞBy analogy with the energy of pure components, we define

−E0=rN ¼ p⁎v⁎=v ¼ ckT=v ð19Þon comparing Eqs. (18) and (19) the characteristic pressure for aquaternary liquid mixture is obtained as

P⁎ ¼X4i¼1

WiP⁎i −

X4i¼1

X1j¼4

Wihjxij

!" #; ipj ð20Þ

Eqs. (19) to (20) yield the characteristic temperatureexpression for quaternary liquid mixture

T⁎ ¼ P⁎=X4i¼1

WiP⁎i =T

⁎i ð21Þ

Assuming the volume reduction parameter of a quaternaryliquid mixture to be linear in mole fractions of the components,we have

v⁎ ¼X4i¼1

xiv⁎i and v ¼ v=

X4i¼1

xiv⁎i ð22Þ

where v is the molar volume of quaternary mixture. By adoptingthe familiar relationship of Berthelot, nij=(niinjj)

1/2 for homo-polar species, the interaction parameter xij can be readilycomputed from Eqs. (15) and (17)

xij ¼ P⁎i ½ð1−ðP⁎j =P⁎i Þ1=2 ðSi=SjÞ1=2Þ�2; ipj: ð23Þ

4. Result and discussion

Molecular interaction study plays a vital role in elucidatingthe complete picture of the multi component systems. Natureand extent of molecular interactions are generally expressed interms of excess functions i.e. ηE and ε derived from statisticalequations of Flory and Gunberg–Nissam [26] equations,respectively. The values of excess viscosity, ηE, have beenevaluated using the following relation

gE ¼ g−X4i¼1

xigi

" #ð24Þ

Guberg–Nissam equation for a quaternary liquid mixture can beexpressed as

lngtheo ¼"X4

i¼1

xilngi þ eðX1X2 þ X2X3 þ X3X4 þ X4X1

þ X2X4 þ X3X1 þ X1X2X3X4Þ#

ð25Þ

Table 1Thermal expansion coefficient (α), isentropic compressibility (KT), reduced volume (v), characteristic volume (v⁎) and dynamic viscosity (η) of pure componentliquids at 298.15 K

Component liquids α a (×103 K−1) KTb, c (T Pa−1) v (cm3 mol−1) v⁎ (cm3 mol−1) ηtheo

a (kg m−1 s−1) ηexp (kg m−1 s−1)

Hexadecane 0.8677 86.7 1.2199 240.8232 0.3012 0.3062Carbon tetrachloride 1.2171 106.67 1.2906 75.2423 0.8631 0.8989Benzene 1.2265 96.7 1.2934 69.2198 0.6002 0.6042Hexane 1.3898 170.9 1.3225 99.4560 0.2937 0.2933a Ref. [31].b Ref. [32].c Ref. [33].

36 R.K. Shukla et al. / Journal of Molecular Liquids 133 (2007) 33–38

where ε is the measure of non-ideality in quaternary liquidmixture and is evaluated by the above equation. Exhaustivestudy of non-ideality parameter and excess viscosity, ηE,reveals the rate of interactions which seems to be weakened bythe addition of third and fourth component in multicomponentsystems which is supported by the work of Rastogi and others[27–29]. According to Nigam et al. [29] ε>0 and higher inmagnitude, there will be strong specific interaction in themixture and ε<0 the weak interaction is indicated.

In order to support the interpretation of molecular interactioninvolved, the expressions for the theoretical evaluation ofexcess free energy of mixing, αΔFm, and interaction energy,Wvis, have been extended for quaternary liquid mixtures. Theresulting equations may be written as

aDFm ¼ −RTðlngtheo−lngidlÞ ð26Þ

Table 2Reduced volume (v), residual free energy of mixing (ΔGM

R ) experimental (ηexp),hexadecane (x1), carbon tetrachloride (x2), benzene (x3), hexane (x4) quaternary syst

SI no. x1 x2 x3 v (cm3 mol−1) ΔGMR (GPa) ηexp

1. 0.0245 0.0979 0.1039 1.3128 1.2960 0.3602. 0.0399 0.1499 0.1367 1.3076 0.6497 0.4043. 0.0397 0.4704 0.1559 1.2977 1.4704 0.5684. 0.0307 0.1033 0.5043 1.3058 1.7838 0.4535. 0.0530 0.2047 0.2157 1.3012 1.9948 0.4646. 0.0280 0.7538 0.1170 1.2911 0.6741 0.7737. 0.0687 0.2565 0.2394 1.2961 2.0844 0.5268. 0.0673 0.4077 0.2382 1.2920 1.7093 0.6189. 0.0289 0.4056 0.4717 1.2924 0.7927 0.67210. 0.0449 0.6744 0.1523 1.2897 0.9333 0.75911. 0.0653 0.2243 0.4231 1.2936 1.3124 0.57012. 0.0709 0.3385 0.3326 1.2909 1.7150 0.62313. 0.0154 0.0674 0.8460 1.2946 0.5915 0.57714. 0.0662 0.5369 0.2370 1.2878 1.2624 0.73615. 0.0648 0.2098 0.5662 1.2900 1.3568 0.63116. 0.0376 0.0967 0.7809 1.2919 0.8059 0.60817. 0.0499 0.1559 0.6857 1.2905 1.0167 0.62818. 0.1059 0.3654 0.2579 1.2853 1.9890 0.69019. 0.1045 0.2304 0.4075 1.2861 2.0048 0.65220. 0.1030 0.2856 0.3697 1.2856 1.9132 0.67821. 0.1318 0.2700 0.2672 1.2828 2.4031 0.68022. 0.1153 0.2568 0.3490 1.2815 1.7079 0.76123. 0.1296 0.3986 0.2776 1.2798 1.8415 0.79124. 0.1941 0.1855 0.1562 1.2759 3.2328 0.71725. 0.1567 0.1345. 0.5734 1.2757 1.6809 0.78926. 0.1718 0.5529 0.1503 1.2716 1.6974 1.00827. 0.2048 0.2877 0.2946 1.2704 2.4277 0.89528. 0.5779 0.1432 0.1108 1.2785 2.2582 1.752

and

Wvis ¼ RT=dlnvP4

i¼1vixi

þ eRT ð27Þ

The values of excess free energy of mixing and interactionenergy, obtained using the above relation, have been incorpo-rated in Table 3. Negative values of interaction energy andpositive values of excess free energy of mixing indicate theweakening of interactions of quaternary liquid mixtures, asevident by the careful observations of Table 3. Table 1 lists theproperties of pure components while Tables 2 and 3 contain theproperties of quaternary liquid mixtures.

Molecular shape and size, condensation effect, sterichindrance contribution, etc., can be explained in terms of

ideal (ηidl) and theoretical (ηtheo) viscosities, percentage deviations (Δ%) ofem at 298.15 K

(kg m−1 s−1) ηidl (kg m−1 s−1) ηtheo (kg m−1 s−1) Δ% (kg m−1 s−1)

5 0.4550 0.3551 1.515 0.5392 0.4113 −1.683 0.7382 0.5715 −0.561 0.5988 0.4425 2.335 0.6328 0.4548 2.090 0.8651 0.7733 −0.039 0.7119 0.5112 2.989 0.8020 0.6101 1.420 0.7665 0.6667 0.795 0.8747 0.7532 0.824 0.7441 0.5570 2.341 0.7992 0.6101 2.091 0.6401 0.5779 −0.158 0.8767 0.7206 2.196 0.7765 0.6175 2.228 0.6991 0.6014 0.891 0.7396 0.6175 1.706 0.8893 0.6602 4.405 0.8499 0.6235 4.448 0.8675 0.6469 4.696 0.9061 0.6393 6.078 0.9381 0.7268 4.599 0.9809 0.7678 3.035 0.9932 0.6612 7.836 0.9874 0.7488 5.163 1.1517 0.9363 7.149 1.1272 0.8208 8.371 2.0152 1.5526 11.38

Table 3Experimental (ηexp

E ) and theoretical (ηtheoE ) excess viscosities, (Δlnηexp−Δlnηtheo) non-ideality parameter (εtheo), excess free energy of mixing (αΔFm) and interaction

energy (Wvis)theo of n-hexadecane–carbon tetrachloride–benzene–hexane at 298.15 K

SI no. x1 x2 x3 ηexpE (kg m−1 s−1) ηtheo

E (kg m−1 s−1) Δlnηexp−Δlnηtheo(kg m−1 s−1)

εtheo (cal) αΔFm (cal) (Wvis)theo (cal)

1. 0.0245 0.0979 0.1038 −0.0945 −0.0999 0.0150 −0.2887 146.86 −109.452. 0.0399 0.1499 0.1367 −0.1314 −0.1279 −0.0167 −0.1048 160.41 6.523. 0.0397 0.4704 0.1559 −0.1699 −0.1667 −0.0056 −0.2225 151.63 −134.814. 0.0307 0.1033 0.5043 −0.1457 −0.1563 −0.0237 −0.4387 179.20 −259.895. 0.0530 0.2047 0.2157 −0.1683 −0.1780 0.0211 −0.2151 195.67 −54.186. 0.0280 0.7538 0.1170 −0.0921 −0.0918 0.0004 −0.1259 66.46 2.597. 0.0687 0.2565 0.2394 −0.1880 −0.2037 0.0303 −0.1812 198.69 −24.848. 0.0673 0.4017 0.2382 −0.1831 −0.1919 0.0143 −0.1453 162.02 −1.479. 0.0289 0.4056 0.4717 −0.0945 −0.0998 0.0079 −0.1323 82.64 −21.5610. 0.0449 0.6744 0.1523 −0.1152 −0.1215 0.0083 −0.1088 88.60 24.2111. 0.0653 0.2243 0.4281 −0.1737 −0.1871 0.0238 −0.1745 171.57 −14.3312. 0.0709 0.3385 0.3326 −0.1761 −0.1891 0.0211 −0.1400 159.95 5.6013. 0.0154 0.0674 0.8460 −0.0630 −0.0622 −0.0014 −0.3188 60.56 −103.9814. 0.0662 0.5369 0.2370 −0.1399 −0.1561 0.0223 −0.0893 116.16 42.3515. 0.0648 0.2098 0.5662 −0.1449 −0.1590 0.0227 −0.1581 135.73 11.5916. 0.0376 0.0967 0.7809 −0.0923 −0.0977 0.0090 −0.2207 89.18 −9.8617. 0.0499 0.1559 0.6857 −0.1115 −0.1221 0.0171 −0.1646 106.89 13.5718. 0.1059 0.3654 0.2579 −0.1987 −0.2291 0.0450 −0.0868 176.48 58.1919. 0.1045 0.2304 0.4075 −0.1974 −0.2264 0.0455 −0.1135 183.51 47.9520. 0.1030 0.2856 0.3697 −0.1887 −0.2206 0.0482 −0.0983 173.83 54.3221. 0.1318 0.2700 0.2672 −0.2255 −0.2668 0.0626 −0.0685 206.62 78.7522. 0.1153 0.2568 0.3490 −0.1768 −0.2113 0.0470 −0.0421 151.19 99.4123. 0.1296 0.3986 0.2776 −0.1891 −0.2132 0.0310 0.0214 145.11 142.9524. 0.1941 0.1855 0.1562 −0.2757 −0.3319 0.0817 0.0812 241.04 186.1425. 0.1567 0.1345 0.5734 −0.1978 −0.2386 0.0531 0.0091 163.87 195.2626. 0.1768 0.5529 0.1503 −0.1434 −0.2154 0.0741 0.0783 122.66 224.4227. 0.2048 0.2877 0.2946 −0.2313 −0.3064 0.0876 0.0124 187.92 165.1528. 0.5779 0.1432 0.1108 −0.2631 −0.4625 0.1209 0.2033 154.49 324.93

37R.K. Shukla et al. / Journal of Molecular Liquids 133 (2007) 33–38

Δlnηexp, Δlnηtheo, and (Δlnηexp−Δlnηtheo) factors which arelisted in Table 3. It was found that with the van der Waals modelfor energy, the difference and (Δlnηexp−Δlnηtheo) is relativelysmall for mixtures of molecules not having a large sizedifference. However, for the systems having a large sizedifference, the difference and (Δlnηexp−Δlnηtheo) should berelatively large due to increasing function of the size difference.The values of Δlnηtheo can be adjusted to Δlnηexp by adding aterm and lnηΔv which is expressed as

lngDv ¼X4i¼1

X1j¼4

Cijðv⁎1=2i −v⁎1=2j Þ2; ipj ð28Þ

where v⁎ is the core volume and Cij is an adjustable parameter.Based on the above fact, it can be concluded that the moleculesconstituting the system have moderately small size differencedue to low values. The size effect marks the shape effect, too. Itis very possible that the liquid viscosity is increased when themolecules have large difference, because the probability of asuitable empty site near to the molecule diminishes. This islikely due to a good filling of the small molecules in betweenthe spaces left by the large ones and it may be the only reasonfor the increased viscosity and decreased volume [30]. Thus, thestructural orientations and shapes of the molecules are alteredcompletely.

The condensation effect is related to some kind of couplingbetween the motions of the condensing and condensated

molecules. If the molecules have the same shape, it is possiblethat the maximum of the effect does not happen.

Steric hindrance contribution is associated with the differencebetween the experimental and theoretical excess data, significantsteric hindrance contribution is supposed to occur either when amolecule in the mixture has a crowded central atom, such as thehighly branched alkanes or when it has special flat shape such ascyclopentane. A careful perusal of Table 2 shows that areasonable agreement has been achieved between theory andthe experiment which provides a better understanding of thestatistical mechanical theory of Flory. The average percentagedeviation has been found to be ±3.14. The results obtained fromFlory's statistical theory can be improved further by consideringthree and four body effects also. In defining the segment and sitefractions, the spherical shape of molecule, i.e. the minimum areaof contact has been assumed. The possibility of only two bodyinteractions has been considered during the extension of thetheory. However, there is every possibility of three and four bodyinteractions also, and these have been ignored in order tosimplify the theoretical procedure. Although, three and fourbody interactions contribute very little to the energy of thesystems yet they probably cannot be ignored in spite of thespherical nature of the molecules. The possibility of three andfour body collisions increases as the chain length increases, i.e.the area of contact increases.

Therefore in order to get a comparable result, a correctionterm is needed to include three and four body effects in the

38 R.K. Shukla et al. / Journal of Molecular Liquids 133 (2007) 33–38

evaluation of characteristic and interchange energy parameters.If the molecule of the larger area of contact constitutes themulticomponent system, the expression for the characteristicpressure of a quaternary liquid system will be of the form

P⁎ ¼"X4

i¼1

WiP⁎i −

( X4i¼1

X1j¼4

Wihixij

!

þX4i¼1

X4j¼1

X4k¼1

WijkhijkXijk

!

þX4i¼1

X4j¼1

X4k¼1

X4l¼1

WijklhijklXijkl

!)#; ipjpkpl

ð29Þwhere ψa, θa and X, respectively, are the segment fraction, sitefraction and interchange energy parameter average of theirpossible contributory liquid components. This relation canultimately be used to evaluate the characteristic temperature ofthe multicomponent liquid mixture.

Acknowledgement

Authors are very thankful to VERSTEHEN, a centre forunderstanding and research for there kind cooperation and help.

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