ds 140 (2008) 117–122www.elsevier.com/locate/molliq

Journal of Molecular Liqui

Density, ultrasonic velocity, surface tension, excess volume andviscosity of quaternary fluid solutions

R.K. Shukla ⁎, Atul Kumar, Alok Shukla, Kirti Srivastava

Department of Chemistry, V.S.S.D. College, Kanpur-208002 India

Received 6 July 2007; accepted 5 February 2008Available online 14 February 2008

Abstract

Density, ultrasonic velocity, surface tension, excess volume and viscosity of quaternary non-electrolyte solution n-pentane+tolune+n-heptane+cyclohexane has been experimentally determined at 25 °C over a wide range of composition. Assuming a quaternary fluid solution to be made ofsix binaries, a statistical approach of Prigogine–Flory–Patterson (PFP) theory has been extended to develop the expressions for dynamic viscosity,surface tension excess volume and ultrasonic velocity of quaternary fluid system. A reasonable agreement has been achieved between theory andexperiment. An attempt has also bee made to explain the nature of molecular interactions involved in the light of excess thermodynamic functionswhose sign and magnitude depend upon the chain length and size of the component fluids and also studied the role of order breakers (toluene andcyclohexane) on these properties.© 2008 Elsevier B.V. All rights reserved.

Keywords: Quaternary solution; Viscosity; Surface tension; Ultrasonic velocity; Flory theory; Excess volume

1. Introduction

A better understanding about the properties of multi-component fluid system is of considerable physico-chemicalinterest in design calculations involving separations, heattransfer, mass transfer and fluid flow. Multicomponent fluidsystems rather than single component fluid systems are widelyused in processing and product formulations in many industrialapplications. Among the various properties considered inprocess and product design and optimization, these propertiesare of special interest.

A substantial amount of work has been done on binary [1–4]and ternary [5–10] fluid systems and it is still in progress. Inliterature some correlations describing quaternary fluid systemsare available for excess volume [11,12] and activity coefficients[13], these correlations have at least two major defects. First,they are empirically related to any structure or interactions onthe molecular level, second, they are aimed at a particular group

⁎ Corresponding author.E-mail address: rajeevshukla47@rediff.com (R.K. Shukla).

0167-7322/$ - see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.molliq.2008.02.003

of liquid mixtures. Experimental data on surface tension,ultrasonic velocity and their molecular intepretation for qua-ternary fluid systems are very rare except for some measure-ments on viscosity and excess volume [11,12] refractive index[14], heat capacity [15] surface tension [16,17] and liquid–liquid equilibria [18]. Thermodyamic study of ultrasonicvelocity, surface tension, excess volume and viscosity laysstress particularly on the equation of state contribution toexplain molecular size and shape [19,20] in terms of differentparameters i.e. lattice distortion and disorder [21], condensationeffect [21], steric hindrance [19], coupling of tortional oscil-lations [22] and nature and extent of non-ideality [23] arisingfrom the shape factor and molecular interactions in fluids. Thus,these physico-chemical and thermodynamic properties are adirect consequence of the molecular structure of the fluidsolutions having theoretical intercorrelation with variousmacroscopic properties. This is our first attempt to make acomprehensive study for various physico-chemical and thermo-dynamic properties of non-electrolyte quaternary fluid solutioncontaining two simple non-electrolyte components and twoorder breaker components.

118 R.K. Shukla et al. / Journal of Molecular Liquids 140 (2008) 117–122

A quaternary fluid system is more complex than in ternaryones. With this aim, we have undertaken the thermodynamicand physico-chemical study of non-electrolyte quaternary fluidsolution n-pentane+ toluene+n-heptane+cyclohexane at 25 °Cover a wide range of composition. Here, we report the results ofexperimental measurements of density ultrasonic velocity, sur-face tension, excess volume and viscosity of quaternary fluidsystem at 25 °C. Further, we are analysing the experimentalvalues of these properties in the light of Prigogine–Flory–Patterson(PFP) [24–26]. Order breaking due to componentliquids toluene and cyclohexane and molecular interactionswere also studied.

2. Experimental

Component fluids, pentane, toluene, heptane and cyclohex-ane were obtained from the BDH Chemicals Ltd., Poole,England and they were of Anal R grade, again they werepurified and dried in accordance with the usual procedure [27].Densities were measured at 25 °C in a calibrated modifiedbicapillary pyknometer with an accuracy of ±4×10−4 kg m−1.Crystal controlled variable path ultrasonic interferometer sup-plied by M/s Mittal Enterprises, New Delhi (India), operating ata frequency of 2 MHz was used in the ultrasonic measurementswith an accuracy of ±4 ms−1. Surface tension was measured bythe differential capillary rise method with an accuracy of ±7.2×10−4 N m−1. Isentropic compressibilities, ks, were calculatedfrom the relation,

ks ¼ c�2q�1 ð1Þ

where ρ is the density and c is the ultrasonic velocity. Theestimated error in the calculation of isentropic compressibilitywas found to be ±2.5 T Pa−1. The total uncertainty in the molefractions of the mixture was ±3×10−4. The excess volume ofmixing was determined by a four limbed dilatometer which wasplaced in a water thermostat maintained at the requiredtemperature. After attaining a constant temperature, the fourliquids were mixed by tilting the mixing cell. The volumechange on mixing was estimated by noting the change in theheight of the mercury level in the capillary before and aftermixing by using a cathetometer which could read correctly up to±0.001 cm. Accuracy in the measurements of excess volumewas found to be ±0.006 cm3 mol−1 and accuracy of dilatometerwas tested using the method of Rastogi, Nath, and Yadav [28].Viscosity measurements were carried out by calibrated modifiedthree limbed Cannon–Ubbelohde dilution Viscometer which isa suspended level design of Ubbelohde at 25 °C with anaccuracy of ±4×10−3 kg m−1 s−1. All the measurements ofdensity, viscosity, surface tension, ultrasonic velocity and ex-cess volume were carried out in a thermostatic water bathmaintained at (25+0.01) °C.

3. Theoretical

Here it is assumed that a quaternary mixture can be con-sidered to be made up of six binary mixtures. The molecules of

the four components are divided into equal segments. Adoptingthe same procedure employed in the case of binary mixtures, itis possible to evaluate the characteristic and reduced parametersof a quaternary liquid mixture. Complete extension and deri-vation of the Flory theory for a quaternary liquid mixture hasalready been presented in our earlier papers [12–16]. Ultrasonicvelocity of quaternary liquid mixture has been evaluated withthe help of well known relations of Altenburg and Auerbach.

Patterson and Rastogi [25] in their extension of the cor-derivationresponding states theory dealt with the surface tensionby using the reduction parameter.

r4 ¼ k1=3P42=3

T41=3 ð2Þ

Called the characteristic surface tension of the liquid where kis the Boltzmann constant, starting from the work of Prigogineand Sarage [26], they derived a reduced surface tensionequation which in the case of a Vander Waals liquid, can bewritten as

r vð Þ ¼ Mv�5=3 � v1=3 � 1

v2

!ln

v1=3 � 0:5

v1=3 � 1:0

!ð3Þ

where M is the fraction of nearest neighbours that a moleculeloses on moving from the bulk of the liquid to the surface. Thussurface tension of a liquid in terms of the Prigogine–Flory–Patterson (PFP) theory can be described by the expression.

r ¼ r4d r vð Þ ð4ÞAltenburg [29,30] proposed an expression for ultrasonic

velocity of liquids which is expressed as;

c1 Altenburgð Þ ¼ 5:663rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL=q2M

p� �1=6ð5Þ

where L is the Loschmidt constant having a value of (2.68719±0.00001)×1019 cm−3 ) and M is the molecular weight. PFPtheory has also been used to obtain the ultrasonic velocity ofliquids with the aid of well known empirical relation ofAuerbach [31] which is presented as;

c2 Auerbachð Þ ¼r

6:3� 10�4q

� �2=3

ð6Þ

where σ is the surface tension and ρ is the density of fluid.Combining the absolute rate [32] and free volume theories

[33,34] of liquid viscosity, one obtains the expression for theviscosity of liquid mixture.

g ¼ A exp DGf=RT þ rv4=vf� � ð7Þ

where ΔG # is the free energy of activation per mole, R, the gasconstant, v⁎, the enthalpy volume or core volume which must beavailable for a molecular segment jumping to its new site, vf, thefree volume per segment in the mixture, r, a factor of order unityand η, the viscosity of liquid. We have further extended therelationship between viscosity and the activation free energy permole of the solution ΔG#, derived by Litovitz and Macedo [35]

119R.K. Shukla et al. / Journal of Molecular Liquids 140 (2008) 117–122

to quaternary fluid solution by combining the Eyring relation forresidual free energy of mixing ΔGM

R and ΔG#, we get,

DG # ¼X4i¼1

xiDG#i � amDG

RM ð8Þ

whereΔGRM is the residual free energy of mixing closely related

to the excess free energy of mixing and αm is a constant of orderunity. Substituting the value of ΔG # from Eq. (8) into Eq. (7),we get,

lng ¼X4i¼1

xilngi �DGR

M

RTþ v4 1=vf �

X4i¼1

xi=vfi

!" #ð9Þ

Substitution of Gibbs–Helmholtz equation in Eq. (9) and onchanging v⁎ and vf is terms of v, we get,

lng ¼X4i¼1

xilngi �DHM

RTþ DSRM

Rþ 1

v� 1�X4i¼1

xiv� 1

!" #

ð10Þwhere vf =v−v⁎ and v =v /v⁎.

Eq. (10) displays explicitly the various contributions to themixture viscosity; ideal mixture viscosity, the enthalpy andresidual entropy of mixing and the difference in free volumebetween mixture and pure components.

The residual free energy of mixing may be defined as;

DGRM ¼ DGM � DGcomb ð11Þ

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

R becomes identicalwith GE if ΔGcomb can be represented by the ideal mixing law.If however, the component molecules differ in size, at least oneof them being a chain molecule, ΔGcomb and ΔScomb may beexpressed by the polymer solution theory;

DGcomb ¼ �TDScomb ¼ KT lnZcomb

¼ RTX4i¼1

Nilnwið12Þ

This equation includes a contact interaction term andequation of state term. If the molecule of the four componentsis of comparable size and shape, then ΔGcomb should be givenappropriately by the ideal mixing law and the remaining termmay be identified with the excess free energy, ΔGE.

Following assumptions made by Flory, the equation ofreduced partition function directly given the value of residualfree energy for a quaternary fluid solution as;

DGRM ¼ 3 riNvi4

1::4

X4i¼1

wiPi4T iln vi1=3 � 1� �

= v1=3 � 1� �h i

þDHM ð13Þ

Ignoring the difference between energy and enthalpy ofcondensed system at low pressure, we have extended the

expression of enthalpy for quaternary liquid mixture. The re-sulting equation is,

DHM ¼ Eo mixð Þ � Eo 1ð Þ � Eo 2ð Þ � Eo 3ð Þ � Eo 4ð Þ

¼ rNv4ð Þ1::4X4i¼1

wiPi4

vi� Pi4

vð14Þ

where E0 represents the energy term for liquid mixtures andpure components. Eq. (14) allows the results to be expressed inour alternative form,

DHM ¼ rNv4ð Þ1::4X4i¼1

wiPi4 1=vi � 1=vð Þ þX4i¼1

X4j¼4

Xijwihjv

� ��

ð15ÞSubstituting the value ofΔHM from Eq.(15) into Eq. (16) we

get the expression for residual free energy of mixing.

DGRM ¼ ½ 3 rNv4ð Þ1::4

X4i¼1

wiPi4T iln v1=3i � 1� �

= v1=3 � 1� �( )

þf rNv4ð Þ1::4X4i¼1

wiPi4 1=vi � 1=vð Þ

þX4i¼1

X4j¼i

Xij wihj=v g�

ð16Þ

Eq. (16) can be simplified as,

DGRM ¼ ½X4

i¼1

NiPi4 vi4 1=vi � 1=vð Þ

þ3T ilnð v1=3i � 1� �

= v1=3 � 1� �

þX4i¼1

X4j¼i

ðNivi4hjXij=viÞ�

ð17Þ

The value of residual free energy of mixing is substituted inEq. (11) and N, the number of particles is replaced by x to getthe final expression for the viscosity of a quaternary fluidsolution, which is expressed as;

lng ¼ ½X4i¼1

xilngi �fX4i¼1

xiPi4vi4 1=vi � 1=vð Þ

þ3T iln v1=3i � 1� �

= v� 1ð Þ þX4i¼1

X4j¼i

xivi4hjXij=viÞg=RTþ 1= v� 1ð Þ �

X4i¼1

xi= vi � 1ð Þ( )�

ð18Þ

Let v0 represent the ideal reduced volume per segment if novolume change occurred on mixing.

v0 ¼X4i¼1

wivi ð19Þ

Table 1Isobaric thermal expansivity (α), isentropic compressibility (kT), density (ρ), ultrasonic velocity (C), surface tension (σ), viscosity (η), isentropic compressibility (ks)and isobaric molar heat capacity (Cp) of pure component liquids at 25 °C

Component αb kk−1 kTb Tpa−1 ρ g m−1 C m s−1 σ 10−3 N m−1 η kg m−1 s−1 kS Tpa−1 Cp J K

−1 mol−1

Obs Lita Obs Lit Obs Lita Obs Lita

Pentane 1.6226 2123.4 0.6216 0.6213 990.0 991.00 b 15.56 15.50 0.2160 0.2166 1641.4 1638.9 164.85Toluene 1.0740 921.5 0.8627 0.8625 1300.3 1304.00 c 27.24 27.96 0.6021 0.6030 681.7 681.8 156.06Heptane 1.2589 1424.0 0.6791 0.6795 1131.0 1130.23 d 18.69 19.80 0.3832 0.3855 1151.2 1152.5 209.25Cyclohexane 1.2150 1140.0 0.7734 0.7738 1255.3 1253.28 b 24.01 24.40 0.8910 0.8950 824.9 822.7 152.30

a-Ref. [37] b-Ref. [36] c- Ref. [39] d- Ref. [40].

120 R.K. Shukla et al. / Journal of Molecular Liquids 140 (2008) 117–122

where vi represents the reduced volume of the pure componentat the same temperature and pressure. The reduced excessvolume per segment is

vE ¼ v� v0 ¼ v�X4i¼1

wi vi

!" #¼ vEP4

i¼1xivi4

ð20Þ

The reduced excess volume of the mixture v E, can bedefined with the help of the following expression;

vE ¼ Av=ATð Þ T �T0

� �¼ 3 v0� �7=3

4� 3 v0� �1=3� �1=3

T �T0

� �ð21Þ

where T 0 is the ideal reduced temperature corresponding to theideal reduced volume v 0 and it can be obtained if the addivity ofvolume prevailed. Values of T 0 can be evaluated using the relation.

T0 ¼ v0

� �1=3�1

� �v0� ��4=3

ð22Þ

The characteristic and reduced parameters used in theexpression of excess volume can be evaluated by the proceduregiven by Flory.

Table 2Measurement and theoretical ultrasonic velocities (c), surface tensions (σ), viscositheptane (x3) -cyclohexane (x4)

X1 X2 X3 cexpm s−1

C1theo m s−1

(Altenburgrelation)

C2theo m s−1

(Auerbachrelation)

σexp 10−3

N m−1

0.0404 0.6358 0.1544 1270.1 1206.0 1317.4 24.40.0560 0.5737 0.1284 1256.3 1199.9 1314.2 24.20.0735 0.5474 0.1120 1243.5 1194.0 1310.6 23.90.935 0.5292 0.0959 1230.2 1188.4 1308.1 23.90.1141 0.5054 0.0793 1223.7 1181.8 1305.2 23.70.1134 0.4948 0.0660 1232.3 1184.9 1308.3 23.80.1511 0.4602 0.0487 1218.7 1170.6 1299.7 23.10.1709 0.4375 0.0338 1213.2 1163.8 1294.7 23.00.1071 0.4099 0.0783 1234.5 1175.4 1304.3 23.50.1126 0.4267 0.1137 1218.1 1167.8 1301.2 23.20.1783 0.2174 0.1637 1201.5 1118.1 1286.2 22.00.1991 0.2200 0.1674 1199.2 1111.4 1283.3 21.90.1794 0.6020 0.1451 1233.2 1156.1 1293.7 22.90.1351 0.1100 0.1484 1217.2 1125.3 1291.2 21.90.0948 0.3338 0.2524 1227.5 1141.8 1301.3 22.9Average percentage deviation

Values of the excess thermodynamic parameters i.e. excessviscosity, ηE and excess surface tension, σE for quaternaryliquid mixtures were calculated using the following generalequation;

FE ¼ F � Fidl ¼ F−X4i¼1

xiFi

!" #ð23Þ

where, FE is the corresponding excess thermodynamic para-meter. Excess isentropic compressibility of the liquid mixturecan be defined as,

kEs ¼ ks � ks idlð Þ ð24Þwhere ks(idl) is ideal isentropic compressibility which is obtainedfrom the equation,

kS idlð Þ ¼ ½Xi/i ksi þ Tvi aið Þ2=Cpi

n o� T

Xxivi

� � X/iai

� �2=X

xiCpi

� �� ð25Þ

where T is the absolute temperature and ksi, vi, αi, ϕi and Cpi arerespectively the isentropic compressibility, molar volume,isobaric thermal expansivity, volume fraction, and isobaricmolar heat capacity of the component i.

ies (η) and their percentage deviations (Δ%) at 25 °C pentane(x1)-toluene(x2)-

σtheo 10−3

N m−1ηexp kgm−1.s−1

ηtheo kgm−1 s−1

Percentage deviation (%)

Δc1 Δc2 Δσ Δη

24.31 0.5111 0.5077 5.04 −3.72 0.36 0.6624.09 0.5377 0.5325 4.49 −4.60 0.46 0.8723.91 0.5291 0.5309 3.98 −5.39 0.04 −0.3423.74 0.5161 0.5246 3.43 −6.33 0.66 −1.6523.54 0.5111 0.5199 3.42 −6.66 0.67 −1.7223.57 0.5215 0.5295 3.84 −6.16 0.96 −1.5923.19 0.5136 0.5120 3.94 −6.64 0.38 0.3123.01 0.5017 0.5080 4.07 −6.71 −0.04 −1.2723.31 0.5496 0.5528 4.78 −5.65 0.81 −0.5823.17 0.5307 0.5259 4.12 −6.82 0.13 0.9021.79 0.5112 0.5063 4.94 −7.05 0.95 0.9621.62 0.4812 0.4875 7.32 −7.01 1.27 −1.3122.99 0.4132 0.4154 6.25 −4.90 −0.39 −0.5321.90 0.6420 0.6397 6.07 −6.08 0.00 0.3622.51 0.5023 0.5051 6.98 −6.01 1.70 −0.56

±4.98 ±5.98 ±0.59 ±0.91

Table 3Measured density (ρ), reduced volumes (v ), residual free energy of mixing (ΔGR

M), isentropic and excess isentropic compressibilities (kS) and (kES), measured andtheoretical excess volumes (Vexp

E ) and (VtheoE ), excess viscosities (ηexp

E ) and (ηtheoE ) and excess surface tensions (σexp

E and σtheoE ) at 250° C pentane (x1)-toluene(x2)-

heptane (x3)-cyclohexane (x4)

X1 X2 X3 ρ g m−1 v ΔGMR

GPakSTpa−1

kSE

Tpa−1VexpE cm3

mol−1VtheE cm3

mol−1σexpE 10−3

N m−1σtheE 0−3

N m−1ηexpE kg

m−1 s−1ηtheoE kg

m−1 s−1

0.0404 0.6358 0.1544 0.8073 1.2815 0.9974 767.9 −0.4913 0.035 0.037 −0.62 −0.71 −0.0905 −0.09390.0560 0.5737 0.1284 0.8026 1.2805 0.5451 789.4 −0.4057 0.016 0.014 −0.6 −0.71 −0.0845 −0.08970.0735 0.5474 0.1120 0.7995 1.2822 0.6162 808.9 −0.3411 0.015 0.011 −0.75 −0.74 −0.09733 −0.09550.935 0.5292 0.0959 0.7965 1.2840 0.6990 829.6 −0.2741 0.020 0.022 −0.59 −0.75 −0.1102 −0.10170.1141 0.5054 0.0793 0.7927 1.2858 0.7798 842.4 −0.2956 0.010 0.014 0.62 −0.78 −0.1166 −0.10780.1134 0.4948 0.0660 0.7909 1.2859 0.7654 832.6 −0.3538 0.020 0.017 −0.55 −0.78 −0.1165 −0.10850.1511 0.4602 0.0487 0.7857 1.2894 0.9135 856.9 −0.4107 0.025 0.023 −0.9 −0.81 −0.1177 −0.11930.1709 0.4375 0.0338 0.7840 1.2912 0.9820 866.6 −0.4640 0.019 0.021 −0.82 −0.81 −0.1304 −0.12410.1071 0.4099 0.0783 0.7855 1.2880 0.7124 835.3 −0.4365 0.022 0.027 −0.57 −0.76 −0.1109 −0.10770.1126 0.4267 0.1137 0.7837 1.2883 0.7599 860.0 −0.3303 0.026 0.023 0.72 −0.75 −0.1033 −0.10810.1783 0.2174 0.1637 0.7498 1.2989 0.8848 923.9 −0.6914 0.024 0.028 −0.46 −0.67 −0.1135 −0.11840.1991 0.2200 0.1674 0.7465 1.3003 0.9499 931.5 −0.7949 0.031 0.028 −0.38 −0.66 −0.1269 −0.12060.1794 0.6020 0.1451 0.7842 1.2885 1.1321 838.5 −0.9450 0.001 0.000 −0.88 −0.79 −0.1091 −0.10690.1351 0.1100 0.1484 0.7492 1.2987 0.6583 900.9 −0.6710 0.033 0.036 −0.65 −0.65 −0.0507 −0.05300.0948 0.3338 0.2524 0.7615 1.2909 0.6856 871.5 −0.6546 0.015 0.012 −0.23 −0.62 −0.1001 −0.0973

121R.K. Shukla et al. / Journal of Molecular Liquids 140 (2008) 117–122

Table 1 lists all the parameters for the pure componentswhereasTables 2 and 3 contain the properties of quaternary fluid solution.

4. Results and discussion

Density, surface tension, ultrasonic velocity, viscosity andexcess volume were measured for quaternary non-electrolytefluid solution containing two normal non-electrolytes (n-pentane,n-heptane) and twoorder breakers (toluene, cyclohexane) at 25 °Cover a wide range of composition. Various parameters (α, kT) forthe pure components needed for theoretical calculations weretaken from the literature [36,37] and are listed in Table 1. theexperimental and theoretical values of ultrasonic velocity(c),surface tension(σ), viscosity(η) and their percentage deviation ofquaternary fluid solution are listed in Table 2 whereas the valuesofmeasured density (ρ), excess volume (VE) and theoretical valueof isentropic compressibilities (ks), excess isentropic compressi-bilities (kS

E), excess volume (VtheoE ), excess viscosity (ηE) and

excess surface tension (σE) are listed in Table 3.A close perusal of Table 2 indicates a fairly good agreement

between theoretical and experimental findings has been achievedfor ultrasonic velocity evaluated from two relations (Auerbachand Altenburg) with an average percentage deviation (APD)±4.98 and±5.98 respectively, surface tension with an averageAPD±0.59 and viscositywith anAPD±0.91.A careful perusal ofTable 3 shows that values of excess surface tension, excessvolume and excess viscosity agree well with the experimentalfindings both in sign and magnitude thereby indicating poormolecular interaction among the molecules of component liquidsand also there is no appreciable influence of order breakers on thevarious physico-chemical and thermodynamic properties.

Density and ultrasonic velocity rise with increase in thenumber of carbon atoms whereas isentopic compressibility de-clines directly as evidenced by Tables 1–3. In the n-alkanes thedipole is shielded more and this leads to the lowering of co-hesive forces between molecules and a consequent increment in

the compressibility. Cyclohexane has almost negligible quadru-pole moment due to stable chair configuration, toluene has largequadrupole moment due to presence of methyl group onbenzene ring while n-alkanes have no appreciable quadrupolemoment due to long straight c-chain. Cyclohexane and tolueneare order breakers the main effect of the addition of normalalkenes is change in the free volume of the fluid mixture ascompared to that of pure components. Structural disruption andrestriction of rational motion [38] of the components take placewhen the added component comes under the influence of orderbreaker. Interstitial accommodation and orientational order leadto a more compact structure and to an observed decrease in theexcess compressibility. It is primarily the compressibility thatchanges with structure which ultimately affects the ultrasonicvelocity. The n-alkanes and their derivatives form random coilsowing to internal rotation about c–c bonds, and this tendencyincreases with increasing the chain length. In a quaternary n-alkane system, surface layer of the liquid is enriched with thecomponent of the lower surface tension, thereby minimizing thesurface tension of the mixture.

The study of excess volume leads to the two importantstructural aspects which can be assessed by the relation; VE=VsizeE +Vint

E where VsizeE is associated wit the size difference of

molecules and VintE with the interaction forces between

molecules. Theoretical studies of the molecular interactions ina quaternary fluid solution are very complicated due to threeand four body effects, though they contribute very little to theenergy of the system. And that is why the results so obtained arefound to be satisfactory. A complete discussion and interpreta-tion has been given in our earlier papers [8–12].

The results obtained from PFP theory can be improvedfurther by considering three and four body effects also. Indefining the segment and site fractions, a spherical shape ofmolecule i.e. the minimum area of contact has been assumed.The possibility of only two body interactions has been con-sidered during the extension of the theory. However, there is

122 R.K. Shukla et al. / Journal of Molecular Liquids 140 (2008) 117–122

every possibility of three and four body interactions also, andthese have been ignored in order to simplify the theoreticalprocedure. Although, three and four body interactions con-tribute very little to the energy of the system they probablycannot be ignored inspite of the spherical nature of the mole-cules. The possibility of three and four body collisions increasesas the chain length, i.e. area of contact, increases. Therefore, inorder to get a comparable result, a correction term is neededto include three and four body effects in the evaluation ofcharacteristic and interchange energy parameters.

Acknowledgment

We are thankful to the Department of Chemistry V.S.S.D.College, Kanpur for the help and support.

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