oer,

fmplaels.arys med tolu

) and eand into impopecial amolarts partime is ah comp

The PFP model arose from a renement of the PrigogineFlory theory

difference in the degree of thermal expansion between the com-

o study mixtures withs must be considered,d it is based in a well-

Journal of Molecular Liquids 203 (2015) 4751

Contents lists available at ScienceDirect

Journal of Mole

.e lsused for the generation of data with no need of experiments. Littleattention is given tomethods for estimating partialmolar volume, espe-cially when it comes to the prediction of vi by using the same set of

established theory. Those remarks encourage future investigations anddevelopments of the model and from that standpoint, it is quite impor-tant to evaluate the theoretical consistence of themodel. This statementmatical models with different purposes. The application of a modelraises further information about the molecular interactions among thecomponents present in the mixture. Moreover, the model is applied tocheck its ability on correlate experimental data and that ability may be

Generally, the PFP model has been applied tdifferent degrees of success although two remarkthe model has just one adjustable parameter angenerally give poor results [1]motivating alternative developments. De-spite its importance, most papers do not present derivative propertiesand the use of models to predict or to correlate such properties is evenmore limited.

Usually excess thermodynamic properties are correlated to mathe-

ponents involved.(iii) P* contribution. This arises from the differences in the internal

pressures and reduced volumes of components.adjustable parameters used to t vE data.

Corresponding author at: Santa Catarina State UniverE-mail address: alessandro.galvao@udesc.br (A.C. Galv

1 Address: SC 469, km 01, Pinhalzinho, SC 89870-000, B

http://dx.doi.org/10.1016/j.molliq.2014.12.0420167-7322/ 2015 Elsevier B.V. All rights reserved.onent i in the volume ofbe estimated by the ap-ssical equations of state

parameter (12). Flory parameter reects the energy changeupon the formation of contacts between unlike molecules.

(ii) Free volume contribution. This contribution originated from a

a mixture. The volume of a liquid solution canplication of an equation of state although, claThe study of excess thermodynamicvolume (vE), excess molar enthalpy (hE

(gE) plays an important role to understin a liquidmixture. Those studies are alsop theories andmathematicalmodels. SvE because the estimation of the partialponent in a liquidmixture depends on iof mole fraction. The partial molar volurepresents the real contribution of eacxcess molar Gibbs energyermolecular interactionsrtant to test and to devel-ttention shall be given tovolume vi of each com-al derivative as a functionderivative property that

excess thermodynamics properties have been correlated by the PFPmodel for mixtures involving polar, non-polar and ionic components[1217].

The model explains physical interactions as a sum of threecontributions:

(i) Interactional contribution, which is proportional to the Floryproperties such as excessmolar[210] by Patterson andDelmas [11]. Themodelwas originally developedto explain the thermodynamic behavior of non-polar mixtures, although1. IntroductionAbility of the PrigogineFloryPatterson mvolumes of binary liquid mixtures

A.C. Galvo ,1, L.G. Franzosi, A.M. da Luz, R.H. SchneidSanta Catarina State University, UDESC, BrazilDepartment of Food Engineering, DEA, Brazil

a b s t r a c ta r t i c l e i n f o

Article history:Received 25 September 2014Received in revised form 16 November 2014Accepted 30 December 2014Available online 2 January 2015

Keywords:PrigogineFloryPatterson modelExcess molar volumePartial molar volumes

Thermodynamic properties olecular interactions that takeopment of theories andmodpartial molar volumes of binC16) at 298.15 K from excesume data. The results indicatrelate data of partial molar v

j ourna l homepage: wwwsity, UDESC, Brazil.o).razil.del to predict partial molar

W.S. Robazza

ixtures such as partialmolar volumeprovide valuable information about themo-ce in a liquid solution. Such studies are also fundamental for the test and devel-Thiswork presents the ability of the PrigogineFloryPattersonmodel to predictliquid mixtures containing toluene and n-alkanes (C6, C7, C8, C9, C10, C12 andolar volume data. The model was also applied to correlate the partial molar vol-hat themodel, which has just one adjustable parameter, is able to predict or cor-me with high accuracy and robustness.

2015 Elsevier B.V. All rights reserved.

cular Liquids

ev ie r .com/ locate /mol l iqmight be interpreted as: if a model has a set of adjustable parameters tocorrelate a specic excess property, its values should be used to predictthe remaining excess properties and the related derivative propertieswith little deviation. Most research groups are concerned just with theuse of a model to correlate the data and unfortunately the investigation

to show the multiproperty description capability of a model is leftbehind.

This work presents the ability of the PFP model on predicting partialmolar volumes by using the same adjustable parameter that ts themodel to excess molar volume data. Moreover, to check the robustnessof the model, the data of partial molar volume were also applied to becorrelated by the model.

2. Methodology

This study was applied to vE data of binary liquid mixtures containingtoluene and n-alkanes (C6, C7, C8, C9, C10, C12 and C16) at 298.15 K [18,

The partial molar volume of each component in the binary liquidmixture is dened by Eqs. (2) and (3) wherein vi represents the molarvolume, p the pressure and T the temperature.

v1 vE v1 1x1 vE

x1

!p;T

2

v2 vE v2x1vE

x1

!p;T

3

The application of the RedlichKister expansion to Eqs. (2) and (3)allows the calculus of vi by Eqs. (4) and (5).

v1 v1 1x1 2Xnj0

Aj 12x1 j 2x1 1x1 2Xnj0

jA j 12x1 j1 4

v2 v2 x21Xnj0

Aj 12x1 j 2x21 1x1 Xnj0

jA j 12x1 j1 5

The data of partial molar volume are determined by applying the

Table 1Thermal expansion coefcient (), isothermal compressibility () and molar volume (v).

103. (MPa1) 103. (K1) v (cm3 mol1)

Toluene 0.9110a 1.0710b 106.9145c

Hexane 1.7039d 1.3870d 131.5700d

Heptane 1.4380d 1.2450d 147.4500d

Octane 1.3024d 1.1640d 163.5100d

Nonane 1.1754e 1.0900e 178.7900e

Decane 1.1096e 1.0500e 195.9200e

Dodecane 0.9876d 0.9600d 228.5500d

Hexadecane 0.8570d 0.8980d 294.0900d

a [20].b [23].c Calculated from density [23].d [21].e [22].

48 A.C. Galvo et al. / Journal of Molecular Liquids 203 (2015) 475119]. The data were correlated to the RedlichKister expansion of the typerepresented by Eq. (1). The coefcients (Aj) of the linear system wereobtained by least square optimization. Index 1 holds for toluene and 2for n-alkanes throughout all the equations used in this work.

vE x1 1x1 Xnj0

Aj 12x1 j 1Fig. 1. Sequence of calculus for eachcoefcients tted by Eq. (1) into Eqs. (4) and (5).According to the PFP model, the excess molar volume (Eq. (6)) is

described by three contributions.

vE

G IU Y 6

G x1V1 x2V2 7approach applied in this work.

The surface fraction (i) is given by Eq. (17) and the ratio of the con-

tact sites per segment for each component s1

as suggested by Abe and

A2(cm3 mol1)

A3(cm3 mol1)

A4(cm3 mol1)

0.0766 0.0051 0.07500.2259 0.5219 0.13000.1711 0.0606 0.1154

0.1802 0.0013 0.02140.1523 0.0413 0.00980.2372 0.0116 0.04150.5309 0.4982 0.1801

49A.C. Galvo et al. / Journal of Molecular Liquids 203 (2015) 4751The contributions of themodel are known as interactional contribu-tion (Eq. (8)), free volume contribution (Eq. (9)) and characteristicpressure contribution (Eq. (10)).

I eV 131 eV 2343

eV131 2112P1

8

U eV1eV2 2 149

eV131 43

eV13 eV 12 9

Y eV1eV2 P1P2 P12 P21

12 10

The PFP model requires the following properties of each pure com-ponent: thermal expansion coefcient (i), isothermal compressibilitycoefcient (i) and molar volume (vi). The reduced volume of each

pure component eVi is calculated by Eq. (11). Table 1 presents thepure component properties used in this work.

eVi 143

iT

1 iT

0BB@1CCA 11

The hard core volume (Vi) and the characteristic pressure (Pi) of thepure components are calculated by Eqs. (12) and (13) respectively.

Pi iTv

2i

i12

Vi vieVi 13

The volumetric fraction of each component in the mixture ( i) iscalculated as follows:

x V

Table 2Coefcients Aj for the tting of RedlichKister equation.

Binary mixture A0(cm3 mol1)

A1(cm3 mol1)

Toluene + hexane 0.1304 0.1061Toluene + heptane 0.6130 0.1827Toluene + octane 0.9854 0.1207Toluene + nonane 1.3745 0.1732Toluene + decane 1.5653 0.4279Toluene + dodecane 1.8152 0.7426Toluene + hexadecane 2.1167 0.94381 12 1 1x1V1 x2V214

The volumetric fraction is used to calculate the contact energyfraction (i).

1 12 1P1

1P1 2P215

The reduced volume for the mixture () is determined by Eq. (16).

eV 1eV1 2eV2 16s2

Flory [7] is given by Eq. (18).

1 12 1

1 s22s1

17

s1s2

V1

V2

13 18The determination of partial molar volumes by the PFPmodel, using

the same adjusted parameter (12) tted from vE data requires the ap-plication of the model equations to Eqs. (2) and (3). The partial deriva-tive of vE as a function of x1 is analytically determined by Eqs. (19) to(23). Fig. 1 summarizes the sequence of calculus applied in this study.

vE

x1 GI x1

GU x1 GY x1

19

vE

x1 I Gx1

G Ix1

U Gx1

G Ux1

Y Gx1

G Yx1

20

Ix1

IeV

eV1

11 1x1

I1 11

1x1 I2

2x121

Ux1

UeV

eV1

11 1x1

U1 11

1x122

Yx1

Y1 11

1x123

Considering that it has used the adjustable parameter obtained fromvE data to predict partial molar volumes, the robustness of the modelwas evaluated and to accomplish this task the adjustable parameter(12) was tted directly to the vi data by minimization of the objectivefunction represented by Eq. (24). The approach intends to analyze theTable 3Parameter of the PFPmodel (12) and average relative deviation (ARD) for themixtures oftoluene (1) + n-alkane (2): predictive modeling (p) and correlative modeling (c).

12(p) (Jcm3)

12(c) (Jcm3)

ARD1(p) ARD2(p) ARD1(c) ARD2(c)

Hexane 21.83 22.08 0.0597 0.0264 0.0587 0.0283Heptane 18.81 15.17 0.1280 0.0188 0.1062 0.0741Octane 18.30 17.54 0.0177 0.0248 0.0063 0.0210Nonane 17.48 15.25 0.0732 0.0136 0.0254 0.0397Decane 18.16 14.06 0.1241 0.0303 0.0502 0.0275Dodecane 16.27 9.48 0.2303 0.0232 0.0934 0.0475Hexadecane 20.92 11.07 0.3118 0.0323 0.1530 0.0749

Table 4Comparison between partial molar volume data derivate from excess molar volume and those predicted (p) and correlated by the PFP model for the mixture toluene (1) + hexadecane (2).

x1 v1(cm3 mol1)

v1 PFP p (cm3 mol1)

v1 PFP c (cm3 mol1)

v2(cm3 mol1)

v2 PFP p (cm3 mol1)

v2 PFP c (cm3 mol1)

0.1035 108.0265 108.3720 107.8481 294.0932 294.0988 294.09550.2377 107.7161 108.1530 107.7110 294.1268 294.1447 294.12420.2435 107.7036 108.1431 107.7048 294.1295 294.1479 294.12620.4305 107.3315 107.8083 107.4937 294.2859 294.3225 294.23640.6450 106.9963 107.3975 107.2313 294.6966 294.8178 294.55320.6501 106.9905 107.3878 107.2250 294.7119 294.8358 294.56480.6564 106.9836 107.3757 107.2173 294.7313 294.8585 294.57940.6579 106.9819 107.3729 107.2154 294.7360 294.8639 294.58290.7049 106.9375 107.2842 107.1581 294.9053 295.0539 294.70580.7179 106.9276 107.2602 107.1425 294.9606 295.1132 294.74430.7601 106.9025 107.1842 107.0930 295.1728 295.3287 294.88460.7840 106.8933 107.1431 107.0661 295.3196 295.4682 294.97580.7897 106.8916 107.1335 107.0599 295.3580 295.5035 294.99890.8499 106.8857 107.0398 106.9982 295.8603 295.9322 295.28110.8694 106.8877 107.0132 106.9806 296.0688 296.0957 295.3894

50 A.C. Galvo et al. / Journal of Molecular Liquids 203 (2015) 4751order of magnitude of the adjustable parameter and the expectationis to adjust the datawith little change on the value of the original adjust-able parameter.

F:O: XNpi1

v1 exp v1 P FP v1 exp

!2i

XNpi1

v2 exp v2 P FP v2 exp

!2i

24

In order to compare the partial molar volumes correlated andpredicted by the model with the partial molar volumes calculated fromexcess molar volume data an average relative deviation was calculatedfor each set of data. The average relative deviation (ARD) is calculatedby Eq. (25) in which Np represents the number of data points.

ARD 100Np

XNpi1

v exp iv P FP iv exp i

25

3. Results and discussion

The coefcients of the RedlichKister expansion determined bytting the data of excess molar volume for all binary mixtures studiedare presented in Table 2.

Table 3 presents the values of the PFP adjustable parameter obtainedfrom the correlation of excessmolar volume data and applied to predict

partialmolar volumes, the values of the adjustable parameter correlated

Fig. 2. Behavior of partial molar volume as a function of mole fraction for the mixture of tomodeling; - - - zcorrelative modeling.directly to partial molar volumes and the average relative deviationbetween experimental data and those from the PFP model.

The results obtained from the application of the PFP model on theprediction and correlation of partial molar volume data compared withthe data acquired from experimental excess molar volume are presentedin the Supplementary data le. In order to illustrate the results, Table 4presents the data for the mixture of toluene and hexadecane.

In the case of partial molar volume of toluene the model performsbetter for the correlative approach than for the predictive one and forthe partialmolar volume of n-alkanes the correlativemodel gives betterresults for C8 and C10 and for the remaining n-alkanes a better behavioris achieved by the predictive model. Fig. 2 illustrates the behavior ofpartial molar volume as a function of mole fraction for the mixture oftoluene (1) and octane (2).

Generally, the performance of the PFP model for the predictive orcorrelative approach is very close and that similarity is veried by theorder of magnitude of the adjustable parameter. In most cases, it is ob-served that the adjustable parameter correlated directly to partialmolarvolume is numerically lower than the adjustable parameter predictedfrom excess molar volume data although the difference between bothis not large.

Fig. 3 shows thebehavior of the adjustable parameter as a function ofnumber of carbons of alkane for the modeling applied in this work. Itis shown a similar trend of both adjusted curves to a third orderpolynomial. The similarity associated with the close values of 12 indi-

cates the robustness of the PFP model and this observation can be very

luene (1) and octane (2): calculated from experimental data [18,19]; predictive

useful for the development of alternative approaches for modeling liquidphase.

4. Conclusion

The PrigogineFloryPatterson model presents ability to generatedata of partial molar volume of binary liquid mixtures containing non-polar molecules. The results obtained, for both predictive and correlative

volume of a liquid phase must be calculated with a higher order ofprecision.

Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.molliq.2014.12.042.

Acknowledgment

The authors wish to thank FAPESC (Fundao de Amparo Pesquisa eInovao do Estado de Santa Catarina) for nancial support.

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Fig. 3. Adjustable parameter as a function of the number of carbons of the alkane: cor-related to partial molar data; correlated to excess molar volume data.

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Ability of the PrigogineFloryPatterson model to predict partial molar volumes of binary liquid mixtures1. Introduction2. Methodology3. Results and discussion4. ConclusionAcknowledgmentReferences