Theoretical Description of Excess Molar Functions in the Case of Very Small Excess Values. Investigation of Excess Molar Volumes of Propan-2-ol + Alkanol Mixtures

Page 1

J Solution Chem (2008) 37: 1747–1773DOI 10.1007/s10953-008-9334-7

Theoretical Description of Excess Molar Functionsin the Case of Very Small Excess Values. Investigationof Excess Molar Volumes of Propan-2-ol + AlkanolMixtures

Monika Geppert-Rybczynska · Barbara Hachuła ·Monika Bucek

Received: 18 June 2008 / Accepted: 31 July 2008 / Published online: 11 October 2008© Springer Science+Business Media, LLC 2008

Abstract The densities of propan-2-ol + pentan-1-ol, + hexan-1-ol, + heptan-1-ol, +octan-1-ol + nonan-1-ol and speeds of sound in propan-2-ol + pentan-1-ol, + heptan-1-ol,+ nonan-1-ol have been measured over the whole composition range at 298.15 K. Excessmolar functions determined from the experimental data have been plotted as functions ofcomposition. The excess molar volumes have been interpreted on the basis of the Symmet-rical Extended Real Associated Solution Model (S-ERAS).

Keywords Speed of sound · Excess molar volume · Excess molar compressions · S-ERASmodel

1 Introduction

Binary mixtures containing alcohols are characterized by very small excess molar quantities(excess molar volumes, V E, and excess molar compressions, KE

S ) in comparison with the al-cohol + alkane or even alkane + alkane systems. Thus, the excess values are very difficult tomeasure. This is due to effects that contribute to the excess functions in counteracting ways.

A reasonable explanation of the results obtained can be found by comparison of theproperties of mixtures under test with those of similar systems containing alcohols and/ortheir homomorphs [25]. It is also possible to discern between contributions resulting fromthe chemical and physical interactions by applying various theoretical models. New sym-metrical reformulation of the ERAS model (Extended Real Association model), recentlydescribed in the literature [26, 27, 53, 54], seems to be particularly suitable to this end.

This work is a continuation of our studies of binary mixtures containing two alcohols.Our earlier work was devoted first and foremost to systems containing primary chain al-cohols [26]. In the present work, the results obtained for binary mixtures with propan-2-ol as the common compound are presented and discussed. The second compounds were:

M. Geppert-Rybczynska (�) · B. Hachuła · M. BucekInstitute of Chemistry, University of Silesia, Szkolna 9, 40-006 Katowice, Silesia, Polande-mail: [email protected]

M. Geppert-Rybczynskae-mail: [email protected]

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1748 J Solution Chem (2008) 37: 1747–1773

pentan-1-ol, hexan-1-ol, heptan-1-ol, octan-1-ol, and nonan-1-ol. The excess molar volumesof these mixtures were supposed to lie between the negative excesses of the propan-2-ol +propan-1-ol mixture and the positive ones for the propan-2-ol + decan-1-ol system [56, 63].Thus, S-shaped V E(x) curves, rather unusual for binary alcohol mixtures, should occur forsome of those systems.

Additionally, the excess molar compressions, KES , were determined to complete the study.

In our earlier works [25, 26] we reported opposite signs of the V E and KES functions for such

binary mixtures; positive excess volumes and negative excess compressions. That is ratheruncommon for the systems like alcohol + alkane or alkane + alkane. At present, we wouldlike to check if the results obtained from calculations made by the S-ERAS model equationsare consistent with those drawn on the basis of some comparisons.

2 Experimental Section

Chemicals The chemicals: propan-2-ol (Alfa Aesar, 99+%), pentan-1-ol (Lancaster,98+%), hexan-1-ol (Lancaster, 99%), heptan-1-ol (Fluka, ≥99%), octan-1-ol (Lancaster,99%), and nonan-1-ol (Alfa Aesar, 99%) were dried over molecular sieves 3 Å. The watercontent determined by the Karl Fischer method varied from 0.02 to 0.03% (by mass). Thevalues of basic physical properties of the pure components determined in this work and thecorresponding literature data are listed in Table 1.

Density Measurements The mixtures were prepared by mass using an analytical balance(Ohaus AS 200) with the accuracy of 6 × 10−7 kg. The accuracy of the mole fractions wasassessed at ±1 × 10−4.

Density was measured at 298.15 K with an Anton Paar DMA 5000 vibrating tube den-simeter. The accuracy of density measurements was assessed at 5 × 10−5 g·cm−3 whereasthe repeatability was 1 × 10−5g·cm−3.

The standard water used for the calibration of the densimeter was distilled and degasseddirectly before use; its specific conductivity was ca. 1 × 10−4 S·m−1.

Speed of Sound Measurements The speed of sound was measured using the sing-aroundapparatus designed and constructed in our laboratory. The accuracy of the measurementswas 1 m·s−1, whereas the repeatability was better than 1 cm·s−1 [16].

The details concerning the measuring set as well as the calibration procedure were de-scribed previously [16, 27].

All solutions for the density and speed of sound measurements were prepared and de-gassed in an ultrasonic cleaner just before the measurement.

3 Theoretical Section

The S-ERAS Model The Symmetrical Extended Real Associated Solution model (S-ERAS)may be regarded as an expanded version of the ERAS model [53, 54]. The symmetrization ofsome equations of the ERAS model [54] allows the S-ERAS model to reproduce very smallexcess molar quantities, V E and H E, such as those of binary alcohol mixtures [26, 27, 54].

The bases of the ERAS and S-ERAS models have been extensively described previ-ously [5, 27, 30, 54]. Thus, the description given here is limited to some essential relationsneeded for understanding the application of the S-ERAS model to excess molar volumes ofthe systems studied in this work.

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J Solution Chem (2008) 37: 1747–1773 1749

Tabl

e1

Den

sitie

s,ρ

,spe

eds

ofso

und,

c,a

ndre

frac

tive

inde

xes,

nD

,for

pure

com

pone

nts

atT

=29

8.15

K

ρ/(k

g·mol

−1)

c/(m

·s−1)

nD

H2O

Thi

sw

ork

Lit.

Thi

sw

ork

Lit.

Thi

sw

ork

Lit.

%

Prop

an-2

-ol

780.

825

±0.

0578

0.8

[49]

;780

.84

[32]

;78

0.85

[64]

;780

.89

[59]

;78

0.93

[29]

;780

.93

[57]

;78

0.95

[37]

1138

.19

±0.

211

34.2

6[4

2];

1138

.09

[60]

;11

38.7

1[6

8];

1139

[28]

1.37

51±

0.00

021.

3748

7[5

6];

1.37

55[6

6];

1.37

58[3

9]

0.02

Pent

an-1

-ol

810.

796

±0.

0581

0.77

[61]

;810

.80

[57]

;81

0.92

[43]

;810

.93

[8];

810.

97[1

2];8

11.0

1[1

1];

811.

02[2

5]

1274

.73

±0.

212

74.4

2[6

0];

1274

.84

[66]

;12

75.1

8[3

9];

1275

.27

[24]

;12

75.5

[ 8]

1.40

78±

0.00

021.

4079

[59]

;1.

4080

[36]

;1.

4082

[66]

;1.

4085

[13]

0.03

Hex

an-1

-ol

815.

294

±0.

0581

5.25

[15]

;815

.29

[65]

;81

5.3

[52]

;815

.32

[22]

;81

5.34

[57]

;815

.40

[45]

1.41

67±

0.00

021.

4159

5[1

0];

1.41

61[6

7];

1.41

66[5

8]

0.03

Hep

tan-

1-ol

818.

765

±0.

0581

8.75

[55]

;[35

];81

8.76

[23]

;[61

];[4

];81

8.77

[33]

;[34

];81

8.78

[9];

[2]

1327

.01

±0.

213

26.1

[47]

;13

27.2

7[1

4];

1329

[3]

1.42

20±

0.00

021.

4217

[48]

;1.

4222

[17]

;1.

4224

[67]

0.03

Oct

an-1

-ol

821.

572

±0.

0582

1.57

[29]

;821

.59

[31]

;82

1.60

[47]

1.42

76±

0.00

021.

4274

[21]

;1.

4275

[40]

;1.

4275

3[5

5];

1.42

765

[46]

;1.

4278

[25]

0.02

Non

an-1

-ol

824.

271

±0.

0582

4.23

[55]

;824

.22

[2];

824.

28[6

7]13

64.7

0±

0.2

1362

.5[8

];13

64.4

1[4

4];

1364

.64

[14]

1.43

17±

0.00

021.

4315

[50]

;1.

4318

2[5

5];

1.43

19[6

7];

1.43

24[5

9]

0.03

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1750 J Solution Chem (2008) 37: 1747–1773

In the S-ERAS and ERAS models, the excess molar volume is split into two parts. Thechemical part results from the breaking and creating of the hydrogen bonds during mix-ing. Thus, it can be described by the parameters: equilibrium constants of the formationof hydrogen bonds between molecules of the same kind, KA and KB, and between differ-ent molecules, KAB (cross-association constants), self-association molar volumes, �v∗

A and�v∗

B, and the molar volume of complexation, �v∗AB. The physical part results from the con-

cerned processes that can not be described by equilibrium constants.The physical part of the S-ERAS model is based on the Flory’s theory [1, 18–20, 51],

where the properties of each i component of the mixture are represented by characteristicparameters: the characteristic volume (hard-core volume), V ∗, characteristic pressure, p∗,and characteristic temperature, T ∗. All these parameters can be determined from the mo-lar volume, Vi , thermal expansion coefficient, αi , and isothermal compressibility, κT,i ofthe pure components. The characteristic parameters can be used for the determination ofthe appropriate reduced parameters: V = V/V ∗, p = p/p∗, T = T/T ∗. The reduced para-meters of the pure components and of the mixture satisfy Flory’s equation of state (at lowpressures):

T = V 1/3 − 1

V 4/3(1)

The reduced parameters can be calculated from the characteristic parameters using the well-known equations given in the literature, inter alia in [1, 18–20, 30, 51].

The chemical part of the S-ERAS theory is based on the assumption that the two compo-nents (alcohols) associate into linear oligomers according to the following equations:

Am−1 + A1KA� Am, (2)

Bn−1 + B1KA� Bn, (3)

Am + Bn

KAB� AmBn (4)

The association constants, KA,KB and KAB, are independent of the chain lengths, i.e., thevalues of m and n, and depend on temperature according to the van’t Hoff equation. Con-sequently, the equation for describing the excess molar volume, V E, takes the form [26, 27,54]:

V E = (xAV ∗A + xBV ∗

B )(VM − �AVA − �BVB) physical contribution

+ VM[xAKA�v∗A(φA1 − φ0

A1) + xBKB�v∗B(φB1 − φ0

B1)] chemical contribution

+ �v∗AB(xAV ∗

A + xBV ∗B )

φA1

1 − KAφA1

φB1

1 − KBφB1

KAB(KAV ∗A + KBV ∗

B)

(KA + KB)V ∗AV ∗

B

(5)

where: VM is the reduced volume of the mixture; �A and �B are the hard-core volumefractions or segment fractions of the components; φA1, φB1 and φ0

A1, φ0B1 are the volume

fractions of the monomers in the pure state and in the mixture, respectively. The last fourvolume fractions are independent of one another and can be calculated using formulae givenin [26, 27, 54]. It should be noticed that the chemical contribution to V E consists of twoequivalent parts containing the association constants of both the components, KA and KB,and the cross-association constant, KAB, as well as the volumes of formation of hydrogenbonds in the pure state, �v∗

A and �v∗B, and in the mixture, �v∗

AB.

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J Solution Chem (2008) 37: 1747–1773 1751

A new reformulation of the ERAS model equations [54] included changes in the chemicaland physical contributions to the excess molar volume or enthalpy but also for the physicalone. The main change was in the mixing rule for the characteristic pressure of the mixture,p∗

M, resulting from Flory’s theory:

p∗M = �Ap∗

A + �Bp∗B − 1

2(�AϑB + �BϑA)XAB (6)

where XAB is an adjustable energetic parameter; ϑA and ϑB are the surface fractions (or sitefractions) of the components given by the equation:

ϑA = sBφB

sAφA + sBφB= 1 − ϑB (7)

and sA and sB are the surface to volume ratios that may be calculated as sA/sB from the Abeand Flory approximation (sA/sB) = (V ∗

B /V ∗A)1/3. The sA/sB parameter can also be obtained

from Bondi’s group contribution tabulated for the molecular surfaces and volumes. Thismethod was used in the original ERAS calculations [5, 7, 38]. It is also possible to treatthe sA/sB ratio as an adjustable parameter as in the approach of Benson and co-workers [6].It was shown previously [26] that the last method enables a better fit of the excess molarvolumes and excess molar enthalpies. However, the physical meaning of this parameterdiffers from the classic one. Finally, in the routine procedure, the only adjustable parameter,XAB, comes from the physical part of the S-ERAS model.

To sum up, from among all the quantities mentioned above, seven adjustable parameters:�v∗

A,�v∗B,�v∗

AB,KA,KB,KAB and XAB should be determined. A reasonable reduction ofthat number is advisable. Thus, some of the parameters ought to be fixed. For example,�v∗

A and �v∗B are often regarded as independent of the type of the alkanol and assumed

to be −5.6 cm3·mol−1 [30]; next, the KA and KB values, obtained from the enthalpy ofvaporization of the pure alkanols, can be taken from other sources [41].

Finally, after a careful reduction, the remaining parameters: �v∗AB,KAB and XAB should

be obtained from the fitting of the model to the experimental excess quantities. Unfortu-nately, the excess molar volumes do not provide any information on the hydrogen-bondenthalpy in the mixture, �h∗

AB. However, it is obvious that in the theoretical description ofonly the excess molar volumes of a selected class of binary mixtures in terms of the S-ERASmodel gives valuable information about the molecular interactions in the mixtures.

4 Calculations and Interpretation

4.1 The Excess Functions V E and KES

The densities of propan-2-ol + pentan-1-ol, propan-2-ol + hexan-1-ol, propan-2-ol +heptan-1-ol, propan-2-ol + octan-1-ol and propan-2-ol + nonan-1-ol, and the speed ofsound measurements for propan-2-ol + pentan-1-ol, propan-2-ol + heptan-1-ol and propan-2-ol + nonan-1-ol were measured over the whole concentration range at 298.15 K. The mea-sured densities and the speeds of sound are collected in Table 2. In all the calculations, thespeeds of sound for the propan-2-ol + hexan-1-ol and propan-2-ol + octan-1-ol mixtureswere taken from the literature sources [27, 62]. In order to determine the speeds of sound at

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1752 J Solution Chem (2008) 37: 1747–1773

Table 2 Densities and speeds ofsound of mixtures at 298.15 K x1 ρa x1 ub

Propan-1-ol (1) + pentan-1-ol (2)

0.0999 808.637 0.1000 1264.75

0.2004 806.368 0.2026 1254.14

0.3000 803.970 0.3011 1243.24

0.3970 801.469 0.3998 1231.63

0.3993 801.406 0.5000 1218.82

0.5020 798.562 0.5999 1205.22

0.6004 795.621 0.7000 1190.45

0.7004 792.386 0.8001 1174.39

0.7986 788.927 0.8998 1157.00

0.9007 785.016

Propan-1-ol (1) + hexan-1-ol (2)

0.1071 812.964

0.2120 810.484

0.2997 808.244

0.4026 805.379

0.5026 802.300

0.6002 798.992

0.7030 795.110

0.7988 791.072

0.9003 786.240

Propan-1-ol (1) + heptan-1-ol (2)

0.0947 816.725 0.1005 1316.07

0.0983 816.645 0.2257 1300.92

0.1290 815.950 0.3083 1289.84

0.1959 814.367 0.4118 1274.64

0.3107 811.369 0.4946 1261.15

0.4470 807.310 0.5967 1242.59

0.4908 805.856 0.7013 1221.03

0.6072 801.608 0.7748 1203.98

0.7023 797.623 0.8739 1177.77

0.8543 790.045 0.9155 1165.65

0.9189 786.24

Propan-1-ol (1) + octan-1-ol (2)

0.1005 819.464

0.2008 817.128

0.3006 814.526

0.4000 811.604

0.5005 808.260

0.6005 804.438

0.7002 799.986

0.9002 788.469

0.8003 794.736

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J Solution Chem (2008) 37: 1747–1773 1753

Table 2 (Continued)

a(kg·m−3)

b(m·s−1)

x1 ρa x1 ub

Propan-1-ol (1) + nonan-1-ol (2)

0.0997 822.216 0.1003 1353.92

0.2001 819.891 0.2023 1341.45

0.2994 817.314 0.2999 1328.00

0.3993 814.364 0.4001 1312.26

0.4998 810.944 0.5000 1294.33

0.5996 806.982 0.6001 1273.18

0.7000 802.263 0.7001 1248.49

0.7999 796.593 0.8004 1218.76

0.8499 793.304 0.8500 1201.81

0.9002 789.608 0.9000 1182.98

0.9500 785.498 0.9500 1161.78

Fig. 1 Speed of sound isotherms for propan-2-ol (1) + propan-1-ol (2) (!) [62], + butan-1-ol [62] (1),+ pentan-1-ol (P), + hexan-1-ol [62] (E), heptan-1-ol ("), + octan-1-ol [62] (2), + nonan-1-ol (Q)(T = 298.15 K); lines: polynomial (Eq. 8) or drawn on the basis of a polynomial taken form literature [62]

the mole fractions at which the densities were measured, the experimental values were fittedby the following polynomial:

u =n∑

i=0

aixi1 (8)

where x1 is the mole fraction of the first component, ai(given in Table 3 together with stan-dard deviations and the mean deviations from the regression line), are coefficients obtainedby the least-squares method. Figure 1 shows the speed of sound isotherms for all the mix-tures investigated together with the u values taken from the literature for the propan-2-ol +

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1754 J Solution Chem (2008) 37: 1747–1773

Tabl

e3

Coe

ffici

ents

ofth

epo

lyno

mia

lu=

∑n i=

0aix

i 1fo

rm

ixtu

res

with

stan

dard

devi

atio

nsan

dm

ean

devi

atio

nsfr

omth

ere

gres

sion

lines

,δu

a0

aa

1a

a2

aa

3a

a4

aδ u

a

Prop

an-2

-ol(

1)+

pent

an-1

-ol(

2)12

74.7

01±

0.04

1−9

6.74

±0.

37−1

9.83

±0.

90−1

9.97

±0.

590

0.04

6

Prop

an-2

-ol(

1)+

hept

an-1

-ol(

2)13

27.0

68±

0.08

3−1

04.8

4±

0.62

−48.

66±

0.65

0−3

5.27

±0.

590.

095

Prop

an-2

-ol(

1)+

nona

n-1-

ol(2

)13

64.9

81±

0.28

4−1

06.7

±2.

1−5

1.3

±3.

60

−68.

3±

2.0

0.33

a (m

·s−1)

Page 9

J Solution Chem (2008) 37: 1747–1773 1755

propan-1-ol [62], propan-2-ol + butan-1-ol [62], propan-2-ol + hexan-1-ol [27] and propan-2-ol + octan-1-ol [62] mixtures.

From inspection of Fig. 1, one learns that the speeds of sound in the systems under testchange monotonically with composition. As was easy to predict, the most striking changesof the speed of sound with composition are observed for the propan-2-ol + nonan-1-olmixture.

The excess molar volumes, V E, were calculated from the densities using the followingequation:

V E = x1M1 + (1 − x1)M2

ρ− x1

M1

ρ1− (1 − x1)

M2

ρ2(9)

where M1 and M2 are the molar masses of the first and second component, respectively;ρ,ρ1 and ρ2 are the densities of the mixture and the pure components, respectively; and x1 isthe mole fraction of the first component (propan-2-ol). Excess molar volumes are presentedin Table 4. The following equation was fitted to the experimental excess volumes:

V E = x1(1 − x1)∑

i

bi(1 − 2x1)i (10)

The coefficients, bi , of Eq. 10 calculated by the least-squares method, are collected in Ta-ble 5. The isotherms of the excess molar volumes for mixtures containing propan-2-ol as thecommon compound, are shown in Fig. 2 together with the V E data taken from the literaturefor the propan-2-ol + propan-1-ol [56] and propan-2-ol + decan-1-ol [62] systems.

From inspection of Fig. 2, it follows that the V E values obtained in this work changewith the length of the alcohol chain almost regularly. In fact, only the excess molar volumesobserved for the propan-2-ol + propan-1-ol system do not fit exactly into the place deducedfrom remaining isotherms. It is very difficult to find a reason for such behavior, but in ouropinion the quality of propan-1-ol can probably change the position of the V E(x) curve withrelation to those of the other ones. Nevertheless, we consider the latter data to be to a largeextent reliable because of the high precision and accuracy of the dilatometric method usedby Polak and co-worker [56].

A general rule for mixtures containing lower alcohols, such as propan-1-ol, pentan-1-olor hexan-1-ol, is that the V E values are negative in the whole concentration range with min-ima shifted towards higher concentrations of propan-2-ol. The longer the carbon chain themore pronounced is the shift. This points to strong hydrogen bonding between the compo-nents as manifested in the volume. For propan-2-ol + heptan-1-ol, propan-2-ol + octan-1-ol and propan-2-ol + nonan-1-ol, the V E(x) curves become S-shaped with a maximum atlower concentrations of propan-2-ol. What is more to the point the excess molar volumes arevery small, which allows us to conclude that the interactions in these systems cancel eachother out. Finally, for the propan-2-ol + decan-1-ol mixture, the V E value is positive at anyconcentration which means that in this mixture the process of breaking of hydrogen bondspredominates over their creation between the propan-2-ol and decan-1-ol molecules. On theother hand, the physical interactions seem to play an equally important role as the chemicalones.

Another often-discussed thermodynamic quantity is the excess molar adiabatic compres-sion, KE

S . It can be calculated from the densities, ρ, and speeds of sound, u, through theadiabatic compressibility, κS = 1/ρu2, and molar adiabatic compression, KS = κSV , in the

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1756 J Solution Chem (2008) 37: 1747–1773

Table 4 Excess molar volumesand excess molarcompressibilities of theinvestigated mixtures at 298.15 K

x1 106V E a 1015KES

b

Propan-1-ol (1) + pentan-1-ol (2)

0.0999 −0.0032 −0.3075

0.2004 −0.0112 −0.5937

0.3000 −0.0186 −0.8336

0.3970 −0.0240 −1.0102

0.3993 −0.0241 −1.0136

0.5020 −0.0287 −1.1248

0.6004 −0.0308 −1.1440

0.7004 −0.0298 −1.0596

0.7986 −0.0242 −0.8570

0.9007 −0.0155 −0.5061

Propan-1-ol (1) + hexan-1-ol (2)

0.107019 −0.0049 −0.453

0.211981 −0.0109 −0.839

0.299693 −0.0169 −1.115

0.402574 −0.0229 −1.372

0.502636 −0.0260 −1.532

0.600172 −0.0279 −1.585

0.703023 −0.0273 −1.498

0.798815 −0.0244 −1.251

0.900313 −0.0156 −0.763

Propan-1-ol (1) + heptan-1-ol (2)

0.0947 0.0004 −0.442

0.0983 0.0003 −0.458

0.1290 −0.0009 −0.594

0.1959 −0.0046 −0.877

0.3107 −0.0085 −1.304

0.4470 −0.0148 −1.693

0.4908 −0.0158 −1.779

0.6072 −0.0184 −1.887

0.7023 −0.0185 −1.806

0.8543 −0.0152 −1.244

0.9189 −0.0116 −0.796

Propan-1-ol (1) + octan-1-ol (2)

0.1005 0.0019 −0.522

0.2008 0.0023 −1.004

0.3006 0.0017 −1.437

0.4000 0.0005 −1.801

0.5005 −0.0033 −2.069

0.6005 −0.0079 −2.191

0.7002 −0.0086 −2.111

0.8003 −0.0090 −1.771

0.9002 −0.0073 −1.099

Page 11

J Solution Chem (2008) 37: 1747–1773 1757

Table 4 (Continued)

a(m3·mol−1)

b(m5·N−1·mol−1)

x1 106V E a 1015KES

b

Propan-1-ol (1) + nonan-1-ol (2)0.0997 0.0075 −0.5560.2001 0.0139 −1.0970.2994 0.0148 −1.6050.3993 0.0131 −2.0520.4998 0.0100 −2.3900.5996 0.0053 −2.5560.7000 0.0005 −2.4830.7999 −0.0020 −2.0940.9002 −0.0036 −1.3010.8499 −0.0037 −1.7560.9500 −0.0033 −0.724

Fig. 2 Excess molar volumes of propan-2-ol (1) + propan-1-ol (2) [56] (!), + pentan-1-ol (1), +hexan-1-ol (P), + heptan-1-ol (E), + octan-1-ol ("), + nonan-1-ol (2) and + decan-1-ol [63] (Q); points:calculated from Eqs. 11–13 or taken from the literature; solid lines: calculated from Eq. 10, dashed lines:taken from literature [56, 63] (T = 298.15 K)

following way:

KES = KS − K id

S , (11)

K idS = K id

T −(

(Aidp )2T

C idp

)(12)

where K idS and K id

T are the molar adiabatic and isothermal compressions, respectively, Aidp

is the ideal isobaric thermal expansion (Aidp = (∂V id/∂T )p),C id

p is the isobaric molar heatcapacity, and T is the temperature in K. All the quantities for the ideal mixtures (Xid :

Page 12

1758 J Solution Chem (2008) 37: 1747–1773

Tabl

e5

Coe

ffici

ents

ofth

epo

lyno

mia

lVE

=x

1(1

−x

1)∑

ibi(

1−

2x1)i

with

stan

dard

devi

atio

nsan

dm

ean

devi

atio

nsfr

omth

ere

gres

sion

line

(T=

298.

15K

)

Syst

emb

0a×

106

b1

a×

106

b2

a×

106

b3

a×

106

b4

a×

106

δa×

106

Prop

an-2

-ol(

1)+

pent

an-1

-ol(

2)−0

.115

23±

0.00

093

0.06

19±

0.00

370.

0135

±0.

0046

0.03

0±

0.00

10

0.00

04

Prop

an-2

-ol(

1)+

hexa

n-1-

ol(2

)−0

.104

38±

0.00

068

0.05

71±

0.00

26−0

.010

0±

0.00

310.

0363

±0.

0070

00.

0003

Prop

an-2

-ol(

1)+

hept

an-1

-ol(

2)−0

.065

5±

0.00

100.

0490

±0.

0054

00.

059

±0.

012

00.

0006

Prop

an-2

-ol(

1)+

octa

n-1-

ol(2

)−0

.013

6±

0.00

150.

0635

±0.

0029

− 0.0

224

±0.

0067

00

0.00

06

Prop

an-2

-ol(

1)+

nona

n-1-

ol(2

)0.

0388

7±

0.00

072

0.08

25±

0.00

160

0−0

.039

6±

0.00

590.

0004

a (m

3·m

ol−1

)

Page 13

J Solution Chem (2008) 37: 1747–1773 1759

Fig. 3 Excess molar compressibility of propan-2-ol (1) + propan-1-ol (2) [62] (!) , + butan-1-ol [62] (1),+ pentan-1-ol (P), + hexan-1-ol ("), + heptan-1-ol (2), + octan-1-ol [62] ( ), + octan-1-ol (Q), andnonan-1-ol (F); solid lines: calculated from Eq. 14, dashed lines: taken from literature [62] (T = 298.15 K)

Aidp ,C id

p ,K idT ) can be calculated from the following equation:

Xid =∑

i=1

xiX∗i (13)

where the superscript “*” denotes the pure component.The following polynomial was fitted to the excess molar compressions calculated using

Eqs. 11–13, and is presented in Table 4:

KES = x1(1 − x1)

∑

i

di(1 − 2x1)i (14)

The coefficients di of Eq. 14, calculated by the least-square method, are collected in Ta-ble 6. The concentration dependencies of the excess molar compressions for all the systemsunder test and for the KE

S data taken from the literature [62] are shown in Fig. 3. The nega-tive excess molar compressions increase with increasing length of the hydrocarbon chain ofthe alcohols.

We noticed earlier [25, 27] that the excess molar volumes and excess molar compressionsof the primary chain alcohol mixtures had opposite signs. This has not been observed foralkane + alkane and alcohol + alkane mixtures. The respective excesses for binary mixturesof propan-2-ol with heptan-1-ol, octan-1-ol and nonan-1-ol have also opposite signs, eitherin the whole concentration range, or in a part of it, as is illustrated in Figs. 2 and 3. Thisprobably holds also for the mixtures of higher primary alcohols with propan-2-ol.

4.2 Application of the S-ERAS Model

The S-ERAS model has been applied for the description of excess molar volumes at298.15 K for all the mixtures considered in our earlier study, i.e., for the propan-2-ol +

Page 14

1760 J Solution Chem (2008) 37: 1747–1773

Tabl

e6

Coe

ffici

ents

ofth

epo

lyno

mia

lKE S

=x

1(1

−x

1)∑

idi(

1−

2x1)i

with

stan

dard

devi

atio

nsan

dm

ean

devi

atio

nsfr

omth

ere

gres

sion

line

(T=

298.

15K

)

Syst

emd

0a×

1015

d1

a×

1015

d2

a×

1015

d3

a×

1015

d4

a×

1015

δa×

1015

Prop

an-2

-ol(

1)+

pent

an-1

-ol(

2)−4

.494

7±

0.00

151.

3347

±0.

0061

−0.0

688

±0.

0075

−0.0

86±

0.01

60

0.00

07

Prop

an-2

-ol(

1)+

hexa

n-1-

ol(2

)−6

.116

0±

0.00

142.

2811

±0.

0056

−0.7

626

±0.

0066

0.13

5±

0.01

50

0.00

06

Prop

an-2

-ol(

1)+

hept

an-1

-ol(

2)−7

.173

7±

0.00

263.

1996

±0.

0109

−1.0

09±

0.01

00.

168

±0.

024

00.

0012

Prop

an-2

-ol(

1)+

octa

n-1-

ol(2

)−8

. 269

1±

0.00

324.

0300

±0.

0065

−1.1

14±

0.01

50

00.

0014

Prop

an-2

-ol(

1)+

nona

n-1-

ol(2

)−9

.563

4±

0.00

195.

2207

±0.

0062

−1.0

40±

0.01

9−0

.072

±0.

015

−0.2

63±

0.03

30.

0007

a (m

3·Pa

−1·m

ol−1

)

Page 15

J Solution Chem (2008) 37: 1747–1773 1761

Table 7 Properties of the pure components at 298.15 K

Compound V a × 106 αa × 103 κTa × 1012 Ka P ∗ V ∗ sb

m3·mol−1 K−1 Pa−1 J·cm−3 cm3·mol−1 nm−1

Propan-2-ol 76.96 10.64 11.35 131 377.77 62.28 14.87

Propan-1-ol 75.13 10.04 10.26 197 394.23 61.23 14.89

Pentan-1-ol 108.70 9.05 8.84 153 408.54 89.76 14.34

Hexan-1-ol 125.32 8.83 8.24 120 425.07 103.85 14.18

Heptan-1-ol 141.93 8.79 8.00 101 437.46 117.63 14.06

Octan-1-ol 158.34 8.52 7.77 98 435.96 131.71 13.96

Nonan-1-ol 174.97 8.49 7.52 91 451.37 145.52 13.89

Decan-1-ol 191.58 8.18 7.33 88 443.31 160.10 13.82

aKulikov et al., 2001 [41]

bBondi, 1968 [7], calculated on the basis of the surfaces and volumes of methyl, ethyl and hydroxyl groups

propan-1-ol, + pentan-1-ol, + hexan-1-ol, + heptan-1-ol, + octan-1-ol, + nonan-1-ol and+ decan-1-ol systems. For the propan-2-ol + propan-1-ol and propan-2-ol + decan-1-olsystems, the V E values were taken from the literature [56, 62]. The calculations were madeon the assumption that propan-2-ol forms linear oligomers according the Eq. 3.

The properties of the pure components necessary for calculations, i.e., the thermal ex-pansion coefficients, α, and isothermal compressibilities, κT , at 298.15 K were taken fromthe literature [41]. They are collected in Table 7 together with the molar volumes, V , char-acteristic quantities and s-parameters [7].

The characteristic and reduced quantities for the mixtures have been calculated accordingto the mixing rules given in the literature [27, 53, 54].

Like in our earlier work [26], we decided to describe excess molar volumes in two ways.In the first version (A), the theoretical V E values were fitted to the experimental ones toobtain the three fitting parameters: �v∗

AB,KAB and XAB. This procedure is generally thesame as that proposed by Pineiro and co-workers [27, 54, 55].

In the second version (B), �v∗AB was assumed to be −11.2 cm3·mol−1. It is double the

value of �v∗A = �v∗

B = −5.6 cm3·mol−1. Additionally, the sA/sB ratio, which originatesfrom the physical part of V E, was treated as a new adjustable parameter. In this way threefitting parameters KAB,XAB and sA/sB remained. The last procedure was adopted for prac-tical reasons. As mentioned in our earlier work [26], in order to improve the chemical part ofthe S-ERAS equations describing V E, a more complex association model should be appliedthat complicates the calculations. Such an approach is rather inadvisable for very small ex-cess molar functions. On the other hand, the surface to volume ratio, sA/sB, is undoubtedlyresponsible for the symmetry of the excess molar function in the Flory theory, i.e., with thisadjustable parameter, the V E(x) curves can be S-shaped or skewed [6].

However, the second way of finding the S-ERAS parameters leads to changes in theirphysical meaning. Furthermore, if this new approach is correct, the chemical part of theexcess molar volume should remain almost unchanged at least in the qualitative sense. Thus,we decided to apply the two mentioned procedures to check whether the results obtained inthe second procedure are acceptable.

The fits of the excess molar volume using the S-ERAS approach are presented in Figs. 4–18 together with the “physical” and the “chemical” contributions.

Page 16

1762 J Solution Chem (2008) 37: 1747–1773

Table 8 Energetic parameters, XAB, cross-constants of association, KAB, volumes of hydrogen-bond for-mation, �v∗

AB, resulting from the fitting of the S-ERAS model to the excess molar volumes (calculatedaccording Eq. 5) for systems with propan-2-ol as the common compound (T = 298.15 K) (version A)

System XAB KAB �v∗AB δV E/

J·cm−3 cm3·mol−1 %

Propan-2-ol + propan-1-ol −1.10 23.33 −11.20 12.86

Propan-2-ol + pentan-1-ol 1.93 23.57 −11.20 12.59

Propan-2-ol + hexan-1-ol 5.01 23.91 −11.20 13.16

Propan-2-ol + heptan-1-ol 6.23 28.14 −11.64 23.23

Propan-2-ol + octan-1-ol 8.51 24.38 −11.10 35.73

Propan-2-ol + nonan-1-ol 10.45 22.61 −11.20 19.41

Propan-2-ol + decan-1-ol 11.94 20.99 −11.20 8.64

Table 9 Energetic parameters, XAB, cross-constants of association, KAB, and sA/sB ratios resulting fromthe fitting of the S-ERAS model to the excess molar volumes (calculated according Eq. 5) for systems withpropan-2-ol as the common compound (T = 298.15 K) (�v∗

AB = −11.2 cm3· mol−1) (version B)

System XAB KAB sA/sB δV E/%J·cm−3

Propan-2-ol + propan-1-ol −1.01 23.58 2.29 5.33

Propan-2-ol + pentan-1-ol 1.67 22.52 0.40 2.62

Propan-2-ol + hexan-1-ol 4.43 21.94 0.70 2.85

Propan-2-ol + heptan-1-ol 6.10 21.07 0.76 3.81

Propan-2-ol + octan-1-ol 8.27 22.12 0.90 26.46

Propan-2-ol + nonan-1-ol 10.15 21.60 1.02 17.56

Propan-2-ol + decan-1-ol 11.85 20.71 1.06 8.65

The differences between the experimental and theoretical excesses can be expressed byrelative deviations of the excess molar volumes, δV E:

δV E =∑ |V E

exp − V Emodel|∑ |V E

exp|(15)

All the fitting parameters and relative deviations are collected in Tables 8–9.From inspection of Figs. 4, 6, 8, 10, 12, 14, and 16 (version A) it can be concluded

that the S-ERAS model correctly describes the excess molar volumes. A better descriptionwas achieved for mixtures with larger V E values: propan-2-ol + propan-1-ol, + pentan-1-ol, + hexan-1-ol, + decan-1-ol; for these systems, the relative deviations do not exceeded14%. In the most unfavorable case the relative deviation achieved 35.7%. The deviationsare of the same order as the experimental errors. The later, although small if expressed inabsolute numbers, may be quite high when given as the relative ones due to the very smallexcess volumes. For example, the positive excess volume at low octan-1-ol concentrations(Figs. 12 and 13) lies almost within the limits of the measured error estimated ca. 0.002 ×106 m3·mol−1. Thus, it can be concluded that the S-ERAS model seems to provide at leasta semi-quantitative description of the excess molar volumes of the systems considered.

Page 17

J Solution Chem (2008) 37: 1747–1773 1763

Fig. 4 Excess molar volumes of binary mixtures of propan-2-ol (1) + propan-1-ol (2); points: calculatedfrom Eq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashedline: chemical contribution (T = 298.15 K) (version A)

Fig. 5 Excess molar volumes of binary mixtures of propan-2-ol (1) + propan-1-ol (2); points: calculatedfrom Eq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashedline: chemical contribution (T = 298.15 K) (version B)

For the systems propan-2-ol + propan-1-ol, + pentan-1-ol, + hexan-1-ol, + heptan-1-ol,the physical contributions to V E are negative with minima shifted slightly towards higherconcentrations of propan-1-ol. Next, for propan-2-ol + octan-1-ol, + nonan-1-ol and +

Page 18

1764 J Solution Chem (2008) 37: 1747–1773

Fig. 6 Excess molar volumes of binary mixtures of propan-2-ol + pentan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version A)

Fig. 7 Excess molar volumes of binary mixtures of propan-2-ol + pentan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version B)

decan-1-ol, the physical contributions to V E are S-shaped. In that case, V Ephys is negative

with minima at lower concentrations of propan-2-ol and maximum at higher concentrationsof that component.

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J Solution Chem (2008) 37: 1747–1773 1765

Fig. 8 Excess molar volumes of binary mixtures of propan-2-ol + hexan-1-ol; points: calculated from Eq. 9;solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line: chem-ical contribution (T = 298.15 K) (version A)

Fig. 9 Excess molar volumes of binary mixtures of propan-2-ol + hexan-1-ol; points: calculated from Eq. 9;solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line: chem-ical contribution (T = 298.15 K) (version B)

Page 20

1766 J Solution Chem (2008) 37: 1747–1773

Fig. 10 Excess molar volumes of binary mixtures of propan-2-ol + heptan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version A)

Fig. 11 Excess molar volumes of binary mixtures of propan-2-ol + heptan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version B)

The chemical contributions to V E are partly positive and partly negative for almost allthe systems investigated. Thus, the V E

chem(x) curves are S-shaped with maxima at low con-centrations of propan-2-ol and minima at higher concentrations. Only for the propan-2-ol +

Page 21

J Solution Chem (2008) 37: 1747–1773 1767

Fig. 12 Excess molar volumes of binary mixtures of propan-2-ol + octan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version A)

Fig. 13 Excess molar volumes of binary mixtures of propan-2-ol + octan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version B)

hexan-1-ol system, V Echem is negative in the whole concentration range with a minimum at

higher concentrations of propan-2-ol.

Page 22

1768 J Solution Chem (2008) 37: 1747–1773

Fig. 14 Excess molar volumes of binary mixtures of propan-2-ol + nonan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version A)

Fig. 15 Excess molar volumes of binary mixtures of propan-2-ol + nonan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version B)

It is noteworthy that the B procedure (Figs. 5, 7, 9, 11, 13, 15 and 17) leads to the sameconclusions as presented, except for the system propan-2-ol + hexan-1-ol. V E

chem(x) for thatmixture is S-shaped like the chemical contributions to V E for all the others.

Page 23

J Solution Chem (2008) 37: 1747–1773 1769

Fig. 16 Excess molar volumes of binary mixtures of propan-2-ol + decan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version A)

Fig. 17 Excess molar volumes of binary mixtures of propan-2-ol + decan-1-ol; points: calculated fromEq. 9; solid line: result of the fit to the SERAS model (Eq. 5), dotted line: physical contribution, dashed line:chemical contribution (T = 298.15 K) (version B)

Page 24

1770 J Solution Chem (2008) 37: 1747–1773

Fig. 18 The sA/sB ratios for mixtures containing propan-2-ol as a common component

As shown by the data in Table 8, the XAB values increase with lengthening of the hy-drocarbon chain. At the same time, KAB increases up to maximum for the propan-2-ol +heptan-1-ol mixture and then decreases monotonically. The fitted negative values of theinteraction parameter, XAB, for the propan-2-ol + propan-1-ol system are obviously not ac-ceptable from the theoretical point of view. However, we decided to leave the negative valueunchanged instead of substituting: if, for example, by zero for the sake of consistency. Itseems that the negative XAB values appear in fitting the excess molar volumes instead to theexcess molar enthalpies.

The comparison of the parameters given in Table 8, lead to the conclusion that the forma-tion of hydrogen bonds is less important for mixtures of propan-2-ol with the higher alcoholsthan for those containing lower-chain alcohols. It is intuitively acceptable that the “physical”interactions are more important for the alcohols with long hydrocarbon chains. The analy-sis of the adjustable parameters in Table 9 indicates that all the conclusions made earlier(for A version) remain true. Besides, the fixing of �v∗

AB does not affect the results obtainedby the A version. Especially the chemical part was not affected, which is very importantfor the discussion of the properties of binary alcohol systems. As expected, more importantchanges occur in the physical contribution to V E. The advantage of this new treatment wasa better description of the excess molar volumes for almost all of the systems under test. Asbefore, the highest relative deviations are for the propan-2-ol + octan-1-ol and propan-2-ol+ nonan-1-ol systems; however, they do not exceed 26.5% for the former. The remainingδV E values are much lower than 10%. This is evidence of the applicability of the approacheven to the mixtures with small excess volumes.

The two presented approaches to the S-ERAS model may be demonstrated also on thebasis of a comparison of the sA/sB ratios. As mentioned earlier, they can be obtained bythe fitting of this parameter to the experimental V E values or from the Bondi contributionmethod or using the Abe and Flory approximation (Fig. 18). From Fig. 18 one learns thatthe two latter methods result in different values even though their physical basis is similar.Moreover, the sA/sB ratios obtained by fitting to the V E values change in a way that is

Page 25

J Solution Chem (2008) 37: 1747–1773 1771

generally characteristic of the behavior of this parameter calculated by other commonlyapplied procedures.

From all the above given descriptions it may concluded that the results of the excessmolar volumes of binary alcohol mixtures obtained by the S-ERAS model are undoubtedlyconsistent with intuition. Thus, the S-ERAS model seems to provide insight into molecularphenomena in those mixtures.

Acknowledgements Authors are very grateful to Professor Stefan Ernst and Doctor hab. Wojciech Mar-czak for useful discussion and all helpful advices.

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