Isothermal vapor–liquid equilibria of mixtures of (methanol+ethanol+1-propanol or 2-propanol) at 333.15 K

Fluid Phase Equilibria 170 Ž2000. 37–48 www.elsevier.nlrlocaterfluid

Isothermal vapor–liquid equilibria of mixtures of Žmethanolq ethanolq 1-propanol or 2-propanol. at 333.15 K Isamu Nagata) , Kazuhiro Tamura, Kohichi Miyai Department of Chemistry and Chemical Engineering, DiÕision of Physical Sciences, Kanazawa UniÕersity, 40-20, Kodatsuno 2-chome, Kanazawa, Ishikawa 920-8667, Japan Received 15 June 1999; accepted 28 October 1999

Abstract The UNIQUAC associated-solution theory is extended to reproduce systems containing three alcohols and is applied to binary and ternary mixtures of alcohols. The isothermal vapor–liquid equilibrium data of the liquid mixtures of Žmethanolq ethanolq 1-propanol. and Žmethanolq ethanolq 2-propanol. have been measured at 333.15 K using a modified Boublik vapor-recirculating still. The experimental results are analyzed by use of the UNIQUAC associated-solution theory and the UNIQUAC model with binary parameters alone. Both models give comparable results. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Association model; Vapor–liquid equilibrium; Mixture; Pure; Alkanol

1. Introduction It has been well understood that hydrogen-bonded alcohol molecules linearly self-associate and cross-associate with other associated or active nonassociated components. This situation can be described by considering the concept of chemical equilibria according to consecutive chemical reactions to form chemical complexes as chemical models with chemical and physical contribution terms. Many chemical models have been presented to represent the thermodynamic properties of alcohol solutions. One of such chemical models is the UNIQUAC associated-solution theory based on the original UNIQUAC equation w1x. In this laboratory efforts have been made to expand the ability of

)

Corresponding author. Fax: q81-76-234-4829. E-mail address: [email protected] ŽI. Nagata..

0378-3812r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 0 0 . 0 0 3 1 3 - 7

38

I. Nagata et al.r Fluid Phase Equilibria 170 (2000) 37–48

the UNIQUAC associated-solution theory to describe well vapor–liquid equilibria, liquid–liquid equilibria, and excess enthalpies for associated mixtures containing one associated component Žalcohol w2–10x, amine w11,12x, and acid w13,14x. and two associated components Ž two alcohols w15–17x, alcoholq amine w18–20x, alcoholq acid w21,22x.. The UNIQUAC associated-solution theory works well like local composition equations ŽWilson w23x, NRTL w24x, and UNIQUAC w25x. in the calculations of vapor–liquid equilibria for completely miscible alcohol solutions. In the prediction of ternary liquid–liquid equilibria for many methanol solutions, the UNIQUAC associated-solution theory gives usually much better calculated results than the NRTL and UNIQUAC equations. These local composition equations cannot correlate well vapor–liquid equilibria for solutions of acetylacetone with alcohols or organic solvents, because acetylacetone has two tautomers, keto and enol, which cannot be separated. On the other hand the UNIQUAC associated-solution theory can represent binary vapor–liquid equilibria for these tautomeric solutions and predict ternary liquid–liquid equilibria for acetylacetoneq methanolq cyclohexane mixture with good accuracy w9,10x. The chemical theory of Campbell w26x is good for only limited systems composed of alcohols and alkanes Ž or any other inert species. and cannot correlate suitably vapor–liquid equilibria, liquid–liquid equilibria, and excess enthalpy data for many systems of alcohols with unsaturated hydrocarbons and polar solvents such as ketones and esters etc. w27,28x as long as a single adjustable regular solution parameter per binary system is used. Mixtures containing only alcohols show slightly non-ideal behavior. According to chemical models, we must take into consideration all possible chemical complexes of the alcohols present in the mixture and the theory becomes complicated. In this paper, we present an extended version of the UNIQUAC associated-solution theory to ternary mixtures of three alcohols and to test the suitability of the proposed theory for isothermal ternary vapor–liquid equilibrium data using only binary information. Isothermal vapor–liquid equilibrium data for Žmethanolq ethanolq 1-propanol. and Žmethanolq ethanolq 2-propanol. have been measured, since such isothermal ternary data are important from a theoretical point of view for examining solution models and are not available in the literature. Binary isothermal vapor–liquid equilibrium data for the five binary mixtures making up the ternary mixtures are available from the literature: Ž methanolq ethanol. at 313.15 K w29x, Ž methanolq 1-propanol. at 333.15 K w30x, Ž methanolq 2-propanol. at 328.15 K w31x, Žethanolq 1-propanol. at 333.15 K w32x, and Žethanolq 2-propanol. at 313.15 w33,34x.

2. Experimental Analytical reagent grade methanol, ethanol, and propanols were purchased from Wako and were used for experimental work after drying over molecular sieves Ž the purities of the chemicals used were checked by GLC to be at least 99.7%.. The densities and refractive indices of pure components were measured with an Anton–Paar DMA-40 densimeter and an Atago refractometer at 298.15 K. Experimental vapor–liquid equilibrium results were measured using a modified Boublik vapor-recirculating still w35x. Table 1 compares the experimental values of densities, refractive indices and vapor pressures of pure components with literature values w36x. Vapor and liquid samples were analyzed by combining use of a Shimadzu GLC ŽGC-8A. and a Shimadzu Chromatopac Ž C-R6A.. The experimen-

I. Nagata et al.r Fluid Phase Equilibria 170 (2000) 37–48

39

Table 1 Densities, refractive indices, and vapor pressures of substances Densities Žg cmy3 . at 298.15 K

Component Methanol Ethanol 1-Propanol 2-Propanol

Refractive indices at 298.15 K

P ŽkPa. at 333.15 K

This work

Literature w35x

This work

Literature w35x

This work

Literature w35x

0.78667 0.78503 0.79975 0.78129

0.78637 0.78493 0.79960 0.78126

1.32660 1.35926 1.38365 1.37512

1.32652 1.35941 1.38370 1.3752

84.486 46.914 20.258 38.545

84.490 46.912 20.261 38.548

tal errors of the measured variables were mole fraction, "0.002; pressure, "0.013 kPa; and temperature, "0.05 K. Table 2 shows the experimental vapor–liquid equilibrium results for two ternary mixtures at 333.15 K. The activity coefficient g I and fugacity coefficient f I of component I were calculated using the following equations

g I s PyI c Ir x I PI 8c I 8exp Ž ÕI 8 Ž P y PI 8 . rRT

ž

ln c I s 2 Ý y J BI J y Ý J

I

Ý y I y J BI J J

/ PrRT

Ž1. Ž2.

where P is the total pressure, yI is the vapor-phase mole fraction, x I is the liquid-phase mole fraction, PIo , c Io and ÕIo are the pure component vapor pressure, vapor-phase fugacity coefficient, and liquid molar volume calculated from a modified Rackett equation w37x. R is the gas constant and B are the pure and cross second virial coefficients estimated from the method of Hayden and O’Connell w38x. Table 3 shows the vapor pressures, liquid molar volumes, second virial coefficients and molecular structural parameters, r and q, for the pure components and cross second virial coefficients at 333.15 K. The values of r and q for the association model were calculated using the method of Vera et al. w39x and those for the UNIQUAC equation were taken from Prausnitz et al. w40x.

3. Uniquac associated-solution model All alcohols are assumed to form linear chains according to successive chemical reactions: A i q A 1 s A iq1

Ž3.

The association equilibrium constant K A for this reaction is defined by KA s

CA iq1

1

CA i CA 1 rA

Ž4.

I. Nagata et al.r Fluid Phase Equilibria 170 (2000) 37–48

40

Table 2 Vapor–liquid equilibrium data for two ternary systems at 333.15 K y3

P ŽkPa.

g1

g2

g3

c1

c2

c3

x 1 methanolq x 2 ethanolq x 31-propanol 0.331 0.570 0.099 0.485 0.476 0.918 0.050 0.032 0.952 0.034 0.081 0.634 0.279 0.170 0.693 0.864 0.050 0.086 0.936 0.033 0.838 0.093 0.069 0.909 0.064 0.718 0.243 0.039 0.826 0.160 0.450 0.294 0.256 0.662 0.241 0.694 0.261 0.045 0.811 0.175 0.705 0.206 0.089 0.830 0.141 0.633 0.322 0.045 0.763 0.222 0.686 0.266 0.048 0.801 0.184 0.745 0.050 0.205 0.899 0.035 0.446 0.475 0.079 0.607 0.359 0.244 0.434 0.322 0.418 0.429 0.149 0.759 0.092 0.240 0.720 0.905 0.061 0.034 0.942 0.045 0.641 0.273 0.086 0.783 0.180 0.721 0.136 0.143 0.851 0.096 0.831 0.086 0.083 0.921 0.065 0.694 0.261 0.045 0.811 0.175

0.039 0.014 0.137 0.031 0.027 0.014 0.097 0.014 0.029 0.150 0.015 0.066 0.034 0.153 0.040 0.130 0.029 0.053 0.023 0.014

56.689 80.340 43.210 76.994 75.501 72.954 57.155 71.701 71.368 69.514 70.341 70.021 61.608 48.183 50.196 78.967 69.074 70.741 76.674 71.740

0.997 0.988 1.020 0.991 0.974 0.999 1.009 0.998 1.001 0.999 0.962 1.007 1.004 0.995 0.973 0.976 1.006 0.995 1.000 0.998

1.004 1.146 1.008 1.068 1.092 1.011 0.993 1.013 1.029 1.011 1.082 1.033 0.986 1.014 1.013 1.223 1.003 1.052 1.368 1.013

1.078 1.679 1.031 1.328 1.415 1.255 1.046 1.070 1.116 1.112 1.336 1.083 1.277 1.111 1.056 1.444 1.118 1.259 0.663 1.070

0.974 0.963 0.980 0.964 0.965 0.966 0.974 0.967 0.967 0.968 0.967 0.968 0.971 0.978 0.977 0.964 0.968 0.967 0.965 0.967

0.975 0.965 0.981 0.966 0.967 0.968 0.975 0.969 0.969 0.970 0.970 0.969 0.973 0.978 0.978 0.966 0.970 0.969 0.967 0.969

0.968 0.959 0.975 0.961 0.961 0.962 0.969 0.962 0.963 0.963 0.963 0.964 0.966 0.973 0.970 0.960 0.964 0.963 0.961 0.962

x 1 methanolq x 2 ethanolq x 3 2-propanol 0.833 0.090 0.077 0.900 0.057 0.611 0.212 0.177 0.746 0.153 0.910 0.051 0.039 0.946 0.034 0.825 0.135 0.040 0.898 0.083 0.840 0.051 0.109 0.902 0.034 0.703 0.256 0.041 0.805 0.172 0.730 0.065 0.205 0.849 0.045 0.717 0.205 0.078 0.816 0.143 0.743 0.094 0.163 0.852 0.063 0.870 0.091 0.039 0.924 0.057 0.877 0.052 0.071 0.931 0.034 0.096 0.855 0.049 0.148 0.815 0.092 0.709 0.199 0.145 0.694 0.084 0.101 0.815 0.167 0.102 0.329 0.266 0.405 0.490 0.223 0.588 0.330 0.082 0.735 0.220 0.654 0.222 0.124 0.780 0.152 0.578 0.298 0.124 0.780 0.199 0.089 0.344 0.567 0.162 0.353 0.142 0.590 0.268 0.239 0.556

0.043 0.101 0.020 0.019 0.064 0.023 0.106 0.041 0.085 0.019 0.035 0.037 0.161 0.731 0.287 0.045 0.068 0.071 0.485 0.205

77.722 67.565 80.547 77.644 77.219 73.184 71.913 73.229 73.137 79.180 79.038 50.215 48.260 43.317 55.880 68.173 70.714 67.572 45.185 50.253

1.003 0.991 0.999 1.010 0.991 1.003 0.997 0.998 1.004 1.004 1.002 0.937 0.922 1.046 0.987 1.023 1.011 1.025 0.993 1.007

1.032 1.027 1.124 1.001 1.079 1.033 1.035 1.073 1.030 1.039 1.083 1.017 1.004 0.929 1.020 0.957 1.018 0.951 0.987 0.993

1.094 0.977 1.040 0.929 1.143 1.036 1.043 0.972 0.964 0.972 0.982 0.966 0.998 0.999 1.016 0.946 0.981 0.980 0.998 0.980

0.964 0.969 0.953 0.964 0.964 0.966 0.966 0.966 0.966 0.963 0.963 0.977 0.978 0.980 0.974 0.968 0.967 0.969 0.980 0.977

0.966 0.970 0.945 0.966 0.966 0.968 0.968 0.968 0.968 0.966 0.966 0.978 0.979 0.978 0.974 0.970 0.969 0.970 0.979 0.978

0.960 0.964 0.958 0.959 0.960 0.961 0.961 0.961 0.962 0.959 0.959 0.969 0.972 0.978 0.971 0.963 0.963 0.964 0.976 0.972

x1

x2

x3

y1

y2

where the C’s are molar concentrations and rA is the molecular size parameter of the monomeric alcohol molecule.

I. Nagata et al.r Fluid Phase Equilibria 170 (2000) 37–48

41

Table 3 Vapor pressures, liquid molar volumes, second virial coefficients, and molecular structural parameters for the pure components and cross second virial coefficients at 333.15 K Vapor pressure ŽkPa. Molar liquid volume Žcm3 moly1 . Second virial coefficient Žcm3 moly1 . r q qX

Methanol

Ethanol

1-Propanol

2-Propanol

84.488 41.32 y1305 1.15 1.43 1.12 1.43 0.96

46.912 60.83 y1204 1.69 2.11 1.55 1.97 0.92

20.261 74.65 y1493 2.23 2.78 1.98 2.51 0.89

38.548 82.01 y1411 2.23 a 2.78 b 1.98 a 2.51b 0.89 b

Cross second virial coefficient Žcm3 moly1 . Methanolqethanol

Methanolq1-propanol

Methanolq2-propanol

Ethanolq1-Propanol

Ethanolq2-Propanol

y1264

y1365

y1381

y1488

y1497

a b

UNIQUAC associated-solution model. UNIQUAC model.

In a binary alcohol mixture with components A and B, the multisolvation equilibria may be expressed by the following consecutive reactions A iqBjsA iBj

Bi q A j s Bi A j

A iBjqA ksA iBjA k

Bi A j q Bk s Bi A j Bk

A i B j A k q Bl s A i B j A k Bl ... ... ... ... ... ...

Bi A j Bk q A l s Bi A j Bk A l ... ... ... ... ... ...

The multisolvation constant K AB for these reactions is assumed to be independent of the number of associated and solvated alcohol molecules. For the A i B j A k B l forming reaction K AB is defined as K AB s

CA i B jA k B l

1

CA i B jA kC B l rA r B

Ž5.

The molar concentration of each mixed complex may be calculated by multiplication of the proper ) number of K AB s Ž rA r B K AB . and segment mole fractions per segment of the simple complex composed of one type of monomer only. Then in a ternary alcohol mixture whose components A, B, and C, exist many chemical complexes. The general formulae of the chemical complexes are A i , B i , C i , ŽA i B j . k , ŽB i A j . k , A i ŽB j A k . l , B i ŽA j B k . l , ŽA i C j . k , ŽC i A j . k , A i ŽC j A k . l , C i ŽA j C k . l , ŽB i C j . k , ŽC i B j . k , B i Ž C j B k . l , C i ŽB j C k . l , and further ternary chemical complexes formed by A i , B i , and C i , where i, j, k, and l range from 1 to `. Fig. 1 shows a typical example of forming ternary chemical complexes, composed of A i , B i , and C i , starting from A i , B i , C i to third step according to continuous chemical reactions. A total of six equilibrium constants are introduced as K A , K B , K C , K AB , K AC , and K BC .

I. Nagata et al.r Fluid Phase Equilibria 170 (2000) 37–48

42

Fig. 1. Ternary chemical complex forming scheme.

The activity coefficient of any alcohol I in the ternary mixture is given by

lng I s ln

f I1

ž / ž f I18 x I

q rI



q q I 1 y ln

1

1 y

VI 8

V

fI

/ žž / y 5q I ln

uI

uJ t I J

ž Ýu t / y Ý Ýu t J JI

J

K KJ

J

K

0

q1y

fI uI

/ Ž6.

where the nominal segment fraction f I , the overall surface fraction u I , and the adjustable binary parameter t I J related to energy parameter a I J are given by

f I s rI x Ir Ý r J x J

Ž7.

J

u I s q I x Ir Ý q J x J

Ž8.

J

t i j s exp Ž ya i jrT .

Ž9.

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43

The following expressions have been derived by Hofman and Nagata w41x. The true molar volume of the mixture V is expressed by 1 V

`

s sA

`

Ý

EAŽ k . q s B

ks1

`

Ý

E BŽ k . q sC

ks1

Ý ECŽ k .

Ž 10.

ks1

where auxiliary parameters sA , s B , sC , sA , s B , sC , EAŽ k ., E BŽ k ., and ECŽ k . are defined by `

sA s

Ý is1 `

sB s

Ý is1 `

sC s

Ý is1 `

sA s

Ý is1 `

sB s

fA i

s

rA i

f Bi

fA i f B1

s

s

rC i i fA i

Ž 12.

r B Ž 1 y K B f B1 .

r Bi

fC i

Ž 11.

rAŽ 1 y K A fA 1 .

fC1

Ž 13.

rC Ž 1 y K C f C 1 .

fA 1

s

rA i

rAŽ 1 y K A fA 1 .

i f Bi

Ž 14.

2

f B1 2

Ž 15.

2

Ž 16.

) ) EAŽ kq1. s K AB s B EBŽ k . q K AC sC ECŽ k .

Ž 17.

) ) EBŽ kq1. s K AB sA EAŽ k . q K BC sC ECŽ k .

Ž 18.

) ) ECŽ kq1. s K AC sA EAŽ k . q K BC s B E BŽ k .

Ž 19.

EAŽ1. s EBŽ1. s ECŽ1. s 1

Ž 20 .

Ý is1 `

sC s

Ý ks1

s

r Bi

r B Ž 1 y K B f B1 .

i fC i

fC1

s

rC i

rC Ž 1 y K C f C 1 .

The total true mole fraction of a monomer of component A Žboth free and bonded. XA is shown as XA

s

V

sA sA

`

ž

sA

Ý

`

E A

Žk . A q sB

ks1

Ý

`

E A

Žk. B q sC

ks1

ÝE ks1

A

Žk . C

/

Ž 21 .

where E A

Ž kq1 . ) s K AB sB A

E A

Ž kq1 . ) s K AB sA E B

A

Žk. ) A q K BC sC E

A

Žk. C

Ž 23.

E A

Ž kq1 . ) s K AC sA E C

A

Žk. ) A q K BC s B E

A

Žk. B

Ž 24.

E A

Ž1 . A s 1;

E A

žE A

Žk. Žk. B q EB

Ž1 . B sE

A

/ qK

Ž1 . C s0

) AC sC

žE A

Žk. Žk. C q EC

/

Ž 22.

Ž 25 .

I. Nagata et al.r Fluid Phase Equilibria 170 (2000) 37–48

44

Expressions for other parameters, X B and XC , are similarly given. For the binary mixture the expressions for V, XA , and X B are given below. The monomer segment fractions of components, fA 1, f B1, and f C 1, are obtained from the following mass balance equations

fA s Ž XArV . rA

Ž 26.

f B s Ž X BrV . r B

Ž 27.

f C s Ž XCrV . rC

Ž 28.

and at the pure alcohol state the monomeric segment fraction and true molar volume, f I18 and VI 8, are expressed by

f

8 I1 s

2 K I q 1 y Ž4 K I q 1.

Ž 29.

2 K I2

1 s VI 8

0.5

1 y K I f I81

Ž 30.

rA

For a binary mixture ŽA q B. of known stoichiometric mole fractions, Eq. Ž 11. for the true molar volume V and Eqs. Ž26. and Ž27. reduce to 1 s V

fA s

fBs

)2 K AB sA s B )2 1 y K AB sA s B

) q sA q s B . q sA q s B Ž 2rK AB

)2 K AB sA s B sA rA )2 1 y K AB SA B B )2 K AB sA s B s B r B )2 1 y K AB sA s B

ž ž

) 2 q K AB Ž sA q s B . ) )2 K AB sAŽ 1 y K AB sA s B . ) 2 q K AB Ž sA q s B . ) )2 K AB s B Ž 1 y K AB sA s B .

Ž 31.

/ /

q 1 q sA rA

Ž 32.

q 1 q sB r B

Ž 33.

The values of fA 1 and f B1 are obtained by solving simultaneously Eqs. Ž32. and Ž33. and V is found from Eq. Ž31.. The quantities f I1, f I18, V and VI 8 are substituted into Eq. Ž6. to give the activity coefficients.

4. Calculated results The values of the association equilibrium constant K I at 323.15 K were taken from Brandani w42x and that of the molar enthalpy of a hydrogen bond h I is y23.2 kJrmol for all alcohols w43x. The temperature dependence of the equilibrium constants is fixed by van’t Hoff relation. Table 4 gives the

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45

Table 4 Association and solvation parameters Component

K AŽT . a

y hA ŽkJrmol.

Mixture

K ABŽT . a

y hAB ŽkJrmol.

Methanol Ethanol 1-Propanol 2-Propanol

173.9 110.4 87.0 49.1

23.2 23.2 23.2 23.2

Methanol ŽA.qethanol ŽB. Methanol ŽA.q1-propanol ŽB. Methanol ŽA.q2-propanol ŽB. Ethanol ŽA.q1-propanol ŽB. Ethanol ŽA.q2-propanol ŽB.

99 72 70 49 47.6

23.2 23.2 23.2 23.2 23.2

a

T s 323.15 K.

values of the association and solvation parameters used in this work. The solvation constant K AB is treated as an adjustable parameter to obtain the better fit to the experimental data. The same values of the association and solvation parameters were used in the calculations of excess enthalpies of the present ternary mixtures w44x. In fitting the models to experimental vapor–liquid equilibrium data, the molar energy parameters aAB and aBA were obtained using a computer program based on the maximum likelihood principle as described in Ref. w40x. The vapor pressures of pure components were obtained from the Antoine equation whose constants are given in Ref. w36x through all binary and ternary vapor–liquid equilibrium calculations. Table 5 gives binary calculated results. The UNIQUAC equation gives better results than the UNIQUAC associated-solution theory for Ž methanolq ethanol. and Žmethanolq 1propanol. and both models give nearly the same results for the other three mixtures. For a ternary mixture of known compositions, the values of fA 1, f B1, and f C 1 are obtained from simultaneous solution of Eqs. Ž26. – Ž28. , whose sum is up to the index k s 300. Table 6 shows that both models yield comparable results, using the binary parameters.

Table 5 Binary calculated results System ŽAqB. Methanolqethanol

T ŽK.

No. of data points Root-mean squared deviations

313.15 15

Methanolq1-propanol 333.35 26 Methanolq2-propanol 328.15 20 Ethanolq1-propanol

333.15 11

Ethanolq2-propanol

313.15 17

a b

UNIQUAC associated-solution model. UNIQUAC model.

Energy parameters

d P ŽkPa. dT ŽK. d x Ž=10 3 . d y Ž=10 3 . aAB ŽK.

aBA ŽK.

0.063 0.010 0.071 0.034 0.266 0.264 0.111 0.118 0.011 0.016

68.62 a y83.06 b y225.43 a y105.81b 240.19 a y76.05 b y208.84 a y204.02 b 119.89 a 40.29 b

0.00 0.00 0.02 0.01 0.08 0.08 0.01 0.01 0.00 0.00

0.1 0.1 0.3 0.3 0.9 0.9 2.1 2.1 0.0 0.0

1.1 1.0 1.1 1.1 3.8 3.8 9.7 9.4 0.8 0.2

y68.84 115.95 322.26 240.02 y110.66 180.41 301.32 320.86 y40.23 y45.54

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46

Table 6 Ternary predicted results at 333.15 K System Methanol ŽA.qethanol ŽB.q1-propanol ŽC. Methanol ŽA.qethanol ŽB.q2-propanol ŽC. a b

Absolute arithmetic mean deviations d yA

d yB

d yC

d P ŽkPa.

0.0076 0.0081 0.0059 0.0059

0.0054 0.0056 0.0054 0.0053

0.0040 0.0043 0.0027 0.0024

0.388 a 0.428 b 0.285a 0.280 b

UNIQUAC associated-solution model. UNIQUAC model.

We may conclude that the UNIQUAC associated-solution theory, based on the concept multisolvation among different pure alcohol i-mers is valid for predicting the vapor–liquid equilibrium results of the ternary mixtures studied here. List of symbols aI J molar energy parameter for I–J pair B second virial coefficient C molar concentration EsŽ k . coefficient defined by Eq. Ž17.; s s A, B, C EŽ I . Žsk . coefficient defined by Eq. Ž22.; s s A, B, C hI molar enthalpy of a hydrogen bond for component I KI self-association constant defined by Eq. Ž4. K IJ equilibrium constant defined by Eq. Ž5. K I)J rI r J K I J P total pressure PI 8 vapor pressure of pure component I qI molecular surface parameter of pure component I rI molecular size parameter of pure component I R gas constant sI parameter defined by Eqs. Ž11. – Ž 13. ; VsI is equal to total mole fraction of self-associated multimers created by monomers component I only sI parameter defined by Eqs. Ž 14. – Ž 16. T absolute temperature ÕI 8 molar liquid volume of pure component I V true liquid volume of the mixture VI 8 true liquid volume of pure component I xI nominal liquid mole fraction of component I XI total true mole fraction of a monomer of component I Žboth free and bonded. yI nominal vapor mole fraction of component I Greek letters gI activity coefficient of component I uI nominal surface fraction of component I tIJ parameter defined by expŽ ya I JrT .

I. Nagata et al.r Fluid Phase Equilibria 170 (2000) 37–48

cI cI 8 fI f I1 f I18

47

fugacity coefficient of component I at P and T fugacity coefficient of pure component I at PI 8 and T nominal segment fraction of component I true segment fraction of monomeric component I true segment fraction of monomeric component I in pure alcohol

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x w34x w35x w36x w37x w38x w39x

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