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Activity Coefficient Relations in Miscible

and Partially Miscible Multicomponent Systems

Solke Bruin ’

Laboratory for Physical Technology, Eindhoven Uniuersity

of

Technology, Eindhoven, Netherlands

Based on a quasilattice model of a multicomponent solution, three equations are der ived relating component

activity coefficients to the composition of a liquid mixture. The equations also ap ply to systems showing

limited miscibility. Temperature dependence of activity coefficients i s built in. The equations were tested fo r

16 binary systems,

5

of which show limited miscibility, and for 4 ernary systems. The enthalpic Wilson

equation, one of the three, gives the best prediction of binary vapor-liquid equilibrium. Both the enthalpic

Wilson equation and the Orye equations give good results of comparable accuracy for the ternary systems.

PREDICTIONf thermodynamic properties of mixtures from

pure component properties is the main goal of thermo-

dyn amic s of liquid mix tures. Progress in this field has ma de

it possible t o calculate th e properties

of a

mixture

of n

compo-

nents from th e pure component properties supplemented with

experimental vapor-liquid equilibrium (V LE) d ata for all the

binary systems that can be constructed out of the

n

compo-

nents : a tota l of

l / 2n n -

1) (I’rausnitz et al. , 1967). Th e

startin g points of these compu tational methods are thermo-

dynamic models of multicomponent mixtures, which provide

calculational procedures to predict the vapor-liquid equilib-

rium of the mixture from data on the binary sys tems. From

such models relations between the activity coefficients, y

an d th e composition of t he liquid are derived. I n such rela-

tions, parameters characteristic of interactions between

pairs of molecules

of

different kind appear. Formally, such

activ ity coefficient relations have the form

Values for the interaction parameters,

Pi,

a n d

P 3 ( ,

can be

calculated from VL E da ta on th e binary system of compo-

nent s i a n d j .

The best known activity coefficient equations are, un-

doubtedly, the van Laar equations. These equat ions have

proved to be very useful for binary systems b ut less satis-

factory for m ulticomponent systems (Neretnieks, 1968; Orye,

1965; Orye and Prausniti , 1965; Prausnitz

et

al., 1967). Th e

two-parameter Wilson equation gives better results than the

van Laar equat ions for many binary and mult icomponent

mixtures (Nagel and Sinn, 1966, 1967; Prausnitz

et al.,

1967;

Wilson, 1964a, b). No terna ry or higher param eters are needed

in the generalization to multicomponent systems. Tempera-

ture dependence is to some extent bui l t into the Wilson

equations, making them suited to, for example, the isobaric

vapor-liquid equilibrium calculations common ly encountered

in distillation. An undesirab le featu re of th e Wilson equatio ns

is their inapplicability to partially miscible systems. The

modification

of

the equations that Wilson proposed to over-

come this difficulty means introdu ction of a n extra parameter.

I n the present paper three ac tivity coefficient equations

are discussed. These equations follow from the quasilattice

Present address, Western Utilization Research and Develop-

ment Division, Agricultural Research Service, .4lbany, Calif.

94710

model of multicomponent liquid mixtures developed by

Gugg enheim (1935, 1944a, b, 1 952). Th is model yields a ra the r

general relation for the excess Gibbs f~inct ion, n which the

enthalpic and entropic contributions appear as separate

terms. Introduct>ion of van L aar or Wilson parameters i nto

this relation leads to a num ber of possible activity coefficient

equat ions including the van Laar and Wilson

equations

themselves. In Table

I

all poasible combinations of Wilson

paramet,ers (Ai j ) , van Laar parameters

(Aij),

and Gibbs

function are given. The Orye equations (Orye, 1965) result

on introdu ction of Wilson param eters in the general relation

for the excess Gibbs function. Retaining only the enthalpy

part assuming the excess entropy to be ncgligible (regulm

solutions), yields “enthalpic Kilson” equations. The “ex-

tended van Laar” equations (EVL equations) result from

introduction

of

van L aar parameters into the general Gibbs

function,

as

indicated in Table

I.

Introduction

of

van Laar

parameters int’o the e ntropy p art, assuming th e ent,halpy to

be negligible (athermal solutions), gives equations unable to

predict p artial miscibility (Bruin, 1969), and therefore is not

discussed.

The Orye, enthalpic Wilson, and EVL equat ions are

derived and discussed. Th e derivation of th e Orye equations

here differs from th at given originally b y Orye. Subsequently,

the equations are generalized to multicomponent systems.

Th e equations were tested by application to 16 binary an d four

ternary systems, prediction

of

parameters from infinite

dilution activity coefficients, and prediction of temperature

dependence

of

act ivi ty coefficients.

Binary Systems

The activity coefficient,

yl,

in a binary liquid mixture

of

components 1 a n d 2 is given by the exact thermodynamic

relationship

In y1 = [

n l-

N gE/ RT)]

, P , n

1)

where N i s the tota l number

of

moles:

N

= nl +

nz .

T h e

mola r excess Gibbs free energy of m ixing,

g E ,

in Equat io n

1

s

rela ted to the excess molar entrop y and enthalpy according to

=

hE - TsE

= U M +

PvM - TsE

2)

Adopting th e quasilattice model for l iquid mixtures

of

mole-

cules

of

different size, a s developed b y Gu ggenh eim (1935,

Ind. Eng.

Chem.

Fundam.,

Vol.

9 No. 3 , 1970

305

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Table

I.

Summary of Relations

fo r g E

Resulting from Different Combinations

o f

Quasilattice

Equations an d Parameters

Type of Parameter gE = hE

-

T ~ E

A i i A i j hE-term

rE-term

(Wilson)

(van

Laar)

included

included Name o f Resulting Equation

for

y i

+ Wilson

- van Laar

+

Orye

- Enthalpic Wilson

+

t

+

+ +

4-

+

+

+

Extended Van Laar (EVL)

-

1

+

2

3 +

4

+

5

6

-

-

-

-

+

-

1944a, b, 1 952), we may w rite the following equat ion for the

energy of mixing, uM, an d the excess entropy:

sE= -R(zl In {rl / (r lz l

+ r 2 4 )

+

2 2

ln{rz/(rlzl+ r s 2 )

1)

(3)

UM =

x12u12

(4)

In Equa t ion 3 r l is the nu mb er of sites in the qua silattice

structure occupied by a molecule of type 1.

u12

represents an

interaction energy:

5)

N is the Avogadro number, w12 denotes the contribution to

the potential energy by

a

pai r of site s, one of wh ich is occu pied

by an element of

a

molecule of type

1,

the other by an ele-

me nt of a different molecule of ty pe 2; an d

z

s the n um ber of

nearest neighbors to a site, the “coordination number.”

In Equa t ion 4

XI2

is defined as the n um ber of pairs of neigh-

boring s ites occupied by different mo lecules, one of which is

of type

1

an d the other of ty pe 2, divided by the to tal number

of

nearest neighbor sites. To calculate X~Z ,uggenheim

introduced the qu asichemical approxim ation

UIZ= ‘ /2~N(2~12 1 1

W Z Z )

which is the m ass-action law for a “reac tion” where

a

molecule

1 s brought from a pure liquid 1 o a (1-2) liquid mixtu re and

simultaneously a molecule 2 is brought from pure 2-liquid t o

the (1-2) liquid mixtu re. I n th is reaction (zX11) an d (zX,)

bonds are broken up, while

2 (2X12 )

bonds are formed. The

nu mb er of 1-1 pairs and 2-2 pairs can be expressed as

22x11 = z rlz1

-

XIS) ( 6 4

22x22 = z ( r s2 -

X12 )

(6b)

I n th e par ticula r case where all energies of mixing are zerc-

al l u12 -c 0 (all molecules distrib uted a t random)-combina-

tion of

5a

an d 6 gives

(7)

8)

When Equat ions 3 and 8 a re subs t i tu t ed in 2 and U is

assumed t o be zero, one obtains for

g E

X 1 2

=

( r 1 4 ( r z a ) / ( r 1 x 1 +

TZXZ )

UM

=

( r l z J ( r s 2 ) u d ( r l z 1 +

rg2

For UM

one obtains

Renon and Prausnitz (1968) recently developed a relation

for g E in which the condition u12 0 on Equatio n 5a was re-

laxed. Moreover, distinction was made between the transfer

of a molecule

1

into the (1-2) liquid mixture [giving

(zX12’)

bond s] and th e transfe r of a molecule 2 into the (1-2) liquid

mixture [giving

(zX2,’)

bonds]. T he resulting relations for th e

acti vity coefficients’ (N R TL equations) were shown to give

excellent representation of a wide variety of binary and

ternar y liquid mixtures.

In the present paper Equ at ion 9 is used as

a

starting point

to derive activity coefficient equations. Guggenheim (1944b)

showed tha t the inaccu racy of Eq uatio n 9 is theoretically of

the order of ( u ~ ~ / R T ) ~ .

If one assumes in Equat ion 2 th at

h E

= 0 (athermal solu-

tions ; Flor y, 1942; Huggins, 1942) one obtains for

g”

g E / R T = In ( r1

)

+ x z In

(

“ ) (10)

rlxl + rsr2 ~1x1

+

ra2

On the other hand, if s E = 0 (regular solutions, van La ar;

Black, 1959; Black and Derr, 1963a, b; Wohl, 1946)

QE

is

given by

Equat ions 9 t o 11give three relations for

QE

a s a fun ctio n of

liquid composition in terms of th e quasilattice param eters,

rl a n d r2, and the interaction energy,

u12.

To obtain two-

parameter equations for y1 a n d y2 from Equat ions 9 to 11,

rl ,

rz,

a n d

u12

have to be combined to two parameters.

Different suggestions have been made. The most obvious

assumes the ratio ( r l / r2 ) o be equal in

a

first approximation

to th e ratio of m olar volumes (ul/u2) :

r l / r J

- VI/VZ) (12)

Another assumption is due to Carlson an d Colburn, in their

modification of the van Laar parameters:

rZu12/RT m/RT

=

12

rluzl/RT = m/RT =

21

(13)

while

u12

=

u21.

Th e implication is th at

r l / r2 )

equals (A21/A12)

:

( r l / rd +

A d A d (14)

Parameters A12 a n d

A21

are sometimes called “effective

molar volumes” (Hildebrand and S cott, 1950).

Wilson suggested another expression for (n /n ) , aking

into accou nt nonrandomness effects by weighting (uI/u2) with

a Boltzm ann factor containing interaction energies

X

:

r 1 / r 2 ) + (u1/v2) exp

{ -

A l 2

-

hm)/RT)

=

Apl

(15a)

306

Ind. Eng.

Chern.

Fundarn.,

Vol.

9,

No. 3,

1970

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an d similarly for ( r 2 / T 1 )

:

(r2/r1)- v2/u1) exp - A l z - All)/RT] A12

(15b)

One can expect A12,

A l l ,

a n d A22 to be proportional to zw12,

zwl1,

a n d

zws2

in Equat ion

5 .

Therefore

u12/RT

+ 2A12

- All - An) / RT

= -In ( A 1 2 h )

(16)

Equat ions 14 and

15

provide two reasonable choices (van

Laar and Wilson parameters, respectively) to introduce

parameters into Equat ions 9 to

11,

giving six combinations.

In Table

I

the combinations are indicated. Combining Equa-

tion 10 with Wilson parameters gives the Wilson equation.

Introducing v an Laar param eters into Equation 11 gives the

van Laar equations. When van Laar parameters are intro-

duced in Equatio n 10, a relation obtained is unable

t o

predict

phase separation, in much the same way as the Wilson equa-

tion.

Exten ded van Laar (EVL) Equa t ions . Subs t i tu t ing

A12parameters in to Equatio n 9 an d differentiating the result-

ing equation for g according to Equat ion 1 gives for the ac-

tivi ty coefficient,

y l

In y1 = 1

-

n a+ A1222/A21) -

2

+

A122dA21

(17)

1 A z 1 ~ 2 ~

22 + A ~ ~ Z I / A I Z ) ~

Th e relation for In y~ results when indices are rotate d in the

sequence 1

- -

. Equ ation 13 roughly predicts tempera-

ture dependence of A12 a n d A21,

Entha lp ic Wi l son Equa tions . Subs t i tu t ing Ki l son pa ram -

e te rs in E qua t io n 11 g ives

(18)

Postulating proportionality between the number of sites

occupied by a molecule of ty pe 1 and the ratio between its

molar volume a nd a m ean molar volume (weighted by inter-

action energies), one can write

VI

exp ( -Ad RT )

~1 exp ( - A d R T ) + xzvz exp (-A12lRT)

Comparison

of

Equa t ions 19 and 15 i l lust ra tes tha t Equat ion

19 is exact only in the limit

xl

+ 1. Substitution of Equ ation

19 in Equ ation 18 an d differentiating gives

In y1 = -

qE /RT

=

-r121zz

In (A1&)/(A21n

+

22

(19)

1

-

X

2

In (11121121)

21 + A1252)(22

f A z i ~ i )

For

In

y z

the analogous expression results by rotation of

indices (1

+

+

1).

Orye Equa t ions . Subs t i tu t ion of K i l son pa ra mete r s in

Equa tion 9 gives equations for y derived by O rye (1965), who

used a different approach.

In

y1

= 1 - n (zl +

~ ~ 2 x 2

zz{

~ 1 2 / z l

+ A ~ ~ z ~

Again, In y2 follows from rotation

1

2

+ .

The EVL equations (17), enthalpic Wilson equations (20),

an d Orye equations (21) were tested by

Representation of 16 experimental binary vapor-liquid

equi l ibr ium dat a from l i terature

T-

da ta a t cons tan t

P ) .

Prediction of parameter values for binary systems from ac-

tivity coefficients a t infinite dilution.

Prediction of temperature dependence of activity coeffi-

cients a t infinite dilution (an extremely severe test).

Comparison

o f

Equations

by

Fitting Parameters

to Experimental Data on Binary Systems

The EVL equat ions (17), enthalpic Wilson equations (20),

and Orye equations (21) were tested for 16 binary systems

listed in Table

11.

All equilibria were isobaric (1 at m ), in

order to test the built-in temperature dependence of the

equations; five systems show partial miscibility.

The fit t ing procedure proposed by Prausnitz et

al.

(1967)

was used. In this procedure a correction for vapor-phase

nonideality is incorporated. The objective function,

f, t o

be

minimized was defined as

with

m

the num ber of d ata points. A nonlinear multiple

regression subrou tine which adjus ts the param eters of a

function being fit ted to experimental data in such a manner

as to yield a least squares

fit

was written, following the

method of hfa rqu ard t (1959). Details have been given (Bruin,

1969).

The results are summarized in Tables

I11

to VII. Values

Table

I I .

literature Data for Systems Selected to Test

Acti vity Coefficient-Composition Equations

(Isobaric at

1

Atm)

System

1. hIethano1-water

2. Ethanol-water

3. 1-Butanol-water

4. 2-Butanol-water

5. Acetone-water

6. Butanone-mater

7. Methyl acetate-

8.

Furfural-water

water

9. 2-Propanol-water

10. Icetone-methanol

11. Acetone-ethanol

12. Ethanol-2-propanol

13. Ethanol-benzene

( P

= 750 mm

14. Ethanol-methyl-

15. Methylcyclopen-

16. hlethanol-ethanol

Hg)

cyclopentane

tane benzene

Reference

Uchida, 1934

Carey an d Lewis, 1932

Stockhardt and H ul l ,

1931

Altsybeeva and

Belousov, 1964

Othmer

et

al., 1952

Othmer et al., 1952

hlarshall , 1906

Temp.

Range,

K

338-69

351-68

372-84

364-66

330-61

330-61

329-60

Inte rna tion al Critical 371-432

Wilson an d Simon s, 1952 354-68

Hellwig an d Van Winkle, 330-48

Ballard an d Van Winkle, 351-55

Tables, 1928

Uchida et al., 1950 329-37

1953

1952

Tyr er, 1912 342-47

Sinor an d Weber, 1960 339-49

Griswold an d Ludwig 344-52

1943

h m e r

et al.,

1956 338-49

Ind. Eng. Chem. Fundam., Vol.

9, No

3,1970

307

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Table

Ill.

Fitting Results for Wilson Equations to Binary Systems

Parameters, co l /mo le Mo l ar Vo lumes, Cm3/Mole

System A12

-

11 A12 -

A22 v1

V2

F

O K

Methanol-water

198 .113

472.368 42 .11

1 8 . 7 9 3 6 9 . 5 5

Ethanol-water

380.315 910.615 6 3 . 9 8 1 8 . 7 8

368.65

2-Propanol-water 543.845

1 3 5 8 . 3 5 8 2 . 4 8

1 8 , 7 7 3 6 8 . 3 2

Acetone-methanol

-54 .175 472 .74

7 6 . 8 3 4 1 . 2 4

323.15

Acetone-methanol .- 3.454

489.719

7 8 . 4 7 4 2 , 0 7

337.15

Acetone-ethanol

-

4 , 2 3 6 440.50 7 9 . 8 2 6 2 . 2 7 3 4 8 , 2 5

Acetone-water

469 .02 1489.07 81 .42

1 8 . 6 7 3 6 1 . 0

llet 'hanol-ethanol

198 .85 -264 .02 42 .9 3

63 .39 349 .75

Ethanol-2-propanol

215.889

-241.331

62.825 83 .11 355 .05

Ethanol-benzene ( P

=

1540.28

119.80

6 1 . 8 2 9 4 . 6 1

3 4 2 . 6

(isothermal)

750 mm Hg)

Ethanol-methylcyclopentane 2230.23 236.08 61 .58 118.82 339.45

Methylcyclopentane-benzene 13 .68 253.694 120 .66 95.8 9 352.79

Error

in

P

Total,

%

0.3612

1 . 1 5 7 8

2 . 1 5 5

0.3442

0.6474

0 . 8 0 4

2 . 7 0 0

1 . 2 7 9

1 . 2 4 6

0 . 6 2 7

1 . 8 8 4

0.5890

Table

IV.

Fitting Results for van Laar Equations to

Binary Systems

Parameters Error in P

Azi Total, %

ystem

Methanol-water

Ethanol-a a ter

2-Propanol-water

1-Butanol-water

2-Butanol-water

Acetone-methanol

(isothermal)

Acetone-methanol

Acetone-ethanol

Acetone-water

Butanone-water

Methyl acetate-water

Furfural-water

Methanol-ethanol

Ethanol-2-propanol

Ethanol-benzene

( P = 750 mm Hg)

E thanol-methylcylo-

pentane

hfethylcyclopentane-

benzene

Ai2

0 . 8 6 4 9

1 , 6 6 0 1

2.4260

3 . 8 7 7 2

3.5447

0.6166

0.5713

0 , 4 5 4 9

2.3880

3 , 5 0 4 5

3.0565

4 , 6 8 9 5

-0 .5629

-0.01381

1 . 8 3 4 1

2.7404

0.3259

0.5210

0 . 8 6 4 3

1 . 1 6 2 5

1 , 1 7 5 9

1.1387

0.5442

0.5170

0 . 5 8 7 1

1 ,4555

1 ,6525

1

8050

1 . 1 0 0 8

-0 .0580

-

.006945

1.2814

1 . 6 8 8 9

0 . 3 5 0 5

0 . 3 5 2 1

0.7246

1 2026

3 . 3 4 9 1

1 . 4 7 9 5

0 , 3 0 8 3

0 , 6 4 1 0

0 , 7 3 4 0

4 . 4 1 5 1

5 . 1 9 9 6

5 , 5 2 1 2

27.5862

0 . 5 6 9 1

1 . 4 6 2

0 , 3 5 6 5

2 . 7 7 1

0.5796

for

X I Z -

AH),

A ~ z

- XZZ), € 1 2 , and

€21

a re computed a t the

temperatu res indicated in the tables. As a reference, th e Wil-

son an d van Laar equations were also checked. General trends

are that EVL equations (17) give poor results when com-

pared to the van Laar equations. A rem arkable result is tha t

the enthalpic Wilson equation (20) gives very good represen-

tation for most of the systems, in many cases bett,er tha n the

van Laa r or Orye equations (21).

Prediction of Parameter Values from

Infinite Dilution Coefficients

For In yl a t infinite dilution

of

component 1 in a binary

mixture one can derive

F o r

EVL Equations

In ylm

=

1

-

n (AIz/AzI)

- A Z ~ A I Z

AZI

(23)

For Enthalpic Wilson Equations

In

ylm=

-In

( A 1 2 h ) / A n

(24)

F o r Orye

Equations

In ylm= 1 - n 111 2 - AZI - n

hlzA21)/Az1

(25)

Similar relations follow for In

yz

by letting 1 2 1.

If In ylmand In

yZm

re known-e.g., from Pierotti-D eal-Derr

correlations (Pierotti e t al. , 1959) or Helpinstill and Van

Winkle correlations

(1968)-parameter values can be cal-

culated.

Solving Equation 25 and the similar relation for yZm or

A12 a n d A21 is tedious, requiring a double iteration scheme.

Therefore the Orye equ ations were not tested.

For

the EVL and enthalpic Wilson equations convenient

computational methods have been discussed (Bruin, 1969).

Some results for the EVL equation are summarized in Table

VII I . In F igures 1 t o 3 y s and activities, a, are plotted for

acetone and water, 1-butanol and water, and 1-octanol and

System

Methanol-water

Ethanol-water

1-Butanol-water

2-Butanol-water

Acetone-water

Butanone-water

Methyl acetate-water

Furfural-water

Table V.

Ai2

0.6021

0 . 9 8 7 5

1.5696

1 . 5 0 9 3

1.4392

1 . 8 0 4 1

1 . 6 9 7 1

1 ,7366

Fitting Results for EVL Equations to Binary Systems

Parameters

Ail

€21,

cal /mole ell, col /mole r

OK

1 ,0429

442.194

765,917 369 .55

1 . 9 2 2 5

723.459

1408,405 368 .65

4 .6182

1162.336

3419.983 372 .65

4 .2917

1093.13

3 1 0 8 , 2 1

.

3 6 4 . 4 5

2 .5208

1032.47

1808.370 361.00

3 .7793

1282.593

2686.815 361.00

3 .1399

1214,279

2246.585 360.05

5 .4891

1282.762

4054.630 371.71

Error in

P

Total,

%

0.3616

0 , 6 5 0 3

4.0834

1 . 7 8 1 5

5.2401

5 . 5 8 5 8

6.3476

24.1892

308 Ind.

Eng. Chem. Fundom., Vol.

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1970

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System

Methanol-water

Ethanol-water

2-Propanol-water

1-Butanol-water

2-Butanol-water

Acetone-methanol

Acetone-ethanol

Acetone-water

Butanone-water

Methyl acetate-water

Furfural-n-ater

Methanol-ethanol

E

hanol-2-propanol

Ethanol-benzene ( P

=

Ethanol-methylcyclopentane

Methylcyclopentane-benzene

750 mm Hg)

Table VI.

Fitting Results for Enthalpic Wilson Equations

Parameters , CaI /Mol e Mo lar Volumes, Crn3/Mole

XlZ - i1 XlZ - 22

v1

V2

-208.7681 571.3078

44 ,548 18 .792

- 25,7964

897.1059

63 .977 18 ,779

- 23.065 1150.147

82 .48 18 .77

- 57.8748 1229.9084

98.366 18.837

- 79.327 1098,612

85 .668 18 .721

-216.966 506.093

78 .47 42 .07

-116.696 399.150

79 .82 62 .27

-

36.468 1185.436 81 ,420

18 .675

-396.248 1307.900

8 9 . 5 3 3 1 8 , 6 3 3

-419.1093 1290.6923

87 .373 18 .663

-202.367 1158.676 82 .899

18 .823

188.7053

- 290.622

4 2 . 9 4 6 2 . 3 9

-1329.7304

1267.75 62.82

8 1 . 4 3

745.400

- 6.970 61.82

9 4 . 6 1

1006.585 - 82 .718

61 .58 118 .82

- 5.740 238.011

120 .66

95 .89

1, O K

369 .55

368 .65

368.32

372 .65

364 .45

337.15

3 4 8 , 2 5

361.00

361.00

329 .63

371 .71

349.75

3 5 5 , 0 5

342 .6

339 .45

352 .79

Error in

P

Total, %

0.3473

0 .6506

1 .1245

4.1245

1 .6216

0 . 6 5 4

0 .8046

3 , 2 6 9 5

4.1933

3.4580

23.0276

0 ,4790

0 .3871

0 .3938

2.3874

0 .5921

Table VII.

Fitting Results for Ory e Equations to Binary Systems

Parameters , CaI /Mole Mo lar Volumes, Cm3/Mole

Methanol-Tyater

-211.440

450 .860 44 ,548

18 .791

Ethanol-water

-

48.018 747 ,2929 63 ,977 18 .779

2-Propanol-water -435.954

962 .733 82 ,47

1 8 . 7 7

1-Butanol-water -369.779

1043.374 98.366

18 .837

2-Butanol-water - 79,327 921.84

85 .688 18 .721

Acetone-methanol - 35.9306 408.528 78.47

4 2 , 0 7

Acetone-ethanol

243.84

- 4 ,774 79 .82

62 .27

Acetone-water -479.035

1023 .049 81 ,420

18 .675

Butanone-water -435.140 1122.348 89.533

18 .633

Meth yl acetate-water -419.10 93

1290.692 87.373

1 8 , 6 6 3

Furfural-water -231.3797 970 .618

82 .899 18 .823

Methanol-ethanol

- . 4 2 0 5 1 , 2 1 3 3 4 2 , 9 3 5

62 .392

Ethanol-2-propanol

- 79.812

213 ,857 62 .82

8 1 . 4 3

Ethanol-benzene

654.954 - 97 .128 61 .23

9 4 , 6 1

Ethanol-methylcyclopentane 931 ,4907 - 30.439

6 1 , 5 8 1 1 8 . 8 2

Methylcyclopentane-benzene

-

3 ,107 167.237 120 .66 95 .89

System A12 - XI1 A12 - 22 v1 v2

P = 750 mm Hg)

r

OK

369 .55

368 .65

368.32

372.65

364 .45

337 .15

348 .25

361.00

357 .75

329 .63

3 7 1 , 7 1

349.75

355 .05

342.60

339 .45

352 .79

Error in

Total, %

0 .3479

0 .7323

1 .1650

4 , 4 1 6 1

1 , 7 3 3 1

0 .6566

3 . 3 8 0

4.0037

4 .9331

4.4630

23,2258

1 .3748

1 . 3 7 2

0 .4385

2 .7651

0 .6113

~~ ~~

Table VIII. A-Parameters Estimated from Infinite

Dilution Values for Acti vity Coefficients

No.

of

I t era-

System 1,

O K

Ai2 A n t ions

Acetone-\\ atera

373 15 1 5745

2 5228 8

2-Heptanone-waterb 298 15

2 1590 9 8669 7

1-Butanol-water*

298 15 1 6545

4 6425 7

1-Octanol-wa terb 298 15

2 2763 11 6137 7

Acetone-1-octanolb

373 15 3 6702

0

8224 6

l-Butanol-1- 373 15

3 5548 1 6639 6

2-Heptanon e-l- 373 15 1 8025

0

8354 6

octanolc

octanolb

water. Results for acetone and water are compared with the

da t a of Othmer (1952). Phase separation is predicted for 1-

butanol and w ater and 1-octanol and water a t approximately

the right l iquid composit ions, as can be seen from the activity -

composition plots.

The enthalpic Wilson equation was tested using methanol

and water , ethanol and water , and 1-butanol and water .

Table IX gives values for

A l z

-

l l )

a n d ( A 1 2 - 2 2 ) computed

from

ylm

n d

y 2 .

Reasonable agreem ent exists between values

computed from Equation 24 and those given in Table VI.

In Figure 4 a graph is given for binary systems with posit ive

deviations from ideali ty, from which A 1 2 a n d Azl follow

directly when

y lm

a n d

7 2

are known. The procedure is

i l lustrated for methanol

1)

+ water (2): In y lm

=

0.521,

change of indices gives coordinates of point 2:

A21 =

1-00.

Othmer , 1952*

Pierrotti,

and

Derr

In

yz =

0.865. FromFigure 4: A12 = 0.60 (point 1). Inter-

1959.

c

Wilson an d Deal, 1962.

Table IX.

Estimated A Parameters for Enthalpic Wilson Equations (20) from A ct ivi ty Coefficients at Infinite Dilution

(ENTLAM program)

( X i 2

- 221 Temp., O K

System Aiz Azi (A12 -

zz)

369 .55. Methanol-water. (338-72°K) 0.59 90 0.9 94 4 - 57.553

2. Ethanol-w ater. (351-72'K) 0.483 2 0.928 0

-

64.712 952.309 368.65

3. 1-Butanol-waterb (372-80°K) 0.3 04 9 1.0 054 - 44.979 1220.01 372.65

638.017

a

Stock hardt and Hull , 1931. Pierrott i , Deal, an d Derr correlat ions, 1959.

~~

Ind.

Eng. Chem. Fundam., Vol.

9,

No.

3,

1970

309

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6/10

I-Butanol (1)-Water

(2)

A

1 2

1.6545 T

=

298

O K

X,=aons i , =0 .540 xI+

Activity and activity coefficients for the 1-butanol (1)-water

(2

XI

-

Figure

2.

system

Prediction

o f

Temperature Dependence

of ylm

nd

yzm

Th e temperature dependence of activ ity coefficients is given

by the exact thermodynamic relation

where H i o i s the enthalpy

of

component

i

in the s tandard

s t a t e and Ri s the part ia l molar enthalpy of mixing. At

infinite dilution

of

component i

For the EVL equations one obtains by differentiation

of

Equat ion

17

and taking the l imit z l+0,

assuming e12a n d

to be independent

of

temperature.

Th e enthalpic Wilson equation gives

310

Ind. Eng.

Chem.

Fundam.,

Vol. 9 , No. 3 , 1970

8/19/2019 VLE Methanol- water

7/10

2.:

x

C

-

I*.

1.5

I c

0.5

0.0

12Or-

System

AIZ=

2.2763

A21

I

I6137

\

~ T =

298 K

loo--

I \ I System

I

2 k

-t-

I-Octanol(l)+Water(2)

I /

X I

- XI

Figure

3.

system

Activity and activity coefficients for

1

-octanol (1)-water

(2 )

In Yla,

_c

Figure 4.

Nomogram giving

.211

and

for

enthalpic Wilson equation from In

ylm

and In

y2m

Ind. Eng. Chem. Fundom.,

Vol.

9,

No.

3,

1970

3

1

8/19/2019 VLE Methanol- water

8/10

The Orye equation yields

l

@ -

o,8-

Rotat ion

of

indices in the sequence

1

- r

2

-

gives similar

relations for the tem peratu re dependence

of

In 7 2 . Figure 5

VAN LA& EQJATIOK

FOR FOUR

TERNARY SYSTEMS

I : + ccetone;

x

methanol;

e

w t e r

I I : acetone ;

li

methanol: ethanol

a:oethanol ; is0 ptopanol;* water

E ethanol ; * methylcyclopentone

; o

benzene

gives the predicted temperature dependence of In ylm a n d

In yz in the system m ethanol (1) + water 2) for t he EVL,

enthalpic Wilson, and Orye equations. Experimental data

from Per ry (1963) are also given.

It

is evident th at the predicted temperature dependence is

poor for all equations. T he enthalpic Wilson equation ap pears

to give results closer to th e experimentally observed values for

the temperature dependence of In ylm than the Orye equation.

It must be admit ted that the tes t

of

predicting temperature

dependence

of

acti vity coefficients is extremely severe. Th e

conclusion is that

~ I Z A l l

and

(A12

-

A l l )

must have some

dependence on temperature.

WILSON EWATIONS FOR FOUR TERNARY SYSTEMS

I

+

acetone

;

x methanol; wter

I I : acetone

;*

methand;. ethonol

m ethanol ; 1%-popanol: water

E:

uethanol

;

ethylcyclopentane: o benzene

I B

0.6

1

Yexp.

0.5

-

0 4

0.3

2.5

3.0

3.5

4 0

Figure

5

Predicted temperature dependence of In yim

for system methanol (1)-water (2)

- ' / T ) X I O ~

( O K - ' )

--

nthalpic

Wilson

equat ion

- *

-

r y e e q u a t io n

Exper imenta l da ta

_ - _ -

VL

equations

0.6

Yexp.

/

0 0 6 .

//

f

*

0

0 2 3 0.4

0.5

0 6 0 7 0 8 0 9 I0

YCOlC

igure 6.

Fit of van Laar equations to ternary VLE da ta

+

/

Ycolc-

Figure 7. Fit of Wilson equation to ternary VLE data

ENTHALRC WILSON EOUATONS

FOR FOUE

TE3hARY

SYTER'S

1: + acetone: x methanol;

e

water

I I : acetone

: r

methanol; ethanol

E: ethanol

;

iso-propanol;

*

water

E

thonol

; *

methylcyclopentane

:

o benzene

0.67:

Yexp.

0.5

-

0.4

-

0 3

0.2

-

6 I I I I I

0 0.1

0.2

0.3 0.4

0.5

0 6

0.7

0.8

0.9

I

kale

igure 8.

data

Fit of enthalpic Wilson equations to ternary VLE

312

Ind. Chem. Fundam.,

Vol. 9, No. 3, 1970

8/19/2019 VLE Methanol- water

9/10

I

I

.9

CRYE

EQUA; OP.IS FOR FOJR TERN ARY

SYSTEMS

I +

acetzne;

x

re tnono ;

e

wo:er

II ccetoie ;

tr

rne inmol ;

ethanol

m

etccnc

: +

is0

D ~ O ~ G C O ~ : +afer

-

O,s-E .ehcnol

; *

rnethylcyclopentane;

o

benzene

0 7t

G C.

G2

0.3

0 4 0 5 06

G 7

0.8

0.9

k a i c

Figure

9.

Fit of Orye equations to ternary VLE data

Multicomponent Systems

Th e EV L equations (17), enthalpic Wilson equations (20),

and Orye equations (21) can be readily extended to multi-

component systems. In fac t, the quasilattice model of Guggen-

heim (1944a, b, 1952) was developed for multicomponent

systems. In the zeroth approxim ation only two-body inter-

act ions are taken into account and therefore th e Gibbs free

energy is obtained by a summ ation procedure.

Th e EV L equation is not discussed here, as it is clear from

Table

V

that this equation gives rather poor results. The

enthalpic Wilson equation for a multicomponent mixture

becomes

25 111

A1 5A j i )

f f 4 j -

x j xm 1n

AjrnArnj)

a i i m

(31)

a j m

Th e Orye equat ions for a mult icomponent m ixture are

In

Y i

= - n

( ,

A i p x p - Z ( q -

Equat>ions31 and 32 were tested on four ternary systems

listed in Tabl e

IX.

Binary VLE da ta were reduced to p aram-

Table

X.

Ternary Systems Used to Test Enthalpic

Wilson and Orye Equations (Isobaric at 1 Atm)

System Reference

1. Acetone-methanol-water

2. hcetone-methanol-

3.

Ethanol-2-propanol-

4. Ethanol-methylcyclo-

Griswold arid Buford, 1 949

hme r , Paxton, and Van

Kojima, Ochi , and Sa kaw aza,

Sinor and W eber, 1960

ethanol Winkle, 1956

water 1969

pentane-benzene

eters A i j a n d A j i (Tables I11 to VI I) , a nd the values obtai r ied

were used in bubble temperature calculatioiis using the

programs described by Prausnitz et

al.

(1967). Pure compo-

nent properties were t .aken from the li terature. Critical

pressures and temperatures were taken from the li terature or

in some cases calculated with Lyderseri correlations (Reid

and Sherwood, 1958). Acentric factors, dipoles, and vapor

pressure data were taken froni Prausnitz

e t nl.

(1967) and

O'Connell and Prausn itz (1964).

In Figures 6 to 9 the computed vapor compositions for a

num ber of liquid compositions for each syst.ern ar e compared

wit,h t,he experim ental compositions

as

reported in the refer-

ences given in Table IX. As a reference, the results for the

van Laar and Wilson equations for the same four ternaries

are also given.

Conclusions are that the enthalpic Wilsori and

t h e

Orye

equations give

a

considerably bette r

fit

of vapor-liquid equilib-

r ium in ternary sys tems than the va n Laa r equat' ions . Th e

van Laar equation gives a good

fit

for t 'he acetone-methanol-

water system, but poor results for the ethanol-methylcyclo-

pentane -benzen e an d ethanol-2-propanol-water syst.ems.

Th e Wilson equation gives a very good fit

of

VLE d a ta for the

ethanolmethylcyclopentane-benzene system. The results for

the ethanol-2-propanol-water syst.ern, however, are appro xi-

mately

of

the same accuracy as t 'he enthalpic Wilson and

Orye equations. Fo r syst 'ems near t o phase separation o r for

systems showing phase separation, the enthalpic or Orye

equations are recommended. For other systems, the Wilson

equat ions are a t t ract ive t o use .

Acknowledgment

Th e author is grateful to the C omputer Center of the

Eind hov en Univ ersity of T echno logy for th e use of its facili-

ties. Especially th e assistance of M arijke t er Mor sche is grate-

fully acknowledged.

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R. R.,

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A. I.,

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>I.,

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~ ~ ~ . . ~ . , .

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E. L., Papadopoulos, hI. N., Znd. Eng. Chem.

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S.,

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Carey, J.

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P. J . , J .

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41,

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J.,

Ludwig,

E.

E.,

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E. A.,

Proc.

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148,304-12 (t935).

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SOC. 183,203-12

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Guggenheim, E. A,,

Proc. Roy. SOC. 183,213-27 1944b).

Guggenheim, E. A., “Mixtures,” Chap.

XI,

Clarendon Press,

Hellwig, L. R., Van Winkle, hl., Znd. Eng. Chem.

45, 624 1953).

Helpinstill, J. G., Van Winkle, M., Ind. Eng.

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Hildebrand, J. H., Sc ott, R. L., “So lubility of Non-electrolytes,”

Huggins,

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1-32 1942).

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1928.

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Intern. Chem. Eng.

9

2),

Merquardt, D . W.,

Chem. Eng. Progr. 55 6), 65-70 1959).

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J .

Chem.

SOC.

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38

(lo),

265 1966).

Nagel, O., Sinn, R., Chem. Zng. Techn. 39 ( l l ) ,

671-6, 275-82

Neretnieks, I., Ind. Eng. Chem. Process Design Develop.

7, 335-9

O’Connell, J. P., Prausnitz, J.

M.,

IND. NG. HEM.FUNDAM.

,

O’g R. V., Ph.D. dissertation, University

of

California,

Orye, R. V., Prausnitz, J. M.,

Znd.

Eng. Chem.

57

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18 1965).

Othmer, D.

F., Znd. Eng. Chem. 44, 1872 1952).

Othmer, D. F., Chudgar, M. M., Levy,

S.

L., Znd. Eng. Chem.

Perry, J., ed., “Chemical Engineer’s Handbook,” 4th ed., Chap.

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1952.

Design Develop. 7 ,

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1950.

342 1969).

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Pierrotti, G.

J.,

Deal, C. H., Derr, E. L.,

Znd. Eng. Chem. 51 l),

Prausnitz, J. R.I., Eckert, C.

A.,

Orye, R. V., O’Connell, J. P.,

“Com uter Calculations for Multicomponent Vapor-Liquid

Equiligria,” Chap.

1-6,

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N .

J.,

1m 7

95-102 1959).

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Renon. H.. Prausnitz, J.

hI., A.Z.Ch.E.

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35-44 1968).

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7-25,

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1958.

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Data

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S.,

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1438-40

Tyrer, D., J .

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S.,

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23, 1968

ACCEPTED

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Work supported by the

L. E. B.

Foundation, W ageningen.

Thermal Reaction of Propylene

Kinetics

Taiseki Kunugi, Tomoya Sakai, Kazuhiko Soma, and Yaichi Sasaki

Department of Synthetic Chemistry, Faculty of Engineering, University of Tokyo, Hongo, Tokyo, Japan

Kinetics of the thermal reaction of propylene was studied at temperatures ranging from 703’ to 854’C.,

atmospheric pressure, and residence times from 0.078 to

3.3

seconds with and without ni trogen dilut ion.

Main primary products were ethylene, methane, hydrogen, butenes, and butadi ene in the approximate

ratio of

5 :

3 : :

1

: 1 at init ial stages of the reaction. Other primary products were methylcyclopqntene,

hexadienes, acetylene, and ethane. Secondary products were cyclopentadiene, benzene, polycyclic

aromatics, cyclopentene, and toluene. Selectivities of formation of these products, except acetylene and

ethane, showed little dependence on temperature. At higher part ial pressure of propylene, the selec-

tivities of ethylene and methane formation decreased to some extent. The effect of parti al pressure

of propylene

on

the r ate o f propylene disappearance leads to a three-halves-order equation. The rate

constant

i s

given as

k =

10’5*06 cc>/z/(mote’/z sec.).

T H E R M A L REACTION of prop ylene has been studie d exten-

sively (Amano and Uchiyama, 1963; Kallend et aZ., 1967;

Laidler and Wojciechowski, 1960; Sakakibara, 1964; Wheeler

and Wood,

1930).

A few experiments

at

t emperatures f rom

700’

t o

850’

and atmospheric pressure cover the condit ions

of th e industrial m anu factu re of olefins an d aromatics by

cracking hydrocarbon feedstocks. Analyses of th e products

have been l imited to l ighter hydrocarbons below

CCor

t he

products formed in narrow ranges of temperatures and con-

versions. Lack of clear discrimination between the primary

and secondary products is due to these l imited analyses of

the products.

Reaction products were analyzed in d etail to differentiate

the prim ary from the secondary products. The kinetics of th e

therm al reaction of prop ylene was discussed in comparison

with tha t of ethylene. In

a

following paper mechanisms

of

t he

reaction and of formation of higher hydrocarbons above

CS

are to be discussed.

Experimental

Feed propylene was 99.35 mole % pure by gas chromato-

graphic analysis, used without furth er purification. T he im-

purity was propane,

0.65

mole yo.Oxygen content was less

t h a n 1 p.p.m. by weight. Commercially available nitrogen

3

4

Ind. Eng. Chem. Fundam.,

Vol. 9

No.

3, 1970