01.modification_of_van_laar_activity_coeff.model.pdf

Iranian Journal of Science & Technology, Transaction B, Vol. 25, No. B2 Printed in Islamic Republic of Iran, 2001 to Shiraz University

MODIFICATION OF THE VAN LAAR ACTIVITY Yr COEFFICIENT MODEL G. R. VAKILI-NEZHAAD*·1, H. MODARRESS *2AND G. A. MANSOORI 3 IChemical Engineering Department, Faculty of Engineering, University of Kashan, Kaspan, 1 R. of Iran Department, Amirlcabir (polytcclmic) University of Technology, Tehran, I. R. of Iran 3 Chemical Engineering Department, University of illinois, 810 S. Clinton Street, Chicago, USA

2 Chemical Engineering

Abstract - Based on statistical and mechanical arguments, the original van Laar activity coefficient model has been improved by reasonable assumptions. This modification has been done by replacing the van der Waals equation of state with the Redlich-Kwong equation of state in the formulation of van Laar with consistent mixing rules for the energy and volume parameters of this equation of state (a",ix, bmix). Other equations of state, such as the Soave modification of the Redlich-Kwong equation of state, Peng-Robinson and Mohsen-Nia, Modarress and Mansoori equations of state, have been introduced in the formulation of van Laar for the activity coefficients of the components present in the binary liquid mixtures, and their effects on the accuracy of the resultant activity coefficient models have been examined. The results of these revised models have been compared with the experimental data and it was found that the Redlich-Kwong equation of state with the van der Waals mixing rules for the volume and energy parameters of this equation, is the best choice among these equations of state. In addition, it can improve the original van Laar activity coefficient model and, therefore a better agreement with the experimental data is obtained. Keywords -statistical mechanics, phase equilibria, activity coefficients models, non-ideal solutions

1. INTRODUCTION There are two ways for the estimation of fugacities of the species in liquid mixtures: (i) For the mixtures involving onfy hydrocarbons and dissolved gases, simple equations of state such as the Peng -Robinson equation of state [1] may be used for the estimation of species fugacities. (ii) However, for liquid mixtures containing alcohols or electrolytes, in which one or more of the components cannot presently be described by an equation of state in the liquid phase, another procedure for estimating species fugacitieS must be used. The most common starting point can be written as follows: (1)

where r I is the activity coefficient and J/ is the fugacity of the species (i) in the pure state. It is evident from the above equation that in case (ii) for the purpose of studying the behavior of the liquid mixture, one must have the activity coefficient of the species present in the liquid mixtures. Many efforts have been made to present the various activity coefficient models, such as Margules, van Laar, UNIQUAC, etc. In this work, the van Laar model is improved to increase its accuracy in activity coefficient calculations for liquid mixtures. 'Received by the editors November I, 1998 and in final revised form Octohec 4, 1999

·'Corresponding author

282

G. R. Va/cJIl-ne:/uuzd / et Q/.

In the van Laai' theory leading to its activity coefficient mo<;iel, which is based on the application of the van der Waals equation of state, for a two component mixture with mole fractions Xl andx 2 , two main assumptions are involved, which are: ~=O and S£=O. This means that when mixing two pure components no volume change occurs, and the entropy change of mixing is equal to its ideal value. The van Laar theory comprises of three stages as follows [2]: a) Expansion of pure liquids isothermally to a very low pressure. b) Mixing the resultant ideal gases (due to very low pressure). c) Isothermal compression of the mixture of ideal gases. Van Laar used these stages for the formulation of the excess Gibbs free energy of mixing and obtained the following equations for the activity coefficients of components (1) and (2):

(2)

bVDw2 In rVDW2 - RT

[(~ bvDw2 - ra;;;;;;)2/(1 + - - -X2- )2] bVDWI bvDw2 bVDWI

(3)

Xl

where, arow and brow are the parameters obtained from van der Waals equation of state. Van der Waals equation of state is presented as:

P=

RT v-bVDw

aVDW v2

(4)

where, aroWi and browl can be expressed in terms of critical properties of component (i), Pel and Vel, in the following form:

(5) and

(6) On the other hand, according to statistical mechanical arguments, it hlJS been proved that the mixing rules for the cubic equations of state must be consistent with their algebraic form [3]. In this work, we have modified the original formulation of the van Laar activity coefficient model using these concepts.

2. THEORY AND FORMULATION OF THE REVISED VAN LAAR MODEL Based on the works of Leland [4-6], and using the concepts of the statistical mechanics, the theory of the van der Waals mixing rules- bas been proposed by Kwak and Mansoori [3]. They proposed some guidelines for derivation of the van der Waals mixing rules for the various equation of states, which for the Redlich-Kwong equation of state we demonstrate their results as follows [7]: a(RK)mb:

=

nn {77

213 1/3}l.S (n

XiX jaij

bij

Iranilln Journal ofScience & Technology, Volume 25, Number B2

/

77

)112

n

XiX jbij

(7)

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283

ModijiClldon oftire van Laar••• /1 . /1

and

b(RK)mix =

L L XjX jbij j

(S)

j

Where, a and b are the parameters of the Redlich-Kwong equation of state. The Redlich-Kwong equation is;

P=

RT

(9)

v-b RK where, the parameters aRK and b RK can be expressed in terms of clitical properties as:

(10) I

and

_O.OS664RTc b RK-

(11)

Pc

The combining rules for the

aij

and b/j parameters are as follows: a/j

=(1- kij )(a1ia.ii )112

(12)

j

(13)

and 113 bij = L ~bii + bii 1/3 ) / 2

Using the Redlich-Kwong equation of state with the mixing rules presented in Eqs. (7) and (S),

we have reformulated the van Laar theory. Here we demonstrate the steps of the new formulation as follows: According to the thermodynamic relation, (14) and using Eq. (9), we can write (15) Therefore, for the Redlich-Kwong equation of state, similar to the van Laar form'.llation can be obtained: (16) which can be written as, xI{Ujdeal

-U)I =

3aRKI

1

2vT

bRKI

r;;; xI{--)ln

In this stage a reasonable assumption for molar volume Vi ~ bRJ(, therefore,

vi

~

oo

V

v+biUCl

(17) I

VI

of each component is applied, that is

(IS) Similarly for component 2, • 3aRK2 ln2

x2(uideal

~pring2001

-uh = .

r;;; x2

2b RK2 vT

(19)

lranion JourlUll ofScience & Techn%gy, Volume 25, Number B2

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G. It VakJIl..neduuul / et til.

Therefore, the new formulation for internal energy change for the stage (a) as presented by van Laar theory can be expressed as follows: A

aU

3ln2 (aRKIXI + .....:..:;:==--=aRK2X2)

- --

a -

2"fi

b

b

RKI

(20)

RK2

On the other hand, the isothermal mixing of ideal gases has no internal energy change, therefore, for stage (b) we have, (21) For stage (c) which, is the isothermal compression of the mixture of ideal gases, the following equation can be written:

=_3ln 2 a(RK)mix

Ilu c

2"fi

(22)

b(RK)mix

and the total internal energy change, which in this theory is taken as equal to the excess Gibbs free energy, can be written as, (23)

Using Eq. (23), the activity coefficients·of components 1 and 2 are readily obtained; In

3ln2 (a = 2RT 3/2 b RKI

RKI

YRKI

-

M

3/2 I

-"23 M I1/2 M 2 )

(24)

and lny RK2

=

3ln2 (a RK2 -M 3/2 2RT 3/2 b I RK2

-~M 1/2 M 2

I

)

(25)

3

where, M\ , M2 and M3 are given in the appendix. This procedure is repeated with the other equations of state such as the Soave modification of the Redlich-Kwong [9], Peng-Robinson [1] and Mohsen-Nia, Modarress and Mansoori [10] equations of state. Here, for the sake of simplicity, we mention only the results of $ese formulations, the details of which are given in the appendix. The activity coefficients which have been obtained from the SoaveRedlich-Kwong are as follows: 2[Xl (aSRKl - MAlT) + x2 (.,JaSRx 1aSRK 2

-

MA12T)]

(x1b SRK1 + X2 b SRK2) - b SRK 1 [(aSRK 1 - MAIT)xl + -In 2 aSRKl - MAlT InYsRKr-

RT

(aSRK2 - MA2T)x~ + 2x 1X 2 (.Ja SRK 1 aSRK 2

b SRK1

Iranian Journal o/Science & Technology, Volume 25, Number B2

(x\b 1 + X2 b 2)2

-

MAI2T)]

(26)

Spring ZOOI

Modijicllllon o/the wzn Latu•••

285

2[x2(aSRK2 -MA2T) + Xl (,JaSRKlaSRK2 -MA12T)J (x1b SRKI +X2bSRK2)-bSRK2[(aSRKI -MAIT)x; +

In 2 a SRK2 - MA2T

lnrSRK2 = RT

(aSRK2 -MA2T)xi

bSRK2

+2XIX2(~aSRK\~SRK2

-MA12T»)

(27)

+ X2 b SRK2)2

(x1b SRKl

If we use the Peng-Robinson equation of state in the fonnulation of the activity coeffl.cient then the following equations will be obtained:

2[Xl(01 -MTETl)+X2(,JOI02 -MTETl2)] . 2 (x1b pRI +X2bpR2)-bpR1[(Ol -MTETl)Xl +

In

0.6232 01 - MTETl b

rPR1=-RT

(02

-MTET2)x~

+2XIX2(..j8;ii; -MmTl2)]

(28)

(x1b pRI +X2bpR2)2

PRI

and for the component 2 2[X2 (02 - MTET2) + xI (,JOI02 - MTET12)] (x1b pRI

In

0.6232 02 - MTET2 rPR2 =-RT

b

+ X2 bpR2) -

(02 - MTET2)xi

bpR2 [(01 - MTETl)xt

+ 2xlx2 (..[if;ii; - MTETl2)]

(x1b pRl

PR2

+.

+ x 2b pR2)

(29)

2

Finally, if we use the Mohsen-Nia,. Modarress and Mansoori equation of state in the modelling of the activity coefficients, for a binary liquid m~ture the resulting equations are,

2(xl a MMMl + Xl..JaMMMI aMMM2 )M ln r

-~ aMMM1 . ~ M'(Xl aMMMl +X2 b MMM I 1

MMMI. - RT 3!2

2a

1

.J .IJMM2 + 2XI X2 aMMMlaMMM2) r M'

J

(30)

and for the component 4,

In

0.41 rMMM2 = RT 3/ 2

Spring 2001

aMMM2

b

MMM2

2(X2aMMM2

+ x l,JaMMMlaMMM2)M -

S(XraMMMl

+ X~aMlYfM2 + 2xIX1,Jr-a-MMM--,a-MM-'M-2) M2

(31)

Iranian Jourlflll o/Science & Technology, Volume 25, Number B2

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286

V~/etaL

Various parameters which are present iti the above equations, as well as the details for the derivation of these equations, are given in the app~dix. 3. RESULTS AND DISCUSSION . . In order to compare the accuracy of the various models presented in this work, which are given by Eqs. (24) to (31) with the original van Laar model given by Eqs. (2) and (3), we have calculated th~ vapor pressures of some binary mixtures at various temperatures. To calculate the vapor pressure of the mixtures, the following equation may be used:

(32)

Figure 1 shows the variations of vapor pressure for system of Diethyl Ketone/n-HexaJle at 650 C calculated by the new model based on the Redlich-Kwong equation of state and'those obtained from the van Laar original model as well as the experimental data. As it is seen from this figure, the vapor pressures calculated by the new model are closer to the experimental data, which indicates the fact that when the Redlich-Kwong eqUation of state, which is the improved form of van der Waals equation of state, is used along with the appropriate mixing rule.s, the prediction of the van Laar model is improved. The same trend is obtained '.for the systems of acetone/chloroform, chloroformll,4-dioxane, acetone/methanol, methyl ethyl ketone/toluene, nitrogen/methane, and nitrogen/ethane which are presented in Figs. 2-10. It is worth noting that Figs. 6-10 are for high pressure calculations which again indicates that the new formulation is ,more effective in vapor pressure prediction compared to the original van Laar equation.

G,_---------------------,

~,-----------~--------~

II

II ,

II

1~

.

II

II

.L-__

•

~

OJ

__

.

~

____L __ _J __ _ u u

~

Fig. 1. System Diethyl Ketoneln-Hexane at 6SoC

...

,-------------------------,

.

,

u IIoIetactioaal_

OJ

u

Fig. 2. S~rn Acetone/Chloroform at SO°C

~,---------------------,

•

,:~::::

, .. II

i:

j:

to II

.L-__- L_ _ _ _

•

...

~

__

~

____

~

__

~

.L-__ o

~

u

__

~

u

__

~

____

u

~

__

~

u

IIoIeFtll:tlanalllEK

Fig. 3. S~m AcetoneIMethanol at. SSoC lranilm JourlUll ofScience & Technology, Volume 25, NUlllber B2

Fig. 4. S~m MEKlfoluene at SO°C Spring, JOO}

ModijiclltJon oftile v_ lAIzr•••

287

.

'" •

-...

50

.. .. J ..

6

I

~

~

~

J ...

II

••

...

-

..

...

...

••

Fig. 5. System Chlorofonnll, 4-Dioxane at 50"C

...

u

u

u

Fig. 6. System AcetoneJMethanol at loO"C

-.....

- ,...-----------------------------,

, -...

,..-----------------------------,

, ,

,-

i :...

j:

.. ..

.. .

!GO

_ _ 01 '*"III"

_ _ 0I,*-,

Fig. 7. System NitrogenlMethane at 99.82 K

Fig. 8. System NitrogenlMethane at 110.93 K

",...------------------------------,

-,...-----------------------------,

,,-...... ..

f=-==t=..

14

I, :.

, ,

...

....

...

_ _ 01~

i·

..

..,

...

...

_ _ 01,*,-

...

Fig. 10. System Nitrogen/Ethane at 200 K

Fig. 9. System NitrogenlMethane at 122.04K

Table 1. Average absolute deviation (AAO%) for each system using . different equations of state Average Absolute Deviation (AAI)OIo)

System VOW

--

Spring 2001

RI(

SRI(

PR

MMM

-- -- --

--

Diethyl ketone'Hexape

29.12

2.43

2.58

2.65

30.15

AcetoneiChlorofonn

20.01

0.86

0.98

1.03

22.18

AcetonelMetharioi

4.52

1.42

1.58

1.63

4.86

MEKlroluene

5.18

0.46

0.53

0.62

5.88

ChlOrofonn/l,4-Dioxane

23.24

2.02

2.31

2.44

25.20

Nitrogen! Methane

7.23

4.22

4.45

4.53

7.87

Nitrogen I Ethane

20.33

6.75

6.88

6.93

20.86

Iranian JOll17l4l ofScknce & Technology, Volume 25, Nllmber B2

288

G. R. VakIll-neduuu/letaL

The same calculations ·have been performed using Peng-Robinson, Soave-Redlich-Kwong, and Mohsen-Nia-Modarress-Mansoori equations of state. To avoid presenting similar figures as 1-10, we have reported the final results as the Absolute Average Deviation (AAO%) from experimental results for the same systems in Table 1. From Table 1 it can be seen that the Redlich-Kwong equation of state provides the van Laar theory with bettet; vapor pressure prediction for binary liquid mixtures.

4. CONCLUSION It has been shown that the van Laar activity coefficient model, which is based on the van der Waals equation of state, can be improved by replacing this equation of state with other more accurate equations such as Redlich-Kwong, Soave-Redlich-Kwong, and Peng-Robinson equations of state. However, the best results have been obtained using the Redlich-Kwong equation of state . . The new formulation presented in this work has two main advantages. The first advantage is demonstrated by the application of a more accurate equation such as Redlich-Kwong, instead of the van der. Waals equation of state, which is used in the original van Laar theory. The second advantage is shown by application of a mixing rule which has a sound theoretical basis from the statistical mechanical point of view. On the other hand, in spite of the more complex nature of three parametric equations of state such as Peng-Robinson or Soave-Redlich-Kwong equations of state, the results obtained were not as accurate as those of the Redlich-Kwong equation of state which has only two parameters. This can be attributed to two reasons. Firstly, the Redlich-Kwong equation of state is the improved form of the van der Waals equation of state and the reformulation of the van Laar activity coefficient model is completely compatible with the original formulation of van Laar; thus, as one would expect using the Redlich-Kwong equation of state gives a more accurate result. Secondly, the application of the SoaveRedlich-Kwong, and Peng-Robinson equations of state imposes a constraint on the parameters of this equation which is not compatible with the general assumptions of the van Laar model. Similar reason can also be put forward for the Mohsen-Nia, Modarress and Mansoori equation of state, which has a different form of mixing rule which is not compatible with the assumptions of the van Laar theory. Therefore, these equations of state cannot predict vapor pressure close to experimental values when combined with the van Laar theory. Finally, it can be concluded that introducing the Redlich-Kwong equation of state with the new mixing rules in van Laar's formulation provides an activity coefficient model, which is more accurate than the original model, based on the van der Waals equation of state for both low and high vapor pressure calculations for binary liquid mixtures.

NOMENCLATURE

a a"

constant in the equations of state a constant of the Peng-Robinson equation of state which is defined in Eq.(A-IS) volume parameter of the equations of state fugacity of species i in the pure state fugacity of species i in the mixture molar Gibbs free energy interaction parameter inEq. (12) a parameter in the SRK equation of state, defined in Eq. (A-3)

M

parameter of Eqs. (30) and (31), defined in Eq. (A-28) parameter ofEq. (31), defined in Eq.

(~-29)

parameter of Eqs. (24) and (25), defined in Eq. (A-31) IranUtn Journal ofScience & Technology, Volume 25, Number B2

Spring 2001

Modijklzdoll oftlu! WIll Lmzr•••

M2 M3

parame!~

MAlT

parameter of Eqs. (26) and (27), defined in Eq. (A-IO)

MA2T

parameter of Eqs. (26) and (27), defined in Eq. (A-tt)

MAl2T

parameter of Eqs. (26) and (27), defined in Eq. (A-t2)

MTETI

parameter of Eqs. (28) and (29), defined in Eq. (A-19)

MTE12

parameter of Eqs. (28) and (29), defined in Eq. (A-20)

MTETI2

parameter of Eqs.(28) and (29), defined in Eq. (A-2t)

P R

absolute pressure

S T u v

parameter ofEq. (31), defined in Eq. (A-30) and entropy

289

ofEq. (24), defined in Eq. (A-32)

parameter ofEq. (25), defined in Eq. (A-33)

gas constant absolute temperature molar internal energy molar volume mole fraction of the ith compOnent in the mixture compressibility factor

Greek letters

r

Activity coefficient

o

A parameter in the Peng-Robinson equation of state, defined in Eq. (A-14)

CD

Eccentric factor

Subscripts

C

Critical state

1,2,i

Component number

mix

Mixture

MMM

Mohsen-Nia - Modarress - Mansoori equation of state

PR

Peng-Robinson equation of state

PK

Redlich-Kwong equation of state

SRK

Soave modification of the Redlich-Kwong equation of state

VDW

van der Waals equation of state

Superscripts E

Excess property

Sat.

Saturation

REFERENCES 1.

Peng, D. Y. and Robinson, D. B., A new two-constant equation of state.. Ind. Eng. Chem. Fundam., 15, p. 59 (1976).

2.

Prausnitz, J. M., Lichtenthaler, R. N. and Gomes de Azevedo, E., Molecular Thermodynamics of Fluid Phase Equilibria, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ (1986).

3.

Kwak, T .. Y. and Mansoori, G. A., Van der Waals mixing rules for cubic equations of state. Application for supercritical fluid extraction modelling, Chem. eng. Sci., 41, p. 1303 (1986).

4.

Leland, T. W. and Chappelear, P. S., Recent developments in the theory of fluid mixtures, Ind. Eng. Chem., 60, p. 15 (1968a).

5.

Leland, T. W., Rowlinson, J. S., Sather, G. A. and Watson, I. D., Statistical thermodynamics of mixtures of molecules of different sizes, Trans. Faraday Soc., 64, p. 1447 (1968b).

Spring 2001

Irtl11Uz" JounuU ofSciellce & Technology, Volume 25, Numbet' B2

290

6.

Leland, T. W., Rowlinson, J. S., Sather, O. A. and Watson, I. D., Statistical thermodynamics of two-fluid models of mixtures, Trans. Faraday Soc., 65, p. 2034 (1969).

7.

Redlich, O. and Kwong, J. N. S., On the thermodynamics of solutions. V. An equation of state: fugacities of" gaseous solutions, Chern. Rev., 44, p. 233 (1949).

8.

Soave, G. S. Equilibrium constants from a modified Redlich-Kwong equation of state, Chern. Eng. Sci., 27, p. 1197 (1972).

9.

Mohsen-Nia, M., Modarress, H. and Mansoori, G. A., A cubic equation of state based on a simplified hardcore model, Chern. Eng. Cornrn., 131, p. 15 (1995).

APPENDIX A

The detailed derivation ofEqs. (26) to (31) is explained as follows: Derivation ofEqs. (26) and (27) was based on the Soave-Redlich-Kwong equation of state which may be written as: RT

p=

(A-I)

v-bSRJ( where, aSRK and b SRK are as follows: 2

- 0.42747R TJ, P.

Q,"'RK 0>.

~ +m· (1 - ji;) . n

I

J

(A-2)

Ci

where mj is a function of the eccentric factor of the ith component in the mixture, mi

=0.48508 + 1.55171mj -

O.15613ml

(A-3)

and

bi

=0.08664 RTcl

(A-4) i

PCI

I

Similar to the procedure applied for the Redlich-Kwong equation of state, the following equations arc obtained:

)1 -_.JX l (Q SRKI - T GO

Xl (

Uid!Ul1 - U

'"

8a

SRK.lj or dv

(A-5)

v(v+bSRK1 )

Therefore, (A-6)

Which can be written in the following form: xl(uideDl -U)1

Xlln2( mSRK1) =- QSRKI -T---=="-

(A+-7)

mSRK2

(A-8)

bSRKI

OJ'

Similarly, for .component 2 we have, X2(Uideal- u

h

X2ln2( = - - QSRK2 -T---==~·) bSRK2 OJ'

Using similar assumptions which have been applied in the van Laar formulation we get the following result for the excess Gibbs free energy of the 'binary mixture,

lranilln JounuU ofScience &: TecluuJlogy, VoIUlfle 25, NUlflber B2

Sprlng2001

,Jodijktzdon olthe NIl LtIIIr•• •

g

E

=ln2

291

ib SRKI ) -xl ' - + ( aSRK2 -T ib SRK2 ) -X2( aSRKI" -T iJ]' b SRKI iJ]' bSRK2

(A-9)

, CU(SRK)mix ) - ( a(SRK)mix - T iJ]' / b(SRK)mix

Now using this equatien and the following definitions for various parameters, Eqs. (26) and (27) may readily be obtained.

=T cu SRKI

(A-IO)

=T cuSRK2

(A-ll)

MAlT

iJ]'

MA2T

iJ]'

and (A-12) For derivation ofEqs. (28) and (29) , first we write the Peng-Robinson equation of state, P=

RT v - b PR

(}PR

(A-13)

2 v 2 + 2b PR v -. b PR

wliere, (A-14) and (A-IS) _ O.07780RTc bPR Pc

(A-16)

If we repeat the same procedure as applied to the Redich-Kwong equation of state above, the following result will be obtained: E

iJ(}PRI

xliJ(}PR2

iJtJ

g =O.6232[«(}PRI-T--)-+«(}PR2 -T--)-«(}mix - T - -mix )] iJ]'

iJ]'

b PRI

iJ]'

(A-I 7)

Using this equation as well as the following definitions, Eqs. (28) and (29) are obtained, (}mix .

=LLXjX/)PRj(}PRj j -

(A-I 8)

j

and MTETI =T iJ(}PRI

(A-19)

=T

(A-20)

iJ]'

MTET2

iJ(}PR2 iJ]'

(A-21) The last equation which we~pplied was the MMM equation of state which can be written as follows: Spring 2001

lnutitul Jolllfflll olScknce & Teclutology, Volume 25, Number.B 2

292

Go R. VIlkilHle:JuuuJ/etaL

Z = v+0.62bMMM _ v - 0.47bMMM

aMMMv 1I2 RT[T V(V + 0.47bMMM )]

(A-22)

where, aMMM and b MMM parameters are as follows: aMMM

=1.46243RTp2vC

(A-23)

bMMM

=0.41274vc

(A-24)

and This equation of state can be written for the mixtures in the following form:

Zm =

v + 0.62b(MMM)mbe v- 0.47b(MMM)mbe

-

(L LXix jaij(MMM»v I RT [T 1I2 v(v + 0.47Lx b(MMM»]

-:-:::----:.~~---''----

(A-25)

j

where, (A-26) where the combining rules for the volui;ne and energy parameters of this equation of state have the same definitions as Eqs. (12 ) and (13 ). Therefore, if the same procedure as in the above is used, we obtain the following result: a(MMM)mbe]

(A-27)

b(MMM)mbe

Using this equation, as well as the following definitions, Eqs. (30) and (31). can readily be obtained. (A-28) and (A-29) (A-30) Finally, the parameters M1,M 2 and M3 which exist in Eqs. (24) and (25) are as follows: (A-31)

M2

=

Iranian Journal o/Science & Technology, Volume 25, Number B2

(A-32)

Spring 2001