A note on the derivation of mixing rules from excess Gibbs energy models

Fluid Phase Equilibria, 25 (1986) 323-327 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

323

Short Communication A NOTE ON THE DERIVATION

OF MIXING !?!lLESFROM EXCESS GIBBS ENERGY MODELS

J0RGEN MOLLED,UP Instituttet Bygning

for Kemiteknik,

229, 2900 Lyngby

Danmarks

Tekniske

Hajskole

(Denmark)

(Received July 8, 1985; accepted in final form November 4,1985) INTRODUCTION

In a paper from 1979 Huron and Vidal suggested a new method for deriving mixing rules for equations

of state from excess Gibbs energy models. The method

rely

(i) that

on the

assumptions:

equation

of state at infinite

from

liquid

a

parameter volume

phase

an excess

pressure equals

activity

coefficient

b equals the volume v at infinite

is

zero.

Redlich-Kwong

To

illustrate

equation

Gibbs

the

energy, calculated

(ii)

model;

pressure;

procedure,

of state. At infinite

from an

an excess Gibbs energy calculated that

the

covolume

and (iii) that the excess

Huron

and

Vidal

chose

the

pressure they found that

a/b = ):xiai/bi - GL /En 2 i %

model

Prausnitz

GL =

x xi(! i

for

the

excess

Gibbs energy they chose the NRTL model

(Renon and

1968)

J

x.E.. Auji I/(;

JJ’

xkEki)

with

Eji = bj exp(- nji~uji/RT)

The only difference local

compositions

introduction

from the classical as

corrected

of the covolume

A modified derived

(3)

below.

form

we make

point pressure

excess

from

energy

the

natural

as parameters

037%3812/86/$03.50

in existing

0

1986

fractions,

ivoids

which

the assumptions

are that the excess

of a few atmospheres

equation

energy from a liquid phase activity

is in the definition leads

of the to

the

b. in eqn. (3).

(1) that

The assumptions

that we at a boiling Gibbs

volume

parameter

of eqn.

NRTL model

of

state

coefficient

activity

model

(i) and

(ii) is

volume is zero and

or less can equate an with

an excess

Gibbs

model. The latter assumption

coefficient

Elsevier Science Publishers B.V.

models are estimated

is at

324 low to moderate Redlich-Kwong

pressures.

equations

As illustrations

we chose the van der Waals and the

of state.

EXCESS GIBBS ENERGY FROM THE EQUATION

OF STATE

The van der Waals model From the van der Waals model for the residual A'(T,v,x)/RT we obtain

= - an(l-b/v)

an expression

GE/RT = - Ln(l-b/v)

At

low

pressure

are independent neglecting

energy

- a(T)/RTv

(41

for the excess Gibbs energy

- a/RTv + x xian(l-bilvi) i

t z xign(vi/vl i

Helmholtz

t Z xiai/RTvi i

+ PvE/RT

it is reasonable of the pressure.

the pressure,

(51 to assume Solving

that the saturated

the van der Waals

liquid

equation

volumes

of state,

we find that

b/v = f(l + (1 - 4RTb/a)') At saturation

pressures

(61 less than a few atmospheres

eqn. (61 is accurate

to

within one per cent. If we define fi = hi/vi

(7)

f = b/v

(81

and fc = (vi/hi - l)/(v/b

- 1)

(g)

we may solve eqn. (5) with respect a/b = I:xi(ai/bi)(fi/f) i

to (a/b) and obtain

- GE/f t RT(s xian(fcbilb))lf i

The last term of eqn. (101 is a Flory-Huggins (1). At the normal reduced

densities

of the mixture

like term which is absent in eqn.

point f is of the order of 0.8 for liquids.

of the pure species

(hi/vi) are equal to the reduced

If the density

(b/v), then fi = f and fc = 1.

The Redlich-Kwong The residual Ar(T,v,xl/RT -

boiling

(10)

model Helmholtz

energy for a Redlich-Kwong

= - ui(l-b/v) - a(Tl/RTb

en(ltb/vl

fluid is (11)

325 From this function

an equation

similar to eqn. (10) can be derived if we define

fi = an(l+bilvil

(12)

f

(13)

= an(l+b/vl

where b/v = $(l - RTb/a + ((l-RTb/a)'

- 4RTb/a131

(141

and fc is defined by eqn. (91. At

the

normal

boiling

point

f

is

of

the

order

of

0.55

for

Redlich-Kwong

liquids. Eqn. meters

(10) gives

the

relation

between

the equation

a and b and a liquid phase excess Gibbs

energy

of state mixture at moderate

para-

saturation

pressure.

NEW MIXING RULE If we substitute

the excess Gibbs energy in eqn. (10) by the Wilson

19641, the NRTL or the UNIQUAC

(Abrams and Prausnitz

Gibbs energy we obtain the following of

the

equation

of

state

(Wilson

1975) model for the excess

mixing rules which permits the calculation

'a-parameter

from

available

activity

coefficient

parameters. Wilson a/b = c xi[(ai/bi)(fi/f) i

f (RT/f)!,n(c x.EW.1 t (RT/f)en(fcbi/b)] j JJ’

(15)

with EYi = (vj/vi) exp(- asji/RT1

(161

NRTL a/b = z xi[(ai/bil(fi/f) i

- (z x.E!.~u.. ,)/((z x ER.lfl + (RT/f)an(fcbilb) 1 k kkl j J Ji Jl

ER = exp(-ajiAujilRT1 ij

(17)

(181

UNIQUAC a/b = L xi[(ai/bil(fi/fl i

t (RT/fl(qitn(ZS.E'!.) - {z/2)qien(ei/$i) j J J'

- en(Qi/xil + en(fcbi/b)l]

(191

d/here ei = xiqi/z x.q. j JJ

(20)

326

$i = Xiri/L

(21)

.r.

X

j

JJ

and Eyi

(22)

= exp(- zAeji/2RT)

The Wilson and the NRTL equations in the UNIQUAC

eqn.

(19). The last two terms in eqn.

and should

equation

have no.combinatorial

term

in principle

the UNIQUAC equation, Guggenheim

derived

consists

cancel.

term. The combinatorial

of two terms, terms three and four in (191 are both Flory-Huggins

type terms

The second part of the combinatorial

term in

the third term in eqn. (191, is not of general validity.

it

for

r-mer. Guggenheim writes:

a binary

mixture

of a monomer

and an open-chain

"By contrast analogous formulae do not exist for mix-

tures of several kinds of r-mers when some of the r-mers contain closed rings" (Guggenheim

1952).

In the lJNIQlJAC/UNIFAC equation it is applied to any kind of

molecules.

DISCUSSION For pure species f is only a weak function of temperature f is also a function

of composition,

but for practical

and for mixtures

applications

f may be

regarded as a constant. The mixing rules of eqns. (15)-(22) are strictly valid only in the tenperature

and pressure

evaluated,

range where

but the mixing

the

activity

rules for the a-parameter

pressures where the activity coefficient The approach ent model model

by Huron and Vidalis

parameters

from

coefficient

an

Gibbs

of

state

energy

model

parameters

are

may be extended to higher

model approach is not applicable. dubious if available activity coeffici-

are used because one thus equate

equation

excess

coefficient

at

infinite

model

at

an excess Gibbs energy

pressure

moderate

with

pressure.

an

activity

Besides,

the

validity of the equation of state at infinite pressure is not ensured. If Agji

however the

in eqn.

the approach because

by fitting

in eqns.

the equation

rules to experimental suggest

fitting

deficiencies

adapted from the

(161, aji and AUji

are determined energy mixing

parameters

(17) and (18) or

parameters

data, the approach

to

available

A

Helmholtz energy

a,b

Equation of state parameters

by Huron and Vidal and

may turn to be equally good, experimental

in the mixing rule and the approximation,

LIST OF SYMBOLS

eji in eqn. (221,

of state model with the excess Gibbs

in this Short Communication

the

activity coefficient models,

data

corrects

321 f

Defined by eqns. (71-(9) and (12)-(13)

G

Gibbs energy

P

Pressure

q

Pure component

R

Gas constant

r

Pure component

T

Temperature

v

Molar volume

X

Mole fraction

z

Coordination

surface parameter

volume parameter

number

Superscripts E

Excess property

r

Residual

propertv

Subscripts Combinatorial

C

i,j,k

Pure species i, j, k etc.

m

At infinite pressure

Greek letters a

Ji

Aeji Ag.. Ji Au.. Ji $ e

NRTL

parameter

UNIQUAC parameter Wilson

parameter

NRTL

parameter

Volume fraction Surface fraction

REFERENCES Abrams, D.S. and Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: A new expression for the excess Gibbs energy of partly or completely mixible systems, AIChE J., 21:116-128. Guggenheim, E.A., 1952. Mixture, Clarendon Press, Oxford, Ch. 10. Huron, M.-J. and Vidal, J., 1979. Mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilibria, 3:255-271 Renon, H. and Prausnitz, J.M., 1968. Local composition in thermodynamic excess functions for liquid mixtures. AIChE J., 14:135-144. Wilson, G.M., 1964. Vapour-liquid equilibrium. XI. A new expression for the excess free energy of mixing. J.Am.Chem.Soc., 86:127-130.