The document discusses various models for excess Gibbs free energy, primarily used in liquid-phase equilibrium calculations by engineers. It covers several empirical models, including Margules, Redlich-Kister, Van Laar, Wilson, and the Universal Quasi Chemical (Uniquac) equation, detailing their applicability and parameter dependencies. These models simplify vast experimental data into manageable equations for thermodynamic simulations, each offering specific advantages for different liquid mixture characteristics.

1. EXCESS GIBBS FREE
ENERGY MODELS
1

2. CONTENT
• EAXCESS GIBBS FREE ENERGY MODELS
• MARGULES EQUATION
• REDLICH-KISTER EQUATION
• VAN LAAR EQUATION
• WILSON AND “NRTL” EQUATION
• UNIversal QUAsi Chemical equation
2

3. Excess Gibbs Energy Models
Practicing engineers find most of the liquid-phase information
needed for equilibrium calculations in the form of excess Gibbs
Energy models. These models:
 reduce vast quantities of experimental data into a few
empirical parameters,
 provide information an equation format that can be used in
thermodynamic simulation packages (Provision)
“Simple” empirical models
 Symmetric, Margule’s, vanLaar
 No fundamental basis but easy to use
 Parameters apply to a given temperature, and the models
usually cannot be extended beyond binary systems.
Local composition models
 Wilsons, NRTL, Uniquac
 Some fundamental basis
 Parameters are temperature dependent, and multi-
component behaviour can be predicted from binary data.
3

4. Margule’s Equations
212121
21
E
xAxA
xRTx
G
+=
4
While the simplest Redlich/Kister-type expansion is the Symmetric
Equation, a more accurate model is the Margule’s expression:
(11.7a)
Note that as x1 goes to zero,
and from L’hopital’s rule we know:
therefore,
and similarly
∞
→
γ= 1
210x
ln
xRTx
G
lim
E
1
12
0x21
E
A
xRTx
G
1
=
→
∞
γ= 112 lnA ∞
γ= 221 lnA

5. Margule’s Equations
212121
21
E
xAxA
xRTx
G
+=
5
If you have Margule’s parameters, the activity coefficients are
easily derived from the excess Gibbs energy expression:
(11.7a)
to yield:
(11.8ab)
These empirical equations are widely used to describe binary
solutions. A knowledge of A12 and A21 at the given T is all we require
to calculate activity coefficients for a given solution composition.
]x)AA(2A[xln 1122112
2
21 −+=γ
]x)AA(2A[xln 2211221
2
12 −+=γ

6. Redlich-Kister equation
The Margules equation adequately represents the behavior of
liquid solutions, the components of which are of similar size, shape
and chemical nature. For several solutions, the free energy is not
symmetric about x1= 0.5 the activity coefficients do not appear as
mirror images of To behavior solutions, and Kister each other.
represent the of general Redlich expansion an equation where the
parameter A in the equation is replaced with a is given by in (x1-x2)
and introduced additional parameters. The Redlich-Kister equation
The activity coefficients 1, 2 can be obtained form above eq andɣ ɣ
it can be shown that
6
(X)

7. Redlich-Kister equation
7
Substracting eq (2), form eq (1) , we get
(1)
(2)

8. Redlich-Kister equation
8

9. 9
Redlich-Kister equation

10. van Laar Equations
2
/
121
/
21
/
21
/
12
21
E
xAxA
AA
xRTx
G
+
=
10
Another two-parameter excess Gibbs energy model is developed
from an expansion of (RTx1x2)/GE
instead of GE
/RTx1x2. The end
results are:
(1)
for the excess Gibbs energy and:
(2)
(3)
for the activity coefficients.
Note that: as x1→0, lnγ1
∞
→ A’12
and as x2 → 0, lnγ2
∞
→ A’21
2
2
/
21
1
/
12/
121
xA
xA
1Aln
−






+=γ
2
1
/
12
2
/
21/
212
xA
xA
1Aln
−






+=γ

11. van Laar Equations
11

12. Wilson’s Equations for Binary Solution Activity
)xxln(x)xxln(x
RT
G
2112212211
E
Λ+−Λ+=
12
A versatile and reasonably accurate model of excess Gibbs Energy
was developed by Wilson in 1964. For a binary system, GE
is
provided by:
(1)
where
(2)
Vi is the molar volume at T of the pure component i.
aij is determined from experimental data.



−
=Λ


−
=Λ
RT
a
exp
V
V
RT
a
exp
V
V 21
2
1
21
12
1
2
12

13. Wilson’s Equations for Binary Solution Activity






Λ+
Λ
−
Λ+
Λ
+Λ+−=γ
2112
21
1221
12
212211
xxxx
x)xxln(ln
13
Activity coefficients are derived from the excess Gibbs energy using the
definition of a partial molar property:
When applied to equation 11.16, we obtain:
(3)
(4)






Λ+
Λ
−
Λ+
Λ
−Λ+−=γ
2112
21
1221
12
121122
xxxx
x)xxln(ln
jn,P,Ti
E
E
ii
n
nG
GlnRT
∂
∂
==γ

14. Wilson’s Equations for Binary Solution Activity
14

15. UNIversal QUAsi Chemical (UNIQUACK) equation
The to QUAsi Chemical (UNIQUAC) model was
developed by Abrams and for express the excess Gibbs
free energy of a binary mixture. The UNIQUAC equation
The (gE/RT) contains two parts, a part and a residual
The combinatorial part takes into account the
composition, size and shape of the constituent
molecules and contains pure component properties
only. The residual part takes into account the
intermolecular forces and contains two adjustable
parameters. The UNIQUAC equation is given by
15

16. UNIversal QUAsi Chemical (UNIQUACK) equation
16
(1)
(2)
(3)
(4)
(5)
(6)

17. 17
from eq. (1) – (3) One can obtain the activity coefficients
as

18. UNIversal QUAsi Chemical (UNIQUACK) equation
18
where vk is the number of groups of type in a molecule of component
i. The UNI QUAC contains only two adjustable parameters Ʈ12 and
Ʈ 21. The UNIQUAC equation is applicable to a wide variety of liquid
solutions commonly encountered by chemical engineers. Although
UNIQUAC is mathematically more complex.