Fugacity of a component in a mixture
Solution theories and applications
1

2

Be able to:
• Understand the difference between ideal and non-ideal mixtures;
• Understand the concepts of excess properties and activity coefficients;
• Compute fugacity coefficients in vapor and liquid mixtures;
• Compute correlative and predictive activity coefficients.
3

Ideal gas mixtures
Ideal gas mixtures are characterized by:
PV
IGM 
NRT
 j
C 

1
N j



RT

,
1
,..., N
C
  j
C 

1
N U j j
 
4

Partial molar properties in ideal gas mixtures
Partial molar volume and partial molar internal energy:
V
IGM 


 j
C 

1
N j



RT
P

V i
IGM 
RT
P
U i
IGM 
U i
5

Partial pressure
In ideal and non-ideal gas mixtures, the partial pressure is
defined as:
P i
 x P i
Note: the partial pressure is NOT a partial molar property.
P i
IGM  x P i
 j
C 

1
N i
N j




 j
C 

1
N j



RT
V

 
N RT i
V
6

Partial molar properties in ideal gas mixtures
Forming a binary ideal gas mixture at selected conditions:
T, P, N
1
T, P, N
2
T, P, (N
1
+ N
2
)
V
IGM 

N
1

N
P
2

RT

N RT
1
P

N RT
2
P
 
1
V
2
7

Partial molar properties in ideal gas mixtures
There are not heat effects (constant temperature and noninteracting molecules in an ideal gas). The difference in entropy when forming the ideal gas mixtures comes from that the molecules of each gas can now occupy the whole volume:
S
1
IGM 
S
1
IG 
R
IGM ln
V
V
1
IG
R ln

N
1

N
2
 RT
P
RT
N
1
P


R ln
1 x
1
 
R ln x
1
8

Partial molar properties in ideal gas mixtures
For mixtures with any number of components:
S
IGM j

S
IG j
 
R ln x j
Then:
 mix
S
IGM  j
C 

1
 j
IGM j

S
IG j

 
R x ln x j
C 

1 j j
9

Partial molar properties in ideal gas mixtures
Summary:
10

Partial molar Gibbs energy and fugacity
The fugacity of a pure substance was defined in Chapter 7 as: f
 
P

P exp


G
R

,
 

RT

P exp



,
 
G
IG

,
 

RT
The fugacity of a component in a mixture is now defined as : f i
 x i
 i
P
 x P i exp


G i
R 
, ,

RT

  x P i exp

 i

, ,
 
G i
IGM 
, ,
 

RT
11

Partial molar Gibbs energy and fugacity
The fugacity coefficient of a component in a mixture is:
 i
 f i
 x P i exp


G i
R 
, ,

RT

  exp

 i

, ,
 
G i
IGM 
, ,
 

RT
In practice, to compute the fugacity coefficient, you need an equation of state. From the EOS you can have an expression for the departure (residual) Gibbs energy.
12

Partial molar Gibbs energy and fugacity
It then follows that: i

, ,
 
G i
IGM 
, ,
 
RT ln
 i
But:
G i
IGM 
, ,
 
G i
IG

,
 
RT ln x i
And:
G i
IG

,
 
G i
IG

,
0
 
RT ln
P
P
0
13

Partial molar Gibbs energy and fugacity
The overall result is: i

, ,
 
G i
IG

,
0
 
RT ln
P
P
0

RT ln x RT i
 ln
 i i

, ,
 
G i
IG

,
0
 
RT ln x i
 i
P
P
0

G i
IG

,
0
 
RT ln f i
P
0
14

Phase equilibrium criterion
From chapter 8, the phase equilibrium criterion for mixtures in a two-phase system is:
G i
I 
G i
II i

1,..., C
G i
IG

,
0
 
RT ln f i
I
P
0

G i
IG

,
0
 
RT ln f i
II
P
0 f i
I  f i
II
Isofugacity criterion
15

Phase equilibrium criterion
Equivalent forms of writing this criterion are:
G i
I 
G i
II i

1,..., C
G i
IG

,
0
 
RT ln i
I
 I x P i
P
0

G i
IG

,
0
 
RT ln i
II
 II x P i
P
0 ln x i
I x i
II ln x i
II
 i
 x i
I
 i

0
16

• Model the vapor phase as a mixture of ideal gases: f
ˆ i v 
Py i
• Model the liquid phase as an ideal solution f
ˆ i l 
P i sat x i
17

Py
1
Py
2

P
1 sat x
1

P
2 sat x
2
18

• Antoine equations for saturation pressures: ln P
1 sat
/ kPa

14 .
2724

2 , 945
T / o
C

.
47
224 ln P
2 sat
/ kPa

14 .
2043

T
2 , 972 .
64
/ o
C

209
Calculate P vs. x
1 and P vs. y
1 at 75 o C
19

66.72
Bubble line
Dew line
Diagram is at constant T
0.75
20

Knowing T and x
1
, calculate P and y
1
Py
1

Py
2
P
1 sat x
1

P
2 sat x
2
Summing :
P

P
1 sat x
1

P
2 sat
( 1
 x
1
)

( P
1 sat 
P
2 sat
) x
1

P
2 sat
Bubble pressure calculations y y
2
1
 x
1
P
1 sat

P
1
 y
1
21

0.43
Diagram is at constant T
59.74
22

T
y
1
P
x
1
Py
1

P
1 sat x
1

Py
1
P
1 sat
 x
1
Py
2

P
2 sat x
2

Py
2
P
2 sat
 x
2 summing
P

1 y
1
P
1 sat
 y
2
P
2 sat
Dew point calculation
23

78 o C
0.51
0.67
In this diagram, the pressure is constant
24

1
1
Py
1

Py
2

P
1 sat
( T ) x
1
P
2 sat
( T ) x
2
(1)
(2)
Given P and y
1 for T and x
1 solve
T i sat 
A i

B i ln P

C i
Why is this temperature a reasonable guess?
get the two saturation temperatures
Then select a temperature from the range between
T
1 sat and T
2 sat
At the selected T, summing (1) and (2) solve for x
1 25

1
1
Py
1
Py
2
P


P
1 sat x
1

P
2 sat x
2
P
1 sat x
1

P
2 sat x
2
P
P
2 sat

P
1 sat
P
2 sat x
1
 x
2

P
2 sat 
P
P
1 sat
P
2 sat x
1
 x
2
26

1 ln P
1 sat
/ kPa

14 .
2724

2 , 945
T / o
C

.
47
224 ln P
2 sat
/ kPa

14 .
2043

2 , 972
T / o
C

.
64
209
(I) ln
P
1 sat
P
2 sat

0 .
0681

2 , 945 .
47
T

224

2 , 972 .
64
T

209
P
2 sat 
P
P
1 sat
P
2 sat x
1
 x
2
(III)
(II)
Estimate P
1 sat /P
2 sat using a guess T
Then calculate P
2 sat from (III)
Then get T from (I)
Compare calculated T with guessed T
Finally, y
1
= P
1 sat x
1
/P and y
2
= 1-y
2
27

Bubble points
78 o C
Dew points
76.4
In this diagram, the pressure is constant
0.51
0.75
28

• Start from point c last slide (70 kPa and y
1
= 0.6)
Py
1

P
1 sat x
1
 x
1

Py
1
P
1 sat
Py
1

2

P



P
2 sat
P
1 y
1 sat x
2

 x
2

Py
2
P
2 sat y
2
P
2 sat 



P
1 sat 
P


 y
1
 y
2
P
1 sat
P
2 sat 


29

ln P
1 sat / kPa

14 .
2724

2 , 945
T / o C

.
47
224 ln P
2 sat
/ kPa

14 .
2043

2 , 972
T / o C

.
64
209
(I) ln
P
1 sat
P
2 sat

0 .
0681

2 , 945 .
47
T

224

2 , 972 .
64
T

209
P
1 sat 
P

 y
1
 y
2
P
1 sat
P
2 sat

 (III)
(II)
Estimate P
1 sat /P
2 sat using a guess T
Then calculate P
1 sat from (III)
And then get T from (I) x
1
= Py
1
/P
1 sat
30

79.6
0.44
31

K i
= y i
/x i
K i
= P i sat /P
32

Read
Examples
10.4, 10.5, 10.6
33

T and P
1 mol of
L-V mixture overall composition {z i
}
V, {y i
} mass balance:
L + V =1 mass balance component i z i
= x i
L + y i
V for i = 1, 2, …n
L, {x i
} z i
= x i
(1-V) + y i
V
Using K i values, K i
= y i
/x i x i
= y i
/K i
; y i
= z i
K i
/[1 + V(K i
-1)] read and work examples 10.5 and 10.6
SUM {y i
} =SUM{ z i
K i
/[1 + V(K i
-1)]}
34

F=2p
+N
For a binary
F=4p
For one phase:
P, T, x (or y)
Subcooled-liquid above the upper surface
Superheated-vapor below the under surface
L is a bubble point
W is a dew point
LV is a tie-line
Line of critical points
35

36

Each interior loop represents the PT behavior of a mixture of fixed composition
In a pure component, the bubble and dew lines coincide
What happens at points A and B?
Critical point of a mixture is the point where the nose of a loop is tangent to the envelope curve
Tc and Pc are functions of composition, and do not necessarily coincide with the highest T and P
37

At the left of C, reduction of P leads to vaporization
At F, reduction in P leads to condensation and then vaporization (retrograde condensation )
Important in the operation of deep natural-gas wells
At constant pressure, retrograde vaporization may occur
Fraction of the overall system that is liquid
38

39

Minimum and maximum of the more volatile species obtainable by distillation at this pressure
(these are mixture CPs)
40

41

azeotrope
This is a mixture of very dissimilar components
42

The P-x curve in (a) lies below
Raoult’s law; in this case there are stronger intermolecular attractions between unlike than between like molecular pairs
This behavior may result in a minimum point as in (b), where x
1
=y
1
Is called an azeotrope
The Px curve in (c) lies above Raoult’s law; in this case there are weaker intermolecular attractions between unlike than between like molecular pairs; it could end as L-L immiscibility
This behavior may result in a maximum point as in (d), where x
1
=y
1
, it is also an azeotrope
43

Usually distillation is carried out at constant P
Minimum-P azeotrope is a maximum-T (maximum boiling)
Point (case b)
Maximum-P azeotrope is a minimum-T (minimum boiling)
Point (case d)
44

45

When a component critical temperature is < T, the saturation pressure is not defined.
Example: air + liquid water; what is in the vapor phase?
And in the liquid?
Calculate the mole fraction of air in water at 25 o C and 1 atm
T c air << 25 o C
46

For a species present at infinite dilution in the liquid phase,
The partial pressure of that species in the vapor phase is directly proportional to the liquid mole fraction y
 i
P x i
H i
Henry’s constant
47

Calculate the mole fraction of air in water at 25 o C and 1 atm.
First calculate y
2
(for water, assuming that air does not dissolve in water)
Then calculate x
1
(for air, applying Henry’s law)
48

Fugacity vapor
Py
1
Py
2


P
1 sat

1 x
1
P
2 sat

2 x
2
Fugacity liquid
 is the activity coefficient, a function of composition and temperature
It corrects for non-idealities in the Liquid phase
49

the more energetic molecules have enough energy to overcome the intermolecular attractions and escape from the surface to form a vapor.
The smaller the intermolecular forces, the more molecules will be able to escape at any particular temperature.
The same happens for another liquid
50

The trend to escape is the same for both liquids.
That means that the intermolecular forces between two red (or two blue) molecules must be exactly the same as the intermolecular forces between a red and a blue molecule.
51

This is why mixtures like heptane and iso-heptane get close to ideal behavior.
They are similarly sized molecules and similar chemical structure and so have similar van der Waals attractions between them. However, they obviously aren't identical - and so although they get close to being ideal, they aren't actually ideal.
52

Ideal mixtures
The concept of ideal mixture, as the name implies, is an idealization that approximates the behavior of mixtures formed by components whose molecules are similar in size, shape, and intermolecular interactions .
Example: mixtures of n-heptane and iso-heptane
Beyond this physical interpretation, there is a mathematical definition and several consequences that derive from it.
53

When you make any mixture of liquids, you have to break the existing intermolecular attractions (which needs energy), and then remake new ones (which releases energy).
If all these attractions are the same, there won't be any heat either evolved or absorbed.
That means that an ideal mixture of two liquids will have zero enthalpy change of mixing . If the temperature rises or falls when you mix the two liquids, then the mixture isn't ideal.
54

Ideal mixtures
Ideal mixtures can be liquids or gaseous. Mathematically, the following properties define an ideal mixture:
H i
IM 
, ,
  i

,

V i
IM 
, ,
  i

,

From this definition, it follows that:
 mix
H
IM

, ,
  
 mix
V
IM

, ,
  
 i i
IM 
, ,
  i

,



0
 i i
IM 
, ,
  i

,



0
55

Ideal mixtures
Consider the fugacity of a component in an ideal mixture and of a pure component at the same temperature and pressure: f i
IM 
, ,
  x i
 i
IM   i
    i
 
Solving for P from each equation, it results: f i
IM 
, ,
  i i

,

 i
IM
 i
 
 
56

Ideal mixtures
Using the definition of fugacity coefficients from previous slides:
 i
IM
 i
 
 
 exp


G i
IM 
, ,
 
G i
IGM 
, ,

RT

 exp

 i

,
 
G i
IG

,
 

RT
57

Ideal mixtures ln


 i
IM
 i
 
 
 
 
G i
IM 
, ,
 
G i
IGM 
, ,

RT

 

 i

,
 
G i
IG

,
 

RT

P
0 ln


 i
IM
 i
 
 

P

  


0

V i
IM 
RT
, ,




V i

RT
,


1
P

 dP

1
P

 dP


P
0


V i
IM 
RT
, ,


V i

RT
,
 
 dP
58

Ideal mixtures
But, for an ideal mixture:
V i
IM 
, ,
  i

,

It follows that: ln


 IM i
 i
 
 

P

   

0

V i
IM 
RT
, ,


V i

RT
,
 
 dP

0
1 f i
IM 
, ,
  i i

,

 i
IM
 i
 
 
 i i
 
59

Ideal mixtures
Then, in an ideal mixture: f i
IM 
, ,
  i i

,

60

Vapor-liquid equilibrium between ideal phases
With these assumptions:
For an ideal gas mixture: f i
V

, ,

 y f i i
IG

,
  y P i
For an ideal liquid mixture: f , , x f , i
L    i i
L
 
61

Vapor-liquid equilibrium between ideal phases
From Chapter 7, the fugacity of a pure liquid is: f
 f sat exp


P

P sat
V
RT dP

   sat sat
P exp


P

P sat
V
RT dP


Neglecting the fugacity coefficient at saturation and the
Poynting correction : f , , x f , x P T i
L    i i
L
   i i vap
 
62

Vapor-liquid equilibrium between ideal phases
For an ideal gas mixture: f i
V

, ,

 y f i i
IG

,
  y P i
For an ideal liquid mixture: f , , x P T i
L    i i vap
 
63

Vapor-liquid equilibrium between ideal phases
VLE: f i
L 
, ,
  x P i i vap
   f
V i

, ,

 y P i x P i i vap
   y P i
Known as Raoult’s law
64

Ideal mixtures
Summary of the relationships for ideal mixtures (please refer to the book for the proofs):
U
IM

, ,
  i
C 

1 i i
 
H
IM

, ,
  i
C 

1 i i
 
V
IM

, ,
  i
C 

1 i i
 
65

Ideal mixtures
Summary of the relationships for ideal mixtures (please refer to the book for the proofs):
S
IM

, ,
  i
C 

1 i i

,
 
R i
C 

1 x i ln x i
A
IM

, ,
  i
C 

1 i i

,
 
RT i
C 

1 x i ln x i
G
IM

, ,
  i
C 

1 i i

,
 
RT i
C 

1 x i ln x i
66

Excess mixing properties
An excess mixing property is the difference between the property of the real mixture and that of the ideal mixture, both of same temperature, pressure, and composition.
 ex
        IM
 
 ex
  


    i
C 

1 x i
 i

,






 IM
   i
C 

1 x i
 i

,



 ex
    mix
     mix
 IM
 
67

Excess mixing properties
An excess mixing property is the difference between the property of the real mixture and that of the ideal mixture, both of same temperature, pressure, and composition.
 ex
    mix
     mix
 IM
 
 ex
   i
C 

1 x i
 i

, ,
  i
C 

1 x i
 i
IM  
 ex
   i
C 

1 x i

 i
    i
IM  

68

Excess mixing properties
An excess mixing property is the difference between the property of the real mixture and that of the ideal mixture, both of same temperature, pressure, and composition.
 ex
   i
C 

1 x i

 i
    i
IM  

 i ex     i
    i
IM  
G i ex 
, ,
  i

, ,
 
G i
IM 
, ,

69

Excess mixing properties and activity coefficients
Define the activity coefficient of component i as:
RT ln
  i
G i ex 
, ,
  i

, ,
 
G i
IM 
, ,

Then: i

, ,
 
G i
IM 
, ,
 
RT ln
 i
But:
G i
IM 
, ,
  i

,
 
RT ln x i
70

Excess mixing properties and activity coefficients
And: i

,
 
G i
IG

,
0
 
RT ln i
 
P
0
Then, collecting all the terms: i

, ,
 
G i
IG

,
0
 
RT ln i
 

RT ln x RT i

P
0 ln
 i i

, ,
 
G i
IG

,
0
 
RT ln x
 i i i

,

P
0
71

Excess mixing properties and activity coefficients
The fugacity of component i in the mixture is: f i

, ,
  x
 i i i

,

Note: the activity coefficient accounts from deviations from ideal mixture behavior.
72

Excess mixing properties and activity coefficients
To obtain an expression for the activity coefficient of a certain species, you need an expression for the excess mixing Gibbs energy: ln
 i





 ex
N G RT

N i



Several expressions (models) exist.
73

Example
In a binary mixture, the excess Gibbs energy of mixing is given
G

Ax x
1 2 activity coefficients of components 1 and 2.
74

Example 7
In a binary mixture, the excess Gibbs energy of mixing is given
G

Ax x
1 2 activity coefficients of components 1 and 2.
Solution: ln

1





NG ex
  

N
1

 ln

1

A
RT





N
N N
1 2

1
N

N
1
2




2

 


NAx x
1 2
  

N
1



, ,
2

AN
2
RT


N
1

N
2
 
N
1

N
1

N
2

2

2
2
AN
2

1

N
2

2

A
RT x
2
2
75

Example 7
Note that: ln

1





NG ex
  

N
1



, ,
2





G ex
  
 x
1


76

For component 2, the procedure is analogous, leading to: ln

2





NG ex
  

N
2


, ,
1

 


NAx x
1 2
  

N
2



, ,
1 ln
 
2
A
RT x
1
2
These are the simplest formulas for activity coefficients, but generally give poor description of liquid phase behavior.
77

Benzene (1) +2,2,4-trimethyl pentane at 55 o C.
78

Activity Coefficient Models
Expressions for activity coefficients are obtained from expressions for the molar excess Gibbs energy of mixing using the steps outlined in the previous example.
The molar excess Gibbs of energy of mixing can show very diverse behavior depending on the liquid mixture and its conditions of temperature and composition.
79

Activity Coefficient Models
Trimethyl methane (1) + benzene (2) at 100 o C
Trimethyl methane (1) + carbon tetrachloride (2) at 0 o C methane (1) + propane (2) at 100 K
Water (1) + hydrogen peroxide (2) at 75 o C
80

Activity Coefficient Models
Redlich-Kister expansion:
G ex  x x
1 2
  
1
 x
2
  
1
 x
2

2 
...

For A and B different from zero with C, D, and other parameters equal to zero: ln

1
  x
1 2
2   x
3
1 2 ln

2
  x
2 1
2   x
2 1
3
 i
A 3
  i

1
B
 i

4
  i
B
81

Activity Coefficient Models
Van Laar equations:
G ex
RT

2 a x q x q
12 1 1 2 2 x q
1 1
 x q
2 2 q q
1 2
: size parameters a
12
: molecular interaction between unlike molecules ln

1



1


 x
1 x
2


2
 i
A ln

2


1



 x
2 x
1


2
3
  i

1
B
 i

4
  i
B
82

Activity Coefficient Models
Flory-Huggins model (for molecules very different in size, as in solvent+polymer solutions):
G ex
RT


 x
1 ln

1 x
1
 x
2 ln
 x
2
2


   x
1
 mx
  
2 1 2 ln

1
 ln

1 x
1
 
1 x v
1 1 x v
1 1
 x v
2 2


1 v v
1 2
1 m 

2
 
2
2 ln

2
 ln

2 x
2
  m

1
 
1
 m

1
2
 
2 x v
2 2 x v
1 1
 x v
2 2 m
 v
2 v
1

: molecular interaction between unlike molecules
83

• There are cases where the cross-parameter may be a function of composition.
A
12
= A
12
(x)
So, there could be “local” compositions different than the overall “bulk” compositions. For example (if coordination number is 8)
AAAAAAA
AABBAAA
AAAAAAA x
AB
= ; x
BB
=
“A around B” or “B around B”
84

• Specific interactions such as H-bonding and polarity
85

•
• x
21
• x
11
• x
11 x
• x
12
22
• x
22
= mole fraction of “2” around “1”
= mole fraction of “1” around “1”
+ x
+ x
21
12
=1
=1
112211
= mole fraction of “1” around “2”
= mole fraction of “2” around “2”
111111
111111
• Local compositions are related to overall compositions: x
21 x
11
 x
2 x
1

21
; x
12 x
22
 x x
2
1

12
If the weighting functions are =1 random solutions
86

 ij
x
11
 x
21

1 and x
21
 x
2

21
 x
21
 x
11 x
11 x
1 x
11


1
 x x
21
12

 x x
2
1

21 x x
2
1 x
2

 x
2
21

21 x
1

12
 x
1

12



1 x
2 x
1

21
If
 ij
=1 => random mixture
87

• Wilson assumes that the weighting functions are functions of size and energetic interactions:
 ij
  ij

V j
V i exp



N
A z (
 ij
2 RT
  jj
)



V j
V i exp
 
 a
RT ij


 ii
 ij
  jj
  ji

1 z is the coordination number for atom i even if
 ij
=
 ji
(this is not always the case), the
 ij may be different, why?
parameters
88

U ij
 ij
89

G
E
  x
1 ln( x
1
 x
2

12
)
 x
2 ln( x
2
 x
1

21
)
RT ln

1
  ln( x
1
 x
2

12
)
 x
2

 x
1


12 x
2

12
 x
2


21 x
1

21

 ln

2
  ln( x
2
 x
1

21
)
 x
1

 x
1


12 x
2

12
 x
2


21 x
1

21


For infinite dilution:
90

G
E x
1 x
2
RT
 x
1
G
21


21 x
2
G
21

G
12
 x
1
G
12

12 x
2 ln

12
G
12

1



(
 x
2
2
12 exp(




 
21


22
) x
1
/
 
12

G
21 x
2
G
21


2

(
G
12
 x
1
G
12

12 x
2
RT
); G
21
 b
12

/ RT exp(
;

 
21

21
)
(

21
)
2





11
) / RT
 b
21
/ RT
Actual parameters:

, b
12 and b
21
See Table 12.5, next slide
91
Renon and Prausnitz, 1968

92

• UNI versal QUA si C hemical model (Abrams and
Prausnitz, AIChE J. 21:116 (1975)
 ij
 q q j i exp



N
A z (
 ij
2 RT
  jj
)


 q i q j exp
 
 a
RT ij


 q i q j
 ij
Uses surface areas (q i
) to represent shapes q i is proportional to the surface area of i z is the coordination number
93

• coordination number, z = 10
• q j accounts for shape, r j accounts for size
Energetic parameters


G
E
RT

 residual
  x
1 q
1 ln(

1
 
2

21
)
 x
2 q
2 ln(

1

12
 
2
)

1
 x
1 q
1 x
1 q
1
 x
2 q
2
 ji
=exp-(
 ji
-
 ii
)/RT= exp [(-a ji
)/RT]


G
E
RT

 combinator ial
 x
1 ln
 
 x
1
1


 x
2 ln
 
 x
2
2



5


 q
1 x
1 ln



1

1


 q
2 x
2 ln




2
2






1
 x
1 r
1 x
1
 r
1 x
2 r
2
Pure species molecular parameters (in tables): r
1
, r
2
, q
1
, q
2 r i are molecular size parameters relative to –CH
2
94

ln
 k
 ln
 k comb  ln
 k residual ln
 k comb 
1

 k x k
 ln
 k x k

5 q k


 ln


 
 k k







1


 k k ln
 k residual  q k





1
 ln



 i
 i
 ik 


   i
 j

 i
 kj ij











95

96

97

UNI
F
A
C
• The solution is made of molecular fragments
(subgroups)
• New variables (R k and Q k
)
• Combinatorial part is the same as UNIQUAC ln
 k comb 
1

 k x k
 ln
 k x k

5 q k

 ln
 

 k k




 1


 k k



 where
 k and
 k are the volume fractions and surface fractions
98

99

ln
 i
R i identify species r i
  k k

 q i k
( i )



1
R k
 k
( i )
Q k k (
 subgroups )

# of subgroups k in molecule i q i
  


 k
 ik s k
 k ik


 e ki ln
 s k ik

 m i
 j e mi
 x i x j q i q e ki mk j





Be careful, this
 is different than the surface fraction !!
e ki
 mk

 k
( i )
Q k q i
 exp

s k a mk
T
  m
 m
 mk
100

101

Activity Coefficient Models
Wilson model (local composition; expandable to any number of components):
102

Activity Coefficient Models
NRTL model (non-random two-liquid) (local composition; expandable to any number of components):
103

Activity Coefficient Models
UNIQUAC model (universal quasi-chemical) (local composition; expandable to any number of components):
104

Activity Coefficient Models
UNIQUAC model (universal quasi-chemical) (local composition; expandable to any number of components):
105

Activity Coefficient Models
A common feature of the models presented in the previous slides is the need for experimental data to fit the model parameters to represent a system of interest. They are correlation-based models.
A few models are predictive , i.e., they predict activity coefficients in the absence of experimental data for the system of interest.
106

q q
1 2
: size parameters
G ex
RT

2 a x q x q
12 1 1 2 2 x q
1 1
 x q
2 2 ln

1



1


 x
1 x
2


2
 i
A ln

2


1



 x
2 x
1


2
3
  i

1
B
 i

4
  i
B
107

• Species have similar sizes and interaction energies
V ex 
0 S ex 
0
G ex 
U ex 
T S ex 
PV ex 
U ex
• How to calculate the excess Gibbs free energy: assume a thermodynamic cycle and Van der
Waals equation is valid for both phases
108

Very low P
Ideal gas
Mix ideal gases (step II)
Ideal gas mixture
Isothermal
Vaporization
Step I
Isothermal
Compression
(liquefaction)
Step III
Pure liquid at P
Liquid mixture
Formation of a liquid mixture from the pure liquids at constant T
109




 
U

V 


T

T


 
P

T



V

P
And using the Van der Waals EOS for both phases we can evaluate

U at each step
110

G ex  
U
 x
1 a
1
V
1
 x
2 a
2
V
2
 a
V mix mix
Because of liquid incompressibility:
G ex  
U
 x
1 a
1 b
1
 x
2 a
2 b
2
 a mix b mix
111

G ex  
U
 x
1 a
1 b
1
 x
2 a
2 b
2
 a mix b mix
We get the Van Laar activity coefficients ln

1



1


 x
1 x
2


2 ln

2


1



 x
2 x
1


2
112

• Hildebrand (1929) found that the properties of iodine solutions in various nonpolar solvents in agreement with Van Laar model.
Hildebrand called these REGULAR solutions
(no excess entropy and no change of volume due to mixing)
• Both Hildebrand and Scatchard working independently improved over the Van Laar model
113

Activity Coefficient Models
Regular solution model (Scatchard-Hildebrand)
V ex 
0 S ex 
0
G ex 
U ex 
T S ex 
PV ex 
U ex
U ex
Goes beyond the limitations of the use of the Van der Waals EOS
114

c

 vap
U
V l experimental
For the mixture,
Using the concept of dispersion forces where c
12
=(c
11 c
22
) 1/2
 vap
U mixture
 x
1
V
1
 vap
U
1  x
2
V mix
V
2
 vap
U
2
V mix
2
115

Activity Coefficient Models
Regular solution model (practical formulas for a binary mixture) ln

1

V
1
RT
 2
2
  
1 2

2 ln

2

V
2
RT
 2
1
  
1 2

2
 
1
,
2
: volume fractions
 i
 x i
V
V mix i
 
1 2
: solubility parameters (available in tables)
RT ln

1

V
1

2
2


1
 
2

2
RT ln

2
 i

 

V
U i i


1 / 2

V
2

1
2


1
 
2

2
116

• Can give activity coefficients only > 1
• That is positive deviations of Raoult’s law
• Can be applied to certain nonpolar mixtures
• Improvements:
– The Flory-Huggins theory of polymer solutions
– Gonsalves and Leland (1978) modified the equations for mixtures with appreciable differences in size and shape
• HW: Discuss Gonsalves and Leland theory and more recent applications of regular solution theory
117

• Liquid state intermediate between gas and solid
• In a quasicrystalline picture of a liquid, the molecules are arranged in a lattice
• Typical statistical mechanical models
• Nonidealities may arise from:
– attractive forces between unlike molecules(enthalpy of mixing),
– differences in size and shape between unlike molecules
(entropy of mixing)
– Differences in attractive forces between the three different pair of interactions
118

• After mixing, there will be some interchange energy, w
• Excess volume is zero
• Concept of coordination number (z)
• Total number of nearest neighbors = z/2(N
1
+N
2
)=N
11
+N
22
+N
12
• Picture of interchange energy
119

U
1

N
1

11

N
2

22
 w
N
12
2
Interchange energy w
 z



12

1
2


11
 
22
 

120

Q lattice
 
N
12 g ( N
1
, N
2
, N
12
) exp(

U t
/ kT )
121

a) Random distribution
12
122

123

124

125

126

127

128

12
129

12
130

At x1=x2=0.5
131

132

133

134

• The random approximation becomes satisfactory as the exchange energy w between pair of molecules becomes small relative to the thermal energy (kT)
• For a given mixture, randomness increases with temperature.
• At fixed T, randomness increases as the interchange energy w falls.
• The excess entropy is never > 0, thus the entropy of mixing is maximum for the random mixture
135

• The excess G and excess H can be either positive or negative, depending on the sign of w
• When w/zkT is not very large (totally miscible mixtures), G ex calculated by the random or nonrandom approximations do not differ much.
• However when w/zkT is large enough to induce limited miscibility of the two components, deviations from random mixing can be significant.
136

137

138

139

140

• Corresponding states theory applied to a mixture:
– One fluid theory: mixture is assumed to be a hypothetical fluid with molecular size and potential energy comparable with the average of the mixture components.
– m-fluid theory: (example NRTL, non-random two liquid) local composition ideas
141

Liquid 1 (molecule 1 at the center)
Liquid 2 (molecule 2 at the center)
Fluid 1 has “cells” of type 1; fluid 2 has “cells” of type 2
M mix
= x
1
M (1) + x
2
M (2)
M is an extensive configurational property
Using these assumptions we can get to UNIQUAC derivation, see Maurer and Prausnitz,
Fluid Phase Equilibria, 2, 91 (1978)
142

P
 kT
 ln Q

V
T , V , N 1 , N 2
For a simple pure fluid, Vera and Prausnitz (1972) proposed:
143

144

145

It could be assumed that q(rot, vib) = q ext
(V) q int
(T) density dependent T-dependent how to include the effect of density
1. a large rigid molecule with r segments, bond lengths, bond angles and torsional angles are fixed – 3 translational DOF; 2 (linear) or 3 (nonlinear) rotational DOF; total= 5or 6
2. a large flexible molecule with r segments (no restrictions in bond lengths and angles)
3r DOF (each segment has 3) real molecule will be intermediate between these limits
146

• introduce a parameter c, such that 1 < c < r
• for a small molecule, c =1
• for more complex molecules, c >1 for example, for n-decane c=2.7
therefore for isomers of decane, c < 2.7
(branched paraffin less flexible)
147

• Donohue (1978)
148

• Van der Waals assumption V f
= V- N b
/N
A has serious problems.
Percus Yevick theory (1958) developed an “integral equation” theory based on molecular structure and pair (and higher order) correlation functions. This theory and the development of molecular simulations improved the description of the free volume.
Carnahan and Starling (1969)
149

• perturbation theory
• z = PV/RT = z ref
+ z pert
• Reference fluid:
– hard spheres (each sphere moves independent from each other)
– hard spheres chains (each segment is connected to at least one sphere; chain connectivity)
150

• Chapman and Gubbins (1989, 1990)
A Res (T, V, N) = A (T, V, N) – A IG (T, V, N) = A
HS
A dispersion
+ A chain
+ A association/solvation
+
A
HS short-range repulsions
A dispersion long-range dispersions
A chain chemically stable chains
A association/solvation example H-bonding
151

• A repulsion
• (Huang and Radosz, 1990)
• A chain
152

• A dispersion
• A association
(Wertheim’s association theory)
– # association sites unlimited, but needs to be specified
– Location of association sites not specified
– There could be steric hindrance
153

VLE for propanol-n-heptane at 323 K (Fu and Sandler, 1995)
154

VLE for CO
2
/2-propanol at two temperatures
155

• solutions of polymers in liquid solvents
• regular solutions S ex = 0, H ex is described
• for mixtures of components of very different sizes, S ex needs to be described.
– 
H mix
= 0 athermal solutions, similar energetic interactions, different sizes. example polystyrene and toluene
– 
S mix
=

S comb
+

S res
156

• N
1 molecules of type 1: spheres (solvent)
• N
2 molecules of type 2: flexible chains
(polymers)
• each segment (r) has the same size as the solvent molecule
• # lattice sites = N
1
+ r N
2
• fractions of occupied sites:
157

158

dependence on molecular shape q/r = 1 for Flory-Huggins
159

160

Dependence of the activity coefficient of the polymer on the number of segments, r
161

Flory Huggins model including the interaction energy parameter

162

163

164

Activity Coefficient Models
UNIFAC (UNIQUAC functional activity coefficient)
165

Summary: equilibrium criterion
Fugacities are essential for phase and chemical equilibrium calculations. When a component is present in phases I, II, III, IV, etc…, it is valid that: f
I i
 f i
II  f i
III  f i
IV 
...
These phases I, II, III, IV, etc, can vapor, liquid, or solid.
166

Summary: vapor phase fugacities f
V i
 y i
 i
P
Expressions for the fugacity coefficient of component i in the mixture can be derived from equations of state (usually long and cumbersome derivations).
Many EOS exist: the next slide has some recommendations.
167

Summary: vapor phase fugacities
168

Summary: liquid phase fugacities
There are two major paths, using equations of state or excess Gibbs energy of mixing models.
Path 1: equations of state f i
L  x i
 i
P
Expressions for the fugacity coefficient of component i in the mixture can be derived from equations of state (usually long and cumbersome derivations).
Many EOS exist: the next slide has some recommendations.
169

Summary: liquid phase fugacities
170

Summary: liquid phase fugacities
There are two major paths, using equations of state or excess Gibbs energy of mixing models.
Path 2: excess Gibbs energy of mixing models f i
L  x
 i i i

,
 Fugacity of pure liquid
OR
Henry’s constant
Expressions for the activity coefficient of component i in the mixture can be derived from models for the excess Gibbs energy of mixing (usually long and cumbersome derivations).
171

Summary: liquid phase fugacities
There are two major paths, using equations of state or excess Gibbs energy of mixing models.
Path 2: excess Gibbs energy of mixing models f i
L  x
 i i i

,

Fugacity of pure liquid i
    i sat sat
P i exp


P i

P sat
V
RT i dP


Fugacity of pure liquid
OR
Henry’s constant
172

Summary: liquid phase fugacities
173

Summary: liquid phase fugacities
174

Recommendation
Read chapter 9 and review the corresponding examples.
175

Example 1
Use the XSEOS package to evaluate the fugacity of ethane and n-butane in an equimolar mixture at 373.15 K at 1, 10, and 15 bar with the Peng-Robinson equation of state.
176

Example 2
Use the XSEOS package to compute vapor-liquid equilibrium for various compositions of the mixture of propane and n-butane at an 313.15 K with the Peng-Robinson equation of state. Plot the
Pxy diagram and identify the bubble and dew point curves.
177

Example 3
Use the XSEOS package to compute vapor-liquid equilibrium for various compositions of the mixture of propane and n-butane at
10 bar with the Peng-Robinson equation of state. Plot the Txy diagram and identify the bubble and dew point curves.
178

Example 4
A methane(1) + ethane(2) mixture is to be continuously separated by reverse osmosis at 320 K using a rigid membrane only permeable to methane. On the mixture side of the membrane, the mole fractions are kept constant (x
1
=0.3). The pressure on the pure methane side of the membrane is 2 bar.
Find the minimum pressure to be imposed on the mixture side of the membrane for operating this reverse osmosis setup.
Make your calculations using the with the Peng-Robinson equation of state.
179

Example 5
Assuming Raoult’s law is valid, compute the bubble point pressure of an-hexane(1) + n-heptane(2) mixture with x
1
=0.6 at
373.15 K and the mole fractions in the vapor phase. Compute the vapor pressure of each component using the Antoine equation with parameters from NIST Chemistry Webbook.
180

Example 6
Assuming Raoult’s law is valid, compute the dew point temperature of an-hexane(1) + n-heptane(2) mixture with y
1
=0.6 at 3 bar and the mole fractions in the liquid phase.
Compute the vapor pressure of each component using the
Antoine equation with parameters from NIST Chemistry
Webbook.
181

Example 8
Plot the values of the activity coefficients methyl ethyl ketone and toluene in their liquid binary mixture at 323.15 K as function of composition, as predicted by the Margules 3-suffix formula. Use the XSEOS package.
182

Example 9
Knowing the vapor pressures of methyl ethyl ketone and toluene at 323.15 K are respectively equal to 36.09 kPa and
12.30 kPa, plot the Pxy diagram of the vapor-liquid equilibrium of this mixture as function of composition assuming the vapor phase is an ideal gas mixture and the liquid phase can be described using the Margules 3-suffix formula. Use the XSEOS package.
183

Example 10
Use the Flory-Huggins model in the XSEOS package and vary the size of component 2 to have a sense of its effect on non-ideal solution behavior.
184

Example 11
Fit the binary interaction parameters of the NRTL for the best possible representation of the vapor-liquid equilibrium of a methanol (1) + water (2) mixture at 333.15 K. Data and additional details are available in the “NRTL” Excel file.
185

Example 11
Use the UNIFAC model to compute the activity coefficients of the components of the binary mixture acetone (1) + n-pentane
(2) at 307 K, with a mole fraction of acetone equal to 0.047.
Data and additional details are available in the “UNIFAC” Excel file.
186