Transcript Nonideal Behavior

ITK-234 Termodinamika Teknik Kimia II
Nonideal Behavior
Dicky Dermawan
www.dickydermawan.net78.net
[email protected]
Nonideal Behavior, Outline




Introduction: Effect of Nonideality
Partial Molar Properties
Residual Properties
 Fugacity & Fugacity Coefficient
Excess Properties
 Activity & Activity Coefficient
Intorduction: Effect of Nonideality:
Tetrahydrofuran(1)/Carbon-tetrachloride(2)
30oC
txy diagram
Acetonitril(1)/Nitrom ethane(2)
1 atm
90
@ P = 70 kPa
t, oC
85
80
75
70
65
0
P - xy D ia gra m
A c e t o nit ril( 1) / N it ro m e t a na ( 2 ) @
75 o C
85
80
75
P
70
65
60
55
P-x-y diagram
50
45
40
0
0.2
0.4
0.6
x1, y1
0.8
1
t-x-y diagram
0.2
0.4 0.6
x1, y1
0.8
1
Effect of Nonideality:
Chloroform(1)/Tetrahydrofuran(2)
txy diagram
Acetonitril(1)/Nitrom ethane(2)
90
1 atm
@ P = 70 kPa
85
t, oC
30oC
80
75
70
65
0
P - xy D ia gra m
A c e t o nit ril( 1) / N it ro m e t a na ( 2 ) @
75 o C
85
80
75
P
70
65
60
55
50
45
40
0
0.2
0.4
0.6
x1, y1
0.8
1
P-x-y diagram
t-x-y diagram
0.2
0.4 0.6
x1, y1
0.8
1
Effect of Nonideality:
Furan(1)/Carbontetrachloride(2)
1 atm
90
@ P = 70 kPa
85
t, oC
30oC
txy diagram
Acetonitril(1)/Nitrom ethane(2)
80
75
70
65
0
P - xy D ia gra m
A c e t o nit ril( 1) / N it ro m e t a na ( 2 ) @
75 o C
85
80
75
P
70
65
60
55
P-x-y diagram
50
45
40
0
0.2
0.4
0.6
x1, y1
0.8
1
t-x-y diagram
0.2
0.4 0.6
x1, y1
0.8
1
Effect of Nonideality:
Ethanol(1)/Toluene(2)
90
1 atm
@ P = 70 kPa
85
t, oC
65oC
txy diagram
Acetonitril(1)/Nitrom ethane(2)
80
75
70
65
0
P - xy D ia gra m
A c e t o nit ril( 1) / N it ro m e t a na ( 2 ) @
75 o C
85
80
75
P
70
65
60
55
P-x-y diagram
50
45
40
0
0.2
0.4
0.6
x1, y1
0.8
1
t-x-y diagram
0.2
0.4 0.6
x1, y1
0.8
1
Effect of Nonideality:
x – y Diagram at Constant P = 1 atm
a. Tetrahydrofuran(1)/Carbon-tetrachloride(2)
b Chloroform(1)/Tetrahydrofuran(2)
c. Furan(1)/Carbontetrachloride(2)
d. Ethanol(1)/Toluene(2)
Partial Molar Properties
Pure-species Properties:
Mi  U i , H i , Si , Vi , or G i , etc.
Solution Properties:
M
NOT:
Partial Properties:
 x i  Mi
M   xi  Mi
Mi  Ui , Hi , Si , Vi , or Gi , etc.
….are properties of component i in the state of mixtures,
which, in general different from that in the state of pure
species
What physical interpretation can be given for, viz.
partial molar volume ?
Methanol – Water Mixture, An Example
For pure species at 25oC:
Methanol (1) : V1 = 40.727 cm3/mol
Water (2)
: V2 = 18.068 cm3/mol
M
 x i  Mi
What is the volume of 10 moles of methanol/water solution containing
30% mol of methanol?
Most people would think, logically:
Mol of methanol
: 0.3
x 10 moles = 3 moles
Mol of water
: (1-0.3) x 10 moles = 7 moles
Volume of methanol
: 3 moles x 40.727 = 122.181 cm3
Volume of water
: 7 moles x 18.068 = 126.476 cm3
Thus, the total volume : 122.181 + 126.476= 248.657 cm3
Wrong answer!
The correct answer is 240.251 cm3
Thus there is 240.251 – 248.657 = -8.406 cm3 deviation from expected value
More on Partial Molar Properties
nM  M(T, P, n1 , n 2 , n 3 ,......... .)
 nM 
dnM  
 dT

 T  P,n
 nM 

 dP

 P  T,n
 nM 
 nM 

 dn1  
 dn2  .....


 n1  T,P,n 2 ,n 3 ,...
 n 2  T,P,n1 ,n 3 ,...
 nM
Mi  


n
i  T , P, n j

 nM
 nM
dnM  

dT

 dP 



 T  P,n
 P  T,n
NOT: M   x i  M i

 nM
 dni .


 n i  T,P,n j
M
 x i  Mi
Chemical Potential as Partial Molar Property
Criteria for Vapor - Liquid Equilibria
Tg  T
P P
g
g

i  i

i
The chemical potential of i-th
component is defined as:
 (nG) 
i  

 n i  T,P,n j
Chemical Potential as Partial Molar Property
If we set M = G:
 nG
 nG
dnG  
 dT  
 dP 


 T  P,n
 P  T,n

 nG
 dni .


 n i  T,P,n j
 nG
Gi  


n
i  T,P,n j

The definition of chemical potential:
Thus:
 (nG) 
i  


n
i  T,P,n j

i  Gi
Evaluation of Partial Molar Properties
Methanol – Water Mixture Example
 x i  Mi
M1  M  x 2 
M 2  M  x1 
Methanol
Molar volume,
mol fraction
mL/mol
0
18.1
0.114
20.3
0.197
21.9
0.249
23.0
0.495
28.3
0.692
32.9
0.785
35.2
0.892
37.9
1
40.7
M
x1
M
x1
40
M1
36
Mixture Property M
M
32
28
M
24
20
M2
16
0
0.2
0.4
0.6
x1
0.8
1
Exercise
A group of students came across an unsuspected supply of laboratory
alcohol, containing 96 mass-percent ethanol and 4 mass-percent water.
As an experiment they decided to convert 2 L of this material into vodka,
having a composition of 56 mass-percent ethanol and 44 mass-percent
water. Wishing to perform the experiment carefully, they search the
literature and found the following partial-specific volume data for
ethanol – water mixtures at 25oC and 101.3 kPa.
In 96% ethanol
In vodka
V H 2O , L/kg
0.816
0.953
V EtOH , L/kg
1.273
1.243
The specific volume of water at 25oC is 1.003 L/kg. How many L of water
should be added to the 2 L of laboratory alcohol, and how many L of
vodka result?
Fugacity, f
dG  S  dT  V  dP
At constant T
Ideal gas
:
dGig  R  T  dln P
Real gas
:
dG  R  T  dln f 
dG  R  T  dln 
R
Residual Gibbs energy : G R  G  G ig
Fugacity coefficient
:
f

P
GR
 ln 
R T
Residual Property
V R  V  V ig
V R  (Z  1) 
RT
P
Evaluation of Pure Component
Fugacity, fi
At constant T:
Real gas
dGi R  Vi R  dP
:
Vi R  ( Zi  1) 
RT
P

dGi R

RT
P

0
Vi R
 dP
RT
P
GiR
dP
 ( Z i  1) 
RT
P

0
GiR
 ln  i
R T
Pure Component Fugacity Coefficient:
P

ln  i  ( Z i  1) 
0
The fugacity :
f i  i  P
dP
P
Evaluation of Pure Component
Fugacity, fi
From the following compressibility data for hydrogen at 0oC,
determine the fugacity of hydrogen at 950 atm
P, atm
100
200
300
400
500
Z
1.069
1.138
1.209
1.283
1.356
P, atm
600
700
800
900
1000
Z
1.431
1.504
1.577
1.649
1.720
Evaluation of Pure Component
Fugacity, fi
From the following compressibility data for isobutane,
determine the fugacity of butane at various temperature and pressure
P/bar
340 K
350 K
360 K
370 K
380 K
Evaluation of Pure Component
Fugacity, fi from Equation of State
Virial :
PVi
Bi P
 Zi  1 
RT
RT
B  Pc
 B0    B1
R  Tc
B0  0.083
0.422
B1  0.139
0.172
Tr1.6
Tr 4.2
Bi P
ln i 
RT

Pr 0
ln i 
B    B1
Tr
T
Tr 
Tc
Pr 
P
Pc

Critical Constants & Accentric Factors:
Paraffins
Tc/K
Pc/bar
-6
3.
Vc/10 m mol
-1
Zc

Critical Constants & Accentric Factors:
Olefin & Miscellaneous Organics
Tc/K
Pc/bar
-6
3.
Vc/10 m mol
-1
Zc

Critical Constants & Accentric Factors:
Miscellaneous Organic Compounds
Tc/K
Pc/bar
-6
3.
Vc/10 m mol
-1
Zc

Critical Constants & Accentric Factors:
Elementary Gases
Tc/K
Pc/bar
-6
3.
Vc/10 m mol
-1
Zc

Critical Constants & Accentric Factors:
Miscellaneous Inorganic Compounds
Tc/K
Pc/bar
-6
3.
Vc/10 m mol
-1
Zc

Evaluation of Pure Component Fugacity, fi
from Virial Equation of State, Example
Using virial equation of state,
calculate the fugacity and fugacity coefficient of:
1.
Pure methyl-ethyl-ketone
2.
Pure toluene
at 50oC and 25 kPa.
The required data:
ij
Tcij/K
Pcij/bar
Vcij/cm3.mol-1
Zcij
wij
11=MEK
12=Toluene
535.6
591.7
41.5
41.1
267
316
0.249
0.264
0.329
0.257
Evaluation of Pure Component
Fugacity, fi from Equation of State
Redlich-Kwong:
0.42748 R 2  Tci 2.5
ai 
Pci
h
bi  P
Z R T
ai
1
h
Z


1  h b i  R  T1.5 1  h
bi 
0.08664 R  Tci
Pci
}
to be solved simultaneously


a

ln   Z  1  ln(1  h )  Z  
1.5 
 bR T 
Evaluation of Pure Component Fugacity, fi
from Redlich-Kwong Equation of State
Using Redlich - Kwong equation of state,
calculate the fugacity and fugacity coefficient of:
1.
Pure methyl-ethyl-ketone
2.
Pure toluene
at 50oC and 25 kPa.
The required data:
ij
Tcij/K
Pcij/bar
Vcij/cm3.mol-1
Zcij
wij
11=MEK
12=Toluene
535.6
591.7
41.5
41.1
267
316
0.249
0.264
0.329
0.257
Evaluation of Pure Component Fugacity, fi :
Pitzer’s Generalized Correlation
   
i  i
i  f Pr , Tr 
0
P
Pr 
Pc
0
i1

i1  f Pr , Tr 
T
Tr 
Tc
Evaluation of
Pure
Component
Fugacity, fi :
Pitzer’s
Generalized
Correlation
Tr 
i 0
Pr 
P
Pc
T
Tc
Tr 
T
Tc
Evaluation of
Pure
Component
Fugacity, fi :
Pitzer’s
Generalized
Correlation
i 0
Pr 
P
Pc
Evaluation of
Pure
Component
Fugacity, fi :
Pitzer’s
Generalized
Correlation
Tr 
i1
Pr 
P
Pc
T
Tc
Tr 
i1
T
Tc
Evaluation of
Pure
Component
Fugacity, fi :
Pitzer’s
Generalized
Correlation
Pr 
P
Pc
Evaluation of Pure Component Fugacity,
fi : Pitzer Correlation
Using Pitzer Correlation,
calculate the fugacity and fugacity coefficient of:
1.
Pure methyl-ethyl-ketone
2.
Pure toluene
at 50oC and 25 kPa.
The required data:
ij
Tcij/K
Pcij/bar
Vcij/cm3.mol-1
Zcij
wij
11=MEK
12=Toluene
535.6
591.7
41.5
41.1
267
316
0.249
0.264
0.329
0.257
Evaluation of Liquid
Pure Component Fugacity, fi
fi 
1
ln


sat
R T
fi
P

Vi   dP
Pi sat
Since Vl is a weak function of P at temperatures well below Tc:


sat

V

P

P
i
f i   i sat  Pi sat  exp  i
R T

Poynting factor
Fugasity of saturated vapor,
calculated exactly as calculating gas phase fugacity


Estimation of Liquid Density
Rackett Equation:
r 
sat
V
 Vc  Zc
Vc
V
(1Tr )0.2857
Tr 
Pr 
P
Pc
T
Tc
Examples of
Evaluation of Liquid Pure Component Fugacity, fi
11.5
Estimate the fugacity of liquid acetone at 110oC and 275 bar.
At 110oC the vapor pressure of acetone is 4.36 bar and the
molar volume of saturated-liquid acetone is 73 cm3.mol-1
11.6
Estimate the fugacity of liquid n-butane at 120oC and 34 bar.
At 120oC the vapor pressure of n-butane is 22.38 bar and the
molar volume of saturated-liquid n-butane is 137 cm3.mol-1
Examples of
Evaluation of Liquid Pure Component Fugacity, fi
11.10
The normal boiling point of n-butane is 0.5oC.
Estimate the fugacity of liquid n-butane at this
temperature and 200 bar.
11.11
The normal boiling point of 1-pentene is 30.0oC.
Estimate the fugacity of liquid 1-pentene at this
temperature and 350 bar.
11.12
The normal boiling point of isobutane is -11.8oC.
Estimate the fugacity of liquid isobutane at this
temperature and 150 bar.
Examples of
Evaluation of Gas & Liquid Pure Component
Fugacity, fi
11.13
Prepare plots of f vs P and  vs P for isopropanol at 200oC for the
pressure range from 0 to 50 bar. For the vapor phase, values of
Z are given by:
Z  1  9.86 10 3  P  11.41 10 5  P 2
Where P is in bars. The vapor pressures of isopropanol at 200oC
is 31.92 bar, and the liquid-phase isothermal compressibility k
at 200oC is 0.3.10-3 bar-1, independent of P.
Hint:
1  V 
k   

V  P  T
Critical constants:
Vc = 219 cm3/mol
Pc = 53,7 bar
Tc 508,8 K
Zc = 0,278
Examples of
Evaluation of Gas & Liquid Pure Component
Fugacity, fi
11.14
Prepare plots of f vs P and  vs P for 1,3-butadiene at 40oC for
the pressure range from 0 to 10 bar. At 40oC The vapor
pressures of 1,3-butadiene is 4.287 bar.
Assume virial model to be valid for the vapor phase.
The molar volume of saturated liquid 1,3-butadiene at 40oC is
90.45 cm3.mol-1
Fugacity of Steam and Water,
Using Steam Table
Up to Pisat, i.e. gas phase water (steam):
*

 f i  1  Hi  Hi
*
ln    
 (Si  Si )
T
 P *  R 

P* : lowest value of P in steam table
At P >= Pisat, i.e. liquid phase water:

f i  i
sat
 Pi
sat

 V   P  P sat
i
 exp  i
R T



Example of Steam and Water Fugacity
Calculation Using Steam Table
11.7
From data in the steam tables, determine a good estimate for f/fsat
of liquid water at 100oC and 100 bar, where fsat is the fugacity
of saturated liquid at 100oC.
11.8
Steam at 13000 kPa and 380oC undergoes an isothermal change of
state to a pressure of 275 kPa. Determine the ratio of the
fugacity in the final state to that in the initial state
11.9
Steam at 1850 psia and 700oF undergoes an isothermal change of
state to a pressure of 40 psia. Determine the ratio of the fugacity
in the final state to that in the initial state
Fugacity of Mixtures
Virial :
Are formulated exactly as calculation for pure component,
Bi P
ln i 
but we use Mixing Rules to obtain the parameters
RT
B
 yi  y j Bij
i
Bii  Bi  B of purecomponenti
j
For binary mixtures, i = 1,2 and j = 1,2
B  y12  B11  2  y1  y 2  B12  y 2 2  B22
Bij  B ji 
Zcij 
R  Tcij
Pcij
Zci  Zcj
2

 B0  ij  B1

1
1


3
V V 3 
ci
cj 
Vcij  


2




ij 
3
i   j

2
Tcij  Tci  Tcj
Pcij 

1
2
Z cij  R  Tcij
Vcij
 (1  k ij )
Example of Calculation for
Fugacity of Mixtures Using Virial Equation
Estimate the fugacity and fugacity coefficient of an equimolar mixture
of methyl-ethyl-ketone (1) and toluene (2) at 50oC and 25 kPa
The required data are as follows:
ij
Tcij/K
Pcij/bar
Vcij/cm3.mol-1
Zcij
wij
11=MEK
12=Toluene
535.6
591.7
41.5
41.1
267
316
0.249
0.264
0.329
0.257
Fugacity of Components in Mixture
ˆ
ˆ  f i
i
yi  P
GiR
 ln  i
R T
Thus:
GiR
 ln ˆ i
R T
ln(ˆ i ) is partial molar property of ln( i )
 n  ln 
ln ˆ i  


n
i

 T, P,n j
Virial, binary mixtures:
P
 (B11  y 2 2  12 )
RT
P
ln ˆ 2 
 (B22  y12  12 )
RT
ln ˆ 1 
12  2  B12  B11  B12
Fugacity of Components in Binary Mixtures,
Example using Virial Eqn.
Estimate the fugacity and fugacity coefficient of methyl-ethyl-ketone
(1) and toluene (2) for an equimolar mixture at 50oC and 25 kPa.
Set all kij = 0
The required data are as follows:
ij
Tcij/K
Pcij/bar
Vcij/cm3.mol-1
Zcij
wij
11=MEK
12=Toluene
535.6
591.7
41.5
41.1
267
316
0.249
0.264
0.329
0.257
11.18
Estimate the fugacity and fugacity coefficient of ethylene (1) and
propylene (2) for a binary mixture of 25% ethylene as a gas at 200oC
and 20 bar.
Set all kij = 0
More on Virial Eqn:
Fugacity of Ternary and Multicomponent Mixtures
Mixing Rules :
B
 yi  y j Bij
i
j
For ternary mixtures, i = 1,2,3 and j = 1,2,3
Bii  Bi  B of purecomponenti
Bij  B ji 

P
1
ˆ
B kk  
ln  k 
R T 
2
i

R  Tcij
Pcij

 B0  ij  B1


y i  y   (2   ik   i )



 ik  2  B ik  Bii  B kk
ii  0
 i  2  B i  Bii  B
 kk  0
 ki  ik
More on Virial: Fugacity of
Ternary & Multicomponent Mixtures Example
11.19
Estimate the fugacity and fugacity coefficient of each component in a
ternary mixture of methane (1) / ethane (2) / propane (3) at 40oC
and 20 bar with the composition of 17% methane and 35% ethane
Set all kij = 0
Evaluation of Mixture Fugacity, f,
from Equation of State
Redlich-Kwong:
a
 yi  y j  a ij
i
h
Z
b
j
bP
Z R T
1
a
h


1  h b  R  T1.5 1  h
 yi  bi
i
}
to be solved simultaneously


a

ln   Z  1  ln(1  h )  Z  
1.5 
 bR T 
Evaluation of Mixture Fugacity, f , using
Redlich-Kwong Equation of State
Using Redlich - Kwong equation of state,
calculate the fugacity and fugacity coefficient of an equimolar
mixture of methyl-ethyl-ketone (1) and toluene (2) at 50oC and
25 kPa
The required data:
ij
Tcij/K
Pcij/bar
Vcij/cm3.mol-1
Zcij
wij
11=MEK
12=Toluene
535.6
591.7
41.5
41.1
267
316
0.249
0.264
0.329
0.257
Evaluation of Component Fugacity in
Mixture Fugacity, f, from Equation of State
Redlich-Kwong:


a

ln   Z  1  ln(1  h )  Z  
1.5 
 bR T 
b1
a
ˆ
ln 1   ( Z  1)  lnZ  (1  h ) 
b
b  RT1.5

2

b1



 b


 x k  a1k 
b2
a
 ( Z  1)  lnZ  (1  h ) 
b
b  RT1.5

2

b
 2 
 b


 x k  a 2k 
ln ˆ 2 
k
a
k
a
  ln(1  h )



  ln(1  h )



Evaluation of Mixture Fugacity, f , using
Redlich-Kwong Equation of State
Using Redlich - Kwong equation of state,
calculate the fugacity and fugacity coefficient of MEK and
toluene in equimolar mixture of methyl-ethyl-ketone (1) and
toluene (2) at 50oC and 25 kPa
The required data:
ij
Tcij/K
Pcij/bar
Vcij/cm3.mol-1
Zcij
wij
11=MEK
12=Toluene
535.6
591.7
41.5
41.1
267
316
0.249
0.264
0.329
0.257
UTS 1
Excess Gibbs Energy
Pure-species Properties:
Mi  U i , H i , Si , Vi , or G i , etc.
Solution Properties:
M
Partial Properties:
Residual Property
Partial Property of the Excess Property
Excess Property
Partial Property of the Excess Property
 x i  Mi
 n  G 
Mi  


n
i  T,P,n j

G
R
 GG
ig
Gi R  Gi  Gi ig
G E  G  G id
E
Gi  Gi  Gi
id
Excess Gibbs Energy
Pure-species Properties:
Mi  U i , H i , Si , Vi , or G i , etc.
Solution Properties:
M
Partial Properties:
Mi  Ui , Hi , Si , Vi , or Gi , etc.
Residual Property
Partial Property of the Excess Property
Excess Property
Partial Property of the Excess Property
G
R
 x i  Mi
 GG
ig
Gi R  Gi  Gi ig
G  GG
E
E
id
Gi  Gi  Gi
id
Activity Coefficient
Definition of fugacity:
dG  R  T  dln f 
Integration
 fˆi
G i  G i  R  T  ln
 fi





(Ideal solution)
Gi id  Gi  R  T  ln(x i )
G i  G i id

 fˆi
 R  T  ln
 xi  fi

  n  G E / RT

n i






 fˆi
Gi E
 ln
 x i  fi
R T





The definition of activity coefficient
gi
 T,P,n
j
 lng i 
GE

R T
 xi  ln gi
Models for Binary Mixtures Activity Coefficient:
Margules(1856 – 1920)
GE
 A 21  x1  A12  x 2
x1  x 2  RT

  n  G E / RT
lng i   
n i

ln g1  x 2 2  A12  2  (A 21  A12 )  x1 
ln g 2  x12  A 21  2  (A12  A 21)  x 2 

 T,P,n
j
Models for Binary Mixtures Activity Coefficient:
van Laar
A12'  A 21'
GE

x1  x 2  RT A12'  x1  A 21'  x 2

  n  G E / RT
lng i   
n i

'
 A12
 x1 
'
ln g1  A12  1 

'
 A 21  x 2 
2
 A '21  x 2 
'
ln g 2  A 21  1 

'
 A12  x1 
2

 T,P,n
j
Models for Binary Mixtures Activity
Coefficient:
Wilson
12
V2
 a12 

 exp  

V1
 RT 
 21 
V1
 a 
 exp   21 
V2
 RT 
V1 , V2  molar volume of pure liquid1 & 2
a12 , a 21  constan ts, independent of T & x i
Models for Binary Mixtures Activity Coefficient:
Renon: NonRandom Two-Liquid (NRTL)
Models for Multicomponent Mixtures
Activity Coefficient:
Wilson
GE

RT
 xi  ln  x j  ij
i
j
ln g i  1  ln

x j   ij 
j
x k   ki
  x j   kj
k
j
 ij
 ij
 a ij 


 exp 

Vi
 RT 
1
Vj
(i  j)
(i  j)
Vi  molar volume of pureliquidi
a ij  constan ts, independent of T & composition
Models for
Multicomponent
Mixtures Activity
Coefficient:
UNIversal
QUAsi Chemical
(UNIQUAC)
(Abrams & Prausnitz)
Models for Multicomponent Mixtures Activity Coefficient:
ln g i  ln g i C  ln g i R
UNIquac
Functionalgroup
Activity
Coefficient
(UNIFAC)
(Aa Fredenslund,
Rl Jones & JM
Prausnitz)
Models for Multicomponent Mixtures Activity Coefficient:
UNIFAC: Rk & Qk
Models for Multicomponent Mixtures Activity Coefficient:
UNIFAC: Rk & Qk
Example
Models for Multicomponent Mixtures Activity Coefficient:
UNIFAC: amk