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Vapor Liquid Equilibrium
Huang Min, PhD
Chemical Engineering
Tongji University
• Phase equilibria
– VLE, VLLE
– Thermodynamic models
• g-f (Activity coefficient models)
• f-f (Equations of state)
• At phase equilibria: – TI = TII = TIII = … = T
– pI = pII = pIII = … = p
– fiI (T, p, xi
I) = fiII (T, p, xi
II) = fiIII (T, p, xi
Iii) = .. for i = 1, 2,.., c
• Criteria for systems at VLE:
– fiV = fi
L for i = 1, 2,…, c
– One-model method:
Both the vapor phase and liquid phase fugacities are calculated from an equation of state.
• Two-model method:
Vapor phase fugacity is calculated from an equation of state and liquid phase fugacity from an activity coefficient model.
pxfpyf i
L
i
L
ii
V
i
V
i ff
oL
iii
L
ii
V
i
V
i fxfpyf gf
The Partial Molar Gibbs Free Energy and Fugacity—Revisit
• Recall
• Then
• So
• Therefore
-C
i
iidNGVdpSdTdG1
jj
ijijij
j
jj
ijijij
j
NT
i
NTNpTi
i
NpTiNpT
NTi
Np
i
NpNpTi
i
NpTiNpT
Npi
p
G
N
G
pV
N
V
p
G
N
T
G
N
G
TS
N
S
T
G
N
,,
,,,,,,
,
,,
,,,,,,,
-
-
jNp
ii
T
GS
,
-
and
jNT
ii
p
GV
,
-
2
1
2
1
,
1121 ),,(),,(p
pi
p
pNT
iii dpVdp
p
GxpTGxpTG
j
for isothermal change at T=T1
The Partial Molar Gibbs Free Energy and Fugacity—Revisit
• Define the fugacity of species i in a mixture
• so that as
• The fugacity coefficient for a component in a mixture
-
-
-
p IG
iii
IG
ii
i
IG
ii
ii
dpVVRT
pxRT
xpTGxpTGpx
RT
xpTGxpTGpxpTf
0
1exp
),,(),,(exp
),,(),,(exp),,(ˆ
-
-
p IG
ii
IG
ii
i
ii
dpVVRT
RT
xpTGxpTG
px
f
0
1exp
),,(),,(exp
ˆf̂
iii ppxfp ˆ ,0
The Partial Molar Gibbs Free Energy and Fugacity—Revisit
for pure component
• To relate the fugacity of pure component i to the fugacity of component i in a mixture
• As
• thus
i
xT
i
xT
i Vp
G
p
fRT
,,
ˆln Vp
G
p
fRT
TT
ln
-
-
-
-
p
ii
i
i
i
ip
i
p
i
p p
T
i
xT
i
dpVV
Tf
pTfRT
xTf
xpTfRTfdfdRT
dpP
fdp
P
fRT
0
00
0 0
,
)(
)0,(
),(ln
),0,(
),,(ˆlnlnˆln
lnˆln
iii pxpfp ˆ ,0 and pfi
-
p
ii
ii
i dpVVpTfx
xpTfRT
0),(
),,(ln
xp
i
IG
iiIG
ii
IG
ii
xp
IG
i
xp
i
IG
ii
IG
ii
xpxp
ii
TRT
HHSSTGG
RT
T
G
T
G
RTRT
GG
RT
xpTGxpTG
TT
pxf
,
22
,,
2
,,
ln1
1
),,(),,()/ˆln(
-----
-
--
-
f
The Partial Molar Gibbs Free Energy and Fugacity—Revisit
• Therefore, for a mixture in which under all conditions,
• true for ideal gas mixtures and ideal solutions.
• Therefore,
ii VV
iii xff ˆ
-
RT
xpTGxpTG
px
fIG
ii
i
ii
),,(),,(exp
ˆf̂
II
i
I
i
II
i
I
iiii
II
i
II
i
I
i
I
i
IIII
i
II
i
pure
i
II
i
I
i
pure
i
IIII
i
IIIG
i
II
i
IIG
iI
iII
i
fRTfRTPfRTPfRTpxf
xRTxRT
xpTRTxRTpTG
xpTRTxRTpTG
xpTRTxpTG
xpTRTxpTGxpTGxPTG
ˆlnˆln/ˆln/ˆln /ˆ
lnln
),,(lnln),(
),,(lnln),(
),,(ln),,(
),,(ln),,(),,(),,(
--
-
---
--
-
f
ff
f
f
f
f
Phase equilibrium criteria in terms of fugacity
• Let • with
where superscripts I and II denotes phases of different composition
• Recall then,
),,(ln),,(),,( xpTRTxpTGxpTG i
IGii f
i
pure
i
IG
i xRTPTGxPTG ln),(),,(
at equilibrium 0),,(),,( -I
i
II
ii xPTGxPTGG II
i
I
i ff ˆˆ
),,(),,(I
iII
ii xpTGxpTGG -
Fugacity and fugacity coefficient for a species in a mixture from an equation of state
• Recall
• Therefore,
• normally volumetric equation of state are pressure explicit. The integral is easier if the integration is based on volume.
- p IG
ii
i
ii dpVV
RTPx
f
0
1exp
ˆf̂
VdV
PdZ
Z
PVd
V
PdZ
V
RTVd
V
PZRTd
VVd
V
PVPd
VdP ---- )(
11
-
p IG
ii
i
ii dpVV
RTpx
f
0
1
ˆlnˆlnf
1,,,,,
-
jj
NVT
i
NTNPTiP
N
V
P
N
V
ijj
j NVTiNTNPTiN
P
V
P
N
V
ij,,,,,
-
VdN
PNdV
N
PdPVdP
N
V
jjNVTiNVTi
i
NPTiij
,,,,,,
-
Fugacity and fugacity coefficient for a species in a mixture from an equation of state
10
ZVd
NVTiN
pN
V
RT
RT
VdV
Vp
RTdZ
ZRT
VpVd
N
pN
RT
VpdV
VpdZ
Z
V
RTVd
N
pN
RT
dPVRT
dPVRT
dPVVRTPx
f
pZRTV
V
ij
i
pZRTV
V
pure
iZpure
ipZRTV
V
ijNVTi
pZRTV
V
IG
iZIG
ipZRTV
V
ijNVTi
p IG
i
p
i
p IG
ii
i
ii
ln
,,
1 ln
111
11
1
11 lnln
/
/
1
/
,,
/
1
/
,,
000
-
-
-
-
-
-
--
-
f
f
=1
=RT
P
RTVV
pure
i
IG
i
fugacity coefficient for pure substance :
)1(ln1
ln/
--
-
ZZVdP
V
RT
RT
pZRTV
Vf
21),,( xaxxPTG
ex 2
211 ln axRTGex
g 2
122 ln axRTGex
g
0)(22 21211221
2
12
2
212211,1
xxdxaxdxdxxax
axdxaxdxGdxGdxGdxexex
PT
C
i
ex
ii
022
)1(2
)1()1(lnln
2121
1
2
1
21
1
,1
2
2
2
,1
2
1
1
,1
2
2
,1
1
1
-
--
-
-
RT
xax
RT
xax
x
x
RT
axx
RT
ax
x
x
RT
ax
x
x
RT
ax
xx
xx
PTPTPTPT
gg
All activity coefficients derived from an excess Gibbs free energy expression
that satisfies boundary conditions of being zero at x1=0 and 1 will satisfy the
Gibbs-Duhem equation.
at x1= 0 and x1=1 0ex
G then the activity coefficients satisfy PT
C
iii
dx,1
ln0
g
Excess Gibbs Free Energy and Gibbs-Duhem Equation
Activity Coefficient Model
• Random mixing assumption (Wohl’s expansion):
– Redlich-Kister model
– Margules model
– van Laar model
RT/Glnex
ii g i
ii
exlnxRT/G g
.....zzzazzaxqRT
Gkji
i j k
ijk
i j
jiij
i
ii
ex
13
Margules’ Equations
•While the simplest Redlich/Kister-type correlation
is the Symmetric Equation, but a more accurate
equation is the Margules correlation:
•let,
•so that
212121
21
xAxAxRTx
GE
210
1 lim1
lnxRTx
GE
x
g
12
0211
AxRTx
G
x
E
112 lngA
221 lngA
Margules’ Equations
•If you have Margules parameters, the activity coefficients can be derived from the excess Gibbs energy expression:
•to yield:
•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.
212121
21
xAxAxRTx
GE
])(2[ln
])(2[ln
2211221
2
12
1122112
2
21
xAAAx
xAAAx
-
-
g
g
•Another two-parameter excess Gibbs energy model was developed from an expansion of (RTx1x2)/GE
instead of GE/RTx1x2. The end results are:
•for the excess Gibbs energy and:
•for the activity coefficients.
•as x10, lng1 A’12 as x2 0, lng2
A’21
Van Laar Correlation
2
'
211
'
12
'
21
'
12
21 xAxA
AA
xRTx
GE
2
2
'
21
1
'
12'
121 1ln
-
xA
xAAg
2
1
'
12
2
'
21'
212 1ln
-
xA
xAAg
•Unfortunately, the previous approach cannot be extended to systems of 3 or more components. For these cases, local composition models are used to represent multi-component systems.
– Wilson’s Theory – Non-Random-Two-Liquid Theory (NRTL) – Universal Quasichemical Theory (Uniquac)
•While more complex, these models have two advantages: – the model parameters are temperature dependent – the activity coefficients of species in multi-component liquids
can be calculated using information from binary data.
Local Composition Models
A,B,C A,B A,C B,C tertiary binary binary binary
Wilson’s Equations for Binary Solution Activity
•A versatile and reasonably accurate model of excess Gibbs Energy was developed by Wilson in 1964. For a binary system, GE is provided by:
Vi is the molar volume at T of the pure component i.
aij is determined from experimental data.
The notation varies greatly between publications. This includes,
– a12 = (12 - 11), a21 = (12 - 22) that you will encounter in Holmes, M.J. and M.V. Winkle (1970) Ind. Eng. Chem. 62, 21-21.
)ln()ln( 2112212211 - xxxxxxRT
GE
-
-
RT
a
V
V
RT
a
V
V 21
2
121
12
1
212 expexp
Wilson’s Equations for Binary Solution Activity
•Recall
•When applied to Wilson’s :
-
-
2112
21
1221
12212211 )ln(ln
xxxxxxxg
-
--
2112
21
1221
12121122 )ln(ln
xxxxxxxg
jnPTi
EE
iin
nGGRT
,,
ln
g
Wilson’s Equations for Multi-Component Mixtures
•The strength of Wilson’s approach resides in its ability to describe multi-component (3+) mixtures using binary data.
– Experimental data of the mixture of interest (ie. acetone, ethanol, benzene) is not required
– We only need data (or parameters) for acetone-ethanol, acetone-benzene and ethanol-benzene mixtures
•The excess Gibbs energy for multicomponent mixtures is written: •and the activity coefficients become:
•where ij = 1 for i=j. Summations are over all species.
)ln( -i j
ijji
E
xxRT
G
--
k
j
kjj
kik
i
ijjix
xxln1lng
Wilson’s Equations for 3-Component Mixtures
•For three component systems, activity coefficients can be calculated from the following relationship:
•Model coefficients are defined as (ij = 1 for i=j):
3322311
33
2332211
22
1331221
11332211 )ln(1ln
xxx
x
xxx
x
xxx
xxxx
i
i
iiiii
-
-
--g
-
RT
a
V
V ij
i
j
ij exp
Non-Random-Two-Liquid Theory (NRTL)
• NRTL model (Non-Random Two-Liquid; Renon and Prausnitz, 1968)
– For binary systems:
– a12 , the so-called non-randomness parameter
– Good for both miscible and partially miscible systems
2
1212
1212
2
2121
2121
2
21Gxx
G
Gxx
Gxln
g
2
2121
2121
2
1212
1212
2
12Gxx
G
Gxx
Gxln
g
--
--
RT
ggexpG;
RT
ggexpGwhere 1121
12212212
1212 aa
Non-Random-Two-Liquid Theory (NRTL)
• For a liquid, in which the local distribution is random around
the center molecule, the parameter α12 = 0. In that case the
equations reduce to the one-parameter Margules activity model
• The NRTL parameters are fitted to activity coefficients that have
been derived from experimentally determined phase
equilibrium data
• Noteworthy is that for the same liquid mixture there might exist
several NRTL parameter sets. It depends from the kind of phase
equilibrium (i.e. solid-liquid, liquid-liquid, vapor-liquid).
2
12112
2
12
2
21221
2
21
ln
ln
Axx
Axx
g
g
Universal Quasichemical Theory
• UNIQUAC (Abrams and Prausnitz, 1975)
• In the UNIQUAC model the activity coefficients of the ith component of a two component mixture are described by a combinatorial and a residual contribution
• The first is an entropic term quantifying the deviation from ideal solubility as a result of differences in molecule shape. The latter is an enthalpic correction caused by the change in interacting forces between different molecules upon mixing.
R
i
C
ii ggg lnlnln
UNIQUAC
• Combinatorial contribution – Vi, is the Volume fraction per mixture mole fraction for the ith
component
– Fi, is the surface area fraction per mixture molar fraction for the ith component
– Z=10
• The excess entropy gC is calculated exclusively from the pure chemical parameters, using the relative Van der Waals volumes ri and surface areas qi of the pure chemicals.
--
i
i
i
iiii
C
iF
V
F
Vq
zVV ln1
2ln1ln g
UNIQUAC
• Residual contribution
Δuij [J/mol] is the binary interaction energy parameter.
Theory defines Δuij = uij - uii, and Δuji = uji - ujj, where uij is the interaction energy between molecules i and j.
• Data is derived from experimental activity coefficients, or from phase diagrams
RTu
ij
j
k
kjkk
ijjj
j
jj
j
ijjj
i
R
i
ije
xq
xq
xq
xq
q
/
ln1ln
-
--
g
The UNIFAC model
residualialcombinatorR
i
C
ii ggg lnlnln
-j
jj
i
i
i
i
i
i
i
i
ilx
xlq
z
xialcombinator
f
f
fg ln
2lnln
with 12/ ---iiii
rzqrl
Same as
UNIQUAC
Model -
K
i
kk
i
kiresidual
)()(lnlnln g
is the number of k groups present in species i )( i
k
)( i
k
is the residual contribution to the activity coefficient of group k in a pure
fluid of species i.
--
m
nnmn
kmm
mmkmkk
Q ln1ln
-
--
T
a
kT
uumnnnmn
mnexpexp
m
surface area
fraction of
group m
nnn
mm
QX
QX
m
X mole fraction of group m in
mixture
Z=10
Example: obtain activity coefficients for the acetone/n-pentane system
at 307 K and xacetone=0.047.
CH3C
O
H3C
Group identification
Molecules (i) name Main No. Sec. No Rj Qj
Acetone (1) CH3 1 1 1 0.9011 0.848
CH3CO 9 19 1 1.6724 1.488
n-pentane CH3 1 1 2 0.9011 0.848
CH2 1 2 3 0.6744 0.540
)( i
j
CH3CH2H3C CH2CH2
calculation of combinatorial contribution
residualialcombinatoriii
ggg lnlnln
-j
jj
i
i
i
i
i
i
i
ilx
xl
z
xialcombinator
f
f
fg ln
2lnln 12/ ---
iiiirzqrl
segment volume for acetone:
047.0Axmole fraction of acetone:
5735.26724.119011.01 A
r
segment volume for pentane: 8254.36744.039011.02 P
r
total volume at xA= 0.047: 7666.38254.3953.05735.2047.0 tot
r
the segment fraction for acetone: 0321.07666.3
5735.2047.0
Af
the segment fraction for pentane: 9679.01 -AP
ff
area for acetone: 336.2488.11848.01 A
q
area for pentane: 316.3540.03848.02 P
q
total area at xA=0.047: 2699.3316.3953.0336.2047.0 tot
q
area fraction for acetone: 0336.02699.3
336.2047.0
A
area fraction for pentane: 9664.01 -AP
3860.0)15735.2(336.25735.22
10)1(
2-------
AAAArqr
Zl
2784.018254.3316.38254.32
10----
Pl
Molecule (i) ri qi fi i li
acetone 2.5735 2.336 0.0321 0.0336 -0.3860
pentane 3.8254 3.316 0.9679 0.9664 -0.2784
combinatorial contribution
-j
jj
i
i
i
i
i
i
i
i
ilx
xlq
z
xialcombinator
f
f
fg ln
2lnln
0403.02784.0953.0386.0047.0047.0
0321.0
386.00321.0
0336.0ln336.2
2
10
047.0
0321.0lnln
----
-A
gfor acetone
for pentane
0007.02784.0953.0386.0047.0953.0
9679.0
2784.09679.0
9664.0ln316.3
2
10
953.0
9679.0lnln
----
-P
g
residual contribution
-K
i
kk
i
kiresidual
)()(lnlnln g
is the number of k groups present in species i )( i
k
)( i
k is the residual contribution to the activity coefficient of group k in a pure fluid of
species i.
--
m
nnmn
kmm
mmkmkk
Q ln1ln
-
--
T
a
kT
uumnnnmn
mnexpexp
m
surface area
fraction of
group m
nnn
mm
QX
QX
m
X mole fraction of group m in
mixture
main group labels
nm
from table 8-22
2119.0307
4.476exp4.476
9,1CH3COCH3,9,1
- aa
9165.0307
76.26exp760.26
1,9CH3CH3CO,1,9
- aa
09,91,1
aa 19,91,1
Calculation of )( i
k
--
m
nnmn
kmm
mmkmk
i
kQ ln1ln
)(
-
--
T
a
kT
uumnnnmn
mnexpexp
for pure acetone, there are only two different kinds of groups: CH3 and
CH3O. Let CH3 be labeled by 1 and CH3O be labeled by 19.
m
surface area
fraction of
group m
nnn
mm
QX
QX
m
X mole fraction of group m in
mixture we are dealing a pure substance!
5.0 ,5011
1 )(
19)(
19
)(
1
)(
1)(
1
A
AA
A
AX.X
637.0 ,363.0848.05.0488.15.0
848.05.0 )(
19
)(
1
AA
--
m
nnmn
kmm
mmkmk
i
kQ ln1ln
)(
409.01637.02119.0363.0
2119.0637.0
9165.0637.01363.0
1363.0848.0
9165.0637.01363.0ln1848.0ln)(
1
-
-A
139.0637.02119.0363.0
2119.0637.0
9165.0637.01363.0
9165.0363.0488.1
637.02119.0363.0ln1488.1ln)(
19
-
-A
1,1
)(
1
A
1,9
)(
19
A
9,1
)(
19
A
1,9
)(
19
A
9,1
)(
1
A
1,1
)(
1
A
9,9
)(
19
A
1,1
)(
1
A
36
for pure pentane, there are two kinds of subgroups, CH3 and CH2 and both
subgroups belong to one main group. Let CH2 be labeled by 2
6.0 ,4.05
22)(
1
)(
1
)(
1
1
(P)
PP
P
(P)XX
Since both subgroups are belong to the same main group
0lnln)(
2
)(
1
PP
Calculation of group residual activity at xA = 0.047
for CH3(labeled 1)
0097.0 ,5884.05953.02047.0
3953.0 ,4019.0
5953.02047.0
2953.01047.01921
XXX
for CH2(labeled 2) for CH3CO
(labeled 19)
5064.0488.10097.0540.05884.0848.04019.0
848.04019.01
now we are dealing a mixture!
4721.0488.10097.0540.05884.0848.04019.0
540.05884.02
214.0488.10097.0540.05884.0848.04019.0
488.10097.019
--
m
nnmn
kmm
mmkmkk
Q ln1ln
3
1
101.45
0214.02119.0)4721.05064.0(
2119.00214.0
9165.00214.04721.05064.0
4721.05064.0848.0
9165.00214.04721.05064.0ln1848.0ln
-
-
-
4
2
1026.9
0214.02119.0)4721.05064.0(
2119.00214.0
9165.00214.04721.05064.0
4721.05064.0540.0
9165.00214.04721.05064.0ln1540.0ln
-
-
-
21.2
0214.02119.0)4721.05064.0(
0214.0
9165.00214.04721.05064.0
9165.04721.05064.0488.1
9165.00214.04721.05064.0ln1488.1ln19
-
-
The residual contributions to the activity coefficients follow
66.1139.021.21409.01045.11ln3
---R
Ag
331068.50.021.230.01045.12ln
----
R
Pg
Finally summing up the combinatorial and residual contributions
62.166.1403.0ln -A
g
331098.41068.50007.0ln
---
Pg
or 01.1 ,07.5 PA
gg
experimental data: 11.1 ,41.4 PA
gg
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