![Page 1: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/1.jpg)
Vapor Liquid Equilibrium
Huang Min, PhD
Chemical Engineering
Tongji University
![Page 2: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/2.jpg)
• 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
![Page 3: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/3.jpg)
• 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
![Page 4: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/4.jpg)
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
![Page 5: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/5.jpg)
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
![Page 6: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/6.jpg)
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
![Page 7: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/7.jpg)
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
![Page 8: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/8.jpg)
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 -
![Page 9: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/9.jpg)
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
,,,,,,
-
![Page 10: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/10.jpg)
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
![Page 11: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/11.jpg)
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
![Page 12: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/12.jpg)
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
![Page 13: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/13.jpg)
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
![Page 14: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/14.jpg)
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
![Page 15: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/15.jpg)
•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
![Page 16: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/16.jpg)
•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
![Page 17: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/17.jpg)
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
![Page 18: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/18.jpg)
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
![Page 19: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/19.jpg)
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
![Page 20: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/20.jpg)
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
![Page 21: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/21.jpg)
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
![Page 22: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/22.jpg)
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
![Page 23: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/23.jpg)
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
![Page 24: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/24.jpg)
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
![Page 25: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/25.jpg)
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
![Page 26: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/26.jpg)
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
![Page 27: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/27.jpg)
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
![Page 28: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/28.jpg)
![Page 29: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/29.jpg)
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
![Page 30: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/30.jpg)
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
![Page 31: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/31.jpg)
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
![Page 32: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/32.jpg)
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
![Page 33: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/33.jpg)
![Page 34: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/34.jpg)
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!
![Page 35: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/35.jpg)
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
![Page 36: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/36.jpg)
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!
![Page 37: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/37.jpg)
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
-
-
-
![Page 38: Vapor Liquid Equilibrium - mestudio.tongji.edu.cnmestudio.tongji.edu.cn/_upload/article/files/f0/0b/974a83264438803b7099d3529511/e09a2e...Vapor phase fugacity is calculated from an](https://reader031.vdocuments.mx/reader031/viewer/2022013008/5e35036a1bc84a5e6b13d586/html5/thumbnails/38.jpg)
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