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