Thermodynamics IIIIIIII

Department of Chemical Engineering

Prof. Kim, Jong Hak

11.5 Fugacity & Fugacity Coefficient : Pure Species11.5 Fugacity & Fugacity Coefficient : Pure Species11.5 Fugacity & Fugacity Coefficient : Pure Species11.5 Fugacity & Fugacity Coefficient : Pure Species

=> · provides fundamental criterion for phase equilibrium

· not easy to apply to solve problem

PlnRT)T(ΓG iig

i +=

Egn (11.27) => valid only for pure species i in the ideal gas

iii flnRT)T(Γ≡G +

i

iig

ii φlnRTP

flnRTG-G == ( : fugacity coefficient)

iµ

Limitation of Egn (11.29) ( G = H - TS)

P -> 0 or yi -> 0 => ∞-→µ i (which is not true for ideal gas)

For real gas, fugacity is introduced instead of P

iφ

i

R

i φlnRTG =∴∴∴∴ (11.33)

set J as zero,

RT

PB1-z

ii

i =

RT

PBdP

RT

BdP

RT

Bφln

iiP

0

iiiiP

0i ∫∫ ===

RT

PBφln∴ iii =

=> : obtained from PVT data or from compressibility factor

1φ&0G,Pf iR

i

ig

i ===

,J)RT

G( 0P

R

i=

=P

dP1)-z(J≡

RT

Gi

P

0

R

i ∫+

For an ideal gas,

In chapter 6,

Jφlnlim)RT

G(lim∴ i

0→P

R

i

0→P==

which means that0)P

fln(limφlnlim

i

0→Pi

0→P== 1

P

flimφlim

i

0→Pi

0→P==

From egn (6.49)P

dP1)-z(φln,

P

dP1)-z(

RT

Gi

P

0ii

P

0

R

i ∫∫ ==

iφ

■ Evaluation of fugacity coeff. from Cubic EOS

iii Iq-)β-ln(Z-1-Zφln∴ =

qI-)β-ln(Z-1-ZRT

GR

=

RT

Pbβ

i

i =

i

R

i φlnRTG = (11.33)

(6.66)

(11.37)

where RTb

)T(aq

i

i

i = )βεtZ

βδtZln(

ε-δ

1I

iii

iii

ii

i =

RTρ

PZ

i

i =

(3.50) (3.51) (6.65b)

(11.37)

If, εδ =

iii

i

iii

ii

iβεZ

β

bβε1

bρI

+=

+=

■ Vapor-Liquid equilibrium for pure species

v

ii

v

i flnRT)T(ΓG +=

l

ii

l

i flnRT)T(ΓG +=

For species i as a saturated vapor, using eqn(11.3)

(11.38a)

(11.38b)

0f

flnRTG-GdG l

i

v

il

i

v

i ===∴∴∴∴

Consider vapor liquid equilibrium

For species i as a saturated liquid

at equilibrium

sat

i

l

i

v

i fff∴ ==sat

if( : either saturated liquid or vapor)

Alternative formulation

sat

i

sat

isat

ip

fφ = sat

i

l

i

v

i φφφ ==∴∴∴∴

=> Criterion of VLE for pure species G, μ, f, Φ

■ Fugacity of a pure liquid

P

dP1)-z(φln vi

P

0

sat

i

sati

∫=

sat

iφ

l

i

v

i ff =

dPVG-G ip

p

sat

ii sati∫=

- Fugacity of pure species i as a compressed Liq.

=> calculated from the product of easily evaluated ratio

CBA

P)P(f

)P(f

)P(f

)P(f

P

)P(f)P(f satisat

i

l

i

l

i

sat

i

v

i

sat

i

l

i

sat

i

sat

i

v

il

i = at const T

Ratio A => vapor phase fugacity coeff., at VLE

Ratio B => 1 since at VLE

Ratio C => the effect of pressure on the fugacity of pure liquid ;

VdPdG = (6.10) at const T

sat

i

l

i PCBA)P(f •••=

l

iV

)RT

)P-P(Vexp(pφf∴

sat

i

l

isat

i

sat

i

l

i =

)dPVRT

1exp(pφf li

P

P

sat

i

sat

i

l

i sati∫∫∫∫=

)dPVRT

1exp(

)P(f

)P(f li

P

Psat

i

sat

i

l

i

sati∫∫∫∫=

Ratio C,

dPVf

flnRT li

p

psat

i

sati∫=∴∴∴∴

from (11-31) sat

i

isat

iif

flnRTG-G =∴∴∴∴iii flnRT)T(ΓG +=

since

Because is a very weak function of P => assumed constant

sat

iφl

iVsat

iP

Poynting factor

,zvi: can be calculated from RTPB

1-Zii

i =

: value for saturated liquid

)RT

PB(EXPφ∴

sat

iiisat

i =

: Antoine eqn.

11.6 Fugacity & Fugacity Coefficient : Species in Solution11.6 Fugacity & Fugacity Coefficient : Species in Solution11.6 Fugacity & Fugacity Coefficient : Species in Solution11.6 Fugacity & Fugacity Coefficient : Species in Solution

For species i in a mixture of real gas or in a solution of liquid

iii f̂lnRT)T(Γ≡µ +

if̂

π

i

β

i

α

i f̂f̂f̂ === …

l

i

v

i f̂f̂ =

: Fugacity of species i in solution (replacing partial pressure)

-> Multicomponent VLE,

(11.46)

at equilibrium, the fugacity of each component is the same in all phases

)Py

f̂φ̂(φ̂lnRTG∴

i

i

ii

R

i ==

ig

ii

R

i M-MM⇒ =

ig

ii

R

i G-GG =

■ Residual property igR M-MM =

jjj n,T,P

i

ig

n,T,P

i

n,T,P

i

R

]n∂

)nM(∂[-]

n∂

)nM(∂[]

n∂

)nM(∂[ =

R

iig

ii

i

iig

ii Gµ-µ,Py

f̂lnRTµ-µ ==

Pyf̂⇒1φ̂&0G iig

i

ig

i

R

i ===

Multiply n and differentiate with respect to ni at constant T, P, nj

From egn(11.29)

(11.46)

Written for G,

ii G≡µ

iii

ii

ig

iig

i

f̂lnRT)T(Γµ

)Pyln(RT)T(ΓGµ

+=

+==

(since )

iφ : fugacity coeff. of species in solution

For ideal gas :

■ The fundamental residual-property relation

dTRT

nG-)nG(d

RT

1≡)

RT

nG(d 2

∑i

iidnµ(nS)dT-dP)nV()nG(d +=

i

i

i

2 dnRT

GdT

RT

nH-dP

RT

nV)

RT

nG(d ∑+=

Alternative form

(11.54)

Special case of 1 mol of a constant-composition

SdT)-VdPdG

TS-HG(

=

=

→)RT

nG(∴ Function of all of i canonical variable (T, P, nj)

=> Allow evaluation of all other thermodynamic properties

compare (6.37) dTRT

H-dP

RT

V)

RT

G(d

o

=

=> Since we cannot evaluate absolute value of thermodynamic propertyes

=> use residual property

For ideal gas

∑i

i

ig

i

2

igigig

dnRT

G

RT

nH-dP

RT

nV)

RT

nG(d +=

)φ̂lnRTG( iR

i =

∑i

ii2

RRR

dnφ̂lndTRT

nH-dP

RT

nV)

RT

nG(d +=

Subtracting this eqn from (11.54)

∑ iR

i

2

RRR

dnRT

GdT

RT

nH-dP

RT

nV)

RT

nG(d +=

=> The fundamental residual-property relation

by introducing fugacity coeff.

x,P

R

x,T

RR

]T∂

)RT/(∂-T[

RT

H,]

P∂

)RT/G(∂[

RT

V⇒

RG

==

=> Use eqn (6.46), (6.48), (6.49) for the calculation

jn,T,P

i

R

]n∂

)RT/nG(∂[φ̂ln = iφ̂ln⇒ RT/G

R: partial property of

For n mol of constant – composition mixture

P

dPn)-nZ(

RT

nG P

0

R

∫∫∫∫=

1n∂

n∂,Z

n∂

)nZ(∂

i

i

i

==

Differentiation with respect to nj at constant T, P, nj

P

dP]

n∂

n)-nZ(∂[φ̂ln

jn,T,P

j

P

0i ∫=

since,

P

dP1)-Z(φ̂ln∴ i

P

0i ∫= (11.60)

■■■■ Fugacity coefficients from the Fugacity coefficients from the Fugacity coefficients from the Fugacity coefficients from the virialvirialvirialvirial equation of stateequation of stateequation of stateequation of state

iφ̂

RT

BP1z +=

∑∑i j

ijji ByyB =

Evolution of value of from EOS

Simplest form of virial eqn

For binary mixture

B = y1y1B11 + y1y2B12 + y2y1B21 + y2y2B22

= y21B11 + 2y1y1B12 + y22B22

(B11, B22) : virial coeff. of pure species

(B12) : cross coeff (mixture property)

(11.62)

P

dP1)-z(φln

P

0∫= (11.60)

(3.38)

where Bij : characterize bimolecular interaction

b/w i and j (Bij = Bji)

(11.61)

where B=f(T) composition second virial coeff.

For n mol of binary gas mixture eqn(3.38) becomes

RT

nBPnnZ +=

Differentiation with respect to n1

From eqn (11.60)

Second virial coeff. can be written

22 n,T

1

n,T,P

1

1 ]n∂

)nB(∂[

RT

P1]

n∂

)nz(∂[≡z +=

22 n,T

1

n,T

1

P

0i ]

n∂

)nB(∂[

RT

P⇒dP]

n∂

)nB(∂[

RT

1φ̂ln ∫=

B = y1(1 - y2)B11 + 2y1y2B12 + y2(1 - y1)B22

= y1B11 - y1y2B11 + 2y1y2B12 + y2B22 – y1y2B22

= y1B11 + y2B22 + y1y2δ12, δ12 ≡ 2B12 – B11 – B22

since yi = ni / n, multiplying by n 1221

222111 δn

nnBnBnnB ++=

122

2

1

11n,T

1

δn)n

n-

n

1(B]

n∂

)nB(∂[

2+=

12

2

211122111 δyBδy)y-1(B +=+=

by differentiation

)δyB(RT

Pφ̂ln∴ 12

2

2111 +=

)δyB(RT

Pφ̂ln 12

2

1222 +=by similar method

For multicomponent gas mixture, the general eqns.

])δ-δ2(yy2

1B[

RT

Pφ̂ln ∑∑

I J

ijikjikkk +=

0δB-B-B2≡δ

δδ0δB-B-B2≡δ

kkjjiiijij

ikkiiikkiiikik

=

==

11.7 Generalized Correlations for the Fugacity Coefficient11.7 Generalized Correlations for the Fugacity Coefficient11.7 Generalized Correlations for the Fugacity Coefficient11.7 Generalized Correlations for the Fugacity Coefficient

To calculateΦ, use Generalized methods for compressibility factor Z

rcrc dPPdP,PPP ==

P

dP1)-z(φln

P

0i ∫=

10zωzz +=

r

r1P

0r

r0Pr

0 P

dPzω

P

dP1)-z(φln

r

∫∫ +=

Correlation for (3.57)

where 7.0Tr

sat

r )log(P-1.0-≡ω =

Eqn (11.35)

r

r

i

P

0i

P

dP1)-z(Φln⇒

r

∫=∴∴∴∴

(const T) transformed into generalize form

using (11.65)

Using eqn (3.57)

⇒φlnωφlnφln⇒ 10 += Three parameter generalized correlation for Φ

⇒⇒⇒⇒=∴∴∴∴ ω10 )φ)(φ(φ use table E13 ~ E14

10 BωBB̂ +=

)BωB(T

P1-Z 10

r

r+=∴∴∴∴

)]BωB(T

Pexp[φ⇒)BωB(

T

Pφln 10

r

r10

r

r+=+=

r

r

T

PB̂1Z +=

Using Pitzer correlations for the second virial coeff.

=> Insert into eqn(11.15) and integrate

,T

0.422-083.0B 1.6

r

0= 4.2

r

1

T

0.172-139.0B =

(3.61) (3.63)

where

■ Generalized correlations for fugacity coefficient in gas mixture

])δ-δ2(2

1B[

RT

Pφ̂ln

i j

ijikkkk ∑∑∑∑∑∑∑∑+=

More general form for coeff.1

ij

0

ij BωBB̂ +==> B0, B1 -> function of Trij

Cijrij T/TT =

Cij

Cijij

ijRT

PB≡B (11.69) =>use Prausnitz’s combining rule for the

calculation of Cij,Cijij RT,ω

)k-1()TT(T ij2/1

cjcicij =

cij

cijcij

cijV

RTzP =

iφ̂

2

ωωω

ji

ij

+=

2

zzz

cjci

cij

+=

(11.70) (11.71)

(11.72) (11.73)

Where kij = empirical interaction parameter

if i=j and for chemically similar species kij=0 -> all eqns reduce

to value for pure species

Procedure to obtain

① Find value of Bij from (11.69)

② insert Bij into eqn(11.61)

③ using eqn(11.14)

obtain value of

value of the pure-species virial coeff. Bkk, Bii for eqn

∑∑i j

ijji ByyB =

])δ-δ2(yy2

1B[

RT

Pφ̂ln ∑∑

i j

ijikjikkk +=

φiφ̂ln

10

C

CBωB

RT

BPB̂ +==

=>find B

10101010----8. The ideal solution model8. The ideal solution model8. The ideal solution model8. The ideal solution model

Definition : · all molecules are of the same size

· all forces b/w molecules (like and unlike) are equal

ii

id

i

id

i xlnRT)P,T(GG≡µ +=

The chemical potential from ideal-gas mixture model

i

ig

i

ig

iig

i ylnRT)P,T(GG≡µ += (11.24)

=> Only applicable for ideal gas

replace )P,T(G⇒)P,T(G iig

i(Gibbs energy of pure i in its real physical state

of gas liquid, solid)

■ chemical potential of an ideal solution

(11.75)

=> applicable gas, liquid, solid

From eqn(11.18)

iT

i

x,T

id

iid

i V)P∂

G∂()

P∂

G∂(V ===

ix,T

id

iV)

P∂

G∂( =

i

id

i VV∴ = (11.76)

,Hid

i

iiip

i

x,P

id

iid

i Rlnx-SRlnx-)T∂

G∂-()

T∂

G∂-(S ===

∑

∑

∑∑

∑∑

i

ii

id

i

ii

id

i

ii

i

ii

id

i

ii

i

ii

id

VxV

VxV

xlnxR-SxS

xlnxRTGxG

=

=

=

+=

id

i

id

i

id

i TSGH +=

iiii

id

i RTlnx-TSxlnRTGH ++=

iii HTSG =+=

From eqn(11.19) ix,Pi

S-)T∂

G∂( =

(11.77)

For , using (11.75) (11.77)

by summability relation, ∑i

ii MxM =

∑i

ii

id HxH, =

■■■■ The Lewis / Randall RuleThe Lewis / Randall RuleThe Lewis / Randall RuleThe Lewis / Randall Rule

From eqn (11.46)

Alternative form of Lewis/Randall Rule :

(divide 11.83 with Pxi)

ii

id

i fxf̂ =

i

id

i φφ̂∴ =

Lewis/Randall Rule : fugacity of each species ∝ mole fraction

proportional constant = fugacity of pure species i

(11-81)

iii flnRT)T(Γ≡G +iii f̂lnRT)T(Γ≡µ + (11.31)

(11.46-11.31) =>

)f

f̂ln(RTGµ

i

i

ii +=

)f

f̂ln(RTGGµ

i

id

i

i

id

iid

i +==

For the special case of an ideal solution

If compared with eqn(11.75) iiid

i xlnRTGµ +=

(11.83) => Show the composition dependence of the

fugacity in an ideal solution (Lewis/Randall rule)

Fugacity coeff. of species i in an ideal solution

= fugacity coeff. of pure species i

11.9. Excess Properties11.9. Excess Properties11.9. Excess Properties11.9. Excess Properties

Mathematical formulationidE M-M≡M

,G-GG idE =

)M--(MM-M igidRE =

∑ ∑i i

ii

ig

ii

igxlnxRTGxG +=

-> obtain thermodynamic property using Residual properties

In the case of liquid, measure the departure from ideality not from ideal gas, but

from ideal solution => excess property

(ideal gas mixture => ideal solution of ideal gas)

,H-HH idE = ,S-SSidE

=EEE TS-HG =

igR M-MM = 0ME

=

Recall the relation b/w ii

R φ̂,φ,G and PVT data

P

dP1)-z(

RT

Gi

P

0

R

i ∫=

P

dP1)-z(φln i

P

0i ∫=

P

dP1)-z(φ̂ln i

P

0i ∫= (11.60)(11.35)

(6.49)

(Difference b/w actual property and ideal solution)

For example,

considering Pure species

∑ ∑ ∑i i i

R

ii

ig

iiii

igid MxMx-MxM-M∴ ==

∑i

R

ii

RE Mx-MM∴ =

i

i

E

i

2

EEE

dnRT

GdT

RT

nH-dP

RT

nV)

RT

nG(d ∑+=

Compare and

※ Fundamental excess-property relation

※ Excess Property : applied to only mixture

Residual property : applied to both pure species & mixture

For partial excess property :

=> Similar to fundamental residual-property relation

id

i

E

i

E

i M-MM =

igM idM

∑ igiiig HyH = ∑ iiid HxH =

∑∑ iiigiiig ylnyR-SyS = ∑∑ iiiiid xlnxR-SxS =

∑∑ iiigiiig ylnyRT-GyG = ∑∑ iiiiid xlnxRT-GxG =

ERigid M-MM-M =>

■■■■ The Excess Gibbs Energy and the Activity CoefficientThe Excess Gibbs Energy and the Activity CoefficientThe Excess Gibbs Energy and the Activity CoefficientThe Excess Gibbs Energy and the Activity Coefficient

iii f̂lnRT)T(ΓG +=

iii

id

ii

id

i fxlnRT)T(Γf̂lnRT)T(ΓG +=+=

ii

iE

i

id

iifx

f̂lnRTGG-G ==

ii

i

ifx

f̂≡)γ(

From Lewis/Randall rule, for an ideal solution

Difference

activity coefficient (11.90)

From eqn (11.46) iii

f̂lnRT)T(Γµ +=

i

E

i γRTln≡G∴ (11.91)

i

R

i φ̂lnRTG = 1γ,0G iE

i ==, for ideal solution

To device chemical potential of mixture

,γlnRTG-G iid

ii = ii

id

i xlnRTGG += (11.75) iiii xγlnRTGG += (11.92)

Comparison of three equations defining chemical potential

ii

i

i

i

id

i

i

igig

i

xγlnRTGµ

xlnRTGµ

ylnRTGµ

i

i

i

+=

+=

+= (11.24)

(11.75)

(11.92)

1st Eqn : ideal gas mixture model2nd Eqn : ideal solution model

■ Excess-property Relation

i

i

E

i

2

EEE

dnRT

GdT

RT

nH-dP

RT

nV)

RT

nG(d ∑+=

x,T

EE

]P∂

)RT/G(∂[

RT

V∴ =

x,P

EE

]T∂

)RT/G(∂[-T

RT

H=

jn,T,P

i

E

i ]n∂

)RT/nG(∂[γln =

In fundamental excess property relation

(11.91)(11.90),

(11.96)

)γlnRTG(dnγlndTRT

nH-dP

RT

nVi

E

i

ii2

EE

∑ =+=

=> Effect of T, P on the EG

=> : partial properties of iγln

i

R

n

G

=> Similar to eqn for Residual property

※ Difference : in the case of relation to GR -> we can use experimental PVT

data ~ EOS to calculate Residual property

i

EE γ,H,V => Can be obtained by experiment

iγ

EE H,V

: from VLE data

: from mixing experiment

x,T

i

E

i)

P∂

γln∂(

RT

V∴ =

x,P

i

2

E

i)

T∂

γln∂(

RT

H- =

∑ 0Mdx ii =

∑ ii MxM =

∑i

ii

E

γlnxRT

G=

∑i

ii 0γlndx =

=> Effect of T, D on the

(11.98)(11.97),

iγln ,RT

GE

RT

Gγln

E

i =∴∴∴∴: partial property of

iγ

From the summability eqn

From Gibbs/Duhem eqn (at const T, P)

(11.99)