thermodynamics iiiiiiii · 2006. 10. 9. · in chapter 6,) lim ln φ j rt g ∴lim (i p→0 r i...
TRANSCRIPT
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Thermodynamics IIIIIIII
Department of Chemical Engineering
Prof. Kim, Jong Hak
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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)
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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
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■ 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
+=
+=
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■ 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
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■ 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
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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.
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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
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)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 :
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■ 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
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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
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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)
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■■■■ 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
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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
=
==
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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
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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
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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, =
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■■■■ 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
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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)