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October 14, 2008 Tan - ChE3015 1
Definition:
RTG
nnG
RTi
nPTii
ij
R
,,
R1ˆln =
∂∂
=≠
φRTGR
ln =φ
To probe the physical phase of pure substance, we need to minimize G, or equally effective GR or φ.
Fugacity coefficient
Mixture Pure
Two-phase Equilibria Conditions at (T, P)
),,(ˆ),,(ˆ VL yx PTyPTx iiii φφ = φL(T,P) = φV(T,P)
At phase equilibrium of pure substance, the fugacity coefficients are equal in the phases.
Eq (11.33)Eq (11.59)
October 14, 2008 Tan - ChE3015 2
Fugacity coefficient: species i in mixture
Eq (11.60)constant T and {xi}∫=P
ii PdPZ
0
Rˆlnφ
ρρd
ZdZ
PdP
+=
∫∫≠≠
∂∂
+
∂∂
=ρ
ρρρφ
0 ,,
R
1 ,,
Rˆln d
nnZ
ZdZ
nnZ
ijij nTi
Z
nZTii
ijnTii
dZnnZZ
≠
−
∂∂
+−−= ∫,,0
)1(ln)1(ˆlnρ
ρ
ρρφ
RTAR
For EOS: P = f(T, V, ni)
dA = – P dV – S dT
October 14, 2008 Tan - ChE3015 3
∫ −=ρ
ρρ
0
)1(~ dZARwhere constant T and n
Fugacity coefficient: species i in mixture
ijnTii n
AnZZ≠
∂∂
+−−=,,
R~ln)1(ˆln
ρ
φ
nT
RAZ,
~1
∂∂
+=ρ
ρ
Common EoSStatistical-mechanics-based EoS
October 14, 2008 Tan - ChE3015 4
Cubic EoS:
Fugacity coefficients: cubic EOS
Eq (3.42))]()][([
),()( xx
xx bVbV
TabV
RTPσε ++
−−
=
where constant T and n
ijnTii n
AnZZ≠
∂∂
+−−=,,
R~ln)1(ˆln
ρ
φ
∫ −=ρ
ρρ
0
)1(~ dZAR
))(1)()(1(/),(
)(11
ρσρερ
ρ xxx
x bbRTTa
bZ
++−
−=
October 14, 2008 Tan - ChE3015 5
Fugacity coefficient calculation:
∫
++
−−
=ρ
ρρσρερ0 ))(1)()(1(
/),()(1
)(~ dbb
RTTabbAR
xxx
xx
ρερσ
σερ
)(1)(1ln
))((/),())(1ln(
xx
xxx
bb
bRTTab
++
−+−−=
ijnTinAn
≠
∂∂
,,
R~
ρ
{ }[ ] { }{ }
{ }{ }ρε
ρσσε
ρ)()(ln
)()(/),()(lnln
2
xx
xxx
nbnnbn
nbRTTannbnnnn
++
−+−−=R~An
constant T and x
October 14, 2008 Tan - ChE3015 6
iji j
ji axxa ∑∑=
iji j
ji annan ∑∑=2
∑=
∂∂
kikk
nTi
ann
an
j
2,
2
∑=i
iibxb
∑=i
iibnnbi
ni
bnnb
j
=
∂∂
For the partial derivative we need:
Fugacity coefficient calculation:
October 14, 2008 Tan - ChE3015 7
Fugacity coefficient
=),,(ˆln xρφ Ti
Mixture:
Pure:
++
−+
−−−=
ii
iiiiii
i
iiiii bV
bVRT
bTaZV
bVZVTεσ
σεφ ln
)(/)(ln)1(),(ln
( )
++
−
−+−−− ∑
ρερσ
σερ
)(1)(1ln
)(),(),(2
)()(/),()(1ln)1(
)( xx
xxxxxx
x bb
bb
TaTax
RTbTabZZ
bb iikki
++
−
−+
−−−= ∑
)()(ln
)(),(),(2
)()(/),()(ln)1(
)(),,(ˆln
xx
xxxxxx
xx
bVbV
bb
TaTax
RTbTaZ
VbVZ
bbVT iikki
i εσ
σεφ
where V ≡ V(T, P, x)
where Vi ≡ Vi(T, P)
σ = 1 + √2 ε = 1 – √2Peng-Robinson cubic EOS
October 14, 2008 Tan - ChE3015 8
)]()][([),(
)( xxx
x bVbVTa
bVRTP
σε ++−
−=
Molar volume calculation:
[ ] [ ] 0)(),()]()][([)]()][([)( =−+++−++− xxxxxxx bVTabVbVRTbVbVbVP σεσε
=
3
2
1
0
:),,(
cccc
polyrootsPTV x
MathCad code: σ = 1 + √2 ε = 1 – √2Peng-Robinson cubic EOS
