Now we introduce a new concept: fugacity
• When we try to model “real” systems, the
expression for the chemical potential that we
used for ideal systems is no longer valid
• We introduce the concept of fugacity that for
a pure component is the analogous (but is not
equal) to the pressure
We showed that:
Giig  i (T )  RT ln P
Pure component i, ideal gas
Gi  i  i (T )  RT ln( yi P) Component i in a mixture
ig
ig
of ideal gases
Let’s define:
Gi  i (T )  RT ln f i
For a real fluid, we define
Fugacity of pure species i
Residual Gibbs free energy
fi
G  Gi  G  RT ln
P
R
i
ig
i
G  RT ln i
R
i
Valid for species i
in any phase and
any condition
Since we know how to calculate residual
properties… (section 6.2)
G  RT ln i
R
i
R
i
P
G
dP
 ln i   ( Z i  1)
0
RT
P
Zi from an EOS, Virial, van der Waals, etc
Eqn. 6.49)
examples
• From Virial EOS
Bii P
ln i 
RT
• From van der Waals EOS
bi P 
ai P

ln i  Z i  1  ln  Z i 
 2 2
RT  R T Z i

General form, see eqn. 11.37 for cubic EOS.
First solve for Zi in the vapor or in the liquid phase
For cubic EOS
For the vapor phase:
Z 
Z  1    q
( Z   )( Z   )

bP
P
 r
RT
Tr
q
3.52
a (T )  (Tr )

bRT
Tr
For the liquid phase:
1   Z 

Z    ( Z   )( Z   )
 q 
See Table 3.1 for parameters
3.56
Page 98
Fugacities of a 2-phase system
G  i (T )  RT ln f i
v
i
G  i (T )  RT ln f i
l
i
v
l
One component, two phases:
saturated liquid and saturated vapor at Pisat and Tisat
What are the equilibrium conditions for a pure component?
Fugacity of a pure liquid at P and T
v
sat
l
sat
l
f i ( Pi ) f i ( Pi ) f i ( P) sat
f i ( P) 
Pi
sat
v
sat
l
sat
Pi
f i ( Pi ) f i ( Pi )
l
Fugacity of a pure liquid at P and T
f i ( P)   P
l
sat sat
i
i
1
exp
RT

P
Pi
l
V
dP
i
sa t
example
• For water at 300oC and for P up to 10,000 kPa (100 bar)
calculate values of fi and i from data in the steam tables
and plot them vs. P
*

fi
1
1 Hi  Hi
*
* 
ln * 
(Gi  Gi )  
 ( Si  Si ) 
RT
R T
fi

At low P, steam is an ideal gas => fi* =P*
Get Hi* and Si* from the steam tables at 300oC and the lowest P, 1 kPa
Then get values of Hi and Si at 300oC and at other pressures P and calculate fi (P)
Problem
• For SO2 at 600 K and 300 bar, determine good
estimates of the fugacity and of GR/RT.
SO2 is a gas, what equations can we use to calculate
f = /P
Find Tc, Pc, and acentric factor, w, Table B1, p. 680
Calculate reduced properties: Tr, Pr
Tr=1.393 and Pr=3.805
What equations can we use to determine i (gas phase)
Generalized correlations: fugacity coefficient
GiR  RT ln i
P
GiR
dP
 ln i   ( Z i  1)
0
RT
P
P  Pc Pr
ln i  
Pr
ln i  
Pr
0
0
dPr
( Z i  1)
Pr
Pr
dPr
1 dPr
( Z  1)
 w Z
0
Pr
Pr
0
ln i  ln   w ln 
0
   0 ( 1 )w
1
Tables E13 to E16
Lee-Kessler
High P, high T, gas: use Lee-Kessler
correlation
• From tables E15 and E16 find 0 and 1
• 0 = 0.672; 1 = 1.354
•  = 0 1w  0.724
• f =  P = 0.724 x 300 bar = 217.14 bar
• GR/RT = ln   0.323
Problem
• Estimate the fugacity of cyclopentane at 110oC and
275 bar. At 110 oC the vapor pressure of
cyclopentane is 5.267 bar.
• At those conditions, cyclopentane is a high P liquid
1
l
sat sat
f i ( P)  i Pi exp
RT

P
Pi
sa t
l
Vi dP
Find Tc, Pc, Zc,, Vc and acentric factor, w, Table B1, p.
680
Calculate reduced properties: Tr, Prsat
Tr = 0.7486 and Prsat = 0.117
At P = Psat we can use the virial EOS to calculate isat
 Pr 0
1 
Eqn. 11.68
  exp  ( B  wB )
 Tr

0.422 1
0.172
0
B  0.083  1.6 ; B  0.139  4.2
Tr
Tr
sat
i
Eqns. 3.65 and 3.66
isat = 0.9
P-correction term:
Get the volume of the saturated liquid phase, Rackett equation
V
sat
 Vc Z
(1Tr )
c
2/7
Eqn. 3.72, p. 109
Vsat = 107.55 cm3/mol
1
l
sat sat
f i ( P)  i Pi exp
RT
f = 11.78 bar

P
Pi
sa t
l
Vi dP