THERMODYNAMICS II
Fugacity And Fugacity
Coefficient of Pure Species
Recap
RT
dg  v dP 
dp  RTd ln P(const T )
P
ig
i
ig
i
g  i (T )  RT ln P
ig
i
[36]
[37]
  i (T )  RT ln yi P
[38]
g   yi i (T )  RT  yi ln yi P
[39]
ig
i
ig
The Chemical Potential
 The chemical potential µi, provides a criterion for phase
equilibria
 Absolute values for chemical potential do not exist
although they are important for phase & chemical
equilibria.
Where problem solving is concerned μ is deficient for
the following reasons:
Chemical Potential
i)
it is related to primitive quantities, u & s
through g
ii)
from eq. (38): μi
-∞ as P or y
0
iii)as P or species i concentration approaches 0
μ goes
to -∞
(40)
. Fugacity
 Need for a more robust parameter that
accommodates values of P & y down to zero
which is the domain of ideal gases
 A quantity that behaves like μ without its
deficiencies
 G. N. Lewis suggested defining a new
thermodynamic property: fugacity, f
Fugacity
 Tendency to flee (escape) and an exact measure of
a component’s volatility
 Fugacity has units of pressure and plays the same
role in real gases that partial pressures play in
ideal gases.
Starting with the equation:
d i  vi dP
  i 
 vi


 P  T ,n j
At constant T
(41)
(42)
Fugacity
 using the ideal gas relation for this low
pressure situation:
 v
d i  
 ni

RT

dP 
dP
P
T , P ,n j 1
 Choosing a reference state denoted by “o”
P
o
 i   i  RT ln o
P
pi
o
 i   i  RT ln o
pi
ˆ
f
Fugacity is defined as:
 i   io  RT ln io
fˆi
(43)
(44)
(45)
(46)
Fugacity
 Fugacity has units of pressure and is to real gases as
partial pressure is to ideal gases
 Fugacity applies to liquid as well as solids
 Since as pressure tends to zero gases approach ideality
we may complete the fugacity definition, thus:
 fˆi 
lim 0    1 (ideal gas )
p
p 
 i
 Which together with eq. (46) forms the definition for
Fugacity
(47)
Fugacity coefficient
fˆi
pi , sys
fˆi
 ˆ i 
y i Psys
ˆ i  1
Attractive & repulsive forces balance
ˆ i  1
Tendency to escape is less than for an ideal gas
ˆ i  1
Stronger repulsive forces
(48)
Other forms of Fugacity

ˆf   nf  
ˆ
or
f

x
f

i
i
i
 n 
i
i  T .P ,n j  i

 Total solution fugacity
f
lim 0 
p
P

g  g  RT ln
o

 1


f
fo
(50)
(51)
f

Psys
fˆi
g i  g  RT ln o
fˆi
fi
& i 
Psys
Pure species fugacity
 fi 
lim 0    1
p
P
(49)
o
i
(52)
(53)
(54)
Fugacity for chemical equilibrium
mia =mib
(55)
é fˆ a é b ,o
é fˆ b é
mi +RTlné ai ,o é=mi +RTlné ib ,o é
éfi é
éfi é
a ,o
a ,o é
bé
é
é
ˆ
ˆ
f
f
mia ,o-mib ,o=RTlné i b ,o é+RTlné i a é
éfi é
éfi é
éfˆ b é
0=RTlné i a é
éfi é
or fˆi a = fˆi b
(56)
(57)
Definition from(46)
(58)
(59)
Pure gas fugacity
 Vapour-liquid equilibria
ˆf v  fˆ l
i
i
(60)
 For gases the reference state must be low enough
pressure to approximate ideal gas behaviour at the
system temperature
 f iv 
g i  g  RT ln 

P
 low 
o
i
v
f
 iv  i
Psys
(61)
(62)
Pure gases
 Data for fugacity calculation may be obtained
from



1.
2.
3.
tables
E.O.S
Generalized correlations
 Eg.

(a)Determine the fugacity and fugacity coefficient
of saturated steam at 1.0 atm.
Derive an expression for the fugacity of a
pure gas from the van der Waals E.O.S
 Using the viral form of the van der Waals EOS
truncated to the 2nd term:


a 

2
3
PV  n  RT   b 
P

tenms
in
P
,
P
,......


RT




 fiv 
a 
 RT
Plow  P  b  RT  dP  RT ln  Plow 
P

 fiv 
a 

  RT ln  i
b 
 P  RT ln 

RT 

 P 
v
f

a  P  For an ideal gas a=b=0 so φ=1 as
for pure i :  iv  i  exp  b 


P
RT
RT

 expected

Fugacity from generalized correlations
 f iv 
g i  g   vi dP  RT ln 

Dividing
by RT and subtracting ᶴ (1/P)dP (63)
 Plow

Plow
from each side gives:
P
o
i
P
 f iv  P 1
vi
1
dP   dP  ln 
   dP

RT
P
 Plow  Plow P
Plow
Plow
Combining integrals on the
P
(64)
LHS and simplifying the RHS:
 f iv 
1
 v
v

dP

ln

ln

 
i
  RT P 
P
Plow 
P
ln  
v
i
P
 z i  1
Pideal
In terms of compressibility factor (65)
dP
P
In terms of reduced variables
(66)
Pr
ln  iv 
dPr


z

1

Pr , ideal
Pr
(67)
Residual Property
 No experimental method to directly measure g or g/RT
 Residual properties have readily numerical values
 Residual Gibbs energy : gR=g – gig (same T & P)
 Similarly : vR = v- vig = v – RT/P but v = zRT/P
v
R
RT

P
 z  1
m R  m  m ig
Fundamental property relations for
residual properties
 Recall:
v
h
 g 
d
dp 
dT

2
RT
 RT  RT
ig

g
 For an ideal gas : d 
 RT

 Subtracting:

 gR
d 
 RT
ig
 v ig
h
 
dP 
dT
2
RT
 RT
R
 vR
h
 
dP 
dT
2
RT
 RT
Other forms
v R  g R / RT 


RT  P  T
R

h
g / RT 
 T 

RT
 T  P
R
g R  h R  TS R
R
R
R
s
h
g


R
RT RT
g  v
 
d 
dP
 RT  RT
int egrating
R
R
R
R
g
v

dP
0 RP
RT
P
(const P )
(const T )
R
R
g
v
dP


z  1
(const T )
RT 0 RT
P
R
differentiating and substituting for h
P
P  z  dP
h
 T   
0 T
RT
 P P
R
(const T )
P  z  dP
P
s
dP
 T   
  z  1
0 T
0
R
P
P
 P
R
(const T )
Residual Property Relation
 Recall: mR=m-mig
multiplying by n & differentiating w.r.t ni:
  nm R 
  nm 
  nm ig 







n

n

n
i
i
i

 T , P ,n j 
 T , P ,n j 
 T , P ,n j
Each term has the form
of a partial property
(68)
m i  mi  m i
Rewriting for residual
Gibbs energy
(68)
ig
Defines partial residual
Gibbs energy
(68)
ig
R
g i  gi  g i
R
i  
ig
i
fˆi
 RT ln̂
yi P
g i  RT ln  i
R
From μi=gi
(68)
(68)
Residual Property Relation
 The fundamental residual property relation can also
be written for fugacity coefficient as:


  ng R / RT 
ln ˆ i  


n
i

 P ,T ,n j
 Which demonstrates that lnφi is a partial property
w.r.t gR/RT
 Eg.
 Develop a general equation for calculating lnφi
values from compressibility-factor data
(69)
Lee-Kesler EOS
 Lee-Kessler correlations may be used to solve for φ

Log φi =log φ(o) + ωlog φ(1)
 Eg.
 Determine the fugacity and fugacity coefficient of
ethane at a pressure of 50.0 bar and a temperature
of 25.0 oC using generalized correlations.

Pc=48.7 bar, Tc=305.5 K, ω=0.099