Advanced Thermodynamics
Solution Thermodynamics: Theory
Lecturer:Tsiye Tekleyohanis
Compositions
• Real system usually contains a mixture of fluid.
• Develop the theoretical foundation for
applications of thermodynamics to gas mixtures
and liquid solutions
• Introducing
–
–
–
–
–
chemical potential
partial properties
fugacity
excess properties
ideal solution
Fundamental property relation
• The basic relation connecting the Gibbs energy to
the temperature and pressure in any closed system:
 (nG ) 
 (nG ) 
d (nG )  
dP  
dT  (nV )dP  (nS )dT


 P  T ,n
 T  P ,n
– applied to a single-phase fluid in a closed system
wherein no chemical reactions occur.
• Consider a single-phase, open system:
 (nG) 
 (nG) 
 (nG) 
d (nG)  
dP  
dT   
dni



 P T ,n
 T  P ,n
i  ni  P ,T , n
j
 (nG) 
Define the chemical potential: i  


n
i  P ,T , n j

The fundamental property relation for single-phase fluid systems of constant or
variable composition:
d (nG)  (nV )dP  (nS )dT   i dni
i
When n = 1, dG  VdP  SdT 
  dx
i
i
i
G  G( P, T , x1 , x2 ,..., xi ,...)
 G 
V 


P

T , x
Solution properties, M
Partial properties, M i
Pure-species properties, Mi
 G 
S 


T

 P,x
The Gibbs energy is expressed as a
function of its canonical variables.
Chemical potential and phase equilibria
• Consider a closed system consisting of two phases
in equilibrium:
d (nG)  (nV ) dP  (nS ) dT   i dni d (nG)   (nV )  dP  (nS )  dT   i dni
i
i
nM  (nM )  (nM ) 
d (nG)  (nV )dP  (nS )dT   i dni   i dni
i
Multiple phases at the same T and P are in
equilibrium when chemical potential of
each species is the same in all phases.
i
Mass balance:
dni   dni
 i   i
Partial properties
• Define the partial molar property of species i:
 (nM ) 
Mi  


n
i

 P ,T ,n j
– the chemical potential and the particle molar Gibbs
energy are identical: i  Gi
– for thermodynamic property M: nM  M ( P, T , n1 , n2 ,..., ni ,...)
 M 
 M 
d (nM )  n 
dP

n
dT   M i dni



 P  T ,n
 T  P ,n
i
 M 
 M 
d (nM )  n 
dP

n
dT   M i dni



 P  T ,n
 T  P ,n
i
 M 
 M 
ndM  Mdn  n 
dP

n
dT   M i ( xi dn  ndxi )



 P  T ,n
 T  P ,n
i




 M 
 M 
dM

dP

dT

M
dx
n

M

x
M







i
i
i
i  dn  0

 P T ,n
 T  P ,n
i
i




 M 
 M 
dM  
 dP  
 dT   M i dxi  0
 P T ,n
 T  P ,n
i
dM   xi dM i   M i dxi
i
i
and
M   xi M i  0
i
nM   ni M i  0
i
 M 
 M 

 dP  
 dT   xi dM i  0
 P T ,n
 T  P ,n
i
The Gibbs/Duhem equation
Calculation of mixture
properties from partial
properties
Partial properties in binary solution
• For binary system
M  x1M1  x2 M 2
dM  x1dM1  M1dx1  x2 dM 2  M 2 dx2
Const. P and T, using Gibbs/Duhem equation
dM  M1dx1  M 2 dx2
x1  x2  1
dM
 M1  M 2
dx1
M 1  M  x2
dM
dx1
M 2  M  x1
dM
dx1
The need arises in a laboratory for 2000 cm3 of an antifreeze solution consisting of 30
mol-% methanol in water. What volumes of pure methanol and of pure water at 25°C
must be mixed to form the 2000 cm3 of antifreeze at 25°C? The partial and pure molar
volumes are given.
V  x1V1  x2V2
V  (0.3)(38.632)  (0.7)(17.765)  24.025 cm3 / mol
Vt
2000
n

 83.246 mol
V
24.025
n1  (0.3)(83.246)  24.974 mol
n2  (0.7)(83.246)  58.272 mol
V1t  n1V1  (24.974)(40.727)  1017 cm3 V2t  n2V2  (58.272)(18.068)  1053 cm3
Fig 11.2
Fig 11.2
The enthalpy of a binary liquid system of species 1 and 2 at fixed T and P is:
H  400 x1  600 x2  x1 x2 (40x1  20x2 )
Determine expressions for H1 and H 2 as functions of x1, numerical values for the
pure-species enthalpies H1 and H2, and numerical values for the partial enthalpies at
infinite dilution H1 and H 2 
H  400 x1  600 x2  x1 x2 (40x1  20x2 )
x1  x2  1
H  600  180 x1  20 x
3
1
x1  x2  1
H 1  H  x2
dH
dx1
H1  420  60 x12  40 x13 H 2  600  40 x13
x1  0

H1  420
x1  1
J
mol

H 2  640
J
mol
Relations among partial properties
d (nG)  (nV )dP  (nS )dT   Gi dni
i
• Maxwell relation:
 V 
 S 

   
 T P ,n
 P T ,n
 Gi

 T
 (nS ) 

  


n
P ,n
i  P ,T , n j

 Gi

 T

   Si
 P ,x
 Gi

 P
 (nV ) 

  


n
T ,n 
i  P ,T , n j
 Gi

 P
dGi  Vi dP  Si dT
H  U  PV
H i  U i  PVi

  Vi
T ,x
Ideal-gas mixture
• Gibbs’s theorem
– A partial molar property (other than volume) of a
constituent species in an ideal-gas mixture is equal to
the corresponding molar property of the species as a
pure ideal gas at the mixture temperature but at a
pressure equal to its partial pressure in the mixture.
M iig  Vi ig
M (T , P)  M (T , pi )
ig
i
ig
i
For those independent of pressure, e.g.,
H ig   yi H iig U ig   yiU iig
H (T , P)  H (T , pi )  H (T , P)
ig
i
ig
i
ig
i
i
i
For those depend on pressure, e.g., dSiig   Rd ln P const.T
Siig (T , P)  Siig (T , pi )   R ln
P
P
  R ln
 R ln yi
pi
yi P
M iig (T , P)  M iig (T , pi )
S (T , P)  S (T , P)  R ln yi
ig
i
Gi ig  H iig  TS iig
ig
i
H iig (T , P)  H iig (T , P)
Siig (T , P)  Siig (T , P)   R ln yi
iig  i (T )  RT ln yi P
G ig   yi i (T )  RT  yi ln yi P
i
i
S ig   yi Siig  R yi ln yi
i
i
Gi ig  H iig  TSiig  RT ln yi
iig  Gi ig  Giig  RT ln yi
or
Giig  i (T )  RT ln P
From integration of dGiig  Vi ig dP 
RT
dP  RT ln P
P
Fugacity and fugacity coefficient
• Chemical potential:
 (nG) 
i  


n
i  P ,T , n

j
– provides fundamental criterion for phase equilibria
– however, the Gibbs energy, hence μi, is defined in
relation to the internal energy and entropy - (absolute
values are unknown).
• Fugacity:
Gi  i (T )  RT ln f i
– a quantity that takes the place of μi
With units of pressure
Gi  i (T )  RT ln f i
Giig  i (T )  RT ln P
fi
ig
Gi  Gi  RT ln
P
fi
i 
P
GiR  RT ln i
Residual Gibbs energy
ln i  
P
0
Fugacity coefficient
dP
( Z i  1)
P
(const . T )
VLE for pure species
• Saturated vapor:
• Saturated liquid:
Giv  i (T )  RT ln f i v
Gil  i (T )  RT ln f i l
fi v
G  G  RT ln l
fi
v
i
l
i
VLE
v
f
Giv  Gil  RT ln i l  0
fi
iv  il  isat
f i v  f i l  f i sat
For a pure species coexisting liquid and vapor phases are in equilibrium when they
have the same temperature, pressure, fugacity and fugacity coefficient.
Fugacity of a pure liquid
• The fugacity of pure species i as a compressed
liquid:
Gi  Gisat  RT ln
ln
fi
fi
sat
1

RT

fi
f i sat
P
Pi sa t
Gi  G
sat
i
P
  sa t Vi dP (isothermal process )
Pi
Vi dP
Since Vi is a weak function of P
ln
fi
f i sat
Vi l ( P  Pi sat )

RT
fi
sat
 P
sat sat
i
i
fi   P
sat sat
i
i
Vi l ( P  Pi sat )
exp
RT
For H2O at a temperature of 300°C and for pressures up to 10,000 kPa (100 bar)
calculate values of fi and φi from data in the steam tables and plot them vs. P.
For a state at P:
Gi  i (T )  RT ln f i
For a low pressure reference state:
Gi*  i (T )  RT ln f i *
fi
1
*
fi
1  H i  H i*
* 
ln

(
G

G
)
i
i
ln *  
 ( Si  Si )
*
fi
RT
fi
R T
 Gi  H i  TSi
The low pressure (say 1 kPa) at 300°C: f i *  P*  1 kPa S *  10.3450 J
i
gK
H i*  3076.8 J
g
For different values of P up to the saturated pressure at 300°C, one obtains the
values of fi ,and hence φi . Note, values of fi and φi at 8592.7 kPa are obtained
Values of fi andφi at higher pressure:
fi   P
Fig 11.3
sat sat
i
i
Vi l ( P  Pi sat )
exp
RT
Fig 11.3
Fugacity and fugacity coefficient:
species in solution
• For species i in a mixture of real gases or in a solution of
liquids:
i  i (T )  RT ln fˆi
Fugacity of species i in solution
(replacing the particle pressure)
• Multiple phases at the same T and P are in equilibrium
when the fugacity of each constituent species is the same
in all phases:



fˆi  fˆi  ...  fˆi
The residual property: M R  M  M ig
The partial residual property: M R  M  M ig
i
i
i
Gi R  Gi  Gi ig
i  i (T )  RT ln fˆi
iig  i (T )  RT ln yi P
fˆi
i    RT ln
yi P
ig
i
 (nG) 
i  
 Gi

 ni  P ,T ,n
j
Gi  0
R
fˆi
ˆ
i 
1
yi P
For ideal gas,
fˆi  yi P
fˆi
Gi R  RT ln ˆi ˆ
i 
yi P
The fugacity coefficient of
species i in solution
Fundamental residual-property relation
1
nG
 nG 
d

d
(
nG
)

dT

2
RT
 RT  RT
d (nG)  (nV )dP  (nS )dT   i dni
i
G  H  TS
Gi
nH
 nG  nV
d

dP

dT

dni


2
RT
 RT  RT
i RT
nG
 f ( P, T , ni )
RT
G/RT as a function of its canonical variables allows evaluation of all other
thermodynamic properties, and implicitly contains complete property information.
The residual properties:
R
 nG R  nV R
 nG R  nV R
nH R
Gi
nH R
 
 
dP 
dT   ln ˆi dni
d 
dP 
dT  
dni or d 
2
2
RT
RT
i
i RT
 RT  RT
 RT  RT
R
 nG R  nV R
Gi
nH R
 
d 
dP 
dT  
dni
2
RT
i RT
 RT  RT
 nG R  nV R
nH R
 
d 
dP 
dT   ln ˆi dni
2
RT
i
 RT  RT
Fix T and composition:
V R   (G R / RT ) 

 
RT 
P
T , x
Fix P and composition:
  (G R / RT ) 
HR

 T 
RT
T

 P,x
Fix T and P:
R


(
nG
/ RT ) 

ln ˆi  
ni

 P ,T ,n j
Develop a general equation for calculation of ln ˆi values form compressibilityfactor data.
R


(
nG
/ RT ) 

ln ˆi  
ni

 P ,T ,n j
P
nG R
dP
  (nZ  n)
0
RT
P
P   (nZ  n) 
dP
ˆ
ln i   

0

n
i

 P ,T ,n j P
n
 ( nZ )
1
 Zi
ni
ni
P
dP
ˆ
ln i   ( Z i  1)
0
P
Integration at constant temperature and composition
Fugacity coefficient from the virial
E.O.S
• The virial equation:
Z  1
BP
RT
B   yi y j Bij
– the mixture second virial coefficient B:
i
j
– for a binary mixture: B  y1 y1B11  y1 y2 B12  y2 y1B21  y2 y2 B22
• n mol of gas mixture:
nBP
nZ  n 
RT
  (nZ ) 
P   (nB) 
Z1  
 1




n
RT

n
1  P ,T , n2
1  T , n2


1
ln ˆ1 
RT

P
0
  (nB) 
P   (nB) 

 dP 



n
RT

n
1  T , n2
1  T , n2


1
ln ˆ1 
RT

P
0
  (nB) 
P   (nB) 
dP






n
RT

n
1  T , n2
1  T , n2


B  y1 y1B11  y1 y2 B12  y2 y1B21  y2 y2 B22
12  2B12  B11  B22 yi  ni / n

P
ˆ
ln 1 
B11  y2212
RT


P
ˆ
B22  y1212
Similarly: ln 2 
RT
For multicomponent gas mixture, the general form:

P 
1
ˆ

ln k 
Bkk   yi y j (2 ik   ij ) 

RT 
2 i j

where   2 B  B  B
ik
ik
ii
kk

Determine the fugacity coefficients for nitrogen and methane in N2(1)/CH4(2) mixture
at 200K and 30 bar if the mixture contains 40 mol-% N2.
3
12  2 B12  B11  B22  2(59.8)  35.2  105.0  20.6 cm mol




P
30
ln ˆ1 
B11  y2212 
 35.2  (0.6) 2 (20.6)  0.0501
RT
(83.14)( 200)
ˆ1  0.9511




P
30
2
ˆ
ln 2 
B22  y1 12 
 105.0  (0.4) 2 (20.6)  0.1835
RT
(83.14)( 200)
ˆ2  0.8324
Generalized correlations for the
fugacity coefficient
Z 1 
ln i  
Pr
0
dPr
( Z i  1)
Pr
(const . Tr )
Pr 0
( B  B 1 )
Tr
Z  Z 0  Z 1
ln   
Pr
0
Pr
dP
dP
( Z  1) r    Z 1 r
0
Pr
Pr
0
  PHIB (TR, PR, OMEGA)
(const . Tr )
or
ln   ln  0   ln  1
ln   
0
Pr
0
with
Pr
dPr
dP
1
( Z  1)
ln    Z 1 r For pure gas
0
Pr
Pr
0
Table E1:E4
or
Pr 0
ln   ( B  B1 )
Tr
Table E13:E16
For pure gas
Estimate a value for the fugacity of 1-butene vapor at 200°C and 70 bar.
Tr  1.127
Pr  1.731
 0  0.627
  0.191
Table E15 and E16
ln   ln  0   ln  1
and
 1  1.096
  0.638
f  P  (0.638)(70)  44.7 bar
For gas mixture:

P 
1
ˆ
 Bkk   yi y j (2 ik   ij ) 
ln k 

RT 
2 i j

Bij 
RTcij
Pcij
( B 0  ij B1 ) ij 
Prausnitz et al. 1986
i   j
Empirical interaction parameter
Tcij  (TciTcj ) (1  kij )
2
1/ 3
1/ 3 3
Z cij RTcij
Z ci  Z cj
 Vci  Vcj 
Pcij 
Z cij 

Vcij  
Vcij
2


2


ˆ
ˆ
Estimate 1 and 2 for an equimolar mixture of methyl ethyl ketone (1) /
toluene (2) at 50°C and 25 kPa. Set all kij = 0.
ij 
Bij 
i   j
2
RTcij
Z cij 
Z ci  Z cj
( B 0  ij B1 )
Pcij
Z cij RTcij
3
V V 

Vcij  
Vcij
2


2


Tcij  (TciTcj ) (1  kij ) 12  2B12  B11  B22
Pcij 


ˆ1  0.987


ˆ2  0.983
P
ln ˆ1 
B11  y2212  0.0128
RT
P
ln ˆ2 
B22  y1212  0.0172
RT
1/ 3
ci
1/ 3
cj
The ideal solution
• Serves as a standard to be compared:
M id   xi M iid
G id   xi Gi  RT  xi ln xi
i
Gi  Gi  RT ln xi
id
i
i
ig
ig
cf. Gi  Gi  RT ln yi
S
id
i
 Gi id
 
 T
Vi
id
H
id
i

 G
   i
 P,x
 T
 Gi id
 
 P

 G
   i
T , x
 P
 Gi  TS
id

  R ln xi
P
id
i


T
S
id
i
id
S
  xi Si  R xi ln xi
 Si  R ln xi
i
Vi
id
i
id
V
  xiVi
 Vi
 Gi  RT ln xi  TSi  RT ln xi
i
H
id
i
id
H
  xi H i
 Hi
i
The Lewis/Randall Rule
• For a special case of species i in an ideal solution:
i  i (T )  RT ln fˆi
iid  Giid  i (T )  RT ln fˆi id
Gi id  Gi  RT ln xi
Gi  i (T )  RT ln f i
The Lewis/Randall rule
fˆi id  xi fi
ˆiid  i
The fugacity coefficient of species i in an ideal solution is
equal to the fugacity coefficient of pure species i in the same
physical state as the solution and at the same T and P.
Excess properties
• The mathematical formalism of excess properties
is analogous to that of the residual properties:
M E  M  M id
– where M represents the molar (or unit-mass) value of
any extensive thermodynamic property (e.g., V, U, H, S,
G, etc.)
– Similarly, we have:
 nG E
d 
 RT
E
 nV E
Gi
nH E
 
dP 
dT  
dni
2
RT
i RT
 RT
The fundamental excess-property relation
Table 11.1
(1) If CEP is a constant, independent of T, find expression for GE, SE, and HE as
functions of T. (2) From the equations developed in part (1), fine values for GE , SE,
and HE for an equilmolar solution of benzene(1) / n-hexane(2) at 323.15K, given the
following excess-property values for equilmolar solution at 298.15K:
CEP =-2.86 J/mol-K, HE = 897.9 J/mol, and GE = 384.5 J/mol
  2G E
From Table 11.1: C  T 
2
 T
E
P


 P , x CPE  a  const .
  2G E

2
 T

a
  
T
 P, x
integration
 G E
From Table 11.1: S  
 T
E


 P,x
integration
S E  a ln T  b
 G E

 T

   a ln T  b
 P,x
integration
G E  a(T ln T  T )  bT  c
H E  G E  TS E  aT  c
CPE  a  2.86
897.9  a(298.15)  c
384.5  a(( 298.15) ln( 298.15)  (298.15)  b(198.15)  c
We have values of a, b, c
and hence the excessproperties at 323.15K
The excess Gibbs energy and the
activity coefficient
• The excess Gibbs energy is of particular interest:
G E  G  G id
Gi  i (T )  RT ln fˆi
Gi id  i (T )  RT ln xi f i
fˆi
Gi  RT ln
xi f i
E
fˆi
i 
xi f i
Gi E  RT ln  i
c.f. Gi R  RT ln ˆi
The activity coefficient of species i in solution.
A factor introduced into Raoult’s law to
account for liquid-phase non-idealities.
For ideal solution, G E  0,   1
i
i
 nG E
d 
 RT
E
 nV E
Gi
nH E
 
dP 
dT  
dni
2
RT
RT
RT
i

 nG E
d 
 RT
 nV E
nH E
 
dP 
dT   ln  i dni
2
RT
RT
i

V E   (G E / RT ) 

 
RT 
P
T , x
  (G / RT ) 
H

 T 
RT
T

 P,x
E
E
Experimental accessible values:
activity coefficients from VLE data,
VE and HE values come from mixing experiments.
 (nG E / RT ) 

ln  i  
ni

 P ,T ,n j
GE
  xi ln  i
RT
i
 x d ln 
i
i
i
 0 (const. T , P)
Important application in phase-equilibrium
thermodynamics.
The nature of excess properties
• GE: through reduction of VLE data
• HE: from mixing experiment
• SE = (HE - GE) / T
• Fig 11.4
– excess properties become zero as either species ~ 1.
– GE is approximately parabolic in shape; HE and TSE exhibit
individual composition dependence.
– The extreme value of ME often occurs near the equilmolar
composition.
Fig 11.4