Chemical Engineering
Thermodynamics-II
Solution Thermodynamics: Theory
Lecturer by Dr.S.R.Patel
Thermodynamics of Mixtures
• Thermodynamics II is concerned with the properties of
mixtures
1. Quantifying phase equilibrium behaviour
 At a given pressure and temperature, how many phases exist in a system?
 What is the composition of each phase?
 What are the thermodynamic properties (U,S,Cp,V m,…) of each phase and the
system as a whole?
2. Describing systems that undergo chemical reactions
 Under specified conditions, to what extent does a reaction take place?
 What is the equilibrium composition of the system?
 How much heat is evolved/absorbed by the reaction and the mixing of reactants?
Main Topics
Vapour-Liquid Equilibria (VLE)
• Solution Thermodynamics: chemical potential, fugacity, activity coefficients
• Application of Solution Thermodynamics to VLE (ideal and non-ideal systems)
• Applications of VLE to the separation of mixtures
• Purpose: Design of Separation processes
Chemical Reaction Equilibria
• Calculation of the equilibrium composition
• Effect of T, P, and composition on the reaction equilibrium
Purpose: Maximise the conversion of a chemical reaction
Apart from above overview of LLE,SLLE, SLE
CHEE 311
Lecture 1
3
Thermodynamics in Separation Processes (VLE)
Flash Evaporation
T 2, P 2, y2, out,
T1, P1, x1,in …
T 2, P 2, x2, out, …
Source: https://en.wikipedia.org/wiki/Vapor%E2%80%93liquid_separator
Distillation
T 2, P 2, y2, out,
T1, P1, x1,in …
T 2, P 2, x2, out, …
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
The Chemical Potential
• The basic relation connecting the Gibbs energy to the
temperature and pressure in any closed system:
  ( nG ) 
  ( nG ) 
d ( nG )  
dP

dP  ( 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:
•
For a single phase (gas or liquid), close system, the total Gibbs energy (nG) is a function of T, P
and the numbers of moles of the chemical species (n1, n2, …, ni, ..) present in the 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
G  G ( P, T , x1 , x2 ,..., xi ,...)
i
i
 G 
V 


P

T , x
 G 
S 


T

 P,x
The Gibbs energy is expressed as a
function of its canonical variables.
Overall molar properties for the solution,M
Partial molar properties for species i in the solution, M i
Molar properties for pure compound of species i, Mi
M can be V, U, H,S, G (all are molar properties)
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
Mixture or Solution
•
Consider mixing 1000 cm3 methanol and 1000 cm3 water at 25oC.
•
1000 cm3 MeOH + 1000 cm3 H2O ≠ 2000 cm3 solution !!!!!
•
•
In fact
1000 cm3 MeOH + 1000 cm3 H2O = 1970 cm3 solution
•
Discuss with a person next to you why the volume of the solution is less than
2000 cm3?
What problems we are solving in this chapter?
Variation of thermodynamic properties (e.g., V, G, H, U,
S,etc.) of a multi-component system (or a solution) with
varying compositions (ni or xi)
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
 M 
 M 

 dP  
 dT   xi dM i  0
 P T , n
 T  P , n
i
The Gibbs/Duhem equation
and
M   xi M i  0
i
nM   ni M i  0
i
Calculation of mixture
properties from partial
properties
Partial properties in binary solution
• For binary system
M  x1M 1  x2 M 2
dM  x1dM 1  M 1dx1  x2 dM 2  M 2 dx2
Const. P and T, using Gibbs/Duhem equation
dM  M 1dx1  M 2 dx2
x1  x2  1
dM
 M1  M 2
dx1
M 1  M  x2
dM
dx1
M 2  M  x1
dM
dx1
Graphical Method
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 cm 3 / 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
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 ( 40 x1  20 x2 )
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 H 1 and H 2 
H  400 x1  600 x2  x1 x2 ( 40 x1  20 x2 )
x1  x2  1
H  600  180 x1  20 x
3
1
x1  x2  1
H 1  420  60 x12  40 x13 H 2  600  40 x13
x1  0
H 1  H  x2
dH
dx1

H 1  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

   S i
 P ,x
 Gi

 P
  ( nV ) 

   


n
T ,n
i  P ,T , n j

 Gi

 P
dGi  Vi dP  S i 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 (T , P )  H (T , pi )  H (T , P )
ig
i
ig
i
ig
i
H ig   yi H iig U ig   yiU iig
i
i
For those depend on pressure, e.g., dS iig   Rd ln P const .T
dS iig (T , P )  S iig (T , pi )   R ln
P
P
  R ln
  R ln yi
pi
yi P
M iig (T , P )  M iig (T , pi )
S iig (T , P )  S iig (T , P )   R ln yi
Gi ig  H iig  TS iig
H iig (T , P )  H iig (T , P )
S iig (T , P )  S iig (T , P )   R ln yi
S ig   yi S iig  R  yi ln yi
i
i
Gi ig  H iig  TS iig  RT ln yi
 iig  Gi ig  Giig  RT ln yi
G   yi i (T )  RT  yi ln yi P
ig
i
i
or
 iig  i (T )  RT ln yi P Giig  i (T )  RT ln P
Find expressions of the fugacity coefficient φ and the fugacity f versus the pressure
P for chloroform (CHCl3) at 200°C for the pressure range from 0 to 40 bar. At
200°C, the saturated vapor pressure of chloroform is 22.27 bar.
For CHCl3: TC=536.4 K, PC=54.72 bar, ZC=0.293, VC=239 cm3/mol and ω =
0.222.
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
)
ln *  
 ( Si  Si )
i
i
*
fi
RT
fi
R T
 Gi  H i  TS i
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
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
Gi  0
R
fˆi
ˆ
i 
1
yi P
For ideal gas,
fˆi  yi P
j
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
nH
Gi
 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:
 nG R
d 
 RT
R
 nV R
nH R
Gi
 
dP 
dT  
dni
2
RT
i RT
 RT
 nG R
or d 
 RT
 nV R
nH R
 
dP 
dT   ln ˆi dni
2
RT
i
 RT
 nG R
d 
 RT
R
 nV R
Gi
nH R
 
dP 
dT  
dni
2
RT
i RT
 RT
 nG R
d 
 RT
 nV R
nH R
 
dP 
dT   ln ˆi dni
2
RT
i
 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:
  (nG R / RT ) 
ˆ

ln i  
ni

 P ,T ,n j
Develop a general equation for calculation of ln ˆi values form compressibilityfactor data.
R


(
nG
/ RT ) 
ˆ

ln i  
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 y1 B11  y1 y 2 B12  y 2 y1 B21  y 2 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 y1 B11  y1 y 2 B12  y 2 y1 B21  y 2 y 2 B22
 12  2 B12  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  y 22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 for pure gas
ln i  
Pr
0
dPr
( Z i  1)
Pr
Z 1 
(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
 1  1.096
and
  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
Pcij 
i   j
Empirical interaction parameter
2
Tcij  (TciTcj ) (1  kij )
Z cij RTcij
Z ci  Z cj
Vcij
Z cij 
2
V
Vcij  


1/ 3
ci
V
1/ 3
cj
2




3
Estimate ˆ1 and ˆ2 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  
2
Vcij


2


Tcij  (TciTcj ) (1  kij )  12  2 B12  B11  B22
Pcij 


ˆ1  0.987


ˆ2  0.983
P
ln ˆ1 
B11  y 2212  0.0128
RT
P
ln ˆ2 
B22  y1212  0.0172
RT
1/ 3
ci
1/ 3
cj
Ideal Solution
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 S i  R  xi ln xi
 S i  R ln xi
i
Vi
id
i
id
V
  xiVi
 Vi
 Gi  RT ln xi  TS i  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  Gi id  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 f i
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
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
i RT
 RT
 nG E
d 
 RT
 nV E
nH E
 
dP 
dT   ln  i dni
2
RT
i
 RT
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.