Thermodynamics II
Solution Thermodynamics: Theory
Dr.-Eng. Zayed Al-Hamamre
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Content
 Introduction
 Thermodynamics of gas mixtures and liquid solutions
 Chemical potential
 Partial properties of mixtures
 The Ideal-Gas Mixture Model
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Introduction
 In the chemical, petroleum, and pharmaceutical industries multicomponent gases or liquids
commonly undergo composition changes as the result of mixing and separation processes, the
transfer of species from one phase to another, or chemical reaction.
 The property of such systems depend strongly on composition as well as on temperature and
pressure
 The purpose in this chapter is to develop the theoretical foundation for applications of
thermodynamics to gas mixtures and liquid solutions.
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Fundamental property relation
 What is the most important property ?
……G……….
 For pure component;
G = G (T, P)
 For a homogeneous mixture e.g. containing i components mixture;
G = G (T, P, n1, n2, …, ni)
 Also, for closed system: no mass transfer across boundary or in a single-phase fluid in a
closed system wherein no chemical reactions occur
d (nG)  (nV)dP  (nS)dT
 Since n is the total number of moles of the system (= constant);
G = G (T, P)
dG  VdP  SdT
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Fundamental property relation
 G 
 P   V
  T ,n
 G 
 T   S
  P,n
 Open system, single phase
G  nG  g P, T , n1 , n2 ,, ni ,
ni is the number of moles of species i
  (nG ) 
  (nG ) 
  (nG ) 
d (nG )  
dP

dT

i  n  dni

 T 
 P  T ,n

 P ,n
i

 T , P ,n j
all mole numbers held constant
all mole numbers except ni held constant
 The fundamental property relation for single phase fluid systems of variable mass and
composition
d (nG)  (nV)dP  (nS)dT   i dni
i
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Fundamental property relation
 G 
V 
 P  T , n
 G 
S   
 T  P , n
 The chemical potential of species i
 (nG) 
i  

 ni  P,T ,n
j
 This equation forms the basis for the definition of partial properties
 For pure species
n  ni
 n G 
G
i   i   G  ni
 G  ni  0
ni
 ni  P,T
i  G
molar Gibbs energy,
G is intensive property independent of the system size ni
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The Chemical Potential and Phase Equilibria
 Consider the following:
o A closed system and multicomponent
o Containing two phases in equilibrium.
o Mass transfer occurs if the equilibrium is disturbed
o Each individual phase is an open system, free to transfer
mass to the other
d (nG)  (nV) dP  (nS) dT   i dni

d (nG)   (nV)  dP  (nS)  dT   i dni


i

i
nG  (nG)  (nG) 
 The total Gibbs energy of the two-phase system
d (nG)  (nV) dP  (nS) dT   i dni   i dni



i

i
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The Chemical Potential and Phase Equilibria
 compare with d (nG )  (nV )dP  (nS)dT
for the whole closed system
   dn     dn   0
i
i
i
i
i
i
 Mass conservation requires
ni  ni  constant
dni  dni  0
      dn  0
i
i
i
i
 Quantities dni
Hence
are independent and arbitrary (never be zero).
 i   i
(i  1,2,  , N )
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The Chemical Potential and Phase Equilibria
Chemical Potential ( )
 Is an extensive property,
 Provides a measure of the work of a system is capable when a change in mole numbers occurs
e.g. chemical reaction or a transfer of mass.
 For π phases at equilibrium, and N is the number of species, generalization to multiple phases
in equilibrium
i  i    i
i  1, 2,, N 
N ( – 1) equations
In addition to thermal and mechanical equilibrium criteria
T  T   T
P  P     P 
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Partial (molar) properties
 Partial molar properties are defined as partial derivatives with respect to moles
 M 
Mi  

 ni  P,T , n j
M denotes for any extensive properties
 It is a response function, i.e., a measure of the response of total property nM
to the addition at constant T and P of a differential amount of species I to a
finite amount of solution.
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Partial (molar) properties
Notations
 Solution properties
M
V ,U , H , S , A, G
 Pure-species properties
Mi
Vi ,Ui , Hi , Si , Ai , Gi
 Partial molar properties
Mi
Vi ,U i , Hi , Si , Ai , Gi

 G 

Gi   i  

n
 i  P ,T , n j
Re-call
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Partial (molar) properties
 In general, for a homogenous mixture
nM  M(T,P,x1,x 2 ,....xi )
 The total differential of M is
 M 
 (nM)
 (nM)
d(nM)  


dP
dT
i  n  dni
 T 
 P T,n
P,n
 i  P,T ,n j
 Could also be written as
 (M) 
 (M) 
d(nM)  n 
dP  n 

 dT   M i dni (11.9)
 T  P,x
 P  T,x
differentiation at constant composition
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Partial (molar) properties
dni  xi dn  ndxi
Since ni = xin
d(nM)  ndM  Mdn
And
Substitute these terms to Eq. (11.9), and then rearrange:
Rearrange,
=0.0
=0.0
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Partial (molar) properties
 M 
 M 
dM-
 dP-
 dT- M i dxi  0
 P T,x  T  P,x
i
 M 
 M 
dM  
dP



 dT   M i dxi
 P T,x
 T  P,x
i
(11.10)
M   xi M i  0
i
M   xi M i
(11.11)
nM   ni M i
(11.12)
i
i
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Partial (molar) properties
Differentiating Eq 11.11
dM   xi dM i   M i dxi
i
i
 Comparison of this equation with Eq. (11.10) (Subtraction gives) yields
 M 
 M 

 dP  
 dT   xi dM i  0
 P T ,x
 T  P,x
i
 This equation must be satisfied for all changes in P, T, and the Mi caused by changes
of state in a homogeneous phase
 As a special case at constant T and P:
 x dM
i
i
0
i
Gibbs-Duhem equation at constant T and P
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Partial Properties in Binary Solutions
M   xi M i
i
 For a binary solution
M  x1 M1  x2 M 2
A
dM  x1 dM1  M1dx1  x2 dM 2  M 2 dx2
B
 Gibbs-Duhem equation is
 x dM
i
i
0
i
x1 dM1  x2 dM 2  0
 For a binary solution
C
Dividing by dx1, we have the Gibbs-Duhem equation in derivative forms
x1
dM1
dM 2
 x2
0
dx1
dx1
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Partial Properties in Binary Solutions
Since
x1  x2  1
dx1  dx2
dM
 M1  M 2
dx1
Eq. B becomes
D
 From Eq A and D
M1  M  x2
dM
dx1
M 2  M  x1
dM
dx1
These equations can be used to obtain partial
molar properties from solution property.
 As a solution becomes pure in species i, both properties approach pure species
property
lim M  lim M i  M i
xi 1
xi 1
 In the limit of infinite dilution
lim M i  M i
xi 0

 Mi
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Partial Properties in Binary Solutions
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Partial Properties in Binary Solutions
 For a binary (For two components) solution
dH  TdS  VdP  (μ1 )S,Pdn1  (μ2 )S,Pdn2
dU  TdS  PdV  (μ1 )V,S dn1  (μ2 )V,S dn2
dG  VdP -SdT  (μ1 )T,Pdn1  (μ2 )T,Pdn2
dA  PdV -SdT  (μ1 )T,V dn1  (μ2 )T,V dn2
 H 
 U 
 G 
 A 




 
 
 
 n1  S , P,n2  n1 V ,S ,n2  n1 T , P,n2  n1 T ,V ,n2
1  
 Also,
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Example
The need arise 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 of antifreeze, also at 25 C ? Partial molar volumes for
methanol and water in a 30 mol % methanol solution and their pure-species molar
volume, both at 25 C , are:
Methanol (1) and water (2):
V1  38.632 cm 3 mol 1
V1  40.727 cm 3 mol 1
V2  17.765 cm 3 mol 1
V2  18.068 cm 3 mol 1
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Example Cont.
V  x1V1  x2V2  (0.3)(38.632)  (0.7)(17.765)
V  24.025 cm3mol 1
Vt
2000 cm3
3
n


83
.
246
cm
V 24.025 cm3mol 1
n1  x1n  (0.3)(83.246 )  24.974 mol
n2  x2 n  (0.7)(83.246 )  58.272 mol
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Example Cont.
Solution
The line drawn tangent to the V-x1 curve
at x1=0.30, illustrates the values of
V1=40.272 cm3 mol-1 and V2=18.068 cm3
mol-1.
V1t  (24.497 mol )(40.727 cm3 mol 1 )
V1t  1017 cm3
V2t  (58.272 mol )(18.068 cm3 mol 1 )
V2t  1053 cm3
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Relations among Partial Properties
We show now how partial properties are related to one another. By Eq. (11.8), μi ≡ Gi,
and Eq. (11.20 may be written:
d(nG)  (nV)dP  (nS)dT   Gi dni
( 11.17 )
Application of the criterion of exactness, Eq. (6.12) , yields the Maxwell relation,
 V 
 S 

  - 
 T  P,n
 P T,n
 Gi 


 T  

P,n
( 6.16 )
 G 
 (nS)

-
and  i  
 ni P,T,nj
 P T,n
 (nV)



n
 i P,T,nj
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Relations among Partial Properties
 One can write the RHS in the form of partial molar, and change the composition
from n to x.
 Gi 


 T 

 P,x
  i 
 Si  

  T  P,x
 Gi 


 P 

 T,x
  i 
 Vi  

  P  T,x
 Every equation that provides a linear relation among thermodynamic properties of a
constant-composition solution has as its counterpart an equation connecting the
corresponding partial properties of each species in the solution.
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Relations among Partial Properties
 U  PV
H
For n moles ,
 nU
nH
 P(nV)
  ( nH ) 
  ( nU ) 
  ( nV ) 

 P




  n i  P, T, n j   n i  P, T, n j
  n i  P, T, n j
H i  U i  P Vi
 Gi
d G i  
 P

 Gi
 dP  

 T
T ,x


 dT

 P ,x
d G i  V i dP  S i dT
 This may be compared with Eq. (6.10). These examples illustrate the parallelism that exists
between equations for a constant composition solution and the corresponding equations for
the partial properties of the species in solution. We can therefore write simply by analogy
many equations that related partial properties.
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Example
H  400x1  600x2  x1 x2 40x1  20x2 
x2 1  x1
3
H  600  180x1  20 x1
dH
2
  180  60 x1
dx1
H1  H  x2
H 2  H  x1

H1  420
dH
dx1
dH
dx1
2
3
H1  420  60 x1  40 x1
3
H 2  600  40 x1

H 2  640
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The Ideal-Gas Mixture Model
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The Ideal-Gas Mixture Model
 Dalton Law: Every gas has the same V and T.
Pi  yi Pt
Pt   Pi
where yi 
ni
nt
i
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Molar Volume and Partial Molar Volume
 Application of partial properties to molar volume
 (nRT)/P
 (nVi ig ) 
RT  n 
 

Vi  




n
n
P


i
i
 P,T,nj
 ni n j
 P,T,nj 

ig
n  ni   n j
j
Vi ig  Vi ig  V ig 
RT
P
 partial molar volume = pure species molar volume= mixture molar volume
 Note: Partial pressure of species i (It is not partial molar property)
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Partial molar properties
 Properties of each component species are independent of the presence of other
species.
 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.
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Partial molar properties
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Partial molar entropy
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Partial molar Gibbs energy
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Property Change of Mixing
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Property Change of Mixing
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Property Change of Mixing
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Property Change of Mixing
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The Ideal-Gas Mixture Model
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Summary
M iig (T,P)  M iig (T,p i )
(11.21)
H ig   yi H iig
(11.23)
U ig   yiU iig
(11.23)
i
i
H ig   yi H iig  0
i
S ig   yi S iig -R  yi ln yi
i
(where yi  pi /P )
(11.25)
i
S iig is the pure - species value at the mixture T and P.
The entropy change of an ideal gas mixing is
1
S ig   yi S iig  R  yi ln
yi
i
i
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Summary
μiig  Giig  Giig  RT ln yi
( 11.26 )
RT
dP  RTd ln P
P
Giig  Γ i(T)  RT ln P
(11.27 )
From dGiig  Vi ig dP 
Γ i(T) is the integration constant
μiig  Γ i(T)  RT ln yi P
G ig   yi Γ i(T)  RT  yi ln yi P
i
(11.28 )
( 11.29 )
i
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Example
What is the change in entropy when 0.7 m3 of CO2 and 0.3 m3 of N2 each at 1 bar and
25 C blend to form a gas mixture at the same condition? Assume ideal gases.
The entropy change of an ideal gas mixing is
1
Smixing  S ig   yi Siig  R yi ln
yi
i
i
 8.314 J (g mol)-1 K -1 (0.3 ln
1
1
 0.7 ln )
0.3
0.7
 5.079 J (g mol)-1 K -1
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