Chapter 8:
The Thermodynamics
of Multicomponent
Mixtures
1
Important Notation
2
Learning objectives
Be able to:
• Understand the difference between partial molar
properties and pure component properties;
•Use the mass, energy, and entropy balance for mixtures;
•Compute partial molar properties from experimental data;
•Derive the criteria for phase and chemical equilibria in
multicomponent systems.
3
Thermodynamic description of mixtures
In previous chapters (which dealt with pure components), it was
identified that properties such as volume, internal energy,
enthalpy, Helmholtz and Gibbs energies are extensive, i.e., they
depend on the amount of substance present.
V  V  P, T , N 
U  U  P, T , N 
H  H  P, T , N 
A  A  P, T , N 
G  G  P, T , N 
4
Thermodynamic description of mixtures
For mixtures, it is reasonable to expect that these properties will
depend on the amount of each of the C components present,
i.e.:
V  V  P, T , N1, N2 ,..., NC 
U  U  P, T , N1 , N2 ,..., NC 
H  H  P, T , N1 , N2 ,..., NC 
A  A  P, T , N1, N2 ,..., NC 
G  G  P, T , N1 , N2 ,..., NC 
5
Thermodynamic description of mixtures
For mixtures, it is reasonable to expect that the corresponding
molar properties will depend on the mole fractions of the C
components present.
But, there is a subtle point to note:
Consider an open system. Depending on the type of interaction
with the surroundings, the number of moles of each component
may vary independently: N1 , N2 ,..., NC  can each be treated as
an independent variable.
The same does not happen with mole fractions. Why?
6
Thermodynamic description of mixtures
The summation of the mole fractions of all C components is
always equal to 1.
C
x
i 1
i
1
Only (C-1) mole fractions can be treated as independent. One of
them, say that of component C, can be calculated as function of
the other mole fractions
C 1
xC  1   xi
i 1
7
Thermodynamic description of mixtures
For the molar properties:
V  V  P, T , x1, x2 ,..., xC 1 
U  U  P, T , x1 , x2 ,..., xC 1 
H  H  P, T , x1 , x2 ,..., xC 1 
A  A  P, T , x1, x2 ,..., xC 1 
G  G  P, T , x1 , x2 ,..., xC 1 
8
Thermodynamic description of mixtures
Would a relationship such as this one be true?
C
V T , P, x1 , x2 ,..., xC 1    xi V i T , P 
i 1
9
Thermodynamic description of mixtures
Would a relationship such as this one be true?
C
V T , P, x1 , x2 ,..., xC 1    xi V i T , P 
i 1
Let us watch a movie to help answer this question.
http://www.youtube.com/watch?v=84k206qaVRU
10
Thermodynamic description of mixtures
We see that:
C
V T , P, x1 , x2 ,..., xC 1    xi V i T , P 
i 1
11
Thermodynamic description of mixtures
We see that:
C
V T , P, x1 , x2 ,..., xC 1    xi V i T , P 
i 1
Can we quantify this difference, perhaps give it a name?
12
Thermodynamic description of mixtures
Volume change of mixing:
C
 mix V  V T , P, x    xi V i T , P 
i 1
Enthalpy change of mixing:
C
 mix H  H T , P, x    xi H i T , P 
i 1
13
Thermodynamic description of mixtures
Volume change of mixing:
C
ˆ T , P 
 mixVˆ  Vˆ T , P, w    wV
i i
i 1
Enthalpy change of mixing:
C
 mix Hˆ  Hˆ T , P, w    wi Hˆ i T , P 
i 1
14
Thermodynamic description of mixtures
Methyl formate+ethanol
298.15 K
Methyl formate+methanol
Benzene+C6F5Y at 298.15 K
15
Example 1
What is the density of an equimolar liquid solution of methyl
formate and ethanol at 298.15 K, 1 atm?
Data
Density of pure ethanol at 298.15 K: 0.789 g/cm3
Density of pure methyl formate at 298.15 K: 0.977 g/cm3
Molar mass of ethanol: 46.07 g/mol
Molar mass of methyl formate : 60.1 g/mol
16
Example 2
A continuous mixer operating at steady-state blends has two
input streams (streams 1 and 2) to produce one output stream
(stream 3), all of them liquid binary mixtures of water and
sulfuric acid. Stream 1 has a temperature of 21.1oC and a
sulfuric acid mass fraction of 0.3 and flow rate of 1 kg/s. Stream
2 is at the same temperature and a water mass fraction of 0.2
and flow rate of 2 kg/s.
a) What is the molar enthalpy change of mixing of stream 1?
b) What is the mass fraction of sulfuric acid in stream 3?
c) What is the heat transfer rate to/from the mixer if the
temperature of output stream is 21.1oC?
d) What is the temperature of the output stream if the mixer
operation is adiabatic?
17
Sulfuric acid + water
18
Thermodynamic description of mixtures
Partial molar properties
The discussion that follows is applicable to V, U, H, S, A, G,
but it will be developed based on volume.
Consider a mixture:
NV  V T , P, N1 , N2 ,..., NC 
V  V T , P, x1 , x2 ,..., xC 1 
N  N1  N2  ...  NC
19
Thermodynamic description of mixtures
Partial molar properties
If the amounts of all components double:
2 NV  V T , P, 2 N1, 2 N2 ,..., 2 NC   2V
V  V T , P, x1 , x2 ,..., xC 1 
20
Thermodynamic description of mixtures
Partial molar properties
If the amounts of all components are multiplied by a positive
constant  :
 NV  V T , P,  N1,  N2 ,...,  NC   V
V  V T , P, x1 , x2 ,..., xC 1 
The total volume is a homogeneous function of degree 1.
21
Thermodynamic description of mixtures
Partial molar properties
Now differentiate this expression with respect to  :
C 
d  V 
  V  
 d   Ni  
 V   



d
d

i 1     N i  


T , P ,  N j i
 V 


 Ni 



i 1  N i T , P , N
j i


C
22
Thermodynamic description of mixtures
Partial molar properties
 V 
V   Ni 

i 1
 Ni T , P , N ji
C
C
Ni  V 
V
 

N i 1 N  Ni T , P , N
j i
 V 
V   xi 

i 1
 Ni T , P , N ji
C
23
Thermodynamic description of mixtures
The partial molar volume is defined as:
 V 
Vi 

 Ni T , P , N ji
C
V   xi Vi
i 1
The molar volume of the mixture is the average weighted by
the mole fractions of the partial molar volumes (and not of
the pure component molar volumes).
24
25
Gibbs-Duhem equation
The Gibbs-Duhem equation can be developed in generalized
form (see the textbook). Here, it is developed based on the
Gibbs energy:
G  G T , P, N1 , N2 ..., NC 
C
 G 
 G 
 G 
dG  
dNi

 dT  
 dP   
 T  P, N
 P T , N
i 1  Ni T , P , N
j i
C
dG   SdT  VdP   G i dN i
i 1
26
Gibbs-Duhem equation
Side note:
Gi  i
Chemical potential of component i.
27
Gibbs-Duhem equation
From the discussion of partial molar properties, we also have
that:
C
G   N i Gi
i 1
C
C
i 1
i 1
dG   N i dGi   Gi dN i
28
Gibbs-Duhem equation
The previous slides have two expressions for changes in Gibbs
energy:
C
dG   SdT  VdP   G i dN i
i 1
C
C
i 1
i 1
dG   N i dGi   Gi dN i
Subtracting them:
C
 SdT  VdP   N i dG i  0
i 1
29
Gibbs-Duhem equation
This is the Gibbs-Duhem equation:
C
 SdT  VdP   N i dG i  0
i 1
C
 SdT  VdP   xi dG i  0
i 1
Its interpretation is that the changes in temperature, pressure,
and chemical potentials are interrelated. In other words, they
cannot all change independently.
30
Generalized Gibbs-Duhem equation
For a molar thermodynamic property  (other than
temperature, pressure, and mole numbers):
C






 

 dT    dP   xi d i  0
 T  P , x
 P T , x
i 1
Please refer to the proof in the textbook.
31
Experimental determination of partial
molar volume and enthalpy
The next slide shows a table of experimental density data for
liquid mixtures of water and methanol at 298.15 K.
32
33
Experimental determination of partial
molar volume and enthalpy
How to use the information available in this table to determine
the partial molar volumes?
34
Binary solutions
dM
M 1  M  x2
dx1
dM
M 2  M  x1
dx1
Obtain dM/dx1 from (a)
Example 11.3
• We need 2,000 cm3 of antifreeze solution: 30
mol% methanol in water.
• What volumes of methanol and water (at 25oC)
need to be mixed to obtain 2,000 cm3 of
antifreeze solution at 25oC
• Data:
V1  38.63cm / mol
V1  40.73cm / mol methanol
V2  17.77cm / mol
V2  18.07cm / mol water
3
3
3
3
solution
• Calculate total molar volume of the 30% mixture
• We know the total volume, calculate the number of
moles required, n
• Calculate n1 and n2
• Calculate the total volume of each pure species needed
to make that mixture
Note curves for partial molar volumes
40
Experimental determination of partial
molar volume and enthalpy
How to use the information available in this table to determine
the partial molar volumes?
41
+
42
Experimental determination of partial
molar volume and enthalpy
The molar volume change of mixing is:
C
 mix V  V T , P, x    xi V i T , P 
i 1
C

 mix V   xi V i  V i
i 1

For a binary mixture:



 mix V  x1 V 1  V 1  x2 V 2  V 2

43
Experimental determination of partial
molar volume and enthalpy
Let’s evaluate the slope of the curve in the binary diagram (at
constant temperature and pressure):
 V 1 
  mix V 
 x1 
 

  V 1 V 1 
  x1 
 x1 T , P
 x1 T , P
 x1 T , P

V

2
V 2

 V 2 
 x2 


  x2 
 x1 T , P
 x1 T , P
44
Experimental determination of partial
molar volume and enthalpy
  mix V 

  V 1 V 1  V 2 V 2 
 x1 T , P

 

 V 1 
 V 2 
x1 
  x2 

 x1 T , P
 x1 T , P
0: Why?
45
Experimental determination of partial
molar volume and enthalpy
  mix V 

  V 1 V 1  V 2 V 2 
 x1 T , P

 

 V 1 
 V 2 
x1 
  x2 

 x1 T , P
 x1 T , P
Generalized Gibbs-Duhem equation
46
Experimental determination of partial
molar volume and enthalpy
  mix V 

  V 1 V 1  V 2 V 2
 x1 T , P

 

47
Experimental determination of partial
molar volume and enthalpy
Note that:
  mix V 
 mix V  x1 
 
 x1 T , P








x1 V 1  V 1  x2 V 2  V 2  x1 V 1  V 1  x1 V 2  V 2 
V
2

V 2  A
48
Experimental determination of partial
molar volume and enthalpy
Note that:
  mix V 
 mix V  x2 
 
 x1 T , P








x1 V 1  V 1  x2 V 2  V 2  x2 V 1  V 1  x2 V 2  V 2 
V
1

V 1  B
49
+
50
Experimental determination of partial
molar volume and enthalpy
The experimental data are often fitted to a polynomial form,
known as Redlich-Kister expansion:
n
 mix V  x1 x2  ai  x1  x2 
i
i 0
For water(1) + methanol(2) at 298.15K with volume in m3/mol:
a0  4.0034 106
a1  0.17756 106
a2  0.54139 106
a3  0.60481106
51
Equilibrium in multicomponent systems
The discussion extends that of Chapter 7, which
dealt with pure components.
All the steps are very, very similar.
52
Equilibrium in an isolated system
No mass in or out; neglect changes in kinetic and potential
energies. The energy and entropy balances are:
dU
dV
QP
dt
dt
dS Q
  S gen
dt T
with Sgen  0 because of the 2nd law of thermodynamics.
53
Equilibrium in an isolated system
0, no heat transfer to an isolated system
dU
dV
QP
 0  U  constant
dt
dt
0, no change in volume in an isolated system
0, no heat transfer to an isolated system
dS Q
  S gen  S gen  0
dt T
54
Equilibrium in an isolated system
dS
 S gen  0
dt
for systems in which M, U, V are constant.
Away from equilibrium, the system conditions change,
increasing its entropy.
After a long enough wait (from a fraction of a second to many
years, depending on the system), the system attains a
condition in which its state properties no longer change,
including the entropy. At this state, the equilibrium state, the
entropy has a maximum value.
55
Equilibrium in multicomponent systems
Chemical reactions may occur in a
multicomponent system.
Let us begin with a situation in which chemical
reactions do not occur.
56
Equilibrium in multicomponent systems
Consider an isolated system composed of two subsystems
containing a non-reactive multicomponent mixture:
Ni  N  N
I
i
II
i
i  1,..., C
U  U I  U II
V  V I  V II
S  S I  S II
57
Equilibrium in multicomponent systems
Consider an isolated system composed of two subsystems
containing a non-reactive multicomponent mixture:
Ni  N  N
I
i
II
i
i  1,..., C
U  U I  U II
V  V I  V II
S  S I  S II
Find the equilibrium condition if the internal wall is diathermal, rigid, and impermeable.
Find the equilibrium condition if the internal wall is diathermal, moveable, and
impermeable.
58
Equilibrium in multicomponent systems
Consider an isolated system composed of two subsystems
containing a non-reactive multicomponent mixture:
Ni  N  N
I
i
II
i
i  1,..., C
U  U I  U II
V  V I  V II
S  S I  S II
Find the equilibrium condition if the internal wall is diathermal, moveable, and
permeable.
59
Equilibrium in multicomponent systems
S  S I  S II
dS  dS I  dS II
I
I
C
 1

P
Gi
I
I
I
dS   I dU  I dV   I dN i  
T

T
i 1 T


II
II
C
 1

P
Gi
II
II
II
 II dU  II dV   II dN i 
T

T
i 1 T


dU II  dU I
dV II  dV I
At constant U, V, Ni
dNiII  dN iI
60
Equilibrium in multicomponent systems
I
II

P P  I
1 
Gi Gi  I
 1
I
dS   I  II  dU   I  II  dV    I  II dNi


T 
T
T T 
i 1 T
T


I
II
C
At equilibrium:
1
1
 II
I
T
T
I
II
P
P
 II
I
T
T
I
i
II
i
G
G
 II
I
T
T
i  1,..., C
61
Equilibrium in multicomponent systems
In summary, at equilibrium:
T I  T II
P I  P II
I
i
G G
II
i
i  1,..., C
62
Equilibrium in multicomponent systems
Find the two-phase equilibrium conditions of a non-reactive
multicomponent system with specified temperature, pressure,
and amounts of each component.
63
Closed system, constant T and P
dG  dG  dG  0
I
II
C
dG |T , P   G i dN i
At constant T and P
C


i 1
dG   G  G dN  0
i 1

I
i
II
i
I
i
G G
I
i
II
i
64
Recommendation
Read chapter 8 and review the corresponding examples.
65