Chemical Engineering
Thermodynamics
Chapter 9
Solution of Thermodynamics:
Theory and applications
Chapter Outline
9.1
9.2
9.3
9.4
9.5
Fundamental Property Relation
The Chemical Potential and Phase Equilibria
Partial Properties of Solution
Ideal Gas Mixture
Fugacity and Fugacity Coefficient: Pure
Species and Species in Mixture/Solution
9.6 Fugacity Coefficient of Gas Mixture from the
Virial Equation of State
9.7 Ideal Solution and Excess Properties
9.8 Liquid Phase Properties from VLE data
9.9 Property Changes of Mixing
9.10 Heat Effects of Mixing Process
Multi-component gases and liquids commonly
undergoes composition changes by separation
and mixing processes.
This chapter gives the thermodynamics
applications of both gas mixtures and liquid
solutions.
9.1 Fundamental Property Relation
The definition of the chemical potential of
species i in the mixture of any closed system:
 nG 
i  


n

 P ,T ,n
i
j
the Gibbs energy which is the function
of temperature, pressure and number of moles
of the chemical species present.
9.2 The Chemical Potential and Phase
Equilibria
For a closed system consists of 2 phase in
equilibrium, the mass transfer between phases
may occur.
At the same P and T, the chemical potential of
each species of multiple phases in equilibrium
is the same for all species.
    .......  



i
i
i
9.3 Partial Properties in Solution
M i  Vi , H i , Si , Gi
Pure-species properties,
In a mixture solution:
1) A solution properties, M  V , H , S , G
2) The partial properties base on components
in a solution, M  V , H , S , G
i
i
i
i
i
Partial molar property M i of species i in solution:
 nM 
Mi  


n

 P ,T ,n
i
j
In a solution of liquids, its properties:
M  i xi M i
M  V , H , S,G
M i  Vi , H i , Si , Gi
For a binary solution, its properties:
M  x1M 1  x2 M 2
Similarly, for separate x1 and x2;
M  M 1  x2 M 1  M 2 
M  x1 M 1  M 2   M 2
(See Example 11.3)
9.4 Ideal Gas Mixture
In an ideal gas mixture, partial molar properties
of a species (except volume) is equal to its
molar properties of the species as a pure ideal
gas when the temperature is the mixture
temperature and the pressure equal to its
partial pressure in the mixture.
M ig T , P   M iig T , pi 
Partial pressure of a species i in ideal-gas
mixture:
yi RT
pi  ig  yi P
V
Hence, for enthalpy; H  i yi H i
ig
ig
For entropy; S ig   yi Siig  R  yi ln yi
i
For Gibbs energy;
i
integration constant
G  i yi i T   RT i yi ln yi P 
ig
9.5 Fugacity and Fugacity Coefficient:
Pure Species and Species in Gas
Mixture or Solution of Liquids
For pure species in ideal-gas state;
Gi  i T   RT ln P
ig
For pure species in real-gas state;
Gi  i T   RT ln f i
fi
R
Gi  Gi  RT ln  Gi
P
ig
where
fi
i 
P
is called fugacity coefficient of pure species.
For species i in a mixture of real gases or in a
solution of liquids, in equilibrium;
fˆi   fˆi   ....  fˆi 
The fugacity of each species is the same in
all phases.
For vapor-liquid equilibrium,
fˆi v  fˆi l
For species in gas mixture or solution of
liquids,
Fugacity coefficient of species i in gas mixture;
ˆi
f
ˆi 
yi P
Fugacity coefficient of species i in solution;
ˆi
f
ˆi 
xi P
ig
ˆ
For species i in ideal-gas mixture, i  1
9.6 Fugacity Coefficient for Gas
Mixture from the Virial Equation of
State
P 
1
ˆ

ln k 
B

kk
RT 
2i
 yi y j
j
2
ik

  ij 

i, j, k are run over all species in gas mixture.
  2B  B  B
ik
ik
ii
kk
 0  0  0
kk
jj
ii
  2B  B  B
ij
ij
ii
   , etc.
ki
ik
jj
RTcij Bˆ ij
Bij 
Pcij
(Examples 11.7,
0
1
ˆ
Bij  Bij  ij Bij
11.8 & 11.9)
0.422
Bij  0.083  1.6
Trij
0.172
Bij  0.139  4.2
Trij
0
 
ij
 
i
2
j
1
Tcij  TciTcj  1  kij 
Z ci  Z cj
Z cij 
2
1/ 2
 Vci  Vcj 
Vcij  

2


Z cij RTcij
Pcij 
Vcij
1/ 3
1/ 3
3
9.7 Ideal Solution and Excess Properties
E
M is defined as the difference between the
actual value of solution and value from ideal
solution;
M  M M
E
id
9.8 Liquid Phase Properties from VLE data
In a vapor which a gas mixture
and a liquid solution coexist in
vapor/liquid equilibrium,
For species i in vapor mixture,
ˆf i v  yiˆiv P
Similar for species i in solution,
ˆf i l  yiˆiv P
In vapor-liquid equilibrium, vapor is assumed
ideal gas, hence,
ig
ˆ
i  1
ˆf i l  fˆi v  yi P
Thus, fugacity of species i (in both the liquid
and vapor phases) is equal to the partial
pressure of species i in the vapor phase.
fˆ1  y1 P
fˆ2  y2 P
In an ideal solution,
ˆf i id  xi f i
By introducing an activity coefficient;
fˆ i
fˆ i
i 
 id
xi f i fˆi
9.9 Property Changes of Mixing
This is a mixing process
for a binary system.
The 2 pure species both at
T and P initially separated
by a partition, and then
allow to mix.
As mixing occurs,
expansion accompanied by
movement of piston so that
P is constant.
Heat is added or removed to
maintain the constant T.
When mixing is completed,
the volume changed as
measured by piston
displacement.
Property changes of mixing is given by;
M  M   xi M i
M  V , H , S,G
i
Thus, the volume change of mixing, V and
the enthalpy change of mixing H are found
from the measured quantities V and Q .
t
Association with Q , H is called the heat
effect of mixing per mole of solution.
For volume in binary system;
V t
V  V  x1V1  x2V2 
n1  n2
For enthalpy in binary system;
Q
H  H  x1 H1  x2 H 2 
n1  n2
9.10 Heat Effects of Mixing
Heat of mixing per mole of solution;
H  H   xi H i
i
Solving for binary systems;
H  x1H1  x2 H 2  H
This equation provides the calculation of the
enthalpies of binary mixture for pure
species 1 and 2.
Heat of mixing are similar in many respect
to heat of reaction. When a mixture is formed,
energy change occurs because interaction
between the force fields of the molecules.
However, the heat of mixing are generally
much smaller than heats of reaction.
Heats of Solution
When solids or gases are dissolved in liquids,
the heat effect is called the heat of solution.
This heat of solution is based on the
dissolution of 1 mol of solute.
If species 1 is the solute, x1 is the moles of
solute per mole of solution. Since, H is the
~
heat effect of mixing per mole of solution, H
is the heat effect of mixing per mole of solute.
~ H
H 
x1
Mixing processes are presented by physicalchange equations, same like chemical-reaction
equations.
When 1 mol of LiCl(s) is mixed with 12 mol of
H2O, the process;
LiCl( s)  12H2O(l )  LiCl(12H 2O)
LiCl(12H2O) means a solution of 1 mol of LiCl
dissolved in 12 mol of H2O, giving heat effect
of the process at 25°C and 1 bar;
~
H  33,614 J
(Try Example 12.4, 12.5, 12.6, 12.7, 12.9)