Solution thermodynamics theory
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Transcript Solution thermodynamics theory
Solution thermodynamics
theory—Part IV
Chapter 11
When we deal with mixtures of liquids or
solids
• We define the ideal solution model
• Compare it to the ideal gas mixture, analyze
its similarities and differences
i i (T ) RT ln(yi P)
ig
Component i in a mixture
of ideal gases
This eqn. is obtained by combining
Gi i (T ) RT ln P
ig
Gi (T , P) RT ln yi
ig
i
ig
Now we define
Gi (T , P) RT ln xi
id
i
Ideal solution model
Other thermodynamic properties
for the ideal solution: partial molar volume
Gi Gi (T , P) RT ln xi
id
Gi
P
id
Vi
id
V
id
Gi
T , x P
xiVi xiVi
id
i
i
Vi
T
partial molar entropy in the ideal solution
Gi Gi (T , P) RT ln xi
id
Gi
T
id
S
id
i
S xi S
id
i
id
i
Gi
P, x
T
R ln xi Si R ln xi
P
xi Si R xi ln xi
i
i
partial molar enthalpy in the ideal solution
Gi Gi (T , P) RT ln xi
id
H iid Gi id TSiid Gi RT ln xi TSi RT ln xi H i
H id xi H iid xi H i
i
i
Chemical potential ideal solution
i i (T ) RT ln fˆi
Chemical potential component i
in a Real solution
Gi i (T ) RT ln fi
Chemical potential
Pure component i
Subtracting:
For the ideal solution
fˆi
i Gi RT ln
fi
i
id
ˆf id
Gi RT ln i
fi
Lewis-Randall rule
i Gi RT ln xi
id
i
id
ˆf id
i
Gi RT ln
fi
ˆf id x f
i
i i
id
ˆ
i i
Lewis-Randall rule
(Dividing by Pxi each side of the equation)
When is the ideal solution valid?
• Mixtures of molecules of similar size and
similar chemical nature
• Mixtures of isomers
• Adjacent members of homologous series
Virial EOS applied to mixtures
BP
Z 1
RT
B yi y j Bij
i
j
How to obtain the cross coefficients
Bij
0
1
ˆ
Bij B ij B
Bˆ ij
Bij Pcij
Tcij
Mixing rules for Pcij, Tcij, ij, 11-70 to 11.73
ˆ
Fugacity coefficient i
from virial EOS
P
2
ˆ
B11 y2 12
ln1
RT
12 2B12 B11 B22
For a multicomponent mixture, see eqn. 11.64
problem
• For the system methane (1)/ethane (2)/propane (3)
as a gas, estimate
fˆ , fˆ , fˆ and ˆ ,ˆ ,ˆ
1
2
3
1
2
3
at T = 100oC, P = 35 bar, y1 =0.21, and y2 =0.43
Assume that the mixture is an ideal solution
Obtain reduced pressures, reduced temperatures, and calculate
k
id
Prk 0
1
exp ( Bkk Bkk )
Trk
id
fˆkid k yk P
Results: methane (1) ethane (2) propane (3)
ˆ1 1.02;ˆ2 0.88;ˆ3 0.78
Virial model
fˆ1 7.49; fˆ2 13.25; fˆ3 9.76bar
1 0.98;2 0.88;3 0.76
Ideal solution
ˆf id 7.18; fˆ id 13.25; fˆ id 9.57bar
1
2
3
Now we want to define a new type of
residual properties
• Instead of using the ideal gas as the reference,
we use the ideal solution
Excess properties
M M M
E
id
The most important excess function is
the excess Gibbs free energy GE
Excess entropy can be calculated
from the derivative of GE wrt T
Excess volume can be calculated
from the derivative of GE wrt P
And we also define partial molar excess properties
ˆ
Gi i (T ) RT ln f i
Gi i (T ) RT ln xi f i
id
subtracting :
ˆf
i
i
xi f i
Definition of activity coefficient
Summary
Gi RT ln i
R
ˆ
G RT ln
E
i
i
Summary
G RT ln yi
ig
i
ig
i
Gi RT ln xi
id
i
i Gi RT ln i xi
Note that:
G RT ln yi i (T ) RT ln yi P
ig
i
ig
i
Gi RT ln xi i (T ) RT ln xi f i
id
i
id
ˆ
i (T ) RT ln f i
i Gi RT ln i xi i (T ) RT ln i xi f i
i (T ) RT ln fˆi
problem
• The excess Gibbs energy of a binary liquid mixture
at T and P is given by
G / RT (2.6x1 1.8x2 ) x1 x2
E
a) Find expressions for ln 1 and ln 2 at T and P
Using x2 =1 – x1
GE/RT= x12 -1.8 x1 +0.8 x13
Since i comes from
Gi RT ln i
E
We can use eqns 11.15 and 11.16
E
dG
G G x2
RT ln 1
dx1
E
1
E
E
dG
G G x1
RT ln 2
dx1
E
2
E
then
E
dG / RT
2
1.8 2 x1 2.4 x1
dx1
And we obtain
ln 1 1.8 2 x1 1.4 x 1.6 x
2
1
ln 2 x 1.6 x
2
1
3
1
3
1
If we apply the additivity rule and the
Gibbs-Duhem equation
G
E
RT
xi ln i
i
x d ln
i
i
At T and P
0
i
(b and c) Show that the ln i expressions satisfy
these equations
Note: to apply GibbsDuhem, divide the equation
by dx1 first
Plot the functions and show their
values
0
0
0.2
0.4
0.6
-0.5
-1
-2.5
-3
1
GE/RT
ln 1
-1.5
-2
0.8
ln 2
GE/RT
ln g2
ln g1
x1