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Chapter 6
Thermodynamics of Mixtures
A mixture is a gaseous, liquid or solid phase containing more than one component. The chemical
process industries mainly deal with the separation of mixtures and their properties are needed
in design calculations. Mixture properties, however, cannot be determined from the weighted
average of the properties of the pure components comprising the mixture. The purpose of this
chapter is to show how to calculate the mixture properties.
6.1 EQUATIONS OF STATE FOR MIXTURES
6.1.1 Virial Equation of State
At low to moderate pressures, the virial equation of state, truncated after the second term, for
a gas mixture consisting of
k
species is given by
Z
mix
=1+
B
mix
P
RT
(6.11)
where the parameter
B
mix
is de
f
ned as
B
mix
=
k
X
i
=1
k
X
j
=1
y
i
y
j
B
ij
(6.12)
in which
y
i
represents the mole fraction of species
i
.When
i
=
j
, the terms
B
ii
and
B
jj
represent
the second virial coe
ﬃ
cients corresponding to the pure components
i
and
j
, respectively. When
i
6
=
j
, the unlike (or cross) virial coe
ﬃ
cients are symmetric, i.e.,
B
ij
=
B
ji
. In engineering
calculations involving normal
F
uids, the second cross virial coe
ﬃ
cient,
B
ij
, can be estimated
from Eqs. (3.110)(3.112) with the following modi
f
cations of
T
c
,
ω
,and
P
c
T
c
ij
=(1
−
k
∗
ij
)
q
T
c
i
T
c
j
ω
ij
=
ω
i
+
ω
j
2
P
c
ij
=
4
T
c
ij
³
e
V
1
/
3
c
i
+
e
V
1
/
3
c
j
´
3
Ã
P
c
i
e
V
c
i
T
c
i
+
P
c
j
e
V
c
j
T
c
j
!
(6.13)
where
e
V
c
i
and
e
V
c
j
are the critical molar volumes of components
i
and
j
, respectively
1
.T
h
e
values of the binary interaction parameter
2
,
k
∗
ij
, have been reported by Chueh and Prausnitz
(1967a), Tsonopoulos (1979), Nishiumi
et al.
(1988), and Meng
et al
. (2007). For binary
1
In the absence of data, critical molar volume can be estimated from the following formula
e
V
c
=
RT
c
P
c
(0
.
290
−
0
.
085
ω
)
2
Keep in mind that interaction parameters have no theoretical basis, they are simply adjustable parameters
to
f
t experimental data.
123
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View Full Document mixtures of chemically similar molecules, i.e., similar size, shape, and chemical nature,
k
∗
ij
may
be considered zero. Chueh and Prausnitz (1967b) proposed the following equation to estimate
k
∗
ij
k
∗
ij
=1
−
8
q
e
V
c
i
e
V
c
j
(
e
V
1
/
3
c
i
+
e
V
1
/
3
c
j
)
3
(6.14)
In the absence of reliable data, the second virial cross coe
ﬃ
cient,
B
ij
,maybeassumedto
be the arithmetic mean of
B
ii
and
B
jj
,i
.e
.
,
B
ij
'
(
B
ii
+
B
)
/
2
.
Example 6.1
A rigid tank contains
3
moles of ethane (1) at
373 K
and
4bar
.E
s
t
im
a
t
e
thepressureinthetankwhen
5
moles of
n
butane (2) is added isothermally to the tank. The
mixture is described by the virial equation of state and the following critical properties are
provided
Component
e
V
c
(cm
3
/
mol)
Ethane
145
.
5
n
Butane
255
.
0
Solution
From Appendix A
Component
T
c
(K)
P
c
(bar)
ω
Ethane
305
.
34
9
0
.
099
n
Butane
425
.
03
8
0
.
199
The
f
nal pressure in the tank can be estimated from Eq. (6.11), i.e.,
(
Z
mix
)
f
=
P
f
(
e
V
mix
)
f
RT
=1+
(
B
mix
)
f
P
f
RT
=
⇒
P
f
=
RT
(
e
V
mix
)
f
−
(
B
mix
)
f
(1)
where the subscript "f" represents the
f
nal state. The term
B
mix
is calculated from Eq. (6.12)
as
B
mix
=
y
2
1
B
11
+2
y
1
y
2
B
12
+
y
2
2
B
22
(2)
The total tank volume,
V
, can be calculated from the initial conditions, i.e., when the tank is
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This note was uploaded on 04/06/2010 for the course CHE 327 taught by Professor Ozbelge during the Fall '10 term at Middle East Technical University.
 Fall '10
 ozbelge
 pH

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