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In-Depth Information
whose corresponding volumetric, enthalpic, and entropic expressions are given in
full details elsewhere (Chialvo et al. 2008).
8.3.3.2 Mixtures of Imperfect Gases
Low-density gaseous systems are frequently described by virial-based equations of
state, such as the case of the mixtures of imperfect gases whose behavior can be
described accurately at the level of their second virial coefficients (Van Ness and
Abbott 1982). These systems become an instructive example to illustrate that the
consistency of the derived expressions in Section 8.3.1 is independent of the type
of intermolecular forces involved. In fact, within the density range where
Z
= 1 +
Bp
/
k
B
T
is an adequate representation of the system behavior, with
B
(
T
) = Σ
i,j
x
i
x
j
B
ij
(
T
)
the total correlation function integrals
G
ij
(
T
,
p
,
x
k
) become proportional to the second
virial coefficients,
B
ij
(
T
) (Hansen and McDonald 1986; Ben-Naim 2006), that is,
∞
∫
=−
BT
(
)
−
2
lim
GpTx
(, ,
)
=
4
π
exp
−
β
ur
( )
1
rdr
ij
k
ij
ρ
→
0
(8.58)
0
2
ij
()
where we have invoked the definition
g
ij
(
r
;
p
,
T
,
x
k
) = exp(−βω
ij
(
r
;
p
,
T
,
x
k
)), and the fact
that the zero-density limit of the potential of mean force,
lim
rpTx ur
is the corresponding intermolecular pair potential
u
ij
(
r
). Under these conditions, the
molecular expression of the expansion coefficient
k
ij
(
T
,
p
), Equation 8.48, reduces to
the following simple combination of second virial coefficients, that is
,
ω
(; ,, )
=
( )
ij
k
ij
ρ
→
0
(
)
kTp
(,)
=−
2
p BBBBkT
+− −
ij
,
=
2
,, 3
(8.59)
ij
11
ij
1
i
1
j
B
Moreover, by invoking the statistical mechanic expression for the second virial coef-
ficient of the mixture written in Van Ness' compact form (Van Ness and Abbott 1982),
BT
Σ Σ δ
with δ
ij
(
T
) = 2
B
ij
−
B
ii
−
B
jj
, the expansion coeffi-
cients in Equation 8.38 for ternary mixtures of imperfect gases reduce to
()
=
xB T
()
+
x xT
()
ii ii
ij ijij
>
(
)
kTpp
(,)
=
δδ δ
1
+
−
kT
ij
,
=
23
,
(8.60)
ij
i
1
j
ij
B
Consequently, by substitution of Equation 8.60 into Equation 8.38, we obtain the
desired expressions for the mixture of imperfect gases, that is,
+
(
)
(
)
o
2
2
ln
φ
(,) n
xx
=
φ
pkT
05
.
δ
x
+
05
.
δ
x
+
δδδ
12
+−
xx
123
1
B
12
2
13
3
13
23
23
(
)
(
)
∞
ln
φ
(,) n
xx
=
φ
−
p
/
kkT
δ
x
++−
δ
δδ
x
−
(8.61)
223
2
B
12
2
12
13
233
(
)
2
2
0.5δ
x
−
05
.
δ
x
−
δδ
+
−
δ
xx
12
2
13
3
12
1
3
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