Chemistry Reference
In-Depth Information
−
(
)
(
)
∞
ln
φ
(,) n
xx
=
φ
pkT
δδδ
+−
x
+
δ
x
−
323
3
B
12
13
23
2
13
3
(
)
2
2
0.5δ
x
−
05
. δ
x
−
δ
+−
δδ
x x
12
2
13
12
13
23
23
Note that, after some straightforward manipulation, for example, Appendix E
of Chialvo et al. (2008), these equations are precisely the expressions (4-121a-c)
derived by Van Ness and Abbott (1982) for ternary mixtures obeying the equation of
state
Z
= 1 +
Bp
/
k
B
T
. Likewise, we can derive the corresponding volumetric, enthal-
pic, and entropic expressions for the mixture of imperfect gases after considering
that (Chialvo et al. 2008)
(
)
=+−
(
)
∂∂
kp
δδ δ
kT
ij
1
i
1
j
ij
B
T
(8.62)
(
)
=
(
)
(
)
2
∂∂
kT
pd
δδ
+
−
δ
dT
kT
−
p
δ
+
δδ
−
k T
ij
1
i
1
j
ij
B
1
i
1
j
ij
B
p
8.3.4 h
ighlighTs
and
d
iscussion
on
The
c
omPosiTion
e
xPansions
The proposed second-order composition expansion highlights two features; namely,
(a) that the expressions for the solvent's partial molecular properties do not contain
irst-order terms (e.g.
,
Equation 8.38, Equation 8.42, and Equation 8.45), and conse-
quently, (b) that the quadratic composition dependence of the corresponding solute
and cosolute properties actually comes from that of the solvent, that is, the first
expression in each of Equation 8.38, Equation 8.42, and Equation 8.45. These fea-
tures are transferred to the special system cases, such as binary systems and the cor-
responding mixtures of imperfect gases analyzed in Section 8.3.3.1 to Section 8.3.3.2,
and point to often-overlooked inconsistencies in irst-order composition expansions.
Numerous publications have dealt with solvation phenomena involving dilute solutes
in compressible media (Chialvo 1993b; Munoz and Chimowitz 1993; Li, Chimowitz,
and Munoz 1995; Munoz, Li, and Chimowitz 1995; Ruckenstein and Shulgin 2001b,
2002a; Shulgin and Ruckenstein 2002a), based on the irst-order truncated compo-
sition expansion for the solute partial molecular fugacity coefficients developed by
Jonah and Cochran (1994) as a multicomponent generalization of Debenedetti and
Kumar analysis for binary systems (Debenedetti and Kumar 1986). The common
denominator in all these studies is the following irst-order approximation,
ln
φ
(,) n
xx
φ
∞
−
kx
−
kx
223
2
22
2
233
(8.63)
∞
−
ln
φ
(,) n
xx
φ
3
kx
−
kx
323
232
33
3
which, according to Equation 8.40, would indicate that the validity of Equation 8.63
is tied to the condition
ϕ
o
. The immediate consequences of this condition
are: (a) the Maxwell relations are only satisfied when
k
22
=
k
33
=
k
23
, that is, when the
ϕ
1
(
x
2
,
x
3
) =
