Chemistry Reference
In-Depth Information
attempt to determine the properties of the real system by a perturbation expansion
around the chosen reference. For the solvation behavior of sparingly soluble sol-
utes, the obvious choice for that reference is the infinitely dilute solution at the same
temperature and pressure. This is a natural choice in that the systems of interest are
dilute solutions, whose properties can be obtained from suitable free-energy com-
position expansions, based on fugacity or activity coefficients (Van Ness and Abbott
1982; Prausnitz, Lichtenthaler, and Gomes de Azevedo 1986), around the infinite
dilution counterpart, for which we have a full characterization based on the solvation
formalism discussed in Section 8.2.
Truncated composition expansions have been frequently used in fluid phase equilib-
rium calculations, especially for binary mixtures (O'Connell and Haile 2005) for which
compliance with the Gibbs-Duhem equation also implies exactness of the correspond-
ing differentials. However, care must be exercised when dealing with multicomponent
systems because they might become thermodynamically inconsistent whenever the trun-
cated expressions do not fulfill the condition of exactness, also known as the
Maxwell
relations
(O'Connell and Haile 2005). In other words, if the truncated expansions are
not state functions, their predictions will depend on the path used for the integration of
the differential expressions; consequently, the validity of any related theoretical devel-
opments and corresponding conclusions will become questionable (e.g., see Appendix F
of Chialvo et al. [2008]). Moreover, the identification and calculation of the expansion
coefficients, including their precise microscopic meanings in the context of the refer-
ence system, are as significant as the compliance of the thermodynamic consistency.
In what follows, we describe succinctly the development of isothermal-isobaric
second-order truncated composition expansions of the relevant partial molecular proper-
ties of species in ternary systems, provide a molecular-based interpretation of the expan-
sion coefficients, derive some special case systems, show how the expansions reduce to
well-known results for the zero-density limit, and discuss some modeling implications.
8.3.1 s
econd
-o
rder
T
runcaTed
e
xPansions
For
T
ernary
d
iluTe
s
oluTions
Our goal here is to develop a self-consistent second-order truncated expansion of the
residual chemical potential of species in ternary systems involving a compressible
solvent and two solutes at high (but finite) dilution. For the sake of simplicity, we con-
sider here a ternary mixture at constant pressure and temperature that comprises
N
1
solvent molecules,
N
2
solute molecules of species 2, and
N
3
cosolute molecules of spe-
cies 3, respectively, for which the corresponding residual chemical potentials, that is,
partial molecular fugacity coefficients (O'Connell and Haile 2005) at finite composi-
tion are expanded around the infinite dilution condition for the two solutes as follows,
o
2
2
ln
φ
(,) n
xx
=
φ
+
ax
++++
bx
cx
dx
ex
x
123
1
12
13
12
13
1
223
∞
2
2
ln
φ
(,) n
xx
=
φ
+
ax
+++
bx
cx
dx
+
exx
(8.37)
223
2
22
23
22
23
223
∞
2
2
ln
φ
(,) n
xx
=
φ
+
ax
+++
bx
cx
d
xexx
+
323
3
32
33
32
333
323
where we have chosen
x
2
and
x
3
as the independent variables, μ
i
r
,⊗
(
x
2
,
x
3
)
p,T
=
k
B
T
ln
ϕ
i
⊗
(
x
2
,
x
3
) is the isothermal-isobaric residual chemical potential of species
i
,
