Chemistry Reference
In-Depth Information
0
2
lnγ
1
=
∆
x
…
+
(9.28)
12
2
O'Connell (1971a) also gave these formulas in terms of direct correlation function
integrals with
(
)
2
0
1
1
−
C
C
1
2
(
)
−
12
∆
1
0
0
=
1
−
C
(
)
(9.29)
11
0
−
22
and an equivalent for Δ
112
in terms of pair and triplet DCFI. Expressions were also
given for the partial molar volumes and the reduced bulk modulus to lowest order of
the expansion in mole fraction. It is expected that this correlation, with an empirical
value of Δ
12
, would be adequate up to solute mole fractions of
x
2
~ 0.1. Brelvi (Brelvi
and O'Connell 1975c) developed a correlation for Δ
12
in the spirit of his prior work
(Brelvi and O'Connell 1972).
If models for TCFIs or DCFIs are available, the complete expression for both ln γ
1
and ln γ
*
can be obtained through appropriate integration. The expression is simpler
for DCFIs,
x
ρ
j
n
c
1
−
CT
(, )
ρ
∑
=
∫
ij
i
*
ln
γ
=
d
ρ
(9.30)
i
j
ρ
j
1
00
x
ρ
j
If the solution state is specified by pressure rather than density, the same model for
DCFI is used in both Equation 9.5 and Equation 9.30 to obtain ρ and γ
1
*
for a speci-
fied state of
T
,
p
, and {
x
} relative to the state
T
,
p
0
, ρ
0
, and {
x
0
} where γ
*
is unity. The
methodology for this approach is fully described by O'Connell (1981, 1994, 1995).
This method was used for gas solubility and solution densities of liquids by Mathias
and O'Connell (Mathias and O'Connell 1979; O'Connell 1981) and expanded by
Campanella, Mathias, and O'Connell (1987) for a wide variety of systems includ-
ing hydrocarbons, organics, and aqueous solvents. The method has recently been
extended by Ellegaard, Abildskov, and O'Connell
(2009) and Abildskov, Ellegaard,
and O'Connell (2010b) for use in IL systems. For a model of the form of Equation 9.4,
this becomes
(
)
=
HS
(
)
{}
{}
0
{}
{}
0
0
0
ln
γ
Tx
,,
ρ ρ
;,
x
ln
γ
T
,,
ρρ
x
;,
x
i
i
(9.31)
n
c
2
(
)
(
)
00
HS
+
x
ρρ
−
x
bb
−
j
j
ij
ij
j
=
1
