Chemistry Reference
In-Depth Information
Clearly, at this limit, the pair correlation functions are given by
()
≈
()
gr
exp
−
β
Ur
(2.71)
αβ
αβ
Hence, the KBIs in Equation 2.70 are all negative, and are due to the repulsive part
of the hard core potential function Equation 2.57.
The other limit is of very high densities. There is a limit on the total density,
which can be obtained from the equation of state (Equation 2.64). The limiting den-
sity can be obtained either at
p
→ ∞ or at
T
→ 0, and is
[
]
−
1
(
)
−
1
ρ
ax
→+
x
σ
1
−
x
σ
=
σ
(2.72)
m
A
AA
A
B
At this limit we have
σ
σ
2
G
=
−
2
σ
AA
AA
2
σ
σ
G
=
−
2
σ
(2.73)
BB
BB
2
σ
σ
G
=
−
σσ
−
AB
AA
BB
Figure 2.5 shows
G
AA
,
G
BB
, and
G
AB
as a function of
x
A
for three total densities
(all the
G
αβ
are measured in units of σ
AA
= 1). Note that the maximum density of
pure A is ρ
max
,
A
= 1, and for B ρ
max
,
B
= ½. Therefore, we plotted the values of
G
αβ
for
densities below the maximal density of the mixture. All the values of the
G
αβ
are
negative. At very low total densities,
G
αβ
is equal to minus twice the distance of clos-
est approach between α and β.
The limiting coefficients of the preferential solvation are
(
)
=−
(
)
0
δ
=
xx GG
−
σ
σ
xx
A,A
AB
AA
AB
B
AAB
(2.74)
(
)
=−
(
)
0
δ
=
xx G
B
−
G
σσ
x x
B,B
AB
B
B
A
BAB
and
(
)
+−
(
)
=
∆
AB
=+−
GG
2
=
σσ
−
σσ
0
(2.75)
AA
BB
AB
B
A
AB
Clearly, since the system is an SI solution, Δ
AB
= 0, but from
G
αβ
we see that the
ideality arises from the cancellation of the two terms
δ
AA
0
0
and
δ
BB
, which in general
are nonzero, and in our case σ
B
− σ
A
= 1 and σ
B
− σ
A
= −1.
