Chemistry Reference
In-Depth Information
R
cor
→ ∞, rendering all
G
αβ
independent of
V
cor
. The only dependence on
V
cor
is
through the denominator in Equations 2.6 and 2.7. If one wishes to calculate the PS
in a small
V
cor
region around
A
or
B
, then one needs to take also the finite limit of the
KBI as in Equation 2.4.
The third quantity is a measure of the deviation of the mixture from SI solution
behavior (see Section 1.3.7 in Chapter 1). This quantity is defined as
∆
AB
=+−
GG
2
(2.10)
AA
BB
AB
and depends on all three KBIs (note that
G
AB
=
G
BA
). From the KB theory, one
obtains the exact relationship for two-component mixtures of A and B (Kirkwood
and Buff 1951; Ben-Naim 2006),
∂
∂
µ
1
x
ρ
∆
A
B
B
=
kT
−
(2.11)
B
x
x
1
+
ρ
xx
∆
A
A
AB AB
pT
,
The SI solution is defined as a solution for which the chemical potential of, say, A is
o
µ
AA
=+
kT
ln
x
for
0
≤≤
x
1
(2.12)
B
A
A
Equation 2.12, combined with the Gibbs-Duhem relationship leads to
o
µµ
B
=+
kT
ln
x
for
0
≤≤
x
1
(2.13)
B
B
B
B
Clea rly, Δ
AB
= 0 is a necessary and sufficient condition for SI behavior (Ben-Naim
2006). Therefore, any finite Δ
AB
value represents a measure of the extent of deviation
from SI behavior. More precisely, in the general case we write
o
SI
SI
E
µµ
=+
kT
ln
xkT
+
ln
γ
=
µµ
+
(2.14)
AA
B
A
B
A
AA
where
x
ρ
ρ
x
xx
′
∆
B
∫
E
SI
BAB
AB AB
µ
=
kT
ln
γ
=
kT
dx
′
(2.15)
A
B
A
B
B
1
+
∆
′′
0
with
x
B
= 1 -
x
A
, Δ
AB
being regarded as a function of
x
B
. Thus, knowing Δ
AB
(
x
B
),
one can estimate the deviation from SI, either in terms of an activity coefficient or
in terms of an excess chemical potential.
The condition for a solution to be SI, Δ
AB
= 0, does not necessarily imply anything
about the PS, for finite volumes
V
cor
; the reason is that the condition Δ
AB
= 0 involves
use of the infinite limit in the integration, as in Equation 2.2, whereas the PS values
