Environmental Engineering Reference
In-Depth Information
where
V
i
is the molar volume of solute (A, B, or AB
†
)
,
2
δ
i
= Δ
E
i
/V
i
is the
internal
pressure
or the
cohesive energy density
, and
δ
is the same for both solvent and solute, solubility maximum is observed. Values for
δ
δ
s
is the
solvent internal pressure
.If
are listed elsewhere (e.g., Moore and Pearson, 1981). Utilizing the above we can
write
V
AB
†
V
A
RT
(
V
B
RT
(
δ
s
− δ
A
)
2
δ
s
− δ
B
)
2
δ
s
− δ
AB
†
)
2
.
ln
k
sol
=
ln
k
0
+
+
+
RT
(
(5.86)
Since
V
A
,
V
B
, and
V
AB
are similar, it is the
term that determines the value of ln
k
sol
.
If the internal pressures of A and B are similar to that of the solvent s, but different
from that of AB
†
, then the last term dominates and ln
k
sol
will be lower than the ideal
value. The value of ln
k
sol
will be high if the internal pressure for eitherA or B differs
considerably from s, but is similar to that for AB
†
.
δ
5.4.3 L
INEAR
F
REE
E
NERGY
R
ELATIONSHIPS
There exists a relationship between equilibrium constant
K
eq
(
=
−Δ
G
0
/RT)
and
exp
−Δ
G
†
/RT)
for several reactions. The correlation is linear and
is evident for reactions in solution of the type
R
rate constant
k(
=
exp
products, where
R
is the
reactive site and
X
a substituent that does not directly participate in the reaction.When
ln
k
is plotted against ln
K
eq
for different reactions that involve only a change in
X
,a
linear correlation is evident. The linearity signifies that as the reaction becomes ther-
modynamically more favorable, the rate constant also increases. These correlations
are called LFERs and are similar to the LFERs between equilibrium constants that
we encountered in Chapters 3 and 4 (e.g.,
K
oc
versus
K
ow
,
C
i
versus
K
ow
)
. LFERs
are particularly useful in estimating
k
values when no such data are available for a
particular reaction. The correlations are of the type
−
X
+
A
→
K
eq
K
eq
α
,
k
k
=
(5.87)
where
k
and
K
are for the reaction with substituent X
. The relationship between
free energies is therefore
G
†
G
0
Δ
= αΔ
+ β
.
(5.88)
In environmental chemistry, several LFERs have been obtained for the reactions of
organic molecules in the water environment.These relationships are particularly valu-
able in environmental engineering since they afford a predictive tool for rate constants
of several compounds for which experimental data are unavailable. Brezonik (1990),
Wolfe et al. (1980a,b), Betterton, Erel, and Hoffmann (1988), and Schwarzenbach et
al. (1988) have reported LFERs for the alkaline hydrolysis of several aromatic and
aliphatic compounds. The hydrolysis of triaryl phosphate esters follows the Hammett
LFER between log
k
and log
k
0
most readily. The organophosphates and organophos-
phorothionates appear to obey the LFER between log
k
and log
K
a
. The primary
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