Environmental Engineering Reference
In-Depth Information
E
XAMPLE
5.9 A
CTIVATION
P
ARAMETERS FOR A
G
AS
P
HASE
R
EACTION
The bimolecular decomposition of NO
2
by the following gas-phase reaction is of sig-
nificance in smog formation and in combustion chemistry of pollutants: 2NO
2
→
2NO
+
O
2
. The reaction rate is
r
=
k
∗
P
NO
2
. An experiment was designed to measure
the rate constant as a function of
T
:
k
∗
T
(K)
(L/mol s)
600
0.46
700
9.7
800
130
1000
3130
A plot of ln
k
∗
versus 1
/T
gives a slope of
−
13,310 and an intercept of 21.38
with a correlation coefficient of 0.9994. Therefore,
E
a
=
110 kJ/mol and
A
=
1.9
×
10
9
L/mol s.Forabimoleculargas-phasereaction
Δ
H
†
=
E
a
−
2
RT
.At600 K,
Δ
H
†
=
110
−
2
(
8.314
×
600
)/
1000
=
100 kJ/mol. Assume that the transmission coefficient
κ =
1. Then, we have from the equation for rate constant,
k
∗
=
0.46
=
e
2
(
1.2
×
10
13
)
e
Δ
S
†
/R
(
2.65
×
10
−
10
)
, from which
Δ
S
†
=−
90 J/mol K. Note that in general,
if
Δ
S
†
is negative the formation of the activated complex is less probable and the reac-
tion is slow. If
Δ
S
†
is positive the activated complex is more probable and the reaction is
faster. This is so since exp(
Δ
S
†
/
R)
is a factor that determines whether a reaction occurs
faster or slower than normal.
At 600 K,
Δ
G
†
= Δ
H
†
−
T
Δ
S
†
=
164 kJ/mol. This is the positive free energy
barrier that NO
2
must climb to react. The value of
A
calculated from
Δ
S
†
is
A
=
e
(kT/h)
exp
(
−Δ
S
†
/R)
=
6.6
×
10
8
L/mol s. The value of
K
†
=
exp
(
−Δ
G
†
/RT)
=
5.3
×
10
−
15
L/mol. Note that the overall equilibrium standard free energy of the
reaction will be
Δ
G
0
=
2
(
86.5
)
−
2
(
51.3
)
=
71 kJ/mol. Hence,
Δ
G
†
≈
2.3
Δ
G
0
.
5.4.2 E
FFECT OF
S
OLVENT ON
R
EACTION
R
ATES
The ACT can also be applied to reactions in solutions, but requires consideration of
additional factors. Consider, for example, the reaction A
D that occurs
both in the gas phase and in solution. The ratio of equilibrium constants will be
+
B
C
+
RT
V
0
Δ
n
K
sol
eq
K
gas
K
HA
K
HB
K
HC
K
HD
γ
A
γ
B
γ
C
γ
D
,
=
(5.79)
eq
where
K
H
i
is Henry's constant
(
=
P
i
/x
i
)
,
V
0
is the volume per mole of solution, and
Δ
denotes the activity
coefficient in solution. It can be argued that as per the above equation, reactions in
solution will be favored if the reactants are more volatile than products and vice versa
if products are more volatile. The primary interaction in the solution phase is that
with the solvent molecules for both reactants and products on account of the close-
packed liquid structure; whereas in the gas phase no such solvent-mediated effects
are possible.
n
is the change in moles of substance during the reaction.
γ
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