Chemistry Reference
In-Depth Information
C
2T/
C
T
2
a.1
C
r.1
Nh//
a.1
C
r/T
2
3
T
2
C
2
.a.r
C
1/T
2
C
2T
C
a.1
C
r.1
Nh//
C
2T//
2
a.1
C
r.1
Nh//
a.1
C
r/T
2
D
.T
2
C
.a.r
C
1/T
2
C
D
0:
C
2T
Therefore the eigenvalues are
s
a.1
C
r.1
Nh//
a.r
C
1/T
2
1;2
D˙
i
C
2T
and
a.r
C
1/T
2
C
2T
3
D
<0:
T
2
So we have a pair of pure complex eigenvalues, and the third eigenvalue is negative.
Differentiating equation (5.40) implicitly with respect to T and using the notation
P
d
dT
D
we have
P
T
2
C
2
3
T
C
2
P
.a.r
C
1/T
2
3
2
C
2T/
C
2
.2a.r
C
1/T
C
2/
C
P
.1
C
2a.r
C
1/T/
C
2a.r
C
1/
D
0
implying that
2
3
T
2
.2a.r
C
1/T
C
2/
2a.r
C
1/
3
2
T
2
P
D
C
2T/
C
.1
C
2a.r
C
1/T/
:
(5.44)
C
2.a.r
C
1/T
2
For the sake of simplicity introduce the notation
a.1
C
r.1
Nh//
a.r
C
1/T
2
1
C
2aT.r
C
1/
T
2
˛
2
D
D
C
2T
then
1;2
D˙
˛i and at these values
˙
2˛
3
iT
C
˛
2
.2a.r
C
1/T
C
2/
2a˛.r
C
1/i
3˛
2
T
2
P
D
˙
2˛i.a.r
C
1/T
2
C
2T/
C
.1
C
2a.r
C
1/T/
˛
2
.2a.r
C
1/T
C
2/
C
.
˙
2˛
3
T
2a˛.r
C
1//i
2˛
2
T
2
D
˙
2˛i.a.r
C
1/T
2
C
2T/
with real part
4˛
4
T
2
4a
2
˛
2
.r
C
1/
2
T
2
8a˛
2
T.r
C
1/
Re
P
D
:
(5.45)
.
2˛
2
T
2
/
2
C
4˛
2
.a.r
C
1/T
2
C
2T/
2
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