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It is also possible to look at the changes in types of degrees
of freedom, and
changes in internal motions on forming the activated complex.
Considerable insight
into trends in A factors, and interpretations of these in terms
of changes in internal
motions, can be made using rough order of magnitude calculations
without having
recourse to detailed accurate values for the partition
functions. The approximations
made are acceptable because contributions from the internal
motions do not vary
much with mass, moment of inertia or vibrational frequency.
Typical values of these contributions:
for each translation: 109 to 1010 dm1
for each rotation: 10 to 102
for each vibration: 1 to 10
Worked Problem 4.13
Question. At 298 K, and assuming 1 and kT=h 6:2 1012 s1, and
usingtypical values (in the middle of the ranges above) for the
contributions from
translation, rotation and vibration,
for each translation: 5 109
dm
1
for each rotation: 50
for each vibration 5
calculate the predicted A factor for the following reaction
types:
(a) atom atom! activated complex;
(b) atom linear molecule! non-linear activated complex;
(c) two non-linear molecules! non-linear activated complex.
Answer.
(a) Each atom has three translations; the activated complex has
three translations,
two rotations, one internal free translation and ;3 2 6
vibrations i.e.zero. The contribution from the internal translation
has already been incorpo-
rated in the derivation, so that
contributions to Q6
QXQYZare 3trans; 2rots; 0vib
3trans; 3trans 2rots
3trans
138 THEORIES OF CHEMICAL REACTIONS
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value of kT
h
Q6
QXQYZ
6:2 1012 50 50
5 109 5 109 5 109dm3 s1
1:2 1013 dm3 s1
(b) An atom has three translations; a linear molecule has three
translations, two
rotations and 3N 5 vibrations; the non-linear activated complex
has threetranslations, three rotations, one internal free
translation and ; 3N 1 7vibrations i.e. 3N 4 vibrations.
Contributions toQ6
QXQYZare
3trans; 3rots; 3N 4vibs
3trans; 3trans; 2rots; 3N 5vib
1rot; 1vib
3trans
value of kT
h
Q6
QXQYZ
6:2 1012 50 5
5 109 5 109 5 109dm3 s1
1:2 1014 dm3 s1
(c) A non-linear molecule, A, has three translations, three
rotations and 3NA 6vibrations. Likewise non-linear molecule B has
three translations, three
rotations and 3NB 6 vibrations. Total for reactants six
translations, sixrotations and 3NA 3NB 12 vibrations.
The non-linear activated complex has three translations, three
rotations, one
internal free translation and 3NA 3NB 7 vibrations.
Contributions toQ6
QXQYZare
3trans; 3rots; 3NA 3NB 7vibs
3trans; 3trans; 3rots; 3rots; 3NA 3NB 12vibs
5vib
3trans; 3rots
kT
h
Q6
QXQYZ
6:21012 555 5 5
510951095109 50 50 50dm3 s1
1:2 1018 dm3 s1
The results from Problem 4.13 are instructive. They are
(a) atom atom! activated complex, A 1:2 1013 dm3 s1;
(b) atom linear molecule! non-linear activated complex, A 1:2
1014 dm3
s1;
(c) two non-linear molecules! non-linear activated complex, A
1:2 1018 dm3
s1.
These values of A are in molecular units, and cannot be compared
directly with a
typical value ofZ. This is because partition functions are
molecular quantities and the
contributions to kT=hQ6=QXQYZ are in molecular units.
Multiplying by
TRANSITION STATE THEORY 139
