Assumption: statistically equal distribution in final states
Involves neither intermediate nor transition state
0th order approximation
cf.) Photodissociation of diatomic molecules (HCl, HI, etc.):
(excess energy)
distributes only in translation
ex.) | Photodissociation of HCN at 215 nm (46512
![]() | |||
HCN + ![]() ![]() |
![]() ![]() ![]() |
[channel-0] | ||
![]() ![]() ![]() |
[channel-1] | |||
![]() ![]() |
Probability of a specific rovibrational state
(3.1.1)
ex.) | Rotational distribution of CN via channel-1
![]() | |
![]() ![]() |
Fig. 3.1 Rotational distribution of CN formed via HCN +
(channel-1)
Probability of a specific vibrational state (indistinctive of
rotational states) Rotational sum:
,
(3.1.2)
(summation integration)
(3.1.3)
ex.) | Branching ratio of channel-0 : channel-1 = (3211)3/2 :
(1151)3/2 ![]() |
Problem-3.1 [OPTION] Derive an equation similar to 3.1.3 for the case that an atom and a non-linear molecule (3-D rotation) are formed. |
[Rotational Sum]
Rotational sum for the case that two fragments are formed
(i.e., Branching probability for the specific vibrational state) :
(3.1.4)
(3.1.5)
: Energy partitioned into
translation and rotation
(
: Total excess energy)
: Translational degree of
freedom (= 3),
: Number of rotators,
: Total rotational degree of
freedom (
)
for integer :
,
for half integer :
,
, ...
[Coefficients for rotational sum (eq. 3.1.5) for specific cases]
Problem-3.2 1) Calculate the prior vibrational energy distribution of OH formed via the reaction O( ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 2) [OPTION] Calculate the prior rotational distribution OH in its vibrational ground state ( ![]() ![]() |
Branching probability for a reaction channel
(vibrational sum)
ex.) | O(![]() |
![]() | channel-a | ||
![]() |
channel-b | ||||
![]() ![]() ![]() ![]() Vibrational frequencies ( ![]() Rotational constants ( ![]() (after ZPE & thermal corr. @ 298 K) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |||||
Branching ratio =
![]() ![]() |
cf.) | summation ![]() | |
![]() ![]() note) ![]() ![]() |
[Vibrational sum]
General formula by replacement of summation by integration :
(3.2.2)
,
: vibrational degree of freedom,
i : vibrational frequency
: excess energy
measured from the classical origin
ex.) | O(![]() ![]() ![]() ![]() - Prior vibrational distribution of OH (298 K) | ||
Motion other than OH-vibration ![]() ![]() ![]() ZPE(CH3) = 76.1 ![]() ![]() ![]() ![]() ![]() cf.) ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() Vibrational distribution ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Fig. 3.2 Vibrational distribution of OH formed via
O() + CH4
Problem-3.3 Calculate the prior vibrational distribution of OH formed by O( ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
[Atoms] (except for the excited states of rare gas atoms)
(spectrum) Term :
: electron spin q. n.
: electron orbital angular momentum
q. n.
: total angular momentum q. n.
[] : symbolic representation of
- S, P, D, F, G, H, ... for
= 0, 1, 2, 3, 4, 5, ...
Total degeneracy :
Degeneracy of the fine-structure state:
ex.)
term | ![]() |
![]() |
![]() |
![]() |
![]() |
fine str. term |
![]() | 2 | 0 | 5 | 5 | 2 | |
![]() | 0 | 1/2 | 2 | 2 | 1/2 | |
![]() | 1 | 1/2 | 6 | 2 | 1/2 | ![]() |
4 | 3/2 | ![]() | ||||
![]() | 1 | 1 | 9 | 1 | 0 | ![]() |
3 | 1 | ![]() | ||||
5 | 2 | ![]() |
ex.) | branching ratio | ||||
HCl + ![]() |
![]() ![]() |
(a) | 4 ![]() ![]() | ||
![]() ![]() |
(b) | 2 ![]() ![]() |
[Linear Molecules]
(spectrum) Term :
: projection of
to the molecular axis
: projection of
to the molecular axis
[]: symbolic representation of
-
,
,
,
, ... (for
=
0, 1, 2, 3, ...)
(+-): parity (+ or -, only for S states)
Total degeneracy :
Degeneracy of the fine-structure state :
ex.)
term | ![]() |
![]() |
![]() |
![]() |
![]() ![]() |
fine str. term |
![]() | 2 | 0 | 2 | 2 | 2 | |
![]() | 0 | 1/2 | 2 | 1 | ![]() | |
![]() | 1 | 1/2 | 4 | 2 | 1/2 | ![]() |
2 | 3/2 | ![]() | ||||
![]() | 1 | 1 | 6 | 2 | 0 | ![]() |
2 | 1 | ![]() | ||||
2 | 2 | ![]() |
ex.) | branching ratio | ||||
NH3 | ![]() ![]() |
(X) | 3 ![]() ![]() | ||
![]() ![]() |
(a) | 2 ![]() ![]() |
[Non-Linear Molecules]
(spectrum) Term :
: symmetry species of the electronic
state ...
,
',
2,
1,
,
, etc.
Total degeneracy :
Degeneracy of the fine-structure state :
ex.)
term | ![]() |
![]() |
![]() |
fine str. term |
![]() | 0 | 3 | 3 | |
![]() | 1/2 | 2 | 1 | |
![]() | 1/2 | 4 | 2 | ![]() |
2 | ![]() |
ex.) | branching ratio | ||||
CH3OH | ![]() ![]() |
(X) | 4 ![]() ![]() | ||
![]() ![]() |
(A) | 2 ![]() ![]() |
[Number of optical isomers]
ex.) | branching ratio | ||||
CFCl2Br +
![]() |
![]() ![]() |
(a) | 2 ![]() ![]() | ||
![]() ![]() |
(b) | 2 ![]() ![]() |
channel-a two optical isomers for CFClBr
(= two reaction pathway = inversion doubling)
[Rotational distribution and nuclear spin statistics]
- Nuclear spin :
- Resultant total nuclear spin of a molecule
:
- Bose/Fermi particle-Bose/Fermi statistics
(symmetric/assymmetric to permutation)
ex.) |
H2(![]() - H nucleus : ![]() ![]() ![]() - ![]() ![]() ![]() ![]() |
ortho-H2 | para-H2 | ||
![]() |
1 (![]() | 0 (![]() | |
![]() |
Sym. | ![]() | Asym. |
![]() |
Asym. | ![]() | Sym. |
![]() |
1, 3, ... (odd) | ![]() |
0, 2, ... (even) |
![]() ![]() ![]() |
3 ![]() ![]() |
1 ![]() ![]() |
ex.) |
N2 or D2(![]() - N or D nucleus : ![]() ![]() ![]() |
ortho-N2/D2 | para-N2/D2 | ||
![]() |
0 (![]() ![]() |
1 (![]() | |
![]() |
Sym. | ![]() | Asym. |
![]() |
Sym. | ![]() | Asym. |
![]() |
0, 2, ... (even) | ![]() |
1, 3, ... (odd) |
![]() ![]() ![]() |
(1+5) ![]() ![]() |
3 ![]() ![]() |
- = 2
either in
sym. or asym. rotational state, rotational density of states is 1/2
Problem-3.4 Calculate the prior branching fractions for the reaction of O( ![]()
1 ![]() ![]() ![]() ![]() |