(c) 2000-2003 by Akira Miyoshi. All rights reserved.
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Reaction Dynamics 2002 - section-3

3. Prior Distribution

Assumption: statistically equal distribution in final states
  Involves neither intermediate nor transition state 0th order approximation


3.1 Rotational Distribution and Rotational Sum

cf.) Photodissociation of diatomic molecules (HCl, HI, etc.):
  (excess energy) distributes only in translation

  ex.) Photodissociation of HCN at 215 nm (46512 ) :
HCN + (46512 )   H + CN( = 0) + 3211     [channel-0]
  H + CN( = 1) + 1151     [channel-1]
rel. translation + CN rotation

Probability of a specific rovibrational state
      (3.1.1)

  ex.) Rotational distribution of CN via channel-1
  (max = 24, fig. 3.1)


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 4.66 : 1

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() + H2 OH + H ( = -182.2 ) at 298 K.   Considering the thermal energy of the relative translation, , and that of rotation, , (note that = 2 for H2), the total excess energy at temperature is;   .   The vibrational frequency of OH is 3568 .
2) [OPTION] Calculate the prior rotational distribution OH in its vibrational ground state ( = 0) formed in the above reaction.   The rotational constant of OH is 18.51 .


3.2 Vibrational Sum

Branching probability for a reaction channel (vibrational sum)

  ex.)   O() + HD H + OD channel-a  
D + OH channel-b 
[O() + H2 OH + H] = -182.2
Vibrational frequencies () - OH: 3568, OD: 2632, H2: 4162, HD: 3633
Rotational constants () - OH: 18.51, OD: 9.87
(after ZPE & thermal corr. @ 298 K) (a) = 15952, (b) = 15484 ()

max(a) = 6 [15792 ], max(b) = 4 [14272 ]
Branching ratio = = 1 : 1.017

  cf.) summation integration
    (3.2.1)
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() + CH4 OH + CH3 ( = -182.3 )
  - Prior vibrational distribution of OH (298 K)
Motion other than OH-vibration (3.2.2), OH = 3568
ZPE(CH3) = 76.1 = (182.3 + 76.1) [] + 3RT = 22222
cf.) = 182.3 [] + 3 = 15860
max = 4 [14272 ], = (3 + 2 + 1) / 2 = 3, = 6, + = 9
Vibrational distribution (fig. 3.2)
= 0     1
= 1     0.207
= 2     0.0306
= 3     0.00270
= 4     9.60 10-5


Fig. 3.2 Vibrational distribution of OH formed via O() + CH4

Problem-3.3
Calculate the prior vibrational distribution of OH formed by O() + C2H6 OH + C2H5 ( = -210.7 , ZPE(C2H5) = 148.5 ) similarly to above, and compare it with the cases for O() + H2 and O() + CH4.


3.3 Degeneracy of the Electronic States

[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 + H + Cl()     (a)     4 trans(a)
H + Cl()     (b)     2 trans(b)

[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 1/2
1 1/2 4 2 1/2
2 3/2
1 1 6 2 0
2 1
2 2

  ex.) branching ratio
NH3 H2 + NH()     (X)     3 vib-sum(X)
H2 + NH()     (a)     2 vib-sum(a)

[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 H + CH3O()     (X)     4 vib-sum(X)
H + CH3O()     (A)     2 vib-sum(A)


3.4 Number of Optical Isomers and Rotational Symmetry Number

[Number of optical isomers]

  ex.) branching ratio
CFCl2Br + Cl() + CFClBr     (a)     2 vib-sum(a)
Br() + CFCl2     (b)     2 vib-sum(b)

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( = 0) - Rotational state distribution
- H nucleus : = 1/2 (= proton / Fermi particle)
    tot should be Asymmetric
- elec () : Sym.,   vib ( = 0) : Sym.
ortho-H2 para-H2
1 () 0 ()
n.s Sym. Asym.
rot Asym. Sym.
  1, 3, ... (odd)         0, 2, ... (even)  
    3 () 1 ()

  ex.)   N2 or D2( = 0) - Rotational state distribution
- N or D nucleus : = 1 (Bose particle)
    tot should be Symmetric
ortho-N2/D2 para-N2/D2
  0 () or 2 ()   1 ()
n.s Sym. Asym.
rot 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() with CH4,
/
O() + CH4 CH3 + OH   -182.7 channel-a
CH3O + H   -129.1 channel-b
CH2OH + H   -173.2 channel-c
at room temperature (298 K). Two experimental measurements on H-atom reported the sum of branching fractions for channel-b and c as 14% and 25%. Compare these with the prior branching fractions.

Molecular constants
  chemical species H OH CH3 CH3O CH2OH
  mass / amu 1.008   17.003   15.024 31.019   31.019  
  symmetry -
  rot. sym. num.; 1 1 6 3 1
  num. opt. isom.; OPT 1 1 1 1 2
  electronic state    
  deg. of electron. state;   2 4 2 4 2
  rot. deg. of freedom - 2 3 3 3
  rot. consts. /
    (= ) 5.169 6.555
    18.51 9.578 0.9317 1.0061
    4.742 (= ) 0.8879
  vib. deg. of freedom - 1 6 9 9
  vib. freqencies* /
    1 3568 3004 2840 3650
    2 606 1362 3019
    3   3161 (2)   1047 2915
    4 1396 (2)   2774 (2)   1459
    5 1487 (2) 1334
    6 652 (2) 1183
    7 1048
    8 607
    9 420
    * Numbers in parentheses are degeneracy of the vibrational modes.
    1 = 11.96266 , = 8.31451