- Microcanonical form of the Transition State Theory

Fig. 4.1

[]
 

     (4.1.1)

Microcanonical equilibrium : ( : 1-D translational energy of TS)

      (4.1.2)

Half of passes region with length toward products with velocity . rate of reaction :

Rate coefficients for A() :

      (4.1.3)

    [sum of states]     (4.1.4)

by taking adiabatic rotation into account;

      (4.1.5)
      or     (4.1.5')

      (4.1.6)

      (4.1.7)

note:
  Exactly speaking, eq. (4.1.5) is not a correct form of the microscopic rate coefficient.   see appendix for detail.

Problem-4.1 [OPTION] Derive eqs. (4.1.6) and (4.1.7) from eqs. (2.2.7) and (2.2.10).

Fig. 4.2

[]
  - Conservation of angular momentum
 

     (4.1.8)

Microscopic rate coefficients:

      (4.1.9)

note:
  For the case of similar structures of A and TS (i.e., ) -conservation may be ignored.

[Sum of states]

summationintegration (2.3.5)

      (4.1.10)

  ,  

note:
  Classical approximation [(2.3.5) , (4.1.10)] : Not good at low energy (fig. 2)
  Whitten-Rabinovitch Approximation or Direct Count

[Whitten-Rabinovich Approximation]

     (4.1.11)

  ,   ,  

     (4.1.12)

[Direct Count (Beyer-Swinehart algorithm)]

Count states in energy grains (source list 1)
      Stein & Rabinovitch, J. Chem. Phys. 58, 2438 (1973).

Problem-4.2 1) Evaluate (eq. 4.1.5) for CH3CH2I CH2=CH2 + HI at = 19000 and 25000 .   Compare the results of classical approximation [(2.3.5), (4.1.10)] and Whitten-Rabinovitch approximation [(4.1.11), (4.1.12)].   [CH3CH2I]     vib. freq.   3000(5), 1430(3), 1280(3), 1050(2),   940, 760, 500, 240, 200   rot. const.     0.0943, 0.101, 0.956   [TS]   vib. freq. 3000(4), 1770, 1450, 1370(3), 1120, 1090,     1050, 930, 570, 330, 250, 170   rot. const.     0.0609, 0.0644, 0.774   Dissociation threshold energy : = 17800 2) [OPTION] Evaluate the density/sum of states by direct count, and compare with above results.

Fig. 4.3 Lindemann-Mechanism

[Lindemann Mechanism] (review)

Steady-state assumption for []

      (4.2.1)

  High-pressure limit :       (4.2.2)

  Low-pressure limit :       (4.2.3)

      (4.2.4)

  Fall-off pressure (density) :       (4.2.5)

note:
  - Eq. (4.2.4) cannot reproduce the measurements (fig. 4.4) since is not a single state.

Fig. 4.4 Rate constant for CH₃ + O₂ recombination reaction

[Troe's formula]

- semi-empirical / reproduces measurements
    [a] J. Troe, J. Phys. Chem. 83, 114 (1979).
    [b] R. G. Gilbert, K. Luther, and J. Troe, Ber. Bunsenges. Phys. Chem. 87, 169 (1983).

      (4.2.6)

  ,       [a] (4.2.7)

  ,
  ,   ,       [a] (4.2.8)

  (4.1.5), (4.1.9) Microcanonical form of RRKM theory

['Lindemann' to RRKM]

,   ,   ,   [ : a constant independent of ]

  Detailed balancing :       (4.3.1)

steady-state assumption to

  distribution function :       (4.3.2)

      (4.3.3)

[High-pressure limit] = canonical average = TST (transition-state theory)

Canonical (Boltzmann) distribution of

      (4.3.4)

(4.3.2) ...

  = TST     (4.3.5)

Problem-4.3 Derive the eq. (4.3.5) by using in eq. (4.1.5).

[Low-pressure limit] = excitation is rate-determining

(4.3.3) :
      (4.3.6)

note :
- Fall-off region low-pressure limit :   Vibrational distribution is non-Boltzmann

[Strong-collision RRKM theory]

assumption : of deactivates to by a single collision with M.

  Lennard-Jones collision frequency :       (4.3.7)

  Strong-collision deactivation rate coefficient :       (4.3.8)

    or  

    ,   ,  

:
    (4.3.3) ,   (4.3.6) ,   (4.3.2)

[Weak-collision correction]

- Strong collision RRKM > measurements @ fall-off low-pressure limit

      (4.3.9)

    : weak collision parameter 0.1 1 (depending on temperature, molecule, M)

:
    (4.3.3) ,   (4.3.6) ,   (4.3.2)

Problem-4.4 [OPTION] 1) By using the weak collision corrected RRKM theory, calculate , , and at at 800 K for the reaction in the problem-4.2.   Calculate and by the Whitten-Rabinovich approximation or by the direct count.   Assume = 0.2 and M = Ar, and use (CH3CH2I) = 5.3 , (Ar) = 3.5 , (CH3CH2I) / B = 320 K, and (Ar) / B = 93 K. 2) Plot the weak collision vibrational energy distribution, , at against , and compare it with the Boltzmann distribution, .

[Master Equation-RRKM]

WC-RRKM :
    - Satisfactory in many cases, but has little physical meaning.
    - Cannot describe multi-channel / multi-well problems.

Master equation system describing the problem :

      (4.3.10)

    : (steady-state) unimolecular reaction rate coefficients,
    : collision frequency; [= (4.3.7)],
    or : probability of energy transfer (from to , or to )

By using energy grain matrix form :

      (4.3.11)

    ... eigenvalue problem of matrix