- Microcanonical form of the Transition State Theory
Microcanonical equilibrium : ( : 1-D translational energy of TS)
Half of passes region with length toward products with velocity . rate of reaction :
Rate coefficients for A() :
[sum of states] (4.1.4)
by taking adiabatic rotation into account;
Exactly speaking, eq. (4.1.5) is not a correct form of the microscopic rate coefficient. see appendix for detail.
Derive eqs. (4.1.6) and (4.1.7) from eqs. (2.2.7) and (2.2.10).
- Conservation of angular momentum
Microscopic rate coefficients:
For the case of similar structures of A and TS (i.e., ) -conservation may be ignored.
[Sum of states]
Classical approximation [(2.3.5) , (4.1.10)] : Not good at low energy (fig. 2)
Whitten-Rabinovitch Approximation or Direct Count
[Direct Count (Beyer-Swinehart algorithm)]
Count states in energy grains (source list 1)
Stein & Rabinovitch, J. Chem. Phys. 58, 2438 (1973).
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)].
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 
High-pressure limit : (4.2.2)
Low-pressure limit : (4.2.3)
Fall-off pressure (density) : (4.2.5)
- 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
- 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).
, [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)
[High-pressure limit] = canonical average = TST (transition-state theory)
Canonical (Boltzmann) distribution of
= TST (4.3.5)
Derive the eq. (4.3.5) by using in eq. (4.1.5).
[Low-pressure limit] = excitation is rate-determining
- 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)
(4.3.3) , (4.3.6) , (4.3.2)
- Strong collision RRKM > measurements @ fall-off low-pressure limit
: weak collision parameter 0.1 1 (depending on temperature, molecule, M)
(4.3.3) , (4.3.6) , (4.3.2)
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, .
- Satisfactory in many cases, but has little physical meaning.
- Cannot describe multi-channel / multi-well problems.
Master equation system describing the problem :
(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 :
... eigenvalue problem of matrix