- 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
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 ![]() |
[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 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 2) Plot the weak collision vibrational energy 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