1
Molecular Size Dependent
Fall-off Rate Constants for the
Recombination Reactions of
Alkyl Radicals with O2
Department of Chemical Systems Engineering, University of Tokyo
Akira Miyoshi
7th ICCK (MIT, Cambridge) July 11, 2011
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Introduction
— R (alkyl) + O2
• key reactions that lead to chain branching in low-temperature
oxidation of hydrocarbons
— Challenges
• resolution of complicated pressure- and temperature- dependent
product specific rate constants including second O2 addition
reactions to QOOH
— Objectives
• evaluation of universal fall-off rate expression for recombination
• master equation analysis for the dissociation/recombination steadystate
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Computational
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Computational
— Quantum Chemical Calculations
• B3LYP & CBS-QB3 calculations by Gaussian 03
• CASPT2 calculations by MOLPRO 2008.1
— TST and VTST Calculations
by GPOP* including:
• Pitzer-Gwinn approximation for hindered rotors, qPG
(after analysis by BEx1D*)
• 1D tunneling correction (asymmetric Eckart), κtun
• rotational conformer distribution partition function, qRCD
— RRKM/ME Calculations
• ρ(E) and k(E) accounting for all TST feature (qPG, κtun, and qRCD) by
modified UNIMOL RRKM program
• steady-state & transient master equation calculations by SSUMES*
* http://www.frad.t.u-tokyo.ac.jp/~miyoshi/tools4kin.html
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Hindered Rotor (carbon-centered radical)
— Pitzer-Gwinn Approximation
• partition function calculated from eigenstate energies, qexact, is well
approximated by qPG(V0 = 100 cm–1) or qFR (free rotor)
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Hindered Rotor (RO2)
— Taken into Account as Rotational Conformers
• partition function calculated from eigenstate energies, qexact, is well
approximated by 2qHO+qHO' or qHOqRCD
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Rotational Conformers
— Taken into Account via Partition Function
• rotational conformer distribution partition function, qRCD

qtot   qi exp   i 
 kT 
i
by assuming qi  q0
qtot  qRCD q0
  
qRCD   gi exp   i 
 kT 
i
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Molecular Size Dependent
Fall-off Rate Constants
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Potential Energy Curves
• CASPT2(7,5)/aug-cc-pVDZ //
B3LYP/6-311G(d,p) potential
energy well reproduced
experimental k(300 K)
within ± 25%
R (alkyl) + O2  RO2
• B3LYP/6-311G(d,p) potential
energy systematically
underestimated k(300 K)
Rate Constants for R + O2  RO2
R
C2H5
i-C3H7
n-C4H9
s-C4H9
t-C4H9
k(300 K) / 10–11 cm3 molecule–1 s–1
exp.
CASPT2 (%err) B3LYP (%err)
0.780
0.728
(–7)
0.411 (–47)
1.41
1.25
(–11)
0.829 (–41)
0.750
0.921 (+23)
0.354 (–53)
1.66
1.26
(–24)
0.426 (–74)
2.34
2.50
(+7)
1.73
(–26)
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High-Pressure Limiting Rate Constants, k
— Size-Independent
 same for secondary R's
 same for primary R's
— Class-Specific
• class (primary,
secondary, or tertiary)
determines the rate
constant
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Fall-off Calculations
— Energy Transfer
Model
• experimental data for
C2H5 + O2 in fall-off
region were well
reproduced by the
exponential-down
model with:

 T 

400


1
1000
K


cm
Plumb & Ryan, Int. J. Chem. Kinet., 1981, 13, 1011; Slagle et al., J. Phys.
Chem., 1984, 88, 3648; Wagner et al., J. Phys. Chem., 1990, 94, 1853.
0.7
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Low-Pressure Limiting Rate Constants, k0
— Size-Dependent
— Class-Independent
 same for three C4 R's
irrespective of class
(primary, secondary,
or tertiary)
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Size-Dependent Expression for k0
Parameters for modified Arrhenius Expression:
k0 = A T b exp(–Ea / RT )
nHA = number of heavy (non-hydrogen) atoms
— Universal Fall-off Rate Constants for R + O2
• class-specific k + size-dependent k0
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Collapse of Steady-State
Assumption?
?
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Steady-State Distribution of Large RO2
• steady-state distribution for
dissociation?
 rump distribution after
major part has gone
• steady-state distribution for
chemical-activation
 Boltzmann distribution
Collapse of steady-state
assumption or LindemannHinshelwood type mechanism
RO2
R  O2
RO2 *  R  O2
RO2 *  RO2
• k  k at high temperatures
(Miller and Klippenstein, Int. J. Chem.
Kinet., 2001, 33, 654–668)
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Dissociation/Recombination
Steady-State
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 RO Partial Equilibrium
R + O2 
2
— Dissociation/Recombination Steady-State
• more general condition where near F(E) is established
Chemical activation steady state
d
n  Jn  kin r  0
dt
where
r(E) 
1
kdis,
kd ( E ) F ( E )
When other channels are not present, there is trivial solution
n( E ) 
kin
kdis,
F (E)
= Boltzmann distribution
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Dissociation/Recombination Steady-State
— Near Boltzmann Distribution
• rate constants for subsequent isomerization/dissociation reactions
of RO2 can be estimated to be in near high-pressure limit
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Three "Steady-States"
"delayed"
"prompt"
Miller and Klippenstein, Int. J.
Chem. Kinet., 2001, 33, 654–668.
Clifford, Farrell, DeSain and Taatjes, J.
Phys. Chem. A, 2000, 104, 11549–11560.
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 C H O
HO2 formation in C2H5 + O2 
2 5 2
• k(HO2)  k(HO2)
at moderate T
but in partial equilibrium of
R + O2  RO2
Experimental data by Clifford, Farrell,
DeSain and Taatjes, J. Phys. Chem. A,
2000, 104, 11549–11560.
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Time Dependent Solution
Time-dependent solution for
d
n  Jn  kin r
dt
with n0 = 0 and kin = const.
• Nearly the same with and
without concerted HO2
elimination channel
Build-up time  kdis,FO–1
In Autoignition Modeling
T/K
near partial equilibrium
mole fraction
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3000
2000
1000
10 -2
10 -3 n-C7 H16 /air  = 1,
10 -4 720 K, 20 atm
10 -5
10 -6
3-C7H15O2
10 -7
10 -8
10 -9
10 -10
3-C7H15
-11
10
10 -12
OH
10 -13
0
1
2
t / ms
transient
3
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Building-Up Transient for C8H17O2
build-up of F(E) with
bu–1  kdis,FO  kdis,
build-up of F(E) with
• bu–1  kdis,FO  kdis, (0.01atm)
• bimodal build-up (10–6 atm)
collision-free build-up of F(E)
with bu–1  kdis, >> kdis,FO
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Summary
— Size-Dependent Fall-Off Rate Constants for R + O2
• VTST and RRKM/ME calculations for
R = C2H5, i-C3H7, n-C4H9, s-C4H9, t-C4H9, n-C6H13, and i-C8H17
• k is class-specific but size-independent
• k0 is size-dependent but class-independent
• Universal fall-off rate expression for arbitrary R + O2
— Collapse of Steady-State Assumption
• For large RO2 at high temperatures
— Dissociation/Recombination Steady-State
• nss(E)  F(E) for RO2 in partial equilibrium with R + O2
• HPL(k) can be assumed for subsequent reactions of RO2
• build-up time  kdis,FO–1 at low T
 kdis,–1 at high T irrespective of P
bimodal build-up at midium T especially at low P