Unimolecular Decomposition of n-C4H9 and iso-C4H9 Radicals

Article abstract of DOI:10.1021/jp952229k

The kinetics of the unimolecular decomposition of the n-C4H9 radical has been studied experimentally in a heated tubular flow reactor coupled to a photoionozation mass spectrometer.Rate constants for the decomposition were determined in time-resolved experiments as a function of temperature (560-620 K) and bath gas density ((3-18)E16 molecules cm^-3) in two bath gases, He and N2.The rate constants are in the falloff region under the conditions of the experiments.Structures, vibrational frequencies, and barriers for internal rotations of n-butyl and iso-butyl radicals and their decomposition transition states were obtained by ab initio calculations using UHF/6-31G* and MP2/6-31G* methods.The results of ab initio calculation, together with the reanalysis of earlier studies of the reverse reactions, were used to create transition-state models of the reactions of unimolecular decomposition of n-butyl (1) and iso-butyl (2) radicals.Falloff behavior of reaction 1 was reproduced using master equation modeling with the energy barrier height for decomposition obtained from optimization of the agreement between experimental and calculated rate constants.The values of <ΔE>_all = -28 cm^-1 (He) and -40 cm^-1 (N2) for the average energy loss per collision were obtained using an exponential-down model.The resulting models of the reactions provide the high-pressure limit rate constants for the decomposition reactions (k₁(n-C4H9 -> C2H5 + C2H4) = 1.06E13 exp(-14005 K/T), k₂(iso-C4H9 -> CH3 + C3H6) = 2.14E12TE0.65 exp(-15529 K/T) s^-1) and the reverse reactions (k_-1(C2H5 + C2H4 -> n-C4H9) = 6.59E-21TE2.44 exp(-2697 K/T), k_-2(CH3 + C3H6 -> iso-C4H9) = 1.66E-20TE2.57 exp(-3879 K/T) cm³ molecule^-1 s^-1).Parametrization of the temperature and pressure dependence of the unimolecular rate constants for the temperature range 298-900 K and pressures 0.001-10 atm in He and N2 is provided using the modified Lindemann-Hinshelwood expression.

Full text of DOI:10.1021/jp952229k

5
318
J. Phys. Chem. 1996, 100, 5318-5328
Unimolecular Decomposition of n-C4H9 and iso-C4H9 Radicals
Vadim D. Knyazev* and Irene R. Slagle*
Department of Chemistry, The Catholic UniVersity of America, Washington, D.C. 20064
X
ReceiVed: August 2, 1995; In Final Form: December 11, 1995
4 9
The kinetics of the unimolecular decomposition of the n-C H radical has been studied experimentally in a
heated tubular flow reactor coupled to a photoionization mass spectrometer. Rate constants for the
decomposition were determined in time-resolved experiments as a function of temperature (560-620 K) and
1
6
-3
bath gas density ((3-18) × 10 molecules cm ) in two bath gases, He and N
2
. The rate constants are in
the falloff region under the conditions of the experiments. Structures, vibrational frequencies, and barriers
for internal rotations of n-butyl and iso-butyl radicals and their decomposition transition states were obtained
by ab initio calculations using UHF/6-31G* and MP2/6-31G* methods. The results of ab initio calculation,
together with the reanalysis of earlier studies of the reverse reactions, were used to create transition-state
models of the reactions of unimolecular decomposition of n-butyl (1) and iso-butyl (2) radicals. Falloff
behavior of reaction 1 was reproduced using master equation modeling with the energy barrier height for
decomposition obtained from optimization of the agreement between experimental and calculated rate constants.
The values of 〈∆E〉all ) -28 cm (He) and -40 cm (N ) for the average energy loss per collision were
2
obtained using an exponential-down model. The resulting models of the reactions provide the high-pressure
limit rate constants for the decomposition reactions (k (n-C
-
1
-1
∞
1
13
4
H
9
f C
2
H
5
+ C
2
H
4
-1
) ) 1.06 × 10 exp(-14005
∞
2
12 0.65
K/T), k (iso-C
4
H
9
f CH
f n-C
3
+ C
) ) 6.59 × 10
-1 -1
3
H
6
) ) 2.14 × 10 T
exp(-15529 K/T) s ) and the reverse reactions
∞
-21 2.44
∞
-2
(
k (C
2
H
5
+ C
2
H
4
4
H
9
T
exp(-2697 K/T), k (CH
3
3 6 4 9
+ C H f iso-C H ) ) 1.66
-
1
-
20
3
×
10 T2.57 exp(-3879 K/T) cm molecule s ). Parametrization of the temperature and pressure dependence
of the unimolecular rate constants for the temperature range 298-900 K and pressures 0.001-10 atm in He
and N is provided using the modified Lindemann-Hinshelwood expression.
2
I. Introduction
pressure limit can be significant.7,8 Rate constants are measured
as a function of temperature and bath-gas density. Falloff
behavior is analyzed using a master equation approach to
characterize weak-collision effects. It has been possible in
favorable cases to obtain not only a temperature-averaged value
of 〈∆E〉down, the average internal energy lost in deactivating
collisions, for the conditions of the experiments but also an
The unimolecular decomposition of molecules and polyatomic
free radicals is an important class of elementary reactions in
high-temperature pyrolysis and combustion processes.1 In any
quantitative combustion modeling study, the proper inclusion
of the kinetics of these reactions, including the pressure and
temperature dependencies of their unimolecular rate constants,
-3
9
,10
indication of the temperature dependence of this parameter.
4
,5
is essential.
The ultimate products formed by combustion
Such temperature dependencies are required to extrapolate weak-
collision effects observed under the relatively mild experimental
conditions used in laboratory studies to the harsher conditions
of combustion processes. To date the unimolecular decomposi-
tion of the following radicals has been studied using the above
processes, rates of burning, and temperature dependencies of
flame properties are sensitive functions of the relative impor-
tances of the competing reactions of decomposition and other
elementary reactions involving the same intermediates (e.g.,
reactions with atoms, O2, and other free radicals).
Most of the current knowledge of the rate constants of these
unimolecular reactions is derived from kinetic data on the
reverse addition reactions combined with thermodynamic prop-
erties of the reactants and products.6 In general, the kinetic
data on the reverse reactions has been obtained at much lower
temperatures than those of combustion processes and under
conditions which are at or near the high-pressure limit. Major
temperature and density extrapolations of the rate constants and
the falloff characteristics of the unimolecular rate constant so
derived are required to obtain rate constants that can be used
for modeling high-temperature chemical processes.
The method employed in the current study has been success-
fully used before in direct studies of the unimolecular decom-
position of polyatomic hydrocarbon radicals. The experimental
procedure makes it possible to isolate and quantitatively study
the kinetics of the unimolecular decomposition of free radicals
at elevated temperatures (up to ≈1100 K in selected cases) and
at low pressures (1-10 Torr) where falloff from the high-
7
8
9
10
11
experimental method: HCO, n-C5H11, C2H5, i-C3H7, n-C3H7,
1
2
13
14
15
CH3CO, CH3CHCl, t-C4H9, and s-C4H9.
The current study addresses the reactions of unimolecular
decomposition of the two remaining isomers of the butyl
radical: n-C4H9 and iso-C4H9. Rate constants for the reaction
n-C H a C H + C H
4
(1,-1)
4
9
2
5
2
1
6
were measured at four densities of He ([He] ) (3-18) × 10
-3
molecules cm ) and at two densities of N2 ([N2] ) (6-12) ×
16
-3
10 molecules cm ) in the temperature range 560-620 K. Data
analysis involved creation of a transition-state model which
reproduces the temperature dependence of the high-pressure
limit rate constants of the reverse reaction (-1). The thermo-
chemistry of the reaction (1,-1) was analyzed based on the
data available in the literature as well as on modeling of our
experimental results on reaction 1. A transition state model
provided k(E) values which were then used in the master
equation analysis of the weak collision effects in the observed
falloff behavior of the experimentally measured rate constants.
X
Abstract published in AdVance ACS Abstracts, February 15, 1996.
0022-3654/96/20100-5318$12.00/0 © 1996 American Chemical Society

Decomposition of n-C4H9 and iso-C4H9 Radicals
J. Phys. Chem., Vol. 100, No. 13, 1996 5319
The unimolecular decomposition of the iso-C4H9 radical
the ion source, a portion was photoionized and mass selected.
n-Butyl radicals were ionized using the light from a chlorine
resonance lamp (8.8-8.9 eV) with a CaF2 window. Temporal
ion signal profiles were recorded on a multichannel scaler from
a short time before each laser pulse up to 20 ms following the
pulse. Data from 1000-30000 repetitions of the experiment
were accumulated before the data were analyzed.
The gases used were obtained from Aldrich (1-bromobutane,
99%) and Matheson (He, >99.995%; N2, >99.998%). 1-Bro-
mobutane was purified by vacuum distillation prior to use.
Helium and nitrogen were used as provided.
iso-C H a CH + C H
6
(2,-2)
4
9
3
3
could not be isolated for direct experimental study due to a
contribution of other butyl radical isomers to the signal (see
section II). Analysis of the data on the kinetics of nonterminal
addition of methyl radicals to propylene (reaction -2) and the
thermochemistry of reaction (2,-2) available in the literature,
together with our ab initio study, resulted in a recommendation
for the rate constants of this reaction.
The unimolecular decomposition of the n-C4H9 and iso-C4H9
radicals has not been isolated for direct study prior to this
investigation. However, the rate constant of dissociation relative
to that of recombination has been determined by both Kerr and
The n-butyl ion signal profiles were fit to an exponential
function ([n-C4H9]t ) [n-C4H9]0e- ) using a nonlinear least-
squares procedure. Experiments were performed to establish
that the decay constants did not depend on the concentration of
precursor (provided that the concentration was kept low enough
to ensure that radical-radical reactions had negligible rates) or
the laser intensity. The exponential decay constants depended
only on temperature and bath gas density.
k′t
16
17
Trotman-Dickenson and Morganroth and Calvert for n-C4H9
18
radical and by Slater et al. and Metcalfe and Trotman-
1
9
Dickenson for iso-C4H9 radical. The reverse reactions (-1
and -2) have been studied by several groups.2
0-25
Each of
these investigations of both reactions 1 and -1 involved the
study of complex reaction systems and will be discussed in detail
in Section IV.
Unimolecular decay of n-C4H9 was observed only above 500
K. Below this temperature a slow decay of n-C4H9 is observed
due to a first-order heterogeneous wall-loss process:
Recommendations for the temperature dependence of the rate
constants of decomposition reactions have been provided by
n-C H f heterogeneous loss
(4)
4
9
1
26
Warnatz (reactions 1 and 2) and Tsang (reaction 2) and of
27
Above 500 K the decay constant increased rapidly with rising
temperature due to the increasing importance of the thermal
decomposition of the n-butyl radical, reaction 1. The n-C4H9
decay constants were analyzed assuming that the n-butyl radicals
were consumed only by two elementary reactions, 1 and 4. At
low temperatures, only the heterogeneous loss is observed (k′
reaction -1 by Kerr and Parsonage (based on the results of
Kerr and Trotman-Dickenson22 and Watkins and O’Dean23).
We have reanalyzed these relative studies of both the direct
and reverse reactions (1,-1) and (2,-2) using new data available
on the kinetics of the corresponding reference reactions and
appropriate falloff corrections. This analysis together with the
modeling of our experimental data resulted in recommendations
for the high-pressure limit rate constants of these reactions as
well as in information on weak collision effects in reaction
)
k4). Above 500 K the sum of the two loss processes is
observed (k′ ) k1 + k4).
Calculations of k1 from measurements of k′ require knowledge
of k4 above 500 K. While k4 was directly determined below
(1,-1).
-1
5
00 K (15-38 s , independent of temperature), it could not
II. Experimental Study
be measured above this temperature due to the additional loss
of radicals by unimolecular decomposition. Values of k4 above
Details of the experimental apparatus28 and procedures used
have been described before and so are only briefly reviewed
here. The n-butyl radicals were produced homogeneously at
elevated temperatures in a heatable quartz tubular reactor by
pulsed laser photolysis and their unimolecular decay subse-
quently monitored in time-resolved experiments using photo-
ionization mass spectrometry. Pulsed unfocused 248-nm ra-
diation (≈5 Hz) from a Lambda Physik EMG 201MSC excimer
laser was directed along the axis of a heatable quartz reactor
8
500 K needed to determine k1 from the measurements of k′ were
obtained by an extrapolation assuming that k4 retains its
temperature independence beyond 500 K up to the highest
temperature of this study, 620 K. To minimize possible errors
in the determination of k1 caused by this assumed temperature
independence of k4 above 500 K, experiments to obtain k1 were
conducted at temperatures sufficiently high to ensure that k′ g
3k4. It was this criterion that established the lowest temperature
used to determine k1 at each bath gas density. The highest
temperature used at each total gas density was determined by
the fact that decay constants above 400 s- could not be
measured accurately.
(
1.05-cm i.d.) coated with boron oxide. Gas flowing through
-1
the tube at ≈4 m s contained the radical precursor (<0.2%)
and the inert carrier gas (He or N2) in large excess (>99.8%).
The flowing gas was completely replaced between laser pulses.
The n-butyl radicals were produced by the pulsed, 248-nm
laser photolysis of 1-bromobutane:
1
A potentially complicating factor in these experiments is the
production of other butyl radical isomers during the photolysis
process. We studied the relative sensitivity of our experimental
ion detection system for the four isomers of butyl radical using
bromine and chlorine ionization lamps. A chlorine lamp with
CaF2 window (used in the current study) provides ionizing
2
48 nm
CH CH CH CH Br
8 n-C H + other products (3)
3
2
2
2
4 9
2
9
Initial conditions (precursor concentration and laser intensity)
were selected to provide low radical concentrations (e10
radiation with photon energy 8.6-8.9 eV. The most intense
1
1
lines in the spectrum of a bromine ionization lamp with a
molecules cm- ). Under these conditions reactions between
radical products (including n-C4H9 + n-C4H9) had negligible
rates compared to that of the unimolecular decomposition of
the n-C4H9 radicals. Gas was sampled continuously through a
conical hole (0.043 cm diameter at the inside of the reactor) in
the side of the reactor and formed into a beam by a skimmer
before the gas entered the vacuum chamber containing the
photoionization mass spectrometer. As the gas beam traversed
3
sapphire window lie in the region 7.59-7.87 eV, although
29
there are several weak components at higher energies. The only
two isomers of butyl radical that have ionization potentials (IPs)
low enough to be detected with a sensitivity sufficient for kinetic
measurements using a bromine lamp are t-C4H9 (IP ) 6.70 eV)
30
and s-C4H9 (IP ) 7.25 eV). The methods to generate the tert-
1
4,15
butyl and sec-butyl radicals were described earlier.
iso-
C4H9 radicals were generated using the 248-nm photolysis of

5
320 J. Phys. Chem., Vol. 100, No. 13, 1996
Knyazev and Slagle
TABLE 1: Conditions and Results of Experiments To
1-bromo-2-methylpropane. The detection sensitivities obtained
1
Measure the Unimolecular Rate Constants (k ) of the
with the bromine and chlorine ionization lamps differed only
by a factor of 2 for both t-C4H9 and s-C4H9 isomers (with better
sensitivity obtained using the chlorine lamp). Although both
n-C4H9 and iso-C4H9 radicals could also be detected using the
bromine lamp, the sensitivity in this case was approximately
2
Thermal Decomposition of n-Butyl Radicals in He and N as
Bath Gases
master
0-
13
equation
analysis
master
1
0-16[M]
1
[n-C4H9Br]
equation
analysis
o
20 times lower than that obtained if the more energetic chlorine
(molecules
(molecules k4
k1
〈∆E〉 down
-3
-3
-1
-1
-1
-〈∆E〉all (cm-1)
cm
)
T (K)
cm
)
(s
)
(s
)
(cm )
lamp was used. The absence of a strong ion signal at the mass
of C4H9 if a bromine ionizing lamp is used during the 248-nm
photolysis of 1-bromobutane indicates the absence of significant
concentrations of t-C4H9 or s-C4H9.
He as Bath Gas
18.8 53.5
18.8 83.6
18.8 114
26.0 144
26.0 197
26.0 256
24.8 72.6
21.0 78.9
18.3 72.8
24.8 112
18.3 101
32.2 144
18.3 138
32.2 180
24.3 189
32.2 253
24.7 308
19.6 60.2
15.4 67.4
19.6 93.3
17.8 93.3
19.6 132
15.4 140
19.6 179
17.8 169
35.6 234
37.7 232
35.6 340
16.6 80.0
16.6 114
16.6 156
16.6 220
24.2 281
3
3
3
3
3
3
6
6
6
6
6
6
6
6
6
6
6
2
2
570
580
590
600
610
620
570
570
570
580
580
590
590
600
600
610
620
560
560
570
570
580
580
590
590
600
600
610
560
570
580
590
600
3.6
3.6
3.6
3.6
3.6
3.6
6.6
6.6
2.3
6.6
2.3
11.0
2.3
11.0
11.0
11.0
6.1
8.1
8.1
8.1
2.3
8.1
8.1
8.1
2.3
5.4
5.4
5.4
5.8
5.8
5.8
5.8
5.8
190
34
41
38
31
32
30
32
40
33
38
30
31
28
25
27
25
21
27
37
30
30
28
33
26
22
22
21
24
42
37
29
28
22
214
209
193
200
194
187
210
187
208
183
188
178
168
178
176
160
165
196
181
181
177
194
169
156
156
155
170
213
201
181
180
159
Ethyl radical (C2H5) was detected as a primary product of
the unimolecular decomposition of the n-C4H9 radical with the
rise time of C2H5+ matching that of the decay of the n-butyl
ion signal. No formation of C3H6 (which is a primary product
1
5
of the unimolecular decomposition of the s-C4H9 isomer and
is expected to be the primary product of the thermal decomposi-
tion of iso-C4H9 as well) was observed in the current study.
Under our experimental conditions the four isomers of the
butyl radical thermally decompose to a measurable extent at
different temperatures. We studied the unimolecular decom-
position of t-C4H9 earlier using the same experimental technique
14
at T ) 712-779 K. The decomposition of the s-C4H9 radical
1
1
1
5
was studied at T ) 600-680 K, and the iso-C4H9 isomer
decomposes within an approximate temperature range 580-
12
1
1
1
1
2
2
2
2
12
12
2
2
8
8
6
50 K. The current study of the thermal decomposition of the
n-C4H9 radical was limited to the 560-620 K temperature range.
The absence of any significant ion signal if the bromine
ionization lamp is used, together with the different temperature
ranges in which the four butyl radical isomers decompose, and
the absence of propene among the products of the radical
decomposition permits us to eliminate the potential contribution
of other butyl radical isomers to the n-C4H9 signal under the
conditions of our experimental study.
It was, however, not possible to eliminate such contributions
to the signal of the iso-butyl radical. At 650 K, the temporal
profile of the iso-C4H9 ion signal exhibited a noticeable
contribution of a slow-decaying (compared to a fast decay due
to the decomposition) component. The relative contribution of
this component significantly increased if a bromine ionization
lamp was used. This indicates the possibility of the formation
of some amount of tert-butyl radicals (which decompose at
higher temperatures) in the photolysis of 1-bromo-2-methyl-
propane. Hence, the rate constants of the unimolecular decom-
position of iso-C4H9 could not be measured directly.
1
1
1
1
18
8
8
1
1
N2 as Bath Gas
23.6 123
31.9 112
31.9 127
31.9 175
31.9 236
32.3 96.6
37.4 136
37.4 166
37.4 249
6
6
6
6
6
2
2
2
580
580
590
600
610
570
580
590
600
10.8
10.8
10.8
10.8
10.8
10.8
11.0
11.0
11.0
283
251
191
193
192
225
218
180
200
67
54
32
31
30
45
42
28
33
1
1
1
12
q
The results of all the experiments and the conditions used to
determine k1 are given in Table 1. The unimolecular rate
constants for reaction 1 obtained from these sets of experiments
conducted at different densities are shown in Figures 1-3.
Estimated uncertainties in the k1 determinations vary from (10%
in the middle of the temperature range to (20% at the extreme
temperatures used.
C H were studied by the MP2/6-31G*//UHF/6-31G* method.
4
9
Geometrical structures corresponding to minima and maxima
of the rotational potential energy surfaces were obtained with
the full optimization at the UHF/6-31G* level and energy was
calculated at the MP2/6-31G* level. The rotational energy
barriers of the -CH3 torsion and rotation about the C2-C3 bond
are listed in Table 3S. The torsional motion of the -CH2 group
of the n-butyl radical was found to have a very small rotational
barrier (6-fold barrier with the highest maximum only 0.8 kJ
III. Ab Initio Study of the n-Butyl and iso-Butyl
Radicals and Decomposition Transition States
-1
mol above the lowest minimum, correction for the zero-point
vibrational energy included). Hence, it was treated as free
rotation. The -CH2 group torsional motion in the iso-C4H9
radical also has a very small barrier (1.2 kJ mol- ) and,
therefore, was also treated as free.
We studied the geometries and harmonic vibrational frequen-
cies of n-C4H9 and iso-C4H9 radicals and decomposition
1
q
q
transition states (n-C4H9 and iso-C4H9 ) using the ab initio UHF
method with the 6-31G* basis. Energies were calculated at the
MP2/6-31G* method. Structures and vibrational frequencies
of the most stable conformations of these species are listed in
Tables 1S and 2S (supporting information). The GAUSSIAN
The optimized structure of the n-C4H9 coincides with that
3
2
determined by Pacansky et al. using the same method. The
conclusion about the low barriers for the rotation of the -CH2
group and the shape of the potential of the rotation about the
C2-C3 bond are in an excellent agreement with analogous
results of Pacansky et al. obtained for the case of n-hexyl radical.
9
2 system of programs31 was used in all ab initio calculations.
Internal rotations (-CH2 and -CH3 torsions and rotation
q
about the C2-C3 bond) in n-C4H9, n-C4H9 , iso-C4H9, and iso-

Decomposition of n-C4H9 and iso-C4H9 Radicals
J. Phys. Chem., Vol. 100, No. 13, 1996 5321
Figure 1. Plot of n-C
for different He densities (atom cm ). Lines represent the results of
4 9 1
H unimolecular rate constants (k vs 1000/T)
-
3
Figure 3. Falloff curves, k
1
vs [He] for seven temperatures used in
-
1
the experiments (560-620 K). Lines represent the results of master
master equation simulation using 〈∆E〉all ) -28 cm . Open circles,
equation simulation using 〈∆E〉all ) -28 cm-
.
1
1
6
-3
16
[
He] ) 3 × 10 atoms cm ; closed circles, [He] ) 6 × 10 atoms
-3 16 -3
cm ; open squares, [He] ) 12 × 10 atoms cm ; closed squares,
1
6
-3
[
He] ) 18 × 10 atoms cm
.
calculated assuming separability of vibrational and rotational
including internal) degrees of freedom and using the values of
(
33
Pitzer and Gwinn for thermodynamic functions of the hindered
internal rotors. The calculated values of S°298 (332 and 320 J
-
1
-1
mol
K
for n-butyl and iso-butyl, respectively) coincides
within error limits with the value obtained in the recent
3
4
experimental study of Seetula and Slagle (328 ( 5 and 316
5 J mol- K ).
1
-1
(
The results of the ab initio study of the decomposition
q
q
transition states, n-C4H9 and iso-C4H9 , were also used in
creating the models of reactions (1,-1) and (2,-2). Although
the calculated reaction energy threshold values are not believed
to be sufficiently accurate and are not used here, the geometrical
configurations of the above transition states (Table 1S) and the
barriers for internal rotations (Table 3S) were used in the models.
The calculated sets of vibrational frequencies of the transition
states were used as a basis for the models, with several lowest
frequencies adjusted to fit the known temperature dependencies
of the rate constants (section IV).
Figure 2. Plot of n-C
for different densities of N
4
H
9
unimolecular rate constants (k
1
vs 1000/T)
-3
2
(molecule cm ). Lines represent the results
IV. Data Analysis
-1
of master equation simulation using 〈∆E〉all ) -40 cm . Closed circles,
1
6
-3
16
[
M] ) 6 × 10 molecules cm ; open squares, [M] ) 12 × 10
Transition-state theory formalism was used to model the high-
pressure limit rate constants of reactions (1,-1) and (2,-2).
Weak collision effects in reaction (1,-1) (falloff from the high-
pressure limit) were analyzed using master equation modeling.
In this section knowledge of the high-pressure limit rate
-
3
molecules cm
.
The models of the n-C4H9 and iso-C4H9 radicals used in the
data analysis (next section) were built on the basis of the current
ab initio study. The moments of inertia for the overall rotations
and reduced moments of inertia for the internal rotations in
radicals and decomposition transition states were calculated
using the data of Table 1S. The calculated vibrational frequen-
cies (scaled by a factor of 0.89) of the lowest-energy conforma-
tions of n-butyl and iso-butyl radicals were used in these models.
The potential energy surface of the C2-C3 torsional motion in
the n-C4H9 radical has a complex form with three minima
∞
1
∞
-1
∞
2
∞
-2
constants (k , k , k , k ) is first reviewed. Then, transition-
state models are created which reproduce the high-pressure limit
∞
-
rate constants of the reverse (association) reactions k (T) and
1
∞
-
k (T). The density of states and sum of states functions of
2
the n-butyl and iso-butyl radicals and the transition states, needed
for the calculation of k(E) values and further analysis, are
obtained from the rovibrational models of these species. The
hindered internal rotors are treated classically using the simple
(conformations anti n-C4H9 and gauche n-C4H9 in Table 3S),
1
4
two of them being identical (gauche n-C4H9). In our model of
the n-butyl radical, we assume a triple sinusoidal well with the
barrier height of 18.6 kJ mol sthe average of the barriers
between the anti and gauche conformations and between the
two gauche conformations.
One test of the appropriateness of the selected parameters of
the n-butyl and iso-butyl radicals is to compare the calculated
entropy with the values obtained from experiment. The room-
temperature entropies for the two butyl radical isomers were
formalism introduced by us earlier.
The transition-state
models provide the values of the microcanonical rate constants,
k(E), needed in the master equation modeling. The energy
barrier height for the n-butyl decomposition reaction is obtained
by an optimization. In this optimization procedure, the experi-
mental rate constants are reproduced by master equation/RRKM
modeling and the deviation between the experimental and
calculated values is minimized. The selected model of reaction
(1,-1) is used to obtain quantitative information on the weak
-
1

5
322 J. Phys. Chem., Vol. 100, No. 13, 1996
Knyazev and Slagle
∞
1
∞
2
Figure 4. Plot of experimental and calculated values of k vs 1000/T.
Figure 5. Plot of experimental and calculated values of k vs 1000/T.
1
7
18
19
Open circles, Morganroth and Calvert; closed circles, Kerr and
Trotman-Dickenson. Solid line is obtained from the TST calculations
Squares, Slater et al.; circles, Metcalfe and Trotman-Dickenson.
Open symbols, experimental values of k ; closed symbols, “experi-
16
2
using the model described in text, section IV.2. Dashed line represents
mental” values corrected for the falloff effects (see text). Solid line is
obtained from the TST calculations using the model described in text,
section IV.4. Dashed line, recommendation by Tsang.
the k
values obtained using this model with 〈∆E〉°down ) 500 cm-
1
1
2
6
under the conditions of the experiments of Kerr and Trotman-Dickenson
see text).
(
radicals, k5, is required. No experimental data are available on
this reaction. We assume here that k5 equals the rate constant
of the recombination of n-propyl radicals. Taking the recom-
collision effects. The transition state model of reaction -2, in
combination with the known thermochemistry of reaction (2,-
2
), provides the temperature dependence of the high-pressure
36
mendation of Tsang for the latter value, we obtain k5 ) 1.7
∞
2
limit rate constant of reaction 2, k (T).
-11
3
-1 -1
× 10
cm molecule s . Using this expression and the
1
. Experimental High-Pressure Limit Rate Constants of
16
results of Kerr and Trotman-Dickenson and of Morganroth
and Calvert, values of k1 are obtained which can be fitted with
Reactions (1,-1) and (2,-2). Present knowledge of the
kinetics of reaction (1,-1) and (2,-2) is based on relative rate
studies. In this section we review the relevant literature and
reanalyze the relative rate measurements in light of current
kinetic data available on the reference reactions in order to
develop transition-state models of reactions (1,-1) and (2,-
17
simple Arrhenius expressions in the experimental temperature
ranges of these authors. We obtain
10
-1
k (KTD) ) 7.73 × 10 exp(-11349 K/T) s
(I)
1
2
).
Decomposition Reaction n-C4H9 f C2H5 + C2H4 (1).
from the results of Kerr and Trotman-Dickenson and
13
-1
3
5
Gierczak et al. studied chemically activated s-C4H9 from the
photolysis of H2S/cis-2-butene mixtures with final product
analysis. Kinetic modeling was used to determine the rates of
unimolecular decomposition and isomerization of the sec-butyl
and n-butyl radicals. No details of the model are reported, and
k (MC) ) 2.58 × 10 exp(-14481 K/T) s
(II)
1
from the results of Morganroth and Calvert (Figure 4).
Decomposition reaction iso-C4H9 f CH3 + C3H6 (2).
1
9
18
Metcalfe and Trotman-Dickenson and Slater et al. deter-
mined the iso-C4H9 decomposition rate constants relative to that
of the recombination reaction 6:
1
3.65 -1
it is not clear to what extent the results (A ) 10
s , threshold
-
1
energy ) 121.8 kJ mol ) are affected by the assumptions and
details of the model. We therefore did not consider the results
3
5
iso-C H + iso-C H f C H
18
(6)
of Gierczak et al. in creating our transition-state model.
Kerr and Trotman-Dickenson16 and Morganroth and Calvert17
determined the decomposition rate constant relative to that of
4
9
4
9
8
Metcalfe and Trotman-Dickenson used the photoinitiated chain
decomposition of isovaleraldehyde with product analysis by gas
chromatography. Their experiments were conducted at tem-
peratures in the range 552-691 K and concentrations of
isovaleraldehyde (the main component of the bath gas) (4.0-
1
6
the recombination reaction 5. Kerr and Trotman-Dickenson
n-C H + n-C H f C H (5)
4
9
4
9
8
18
1
7
-3
used medium-pressure mercury arc photolysis of n-valeralde-
hyde as a n-C4H9 source. The experiments were conducted at
7.2) × 10 molecules cm . Slater et al. used the 366-nm
photolysis of 1,1′-azoisobutane with gas chromatographic
analysis of products at T ) 543-598 K and an average pressure
of 1,1′-azoisobutane of 5.4 Torr.
17
an average concentration of valeraldehyde of 4 × 10 molecules
-
3
cm and temperatures in the range 477-689 K using final
product analysis. Morganroth and Calvert17 used 366-nm
photolysis of azobutane with mass spectrometric and chromato-
graphic analysis of products at T ) 432-520 K and an average
The rate constants of reaction 6 have not been determined
26
experimentally. Following the recommendation by Tsang, we
use the value of the rate constant of the recombination of n-C H
7
3
1
7
-3
36
azobutane concentration of 1.5 × 10 molecules cm .
radicals for reaction 6. The resultant values of k obtained
2
∞
1
To obtain k values from these relative measurements,
from the relative measurements of Metcalfe and Trotman-
1
9
18
knowledge of the rate constant for the recombination of n-butyl
Dickenson and Slater et al. are shown in Figure 5. As can

Decomposition of n-C4H9 and iso-C4H9 Radicals
J. Phys. Chem., Vol. 100, No. 13, 1996 5323
2
7
TABLE 2: Models of the Molecules and Transition States
Used in the Data Analysis
These results were analyzed by Kerr and Parsonage who, based
22
on the results of Kerr and Trotman-Dickenson and of Watkins
2
3
-13
Energy Barriers
and O’Dean, recommended k-1 ) 2.63 × 10
exp(-3674
E
E
1
(reaction 1) ) 115.8 kJ mol- ; E
1
2
(reaction 2) ) 126.6 kJ mol-1
K/T) cm molecule s . This expression was obtained by
assuming the rate constant of C2H5 recombination, k7, to be
3
-1 -1
-
1
-1 (reaction -1) ) 28.0 kJ mol ; E-2 (reaction -2) )
-
1
37.2 kJ mol
-11
3
-1 -1
equal to 4.2 × 10
cm molecule s . Using the value
Vibrational Frequencies (cm-1)
250, 398, 433, 702, 788, 836, 928, 977, 1020, 1059,
37
-11
3
recommended by Baulch et al., k7 ) 1.9 × 10
cm
n-C
n-C
4
4
H
9
9
:
-1 -1
molecule s , we obtained the expression at T ) 348-482 K
1
1
2
135, 1223, 1284, 1290, 1373, 1394, 1425, 1450,
458, 1463, 1472, 2789, 2841, 2846, 2849, 2871,
901, 2903, 2944, 3030
(
combined temperature interval of experimental measurements
of Kerr and Trotman-Dickenson and of Watkins and O’Dean)
q
H
:
374, 400, 500, 700, 800, 900, 900, 900, 900, 950,
k-1 ) 1.78 × 10 exp(-3674 K/T) cm molecule s-1
-13
3
-1
1
1
2
000, 1047, 1137, 1187, 1193, 1384, 1420, 1438,
455, 1458, 1589, 2825, 2876, 2901, 2928, 2951,
961, 3000, 3020, 3042
(III)
iso-C
iso-C
4
4
H
9
:
345, 352, 383, 489, 771, 877, 912, 932, 940, 1064,
1
1
2
142, 1166, 1306, 1318, 1383, 1394, 1428, 1454,
458, 1466, 1474, 2778, 2843, 2848, 2898, 2900,
903, 2906, 2941, 3027
which was used in further analysis.
Association Reaction CH3 + C3H6 f iso-C4H9 (-2). The
recommendation for the rate constants of the nonterminal
addition of methyl radical to propylene (reaction -2) is based
on our analysis of the reaction -2′, the terminal addition of
q
H
9
:
200, 220, 360, 470, 531, 544, 696, 820, 875, 905,
9
1
2
38, 1021, 1128, 1181, 1359, 1397, 1399, 1402,
454, 1459, 1474, 2844, 2894, 2898, 2917,
951, 2966, 3024, 3030, 3032
1
5
CH3 to C3H6:
C
2
H
5
3112, 3033, 2987, 2920, 2842, 1511, 1496, 1440,
1
366, 1175, 1138, 1000, 821, 540
CH + C H f sec-C H
9
(-2′)
3
3
6
4
Rotational Constants (cm- ),Symmetry Numbers,
1
-
1
and Rotational Barriers (kJ mol
)
The relative importance of terminal (reaction -2′) and
nonterminal (reaction -2) addition of methyl radical to propyl-
Overall Rotations
A ) 0.6566; σ ) 1 (active)
n-C
n-C
4
H
9
:
25
ene was analyzed by Baldwin et al. Their recommendation
(
BC)1 ) 0.1009; σ ) 1 (inactive)
/2
(
k-2′/(k-2 + k-2′) equal to 0.735 at 753 K and 0.9 at 403 K)
was based on their experimental study of the addition of CH3
to propylene relative to the reaction of CH3 with H2 and on the
q
H
4 9
:
A ) 0.847; σ ) 1 (active)
BC)1 ) 0.122; σ ) 1 (inactive)
/2
(
iso-C
iso-C
4
H
9
9
:
A ) 0.2805; σ ) 1 (active)
(
BC)1 ) 0.2023; σ ) 1 (inactive)
/2
results of the relative rates study of Cvetanovic and Irwin.
24
q
4
H
:
A ) 0.2626; σ ) 1 (active)
Following this recommendation and using our Arrhenius
(
BC)1 ) 0.1600; σ ) 1 (inactive)
/2
-12
3
dependence of k-2′ (k-2′ ) 1.81 × 10
exp(-4239 K/T) cm
1
/3
C
2
H
5
:
(ABC) ) 1.227; σ ) 1
molecule- s ), we obtain
1
-1 15
Internal Rotations
n-C
4
H
9
:
a
a
a
a
a
a
a
a
a
1
2
3
1
2
(CH
(CH
(CH
(CH
(CH
3
3
3
3
3
-CH
2
CH
-CH
CH
CH
‚‚‚CH
-CH(CH
CH(CH )-CH
CH(-CH )CH
‚‚‚CH(CH
) ) 15.16; σ ) 6; V
2
CH
2
2
2
2
) ) 6.256; σ ) 3; V
) ) 1.644; σ ) 1; V
) ) 10.47; σ ) 2; V
) ) 5.919; σ ) 3; V
0
0
0
0
) 14.25
) 18.61
) 0
k-2 ) 2.53 × 10 exp(-5260 K/T) cm molecule s-1
-
12
3
-1
CH
CH
2
2
CH
2
2
-CH
CH
(IV)
q
n-C
4
H
9
:
-CH
CH
2
2
) 7.82
) 2.32
2
2
CH
2
) ) 1.606; σ ) 1; V
0
at T ) 403-753 K.
iso-C
iso-C
4
H
9
:
1,2(CH
3
3
)CH
2
) ) 5.703; σ ) 3; V ) 16.84
0
(CH
(CH
(CH
3
3
3
3
2
) ) 9.979; σ ) 2; V
) ) 5.643; σ ) 3; V
0
) 0
) 7.39
) 3.86
2. Transition-State Model of Reaction (1,-1). The
development of a transition-state model for reaction (1,-1) was
based on expression III for the temperature dependence of the
rate constant of reaction -1.
3
1
2
q
4
H
9
:
3
2
0
3
)CH
2
) ) 5.315; σ ) 3; V
0
C
2
H
5
:
a(CH
3
-CH
2
0
) 0
Lennard-Jones Parametersa
∞
1
∞
-1
Calculation of k and k requires knowledge of the struc-
ture and vibrational frequencies of the involved molecules. The
geometric structure and vibrational frequencies of the ethylene
gas
ꢀ/k (K)
σ (Å)
He
10.22
71.4
531.4
330.1
2.551
3.798
4.687
5.278
38,39
molecule are known.
The moments of inertia of C2H5 were
N
2
4
0
b
taken from an ab initio study of Chen.
Experimentally
observed vibrational frequencies were used together with the
n-C
iso-C
4
H
9
c
41
4 9
H
4
2
a
From ref 44. b Taken to be the same as for n-C
the same as for iso-C
10. c Taken to be
H
UMP2/6-311G** values of Pacansky et al. for modes for
which no experimental determinations were available. The
4
4 10
H .
43
internal rotation about the C-C bond was treated as free. Table
lists the molecular parameters of C2H5 used in the current
study.
The model of the n-C4H9 radical was taken from the results
2
be seen from Figure 5, the results of these two studies are in
significant disagreement. The rate constants of Slater et al. are
more than an order of magnitude lower than those of Metcalfe
and Trotman-Dickenson. It has been noted by Tsang26 that the
of the ab initio study (section III). All hindered rotations were
treated classically in order to maintain compatibility with
classical expressions for the density-of-states and sum-of-states
methodology used by Slater et al. results in an unacceptably
low value for the rate of the isobutyl radical reaction with O2.
It will be shown later (Section IV.5) that the results of both
these studies are in the falloff region.
1
4
functions of a hindered internal rotor which are required for
later calculation of energy-specific rate constants, k(E). The
errors in the values of rate constants due to this classical
Association (ReVerse) Reaction C2H5 + C2H4 f n-C4H9 (-1).
∞
∞
1
The addition of ethyl radical to the double bond of ethylene
approach are less than 4% for k-1 and less than 6% for k
at
has been studied by four groups.2
0-23
In all cases ratios of k-1
temperatures higher than 298 K, as determined by comparing
3
3
to the square root of the C2H5 recombination rate constant were
determined by product analysis in complex reaction systems.
classical partition functions with exact tabulated values.
∞
is given by45
k
-
1

5
324 J. Phys. Chem., Vol. 100, No. 13, 1996
Knyazev and Slagle
q
k T
E-1
The energy-dependent rate constants k(E) were calculated
∞
-
B
Q
k
)
exp -
45,50
1
using the RRKM method.
The density of states and sum
(
)
h Q
Q
C H C H
2 5
k T
B
2
4
of states functions of the n-butyl radical and the transition state,
needed for the calculation of k(E) values and further analysis,
are obtained from the models of these species described above.
Hindered internal rotors were treated classically using the simple
q
where Q , QC H , and QC H are partition functions of the
2
5
2 4
transition state, ethyl radical and ethylene molecule respectively,
and E-1 is the energy barrier height for association. There exists
an addition barrier in reaction -1, and hence the transition state
is tight. The moments of inertia of the transition state and the
barriers for hindered internal rotations were taken from the
results of the ab initio study (section III, Tables 1S and 3S).
The lowest vibrational frequencies of the transition state and
the addition energy barrier height for the association reaction
E-1 were varied to fit Arrhenius expression III at temperatures
in the range 348-482 K. Frequencies and other parameters of
the transition state obtained through this fitting procedure are
listed in Table 2. Since hindered internal rotations in n-C4H9
and C2H5 and in the transition state are treated classically, the
energies of these rotations are calculated relative to the minimum
of the potential energy of the torsional motion.
14
formalism introduced by us earlier. The density of states
function was calculated separately for each hindered internal
rotor of the n-C4H9 radical. The total rotational density of states
function was calculated by numerical convolution of these
individual functions. The density of states of vibrational degrees
of freedom was included using the modified Beyer-Swinehart
51
algorithm. The sum of states function for the transition state
was similarly calculated.
To account for the angular momentum conservation effects,
52
a method introduced by Marcus was used. In this method,
rate constants obtained originally without accounting for
q
adiabatic rotations are multiplied by a factor I /I, and the low-
pressure limit rate constant is also multiplied by a factor
The model of the transition state described above results in
the following expression for high-pressure limit rate constant
of reaction -1 at T ) 298-1500 K:
‡
Frot ) ρ[E + (1 - I /I)k T]/ρ[E ]
1
B
1
where F(E) is the density of states function of the excited
molecule at energy E, I and I are the moments of inertia of the
adiabatic rotational degrees of freedom of the transition state
and n-C4H9 molecule. Thus, falloff curves are “shifted” along
∞
-
) 6.59 × 10-21T2.44 exp(-2697 K/T)
q
k
1
cm molecule s-1 (V)
3
-1
50,53
the ln(P) axis by ln(Frot). It has been shown
that this simple
The barrier to dissociation of the n-C4H9 radical was obtained
from the master equation/RRKM fitting of our experimental data
section IV.3). The n-butyl radical has no optical isomers, no
overall rotational symmetry, and three internal rotors (-CH3,
CH2, and C2-C3) with symmetry factors of 3, 2, and 1stotal
symmetry factor 6. The transition state has two internal rotors
-CH3 and C2-C3) with symmetry factors of 3 and 1. Since
q
q
treatment is accurate at I /I e 3. I /I equals 1.22 for n-C4H9
and 1.26 for iso-C4H9. Frot for both molecules equals ap-
proximately 0.9 at 600 K.
(
4
5,54
The method of Gaynor et al.
for solving the master
-
equation was employed. The energy bin size used in the fitting
-1
procedures was 30 cm . By comparing the exact values of
the partition functions and high-pressure limit rate constants with
those obtained by numerical integration, it was found that errors
resulting from numerical convolution of the density of states
and sum of states functions, as well as from numerical
integration, are not more than 7% of the exact values at the
temperatures of the experiments.
(
45
no optical isomerism is involved, we obtain the reaction path
degeneracy of 2 for the decomposition reaction.
3
. Master Equation Calculations of k1(T,[M]) and De-
terminations of the Energy Barrier for Decomposition of
n-C4H9, 〈∆E〉°down, and 〈∆E〉all. The measured rate constants
of reaction 1 (k1(T,[M])) were reproduced using master equation
simulations. The “exponential down” model4
5,46
The optimum value of the energy barrier for the decomposi-
tion, E1, was obtained using a grid search with a step size of 1
of collisional
energy transfer was adopted to characterize the probability of
deactivation of the n-C4H9 molecule with a total internal energy
E in active degrees of freedom to a final energy E′ by collision
with a molecule of bath gas:
-1
kJ mol . With each fixed value of E1 a weighted sum of
squares of deviations between calculated and experimental rate
constants was minimized using collisional energy-transfer
parameters (〈∆E〉°down or 〈∆E〉all) as adjustable. Each data point
was assigned a weight of 1/k1. The optimized values of this
weighted sum of squares of deviations were obtained as a
function of E1 and fitted with a parabolic function in the vicinity
of the minimum. This procedure was repeated for the data
obtained with He as bath gas assuming three models of the
temperature dependence of 〈∆E〉°down. First, 〈∆E〉°down was
taken as independent of temperature, 〈∆E〉°down ) constant.
Second, it was assumed to be directly proportional to temper-
ature, 〈∆E〉°down ) RT. The latter model is based on the earlier
studies of the reaction
P(E′,E) ) A exp(-[E - E′]/〈∆E〉down), E′ < E
(VI)
〈
∆E〉down is the average energy lost in deactivating collisions,
and A is a normalization constant.
It has been shown that accounting for the dependence of
47
〈
∆E〉down on E results in a change of shape of the falloff curve.
We express this dependence in the form 〈∆E〉down(E) )
0
.5
〈
〈
∆E〉°down(E/E1)
where E1 is the barrier to dissociation,
o
∆E〉 down is the value of 〈∆E〉down at E ) E1. This square-root
functional form of the dependence of 〈∆E〉down on energy results
in an approximate dependence of 〈∆E〉all ∝ E (〈∆E〉all is the
C H a H + C H
(7)
average energy lost in both activating and deactivating colli-
2
5
2
4
sions) which is appropriate for large molecules.4
8,49
To estimate
9
the importance of inclusion of the energy dependence of
∆E〉down in the modeling method, we compared the values of
The analysis of this system demonstrated a strong dependence
-
1
〈
of 〈∆E〉down on temperature, increasing from 70-100 cm at
300 K to 280 cm- at 1100 K. This dependence was best fit
with the directly proportional dependence, 〈∆E〉down ) 0.255T
cm . The analysis of the i-C3H7 f H + C3H6 system also
supports a strong temperature dependence of 〈∆E〉down for He
as bath gas.
1
the rate constants of reaction 1 calculated with and without
accounting for this energy dependence. If 〈∆E〉down values are
chosen in such a way that low-pressure-limit rate constants
coincide in these two cases, the calculated rate constant
difference is ≈20% at the conditions of our experiments.
-1
10

Decomposition of n-C4H9 and iso-C4H9 Radicals
J. Phys. Chem., Vol. 100, No. 13, 1996 5325
TABLE 3: Optimized Values of the Energy Barrier Height
the model describes all sections of falloff in helium and nitrogen
equally well in the experimental pressure range.
1
E , 〈∆E〉°down, and 〈∆E〉all of Reaction 1 from the Master
Equation Modeling (Section IV.3)
The 〈∆E〉all ) constant form of the temperature dependence
of collisional parameters for He, obtained as an intermediate
case between two extremes (〈∆E〉down ) constant and 〈∆E〉down
) RT), was advocated by several groups. This independence
of 〈∆E〉all on temperature is consistent with the results of
experimental and trajectory studies of collisional energy transfer
model of 〈∆E〉down
vs T dependence
〈∆E〉°down or 〈∆E〉all/
E
1
/kJ mol-
1
cm
-1
〈
〈
〈
∆E〉°down ) RT
∆E〉°down ) constant
∆E〉all ) constant
115.7
116.1
115.8
〈∆E〉°down ) 0.295T/K
〈∆E〉°down ) 192
〈∆E〉all ) -28
(
accepted in model)
5
5,56
in large polyatomic molecules.
The recent results of Kiefer
5
7
et al. on the thermal decomposition of CF3Cl in Kr at
In the third model of the temperature dependence of
∆E〉°down, it was assumed that 〈∆E〉all is independent of
temperature. This represents an intermediate case of the
∆E〉down vs T dependence, one in which 〈∆E〉down still increases
temperatures of 1521-3059 K were best fit with the constant
〈
-
1
58
value of 〈∆E〉all ) 270 cm . Dashevskaya et al. predict
〈
∆E〉all ) constant dependence for He as a bath gas in the
framework of the sequential direct encounter model.
Transition-State Model of Reaction (2,-2). The
〈
with T but more slowly than in the case of the directly
proportional model.
The resultant optimized values of E1 obtained with these three
models of the temperature dependence of 〈∆E〉down from the
analysis of the experimental data are listed in Table 3. Reaction
4
.
development of a transition-state model for reaction (2,-2) was
based on expression IV for the temperature dependence of the
rate constant of reaction (-2). The same approach was used
as in developing the model of reaction (1,-1), except for the
selection of the E2 value.
1
is close to the high-pressure limit under the conditions of our
∞
experiments (k1([M],T)/k ) 0.15-0.50 if the model with E1
)
1
The geometric structure and vibrational frequencies of the
methyl radical and propylene as well as the barrier for the
115.8 kJ mol- is used) and, hence, the calculated values of
1
the rate constant are relatively insensitive to 〈∆E〉down and,
consequently, to its temperature dependence. As can be seen
from Table 3, the difference between the values of E1 obtained
with the three models of the temperature dependence of 〈∆E〉down
59,60
internal rotation in the propylene molecule are known.
The
q
models of iso-C4H9 and iso-C4H9 were taken from the results
of the ab initio study (section III). The lowest vibrational
frequencies of the transition state and the addition energy barrier
height for the association reaction E-2 were varied to fit
Arrhenius expression IV at temperatures in the range 403-753
K. Frequencies and other parameters of the transition state
obtained through this fitting procedure are listed in Table 2.
The barrier to dissociation of the iso-C4H9 radical, E2, was
obtained from the known thermochemistry of the involved
-1
is only 0.4 kJ mol . We select for further use in our model of
reaction (1,-1) (Table 2) the value of E1 ) 115.8 kJ mol-
obtained assuming 〈∆E〉all independent of temperature (model
1
3
sintermediate case between models 1 and 2). The uncertainty
-1
of E1 obtained from the fitting procedure is 0.6 kJ mol . We
estimate the total uncertainty of E1 as twice the sum of the above
value and the difference between individual values of the energy
barrier for the decomposition of n-C4H9 in Table 3. Hence,
-1 34
species. Using ∆Hf°298(iso-C4H9) ) 72.7 ( 2.2 kJ mol ,
-
1 61,62
∆
2
Hf°298(CH3) ) 146.0 ( 1.0 kJ mol ,
∆Hf°298(C3H6) )
-
1
we obtain E1 ) 115.8 ( 2.0 kJ mol .
-1 63
-1
0.0 ( 0.8 kJ mol , we obtain E2 ) 126.6 kJ mol . The
This model of reaction (1,-1) results in the following
expression for the high-pressure limit rate constant of the n-butyl
radical decomposition at T ) 298-1500 K:
isobutyl radical has no optical isomers, no overall rotational
symmetry, and three internal rotors (2 -CH3 and -CH2) with
symmetry factors of 3, 3, and 2stotal symmetry factor 18. The
transition state has two internal -CH3 rotors with symmetry
factors of 3 each. Taking into account the existence of two
∞
1
13
-1
k ) 1.06 × 10 exp(-14005 K/T) s
(VII)
4
5
optical isomers in the transition state, we obtain a reaction
path degeneracy of 4 for the decomposition reaction.
The model of the transition state described above results in
the following expressions for the high-pressure limit rate
constants of reaction (2,-2) at T ) 298-1500 K:
which is plotted in Figure 4 together with the experimental
determinations of ref. 16 and 17.
Individual values of 〈∆E〉°down can be obtained for each
experiment listed in Table 1 by reproducing the experimental
values of rate constants with the solution of the master equation
using 〈∆E〉°down as an adjustable parameter. Such calculations
were performed with the selected model of the reaction. The
resultant values of 〈∆E〉°down and 〈∆E〉all are listed in Table 1.
∞
-
) 1.66 × 10-20T2.57 exp(-3879 K/T)
k
2
cm molecule s-1 (VIII)
3
-1
-
1
The average value of 〈∆E〉all for He is -30 cm consistent (as
∞
2
12 0.65
-1
-
1
k ) 2.14 × 10 T exp(-15529 K/T) s
(IX)
expected) with the value -28 cm obtained in the “global”
fitting procedure. The average value for N2 as bath gas is 〈∆E〉all
-
1
5. Analysis of Prior Experimental Studies of Reactions
1,-1) and (2,-2). Both experimental determinations of k-1,
)
-40 cm . Due to the scatter of individual values no
(
information on the temperature dependence of 〈∆E〉°down or
2
2
those of Kerr and Trotman-Dickenson and Watkins and
〈
∆E〉all can be extracted from these results.
2
3
O’Dean, were performed under the conditions of the high-
pressure limit, as checked by calculating the values of
The values of the rate constants k1([M],T) calculated using
the accepted model of the reaction (Tables 2 and 3) and values
∞
of 〈∆E〉all ) -28 cm- for He and 〈∆E〉all ) -40 cm for N2
are presented in Figures 1-3 together with the experimental
data.
1
-1
k1(T,[M])/k1 at the temperatures and pressures of these ex-
-1
periments. The values of 〈∆E〉down ) 500 and 800 cm were
assigned to C2H4 and propionaldehyde which were the bath
gases in refs 22 and 23, respectively.
A check of the quality of the model and data analysis
procedure is provided by an inspection of the pressure depen-
dence of 〈∆E〉°down or 〈∆E〉all. The fitted values of these energy-
transfer parameters should exhibit no dependence on pressure.
If the values of 〈∆E〉°down and 〈∆E〉all from Table 1 are plotted
vs pressure, no dependence can be seen which indicates that
As can be seen from the comparison of the experimental
∞
values of k and the calculated results (Figure 4), the results of
1
1
7
Morganroth and Calvert are well reproduced by expression
VII. The results of Kerr and Trotman-Dickenson, although
1
6
being in approximate agreement with formula VII at T ) 500-

5
326 J. Phys. Chem., Vol. 100, No. 13, 1996
Knyazev and Slagle
-
1
6
00 K, are somewhat higher at lower temperatures and
(ref 64) and ∆Hf°298(C2H4) ) 52.5 ( 0.3 kJ mol (ref 59), we
-1
significantly lower at higher temperatures. The experimental
study of reaction 1 by Morganroth and Calvert was practically
in the high-pressure limit (k1/k ) 1 if 〈∆E〉down ) 800 cm is
assigned to 1,1′-azo-n-butane, the bath gas used in ref 17). The
data of Kerr and Trotman-Dickenson on k1 are not exactly in
the high-pressure limit. Assigning 〈∆E〉down ) 800 cm to
n-valeraldehyde (bath gas of ref 16) yields k1/k (689 K) ) 0.7,
obtain ∆Hf°298(n-C4H9) ) 84.1 kJ mol .
17
The uncertainty of this value can be estimated from the
combined uncertainties in the heats of formation of C2H5 and
C2H4, in the value of E1, and the uncertainty induced by possible
∞
1
-1
∞
errors in the k-1(T) dependence used in developing the model
-
1
of reaction 1,-1. The latter value is estimated as follows.
The “global” fitting (section IV.3) of our data on k1(T,[He])
(which are relatively close to the high-pressure limit) is most
sensitive to the high-pressure limit rate constants at the
temperatures of the experiments. Hence, if a different model
of the transition state (and a different high-pressure preexpo-
nential factor) is used, this fitting procedure will try to
compensate for the change of the preexponential factor by
∞
1
-
1
∞
1
and using 〈∆E〉down ) 500 cm , we obtain k1/k (689 K) )
0
.6. The Lennard-Jones collisional parameters used were
estimated from the boiling point and additive volume incre-
ments. Falloff, however, can only partially account for the
difference between the calculated values of k and experimen-
tal results of Kerr and Trotman-Dickenson (Figure 4).
4
4
∞
1
∞
changing E1 in order to get the same values of k1 at the
temperatures of the experiments. In our data analysis, the
vibrational frequencies of the transition state were selected to
1
8
Both experimental determinations of k2 (Slater et al. and
19
Metcalfe and Trotman-Dickenson ) were performed under the
conditions of falloff. Using the above model of reaction (2,-
∞
reproduce the temperature dependence of k , formula III.
) and assigning 〈∆E〉down ) 800 cm-1 to 1,1′-azoisobutane and
-1
2
This dependence is based on the recommendation of Kerr and
isovaleraldehyde (the bath gases used by these authors) we
0.5
Parsonage for k-1/(k7) and the k7 value recommended by
∞
2
obtain the values of k2/k equal to 0.39-0.54 for the condi-
Baulch et al. The recommended37 uncertainty factor for k is
37
7
tions of Slater et al. and 0.48-0.79 for the conditions of
Metcalfe and Trotman-Dickenson. The Lennard-Jones colli-
sional parameters used were estimated from the boiling point
0.3
∞
1
0 , which yields an uncertainty factor of 1.4 for k .
-1
If values of k changed by a factor of 1.4 are used in the
global” fitting procedure (section IV.3), a change of 1.7 kJ
mol will be required in the value of E1 to obtain the same
high-pressure limit rate constant of reaction 1 at 590 K, the
average temperature of our experiments. We use twice this
value (3.4 kJ mol ) as a measure of the uncertainty in the
thermochemistry induced by the uncertainty in the k (T)
∞
-
1
“
44
and additive volume increments. “Experimental” values of
-1
∞
2
k can be determined by dividing the experimental k2 by the
∞
2
∞
2
calculated k2/k values. The values of k2 and k thus obtained
from the results of refs 18 and 19 are shown in Figure 5 together
with our calculated k .
-1
∞
2
∞
-
1
It can be seen from Figure 5 that taking into account the
falloff correction significantly improves the agreement between
the “experimental” values of k2 obtained from the data of
dependence.
Combining this value with the uncertainties of E1 and the
heats of formation of C2H5 and C2H4, we obtain
∞
Metcalfe and Trotman-Dickenson and those predicted by our
model of reaction 2. The calculated and “experimental” values
coincide in the middle of the experimental temperature range,
although the “experimental” results display a slightly lower
activation energy than the calculated ones. The results of Slater
et al. are in significant disagreement with both the results of
Metcalfe and Trotman-Dickenson and with our calculated
-
1
∆H °298(n-C H ) ) 84.1 ( 4.2 kJ mol
f 4 9
and, using ∆Hf°298(n-C4H10) ) -125.6 ( 0.7 kJ mol- (ref 63),
1
we obtain the bond energy
-
1
DH298(n-C H -H) ) 427.7 ( 4.3 kJ mol
4
9
values. Even after correcting for the falloff effects, the values
∞
Our ∆Hf°298(n-C4H9) ) 84.1 ( 4.2 kJ mol-1 is 3.6 kJ mol
-1
of k obtained from the results of ref 18 are more than an order
2
34
higher than the directly determined value of Seetula and Slagle
of magnitude lower than those of ref 19 and our calculated
∞
-1
(
80.5 ( 2.2 kJ mol from the second-law treatment of the
k .
2
results on the kinetics of n-C4H9 + HBr f n-C4H10 + Br
reaction). This disagreement lies within error limits and is not
large considering the indirect nature of the value obtained in
the current study which relies on the results of relative rate
2
6
∞
2
∞
2
Tsang recommended the k (T) dependence (k ) 2 ×
1
3
-1
1
0 exp(-15075 K/T) s ) as a compromise between the results
of Metcalfe and Trotman-Dickenson and values calculated from
thermochemistry assuming 10% nonterminal addition of CH3
constant determinations.
13
-1
to C3H6 and a preexponential factor of 2 × 10 s . Tsang’s
recommendation is lower than that given by formula IX (a factor
of 3 at 600 K). The difference is mainly due to the differences
o
7. Uncertainty in the 〈∆E〉 down and 〈∆E〉all Determination
in Reaction (1,-1). There are sources of uncertainty in the
model of reaction (1,-1) which propagate into possible errors
in the determination of the collisional energy-transfer param-
eters, 〈∆E〉°down and 〈∆E〉all. Since reaction (1,-1) is relatively
∞
-
in approach and in the temperature dependencies of k (T)
2
used.
6
.
Thermochemistry of the n-C4H9 Radical. Kinetic
close to its high-pressure limit at the conditions of these
∞
1
information on reactions 1 and (-1) can be used to obtain the
enthalpy of reaction and, through this, the enthalpy of formation
experiments, a small variation of the calculated values of k at
these temperatures will noticeably affect the fitting parameters
of the n-butyl radical. The recommended value of E1 ) 115.8
(i.e., the determinations of 〈∆E〉°down). In the error assessment,
2.0 kJ mol- together with the fitted value of the barrier for
1
-1
(
we investigate the influence of the (2.0 kJ mol uncertainty
-
1
the reverse reaction, E-1 ) 28.0 kJ mol , gives the energy
difference between reactants and products of reaction (1,-1):
in the barrier for the n-C H decomposition on the fitted values
4
9
of 〈∆E〉°down.
-
1
∆
E1,-1 ) 87.8 kJ mol . Using the models of n-C4H9, C2H5,
The master equation fitting exercise described above was
repeated with no changes of the structure and frequencies of
the transition state but with the barrier height for decomposition
and C2H4 described above and taking into account the zero-
point energies of the torsional rotations (excluded from E1 and
-
1
∞
1
∆
)
E-1), one can obtain for the enthalpy of reaction (1,-1) ∆H°298
changed by (2.0 kJ mol . This provided a change of k at
-1
-1
89.4 kJ mol . With ∆Hf°298(C2H5) ) 121.0 ( 1.5 kJ mol
590 K, the average temperature of our experiments, of a factor

Decomposition of n-C4H9 and iso-C4H9 Radicals
J. Phys. Chem., Vol. 100, No. 13, 1996 5327
4
of 1.5. The resulting individual calculated values of 〈∆E〉°down
are significantly affected by these variations of the model. The
fitted values of 〈∆E〉°down and -〈∆E〉all (in parentheses) are 328-
Fcent ) 1.53 × 10 exp(-10578 K/T) +
0
.190 exp(205 K/T) (reaction 1)
90)/125(14) cm- (He) and 377(113)/146(20) cm (N2) with
1
-1
(
35 -12.92
k2° ) 2.87 × 10 T
exp(-18031 K/T)
the lower values of 〈∆E〉°down corresponding to the lower value
of the energy barrier.
These results can be considered as upper and lower limits of
the 〈∆E〉all parameter obtained from the analysis of our
experimental data. Hence, we obtain
cm molecule s-1 (reaction 2)
Fcent ) 103.2 exp(-6762 K/T) + 0.1186 exp(242 K/T)
(reaction 2)
3
-1
3
1 -11.91
+
62
-14
cm-1 for He as bath gas
k ° ) 3.15 × 10 T
exp(-16237 K/T)
-
〈∆E〉all ) 28
(
)
1
cm molecule s-1 (reaction 1)
3
-1
+
73
cm-1 for N as bath gas
-
〈∆E〉all ) 40
(
)
-20
2
4
Fcent ) 1.46 × 10 exp(-10565 K/T) +
or
0
.189 exp(214 K/T) (reaction 1)
+
144
-59
cm-1 for He as bath gas
3
5 -12.85
〈
∆E〉°down ) 184
(
)
k ° ) 1.05 × 10 T
exp(-17900 K/T)
2
cm molecule s-1 (reaction 2)
3
-1
+
-69
162
cm-1
〈
∆E〉°down ) 215
(
)
for N as bath gas
2
Fcent ) 470 exp(-8187 K/T) + 0.1269 exp(470 K/T)
reaction 2)
at T ) 560-620 K. These average values are comparable to
the values for other alkyl radicals which have now been
by modeling the results of experiments of a
similar type and which cover the range of values 130-360 cm
for He as bath gas and 130-500 cm for N2 as bath gas.
(
8
-15
determined
for nitrogen as bath gas.
-1
The average deviation of the parametrized values of k1 and
k2 from the master equation rate constants over this large range
of conditions is 6%, with maximum deviation of 17%.
-
1
V. Parametrization of Rate Constants
Recently, Bernshtein and Oref66 demonstrated that the steady-
state approximation cannot be used for calculation of unimo-
lecular reaction rates by the master equation solution at high
temperatures and low pressures, where relaxation to the steady-
state population occurs on time scales comparable to the reaction
time. The examples given by these authors indicate that
reasonable agreement between the exact values of rate constants
and those obtained under the steady-state assumption is reached
after several hundred collisions with the bath gas. The
calculations presented here are limited to conditions where the
ratio of collision frequency to the rate constant is higher than
We present here a parametrization of k1 and k2 in helium
and nitrogen which provides rate constant values throughout
the range of temperatures 298-900 K and pressures 0.001-10
atm. The modified Lindemann-Hinshelwood expression in-
troduced by Gilbert et al.65 was used:
0
1
∞
1
k1
k [M]/k
)
∞F(Fcent,[M])
(X)
∞
0
1
k
1 + k [M]/k
1
1
Values of k1([M],T) and k2([M],T) in the above temperature
and pressure intervals were calculated using the master equation/
RRKM approach assuming 〈∆E〉all independent of temperature
for both bath gases. For reaction 1, the values of 〈∆E〉all
400. This requirement and the pressure range of 0.001-10 atm
determine the upper temperature limit (900 K) of the current
parametrization. For the low-pressure limit rate constant
expressions, however, the upper temperature limit determined
by this requirement is lowers820 K.
-
1
obtained from the modeling of experimental data (-28 cm
-1
in He and -40 cm in N2) were used. For reaction 2, the
-
1
value of 〈∆E〉all ) -53 cm was used for both He and N2,
Due to the uncertainty in the parameters of the models
discussed above and in the temperature dependence of 〈∆E〉down
caution is advised in using the results of this extrapolation.
-1
which corresponds to 〈∆E〉down ) 250 cm at 600 K. The latter
8
-15
value was chosen based on the results of previous studies
of decomposition of hydrocarbon radicals. It results in the
-
1
calculated value of k2(650 K, 4 Torr) ) 665 s in He which is
in a qualitative agreement with our experimental observation
made under these conditions: a fast decay of iso-C4H9 was
monitored but could not be measured quantitatively due to
experimental complications (section II). The matrixes of
calculated values of k1(M,T) were fitted with the modified
Lindemann-Hinshelwood expression X and the resulting
parameters are
VI. Summary
The unimolecular decomposition of n-C4H9 in helium and
nitrogen (reaction 1) has been studied in the falloff region. A
transition state model of reaction (1,-1) was developed which
provides the high-pressure limit rate constants. The model was
used to calculate k(E) values for the master equation analysis.
The experimental values of k1 were reproduced by master
equation/RRKM modeling.
∞
1
13
-1
k ) 1.06 × 10 exp(-14005 K/T) s
Literature data on the reaction iso-C4H9 f CH3 + C3H6 (2,-
) were reanalyzed and a transition-state model of this reaction
2
was developed which provides the high-pressure limit rate
constants. Finally, parametrized expressions are presented
which provide values of k1 and k2 over a wide range of pressures
and temperatures.
∞
2
12 0.65
-1
k ) 2.14 × 10 T exp(-15529 K/T) s
3
1 -11.94
k ° ) 3.82 × 10 T
exp(-16292 K/T)
1
cm molecule s-1 (reaction 1)
3
-1
Acknowledgment. This research was supported by the
for helium as bath gas;
Division of Chemical Sciences, Office of Basic Energy Sciences,

5
328 J. Phys. Chem., Vol. 100, No. 13, 1996
Knyazev and Slagle
Office of Energy Research, U.S. Department of Energy under
Grant No. DE-FG02-94ER1446.
(31) Gaussian 92, Revision E.1, Frisch, M. J.; Trucks, G. W.; Head-
Gordon, M.; Gill, P. M. W.; Wong, M. W.; Foresman, J. B.; Johnson, B.
G.; Schlegel, H. B.; Robb, M. A.; Replogle, E. S.; Gomperts, R.; Andres,
J. L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox,
D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. Gaussian, Inc.:
Pittsburgh, PA, 1992.
Supporting Information Available: Tables 1S-3S contain-
ing information on the properties of n-butyl and iso-butyl
radicals and corresponding transition states obtained in the ab
initio study (5 pages). Ordering information is given on any
current masthead page.
(
32) Pacansky, J.; Waltman, R. J.; Barnes, L. A. J. Phys. Chem. 1993,
9
7, 10695.
(33) Pitzer, K. S.; Gwinn, W. D. J. Chem. Phys. 1942, 10, 428.
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(
35) Gierczak, T.; Gawlowski, J.; Niedzielski, J. React. Kinet. Catal.
Lett. 1988, 36, 435.
36) Tsang, W. J. Phys. Chem. Ref. Data 1988, 17, 887.
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