Kinetics of the Reactions of AHyl and Propargyl Radicals with CHb

Article abstract of DOI:10.1021/jp003890d

The allyl-methyl and propargyl-methyl cross-radical reactions were studied by laser photolysis/photoionization mass spectroscopy. Overall rate constants were obtained in direct real-time experiments in the temperature region 301-800 K and bath gas (helium

Full text of DOI:10.1021/jp003890d

3196
J. Phys. Chem. A 2001, 105, 3196-3204
Kinetics of the Reactions of Allyl and Propargyl Radicals with CH3
Vadim D. Knyazev* and Irene R. Slagle*
The Catholic UniVersity of America, Department of Chemistry, Washington, D.C. 20064
ReceiVed: October 20, 2000; In Final Form: January 16, 2001
The allyl-methyl and propargyl-methyl cross-radical reactions were studied by laser photolysis/photoionization
mass spectroscopy. Overall rate constants were obtained in direct real-time experiments in the temperature
1
6
-3
region 301-800 K and bath gas (helium) density (3-36) × 10 atom cm . The observed overall C
CH (1) and C + CH (2) rate constants demonstrate negative temperature dependencies. Master equation
modeling of collisional effects indicates that the C + CH reaction is near its high-pressure limit under all
3 5
H +
3
H
3 3
3
3
H
5
3
experimental conditions used. Minor corrections for the falloff effects (on average, 18% at the highest
temperature) applied to the experimental values result in the high-pressure-limit temperature dependence of
∞
-9
-0.54
3
-1 -1
the rate constant of reaction 1: k
1
) 1.55 × 10
T
exp(117 K/T) cm molecule s . The temperature
∞
dependence of k
reaction of thermal decomposition of 1-C
the C + CH reaction is not near the high-pressure limit and falloff in reaction 2 cannot be neglected at
1
combined with the thermochemistry of reaction 1 results in the rate constants of the reverse
∞
16
-1
4
H
8
k
-1 (T) ) 1.1 × 10 exp(-39100 K/T) s . On the other hand
H
3 3
3
temperatures above 500 K. Falloff corrections applied to reaction 2, on average, reach a factor of 2 at 800 K
and introduce substantial uncertainties in the extrapolated high-pressure-limit rate constant values. The evaluated
∞
-11
high-pressure-limit rate constants of reaction 2 can be represented with the expression k
2
) 6.80 × 10
3
-1 -1
exp(130K/T) cm molecule s . C
, respectively.
4 8 4 6
H and C H were identified as primary products of the reactions 1 and
2
I. Introduction
obtained in the temperature region 301-800 K and at bath gas
1
6
-3
(
helium) densities in the range (3-36) × 10 atom cm .
Radical-radical cross-combination reactions constitute an
integral part of the overall mechanisms of oxidation and
pyrolysis of hydrocarbons.1 Reliable rate and branching data
on this type of reaction are sparse as these reactions are difficult
to study experimentally due to the high reactivity of the chemical
species involved. Stabilized alkenyl and alkynyl radicals play
important roles in the combustion of hydrocarbons. The stability
and low reactivity of stabilized alkenyl radicals has been linked
to the antiknock effects of fuel additives such as ethyl tert-
butyl ether (ETBE).3 Propargyl radicals have been implicated
as important intermediates in the mechanisms of formation of
soot in hydrocarbon flames (see, for example, refs 5-8).
Because of electron delocalization, allyl and propargyl radicals,
the simplest members of these classes of radicals, are relatively
stable and unreactive with respect to thermal decomposition and
reaction with molecular oxygen. This stability results in the
accumulation of these radicals in flames. Thus, other removal
processes, including radical-radical reactions, become important
and accurate knowledge of the rate constants of the reactions
of allyl and propargyl radicals with other reactive intermediates,
such as radicals and atoms, is needed in order to model the
behavior of these species in flames.
Reactions 1 and 2 have been studied experimentally only at
9
room temperature. Garland and Bayes used the laser photolysis/
photoionization mass spectrometry technique to obtain the rate
constant of reaction 1 at 300 K and 4 Torr of argon bath gas,
,2
-
11
3
-1 -1
10
k1 ) (6.5 ( 2.0) × 10 cm molecule s . Fahr and Nayak
studied reaction 2 at room temperature and 50 Torr using laser
photolysis of precursors, final product analysis, and kinetic
modeling. These authors determined the value of k2 ) (1.5 (
cm molecule s .
This article is organized as follows. Section I is an introduc-
-
10
3
-1 -1
0
.3) × 10
,4
tion. Section II presents the experimental method and the results.
Falloff modeling is described in section III and discussion is
given in section IV.
II. Experimental Section
In this section, first, the experimental apparatus used is
described. Second, photolysis routes of free radical precursors
are characterized. The method of determination of rate constants
and the associated kinetic mechanism is explained next, followed
by a detailed description of experimental procedure used.
Finally, the experimental results are presented.
11
Apparatus. Details of the experimental apparatus and
In this work, we present the results of a direct experimental
investigation of the reactions of allyl and propargyl radicals with
methyl radical.
12
method have been described previously. Only a brief descrip-
tion is presented here. Pulsed 193 nm unfocused collimated
radiation from a Lambda Physik 201 MSC ArF excimer laser
was directed along the axis of a 50 cm long 1.05 cm i.d. heatable
tubular quartz reactor coated with boron oxide or poly-
C H + CH f products
(1)
(2)
3
5
3
1
3
C H + CH f products
(dimethylsiloxane). The laser was operated at 4 Hz and at a
fluence of 120-170 mJ/pulse. The energy flux of the laser
radiation inside the reactor was in the range of 6-17 mJ/cm2
per pulse depending on the degree of laser beam attenuation.
3
3
3
Reactions 1 and 2 were studied by means of laser photolysis/
photoionization mass spectrometry. Overall rate constants were
1
0.1021/jp003890d CCC: $20.00 © 2001 American Chemical Society
Published on Web 02/28/2001

Allyl and Propargyl Radicals
J. Phys. Chem. A, Vol. 105, No. 13, 2001 3197
-
1
Gas flowing through the tube at ≈4 m s (in order to replace
the photolyzed gas with a fresh reactant gas mixture between
the laser pulses) contained free radical precursors in low
concentrations and the bath gas, helium. The gas was continu-
ously sampled through a 0.04 cm diameter tapered hole in the
wall of the reactor (gas-sampling orifice) and formed into a beam
by a conical skimmer before it entered the vacuum chamber
containing the photoionization mass spectrometer (PIMS). As
the gas beam traversed the ion source, a portion was photoion-
ized using an atomic resonance lamp, mass selected in an
EXTREL quadrupole mass filter, and detected by a Daly
detector.14 Temporal ion signal profiles were recorded from 10
to 30 ms before each laser pulse to 15-35 ms following the
pulse by using an EG&G ORTEC multichannel scaler interfaced
with a PC computer. Typically, data from 1000 to 15000
repetitions of the experiment were accumulated before the data
were analyzed. The sources of ionizing radiation were chlorine
(>99.9%), allyl bromide (99.0%), 1,5-hexadiene (97.0%), and
1,3-butadiene (>99%)) and TCI (propargyl bromide (>97%))
and were purified by vacuum distillation prior to use. Helium
(>99.999%, <1.5 ppm of O2, MG Industries) was used without
further purification.
Method of Determination of Rate Constants. CH3 and R
radicals (R ) C3H5 or C3H3) were produced simultaneously by
the 193 nm photolysis of a mixture of corresponding precursors
highly diluted in the helium carrier gas (>99.9%). The rate
constant measurements were performed using a technique
analogous to that applied by Niiranen and Gutman to the studies
19
of the SiH3 + CH3 and Si(CH3)3 + CH3 kinetics, which is a
further development of the method used by Garland and Bayes
to study a series of radical cross-combination reactions.9
Experimental conditions (in particular, the two precursor
concentrations) were selected to create a large excess of initial
concentrations of methyl radicals over the total combined
concentration of all the remaining radicals formed in the system.
The initial concentration of methyl radicals was always 22-
180 times higher than that of R. The concentration of R radicals
(
8.9-9.1 eV, CaF2 window, used to detect C3H5 and C3H3),
hydrogen (10.2 eV, MgF2 window, used to detect CH3, C4H8,
C4H6, C3H2, C3H4, C3H5Br, C6H10, and C3H3Br), and argon
11.6-11.9 eV, LiF window, used to detect C3H2) resonance
lamps.
Photolysis of Radical Precursors. Radicals were produced
by 193 nm photolysis of corresponding precursors. The pho-
tolysis of acetone at 193 nm, which was used in this study as
1
1
-3
(
was always less than 2 × 10 molecules cm . Under these
conditions, the self-recombination of methyl radicals was
essentially unperturbed by the presence of the other radi-
cals. At the same time, the kinetics of R decay was com-
pletely determined by the reaction with CH3 and unaffected
either by self-reaction or by reactions with other active species
formed in the system, such as the side products of precursor
photolysis.
1
5
the source of methyl radicals, was shown by Lightfoot et al.
to proceed predominantly (>95%) via channel 3a under
conditions similar to those used in the current work.
Heterogeneous loss was the only additional sink of methyl
and R radicals that had to be taken into account. Thus, the kinetic
mechanism of the important loss processes of CH3 and R in
these experiments is as follows:
1
93 nm
CH C(O)CH3
8 2CH + CO
(3a)
3
3
f H + CH C(O)CH3
2
(3b)
R + CH f products
(1 or 2)
(8)
3
f CH + CH CO
(3c)
4
2
CH + CH f C H
6
3
3
2
Photolysis channels 3b and c are known15 to occur to a minor
degree, <3% and < 2%, respectively. The initial concentration
of CH3 radicals produced by the photolysis can thus be deter-
mined by measuring the photolytic depletion of CH3C(O)CH3
R f heterogeneous loss
CH f heterogeneous loss
(9 or 10)
(11)
3
(
the fraction of acetone decomposed due to photolysis) using
(
Here, reaction 9 is the wall loss of C3H5 and reaction 10 is
time-resolved photoionization mass spectrometry (see below).
Allyl radicals were produced by the photolysis of allyl
bromide or 1,5-hexadiene.
that of C3H3). For this mechanism and for the initial conditions
described above, the system of first order differential equations
can be solved analytically.
1
6
17
1
93 nm
C H Br
8 C H + Br
(4a)
(4b)
3
5
3
5
[CH3]t
k exp(-k t)
11 11
)
(I)
[
CH ]
2k [CH ] (1 - exp(-k t)) + k
8 3 0 11 11
f C H + HBr
3 0
3
4
1
93 nm
[R]
t
C H
8 2 C H5
(5a)
(5b)
)
6
10
3
[R]0
k1[CH3]0/2k8[CH3]0
f other products
k11
k [CH ] (1 - exp(-k t)) + k
11
exp(-k9t)
[2
]
8
3 0
11
Propargyl radicals were produced by the photolysis of propargyl
bromide or 1,3-butadiene.
1
8
18
(II)
1
93 nm
Equations I and II are written for the case R ) C3H5 (k1 and k9
rate constants are used for the R + CH3 and wall loss reactions,
respectively). For the R ) C3H3 case, k1 and k9 need to be
replaced with k2 and k10.
Experimental signal profiles of CH3 and R radicals (see
subsection Procedure below) were fitted with eqs I and II,
respectively, to obtain the values of the k8[CH3]0 and k1[CH3]0
C H Br
8 C H + Br
(6a)
(6b)
3
3
3
3
f C H + HBr
3
2
193 nm
C H
8 C H + CH
(7a)
(7b)
4
6
3
3
3
f other products
(or k2[CH3]0) products. The k1 and k2 rate constants were then
Radical precursors were obtained from Aldrich (acetone
obtained by dividing the experimental k1[CH3]0 and k2[CH3]0

3198 J. Phys. Chem. A, Vol. 105, No. 13, 2001
Knyazev and Slagle
Figure 1. Examples of temporal ion signal profiles obtained in the
Figure 2. Examples of temporal ion signal profiles obtained in the
17
17
experiments to measure k
1
. T ) 305 K, [He] ) 1.20 × 10 atoms
experiments to measure k
2
. T ) 600 K, [He] ) 1.20 × 10 atoms
-3
11
-3
-3
11
-3
cm , [C
×
3
H
5
Br] ) 1.25 × 10 molecules cm , [CH
3
C(O)CH
3
] ) 1.59
cm , [C
× 10 molecules cm , [C
4
H
6
] ) 3.11 × 10 molecules cm , [CH
3 3
C(O)CH ] ) 1.70
1013 molecules cm , [C
-3
e 4.3 × 10 molecules cm , [CH
10
-3
]
3 0
13
-3
11
-3
3 5
H ]
0
H ]
3 3 0
e 1.46 × 10 molecules cm , [CH
3 0
]
1
2
-3
12
-3
)
3.24 × 10 molecules cm . Lines are the results of fits with
) 4.85 × 10 molecules cm . Lines are the results of fits with
formulas I (for CH
3
) and II (for C ).
3
H
5
3 3 3
formulas I (for CH ) and II (for C H ).
values by [CH3]0 determined by measuring the photolytic
depletion of acetone (see below). An important feature of this
method is that exact knowledge of the initial concentration of
R is not required for the determination of the rate constants. In
this respect, the approach is similar to the pseudo-first-order
method frequently applied to studies of kinetics of second-order
reactions.
Procedure. In experiments with only one of the radical
precursors present in the reactor under conditions where
radical-radical reactions are negligible (low precursor concen-
tration or/and low laser intensity), the radical kinetics (CH3,
C3H5, or C3H3) was that of purely exponential decay. The rate
of the decay did not depend on the concentration of the precursor
or the laser intensity but was affected by the wall conditions of
the reactor (such as coating and history of exposure to reactive
mixtures). This decay was attributed to heterogeneous loss
processes. The rate constants of heterogeneous loss of methyl
in the photolysis (reactions 4-7), only upper limit values to
the concentration of R could be obtained.
The procedure of determination of the R + CH3 rate constants
for each set of experimental conditions consisted of the
following sequence of measurements:
1
. Kinetics of heterogeneous loss of R (determination of k9
(or k10)). Only the R radical precursor is present in the reactor
(along with the helium carrier gas which is always present).
2
. Decomposition ratio of the R radical precursor (determi-
nation of an upper limit of [R]0).
. Kinetics of heterogeneous loss of CH3 (determination of
3
k11). Only acetone is in the reactor. The photolyzing laser beam
is significantly attenuated to provide low CH3 concentrations.
4. Decomposition ratio of acetone (determination of [CH3]0).
Both radical precursors are in the reactor, from here to step 6.
Low or no attenuation of the laser bean is used (high CH3
concentrations), from here to step 6.
(k11) and allyl (k9) or propargyl (k10) radicals were determined
5
. Kinetics of methyl radical decay (determination of the
in separate sets of measurements. The wall loss rates of the
k8[CH3]0 product).
C3H5 and C3H3 radicals were in the ranges 3-10 and 3-29
-1
s , respectively, and were minor compared to the rates of
radical decay due to the reactions under study (reactions 1 and
6. Kinetics of R radical decay in the presence of methyl
radicals (determination of the k1[CH3]0 product and k1 (or the
k2[CH3]0 product and k2)).
Measurements 4 and 5 were repeated in reverse order after
monitoring the kinetics of R radicals in the presence of methyl
radicals in order to ensure the stability of initial concentrations
of CH3. Also, the stability of the heterogeneous loss rate
constants during the set of measurements was checked experi-
mentally.
Typical temporal profiles of [CH3C(O)CH3] (photolytic
precursor of CH3 radicals), [CH3], and [R] (R ) C3H5 or C3H3)
are shown in Figures 1 and 2. The lines through the experi-
mental [CH3] and [R] vs time profiles are obtained from fits of
these dependencies with expressions I and II, respectively. In
each experiment (consisting of the set of measurements de-
scribed above), the value of the k8[CH3]0 product was ob-
tained from the fit of the [CH3] vs time dependence (measured
in step 5) using the value of k11 (wall loss of CH3) determined
in step 3. Then the value of the k1[CH3]0 product (or k2[CH3]0)
was obtained from the fit of the [R] vs time dependence using
the k9 (or k10), k11, and k8[CH3]0 values obtained in steps 1, 3,
2
). The wall loss rate constant of CH3 decay was usually in the
-
1
range 0-11 s .
In the experiments to measure the R + CH3 reaction rate
constants, the initial (high) concentration of methyl radicals was
determined by measuring the photolytic depletion of acetone
(the fraction of acetone decomposed due to photolysis). The
value of the decomposition ratio (the relative decrease in the
precursor concentration upon photolysis) was obtained directly
from the acetone ion signal profile (Typical profiles are shown
in Figures 1 and 2.) and corrected for the ion signal background.
The background (less than 10% of the acetone signal) was
mainly due to a low constant concentration of acetone molecules
in the mass-spectrometer vacuum chamber and the interaction
of the scattered UV light from the resonance lamp with the high
voltage target of the Daly detector. The method of correction
for the ion signal background is described in detail in ref 12.
Initial concentrations of R (R ) C3H5 or C3H3) were evaluated
by monitoring the photolytic depletion of corresponding precur-
sors. Since products other than C3H5 or C3H3 were also produced

Allyl and Propargyl Radicals
J. Phys. Chem. A, Vol. 105, No. 13, 2001 3199
TABLE 1: Conditions and Results of Experiments to Determine k
1
/s-1
3.2
[He]a
[prec.]b
[C
H
O]b
[C
H
]
0
b
[CH
]
b
Ic
k
k
11/s-1
k
d
k
d
k/k∞e
k
1
∞ d,f
F f
T/K
3
6
3
5
3
0
9
8
1
g
301
305
305
305
306
306
400
400
600
600
800
800
800
800
12.0
12.0
12.0
12.0
12.0
12.0
12.0
12.0
12.0
12.0
12.0
12.0
3.0
5.99
3.13
1.25
1.25
358.6
164.5
159.3
518.7
291.8
70.1
0.66
1.08
0.43
0.43
1.21
0.37
0.34
0.34
0.70
0.36
1.52
0.45
0.59
0.61
28.7
34.4
32.4
111.0
62.9
14.5
46.2
18.0
68.0
34.3
64.9
8
4.3
0.0
0.0
0.0
3.3
3.3
4.1
4.1
7.3
7.3
4.26 ( 1.13
4.73 ( 1.13
4.39 ( 0.97
3.65 ( 1.40
3.99 ( 1.02
4.62 ( 1.46
3.31 ( 0.98
3.68 ( 0.94
1.80 ( 0.44
1.77 ( 0.50
0.95 ( 0.24
0.88 ( 0.31
0.70 ( 0.15
1.27 ( 0.35
10.43 ( 2.08
9.60 ( 1.22
9.40 ( 1.49
10.47 ( 3.91
10.66 ( 1.65
10.83 ( 2.67
7.89 ( 1.87
8.42 ( 1.54
5.88 ( 1.15
5.55 ( 1.08
4.49 ( 0.59
4.41 ( 0.69
4.12 ( 0.61
4.34 ( 0.66
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.99
0.99
0.92
0.92
0.82
0.96
10.43
9.60
9.40
10.47
10.66
10.83
7.89
8.42
5.95
5.61
4.88
4.79
5.01
4.50
1.20
1.13
1.16
1.37
1.16
1.25
1.24
1.18
1.24
1.24
1.35
1.38
1.57
1.27
h
17
17
16
16
16
16
16
16
6
18
4
16
16
10.1
10.1
10.1
3.2
3.2
5.2
5.2
8.3
8.3
5.5
5.5
9.3
5.3
h
h
6.31
1.94
1.97
1.97
1.91
1.91
169.1
67.8
h
185.6
180.4
162.9
155.9
191.0
160.6
h
5.47
5.47
2.41
2.51
12.6
12.6
9.8
18.23
73.2
55.5
36.0
11.3
a
Concentration of the bath gas (helium) in units of 10 atom cm-3. b Concentrations of the allyl radical photolytic precursor, acetone, C
16
3 5
H
, and
1
1
-3
c
-1
-2
d
-11
CH
3
in units of 10 molecules cm . Concentration of C
3
H
5
is an upper limit (see text). Laser intensity in mJ pulse cm
.
In units of 10
values by the calculated
can be obtained by multiplying or dividing the optimum
Br (allyl
3
-1 -1
. e Calculated falloff correction factor (section III). k
f
∞
cm molecule
s
1
are obtained by dividing the experimental k
1
∞
∞
∞
k/k factors. F is the uncertainty factor of k
1
(i.e., upper and lower limiting values of k
1
value by F). Poly(dimethylsiloxane) reactor wall coating was used. Boron oxide coated reactor was used in all other experiments. h
g
3 5
C H
bromide) was used as a photolytic precursor of C . C 10 (1,5-hexadiene) was used in all other experiments.
3
H
3
6
H
TABLE 2: Conditions and Results of Experiments to Determine k
2
T/K [He] a [prec.]b [C
H
O]b [C
H
3
]
0
b
[CH
]
0
b
Ic
k
10/ s-1
k
11/ s-1
k
d
k
d
k/k∞e
k
2
∞ d,f
Ff
3
6
3
3
8
2
301
304
304
304
305
400
400
400
600
600
600
600
800
800
800
12.0
12.0
12.0
12.0
12.0
12.0
3.0
36.0
12.0
12.0
12.0
12.0
3.0
5.91g,h
1.35
1.35
358.4
196.4
205.1
69.1
0.54
0.53
0.26
0.53
1.23
0.31
0.90
0.90
1.46
0.76
1.46
1.96
0.53
0.44
0.22
28.7
35.8
18.4
11.8
3.46
56.0
41.6
33.8
48.5
25.6
114.4
116.1
53.9
42.5
21.2
8
15
7
2.9
20.8
20.8
20.8
29.0
14.7
19.0
12.0
10.9
10.9
10.9
14.1
14.0
6.4
4.3
1.0
1.0
1.0
0.0
2.4
5.1
0.0
4.3
4.3
4.3
4.3
34.0
2.9
2.9
4.20 ( 1.15 10.71 ( 2.68
4.34 ( 1.59 11.82 ( 3.66
0.98
0.98
0.98
0.98
0.98
0.94
0.85
0.97
0.70
0.70
0.70
0.70
0.24
0.55
0.55
10.87 1.37
12.00 1.37
9.15 1.29
10.14 1.38
10.60 1.30
10.59 1.55
9.52 1.44
8.56 1.33
7.84 1.80
7.39 1.81
7.62 1.88
8.33 1.95i
11.64 2.98
4.19 ( 1.24
4.44 ( 1.69
9.01 ( 2.86
9.99 ( 2.33
1.35
7.93
15
14
16
14
14
15
7
15
15
15
12
6
h
231.3
259.6
195.8
186.2
170.1
171.6
412.9
422.0
184.1
182.6
177.3
4.52 ( 1.67 10.44 ( 3.40
0.99
5.41
5.41
3.42 ( 1.21
2.52 ( 0.64
3.90 ( 1.23
1.53 ( 0.43
1.55 ( 0.49
1.58 ( 0.49
1.54 ( 0.59
0.30 ( 0.14
1.23 ( 0.33
1.56 ( 0.47
9.92 ( 2.61
8.08 ( 1.61
8.32 ( 2.01
5.48 ( 0.96
5.17 ( 0.94
5.33 ( 1.23
5.82 ( 1.60
2.82 ( 0.52
3.94 ( 0.94
4.27 ( 0.86
h
h
3.11
3.11
3.11
h
8.34
1.89
1.55
1.55
i
36.0
36.0
7.18 2.24
12.3
7.78 2.17i
a
Concentration of the bath gas (helium) in units of 10 atom cm-3. b Concentrations of the propargyl radical photolytic precursor, acetone,
16
1
1
-3
c
-1
-2 d
C
3
H
of 10
3
, and CH
3
in units of 10 molecules cm . Concentration of C
3
H
3
is an upper limit (see text). Laser intensity in mJ pulse cm
.
In units
values by the
can be obtained by multiplying or dividing the
-
11
3
-1 -1
. e Calculated falloff correction factor (section III). k
f
∞
cm molecule
s
2
are obtained by dividing the experimental k
2
∞
∞
∞
calculated k/k factors. F is the uncertainty factor of k
optimum value by F). Poly(dimethylsiloxane) reactor wall coating was used. Boron oxide coated reactor was used in all other experiments.
propargyl bromide) was used as a photolytic precursor of C
estimated for the data at 800 K are, most likely, unrealistically large. See section IV for a discussion of the uncertainty range of k
2
(i.e., upper and lower limiting values of k
2
g
h
3 3
C H Br
i
∞
(
3 3 4 6 2
H . C H (1,3-butadiene) was used in all other experiments. Uncertainties of k
∞
2
(T) dependence.
and 5, respectively. Finally, the value of k1 (or k2) was obtained
by dividing the k1[CH3]0 product by [CH3]0 determined in step
all experiments are listed in Tables 1 and 2. It was verified
experimentally that these rate constants did not depend on the
photolyzing laser intensity, initial concentrations of R and CH3,
reactor wall coating, or the nature of the photolytic pre-
cursor of R. The rate constants of reactions 1 and 2 did not
demonstrate any pressure dependence within the experimental
uncertainties.
4
.
The sources of error in the measured experimental parameters
such as temperature, pressure, flow rate, signal count, etc., were
subdivided into statistical and systematic and propagated to the
final values of the rate constants using different mathematical
procedures for propagating systematic and statistical uncertain-
Although the measurement of k8 (CH3 recombination) was
not the goal of the current work, the experiments provided rate
constant values for the CH3 + CH3 reaction. Uncertainty in the
k8 values (Tables 1 and 2) is rather high, up to 40% of the values,
due to the fact that the experimental conditions were optimized
for most accurate determination of k1 and k2, not k8. The results
obtained are in good agreement with those previously meas-
2
0
ties. In particular, the effects of uncertainties in the hetero-
geneous radical decay rates and in the k8[CH3]0 product on the
derived k1 and k2 values were evaluated for all experiments.
The error limits of the experimentally obtained rate constant
values reported in this work represent a sum of 2σ statistical
uncertainty and estimated systematic uncertainty.
1
2,21
ured.
Experimental Results
Arrhenius plots of the C3H5 + CH3 and C3H3 + CH3 rate
constants (k1 and k2) are shown in Figures 3 and 4. The observed
rate constants decrease with increasing temperature. These
temperature dependencies can be represented with parametric
The rate constants of reactions 1 and 2 were determined at
temperatures between 301 and 800 K and bath gas densities
16
-3
[
He] ) (3-36) × 10 atom cm . Conditions and results of

3200 J. Phys. Chem. A, Vol. 105, No. 13, 2001
Knyazev and Slagle
production of C3H5 (C3H5Br and C6H10). The shapes of the C3H4
signal temporal profiles were those of “step” function both in
the presence and absence of CH3 in the reactor. No slowly rising
part of the signal could be detected indicating that the dispro-
portionation channel of reaction 1 can be of only minor (if any)
importance. No signal of C3H2 could be detected in the
experiments on reaction 2.
III. Falloff Correction.
Experimental data on the C3H5 + CH3 f C4H8 (1) and C3H3
CH3 f C4H6 (2) reactions were obtained at low bath gas
+
pressures, where falloff can be of importance. This section
describes the assessment of pressure effects (falloff from the
high-pressure limit) for reactions 1 and 2. First, the method of
obtaining approximate values of the microscopic energy-
dependent rate constants for C4H8 and C4H6 decomposition
based on the inverse Laplace transform of the temperature
Figure 3. Temperature dependencies of the experimentally obtained
∞
values of k
circles). Square represents the k
Solid line is the modified Arrhenius fit of the k
by formula XIII. Dashed and dotted lines are the central and the limiting
1
(filled circles) and extrapolated values of k
1
(open smaller
∞
dependence of the high-pressure-limit recombination rate (k (T))
9
1
value reported by Garland and Bayes.
is presented. Second, the iterative procedure of master equation
modeling performed with the purpose of evaluating the falloff
∞
1
(T) dependence given
∞
∞
corrections (k/k ) is described. Finally, the high-pressure-limit
values of k
1
calculated using the “geometric mean rule” (see text).
rate constants of reactions 1 and 2 are obtained by applying the
falloff corrections to the experimental values and associated
uncertainties are described.
Microscopic Rates via Inverse Laplace Transform of the
High-Pressure-Limit Recombination Rate. The temperature
dependence of a high-pressure-limit rate constant of a dissocia-
∞
tion reaction (kd (T)) can be presented as a Laplace transform
of the product of the microscopic energy dependent rate constant
kd(E) and the density-of-states function of the active molecule
∞
(
see, for example, ref 25). Thus, information on the kd (T) or
the related (via equilibrium constant) rate of the reverse
∞
recombination reaction, kr (T), can be used to obtain the kd(E)
function. The derivation presented here is based on the earlier
2
6
27
28
works by Forst, Pritchard, and Davis et al.
The high-pressure-limit rate constant of a dissociation reaction
AB f A + B
(12)
(V)
Figure 4. Temperature dependencies of the experimentally obtained
can be presented in the form25,29,30
∞
values of k
circles). Square represents the k
Solid line is the Arrhenius fit of the k
2
(filled circles) and extrapolated values of k
2
(open smaller
10
2
value reported by Fahr and Nayak.
(T) dependence given by formula
XIV. Shaded areas demonstrate the uncertainty ranges of k
∞
∞
1
∞
1
k (â) )
∫
kd(E)FAB(E) exp(-âE)dE
∞
d
a
o
1
calculated
Q
(â)
AB
using the “geometric mean rule” and the values of k14 reported by
2
2
Morter et al. (upper shaded area at room temperature), Atkinson and
Here, â ) (kBT)- is the reduced inverse temperature, kB is the
Boltzmann constant, and FAB(E) and Q (â) are the density-
of-states and partition functions of the active
1
2
3
10
Hudgens and Fahr and Nayak (lower shaded area at room temper-
a
AB
2
4
ature), and Alkemade and Homann (shaded area between 623 and
23 K) (see text).
2
5,29,30
7
degrees of
freedom of the AB molecule, respectively. It is assumed here
that all rotational effects due to angular momentum conservation
are averaged over the thermal distribution and incorporated into
an effective kd(E) function. The equilibrium constant of reaction
12 can be given by
fits given by the following expressions:
-
8
-1.08
3
-1 -1
k ) 6.46 × 10 T
exp(- 90K/T) cm molecule
s
1
(III)
-
4
-2.27
exp(-561K/T) cm molecule s-1
3
-1
∞
i
a
AB
k ) 2.91 × 10 T
kr
Q
(â)Q (â)
2
3/2
2
AB
h â
(IV)
K (â) ≡
)
exp(-âE ) (VI)
0
eq
(
)
∞
d
Q (â)Q (â) 2πµ
k
A
B
These expressions have 15% and 20% uncertainties, respec-
tively. Experimental error limits of individual data points are
given in Tables 1 and 2.
i
Here, Q (â) is the partition functions of the inactive, or
AB
2
5,29,30
adiabatic,
degrees of freedom of AB, QA(â) and QB(â)
C4H8 and C4H6 were detected as products of reactions 1 and
, respectively, with product rise times matching those of the
C3H5 and C3H3 decays. Attempts were made to detect the C3H4
and C3H2 products of the possible disproportionation channels
of the C3H5 + CH3 and C3H3 + CH3 reactions. C3H4 was
produced by the photolysis of both precursors used for the
are partition functions of A and B, h is the Planck constant, µ
is the reduced mass, and E0 is the 0 K reaction enthalpy. All
partition functions include only internal degrees of freedom
(electronic, vibrational, and rotational). The term in parentheses
appears from the contribution of the translational degrees of
freedom.
2

Allyl and Propargyl Radicals
J. Phys. Chem. A, Vol. 105, No. 13, 2001 3201
From eqs V and VI one obtains
TABLE 3: Properties of Molecules Used in Models of
Reactions 1 and 2
∞
r
Enthalpies of Formation (kJ mol-1)
k )
0
38
0
39
i
∆fH 298(C3H5) ) 184.5 ( 2.1
∆fH 298(C3H3) ) 339.0 ( 4.0
0 40
2
3/2
Q
(â)
0
48,49
41
AB
h â
∞
∆fH 298(CH3) ) 146.0 ( 1.0
∆fH 298(1-C4H8) ) 0.1 ( 1.0
(
)
∫
kd(E)F (E) exp(-â(E - E ))dE
AB
0
0
o
∆fH 298(1-C4H6) ) 165.2 ( 0.9
Q (â)Q (â) 2πµ
A
B
Vibrational Frequencies (cm-1)
426, 514, 544, 738, 801,
912, 983, 1071, 1182, 1247,
(
VII)
C3H5:42
Let us present the temperature dependence of the high-pressure-
limit recombination rate constant in the modified Arrhenius form
1
3
390, 1464, 1487, 3019,
021, 3054, 3107, 3107
C3H3:43
3462, 3172, 2028, 1475, 1068,
644, 476, 386, 3268, 987, 621, 351
3184 (2), 3002, 1383(2), 580
∞
r
∞
r
n
∞
r
k ) A â exp(-âE )
(VIII)
CH3:44
1-C4H8:45
283, 331, 387, 417, 652, 759, 813,
and assume that the two-dimensional rotation of AB as a whole
is inactive while the remaining one-dimensional AB rotation
together with all internal rotations and vibrations forms the pool
of the active degrees of freedom.25,29,30 Then Q (â) can be
9
1
1
00, 912, 956, 976, 1054, 1160, 1206, 1259,
313, 1394, 1407, 1455, 1467, 1474,
507, 2842, 2859, 2883, 2915, 2920, 2964,
i
AB
2974, 3050
3332, 2988, 2925, 2920, 2116, 1470,
presented as
1-C H :50-52
4
6
1
5
7
446, 1385, 1322, 1070, 1008, 840, 634,
09, 2988, 2939, 1462, 1261, 1090,
82, 630, 344, 213
i
i
-1
Q
(â) ) (B â)
(IX)
AB
AB
i
Rotational Constants (cm-1), Symmetry Numbers
where B
is the rotational constant of the inactive two-
AB
dimensional rotation (with symmetry factor incorporated) and,
combining eqs VII, VIII, and IX, one obtains
(σ, No. of Minima in Parentheses If Different),
-
1
and Rotational Barriers (kJ mol
)
C3H5:46
B ) 0.5658 σ ) 2
47
∞
C3H3:
B ) 0.9787 σ ) 2
B ) 7.6036 σ ) 6
kd(E)FAB(E) exp(-âE)dE )
44
∫
CH3:
o
1
-C4H8:53,54 B ) 0.7522 σ ) 1
(one-dimensional
active)
3
/2
∞
r
i
2πµ
n-1/2
∞
r
A B
Q (â)Q (â)â
exp(-â(E + E )) (X)
AB
A
B
0
(
2
)
h
B ) 0.1369 σ ) 1
(two-dimensional
inactive)
The left side of this equation is a Laplace transform of the
kd(E)FAB(E) function, and therefore, kd(E)FAB(E) can be obtained
by performing the inverse Laplace transform of the right side
B ) 5.849
B ) 2.194
σ ) 3
Vo ) 1104 (CH3 torsion, active)
σ ) 1(3) Vo ) 666 (C2H5 torsion,
active, 3 minima)
-C4H6:5
0-52
B ) 0.9056 σ ) 1
B ) 0.14379 σ ) 1
(1-dimensional active)
(2-dimensional inactive)
Vo ) 1060 (CH3 torsion, active)
1
1
of eq X numerically. An interesting case occurs when n ) /2.
n-1/2
The term with â
disappears and eq X can be presented as
B ) 5.301
σ ) 3
2πµ 3
/2
∞
r
i
∞
r
of the high-pressure-limit and were fitted with expression VIII
L[k (E)F (E)] ) A B
L[F (E - E - E )] (XI)
d
AB
AB
P
0
(
2
)
∞
∞
1
h
using fixed n ) /2. The obtained A and E parameters were
r r
used to evaluate the k(E) functions. These k(E) dependencies
Here, FP(E) is the density-of-states function of a “pseudo-
molecule” with internal degrees of freedom formed from the
combined internal degrees of freedom of molecules A and B
were used in master equation modeling which, in turn, yielded
∞
the calculated k/k falloff factors. Then the experimental kr(T)
∞
values were divided by these k/k falloff factors to obtain the
(
QP(â) ≡ QA(â)QB(â) ) L[FP(E)]).
“
corrected” high-pressure-limit rate constants. The procedure
Thus, kd(E) function can be obtained from
was then repeated until convergence. Convergence was achieved
for reaction 1 after two iterations (within less than 1%) and for
reaction 2 after three iterations (within less than 4%). Molecular
properties such as vibrational frequencies and rotational con-
stants used in the calculations are listed in Table 3.
∞
r
2πµ 3/2
F (E - E - E )
∞
∞i
P
0
k (E) ) A B
AB
(XII)
d
r
(
2
)
FAB(E)
h
Expression XII can be easily evaluated by computing the
densities of states using existing methods for harmonic oscil-
The choice of collisional energy transfer parameter, 〈∆E〉down
(average energy transferred per deactivating collision with the
3
1-33
29,37
lators, free and hindered rotors.
bath gas)
can be important in such modeling if the reaction
Master Equation Modeling and Falloff Corrections. Falloff
factors (k/k ) were calculated for each experimental data point
under study is far from the high-pressure-limit conditions. The
values of 〈∆E〉down for C4H8 and C4H6 used here were based on
analogy with the results of Knyazev and Tsang obtained for a
∞
using the solution of the master equation. The formulation of
master equation for recombination reactions and its relation to
that of reverse decomposition reactions can be found else-
5
5
molecule of similar size, sec-C4H9. These authors modeled
the chemically and thermally activated decomposition of sec-
C4H9 to reproduce the experimental literature data obtained over
a wide range of temperatures (195-680 K). The modeling
yielded 〈∆E〉down proportional to temperature: 〈∆E〉down(sec-
where.2
9,34
The ChemRate program was used in all calcula-
35
tions, including the calculations of the densities of states required
to evaluate the k(E) dependencies via expression XII. The
method of Gaynor et al.36 was used to solve the steady-state
-1
C4H9) ) 0.52 × (T/K) cm . A similar proportional dependence
29,37
56,57
master equation. The exponential-down
energy transfer was used.
model of collisional
was obtained earlier for the decomposition of ethyl radical.
Collisional energy transfer parameters are known to increase
with vibrational energy. References 58-66 can be consulted
for reviews of the current literature. On the basis of the results
presented in these studies, one can expect a proportional
The properties of the model were selected in an iterative
process. First, the experimentally obtained kr(T) temperature
dependencies (Figures 3 and 4) were assumed to represent those

3202 J. Phys. Chem. A, Vol. 105, No. 13, 2001
Knyazev and Slagle
dependence of 〈∆E〉all (the average energy transferred per
collision) on energy for a hydrocarbon molecule of C4Hx size.
This will mean an approximately square-root dependence for
reaction 1 among several other cross-radical reactions using a
laser photolysis/photoionization mass spectrometry method
similar to the one employed in the current investigation. These
authors, however, used literature values of the rate constants of
the CH + CH and C H + C H reactions to evaluate initial
1/2
〈
∆E〉down (〈∆E〉down ∼ E ). The above 〈∆E〉down(sec-C4H9) vs
T dependence was obtained at the energy barrier height for the
3
3
3
5
3
5
-
1
sec-C4H9 decomposition, 31 kcal mol , while the reactions of
radical concentrations. Knowledge of both C H5 and CH3
3
decomposition of C4H8 and C4H6 have barriers of 77.5 and 74.6
concentrations was needed in these experiments. Thus, the
-
1
1/2
-11
3
-1 -1
kcal mol . Scaling in accordance with the 〈∆E〉down ∼ E
resultant k value ((6.5 ( 2.0) × 10
cm molecule s )
1
dependence yields an approximate expression 〈∆E〉down ) 0.8
was dependent on the accuracy of the radical recombination
-
1
×
(T/K) cm for C4H8 and C4H6 decomposition which was
10
rate constants used by the authors. Fahr and Nayak studied
used in master equation modeling.
reaction 2 at room temperature and 50 Torr using laser
photolysis of precursors, GC/MS product analysis, and kinetic
The 〈∆E〉down values estimated by the above, rather crude,
method have large uncertainties. Therefore, to assess the effects
of these uncertainties on the results of falloff modeling,
calculations were also performed with the proportionality
coefficient in the 〈∆E〉down vs T dependence changed by a factor
of 2 in both directions.
-10
3
modeling. Their value of k2 ) (1.5 ( 0.3) × 10
cm
-1 -1
molecule
s
was derived from modeling of the experimentally
obtained final products distributions. These authors identified
the major products of reaction 2 as approximately 60% of
1
-butyne and 40% of 1,2-butadiene. The values of k1 and k2
∞
The values of the k/k falloff correction coefficients obtained
reported in refs 9 and 10 are plotted in Figures 3 and 4 together
with the results of the current work. As can be seen from the
plots, although the results of the current and earlier studies of
reactions 1 and 2 do not coincide exactly, the reported
uncertainty limits overlap for both reactions.
in the modeling are listed in Tables 1 and 2. Relative
uncertainties of extrapolation to the high-pressure limit were
estimated as twice the differences (on a logarithmic scale)
∞
between the k/k values obtained using the 〈∆E〉down ) 0.8 ×
-
1
-1
(
T/K) cm and the 〈∆E〉down ) 0.4 × (T/K) cm formulas for
The high-pressure-limit rate constants of the reaction between
allyl and methyl radicals (1) obtained by a minor extrapolation
of the experimental results (section III) display a pronounced
negative temperature dependence (eq XIII), as can be expected
for a barrierless radical-radical reaction. The temperature
the 〈∆E〉down temperature dependence. The final uncertainty
values of the high-pressure-limit rate constants listed in Tables
1
and 2 (presented as uncertainty factors) were obtained by
adding the extrapolation uncertainties to the experimental ones.
As can be expected, the largest falloff corrections (smallest
∞
∞
dependence of k2 , the high-pressure-limit rate constant of the
k/k values) were obtained at the highest experimental temper-
C3H3 + CH3 reaction, is more uncertain. This uncertainty is
due to the larger extrapolation to the high-pressure limit required
in the case of reaction 2 than in the case of reaction 1. The
difference between reactions 1 and 2 in terms of the relative
positions of the experimental pressure range relative to the falloff
scale is, primarily, due to the differences in the densities of states
of 1-C4H8 (product of reaction 1) and 1-butyne (product of
reaction 2). The density of states of 1-C4H8 is larger than that
of 1-C4H6 since 1-butene has two internal rotational degrees of
freedom while 1-butyne has only one. In the falloff calculations,
ature, 800 K. It was found that, while reaction 1 is very close
to the high-pressure limit, reaction 2 is noticeably further in
the falloff region under the conditions of the experiments. For
∞
∞
reaction 1, the lowest k/k ) 0.82 was calculated. k/k ) 0.24
was obtained for reaction 2 under the same conditions (800 K
1
6
-3
and [He] ) 3 × 10 atom cm ). Thus, the uncertainty of
extrapolation is significantly less for reaction 1 (maximum of
3
7%) than for reaction 2 (maximum factor of 2.5).
The high-pressure-limit rate constants of reactions 1 and 2
obtained by the above extrapolation of experimental results are
displayed in Figures 3 and 4 (smaller open circles). Their
temperature dependencies can be represented by the following
expressions:
1
-butyne was assumed to be the only product of reaction 2.
Although the experimental results of Fahr and Nayak indicate
the importance of the second channel, that of formation of 1,2-
butadiene, the existence of this second channel will not have
any drastic influence on the falloff calculations since the density
of states of 1,2-butadiene can be expected to be similar to that
of 1-butyne (only one internal rotation is present in both of these
molecules).
∞
-9 -0.54
3
-1 -1
k1 ) 1.55 × 10 T
exp(117K/T) cm molecule
s
(XIII)
∞
-11
3
-1 -1
k2 ) 6.80 × 10 exp(130K/T) cm molecule
s
(XIV)
The method of estimating the uncertainty of extrapolation to
∞
the high-pressure limit (twice the difference between the k/k
The estimated uncertainty of expression XIII is 15% at tem-
peratures 300-600 K and increases to 36% at 800 K. That of
expression XIV increases with temperature from 24% at room
temperature to 36% at 400 K, a factor of 1.8 at 600 K, and a
factor of 3 at 800 K. The uncertainty estimated at 800 K using
the above procedure is, probably, too large: the upper limiting
value combined with the data obtained at lower temperatures
-
1
values obtained using the 〈∆E〉down ) 0.8×(T/K) cm and the
-1
〈
∆E〉down ) 0.4×(T/K) cm dependencies, see section III) yields
uncertainty values at 800 K, the highest experimental temper-
ature of this work, that are probably too large. The upper limiting
∞
values of k2 thus obtained for 800 K would result in an
unrealistic positive temperature dependence of k2 . Therefore,
∞
an uncertainty range limited, on the lower end, by the lower
limit to the experimentally obtained (in the falloff) rate constant
values and, on the upper end, by the upper limit to the room-
would result in an unrealistic positive temperature dependence.
A more realistic uncertainty range for k2∞ is proposed in the
next section.
∞
temperature k2 value can be proposed. Thus, the following
IV. Discussion
“central” values and uncertainty range are recommended:
This work presents the first direct determination of the rate
best evaluation:
constants of reactions C3H5 + CH3 (1) and C3H3 + CH3 (2) as
k2 ) 6.80 × 10 exp(130K/T) molecule-1 cm s
∞
-11
3 -1
functions of temperature. Two earlier studies9,10 reported room-
9
temperature values of k1 and k2. Garland and Bayes studied
(XIV)

Allyl and Propargyl Radicals
J. Phys. Chem. A, Vol. 105, No. 13, 2001 3203
∞
-4 -2.27
A similar uncertainty is observed for reaction 2. Four
experimental studies of the self-reaction of propargyl radicals
lower limit:
k2 g 2.33 × 10 T
exp(-561K/T)
molecule- cm s (XVII)
1
3 -1
C H + C H f products
(14)
∞
-10
molecule-1 cm3 s
-1
3
3
3
3
upper limit:
k2 e 1.27 × 10
(
XVIII)
are available in the literature. At room temperature, the value
9
22
Garland and Bayes used their experimental data on radical cross
reactions to test the validity of the “geometric mean rule”
of the rate constant reported by Morter et al., k14 ) (1.2 (
9
,67,68
-10
3
-1 -1
0.2) × 10
cm molecule s , is a factor of 3 larger than
2
3
the values obtained by Atkinson and Hudgens and Fahr and
1/2
1
0
-11
cm molecule s-1 and (4.0 (
3
-1
kAB ) 2(kAAkBB)
(XV)
Nayak, (4.3 ( 0.6) × 10
-
11
3
-1 -1
0
.4) × 10
cm molecule s , respectively. Alkemade and
2
4
-11
3
-1 -1
frequently used to estimate rate constants of cross radical
reactions (kAB) of the type A + B from the values of kAA and
kBB, the rate constants of the A + A and B + B self-reactions.
Such comparison is also performed in the current work. The
rate constants of methyl radical self-reaction
Homann reported k ) 6.54 × 10 cm molecule
s
(no
14
uncertainties were reported) between the temperatures of 623
and 723 K. k values and uncertainty ranges were calculated
using the “geometric mean rule,” eq XV, by the same method
2
as for reaction 1 (see above). Figure 4 illustrates the results. At
room temperature, the uncertainty ranges of the calculated rate
constants obtained with the k14 values of Morter et al., Atkinson
and Hudgens, and Fahr and Nayak are represented by two
shaded areas on the plot. At higher temperatures (623-723 K),
a shaded area represents the upper and lower limits calculated
using k14 reported by Alkemade and Homann (This uncertainty
range is likely to be even wider due to unknown error limits of
the k14 value of Alkemade and Homann). Approximate agree-
ment within the combined uncertainty ranges is achieved
between the experimental k2 values obtained in the current work
CH + CH f C H
6
(8)
3
3
2
are well-known. Two recent “global fits”69,70 of falloff data
provide parametrization for the rate constants that differ very
little (less than 5%) in the high-pressure limit. Most of the
experimental data used in these parametrizations come from the
experimental study of Slagle et al.21 who used the experimental
technique and the apparatus employed in the current work. These
authors reported a ( 20% uncertainty in their experimental rate
constant values. Thus, in the calculations according to the
(
extrapolated to the high-pressure limit at higher temperatures)
“
geometric mean rule,” we used the parametrization of Hessler
23
and the values calculated using k14 of Atkinson and Hudgens,
70
∞
-11
3
-1
and Ogren (k8 ) 8.78 × 10 exp(-T/723 K cm molecule
10
24
Fahr and Nayak, and Alkemade and Homann. The k2 values
-1
s ) with 20% uncertainty.
-10
3
-1 -1
calculated using k14 ) (1.2 ( 0.2) × 10 cm molecule
s
The rate constants of the allyl radical self-reaction
22
reported by Morter et al. are noticeably higher than the
experimental ones. No certain conclusion about the adequacy
of the “geometric mean rule” can be drawn, however, until the
controversy regarding the k14 values is resolved.
C H + C H f products
(13)
3
5
3
5
1
7,71,72
have been reported by several groups.
reported k13 ) (2.65 ( 0.2) × 10
Tulloch et al.
∞
The values of k1 obtained in the current work, combined
-
11
3
-1 -1
cm molecule
s
at
with the known thermochemistry of reaction 1, can be used to
evaluate the high-pressure-limit rate constants of the reverse
2
95 K and the
∞
k (T) ) 1.69 × 10 exp(132 K/T) cm molecule s-1
-
11
3
-1
reaction of 1-C4H8 decomposition. The k-1 values thus obtained
13
can be represented with the expression
(
T ) 293-571 K) (XVI)
∞
16
-1
temperature dependence. Jenkin et al.72 reported the room-
k-1 (T) ) 1.1 × 10 exp(-39100 K/T) s
(XIX)
-11
3
-1
temperature value of k13 ) (3.0 ( 0.5) × 10 cm molecule
-1
71
-11
s . Finally, Boyd et al. obtained k13 ) (2.6 ( 0.2) × 10
evaluated for the experimental temperature range of this work,
301-800 K. The uncertainty of this expression changes from a
factor of 6 at room temperature to the factors of 4 at 400 K, 2.7
at 600 K, and 2.5 at 800 K. This uncertainty originates primarily
3
-1 -1
cm molecule
s
within the temperature range 400-540 K.
In all of these studies, allyl radicals were detected spectroscopi-
cally and the reported k13 values were linearly dependent on
the absorption cross section of C3H5, which had to be determined
independently. In the calculations, we use the temperature
4
0
38
in the error limits of the heats of formation of 1-C4H8, C3H5,
4
8,49
and CH3
(overall uncertainty in the reaction 1 enthalpy is
17
-1
dependence of Tulloch et al. (eq XVI) with the uncertainty of
4.1 kJ mol ). Rate constants of reaction -2, the reverse of the
C3H3 + CH3 recombination reaction 2, cannot be evaluated
using the same approach because of the unknown distribution
of products in reaction 2. Although Fahr and Nayak reported
an approximate ratio of 1-butyne to 1,2-butadiene at room
temperature as 1.5 (60% and 40%, respectively), the temperature
dependence of this value is unknown.
-
11
3
-1 -1
(
0.2 × 10
cm molecule s . However, the upper and
the lower limiting values used were those of all three studies
combined. For example, at room temperature the upper limit
-
11
3
-1 -1
of k13 was taken as 3.5 × 10
cm molecule
s
(from the
-
11
upper limit of Jenkin et al.) and the lower limit as 2.4 × 10
3
-1 -1
cm molecule
s
(from the lower limit of Tulloch et al.).
The resultant k1 values calculated using eq XV (the “geo-
metric mean rule”) using the “central” values of k8 and k13 are
shown in Figure 3 by the dashed line. The upper and lower
limiting values (calculated using the upper and lower limits of
k8 and k13) are shown by the dotted lines. As can be seen from
the plot, the uncertainties of the experimental and the calculated
rate constants overlap. The combined experimental uncertainties
of k1, k8, and k13 do not allow a more certain assessment of the
validity of the “geometric mean rule” as applied to reaction 1.
Acknowledgment. This research was supported by Division
of Chemical Sciences, Office of Basic Energy Sciences, Office
of Energy Research, U.S. Department of Energy under Grant
DE-FG02-94ER1446.
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