Kinetics of the Cyclopentadiene Decay and the Recombination of Cyclopentadienyl Radicals with H-Atoms: Enthalpy of Formation of the Cyclopentadienyl Radical

Article abstract of DOI:10.1002/kin.10005

The recombination of c-C5H5 with H-atoms has been studied behind reflected shock waves. The obtained rate coefficients are almost independent of temperature and were found to be k_rec = 2.6 x 10¹⁴ cm³ mol^-1 s^-1 for pressures around 2 bar in the temperature range between 1150 and 1500 K. Together with rate coefficients for the dissociation, obtained in an earlier work (Roy et al., Proc Combust Inst 1998, 27, 329-336) at pressures and temperatures close to those applied in the present work, we calculated equilibrium constants K_c(T) for C5H6 <-> c-C5H5 + H. A "third law" analysis was performed. Data for the enthalpy of formation of cyclopentadienyl DH_f,0 = 65.4 +/- 1 kcal mol^-1 and DH_f,298 = 62.5 +/- 1 kcal mol^-1 were derived, respectively. The analysis was based upon new results on the entropy of cyclopentadienyl, recently calculated by Kiefer et al. (J phy chem, in press). Finally, our measured data were subjected to a fall-off analysis. The simulation turned out very satisfactorily.

Full text of DOI:10.1002/kin.10005

Kinetics of the
Cyclopentadiene Decay
and the Recombination of
Cyclopentadienyl Radicals
with H-Atoms: Enthalpy
of Formation of the
Cyclopentadienyl Radical
K. ROY, M. BRAUN-UNKHOFF, P. FRANK, TH. JUST
DLR Institut fur Verbrennungstechnik, Bergstrasse, Frankendof, 96155 Buttenheim, Stuttgart, Germany
Received 12 June 2001; accepted 23 August 2001
ABSTRACT: The recombination of c-C5H5 with H-atoms has been studied behind reflected
shock waves. The obtained rate coefficients are almost independent of temperature and were
1
1
found to be krec = 2.6 1014 cm3 mol
s
for pressures around 2 bar in the temperature
range between 1150 and 1500 K. Together with rate coefficients for the dissociation, obtained
in an earlier work (Roy et al., Proc Combust Inst 1998, 27, 329–336) at pressures and temper-
atures close to those applied in the present work, we calculated equilibrium constants Kc(T)
for C5 H6
c-C5 H5 + H. A “third law” analysis was performed. Data for the enthalpy of forma-
1
1
tion of cyclopentadienyl DHf,0 = 65.4 1 kcal mol and DHf,298 = 62.5 1 kcal mol were
derived, respectively. The analysis was based upon new results on the entropy of cyclopentadi-
enyl, recently calculated by Kiefer et al. (J phy chem, in press). Finally, our measured data were
C
subjected to a fall-off analysis. The simulation turned out very satisfactorily. 2001 John Wiley
& Sons, Inc. Int J Chem Kinet 33: 821–833, 2001
decade it became more and more apparent that exact
kinetic data for the benzene/phenyl subsystem play a
key role in the complex chemistry of soot production
as well as of the soot reduction. Phenyl reactions with
molecular oxygen lead to phenoxy and in a further fast
decay of phenoxy to cyclopentadienyl radicals and CO
[1]. Phenol, which is an intermediate appearing on the
route of benzene oxidation, can also be easily converted
INTRODUCTION
Quantitative modeling of soot formation in flames re-
quires rather detailed chemistry models. In the last
Correspondence to: Th. Just; e-mail: thomas.just@t-on-line.de.
2001 John Wiley & Sons, Inc.
c

822
ROY ET AL.
to cyclopentadiene and CO. This happens very likely
through a keto-enol tautomerization at elevated tem-
peratures [2]. According to theoretical work [3] there
might exist a route leading by the association of two
cyclopentadienyl radicals to either naphthalene or to
naphthyl radicals +H-atoms. This route, if existing,
can, under certain fuel-to-oxygen ratios in a flame, con-
tribute to the formation of PAHs. Experimental data on
the formation of cyclopentadiene as an intermediate in
a flame exist [4]; however, a simulation of measured
concentration profiles has not been very successfull up
to now (see e.g. [5,6]). The role of the five-membered
rings in the complex reaction system, typical for a soot-
ing flame, is not fully understood presently.
greatly on new, unpublished entropy data stemming
from Kiefer et al. [14]. A fall-off analysis of our data
is also presented.
EXPERIMENTAL
The thermal decomposition of cyclopentadiene and the
recombination of the cyclopentadienyl radical with H-
atoms
C5H6 → c-C5H5 + H
c-C5H5 + H → C5H6
(R1)
(R-1)
In an earlier study, we performed measurements of
the dissociation of cyclopentadiene to c-C5H5 + H [7].
In the present work, we focus our interest on the cy-
clopentadienyl radical and report here on an experi-
mental investigation on the reverse reaction. To our
knowledge there are no experimental studies yet on
the recombination reaction c-C5H5 + H. But there ex-
ist theoretical calculation studies on the reaction mech-
anism of the c-C5H5 radical, e.g. by Dean [8a] and
Zhong and Bozzelli [8b]. For the benzene and toluene
oxidation mechanism, Emdee et al. [9] estimated a tem-
perature independent value for c-C5H5 + H. The rate
coefficients given in the aforementioned studies are
listed in Table I. The coefficient determined by Dean
is valid for 1038 K and the one by Zhong and Bozzelli
for 1100 K (the same temperature as in the experi-
ment of Lovell et al. [10]). Still, in the work of Zhong
et al. a discrepancy of about a factor 2 is found be-
tween the calculated and measured concentrations of
cyclopentadiene.
To properly interpret experimental flame data it is
absolutely necessary to use the most precise thermody-
namic data available for kinetic calculations. Presently,
there exist larger uncertainties inherent in the published
thermodynamic data. This refers to the enthalpy of for-
mation as well as to the entropy of c-C5H5 [11–13].
From our two sets of rate coefficients, obtained al-
most in the same temperature and pressure range, we
calculated the equilibrium constants and derived from
these the reaction enthalpy. Our computations profited
were studied behind reflected shock waves.
The high purity, heatable stainless steel shock tube
used in this study consists of a test section of 6 m and a
driver section of 4 m in length. The internal diameter is
7.2cm. Thetestsectionisevacuatedtoabout10 7 mbar
before each experiment by a turbomolecular pump. A
detailed description of the apparatus is given elsewhere
[15].
Atomic resonance absorption spectroscopy (ARAS)
was used to monitor the temporal concentration profiles
of hydrogen- and iodine-atoms. The H-atom absorp-
tion was measured at a wavelength = 121.6 nm by
using an oxygen spectral filter, the I-atom absorption
at ␭ = 164.2 nm using a monochromator. The concen-
tration profiles of H-atoms reported here are based on
calibration experiments using the reaction of O-atoms
with H2 to produce H-atoms [15]. In the case of I-
atoms, the dissociation of methyliodide was used for
calibration [16].
The ARAS technique allows high temperature re-
actions to be studied in highly diluted mixtures where
temperature effects caused by the ongoing of the re-
actions are of no importance. Further, these low initial
concentrations strongly reduce, within the time scale of
the experiment (t 800 s), the number of subsequent
reactions that had to be considered for the evaluation
of the measured concentration profiles.
C5H6 → C5H5 + H Dissociation Reaction
The C5H6 decomposition was studied for mixtures
of 0.5 up to 10 ppm at temperatures between 1200
and 1600 K and pressures between 1 and 5 bar [7].
Cyclopentadiene was obtained by distillation from
dicyclopentadiene with a purity of better than 99%
(controlled by mass spectrometry). The initial concen-
trations of cyclopentadiene were routinely measured
in samples drawn from the shock tube. The probes
were analyzed by a gas-chromatograph coupled to
a flame ionization detector. The detection limit was
Table I Published Data on c-C5H5 + H → C5H6
k
T
Reference
5.96 1013
3.2 1014
1.0 1014
1038
1100
a
[8a]
[8b]
[9]
Units are cm3, s, mol, K.
aNo T -range given.

ENTHALPY OF FORMATION OF THE CYCLOPENTADIENYL RADICAL
823
about 0.5 ppm at 100 mbar total pressure. The dilu-
RESULTS
ent in all experiments was highly purified Argon (Ar =
99.9999%).
c-C5H5 + H Recombination Reaction
Quite recently, Backsay and Mackie [17] published
theoretical calculations on the possible decay chan-
nels of C5H6. Additionally to the simple C H bond
fission of the cyclopentadiene they found molecular
channels leading, via hydrogen shifts followed by a
C C fission, eventually to chain molecules. The low-
est barrier was found for CH2CHCHCCH2, whereby
the calculated barrier height is on the order of the
C H bond fission for cyclopentadiene. The formed
CH2CHCHCCH2 molecule then decomposes by a fur-
ther C C fission into C2H2 and methylacetylene. We
performed some model calculations introducing the ad-
ditionalchannels. Withinourexperimentaltemperature
range a practically negligible effect on the H-atom pro-
files for [C5H5]0 10 ppm was observed.
Figure 1 shows an experimental concentration profile
of H-atoms, measured during the reaction of c-C5H5
with H-atoms. In this figure, the simulation of the mea-
sured H profile using the small reaction scheme given
in Table II is demonstrated, in particular with respect
to the rate coefficient krec
.
For modeling two possible reaction channels for the
reaction of c-C5H5 radicals with H atoms were consid-
ered:
c-C5H5 + H → C5H6
(R-1)
c-C5H5 + H → C5H4(1 A1) + H2
(R-1a)
Wang and Brezinsky [12] calculated a value for the
–1
endothermicity of 16 kcal mol for channel R-1a.
This value was used in the present work for model-
ing, assuming no additional barrier for this reaction.
We follow the arguments as outlined in [12] for this
c-C5H5 + H Recombination Reaction
The experiments were conducted with concentrations
of 0.5–1.2 ppm for [C5H5]0 and 0.2–2.5 ppm for
[H]0. Experimental temperatures ranged from 1100 to
1600 K at total pressures of about 2 bar. For measur-
ing the recombination of the cyclopentadienyl radical
with H-atoms, cyclopentadienyliodide (C5H5I) served
as a thermal source for the cyclopentadienyl radical.
H-atoms were produced by the thermal decay of
ethyliodide (C2H5I).
The initial concentration of C5H5I was determined
by measuring the atomic resonance absorption of Io-
dine atoms and calculating the concentration from cali-
bration experiments. The decomposition of C5H5I was
studied in a separate series of experiments [18]. It was
found that above 780 K, C5H5I dissociates completely
into the cyclopentadienyl radical and an Iodine-atom
within only 30 s. Therefore, the derived I-atom con-
centration yields directly the initial c-C5H5 concentra-
tion. The I-atom concentration was correlated to mea-
surements of c-C5H5I by gas chromatography.
reaction. Further, the corresponding A-factor was var-
1
ied between 1014 and 1016 cm3 mol
s
1. Even with
this high A-factor, which is above the upper limit for
a rapid abstraction reaction, no influence on the H-
profiles was found. Additionally, no temperature de-
pendence was detected in the experimental data, which
led us to the assumption that k 1a/k < 0.01. There-
1
fore, R-1a can be safely neglected for our experimen-
tal conditions. Figure 1 shows a strong influence on
the H-disappearance if the rate coefficient for R-1 is
varied by 50%. Thus, a rate coefficient can be eas-
ily derived from the simulation of each of the recorded
Mixtures with varying ratios of C2H5I and C5H5I
were prepared. For each experiment the H-and I-atom-
absorptions were simultaneously detected. The initial
H-atom concentration was then determined from ex-
trapolation of the H-absorption signal to t = 0 s. It
was shown that ethyliodide yields one H-atom for each
I-atom, that splits off the molecule [19]. The I-ARAS
signal gives the absorption of I-atoms from both the
dissociation reactions of C5H5I and of C2H5I. The con-
centration of C5H5I was then derived from the mea-
sured I-absorption. The error in the concentration is
about 20% for the range [C5H5]0 1 ppm.
Figure 1 Modeling of the H-concentration and sensitivity
on the rate coefficient for the reaction R-1 c-C5H5 + H →
C5H6; experimental conditions: T = 1198 K, p = 2.0 bar,
[C5H5]0 = 0.7 ppm, [H]0 = 1.5 ppm (diluted with Ar).

824
ROY ET AL.
Table II Reaction Mechanism Used for Modeling the Measured H Atom Profiles of the Reaction R-1: c-C5H5 + H →
C5H6
Reaction no.
Reaction
A
n
Ea/R
Reference
R-1
R2
R1
R3
c-C5H5 + H→C5H6
2.6 1014
2.8 1013
4.0 1014
2.0 1013
0
0
0
0
0
Present work
[9] [20]
[2]
C5H6 + H → c-C5H5 + H2
C5H6 → c-C5H5 + H
2 c-C5H5 → C10H8 + 2H
1137
38760
2010
[21]
Units are cm3, s, mol, K.
H-absorption profiles to be
by
3
1
= 2.6 1014 cm3 mole 1 s
K
c,exp(T ) = 0.653 exp( 37530/T ) mol cm
k
1
for T = 1100–1400 K at p
2 bar
tot
By a so called third-law analysis, Kc,exp(T ) has to be
compared with the appropriate expression of statistical
thermodynamics:
The error in the rate coefficient is mainly caused by
the determination of the initial c-C5H5 concentration,
which was described above. The resulting scatter is of
the order of 25%. The profiles could be simulated
with the small set of reactions listed in Table II. The
self-combination R3, i.e. 2c-C5H5 → C10H8 + 2H, be-
comes important only for ratios [c-C5H5]0/[H]0 3/1,
when cyclopentadienyl is present in higher amounts.
Q
QH
DHr,0
c-C5H5
Kc(T ) =
exp
QC H
RT
5
6
The Qi are the partition functions for the respective
molecules given in mol cm 3. Provided the molecular
parameters are sufficiently well known, Qi can be
easily calculated. Comparison of Kc(T ) over the given
T -range with Kc,exp(T ) allows then to determine the
enthalpy of reaction DHr,0 and eventually the entha-
lpy of formation DHf,0 for cyclopentadienyl. For QH
and QC H the molecular data from Burcat’s tables
Data Evaluation, Equilibrium Constant
for C5H6
c-C5H5 + H
From the two sets of measurements, decay of C5H6
and recombination of c-C5H5 + H → C5H6, the re-
spective rate coefficients were used to calculate the
equilibrium constants in the temperature range from
1100 to 1600 K. The rate coefficient kdiss was deter-
mined between 1200 and 1600 K (see Roy et al. [7]),
and the recombination coefficient krec between 1100
and 1450 K [22]. The data of kdiss at pressures around
2 bar were subjected to a first-order regression analysis.
The corresponding Arrhenius expression (see Fig. 2)
is
5
6
[23] were used. A complete set of experimental
molecular data for c-C5H5 is not available, in parti-
cular the experimental data for the vibration freque-
ncies are incomplete (see e.g. the few data given in
1
k = 1.7 1014 exp( 37530/T ) s
1200
T
1600 K
1
First, we assumed that krec = 2.6 1014 cm3 mol
s
1 could be linearly extrapolated to the upper limit of
1600 K of our experiments on dissociation (see Fig. 3).
In a second approximation, based upon a fall-off anal-
ysis, we interpolated krec by a regression function of
second order, taking a weak temperature dependence
into account (see discussion later).
The equilibrium constant Kc,exp(T ), defined as
kdiss/krec, is represented in the case of krec = constant
Figure 2 Decay of C5H6. Rate coefficients for the reaction
R1 C5H6 → c-C5H5 + H obtained in the pressure range from
1 to 6 bar [7].

ENTHALPY OF FORMATION OF THE CYCLOPENTADIENYL RADICAL
825
Table III Molecular Parameters
Frequencies (cm 1) and used degeneracy
a
b
c-C5H5
C5H6
3104 (1)
3089 (2)
3072 (2)
1396 (2)
1271 (1)
1146 (2)
1086 (1)
990 (2)
874 (2)
874 (2)
793 (2)
746 (2)
3105 (1)
3091 (1)
3075 (1)
3043 (1)
2900 (1)
2886 (1)
1580 (1)
1500 (1)
1378 (1)
1365 (1)
1292 (1)
1239 (1)
1100 (1)
1106 (1)
1090 (1)
994 (1)
959 (1)
941 (1)
925 (1)
915 (1)
891 (1)
805 (1)
802 (1)
700 (1)
664 (1)
516 (1)
350 (1)
Figure 3 Rate coefficients for the recombination of c-C5H5
with H. p 2 bar.
635 (1)
505 (2)
the NIST Chemistry-Webbook (http//webbook.nist.
gov/chemistry)). Therefore, in case of the cyclopen-
tadienyl radical we rely upon theoretical calculations
performed, e.g., by Wang and Brezinsky [12], by
Moskaleva and Lin [13], and quite recently by Kiefer
et al. [14]. The set of molecular parameters used in
our calculations is given in Table III.
The special problem regarding the cyclopentadienyl
radical is its predicted facile change in space of its ba-
sic C2v structure (valence isomerization see e.g. [25]).
This is caused by the Jahn–Teller effect [26]. Recent
quantum chemical calculations by Wang and Brezinsky
[12] as well as by Moskaleva and Lin [13] have revealed
that for this valence isomerization the pertinent barrier
is almost negligible. This conclusion agrees with ear-
lier calculations by Gu¨nthard and his coworkers [25b].
In both the recent investigations [12,13], it was pro-
posed to model the valence isomerization by assuming
a pseudo free rotor. Both groups ascribed a moment of
inertia to this “pseudo free rotor” being equal to that
belonging to the axis perpendicular to the molecular
plane. The contribution of such a “free rotor” to the
entropy is rather high. There is in our opinion no phys-
ical basis for a model of this type. In the most recent
work of Kiefer et al. [14], the entropy problem was
treated by means of a very high level quantum chem-
ical calculation. The Jahn–Teller distortion potential
was determined and a full set of energy levels for the
nuclear motions on this potential was calculated. The
entropy of cyclopentadienyl in the range from 300 to
2000 K is reproduced from this work in Table IV. We
need for our modeling, equilibrium coefficients in the
temperature range from about 1000 K up to 2500 K.
In particular, data at the highest temperature have to
be used in order to compare the experimental data on
the C5H5 decay obtained by Kern et al. [11] with our
aElectronic degeneracies: gel = 2 (see text); Rotational constants
(cm 1): A = 0.295; B = 0.295; C = 0.147; Symmetry factor = 10.
Source: Kiefer et al. [14]. The two vibrations associated with the
valence isomerisation were semiempirically modeled. The harmonic
1
vibration was taken as 1542 cm and the anharmonic energy levels
2
as Eviso = 600 viso (see text).
bElectronic degeneracies: gel = 1. Rotational constants (cm 1):
A = 0.298; B = 0.275; C = 0.147; Symmetry factor = 2. Para-
˚
˚
meters for ZLJ; ␴AR = 3.76 A; ␴
= 5.2 A; (ε/k)AR = 143 K;
C
˚
H
6
5
(ε/k)C = 400 K;
␴
KR = 4.01 A; (ε/k)KR = 201 K. Source:
H
6
5
Castelucci et al. [24].
fall-off analysis. There are two choices to proceed.
Eitherwecouldextrapolate Kc(T )dataobtainablefrom
the results on H(T )-H(298) and S(T ) of [14] given for
the restricted T -range 300–2000 K, or we could try to
model the partition functions for C5H5 by a semiem-
pirical method. We decided to do the latter, since a sim-
ple extrapolation of Kc(T ) from 2000–2500 K might
cause larger errors which cannot be quantified. For our
purpose, we assumed a harmonic/anharmonic oscil-
lator model adapted to the results obtained with the
full analysis (see e.g. [14]). The Jahn–Teller effect dis-
torts the symmetric E20 in-plane stretch and bending
vibrations of the C5-skeleton, starting with the D5h
symmetry. The resulting distortion is strong only for

826
ROY ET AL.
Ac(T ) = 1.533 105 T 1.1833 exp ( 1813/T )
Table IV Entropies of c-C5H5 as a Function of
Temperature
3
mol cm 700
T
2500 K
1
1
T/K
300
400
500
600
700
S0 (cal mol
K
)
Comparing Kc,exp(T ) as given above with Kc(T ) we
obtain for three selected temperatures the following
DHr,0 values for the reaction set C5H6 c-C5H5 + H:
63.507
70.083
76.270
81.834
86.947
91.717
96.204
100.44
104.45
108.25
111.86
115.29
118.56
121.68
124.65
127.50
130.22
132.83
1200 K: DHr,0 = 80.3 kcal mol 1; 1400 K: DHr,0
81.4 kcal mol 1; 1600 K: DHr,0 = 82.4 kcal mol
=
1
.
The arithmetic mean is DHr,0 = 81.37 kcal mol 1. Pro-
vided that the molecular parameters used in our calcu-
lation are sufficiently accurate, the slight increase in
DHr,0 with temperature likely reflects the combined
experimental inaccuracies of our measurements.
However, some improvement of the data evaluation
seemspossible. Selectingexperimentaldataobtainedat
about 2 bar, it is to be expected that in our experimen-
tal temperature range fall-off effects will be present.
The scatter of the data for the rate coefficients of the
dissociation is rather large. So it was decided to apply
to the 2 bar data a first-order regression analysis only.
On the other hand, the scatter of the data of the re-
combination experiments is considerably smaller. The
trend of these data with increasing temperature seems
to point to the presence of some fall-off effect in the
investigated temperature range. Thus, application of a
second-order regression analysis to the recombination
coefficients may provide some improvement. We ar-
rived at the representation:
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
The Jahn–Teller distortion was fully taken into account by a high
level quantum chemical computation performed by A. Wagner and
his group. For details see Kiefer et al. [14].
the two in-plane stretch motions [14,25(b)] and may
be approximately described by a set of one harmonic
1
vibration with 1542 cm and an anharmonic oscil-
lator with semiempircally modeled energy levels as
2
Eviso = Dviso
[cm 1] (see also [14]). By compar-
iso
ing our entropy data with that given in [14], we deter-
mined the parameter Dviso to Dviso 600 cm 1. The
other vibrational (scaled) frequencies were taken from
[14] obtained for the C5H5 (D5h structure) (Table III).
The electronic degeneracy is given by the electron spin
and is 2. The calculated Jahn–Teller energy levels as
given in [14] are composed of contributions of the two
krec = 2.22 109 exp(2.880 104/T
1
0.1764 108/T 2) cm3 mol 1 s
The regression analysis showed, perhaps just by
chance, the right trend of the data with increasing tem-
peratures, whereby we consider the very slight maxi-
mum in Fig. 3 as not relevant in this context. Compare
Fig. 3 with the Fig. 5. The latter depicts the results of
our fall-off analysis.
2
2
low electronic states: B1 and A2. Thus, the orbital
degeneracy is inherently present in these calculations.
The entropies calculated by us, using the partition func-
tion of cyclopentadienyl agree within less than 0.4%
with the data of Kiefer et al. [14] in the temperature
range from 600 to 2000 K. Almost the same is true
for our calculation of the sensible heat H(T )-H(298).
These data agree within less than 1%.
Combining krec with kdiss gives
Kc(T ) = 7.66 104 exp( 6.633 104/T
3
+ 0.1764 108/T 2) mol cm
With the molecular data for the cyclopentadienyl
radical theoretically obtained, we calculated the tem-
perature dependent preexponential factor Ac(T ) of
Kc(T ):
Now, as one would expect it, the calculated DHr,0 val-
ues turned out to be closer to each other: 1200 K:
DHr,0 = 80.58 kcal mol 1; 1400 K: DHr,0 = 81.21 kcal
mol 1; 1600 K: DHr,0 = 80.71 kcal mol 1. The mean
value is DHr,0 = 80.83 0.35 kcal mol 1 (standard de-
viation). Based upon this enthalpy of reaction, the val-
ues DHf,0 and DHf,298 for cyclopentadienyl were cal-
culated and are given in Table V. It may be noted here
that calculating Kc(T ) directly from the data given in
Q
QH
DHr,0
c-C5H5
Kc(T ) =
exp
QC H
RT
5
6
DHr,0
RT
= Ac(T ) exp

ENTHALPY OF FORMATION OF THE CYCLOPENTADIENYL RADICAL
827
¯
Table V Enthalpy of Formation of c-C5H5
Reaction: C5H6 → c-C5H5 + H; DHr,0
B iscalculatedfromthegeometricmeanofthetwosim-
ilarmomentsofinertia, A and B, giveninTableIII. This
is sufficient for an almost symmetric oblate top as is the
case here. E0(0) = DHr,0 is the exit of a Morse-potential
for J = 0 associated with the dissociating bond C H.
The Morse potential for the H-atom fission was mod-
eled as
=
1
80.8 1 kcal mol
DHf,0
Species (kcal mol
DHf,298
(kcal mol
1
1
)
)
References
C5H6
36.2
37.3
51.63
65.4
32.09a
33.2a
52.10
[16]
[27]
[16]
VM(1r) = DM(1 exp( 1r/ M))2
DM is connected to DHr,0 by
H
c-C5H5
1
1
62.5
1
1
Present work
/Burcat
Present work
/Roth et al.
64.3
61.4
DM = DHr,0 + Ez(C5H6) Ez(c-C5H5)
(2)
Enthalpy of formation at 298 K of c-C5H5 from
literature
Ez(i) is the zero-point energy of the molecule i.
M is the scaling factor for the Morse potential and
is connected to DM and the force constant pertinent to
the vibration of the breaking C H bond.
1
DHf,298 (kcal mol
)
Method
References
58
61
63
2
2
2
Experimental
Experimental
Experimental
Experimental
Theoretical
[28]
[29]
[30]
[11]
[31]
[12]
[13]
p
=
2DM/kv,RC
M
65.3
63.6
62–62.6
62.6
In our case kv,RC = 4.8 105 dyne cm (see [34]).
The C5H6 molecule was considered as a symmetric
top.
1
Theoretical
Theoretical
The effective potential Veff for the dissociating C H
bond for J > 0 can be approximately modeled as (see
[35]):
a No standard deviation available.
I(1r = 0)
Veff(J, 1r) VM(1r) + Erot,B(J)
I(1r)
[14] and applying these to our Kc, exp(T ), resulted in a
reaction enthalpy DHr,298 = 82.2 kcal mol 1. This cor-
responds to a DHr,0 = 80.5 kcal mol 1. The differences
to the data of Table V are practically negligible.
+ (Ez,0 Ez,p) exp( 1r/␣Troe) + Ez,p
(3)
I(1r) is the mean of the two smaller moments of inertia
of cyclopentadiene changing with the lengthening of
the dissociating C H bond
Fall-Off Analysis
We undertook a simple fall-off analysis by using the in-
terpolation scheme of Troe (see [32a,b]). The interpo-
lation scheme needs as input parameters the rate coeffi-
cient for the low pressure limit kd0iss,sc (sc = strong col-
li∞sion case), the coefficient for the high pressure limit
¯
Erot,B(J) = BJ(J + 1)
vRC
Ez,0 = Ez(C5H6)
2
vRC is the frequency of one of the C H stretch vibra-
tions of the CH2 group. RC denotes the reaction coordi-
nate; Ez,p is the zero-point energy of cyclopentadienyl
(Ez,c-C H ). The last two terms in Eq. (3) approximately
kdiss, and the weak collision factor
.
c
Low Pressure Rate Coefficient k0diss,sc. The low pres-
sure rate coefficient was calculated including rotational
effects. The effective dissociation energy E0(J) de-
pends upon the main quantum number J for rotation.
It is assumed (see [33]) that J is conserved while the
activated molecule is dissociating.
5
5
take into account that with increasing C H distance the
two C H bending vibrations of the CH2 group (scis-
sors and twist) eventually go over into rotations of the
cyclopentadienyl radical.
is a scaling parameter
Troe
˚
that was found empirically to be in the order of 1 A by
Quack and Troe [36]. The parameters B1 and p were
simply calculated by assuming that during the length-
ening of one of the C H bonds in the CH2-group, nei-
ther the bond angle nor the structure of the c-C5H5-rest
does change.
p
E0(J) = E0(0) + B1[Erot,B
]
Erot,B
¯
Erot,B(J) = BJ(J + 1)
(1)
√
¯
B = AB

828
ROY ET AL.
mately connected to h1Ei(a0ll) as outlined in [32d]:
With these assumptions we calculated the moments
of inertia dependent on the distance C H. The effec-
tive potential Veff(J, 1r) exhibits maxima at varying
distances. These maxima were determined and the as-
sociated effective dissociation energy E0(J) was mod-
eled accordingly to Eq. (1). The parameters are B1 =
(0)
h1Eiall
c
F(E0)
3
(5)
1/2
c
F(E0)RT
1
1 1
p
0.763 (kcal mol
)
and p = 1.02.
Equation (5) was derived assuming a simple exponen-
tial model for the energy transfer between the bath gas
molecules and the dissociating molecule. The probabil-
ity of the “down” process (energy transfer to the bath
gas) is modeled as [32d]
We modeled the total rotational energy for the
dissociating cyclopentadiene molecule as Erot(J)
2
¯
¯
¯
BJ(J + 1) + (C B)K . Since C < B, the last term
is always negative. The rotational states, associated
with the largest moment of inertia, were convoluted
into the vibrational density of states, v,K (E), follow-
ing a proposal of Troe [32c]. The reason for doing so is
that due to rotation–vibration interaction the K-states
may not stay stable, while J is assumed to be conserved
during the dissociation of the activated molecule.
The densities of states v,K (E) were directly calcu-
P(down) = K(Ei ) e
for E j Ei
(Ei E j )/
i, j
is an empirical model depending parameter that has
to be extracted from comparisons between experiments
and a theoretical model. Generally, for any energy
transfer model the two Eqs. (6) and (7), define h1Eiall:
lated with an energy step size of 20 cm 1. The |K|
J
restriction was regarded. Our calculations were based
upon harmonic vibrations. The frequencies of the cy-
clopentadiene were taken from Castelucci et al. [24].
∞
Z
1 =
P(E j , Ei ) dE j for any given Ei (6)
kd0iss,sc[cm3 mol
s
] was obtained from a some-
1
1
E j =0
what simplified model:
In particular, h1Eiall is defined for Ei = E0 as
J=Jend
X
kd0iss,sc = ZLJ RT
∞
Z
J=0
h1Eiall
=
(E j E0)P(E j , E0) dE j
(7)
¯
(2J + 1) exp( B J(J + 1)/RT )
E j =0
Q
C5H6
By an approximate integration of Eqs. (6) and (7) in
the frame of the exponential energy transfer model it is
shown that
F(E0(J)) v,K (E0(J)) exp( E0(J)/RT )
(4)
2
J
end is a suitably chosen number large enough to assure
(0)
h1Eiall
h1Eiall
(8)
that in the given experimental T -range kd0iss,sc practi-
cally does not depend on J for J > Jend. The factor
F(E0(J)) is defined [33] as:
+ F(E0)RT
if F(E0) < 3 [32d].
Modelling of experimental data exhibits that ei-
ther for a RRKM interpolation or the more sophis-
ticated interpolation scheme of Gilbert et al. [32b],
∞
Z
1
F(E0(J)) =
fv(E)/fv(E0(J)) dE.
(0)
h1Eiall h1Eiall can be assumed to stay approxi-
RT
E=E0(J)
mately constant over a rather large temperature range
([38]; see also the recent review contained in [39]).
For example, for the CH4-dissociation, Cobos and Troe
[40] could model experimental data sufficiently well
fv(E) is the Boltzmann distribution function for vibra-
tion energies between E and E + dE. For the Lennard–
Jones collisions frequency ZLJ [cm3 mol 1 s 1],
the collision parameters determining ZLJ, i , and
(ε/k)i (i = AR, KR, C5H6) were calculated from the
semiempirical formulae given by Cambi et al. [37] (see
Table III).
(0)
with a practically constant h1Eiall and the associated
cs [Eq. (5)] over a temperature range from 1000 to
5000 K. In the CH4 case the values for F(E0) are for all
temperatures <3. h1Eiall depends on the nature of the
collision partner. In many experimental studies on reac-
tion kinetics highly diluted mixtures of the investigated
molecule with noble gases were used. This simplifies
the data reduction, since the energy transfer to and from
The rate coefficient at the low pressure limit in-
cluding weak collision effects is defined as kd0iss,wc
=
ckd0iss,sc. The weak collision coefficient c is approxi-

ENTHALPY OF FORMATION OF THE CYCLOPENTADIENYL RADICAL
829
the molecule is in this case practically determined by
the diluent.
The factor Fwc contains effects associated with the en-
ergy transfer by collisions of the molecule with the bath
Unfortunately, the use of Eq. (5) is for larger
molecules restricted to not too high temperatures.
Equation (5) becomes increasingly inaccurate if the
temperaturedependent F(E0)exceedsalimitofabout3
(see [32b]).
gas; Fwc is a function of EMBED.
c
Troe [32(a), (d)] derived the interpolation scheme
by neglecting effects of external rotation. We assumed
that it should work as well, if we insert for kd0iss,sc the
full expression of Eq. (4). Two parameters were varied:
(0)
Forthe T -rangeofourexperiments F(E0) 3isful-
filled; however, for the T -range of the experiments of
Kern et al. [11] this is not true for all data. For their ap-
plied experimental conditions an improved expression
for c has to be used [see [32b], Eq. (4.7)]. We assumed
that h1Ei(a0ll) stays constant for F(E0) < 3 for both ex-
perimental sets. But for T > 1600 K, F(E0) increas-
ingly exceeds to 3 and so we had to empirically find
data for such that the experimental rate coefficients
of Kern et al. for T > 1600 K can be simulated. We cal-
kr∞ec h1Eiall [cm 1] and (the∞average energy transferred
by collision). The needed kdiss in Eq. (9) was provided
by
kd∞iss = kr∞ec Kc(T )
We arrived at a good representation of our measured
(0)
data with a variety of pairs of kr∞ec and h1E iall (see
Figs. 4 and 5).
T
1600 K:
(0)
culated from Eq. (8) for an assumed constant h1Eiall
kr∞ec = 3.5 1014 cm3 mol 1 s and
1
for T 1600 K. For larger temperatures the parame-
ters were fitted to the experimental rate coefficients
using Eqs. (6) and (7). The obtained (T ) were inserted
(0)
1
h1Eiall = 130 cm
(not given in the figures)
in the improved relation for as given in [32b]:
c
kr∞ec = 4.2 1014 cm3 mol 1 s and
1
2
h1Ei(a0ll) EMBED = 75cm
1
=
c
+ F(E0)RT
kr∞ec = 5.0 1014 cm3 mol 1 s and
1
1
h1Ei(a0ll) EMBED = 25cm
1
h
i
E
0
R
F(E ) RT
0
fv (E) 1
exp( (E0 E)/F(E0)RT ) dE
+ F ( E ) RT
0
0
The reason for this arbitrariness is simply the low sen-
sitivity of our experiments on the low pressure rate co-
efficient, since our experiments were performed rather
close to the high pressure limit.
For further details the reader should refer to the work
of Gilbert et al. [32b].
High Pressure Rate Coefficient k∞diss. We modeled the
needed high pressure rate coefficient by
kd∞iss = kr∞ec Kc(T )
Since it is expected that kr∞ec does not strongly vary with
temperature for simple bond fiss∞ions, we have set, in a
probably good approximation, krec = constant over the
temperature range of our measurements. Appropriate
examples to support this assumption are reviewed in
Baulch et al. ([41], Table III] .
Modeling of Measured Data. The formalism of Troe
[32a] and of Gilbert et al. [32b] was applied to the
dissociation of cyclopentadiene:
kd0iss,wc M/kd∞iss
k
diss(M)
kd∞iss
=
1 + kd0iss,wc M/kd∞iss
Fsc kd0iss,wc M/kd∞iss
Fwc kd0iss,wc M/kd∞iss
Figure 4 Results of the fall-off analysis for kdiss(M).
Experimental data from Roy et al. [7] (see text);
(0)
1
1
♦
1
h1Eiall = 75 cm 1; kr∞ec = 4.2 1014 cm3 mol
s
s
;
(0)
1
1
h1Eiall = 25 cm 1; kr∞ec = 5.0 1014 cm3 mol
.
(9)

830
ROY ET AL.
associated interpolated number is Yip(l). In a very sim-
ple approach we may construct the average relative
deviation for a set of L measured points:
s
L
(Y(l) Yip(l))2
Yi2p(l)
X
1
L
l=1
Application of this approach to kdiss gives 0.28
and to krec 0.035. The relative average devia-
tion of Kc(l) is equal to or smaller than the
sum of both 0.31. This translates eventually into
1
DHr,0 = 80.8 0.66 kcal mol from our measure-
1
ments at 1200 K and DHr,0 = 80.8 0.88 kcal mol
from our measurements at 1600 K. Thus, a conserva-
tive guess of the standard deviation would be DHr,0
=
Figure 5 Results of the fall-off analysis for krec(M)
1
80.8 1 kcal mol
.
(see text); broken line: regression from Fig. 3; solid line:
(0)
1
1
The enthalpy of formation of c-C5H5 can be calcu-
lated by taking the DHf,0 for cyclopentadiene and H
h1Eiall = 25 cm 1; kr∞ec = 5.0 1014 cm3 mol
s
.
1
from Table V. We arrive at DHf,0 = 65.4 kcal mol
for c-C5H5. This value contains the uncertainty of
the DHf,0 of cyclopentadiene. In a recent publica-
tion by Roth et al. [27] these authors propose a
new value, supported by semiempirical calculations:
For the temperature range from 1000 to 2500 K we
have calculated (Eq. (4))
16.788
kd0iss,sc(AR) = 1.07 1080
T
1
1
DHf,298 = 33.2 kcal mol instead of 32.1 kcal mol
exp( 8547.9/T ) cm3 mol 1 s
1
1
as given in Burcat’s tables [23]. The DHf,298 of c-C5H5
wouldthenbereducedto61.4kcalmol 1 (seeTableV).
16.793
kd0iss,sc(KR) = 9.86 1079
T
exp( 8543.3/T ) cm3 mol 1 s
Modeling of the Fall-Off
1.183
kd∞iss = 7.65 1019
T
exp( 42513/T ) s 1 for
Our simulations are based upon the enthalpy of reaction
1
kr∞ec = 5.0 1014 cm3 mol 1 s
.
obtained in the present work for C5H6 → c-C5H5 + H.
(0)
The two parameters, kr∞ec and h1Eiall , were varied in
order to arrive at a satisfying fit of our experimental
data. A good fit can be obtained with pairs of kr∞ec and
DISCUSSION
(0)
h1Eiall , which are bound to a certain range. So we have
to accept some arbitrariness. Our experimental condi-
tions were such that the reaction proceeded rather close
to the high pressure limit. This is in contrast to the mea-
surements by Kern et al. [11]. In those experiments the
reaction was studied closer to the low pressure limit, in
particular, if T exceeds 1500 K. Our experiments are
more sensitive to the selected value for kr∞ec, whereas
Kern’s and Kiefer’s experiments depend stronger on
Enthalpy of Formation
of the Cyclopentadienyl Radical
The average enthalpy of reaction, DHr,0, extracted from
our measured data of kdiss (T , 2 bar) and krec (T , 2 bar) is
80.8 kcal mol 1 for the reaction C5H6 → c-C5H5 + H.
Systematic errors in kdiss as well as on krec may cancel
to some degree, since the main source for these errors is
the determination of the absolute concentration of the
reactants in the mixture filled in the shock tube. This
was considered by our analysis of probes taken from
the shock tube. Remaining random errors are ascribed
to the measured ki s only. We have assumed that the
temperature is accurately given. Consequently, random
errors in T appear in the ki .
(0)
h1Eiall . A simple direct comparison of the respective
modeling of the two experimental data sets, the present
work and that of Kern et al. [11], is not easily possible.
The latter group applied a RRKM analysis not taking
into account effects of external rotation. Their derived
(0)
h1Eiall was found on the order of 18 cm 1. For no-
ble gases as main collision partners, numbers for large
molecules are reported to lie in the range between 50
We take from the regression analysis the average
relative deviation of the measured points from the re-
gression line. If Y(l) is the lth measured value, the
1
and 200 cm (see e.g. [38,42]). We tried to find a
(0)
set of numbers for kr∞ec and h1Eiall , which could be

ENTHALPY OF FORMATION OF THE CYCLOPENTADIENYL RADICAL
831
Figure 7 Optimized parameters for the transition proba-
bilities P(E j , E0) (see text). Dotted lines show the extrapo-
lation of 1(T ) to 0. Note the steep increase of F(E0)with T
Figure 6 Results of the fall-off analysis for kdiss(M).
5 bar. Calculated with the parameters obtained from
p
(0)
(0)
for T > 1600 K and note that h1Eiall approaches h1Eiall
the analysis∞p = 2 bar (see text); dotted line: h1Eiall
=
for low temperatures.
1
75 cm 1; krec = 4.2 1014 cm3 mol
s
1; solid line:
(0)
1
1
h1Eiall = 25 cm 1; kr∞ec = 5.0 1014 cm3 mol
s
.
applying an energy spacing of 10 cm 1. With the
obtained P(E j , E0), which are functions of the
assumed and of T , the integrals
used to reproduce the results of both groups. Starting
(0)
1
the simulation of our data with h1Eiall = 25 cm
,
(0)
which is close to the h1Eiall of Kern et al., led to a
∞
Z
satisfying repro∞duction of our experimental points with
1
a rather high krec of 5.0 1014 cm3 mol
s
1. How-
h1Eiall
=
(E j E0) P(E j , E0) dE j
ever, since we have a barrierless recombination, the
value of kr∞ec, the “capture rate coefficient,” seems plau-
sible. Figure 6 depicts the simulation of our 5 bar ex-
E j =0
were calculated.
We fitted empirically the experimental results of
Kern et al. with by 1(T ). Figure 7 depicts
periments. The result is quite acceptable together with
1
DHr,0 = 80.8 kcal mol
.
=
+
0
(0)
To reduce the range of optimized parameter pairs
as we have extracted them from our experiments, we
should have extended our measurements to much lower
pressures. Helpful in closing this gap are the data from
Kern’s and Kiefer’s groups.
the results. (Note: for low temperatures h1Eiall ap-
proaches EMBED, therefore, 1(T ) → 0 for T → 0;
(0)
h1Eiall as well as h1Eiall are then practically de-
termined by
alone since F(E0) → 1 at low tem-
(0)
peratures and consequently h1Eiall
→
h1Eiall. It
The simulation of the experiments of Kern et al. [11]
clearly turns out that for low temperatures the calcu-
(0)
(0)
cannot be performed with a h1Eiall = const over
lated h1Eiall approaches h1Eiall , but for higher
their total applied T -range, since for T > 1600 K,
F(E0) gets larger than the critical value of about 3.
temperatures h1Eiall contains increasing contribu-
tions from the energy flux from the bath gas into the cy-
clopentadiene molecule with increasing temperature.
Theneeded cswerecalculatedforgiven sbysolv-
ingthemasterequationforthec-C5H5 system(M → 0).
It was verified that indeed the simplified formula for
2
(0)
[32b]. The simple relation h1Eiall
=
does not represent h1Eiall [Eq. (7)] for high+tFem(Ep0)eRrT-
atures and large F(E0). For T > 1600 K the simple
(E)
(E E0)/RT
(E E0 )/F (E0 )/ RT
approximation (E ) e
e
0
E
E0, J = 0, as used by Troe [32c] to calculate
c, as given above, resulted in values within about
10% of the data from the solution of the master equa-
tion. This was already observed by Gilbert et al. [32b].
c
the P(E j , E0) for the c-C5H6 system is inaccurate
and therefore not applicable. We calculated the
P(E j , E0) directly by solving the associated integral
equation
1
1
Our simulation with kr∞ec = 5.0 1014 cm3 mol
s
of Kern’s and Kiefer’s experimental results is shown in
Fig. 8. The lines in Fig. 8, which were taken from [11],
represent the RRKM interpolation for 0.6 and 0.266
∞
Z
bar. Kern et al. [11] give k(M) in units of cm3 mol
1
1 =
P(E j , Ei ) dE j for Ei = E0
s
1. We have converted their data by multiplying them
Ei =0

832
ROY ET AL.
sures ranging from 100 to 450 Torr. However, our re-
sult compares favorably with recently theoretically ob-
tained numbers by Wang and Brezinsky [12] and by
Moskaleva and Lin [13] as well as with older experi-
mental data from Bartmess et al. [29] and DeFrees et al.
[29]. The determination of the enthalpy of formation
of cyclopentadienyl was made possible only by taking
the quite recently calculated entropy data for cyclopen-
tadienyl from Kiefer et al. [14]. Attempts to model the
entropy of c-C5H5 by inadequate assumptions concern-
ing the contribution of the facile valence isomerization
to the entropy ([12,13]) lead to a formation enthalpy of
the cyclopentadienyl radical considerably off the num-
bers presented here.
The fall-off analysis performed depends strongly on
accurately calculated data for Kc(T ). With these new
data satisfying simulations of the dissociation as well
as the recombination coefficients in the investigated
ranges of pressures and temperatures became possible.
Figure 8 Dissociation of C5H6 at low pressures. Data taken
from [11]. The symbols represent results from our fall-
off analysis applied to the experimental conditions of [11]
(0)
(see text).
♦
h1Eiall = 75 cm 1; kr∞ec = 4.2 1014 cm3
1
1
mol
s
(T ) from E∞q. (8) for the whole T -range.
(0)
1
1
1
h1Eiall = 20 cm 1; krec = 5.0 1014 cm3 mol
s
(T ) for T > 1600 K fitted to experimental data.
We are greatly indebted to A. Wagner and J. Kiefer for send-
ing us prior to publication their paper on the effect of the
Jahn–Teller distortion on the thermodynamic functions of the
cyclopentadienyl radical. We also acknowledge stimulating
discussions with J. Troe and Hai Wang.
with M to obtain the ks in units of s 1. The error bars
are guessed from their Fig. 2 in [11]. Since the experi-
mental conditions applied by Kern et al. allow measure-
men∞ts closer to the low pressure limit, the sensitivity
on krec of our modeling is rather low. A satisfying mod-
eling of Kern’s data with the pair kr∞ec = 4.2 1014 cm3
BIBLIOGRAPHY
(0)
1
1
1
mol
s
and h1Eiall = 75 cm for the whole
experimental T -range is not possible. One observes
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