Recombination of benzyl radicals: Dependence on the bath gas, temperature, and pressure

Article abstract of DOI:10.1039/b407074g

The combination reaction C₆H₅CH₂ + C ₆H₅CH₂ (+M) → C₁₄H₁₄ (+M) was studied over the pressure range 0.03-900 bar and the temperature range 250-400 K. Helium, argon, xenon, N₂ and CO₂ were employed as bath gases. Benzyl radicals were generated by H abstraction from toluene by Cl atoms formed via laser photolysis of Cl₂ at 308 nm. Benzyl radicals were monitored by time-resolved transient UV absorption at 253 nm. The observed combination rates were pressure independent over the range 0.04-0.45 bar in CO₂ and 0.03-5 bar in argon which identifies a limiting "high-pressure" value of rate constant within the energy-transfer mechanism, k_∞^ET(T) = (4.1 ± 0.3) × 10-¹¹ (T/300 K)^-0.23 cm³ molecule^-1 s^-1. Although the reaction has clearly reached the "high-pressure limit" (k_∞^ET) at pressures below 1 bar, a further increase of the rate constants was observed when the pressure of the bath gas was raised above ～5 bar. At much higher pressures finally the rate constants decrease when diffusion-controlled kinetics takes over. The degree of enhancement of the combination rates beyond the "high-pressure limit" was found to depend on the bath gas, increasing in the order He < N₂ ≈ Ar < CO₂. The enhancement was most prominent at low temperatures. Measurements of transient UV absorption spectra of benzyl radicals, over the pressure range 5-30 bar of CO₂ at 300 K, confirmed that an only small pressure-dependent solvatochromic shift of benzyl radicals cannot be responsible for the observation of enhanced rate constants. Instead, the results clearly point toward a significant effect of van der Waals clustering, i.e. of "radical-complex" formation, on the combination reaction kinetics in the gas-liquid transition range. An analysis in terms of statistical adiabatic channel/classical trajectory calculation (SACM/CT) appears to provide a consistent description.

Full text of DOI:10.1039/b407074g

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R E S E A R C H P A P E R
Recombination of benzyl radicals: dependence on the bath gas,
temperature, and pressure
Klaus Luther, Kawon Oum,* Kentaro Sekiguchi and Ju
¨
rgen Troe
Institut fur Physikalische Chemie, Universitat Gottingen, TammannstraXe 6 D-37077,
¨
¨
¨
Gottingen, Germany. E-mail: koum@gwdg.de; Fax: þ 49 551 393150; Tel: þ 49 551 3912598
¨
Received 11th May 2004, Accepted 7th July 2004
First published as on Advance Article on the web 21st July 2004
The combination reaction C6H5CH2 þ C6H5CH2 (þM) - C14H14 (þM) was studied over the pressure range
0.03–900 bar and the temperature range 250–400 K. Helium, argon, xenon, N2 and CO2 were employed as bath
gases. Benzyl radicals were generated by H abstraction from toluene by Cl atoms formed via laser photolysis of
Cl2 at 308 nm. Benzyl radicals were monitored by time-resolved transient UV absorption at 253 nm. The
observed combination rates were pressure independent over the range 0.04–0.45 bar in CO2 and 0.03–5 bar in
argon which identifies a limiting ‘‘high-pressure’’ value of rate constant within the energy-transfer mechanism,
kENT(T) ¼ (4.1 ꢀ 0.3) ꢁ 10ꢂ11 (T/300 K)ꢂ0.23 cm3 moleculeꢂ1
s
ꢂ1. Although the reaction has clearly reached the
‘‘high-pressure limit’’ (kENT) at pressures below 1 bar, a further increase of the rate constants was observed when
the pressure of the bath gas was raised above B5 bar. At much higher pressures finally the rate constants
decrease when diffusion-controlled kinetics takes over. The degree of enhancement of the combination rates
beyond the ‘‘high-pressure limit’’ was found to depend on the bath gas, increasing in the order He o N2 E
Ar o CO2. The enhancement was most prominent at low temperatures. Measurements of transient UV
absorption spectra of benzyl radicals, over the pressure range 5–30 bar of CO2 at 300 K, confirmed that an only
small pressure-dependent solvatochromic shift of benzyl radicals cannot be responsible for the observation of
enhanced rate constants. Instead, the results clearly point toward a significant effect of van der Waals clustering,
i.e. of ‘‘radical-complex’’ formation, on the combination reaction kinetics in the gas–liquid transition range. An
analysis in terms of statistical adiabatic channel/classical trajectory calculation (SACM/CT) appears to provide
a consistent description.
hand, our result in the normal gas phase of kE1,TN, ‘‘high-
1. Introduction
pressure limit’’ of the combination, (4.1 ꢀ 0.3) ꢁ 10ꢂ11 cm3
Intermolecular interactions between solute and solvent mole-
cules are of great interest for an understanding of kinetic and
dynamic properties of radical reactions in the gas–liquid transi-
tion region.1–3 Studies of the pressure and temperature depen-
dence of radical combination rate constants in this range can
help to identify contributions of the corresponding van der
Waals complexes to rate constants.4,5 Recently we have ob-
served an unexpected enhancement of combination rates of
polyatomic radicals, such as CCl36 or benzyl7 at pressures higher
than those where the ‘‘high-pressure limit’’ of the standard
energy-transfer (ET) mechanism is established. The reason for
such a pressure- and temperature-dependent increase of combi-
nation rate constants has been not yet completely understood.
However, currently available information suggests that interac-
tions between radicals and solvent molecules may be responsible
for the phenomenon, which then can be described in terms of a
radical-complex (RC) mechanism and/or the density depen-
dence of electronic quenching.6 More experimental studies
appear necessary to separate and clarify, qualitatively and
quantitatively, the influence of radical-solvent molecule interac-
tions on the radical combination kinetics in the gas–liquid
transition region. In the present article, we report new studies
of the combination reaction of benzyl radicals over the wide
pressure range 0.03–900 bar and the temperature range 250–400 K
in the bath gases (M) helium, argon, xenon, N2, and CO2:
moleculeꢂ1 ꢂ1,7 is comparable to two previous direct experi-
s
mental measurements.9,10 On the other hand, at the highest
pressure of our work, reaction (1) is diffusion-controlled and
our rate constants k1 are equivalent to the values from the
work of Claridge and Fischer11 in the liquid solution. In the
medium density region, we have observed a rapid increase of
reaction rates at elevated pressures beyond the normal ‘‘high-
pressure limit’’ before the diffusion control sets in. The inter-
pretation of our experimental observations in terms of a
possible contribution of the radical-complex mechanism is
the central topic of the present article. In continuation of our
earlier Communication,7 in the following we present a full
report of our work.
2. Experimental section
2.1. Time-resolved absorption measurements in the UV
Our high-pressure set up has been described in detail
before,13–15 and only the main features are given here. In brief,
our experiment was carried out in a high-pressure optical flow
cell which can be cooled or heated over the temperature range
200–400 K at pressures of 1–1000 bar. Benzyl radicals were
generated by the laser photolysis of Cl2 at 308 nm using an
excimer laser (Lambda Physik, model COMPEX), and the
subsequent fast bimolecular reaction of chlorine atoms with
an excess amount of toluene (C6H5CH3):
C6H5CH2 þ C6H5CH2 (þM) - C14H14 (þM) k1
(1)
A number of kinetic studies of reaction (1) have been reported
in the gas phase,8–10 in liquid solutions,11 and recently in the
medium density region of supercritical fluids.7,12 On the one
Cl2 þ hn (308 nm) - 2 Cl
Cl þ C6H5CH3 - C6H5CH2 þ HCl
(2)
(3)
T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 4
P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 4 1 3 3 – 4 1 4 1
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Mixtures of Cl2, toluene, and the bath gas were compressed in
an oil-free diaphragm compressor (Nova Swiss), and then
flowed through the high-pressure cell (path length of 10 cm
and optical diameter of 0.9 cm). For experiments o1 bar, a
flow cell made of glass was used (path length of 52 cm and
optical diameter of 3 cm). Flow rates were controlled by flow
meters (Tylan, model FM361 and FM362) at rates such that
reagents and products were removed from the observation
volume between the laser pulses. Total pressures were
measured with high-pressure meters (Burster, model 8201).
Two platinum resistance thermometers were directly attached
to the front and back of the cell to measure the temperature.
The progress of reaction (1) was monitored by recording the
absorption signal of benzyl radicals at 253 nm on time scales of
ms to ms. The light source for the absorption measurements
was a high-intensity Hg–Xe lamp (Ushio, model UXM-200 H,
200W). At 253 nm, the absorption of benzyl radicals dominates
over contributions from other species involved (sbenzyl ¼ 1.3 ꢁ
10ꢂ16 cm2 moleculeꢂ1).16 The absorption signal was detected
by a standard prism-monochromator (Zeiss, model MM3) and
photomultiplier (RCA, 1P28A) arrangement with a bandwidth
of 2 nm, and recorded using a digital storage oscilloscope
(LeCroy, band width 200 MHz). Typically several hundred
shots were averaged. The bath gases of helium, argon, xenon,
N2, and CO2 were of a purity higher than 99.998%. Impurities
in the bath gases, especially oxygen, were carefully removed by
a series of gas cleaning adsorbers (Messer-Griesheim, model
Oxisorb, and Alltech, model Oxytrap) and dust filters. All
chemicals were purified in a pump–thaw–freeze cycle prior
to use.
measured without the laser beam at the same experimental
conditions. Several hundred spectra were usually averaged.
3. Results
Fig. 1 shows typical absorption signals at 253 nm, recorded
after the 308 nm-laser photolysis of Cl2 in the presence of
toluene and the bath gas CO2 at 300 K. All absorption–time
profiles showed clean second-order decays of the benzyl
concentrations and agreed entirely with an assignment to
reaction (1).
Reaction (3) is known to solely proceed through a fast
H-atom abstraction channel (k3 ¼ 6.1 ꢁ 10ꢂ11 cm3 moleculeꢂ1
sꢂ1
) such that chlorine atoms rapidly, completely, and
stoichiometrically lead to benzyl radicals.17 The quantitative
analysis of our experimental observation was carried out
considering the major channel (1) and also accounting for
other possible side reactions such as the recombination of
chlorine atoms and the reaction between chlorine atoms and
benzyl radicals:
Cl þ Cl - Cl2
C6H5CH2 þ Cl - C6H5CH2Cl
(4)
(5)
These two side reactions, however, turned out to be insigni-
ficant because the concentration of toluene was always present
in significant excess. Residual absorptions from products like
dibenzyl and benzyl chloride are also negligible due to their
small absorption coefficients (sdibenzyl E sbenzyl chloride E 1 ꢁ
10ꢂ19 cm2 moleculeꢂ1 18
compared to the strong absorption
)
from benzyl radicals (sbenzyl ¼ 1.3 ꢁ 10ꢂ16 cm2 moleculeꢂ1).16
The residuals of the fits did not show any systematic devia-
tions, giving additional support to the reported values of k1.
Typical concentrations used in our experiments were: [Cl]0 (¼
[benzyl]0) ¼ (1 ꢂ 5) ꢁ 1013 molecule cmꢂ3 and [toluene] ¼
2.2. Transient absorption spectra
Prior to the analysis of the kinetics, it was necessary to check
the pressure dependence of the absorption of benzyl radicals.
In addition, possible contributions to the absorption signals
from other reactants, intermediates, and products (for
example, dibenzyl or benzyl chloride) over the observation
wavelength range were investigated. It turned out that benzyl
absortpion was predominant over the wavelength range
240–270 nm and times up to a few ms.
(0.7 ꢂ 7) ꢁ 1016 molecule cmꢂ3
.
Several other precursor molecules for benzyl radicals were
tested but found to be inadequate for our purpose. For
example, benzyl iodide, ethylbenzene, and benzyl chloride also
yield benzyl radicals in a direct photolytic step. However,
subsequent side reactions of other photolytic products (such
as halogen atoms or ethyl radicals) caused large difficulties of
the analysis.
Pressure-dependent transient spectra of benzyl radicals were
recorded at room temperature in a high-pressure flow cell
closed by two quartz windows (path length of 29.5 cm and
optical diameter of 0.9 cm). A Jobin-Yvon-Spex type of
spectrograph (model SPEX 270M) with a holographic grating
(1200 or 2400 g mmꢂ1, blazed at 500 nm) and a 384 ꢁ 576 pixel
ICCD camera (LaVison, model FlameStar IIF) served as the
detection system (wavelength range of 190–850 nm). The
highest resolution of the spectrograph (0.096 nm) was deter-
mined by an entrance slit width of 12 mm. The minimum gating
time was 1 ns. A high-pressure xenon arc lamp (Osram, model
XBO 150W/1) was used as the light source. A set of lenses
focused the light beam as it emerged from the highcell on to the
entrance slit of the spectrograph. The 308 nm-excimer laser
beam (Lambda Physik, model EMG 101MSC) and the light of
the Xe lamp were directed collinearly through the highcell, via
a set of laser mirrors (Laseroptik, high reflectance at 308 nm
and high transmittance at 253 nm, 451) in a counter-propagat-
ing arrangement. A laser power meter recorded the laser energy
behind the high-pressure cell. A low-pressure mercury lamp
(ORIEL, model 65130, 22–44 W) was used to calibrate the
wavelength of the spectrum before the measurement. A pulse/
delay generator (Stanford Research Systems, model DG535)
was interfaced with the laser controlling system and the data
acquisition system of the ICCD camera. Spectral analysis
software (LaVision, DaVis version 6.0) was used for post-
processing the measured spectra. Prior to obtaining the tran-
sient spectrum of benzyl radicals, a reference spectrum was
Fig. 1 Absorption signals at 253 nm, recorded after the photolysis at
308 nm of mixtures of Cl2, toluene and CO2 at T ¼ 300 K. Main figure:
time-profiles at different CO2 pressures, from the top: 35, 20, 10, and
6 bar. The ratio of [Cl2]/[CO2] was kept constant (1.3 ꢁ 10ꢂ5). Inserted
figure: (K) maximum absorption values as a function of the CO2 bath
gas pressure, (--) ¼ linear fit.
4134
P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 4 1 3 3 – 4 1 4 1
T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 4

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Because of the second order kinetics of reaction (1),
measured reaction rates are sensitive to the absolute initial
concentration of benzyl radicals and therefore the value of
their absorption coefficients. We checked if there was any
influence of the pressure and temperature dependence of sbenzyl
on our evaluation of k1. This was done by the following two
methods. First, in Fig. 1, we kept a constant ratio of [benzyl]0/
[CO2] at different CO2 pressures between 7 and 37 bar. The
inset in Fig. 1 illustrates the maximum absorption values of
benzyl radicals at time ¼ 0 as a function of bath gas pressure.
The linear relationship indicates that under our experimental
conditions we could not see any unusual behaviour in our
absorption signals due to solvent-induced spectral shifts or
unusual changes of absolute absorption intensities. Second, we
investigated the pressure dependence of the UV spectra of
benzyl radicals in the bath gas CO2 at 300 K; the results are
shown in Fig. 2. CO2 was chosen as the bath gas as it is
expected to cause larger spectral shifts than the other bath
gases employed. Our measured benzyl spectra recorded be-
tween 10 ms and 60 ms with a spectral resolution of 0.09 nm)
were calibrated with the Hg line spectra at 253.4 nm, and
compared with the spectrum of benzyl radical measured by Fay
et al. after 193 nm photolysis of ethylbenzene in 5 mbar of
argon (recorded between 1 ms and 3.2 ms).16 We observed no
significant spectral shift over the pressure range 5–30 bar.
Maximum changes of the integrated absorption over the range
(253 ꢀ 1) nm at different CO2 pressures in Fig. 2 corresponded
to maximum errors of 5% in k1, due to the small spectral shift
of the spectrum illustrated in Fig. 2.
similar as observed in our recent study of combination reac-
tions of CCl3 radicals.6 The finally observed decrease of the
rate constants at pressures above 100 bar (in N2 and argon),
and above 60 bar (in CO2), corresponds to a transition into the
regime of diffusion-controlled kinetics such as expected at these
pressures. A modest negative temperature dependence of
ET
k
(T) was observed and measured over the temperature
1,N
range 250–400 K. The results are summarized in Table 2 and
compared in Fig. 4 with values from Lesclaux and co-workers
in the temperature ranges 400–450 K9 and 435–519 K.10 In
order to compare our results with the values from refs. 9 and 10
in Fig. 4, it was necessary to correct their rate constants using
s
benzyl(T) E 1.3 ꢁ 10ꢂ16 (T/300 K)ꢂ0.25 cm2 moleculeꢂ1. The
temperature dependence of sbenzyl(T) was estimated from the
work of Ikeda et al.19 All measurements confirm that there is
only a mild temperature dependence of kE1,TN(T). The tempera-
ture dependence of kE1,TN(T) over the temperature range
250–520 K, including results of refs. 9 and 10, could be
expressed by:
ET,experimental
1,N
k
(T) ¼ (4.1 ꢀ 0.3) ꢁ 10ꢂ11(T/300 K)ꢂ0.45
cm3 moleculeꢂ1 sꢂ1
(6)
The error limits in eqn. (6) represent the scatter of the data and
an estimate of possible systematic errors.
The temperature dependence of the enhancement of k1 over
ET
1,N
k
was investigated in the bath gas CO2 and the results are
illustrated in Fig. 5 over the range 275–350 K. The enhance-
ment of k1 was found to be the more pronounced, the lower the
temperature. We have also investigated the possibility of a
heavy atom effect on the enhancement of k1 which might be the
consequence of electronic quenching effects. Using bath gas
mixtures of argon and xenon, however, we found no notable
increase of k1 compared to measurements in pure argon when
similar density conditions were compared (see Table 1(g)
and 1(h)).
Our results on the pressure dependence of k1 at 300 K are
summarized in Table 1 and Fig. 3. The limiting ‘‘high-
pressure’’ rate constant of the energy-transfer mechanism,
which in this case corresponds to the value of k1 at the lowest
ET
pressure investigated, was determined to be k
(300 K) ¼
1,N
(4.1 ꢀ 0.3) ꢁ 10ꢂ11 cm3 moleculeꢂ1
s
ꢂ1, independent of the
bath gas and considering all data points below 1 bar. At
pressures above 1 bar, clear indications of a further increase
of k1 are recognized in Fig. 3. This pressure-induced enhance-
ment of k1 increases in the order He o N2 E Ar o CO2,
4. Discussion
4.1. Experimental ‘‘high-pressure’’ rate constant, kE1,TN(T)
Reaction (1) has been previously studied in 1 bar of N2 by
Fenter et al.9 in the temperature range 400–450 K and Boyd
et al.10 from 435 to 519 K. Their results of rate constant k1 were
obtained with the reported literature absorption cross-section
s
benzyl (253 nm, 300 K) ¼ 1.1 ꢁ 10ꢂ16 cm2 moleculeꢂ1 by Ikeda
et al.19 and Markert and Pagsberg.20 From this, Boyd et al.
determined k1E,TN to be (2.9 ꢀ 0.3) ꢁ 10ꢂ11 cm3 moleculeꢂ1 sꢂ1
with a total uncertainty of B40%. Boyd et al. furthermore
reduced the previous result of Fenter et al. from (4.6 ꢀ 2.5) ꢁ
10ꢂ11 cm3 moleculeꢂ1 sꢂ1 to (3.9 ꢀ 1.9) ꢁ 10ꢂ11 cm3
moleculeꢂ1 ꢂ1, after they allowed for a temperature depen-
s
dence of sbenzyl(T) recommended by Ikeda et al.19 From a
careful redetermination, Fay16 has recently reported a slightly
higher value of sbenzyl ¼ 1.3 ꢁ 10ꢂ16 cm2 moleculeꢂ1, obtained
from a cold spectrum of benzyl radicals after relaxation to
300 K: We have used this value in our evaluation. Using the
latter value of sbenzyl, the results from refs. 9 and 10 agree with
the present results as shown in Fig. 4.
However, our rate expression in eqn. (6) (both the absolute
value and the temperature coefficient) does not lead to an
agreement with the results of Muller-Markgraf and Troe8 in
¨
their earlier shock wave study at much higher temperatures.
They obtained a rate constant k1 ¼ (6.6 ꢀ 1.5) ꢁ 10ꢂ12(T/
1000 K)10.4 cm3 moleculeꢂ1 sꢂ1 over the range 750 K r T r
950 K. Using eqn. (6), we expect a value of k1 ¼ 2 ꢁ 10ꢂ11 cm3
moleculeꢂ1 sꢂ1 at 1600 K, a value B3 times higher than that
measured in ref. 8. It is unlikely that this disagreement is
caused by uncertainties in sbenzyl(T) as a function of tempera-
ture: in ref. 8, sbenzyl(T) ¼ 4 ꢁ 10ꢂ17 cm2 moleculeꢂ1 was
directly measured at (1160 ꢀ 30) K and (1600 ꢀ 20) K
Fig. 2 Transient spectra of benzyl in the bath gas CO2 over the range
of 5–30 bar. (J): this work, delay time ¼ 10 ms, gating time ¼ 50 ms,
and spectral resolution 0.09 nm; (––– at the top of the figure): cold
spectrum of benzyl radical in 5 mbar of Ar from ref. 16; (––– at the
bottom of the figure): Hg reference line; the arrow points at 253.4 nm.
T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 4
P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 4 1 3 3 – 4 1 4 1
4135

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Table 1 Pseudo-second-order rate constants for the combination reaction of benzyl radicals (k1)
p (He)a
[He]b
p (He)a
[He]b
k1
p (He)a
[He]b
k1
p (He)a
[He]b
c
k1
c
c
c
k1
(a) M ¼ He at 300 K
25
40
60
5.97 ꢁ 1020
9.48 ꢁ 1020
1.41 ꢁ 1021
4.60
3.59
4.76
80
90
1.86 ꢁ 1021
2.09 ꢁ 1021
5.05
4.88
100
120
2.31 ꢁ 1021
2.74 ꢁ 1021
4.85
4.41
150
200
3.39 ꢁ 1021
4.42 ꢁ 1021
5.23
5.06
(b) M ¼ Ar at 300 K
p(Ar)a
0.03
0.05
0.07
0.09
0.15
2
[Ar]b
k1
p(Ar)a
8
10
[Ar]b
k1
p(Ar)a
90
[Ar]b
k1
p(Ar)a
692
756
788
841
842
864
850
900
[Ar]b
k1
c
c
c
c
1.17 ꢁ 1018
1.65 ꢁ 1018
2.23 ꢁ 1018
3.58 ꢁ 1018
4.83 ꢁ 1019
7.24 ꢁ 1019
9.66 ꢁ 1019
1.45 ꢁ 1020
1.93 ꢁ 1020
4.60
3.67
4.06
4.11
4.03
4.97
4.50
5.42
4.61
1.93 ꢁ 1020
2.41 ꢁ 1020
2.90 ꢁ 1020
3.62 ꢁ 1020
6.04 ꢁ 1020
9.66 ꢁ 1020
1.21 ꢁ 1021
1.45 ꢁ 1021
1.93 ꢁ 1021
5.13
4.67
4.57
4.43
5.31
5.30
5.92
5.92
4.56
2.17 ꢁ 1021
2.53 ꢁ 1021
3.04 ꢁ 1021
4.01 ꢁ 1021
5.53 ꢁ 1021
7.61 ꢁ 1021
9.16 ꢁ 1021
1.05 ꢁ 1022
1.15 ꢁ 1022
6.10
5.99
5.85
5.87
4.90
3.93
3.56
3.37
3.14
1.24 ꢁ 1022
1.29 ꢁ 1022
1.32 ꢁ 1022
1.35 ꢁ 1022
1.36 ꢁ 1022
1.37 ꢁ 1022
1.36 ꢁ 1022
1.39 ꢁ 1022
3.22
2.92
3.58
2.96
2.96
3.96
3.50
3.56
100
120
158
220
318
408
507
597
12
15
25
40
3
50
60
4
6
80
(c) M ¼ N2 at 300 K
p(N2)a
[N2]b
k1
p(N2)a
13
15
[N2]b
k1
p(N2)a
50
60
[N2]b
k1
p(N2)a
155
201
[N2]b
k1
c
c
c
c
4
9.66 ꢁ 1019
1.45 ꢁ 1020
1.93 ꢁ 1020
2.41 ꢁ 1020
4.03
4.35
5.18
5.11
3.02 ꢁ 1020
3.62 ꢁ 1020
4.35 ꢁ 1020
9.66 ꢁ 1020
5.17
4.74
5.62
5.17
1.21 ꢁ 1021
1.45 ꢁ 1021
1.93 ꢁ 1021
2.79 ꢁ 1021
5.31
5.48
5.10
5.38
3.62 ꢁ 1021
4.67
4.39
6
4.59 ꢁ 1021
8
10
18
40
80
117
(d) M ¼ CO2 at 300 K
p(CO2)a
0.04
0.07
[CO2]b
k1
p(CO2)a
0.45
3.00
[CO2]b
k1
p(CO2)a
12.00
15.00
18.00
20.00
21.60
25.00
[CO2]b
k1
p(CO2)a
30.00
31.60
35.00
41.60
[CO2]b
k1
c
c
c
c
9.61 ꢁ 1017
1.71 ꢁ 1018
2.36 ꢁ 1018
4.85 ꢁ 1018
7.24 ꢁ 1018
9.75 ꢁ 1018
4.17
3.79
4.09
4.11
3.67
3.90
1.08 ꢁ 1019
7.35 ꢁ 1019
9.85 ꢁ 1019
1.49 ꢁ 1020
2.01 ꢁ 1020
2.54 ꢁ 1020
3.96
4.51
4.76
4.84
5.52
5.40
3.08 ꢁ 1020
3.92 ꢁ 1020
4.79 ꢁ 1020
5.39 ꢁ 1020
5.89 ꢁ 1020
6.97 ꢁ 1020
6.03
6.39
7.33
7.74
6.61
7.79
8.67 ꢁ 1020
9.25 ꢁ 1020
1.05 ꢁ 1021
1.33 ꢁ 1021
8.26
7.07
9.29
8.73
0.10
0.20
4.00
6.00
0.30
0.40
8.00
10.00
(e) M ¼ CO2 at 275 K
p(CO2)a
[CO2]b
k1
p(CO2)a
18
15
[CO2]b
k1
p(CO2)a
12
20
[CO2]b
k1
p(CO2)a
25
30
[CO2]b
k1
c
c
c
c
8
10
2.23 ꢁ 1020
6.81
7.01
5.45 ꢁ 1020
7.97
7.71
3.45 ꢁ 1020
6.60
9.25
8.13 ꢁ 1020
9.88
11.16
2.83 ꢁ 1020
4.42 ꢁ 1020
6.17 ꢁ 1020
1.04 ꢁ 1021
(f) M ¼ CO2 at 350 K
p(CO2)a
[CO2]b
k1
p(CO2)a
[CO2]b
k1
p(CO2)a
12
15
[CO2]b
k1
p(CO2)a
20
25
[CO2]b
k1
c
c
c
c
2
3
4
4.16 ꢁ 1019
6.26 ꢁ 1019
8.37 ꢁ 1019
3.83
4.00
4.52
6
1.26 ꢁ 1020
1.69 ꢁ 1020
2.13 ꢁ 1020
4.28
4.79
4.65
2.57 ꢁ 1020
3.25 ꢁ 1020
3.93 ꢁ 1020
5.26
5.05
5.51
4.40 ꢁ 1020
5.59 ꢁ 1020
6.83 ꢁ 1020
5.64
5.60
5.80
8
10
18
30
(g) M ¼ Ar: Xe (1:1 mixture) at 300 K
pa
20
[M]b
pa
25
[M]b
6.52 ꢁ 1020
k1
5.71
pa
30
[M]b
7.60 ꢁ 1020
k1
6.89
c
k1
c
c
4.98 ꢁ 1020
5.70
(h) M ¼ Ar: Xe (1:1 mixture) at 275 K
pa
20
[M]b
pa
25
[M]b
k1
6.69
pa
30
[M]b
k1
5.49
c
k1
c
c
5.49 ꢁ 1020
5.33
7.23 ꢁ 1020
8.43 ꢁ 1020
a
b
c
Pressure of the bath gas, given in bar. Density of the bath gas, given in molecule cmꢂ3
.
Rate constant, given in 10ꢂ11 cm3 moleculeꢂ1 sꢂ1
.
following the thermal decomposition of several precursors
(benzyl iodide, toluene, and ethyl benzene). This is comparable
to sbenzyl(T ¼ 1600 K) ¼ 3.4 ꢁ 10ꢂ17 cm2 moleculeꢂ1, when one
uses an estimated value recommended by Ikeda et al.19 There-
fore, a possible reason for this disagreement might be the
derivation of k1 from a complicated reaction system using
benzyl iodide as a precursor, as done in ref. 8.
accurate ab initio potentials for reaction (1) are not yet avail-
able, a situation typical for systems involving large radicals. As
a consequence we estimate kE1,TN(T) on the basis of statistical
adiabatic channel/classical trajectory (SACM/CT) calculations
for a ‘‘standard’’ valence potential such as described in the
articles by Maergoiz et al.21,22 for a ‘‘linear þ linear - non-
linear’’ reaction with an adduct angle corresponding to that of
ET
dibenzyl. First, we calculate an upper limit of k
ET,PST
given by
, which is determined by the inter-
1,N
phase space theory, k
4.2. Theoretical analysis of kE1,TN(T)
1,N
action potential between the radicals neglecting the anisotropy.
As usual, the long-range dipole–dipole potential in the present
reaction is irrelevant because of the small dipole moment of
benzyl radicals, which at the UHF level with a 6-31G(d,p) basis
set23 was calculated to be as small as 0.06 D.24 Instead we
characterized the interaction between two benzyl radicals by a
In the preliminary report of the present results,6 we have
suggested that an analysis of kE1,TN, in terms of unimolecular
rate theory is also of major importance for an analysis of the
enhancement of k1. In the following we, therefore, further
elaborate the analysis of kE1,TN(T). Unfortunately, sufficiently
4136
P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 4 1 3 3 – 4 1 4 1
T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 4

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Fig. 4 Temperature dependence of kE1,TN(T): this study (O), ref. 9 (n),
and ref. 10 (’). Data points of our study were measured and averaged
over the pressure range of 1–5 bar in the bath gas argon (see Table 2).
Data points of refs. 9 and 10 are corrected for sbenzyl(T) (see text). The
temperature dependence of the fitted line follows from SACM/CT
calculations (see Section 4.2).
much more than use the experimental f erixgpid at 300 K and fit the
ratio a/b of potential parameters which characterizes the
Fig. 3 Density dependence of the combination rate constant k1 in
helium (this work), argon, N2 and CO2 (ref. 7) at 300 K. Lines indicate
the simulation from the kinetic model described in the text.
anisotropy of the potential.21,22 One then may calculate the
theory
temperature dependence of frigid
and compare that with
the corresponding experimental property of f erixgpid. Following
the treatment of refs. 21 and 22, f erixgpid(300 K) ¼ 0.24 is
theory
Morse potential. The capture rate constant from phase space
theory is then expressed by:21,22
reproduced by a/b ¼ 0.74. Using this value, the theoretical
sffiffiffiffiffiffiffiffiffiffiffi
calculation of f
gives
rigid
8pkT f
m
ET;PST
1;1
k
¼
aspinYðTÞ
ð7Þ
theory
rigid
E 0.24(T/300 K)ꢂ0.6
(11)
b2
f
in very good agreement with the experimental result from
eqn. (10). Combining eqns. (9) and (11) gives
Here m is the reduced mass of the two reactants; aspin ¼ 1/4 is
the spin-statistical factor; the parameter f is equal to 0.5 for
identical reactants; a reduced cross-section Y(T) ¼ ꢂ31.153 ꢂ
18.158X(T) þ 0.8685X(T)2 ¼ 296.2 is estimated21,22 with
X(T) ¼ ln(kT/D0) þ 4 ꢂ bre ¼ ꢂ9.9. The equilibrium distance
of the centre of mass of the two reactants in the adduct is
re ¼ 5.5 A. The Morse parameter b ¼ 1.6 Aꢂ1 is estimated from
the expression:21,22
ET,theory
1,N
k
(T) ¼ 4.1 ꢁ 10ꢂ11 (T/300 K)ꢂ0.23 cm3
moleculeꢂ1 sꢂ1
(12)
where the calculated temperature dependence agrees well with
the experimental result from eqn. (6).
rffiffiffiffiffiffiffiffiffiffi
mCꢂC
2D0
b ¼ 2phcnCꢂC;str
ð8Þ
A value of the dissociation energy of D0/hc ¼ 2.32 ꢁ
104 cmꢂ1 25
and a vibrational frequency in the C–C stretching
,
mode, nC–C,str ¼ 976 cmꢂ1, are used.26 This leads to
ET,PST
1,N
k
¼ 1.7 ꢁ 10ꢂ10(T/300 K)10.40 cm3 moleculeꢂ1 sꢂ1 (9)
The comparison with the experimental results then gives an
experimental rigidity factor of
exp
rigid
f
E 0.24 (T/300 K)ꢂ0.5
(10)
One may also try to rationalize frigid by theory. In the absence
of a higher quality potential energy surface, one cannot do
Table 2 Limiting ‘‘high-pressure’’ rate constants of the energy-trans-
fer mechanism, kE1,TN(T)
s
benzyl(T)/10ꢂ16 cm2
kE1,TN(T)/10ꢂ11 cm3
moleculeꢂ1 sꢂ1
T/K
moleculeꢂ1
250
275
300
325
350
375
400
1.34
1.32
1.30
1.28
1.25
1.22
1.20
5.03 ꢀ 0.36
4.63 ꢀ 0.50
4.10 ꢀ 0.30
3.91 ꢀ 0.21
3.32 ꢀ 0.19
4.47 ꢀ 0.55
3.13 ꢀ 0.05
Fig. 5 Density dependence of the combination rate constant k1 in
CO2 at different temperatures. Experimental data are from ref. 7, and
lines are the results from a fit in terms of radical-complex mechanism
(see text).
T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 4
P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 4 1 3 3 – 4 1 4 1
4137

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4.3. Transition to diffusion-controlled kinetics
At the highest pressure of Fig. 3, k1 passes over a maximum
and starts decreasing, which can be attributed to a transition to
diffusion-controlled kinetics. Before analysing the intermediate
zone of an enhancement of k1, we now look at the high density
range where the recombination is diffusion-controlled. Rate
constants in the gas–liquid transition range traditionally have
been approximated by a relationship:
ꢀ
ꢁ
kgrec
kgrec þ kdiff
krec ¼ kdiff
ð13Þ
with kdiff ¼ 4paspin(M)RD. Here kdiff is the value of k1 in the
range of diffusion control; kgrec is the hypothetical value of the
combination rate constant in the absence of diffusion control,
i.e. the joint contribution from the energy-transfer mechanism
(kET), the mechanism responsible for the enhancement of
k1 (kRC); D is the diffusion coefficient of the recombining
radicals in the bath gas; R is an effective capture distance;
and aspin(M) denotes a possibly density-dependent electronic
weight factor. At this stage we use aspin of 1/4 for reaction (1).
This relates to the simple picture that only one-fourth of the
benzyl þ benzyl encounters lead to singlet electronic ground-
state bibenzyl, whereas three-fourths of the encounters are
unsuccessful because they separate from an excited triplet state
before they can attain the correct spin configuration (see ref. 6
for more detailed explanations). The contact distance R is
usually taken as a more or less undefined fit parameter.
Recently we suggested to relate R to the thermally averaged
capture cross section hsi of two radicals in the ‘‘high-pressure
limit’’ within the energy-transfer mechanism:6
Fig. 6 Diffusion coefficients of benzyl radicals in liquid solvent media
as a function of the inverse of the viscosity of the liquid; in 2-propanol
(., ref. 27), in ethanol (m, ref. 27), in cyclohexane (J, ref. 27, &,
ref. 28, and B, ref. 11), and in hexane (ꢁ, ref. 27). The solid line
represents the calculated values for Dbenzyl in this work (see text).
This approach is based on the rough hard sphere theory which
treats the intermolecular interaction between solute and sol-
vent as being of Lennard-Jones (LJ) type. The result assuming
Dbenzyl E Dtoluene is plotted in Fig. 6 as a function of the
inverse viscosity of the bath gas CO2. The extrapolation of the
derived Dbenzyl to the liquid densities shows good agreement
with the literature data in various liquid solvents mentioned
above, which suggests the quality of the derived Dbenzyl
.
Figs. 7 and 8 employs the calculated rate constants kdiff, k1
from the energy-transfer range, and eqn. (13) for a separate
identification of that part of k1 (denoted by kRC) which is
responsible for the enhancement of k1 at intermediate densities.
Diffusion control in the high-density region is clearly limiting
the kinetics at the highest pressures. We note that in our work
it was not necessary to employ any fit parameters to specify
kdiff in eqn. (13).
rffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffi
ꢂ
ꢃ
1=2
hsi
1
pm
8kT
R ¼
¼
kE1T
ð14Þ
p
aspinðMÞpf
where f is 1/2 for identical reactants and m is the reduced mass
of the two radicals. Following this suggestion, one derives R ¼
5.3 A. It is interesting to note that a similar value (R ¼ 5.9 A)
was estimated by Claridge and Fischer11 in their kinetic studies
of the self-termination of benzyl radicals in liquid cyclohexane,
averaging the results from three methods: (a) from the molar
volume of the corresponding toluene and a space filling factor
for cubic close packing, (b) from a simple volume increment
method, and (c) from a volume increment method using LeBas
increments.
4.4. Contribution kRC from the radical-complex mechanism
Our experimental data for k1 at pressures of about 10–300 bar
in the bath gas argon and of about 3–50 bar in CO2, clearly
differ from a smooth transition between energy-transfer and
Experimental measurements of tracer diffusion coefficients
for radicals in liquids or supercritical fluids are rare. Fortu-
nately there have been efforts to measure D for benzyl radicals
in liquids.27–31 By using the photochemical space intermittency
(PCSI) method, Burkhart et al.28–30 measured D of benzyl
radicals in cyclohexane and determined Dbenzyl ¼ 1.1 ꢁ
10ꢂ5 cm2 moleculeꢂ1 ꢂ1. More recently, Terazima et al.27
s
applied the time-resolved transient grating method and ob-
tained Dbenzyl ¼ 4.1 ꢁ 10ꢂ5, 0.95 ꢁ 10ꢂ5, 1.1 ꢁ 10ꢂ5, 0.64 ꢁ
10ꢂ5 cm2 moleculeꢂ1 sꢂ1 in hexane, cyclohexane, ethanol and
2-propanol, respectively. In Fig. 6, these literature values are
plotted as a function of the inverse viscosity of the studied
liquid solvents. There is, however, no direct measurement of
Dbenzyl in the medium density region of our experiments, such
that we cannot directly use the mentioned experimental results
to our high-pressure data. In order to overcome this problem,
we calculated density-dependent diffusion coefficients of to-
luene, because, at least in liquids, Dbenzyl is known to be very
close to the value of D of toluene.27 Assuming that the ratio
D
benzyl/Dtoluene is independent of the density in the medium
Fig. 7 Density dependence of the combination rate constant k1 in the
bath gas argon (J) at 300 K. (ꢃ ꢃ ꢃ): kET from the energy-transfer
mechanism; (–ꢄ–): limiting diffusion-controlled rate constants; (– – –):
kRC from the radical-complex mechanism; (–ꢄꢄ–): resulting rate con-
density region, we determined density-dependent values of
Dbenzyl on the basis of calculated Dtoluene. Fortunately, several
currently available semi-empirical calculations of tracer diffu-
sion in supercritical fluids have used toluene as a model. We
used the semi-empirical method suggested in refs. 32 and 33.
stants without kRC; and (–––):resulting rate constants including kRC
.
Fitting parameters used in this calculation are summarised in Table 2.
4138
P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 4 1 3 3 – 4 1 4 1
T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 4

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Table 3 Contribution to kRC for the radical-complex mechanism
eq(AM)a
kAM1A
kAM1AM
b
b
M
T/K
K
He
N2
300
300
300
300
275
350
6.6 ꢁ 10ꢂ23
4.9 ꢁ 10ꢂ22
5.8 ꢁ 10ꢂ22
1.1 ꢁ 10ꢂ21
1.4 ꢁ 10ꢂ21
6.7 ꢁ 10ꢂ22
8.08 ꢁ 10ꢂ11
8.36 ꢁ 10ꢂ11
8.36 ꢁ 10ꢂ11
8.85 ꢁ 10ꢂ11
8.95 ꢁ 10ꢂ11
5.64 ꢁ 10ꢂ11
7.22 ꢁ 10ꢂ11
7.52 ꢁ 10ꢂ11
1.19 ꢁ 10ꢂ10
1.20 ꢁ 10ꢂ10
1.17 ꢁ 10ꢂ10
Ar
CO2
CO2
CO2
a
8.67 ꢁ 10ꢂ11
Equilibrium constant, given in cm3 moleculeꢂ1
,
Rate constant,
b
given in cm3 moleculeꢂ1 sꢂ1
.
the ‘‘high-pressure limit’’ of the rate constant in the radical-
complex mechanism. kAM1A and kAM1AM can both be fitted
independently, see Figs. 7 and 8. The solid lines account for
contributions from the usual energy-transfer mechanism and
from the radical-complex mechanism. As compared with the
ET
dashed–dot–dot lines (‘‘k
þ kdiff only’’), the addition of a
1,N
radical-complex component results in a much better represen-
tation of the experimental data. Similar fits were applied to the
N2 and helium data, and the results are summarized in Table 3
and plotted in Fig. 3. The fits of the experiments (see Table 3)
suggest that values of kAM1A and kAM1AM for A ¼ benzyl and
M ¼ bath gas are always larger than kA1A (¼ kE1,TN) and
increase in the order kA1A o kAM1A o kAM1AM. kAM1A and
kAM1AM, on the other hand, increase in the order He o N2 E
Ar o CO2.
Fig. 8 As Fig. 7, at 300 K in CO2 (half-filled diamonds).
diffusion-controlled kinetics (see the dashed–dot–dot line,
‘‘kET þ kdiff only’’ in Figs. 7 and 8, respectively). We have
made a similar observation in the combination of CCl3 radicals
with Br and CCl3.6 However, the effect is much more easily
observed in the present case because the energy-transfer me-
chanism of the recombination is already in the ‘‘high-pressure
limit’’ over the pressure range 0.1–1 bar. The enhancement of
k1, therefore, can be identified and quantified much better. The
additional contribution kRC to k1 shown in Figs. 7 and 8 is the
central and new piece of information derived in this work. We
tentatively as before attribute it to a radical-complex mechan-
ism. In the following we try to provide a quantitative inter-
pretation of kRC within this mechanism.
If kAM1A and kAM1AM would not be larger than kA1A, then
no rate constant enhancement of k1 would have been observed.
It is, therefore, of central importance to rationalize the con-
clusion about the order of the rate constants, i.e. kA1A
kAM1A o kAM1AM. Simple phase space theory does not
o
explain the observed trend. Actually, by the methods described
PST
PST
AþA
in Section 4.2, we obtained values of k
E k
¼ 1.7 ꢁ
AMþAM
10ꢂ10 cm3 moleculeꢂ1 sꢂ1, because the effects of different cross
sections and reduced masses partially compensate. In this
calculation, the distance of the center of mass of the two
reactants, re, was estimated as re,AM1AM ¼ 6.8 A and
re,A1A ¼ 5.5 A following structural optimisation on the
UB3LYP level with a 6-31G(d,p) basis set.23 In addition, we
Analogous to ref. 6, we assume that benzyl radical combina-
tion involves an energy-transfer, eqn. (15), and a radical-
complex, eqn. (16), contribution:
assumed that the dissociation energies D0,A1A E D0,AM1AM
¼
2.32 ꢁ 104 hc cmꢂ1 25
,
and that the change in the vibrational
A þ A # A2* ka, k
frequencies in the C–C stretching modes, nC–C,str, for A þ A
and AM þ AM, are negligible.
ꢂa
A2* þ M - A2 þ M kb
(15)
If the phase space limits of the rate constants are equal, the
differences between kAM1AM and kA1A can only be found in
different rigidity factors. In ref. 6 we suggested that the
presence of a van der Waals complex partner M may shield
A þ M # AM kc, kꢂc, Keq
AM þ A - A2 þ M kAM1A
AM þ AM - A2 þ 2M kAM1AM
(16)
and reduce the anisotropy of the valence potential between A
and A which then results in larger rigidity factors.6
exp
in
rigid
f
Here A denotes the benzyl radical and M is the bath gas.
Neglecting diffusion control, the combined rate constant from
both mechanisms is represented by
k
AM1AM was found to increase in the order He (E0.33) o N2
(E0.43) E Ar (E0.45) o CO2 (E0.70). It appears important
to note that f erixgpid is still below unity. It appears also of interest
to mention that a similarly high rigidity factor as in the bath
ET
kET
k
½Mꢅ
1;1 1;0
Fc
ET
k1;rec ¼kE1 T þ kR1 C
¼
kE1;T1 þ k ½Mꢅ
gas of CO2 was also observed in our subsequent studies of
exp
1;0
ð17Þ
benzyl combination in the bath gases CF3H or SF6.35
kAM1A only showed small variations but was still increasing in
f
in
rigid
2
KeqkAMþA½Mꢅ þ Ke2qkAMþAM½Mꢅ
þ
2
the order He (E0.25) o N2 (E0.26) E Ar (E0.26) o CO2
rigid
ð1 þ Keq½MꢅÞ
exp
(E0.27). It was found to be between the values of f
kAM1AM and f
in
exp
rigid
kE1 T is characterized by the limiting low- and high-pressure rate
constants k and kE1,TN, respectively, and a broadening factor
in kA1A (¼ 0.24). In summary, our inter-
ET
1,0
pretation in terms of the different rigidity factor appears
consistent with the experimental observations. At present these
interpretations still have some hypothetical character and need
further back up from quantitative tests in other experimental
systems and from detailed CT calculations of the capture
process on improved potential surfaces. Both are underway
and a corresponding comprehensive treatment will be given in
the future. However the present study already provides another
piece of experimental evidence for the assumed role of the
of the falloff curve. The equilibrium constant Keq contained in
kR1 C was estimated following refs. 6 and 34, using Lennard-
Jones parameters of sLJ ¼ 5.92 A and eLJ ¼ 410 K for benzyl
radicals such as given by the values of toluene. The values of
Keq at 300 K for different bath gases are summarised in
Table 3; they are in the range 0.6–11 ꢁ 10ꢂ22 cm3 moleculeꢂ1
and show an increasing order Keq(He) o Keq(N2) E Keq(Ar) o
Keq(CO2). kAM1A controls the onset while kAM1AM describes
T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 4
P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 4 1 3 3 – 4 1 4 1
4139

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radical-complex mechanism: A pronounced negative tempera-
ture dependence of the rate constant enhancement at the
elevated densities was observed, just in line with basic expecta-
tions for increasing contributions from radical-complex
mechanism at decreasing temperatures. This is discussed in
the following section.
4.5. Temperature dependence of the radical-complex
mechanism
For a further test, we also investigated the temperature depen-
dence of the rate constant enhancement in the medium density
region. To our knowledge, a quantitative analysis of this effect
is reported here for the first time. Fig. 5 shows a fit to our
experimental data in the bath gas CO2 at the temperatures
275 K, 300 K and 350 K (see Table 3 for the fitted values). The
observation of a larger enhancement at lower temperatures
supports our hypothesis of a contribution from the radical-
complex mechanism. Both the equilibrium constants Keq(T)
and the rate constants kAM1A(T) and kAM1AM(T) are expected
to depend on the temperature.
First, we consider Keq(T). The temperature dependence of
Keq(T) was estimated following the method described in refs. 6
and 34, and the resulting values are summarized in Table 3.
Over the temperature range 250–400 K, they can be repre-
sented as
Fig. 9 (A) Temperature dependence of thermal capture rate constants
of benzyl radicals in the bath gas CO2. (——): kPST(T); (--): kAM1AM
exp
(T); and (–ꢄ–): kA1A(T). Experimental data are: (m) k
(T) at
AMþAM
temperatures of 275, 300, and 350 K in the CO2 bath gas, (J)
kA1Aexp(T) (¼ kE1,TN(T)) in the argon bath gas from Fig. 4.
(B) Temperature dependence of calculated thermal rigidity factors
theory
Keq(T) ¼ 1.1 ꢁ 10ꢂ21 (T/300 K)ꢂ3 cm3 moleculeꢂ1 (18)
Second, we derived expressions for the rate constants of
for reaction (1) in the bath gas CO2. (--): f
(T); and (–ꢄ–):
(T); and (J):
rigid,AMþAM
rigid,AMþAM
theory
rigid,AþA
exp
f
f
(T). Experimental data are: (m): f
(T).
exp
rigid,AþA
exp
AMþA
keAxMp þAM(T) and
Table 3. We obtained
k
(T) from the experiments, see
keAxMp þAM(T) ¼ 1.2ꢁ10ꢂ10(T/300 K)ꢂ0.1 cm3 moleculeꢂ1
sꢂ1
(19)
5. Conclusions
and
The combination reaction C6H5CH2 þ C6H5CH2 (þM) -
C14H14 (þM) was studied at temperatures of 250–400 K over
the pressure range 0.03–900 bar in the bath gases helium,
argon, xenon, N2 and CO2. The fact that the reaction rate
constants reach a pressure-independent range at pressures as
low as 0.01 bar, allowed us to determine the limiting ‘‘high-
pressure’’ rate constant of the energy-transfer mechanism
keAxMp þA(T) ¼ 8.8ꢁ10ꢂ11(T/300 K)ꢂ0.1 cm3 moleculeꢂ1
sꢂ1
.
(20)
It should be noted that the slightly negative temperature
exp
coefficients of keAxMp þAM(T) and k
(T) are similar to those
obtained for keAxþpA(T) in eqn. (6). Third, temperature-depen-
dent rigidity factors of
AMþA
ET
ET
k
mined as
without doing falloff extrapolation. k
could be deter-
1,N
1,N
f erixgpid,AMþAM(T) ¼ 0.70 (T/300 K)ꢂ0.5
f erixgpid,AMþA(T) ¼ 0.27 (T/300 K)ꢂ0.5
(21)
(22)
kE1,TN(T) ¼ (4.1 ꢀ 0.3) ꢁ 10ꢂ11 (T/300 K)ꢂ0.23 cm3 moleculeꢂ1
and
sꢂ1
This value was analysed in terms of SACM/CT theory. The
observed increase of the rate constants at higher pressures,
between about 5 bar and the onset of diffusion-limited kinetics
above about 50 bar, was interpreted in terms of a radical-
complex mechanism. A quantitative analysis of absolute values
and temperature dependences of rate constants in the range of
the radical-complex mechanism was given. A proof of the
suggested mechanism and more quantitative conclusions re-
quires ab initio information on the potential between uncom-
plexed and complexed radicals. As long as this information is
not available, the given interpretation remains tentative. In
particular, the possibility of collision-induced electronic transi-
tions, which could convert triplet benzyl pairs into singlets,
cannot be ruled out. We did not discuss this possibility in the
present article. An empirical investigation of this point will be
described in the future when more experimental data for more
diverse systems will have been collected. For the moment,
however, the hypothesis of a contribution of the radical-
complex mechanism in the pressure range between about 5
and 50 bar could be supported by an intrinsically consistent
quantitative analysis. If the present interpretation is correct,
one has to expect similar phenomena quite generally in radical–
radical recombination reactions.
were derived from the experiments. These results of
exp
keAxMp þAM(T) and f
(T) were then compared with
rigid,AMþAM
theoretical estimates such as described in Section 4.2. Briefly,
we characterised the interaction between two AM (benzyl þ
CO2) by a Morse potential with re,AM1AM ¼ 6.8 A and we
fitted a/b E 0.77 from the value of frigid at 300 K.21 Following
the method as described before, we predicted
theory
rigid,AMþAM
f
E 0.70 (T/300 K)ꢂ0.4
(23)
which is very close to our experimental temperature depen-
exp
dence of f
theoretical estimate of
in eqn. (21). In total, we obtained a
rigid,AMþAM
kAthMeoþryAM(T) ¼ 1.2 ꢁ 10ꢂ10 (T/300 K)ꢂ0.05 cm3
moleculeꢂ1 sꢂ1
(24)
which is again in agreement with the experimental result
from eqn. (19). Fig. 9(a and b) illustrates the derived rate
constants kAM1AM (T), kA1A (T) and k PST(T) as well as the
temperature-dependent rigidity factors for the AM þ AM
and A þ A reactions. The results provide an internally
consistent interpretation in terms of the radical-complex
mechanism.
4140
P h y s . C h e m . C h e m . P h y s . , 2 0 0 4 , 6 , 4 1 3 3 – 4 1 4 1
T h i s j o u r n a l i s & T h e O w n e r S o c i e t i e s 2 0 0 4

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Acknowledgements
Financial support by the Alexander von Humboldt Founda-
tion (‘‘Sofja Kovalevskaja Program’’) and the Deutsche
Forschungsgemeinschaft (Sonderforschungsbereich 357 ‘‘Mo-
lekulare Mechanismen unimolekularer Prozesse’’) is gratefully
acknowledged. Discussions with V. G. Ushakov, T. Lenzer,
D. A. Wild, J. Schroeder, D. Schwarzer and M. Teubner were
also most helpful.
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J. Chem. Phys., 1998, 108, 5265.
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