The role of solvent friction in an orbital symmetry controlled reaction: Ring closure of a carbonyl ylide to cis-2,3-diphenyloxirane

Article abstract of DOI:10.1021/jp9801976

The dynamics of the orbital symmetry controlled ring closure of the trans-ylide, formed upon the 266 nm photolysis of trans-2,3-diphenyloxirane, to produce cis-2,3-diphenyloxirane is examined in a variety of n-alkane solvents as a function of temperature. An unsuccessful attempt was made to model the kinetics within the theoretical framework developed by Kramers for a one-dimensional reaction coordinate. A model developed by Grote and Hynes that employs a frequency-dependent friction was found to give a significantly better fit to the experimental data. The possibility that a multidimensional reaction coordinate is necessary to describe the reaction dynamics is discussed.

Full text of DOI:10.1021/jp9801976

J. Phys. Chem. A 1998, 102, 1691-1696
1691
The Role of Solvent Friction in an Orbital Symmetry Controlled Reaction: Ring Closure of
a Carbonyl Ylide to cis-2,3-Diphenyloxirane
Matthew Lipson and Kevin S. Peters*
Department of Chemistry and Biochemistry, UniVersity of Colorado, Boulder, Colorado 80309-0215
ReceiVed: NoVember 13, 1997; In Final Form: January 9, 1998
The dynamics of the orbital symmetry controlled ring closure of the trans-ylide, formed upon the 266 nm
photolysis of trans-2,3-diphenyloxirane, to produce cis-2,3-diphenyloxirane is examined in a variety of n-alkane
solvents as a function of temperature. An unsuccessful attempt was made to model the kinetics within the
theoretical framework developed by Kramers for a one-dimensional reaction coordinate. A model developed
by Grote and Hynes that employs a frequency-dependent friction was found to give a significantly better fit
to the experimental data. The possibility that a multidimensional reaction coordinate is necessary to describe
the reaction dynamics is discussed.
Introduction
SCHEME 1
The role that orbital symmetry has in controlling the
stereochemical outcome of organic reactions continues to be
an extensively explored subject, from both theoretical and
experimental perspectives. High-level theoretical calculations
have revealed the nature of transition states for a wide array of
pericyclic reactions that include sigmatropic shifts, electrocy-
clizations, cycloadditions, and cheletropic reactions.1 The effect
that solvent has on reaction paths has been explored through
Monte Carlo simulations.2 The nature of dynamics for the
reaction system passing through the transition state is now being
reaction coordinate by Kramers which relates the rate of a
chemical reaction to the frictional coupling between solute and
10
solvent. Qualitatively, at low pressure and hence low friction,
the dynamics of energy transfer from the solvent to the solute
is rate limiting, and the reaction rate increases with pressure.9
At the high pressure-high friction limit, passage along the
molecule’s reaction coordinate becomes rate limiting, and thus
an increase in friction decreases the rate of the reaction. In an
intermediate region of friction, the Kramers turnover region,
small changes in friction have little effect on the rate of the
,3
,11
4
pursued, and the suggestion that the stereoselectivity may be
dynamic in origin and not controlled by the conservation of
5,6
orbital symmetry has recently been presented.
Finally, the
real time dynamics for passage through the transition state for
a pericyclic reaction has now been achieved.7
Our interest in pericyclic reactions stems from our studies of
the effect that solvent has in controlling the dynamics of a wide
variety of organic reactions. We have recently shown that, in
alkane solvents, the thermal ring closure of the carbonyl ylide
11
chemical reaction.
The one great difficulty in the analysis of reaction dynamics
within the Kramer model has been in establishing a measure of
the solvent friction felt by the solute. For the excited-state
isomerization of trans-stilbene and related polyenes, the initial
attempts to relate the solvent friction to the solvent’s shear
viscosity led to a failure of the Kramers model in describing
(trans-ylide, Scheme 1) formed in 400 ps by ultraviolet
photolysis of trans-diphenyloxirane (TDPO) decays solely via
an orbital symmetry predicted pathway to form exclusively cis-
diphenyloxirane (CDPO).8 The decay of the trans-ylide in-
volves a conrotatory processes occurring on the 600 ns time
scale at room temperature.
12-15
the observed solvent dependence of the rate constant.
Microviscosities for a homologous solvent series, derived from
rotational reorientation times for trans-stilbene, were found to
One question that arises regarding the nature of the conro-
tatory process is how the dimensionality of the transition state
should be view as the system passes through the transition state.
Associated with the transition state is one vibrational coordinate
give significantly improved fits of the Kramers model to
15-17
observed reaction rates.
Finally, an excellent fit of the
Kramers model to stilbene’s dynamics in alkane solvents was
achieved when the microfriction coefficients were derived from
the translational diffusion coefficients for toluene in the
5
whose frequency is imaginary. Since there is one vibrational
coordinate that leads to passage through the transition state,
should the reaction coordinate then be viewed as effectively
one-dimensional or with the additional consideration of solvent
does passage through the transition state become inherently
multidimensional?
18-20
n-alkanes.
Troe and co-workers have suggested, on the basis of pressure
and temperature studies of stilbene isomerization, that the
intrinsic barrier for isomerization is solvent dependent for the
n-alkane homologous solvent series and that it is the sensitivity
of the intrinsic barrier to solvent that precludes the global fit of
the kinetic data in the n-alkane solvents to the Kramers model.21
This view that the intrinsic barrier for stilbene isomerization in
the alkane homologous solvent series is solvent dependent has
been challenged by Saltiel and co-workers as they find, on the
For the past 15 years there has been an enormous effort
undertaken in trying to understand the effect that solvent has
upon dynamics of reactions that are viewed as being one-
9
dimensional. The reaction kinetics have been analyzed within
the theoretical framework developed for a one-dimensional
S1089-5639(98)00197-2 CCC: $15.00 © 1998 American Chemical Society
Published on Web 02/12/1998

1692 J. Phys. Chem. A, Vol. 102, No. 10, 1998
Lipson and Peters
basis of the medium-enhanced barrier model, that the intrinsic
potential energy barrier is constant throughout the n-alkane
18
solvent series. This suggestion of solvent-independent isomer-
ization barrier is further supported by the Saltiel-Waldeck study
of the photoisomerization dynamics of trans-4,4′-dimethylstil-
bene, trans-4,4′-dimethoxystilbene, and trans-4,4′-di(tert-butyl)-
stilbene where they find that the intrinsic energy barriers for
the photoisomerization of these various substituted stilbenes are
constant in the n-alkane solvent series.20
Since the reaction pathway for the ring closure for the above-
mentioned ylide to form cis-diphenyloxirane involves the
diffusional motion of two phenyl groups through solution, a
process similar to the isomerization of trans-stilbene, we have
undertaken an investigation of the rate of ring closure for the
trans-ylide as a function of temperature in a series of alkane
solvents. Herein we present the solvent dependence of the rate
constants for the ring closure of the trans-ylide and the varying
success we have had in fitting our data to the Kramers model.
In the various fitting procedures employed in this study, we
assume that the intrinsic barrier for isomerization in the n-alkane
solvent series is independent of solvent.
Figure 1. Decay of the trans-ylide at 23.5 °C in heptane monitored at
470 nm following the 266 nm irradiation of trans-2,3-diphenyloxirane.
TABLE 1: First-Order Rate Constants for the Decay of the
trans-Ylide as a Function of Temperature and n-Alkane
Solventa
temp, °C C5
C6
C7
C8
C9
C10 C12 C14 C16
-
5.9
3.7
.8
3.21 3.23 3.00 2.78 2.56 2.20 2.25
3.45 3.57 3.47 3.21 2.97 2.66 2.39
4.66 4.83 4.15 3.66 3.65 3.51 3.13
6.17 5.50 5.19 4.55 4.54 4.29 4.06
7.91 7.36 6.50 6.16 6.35 5.73 5.40 5.19
9.99 9.38 8.60 7.87 6.96 7.04 7.14 6.56
-
0
4.7
0.0
4.4
1
1
1
2
2
3
3
Experimental Details
8.9 12.7 11.4 11.7 10.5
3.5 15.4 14.4 14.0 13.8 12.3 11.9 11.1 10.8 10.7
7.8 19.6 18.7 17.5 16.9 16.2 15.3 14.1 13.7 12.6
2.5
7.0
9.24 9.66 9.04 8.71
Solvent and Reagents. All alkanes (Aldrich, highest purity
available) used in this study were purified by passage through
silver nitrate impregnated alumina. Other methods of purifica-
tion such as reflux over sulfuric acid followed by distillation
from lithium aluminum hydride lead to irreproducible results.
trans-Diphenyloxirane from Aldrich (TDPO) was recrystallized
from hexanes.
Laser Flash Photolysis. The laser flash photolysis apparatus
is similar to many reported in the literature and will therefore
only be described briefly.22 Samples are excited by the fourth
harmonic of a Nd:YAG laser (Quanta-Ray DCR-2), 0.5 mJ/
pulse, 10 ns, and probed by a pulsed a high-pressure mercury
lamp. Probe light wavelength is selected by monochromator
23.8 21.4 21.4 20.3 19.6 18.6 17.4 17.4
29.6 27.5 25.8 25.0 24.3 23.7 22.8 21.8
38.8 34.3 32.1 30.1 31.0 28.7 27.1 27.5
45.7 45.1 39.4 37.4 38.0 34.7 35.6 33.4
41.8
41.2
45.9
5
5
0.0
4.8
50.5
a
Units are 1 × 10 s-1.
5
TABLE 2: Fitting Parameters from the Arrhenius Analysis
of Kinetic Data in Table 1
E
a
(kcal/mol)
ln A
A (s-1)
1
1
1
1
2
2
2
2
C5
C6
C7
C8
C9
C10
C12
C14
C16
8.63 ( 0.13
8.71 ( 0.15
8.76 ( 0.15
8.86 ( 0.15
8.82 ( 0.15
9.21 ( 0.09
9.16 ( 0.09
9.57 ( 0.11
9.83 ( 0.19
28.92 ( 0.24
29.03 ( 0.26
29.05 ( 0.27
29.15 ( 0.26
29.03 ( 0.27
29.66 ( 0.16
29.51 ( 0.16
30.15 ( 0.20
30.54 ( 0.31
3.64 × 10
4.05 × 10
4.11 × 10
4.58 × 10
4.04 × 10
(Instruments S.A. Inc., model H10) and detected as a function
of time by a photomultiplier (Electron Tubes, Inc., model 9816,
four dynodes wired). The output current of the photomultiplier
is dropped over 50 ohms to ground, and the resulting voltage
is measured by a digital oscilloscope (Tektronix, model
TDS350) and stored and processed on a Power Macintosh 7100/
12
7.59 × 1012
1
1
1
2
3
3
6.56 × 10
1.24 × 10
1.84 × 10
8
0.
For these studies, all kinetic traces were modeled as first-
to the data is less than (1% of the magnitude of the rate
constant. Uncertainty in the temperature for each point is (0.2
order decays, and fits to the data were minimized with the
Levenberg-Marquardt algorithm.
Samples were prepared so as to exhibit optical densities of
°
C, which translates into an uncertainty in rate constant at a
given temperature of approximately 1%. Examining Table 1,
one observes that the rate constant for the closure of the ylide
increases with temperature for a given solvent and decreases
with increasing solvent chain length at constant temperature.
An Arrhenius analysis of the experimental data presented in
Table 1 was undertaken, and the resulting A factor and Ea for
each solvent are given in Table 2. An example of an Arrhenius
plot is shown in Figure 2.
0.3-0.6 at 266 nm and were bubbled with argon for 10 min
prior to experiment to remove oxygen. In agreement with the
literature, atmospheric oxygen has no effect on the lifetimes or
quenching rates that are measured.2 The trans-ylide produced
upon photolysis at TDPO was probed at 470 nm. Typically,
the average of five laser pulses yielded acceptable signal to
noise. Samples were not stirred during the experiment. Sample
temperature was controlled by flowing thermostated water
through an aluminum block that surrounds the sample cuvette.
3
Discussion
Results
Kramers Model. In 1940, Kramers presented a model, based
upon the Langevin equation, for the escape of a Brownian
particle over a one-dimensional barrier. The rate expression
Figure 1 shows the decay of the trans-ylide at 23.5 °C in
heptane, monitored at 470 nm, following the 266 nm irradiation
of TDPO and the corresponding fit of the model assuming first-
order decay. Table 1 lists the first-order rate constants for the
decay of the trans-ylide as a function of solvent and temperature.
Two standard deviations in the uncertainty to the first-order fit
9
for this process takes the form
kobs
)
2
1/2
(
ω/2π)(â/2ω′){[1 + (2ω′/â) ] - 1} exp(-E /RT) (1)
0

Ring Closure of a Carbonyl Ylide
J. Phys. Chem. A, Vol. 102, No. 10, 1998 1693
Figure 2. Arrhenius analysis of the kinetics for the decay of the trans-
ylide in heptane over the temperature range -5.9 °C to 45.9 °C,
Figure 3. Isomicroviscosity analysis of the kinetic data for the decay
of the trans-ylide at 0.3 cP.
1
2
-1
resulting in A ) 4.11 × 10
s
a
and E ) 8.76 kcal/mol.
TABLE 3: Energy of Activation, E
0
, Derived from
Isoviscosity (η) and Isomicroviscosities (η
µ
)
where ω is the frequency of the parabolic reactant well and ω′
is the frequency associated with the curvature of the potential
energy surface at the transition state. The energy of the reaction
barrier is E0. The interaction of the particle with the medium
is described by the reduced friction coefficient, â, which is equal
to the ratio of the friction coefficient êr divided by the particle’s
moment of inertia, I.
One of the great difficulties in the application of the Kramers
model to reacting systems is the determination of the friction
coefficient êr for the system. In some of the earlier studies of
the dynamics of the isomerization of trans-stilbene and related
polyenes, it was assumed, employing the Stokes-Einstein
hydrodynamic equation, that the friction coefficient could be
η (cP)
E0 (kcal/mol)
ηµ (cP)
E0 (kcal/mol)
0
0
0.3
0.5
.1
.2
10.32 ( 0.53
9.30 ( 0.27
9.00 ( 0.21
8.72 ( 0.14
8.59 ( 0.09
8.45 ( 0.04
8.39 ( 0.05
0.1
0.2
0.3
0.5
1.0
2.0
4.0
9.29 ( 0.30
8.67 ( 0.13
8.47 ( 0.11
8.42 ( 0.06
8.29 ( 0.04
8.20 ( 0.05
8.17 ( 0.06
1
2
4
.0
.0
.0
is to examine the rate constant for a given process in a
homologous solvent series at various temperatures where the
temperatures are chosen so that the solvent friction remains
9
constant throughout the temperature range. If the solvent
9
,12,24
related to the solvents shear viscosity ηs, êr ∼ ηs.
However,
friction is assumed to be proportional to the solvent viscosity
or microviscosity, then an isoviscosity plot will yield the intrinsic
energy of activation. A fundamental assumption in this analysis
is that E0 does not change in the homologous solvent series so
that the derived E0 should be independent of the viscosity chosen
for the isoviscosity plot.
this assumption was found to lead to a poor fit of the Kramers
model to the experimental data, leading to the conclusion that
macroscopic shear viscosity is an inadequate description for the
1
3
solute’s interaction with the solvent at the molecular level.
Substantially improved fits for the Kramers model were found
when the friction coefficient êr was related to the friction
coefficients derived from the measurement of the solvent
In our attempt to model the rate of ring closure for the ylide
within Kramers formulation, we assume that solvent friction
can be related to either the shear viscosity or the microviscosity
derived from the translation diffusion of toluene in the n-alkane
solvents, following the method of analysis developed by Saltiel
15,17
dependence of the molecule’s rotational reorientation times.
Saltiel and co-workers have recently developed a microvis-
cosity model for the measure of the solvent friction for the
isomerization of trans-stilbene in alkane solvents which is
derived from translational diffusion coefficients of toluene in
the n-alkane solvents.19 Although the translational diffusion
coefficient is proportional to the solvent’s viscosity, the
proportionality constant will differ from solvent to solvent. Thus,
the microviscosity, ηµ, can be related to the shear viscosity, ηs,
through a proportionality constant f which is obtained from the
ratio of diffusion coefficient from the Stokes-Einstein equation
assuming stick boundary conditions, DSE, and the experimental
diffusion coefficient, Dexp, f ) DSE/Dexp.
19
for trans-stilbene isomerization. The temperature dependence
of the microviscosity ηµ was derived from the temperature
dependence of the shear viscosity ηs and from the assumption
that the f factor (f ) DSE/Dexp) is temperature independent. The
temperature dependence of the shear viscosity is calculated from
9
the Andrade equation,
1/η ) A exp(-E /RT)
(4)
η
η
The results of both the isoviscosity and isomicroviscosity are
given in Table 3, and a plot of the isomicroviscosity for 0.3 cP
is shown in Figure 3. The kinetics for the isoviscosity plots
were obtained from the extrapolation of the temperature
dependence of the kinetic data assuming an Arrhenius type of
behavior. Surprisingly, for the isoviscosity analysis the derived
energy of activation decreases from 10.3 kcal/mol for 0.1 cP to
8.4 kcal/mol for 4.0 cP. Similarly, the isomicroviscosity plots
yield an energy of activation of 9.2 kcal/mol, which decreases
to 8.2 kcal/mol for a change in microviscosity of 0.1 to 4.0.
For a Kramers analysis of a kinetic process that employs a
homologous solvent series, such as the n-alkanes, as a means
of systematically varying the solvent friction, one assumes that
the energy of the reaction barrier does not change with solvent.
Clearly this is not observed in the present study of ylide ring
closure in n-alkane solvents. Therefore, either the viscosity or
microviscosity is not a good measure of the solvent friction felt
ηµ ) fη
s
(2)
Employing microviscosities, they found that the Kramers model
gives an excellent fit to the kinetics for the isomerization of
trans-stilbene in a wide range of n-alkane solvents.
The experimental rate constants can be modeled by the
Kramers equation by assuming that the rate expression has the
form9
kobs ) F(η) exp(-E /RT)
(3)
0
where F(η ) is the dynamical factor, often termed the “reduced
isomerization rate”. Thus, for a Kramers analysis, it is necessary
to isolate the reduced isomerization rate by determining E0. A
common method for obtaining the intrinsic energy of activation

1694 J. Phys. Chem. A, Vol. 102, No. 10, 1998
Lipson and Peters
forces acting on the isomerization coordinate should relax on
the time scale associated with the passage of the reacting
molecule through the transition state so that the friction on the
reacting system becomes frequency dependent; thus, the friction
is a sensitive function of the nature of the intrinsic reaction
2
5
barrier. This would suggest that even though the microvis-
cosity derived from diffusion coefficients of toluene appears to
be a good measure of the friction encountered in the isomer-
ization of trans-stilbene, a process whose intrinsic barrier is 2.8
kcal/mol in the n-alkane solvents, the same microviscosities are
not transferable as a measure of friction in the ring closure of
the trans-ylide, a process whose intrinsic barrier is of the order
1
8
of 8-9 kcal/mol.
The theoretical model employing a frequency-dependent
friction developed by Grote and Hynes has been applied to
several reacting systems. In 1982, Bagchi and Oxtoby used a
hydrodynamic model for the frequency-dependent friction to
fit the experimental values for the rate of photoisomerization
of trans-diphenylbutadiene and DODCI as a function of
2
6
temperature in a variety of solvents. Velsko, Waldeck, and
Fleming have examined the breakdown of Kramers theory for
the photochemical isomerization of DODCI within the frame-
27
work of a frequency-dependent friction. Finally, Hochstrasser
and co-workers have applied the Grote-Hynes model to their
experimental studies of the rate of photoisomerization of trans-
Figure 4. (A) Kramers analysis of kinetic data given in Table 1
employing solvent shear viscosity, η. Resulting parameters E ) 9.06
, and B ) 14.2 P . (B) Kramers analysis
of kinetic data given in Table 1 employing microviscosity, η . Resulting
, and B ) 65.4
0
2
8
1
2
-1
-1
stilbene.
kcal/mol, A ) 6.99 × 10
s
Since the application of the Kramers model, employing a
hydrodynamic model for the friction, fails to account for the
observed solvent dependence of the rate for ring closure of the
trans-ylide, we have analyzed our experimental data within
µ
1
2
-1
parameters E
0
) 8.88 kcal/mol, A ) 5.67 × 10
s
-
1
P
.
by the ring closure process or the Kramers model is not
applicable to the observed kinetic process. It is also possible
that the energy of activation is dependent upon the chain length
of the alkane solvent.21
25
Grote-Hynes theory employing the hydrodynamic model
developed by Bagchi and Oxtoby for the frequency-dependent
26
friction. The rate constant for reaction, k, given by Grote-
Hynes theory is
Instead of deducing the energy of activation, E0, from
isoviscosity plots in order that the reduced isomerization rate
may be obtained, it is feasible to fit the Kramers model given
in eq 1 by not only varying the parameters in the preexponential
term but also allowing E0 to vary as well. It is convenient to
TST
k ) k (λ /ω )
(6)
r
b
where kTST is the rate constant for the reaction predicted by
transition-state theory,
9
reformulate eq 1 so that the observed rate is expressed as
TST
k
) (ω /2π) exp(-E /RT)
(7)
R
0
2
1/2
kobs ) A[(1 + (ηB) ) - ηB] exp(-E /RT)
(5)
0
and ωR is the frequency in the reactant well, ωb is the reaction
-
1
where A ) ω0/2π and ηB ) (2ω′τv) . The term η can be
either the shear viscosity or the microviscosity for the present
analysis, and the velocity relaxation τv is given by τv ) µ/ê
where ê is the friction coefficient and µ is the reduced mass.
Since a single activation barrier is not obtained in the
isoviscosity analysis, we have attempted to fit our data to eq 5
by varying A, B, and E0 to minimize the square of the residuals.
The fit is shown in Figure 4 where the reduced isomerization
rate F(η) is obtained from experimental data and the parameter
E0, F(η) ) kobs exp(E0/RT). Clearly the model cannot account
for the curvature in the correlation of F(η) with η. Also, the
range in the scatter of the data at a given viscosity or
microviscosity is substantially greater that the error in the
experimental data. This scatter presumably results from the
assumption in this last fitting procedure that E0 is independent
of the solvent.
barrier frequency, and λ is reactive frequency given by
r
2
b
λ ) ω /(λ + ú(λ )/µ)
(8)
r
r
r
The term ú(λr) is the Laplace transform of the time-dependent
friction, and µ is the moment of inertia of the twisting group.
For the isomerization of the trans-ylide, the phenyl groups
will experience a friction arising from the sum of two sources:
2
a rotational friction, úr, and a translation friction, (R + l) útr,
where R is the hydrodynamic radius of the phenyl group and l
is the distance between the oxygen atom and the carbon atom
2
6
to which the phenyl group is attached.
An expression for the frequency dependence of the transla-
29
tional friction, útr(p), was derived by Zwanzig and Bixon and
30
modified by Metiu, Oxtoby and Freed. Similarly, an expres-
sion for the frequency dependence of the rotational friction, úr-
31
Grote-Hynes Model. The apparent breakdown of the
Kramers model for describing the frictional dependence of the
rate for ring closure in trans-ylide may possibly be traced to
the assumption that either viscosity or microviscosity is a
measure of the solvent friction felt by the isomerizing molecule
in the transition state. Grote and Hynes have suggested that
(p), was derived by Berne and Montgomery. The expressions
for útr(p) and úr(p) depend on the frequency-dependent shear
viscosity, ηs(p), the frequency-dependent bulk viscosity, ηv(p),
the solvent density, F0, the velocity of sound, c, the slip
parameter, â, and hydrodynamic radius, R. The frequency
dependence of the shear viscosity, ηs(p), and the bulk viscosity,

Ring Closure of a Carbonyl Ylide
J. Phys. Chem. A, Vol. 102, No. 10, 1998 1695
Even though the Grote-Hynes model gives a superior fit to
the experimental data, there is still substantial deviation between
the experiment and the model where the deviation is larger than
the experimental error in the data. One source of error in the
modeling is the assumption of the Maxwell form for the
frequency dependence of the bulk and shear viscosities since
there are no measurements of the viscoelastic response for the
solvents at the reactant frequencies.
Another possibility for the breakdown of the Kramers theory
or Grote-Hynes theory is that the reaction coordinate for the
ring closure of the trans-ylide is inherently multidimensional
so that modeling the reaction as a process corresponding to the
escape of a Brownian particle over a one-dimensional barrier
becomes inappropriate. Models for the dynamics of two-
dimensional diffusional barrier crossings have recently been
3
8
presented. One could imagine that in the absence of solvent
the intrinsic reaction path connecting the trans-ylide to CDPO
has the phenyl rings rotating with the methylene groups such
that the charge/radical character that might otherwise be
localized on the methylene carbons may be delocalized into the
phenyl π-orbitals. When placed in a solvent, this intrinsic
reaction path may require the phenyl rings to move a maximum
number of solvent molecules out of the path, and so, as a
function of solvent, the phenyl rings may rotate so that they
slice more through the solvent, even through this lessens the
conjugation between the methylene groups and the phenyl rings
at the transition-state geometry. Thus, the intrinsic energy of
activation becomes solvent dependent, reflecting different
reaction paths through the transition state.
Figure 5. (A) Grote-Hynes analysis of the kinetic data given in Table
1
where the data for pentane, tetradecane, and hexadecane are removed.
1
3
-1
13 -1
The fitting parameters are ω
R
) 6.6 × 10
s
, ω
b
) 1.2 × 10
s ,
and E
0
) 8.9 kcal/mol. f(ú) and η
v
are defined in the text. (B) Kramers
analysis of kinetic data given in Table 1 employing solvent shear
viscosity, η, where the data for pentane, tetradecane, and hexadecane
are removed. Resulting parameters E
0
) 9.06 kcal/mol, A ) 6.99 ×
1
2
-1
-1
1
0 s , and B ) 14.2 P .
Conclusions
ηv(p), is assumed to be given by the Maxwell forms
The original aim of this study was to ascertain whether the
Kramers model would be effective in describing the reaction
dynamics for ring closure of the trans-ylide to form CDPO.
We found that the data could not be modeled within the Kramers
framework and that it was necessary to employ Grote-Hynes
theory to give a significantly better fit to the experimental data.
However, the fit is still less than ideal, perhaps reflecting that
the reaction coordinate cannot be model as a one-dimensional
process. In the future we plan to examined ring closure of ylides
whose phenyl groups are modified so as to prevent the rotation
of phenyl groups with respect to the methylene centers during
the ring closure process, thus reducing the dimensionality of
the reaction coordinate.
0
s
η (p) ) η /(1 + pτ )
s
s
0
v
η (p) ) η /(1 + pτ )
(9)
v
v
0
0
where ηs is the zero-frequency shear viscosity and ηv is the
zero-frequency bulk viscosity. The times τs and τv are the
viscoelastic relaxation times. For the detail expressions for útr-
p), úr(p), τs, and τv, see the work of Bagchi and Oxtoby.
The values of the solvent parameterssshear viscosity, bulk
viscosity, speed of sound, density, and their associated temper-
ature dependenciesscan be found in refs 32-36. Values for
the solvents pentane, tetradecane, and hexadecane could not be
found, and thus the data for these solvents were removed from
the gobal analysis. The remaining quantities R, µ, and l were
obtained molecular mechanic calculations producing the values
2
6
(
Acknowledgment. This work is supported by a grant from
the National Science Foundation, CHE 9408354.
-
8
2
R ) 2.6 Å, l ) 1.8 Å, and µ ) 9.7 × 10 g cm .
References and Notes
Employing a Simplex nonlinear least-squares routine to vary
37
ωR, ωb, and E0, the three fitting parameters, we found that
(
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13
-1
13 -1
ωR ) 6.6 × 10 s , ωb ) 1.2 × 10 s , and E0 ) 8.9 kcal/
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A where the reduced rate constant defined by
(
(
(
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(
(
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(10)
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r
b
1
18, 8755-8756.
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