Photoinduced Homolysis of Benzhydryl Acetates
A R T I C L E S
Table 2. Electronic Energy of Interaction, ∆Eint, within the
description captures the essence of the Kim-Hynes theoretical
formulation for the electronic structure of the geminate radical
pair.
Geminate Radical Pair for 4-CH3O, 4-CH3, 4-H, 4-CF3, and 4-CN
as a Function of Solvent
solvent
acetonitrile
propionitrile
butyronitrile
The energy difference between the geminate radical pair and
its solvent separated form, defined here as the free radicals, has
yet to be established directly. However, consideration of the
rate constants for the diffusional separation of the GRP to the
FR, kesc, provides insight into the energy of interaction. If a
model for the diffusional separation rate of the GRP, in which
there is no electronic interaction within the GRP, can be
developed, then comparison of this hypothetical rate constant
with the series of observed kesc will provide an estimate of the
energy of interaction. Gardiner developed a theoretical model
for the diffusional separation of a noninteracting geminate
radical pair; the rate constant is given by
∆
Eint
∆
Eint
∆
Eint
4′
-substituent
(kcal/mol)
(kcal/mol)
(kcal/mol)
CH3O
CH3
H
CF3
CN
2.0
1.9
1.9
0.4
0.9
2.1
1.8
1.7
0.4
0.5
1.8
1.5
1.4
0.4
0.3
diffusional separation and assuming a common A factor for the
separation, the ratio of the experimental rate constant, kesc(expt),
to the modeled noninteracting rate constant, kesc(calcd), yields
the energy of interaction, ∆Eint, defined as a positive quantity.
3(DA + DB)
(rA + rB)2
kesc(expt)
kesc
)
(4)
) exp(-∆Eint/RT)
(5)
kesc(calcd)
where DA and DB are the diffusion coefficients for the two
radical fragments, and rA and rB are the radii of the two
fragments.30 This hypothetical rate constant represents an upper
bound for the rate constant associated with the diffusional
separation of a geminate radical pair.
The values of ∆Eint are given in Table 2 for the solvents
acetonitrile, propionitrile (0.42 cps), and butyronitrile (0.58 cps)
where the viscosity dependence of kesc(calcd) has been taken
into account.35 As predicted by the diagram in Scheme 4, when
electron withdrawing substituents are replaced by electron
donating substituents, leading to a decrease in the distance of
the crossing points between the diabatic surfaces, there is an
increase in energy of the well depth associated with the GRP.
Relative to a noninteracting GRP, the energy of interaction
within the GRP is not large, only ranging from 0.4 kcal/mol
(CF3) to 2.0 kcal/mol (MeO) in acetonitrile. The effect of this
small variation in the energy of interaction leads to a change in
kesc by a factor of 15.
Electronic Coupling for the Diabatic Surfaces. The Kim-
Hynes study of the SN1 reaction mechanism cast in terms of
valence bond reaction diagrams reveals that as the ionic surface
is stabilized relative to the covalent surface, the position of the
transition state for bond heterolysis decreases and, importantly,
the electronic coupling between the two diabatic surfaces
increases leading to a larger splitting between the two adiabatic
surfaces, Scheme 4. For the series tert-butyl iodide, tert-butyl
bromide, and tert-butyl chloride in acetonitrile, as the position
of the transition state for bond heterolysis decreases (2.83, 2.65,
and 2.49 Å), the splitting between the two adiabatic surfaces,
2â, increases (8.1, 27.8, and 35.4 kcal/mol).28
Combining the information obtained in the present study with
our prior studies, we calculate the energy separation between
the GRP and transition state for bond heterolysis, the difference
reflecting the value of 2â. In ref 18, we present the methodology
for determining the energy of the contact ion pair (CIP) and
the free radical pair (FR) for 4-CH3O, 4-CH3, and 4-H. The
same procedure is used herein for 4-CF3 and 4-CN. In the same
study, from the temperature dependence of the rate constant
for the collapse of the contact ion pair to form the covalent
bond, the energy of the transition state for bond heterolysis of
4-CH3O, 4-CH3, and 4-H is determined. Finally, the energy of
the GRP is obtained from the calculated energy of the FR and
the stabilization energy given in Table 2. The energies for the
various species in acetonitrile along with the energy for the
The diffusion coefficient of the benzhydryl radical in n-hexane
has been determined. Arita and co-workers measured the kinetics
for the self-coupling of the benzhydrylium radical; analysis of
the kinetics within the Smoluchowski theory for diffusion-
controlled reactions yields a diffusion coefficient of 1.7 × 10-9
m2 s-1 31
. Since the diffusion coefficient is inversely proportional
to the viscosity, the diffusion coefficient in acetonitrile is
estimated to be 1.5 × 10-9 m2 s-1 given the viscosities of
acetonitrile (0.36 cps) and n-hexane (0.30 cps).32 While the
diffusion coefficient for the acetate radical has not been
measured, the diffusion coefficient of acetic acid should be
similar.33,34 In ethyl acetate (0.441 cps), the value is 2.18 ×
10-9 m2 s-1 leading to an extrapolated diffusion coefficient of
2.7 × 10-9 m2 s-1 in acetonitrile. An estimate of the radii for
the two radical species is made by calculating the molecular
volume of the fragments, obtained using Spartan at the AM1
level, and then calculating the radius of a sphere for the same
volume. The calculation leads to radii of 3.5 Å for the
benzhydrylium radical and 2.28 Å for the acetate radical. Based
on eq 4, the hypothetical rate constant for the diffusional
separation of the noninteracting geminate radical pair is 3.9 ×
1010 s-1
.
If it is assumed that the difference between the hypothetical
rate constant for diffusional separation of noninteracting gemi-
nate radical pairs and the experimentally observed rate constants
for diffusional separation is traced to the electronic interaction
within the geminate radical pair, then the energy of interaction
can be estimated. Assuming that there is no electronic interaction
between the radicals within the transition state for GRP
(30) Gardiner, W. C. Rates and Mechanisms of Chemical Reactions; W.A.
Benjamin, Inc: New York, 1969.
(31) Arita, T.; Kajimoto, O.; Terazima, M. J. Chem. Phys. 2004, 120, 7071-
7074.
(32) CRC Handbook of Chemistry and Physics; Lide, D. R., Ed; CRC Press:
Boca Raton, FL, 2001.
(33) Terazima, M.; Tenma, S.; Watanabe, H.; Tominaga, T. J. Chem. Soc.,
Faraday Trans. 1996, 92, 3057-3062.
(34) Lewis, J. B. J. Appl. Chem. 1955, 5, 228-237.
(35) Li, B.; Peters, K. S. J. Phys. Chem. 1993, 97, 7648-7651.
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