Application of non-steady-state kinetics to resolve the kinetics of proton-transfer reactions between methylarene radical cations and pyridine bases

Article abstract of DOI:10.1021/ja982682k

Apparent deuterium kinetic isotope effects (KIE(app)) of four different methylarene radical cation-pyridine base reactions in dichloromethane (0.2 M tetrabutylammonium hexafluorophosphate) were observed to increase toward a constant value with increasing extent of reaction. The reactions were studied by derivative cyclic voltammetry (DCV), and rate constants were assigned by comparing the experimental with the theoretical DCV data. The kinetic results rule out a simple second-order proton-transfer reaction and implicate a mechanism in which a complex is first formed that then undergoes proton transfer, followed by separation of the products. That KIE(app) are extent of reaction-dependent is observed before steady-state is reached. The concurrent analysis of kinetic data for the reactions of both ArCH₃(·+) and ArCD₃(·+) with bases under non-steady-state conditions facilitates the resolution of the apparent rate constant [k(app) = k(f)k(p)/(k(b) + k(p))] into the microscopic rate constants (k(f), k(b), and k(p)) for the individual steps. The KIE(app) observed during proton-transfer reactions need not be the real kinetic isotope effects (KIE(real)). Having access to the microscopic rate constants for the steps in which the proton is transferred allows KIE(real) to be evaluated and compared with the corresponding KIE(app). The present study shows that the KIE(real) are much greater than the KIE(app) derived in the usual way from the rate of the overall reaction.

Full text of DOI:10.1021/ja982682k

12720
J. Am. Chem. Soc. 1998, 120, 12720-12727
Application of Non-Steady-State Kinetics to Resolve the Kinetics of
Proton-Transfer Reactions between Methylarene Radical Cations and
Pyridine Bases
Vernon D. Parker,* Yixing Zhao, Yun Lu, and Gang Zheng
Contribution from the Department of Chemistry and Biochemistry, Utah State UniVersity,
Logan, Utah 84322-0300
ReceiVed July 28, 1998
Abstract: Apparent deuterium kinetic isotope effects (KIEapp) of four different methylarene radical cation-
pyridine base reactions in dichloromethane (0.2 M tetrabutylammonium hexafluorophosphate) were observed
to increase toward a constant value with increasing extent of reaction. The reactions were studied by derivative
cyclic voltammetry (DCV), and rate constants were assigned by comparing the experimental with the theoretical
DCV data. The kinetic results rule out a simple second-order proton-transfer reaction and implicate a mechanism
in which a complex is first formed that then undergoes proton transfer, followed by separation of the products.
That KIEapp are extent of reaction-dependent is observed before steady-state is reached. The concurrent analysis
•+
•+
of kinetic data for the reactions of both ArCH3 and ArCD3 with bases under non-steady-state conditions
facilitates the resolution of the apparent rate constant [kapp ) kfkp/(kb + kp)] into the microscopic rate constants
(kf, kb, and kp) for the individual steps. The KIEapp observed during proton-transfer reactions need not be the
real kinetic isotope effects (KIEreal). Having access to the microscopic rate constants for the steps in which the
proton is transferred allows KIEreal to be evaluated and compared with the corresponding KIEapp. The present
study shows that the KIEreal are much greater than the KIEapp derived in the usual way from the rate of the
overall reaction.
Introduction
comparable magnitude, it is feasible to resolve kapp into the rate
constants for the individual microscopic steps.
The classical work of Brønsted,1 Eigen,2 and Bell3 as well
as diverse contributions of a host of other workers4 over the
past 75 years has provided an immense foundation of knowledge
on the most-studied class of chemical reactions, proton transfer
from acids to bases. A three-step mechanism for proton
transfers(i) the formation of an encounter complex, (ii) proton
transfer to form a new complex, and (iii) separation to productss
generally accepted from the work of Eigen.2 In fact, pointed
out by Eigen, it is not a simple matter to determine which of
the three steps is rate-determining. Resolution of the kinetics
of these reactions, with the assignment of rate constants for all
steps, has never been accomplished.
Proton-transfer reactions involving C-H bond cleavage most
often take place at rates very much lower than expected for the
formation of encounter complexes. Reactions that follow the
Eigen mechanism under these conditions are expected to behave
kinetically as simple second-order reactions, with apparent rate
constants (kapp) reflecting both the equilibrium constant (Keq)
for the formation of the encounter complex and the rate constant
for proton transfer (kp).
Most organic radical cations are exceedingly strong acids and
some are superacids.5-10 This fact suggests that when HA is a
radical cation and B is a relatively strong base, such as pyridine,
the proton-transfer reaction will be thermodynamically favor-
able. The kinetics of several radical cation-base reactions have
been reported.11 Many of these reactions are believed12,13 to pass
(1) Brønsted, J. N.; Petersen, K. Z. Phys. Chim. 1924, 108, 185.
(2) Eigen, M. Angew. Chem., Int. Ed. Engl. 1964, 3, 1.
(3) Bell, R. P. The Proton in Chemistry; (Chapman & Hall: London,
1973).
(4) A review of much of the early work can be found in ref 3.
(5) Nicholas, A. M.; Arnold, D. R. Can. J. Chem. 1982, 60, 2165.
(6) Nicholas, A. M.; Boyd, R. J.; Arnold, D. R. Can. J. Chem. 1982, 60,
3011.
(7) Bordwell, F. G.; Cheng, J.-P. J. Am. Chem. Soc. 1989, 111, 1792.
(8) Zhang, X.; Bordwell, F. G. J. Org. Chem. 1992, 57, 4163.
(9) Zhang, X.; Bordwell, F. G.; Bares, J. E.; Cheng, J.-P.; Petrie, B. C.
J. Org. Chem. 1993, 58, 3051.
(10) Cheng, J.-P. Ph.D. Dissertation, Northwestern University: Evanston,
IL, 1987.
(11) Schlesener, C. J.; Amatore, C.; Kochi, J. K. J. Am. Chem. Soc. 1984,
106, 3567; Schlesener, C. J.; Amatore, C.; Kochi, J. K. J. Am. Chem. Soc.
1984, 106, 7472; Parker, V. D.; Tilset, M. J. Am. Chem. Soc. 1986, 108,
6371; Tolbert, L. M.; Khanna, R. K. J. Am. Chem. Soc. 1987, 109, 3477;
Dinnocenzo, J. P.; Banach, T. E. J. Am. Chem. Soc. 1989, 111, 8646;
Tolbert, L. M.; Khanna, R. K.; Popp, A. E.; Gelbaum, L.; Bettomly, L. A.
J. Am. Chem. Soc. 1990, 112, 2373; Reitsto¨en, B.; Parker, V. D. J. Am.
Chem. Soc. 1990, 112, 4968; Baciocchi, E.; Mattioli, M.; Romano, R.;
Ruzziconi, R. J. Org. Chem. 1991, 56, 7154; Xu, W.; Mariano, P. S. J.
Am. Chem. Soc. 1991, 113, 1431; Steadman, J.; Syage, J. A. J. Am. Chem.
Soc. 1991, 113, 6786; Xu, W.; Zhang, X.-M.; Mariano, P. S. J. Am. Chem.
Soc. 1991, 113, 8863; Anne, A.; Hapiot, P.; Moiroux, J.; Neta, P.; Saveant,
J.-M. J. Am. Chem. Soc. 1992, 114, 4694; Baciocchi, E.; Giacco, T. D.;
Elisei, F. J. Am. Chem. Soc. 1993, 115, 12290; Dinnocenzo, J. P.; Karki,
S. B.; Jones, J. P. J. Am. Chem. Soc. 1993, 115, 7111; Xue, J.-Y.; Parker,
V. D. J. Org. Chem. 1994, 59, 6564.
We now report kinetic studies of a series of reactions of
methylarene radical cations involving C-H bond cleavage that
suggest the formation (kf) of longer-lived complexes that
partition between complex dissociation (kb) and irreversible
proton transfer (kp) as illustrated in eq 1. Unlike the case
kf
kp
9
HA + B y
\
kz HA/B
8 A-/BH+
(1)
b
discussed in the previous paragraph, when kp and kb are of
10.1021/ja982682k CCC: $15.00 © 1998 American Chemical Society
Published on Web 11/25/1998

Non-Steady-State Kinetics in Proton-Transfer Reactions
J. Am. Chem. Soc., Vol. 120, No. 49, 1998 12721
Chart 1
through intermediates, suggested to be π-complexes, within
which proton transfer takes place. Previous work provided
indirect evidence, including negative apparent activation ener-
gies12 and kinetic isotope effects (KIE),13 which are readily
explained by the complex mechanism.
Results
The structures of the radical cations for 9,10-dimethylan-
thracene (DMA•+) and 9-methyl-10-phenylanthracene (MPA•+)
and bases (LUT) and 2,6-diphenylpyridine (DPP) studied are
shown in Chart 1. The overall reactions, in which the proton-
transfer steps are rate determining, are illustrated by general
stoichiometric eq 2.
Figure 1. Experimental DCV data for the reactions of DMA•+ (n)
and DMA-d6 (B) with LUT as a function of voltage sweep rate.
[DMA]o ) 2.0 mM, [LUT]o ) 4.0 mM. The dashed lines are theoretical
data for a simple second-order proton-transfer reaction with kapp ) 4820
M-1 s-1 (DMA•+) and 650 M-1 s-1 (DMA-d6•+).
•+
Table 1. Kinetic Data for the Reaction between DMA•+ and
•+
DMA-d6 with DPP and LUT in Dichloromethane-Bu4NPF6 (0.2
2ArCH3•+ + 2B f ArCH2-B+ + BH+ + ArCH3 (2)
M)a
DPP (50 mM)b
LUT (4.0 mM)c
Products of the reactions between both of the radical cations
and LUT have previously been characterized.13,14
R′I νH (V/s)d νD (V/s)d KIEapp νH (V/s)d νD (V/s)d KIEapp
Our kinetic studies were carried out with derivative cyclic
voltammetry (DCV), a refinement15 of the widely used elec-
trochemical transient technique, cyclic voltammetry.16 DCV has
been used extensively in the study of the kinetics of the reactions
of radical ions in solution.12,13,17 The magnitude of the ratio of
the derivative peaks (R′I ) I′b/I′f) is a measure of the rate of
reaction of the electrode-generated intermediate, ranging from
1.0 for no reaction to 0 for complete reaction during the time
of the experiment. The experimental variable is ν, the voltage
sweep rate. The experimental R′I vs ν-1 response curves may
be compared with theoretical data obtained by digital simula-
tion18 to evaluate rate constants of the homogeneous chemical
reactions coupled to the charge-transfer reaction of the substrate.
Experimental DCV Data for Radical Cation Proton-
Transfer Reactions. DCV data (R′I ranging from 0.85 to 0.50)
were obtained for the reactions of DMA•+ (and DMA-d6•+) with
LUT in dichloromethane-tetrabutylammonium hexafluorophos-
phate (Bu4NPF6) (0.2 M) at 291 K. The two data sets are shown
in Figure 1. The experimental data deviate significantly from
the dashed lines, which are the theoretical data obtained by
digital simulations for a simple second-order proton-transfer
mechanism. Data for the reactions of these radical cations with
both LUT and DPP are summarized in Table 1. Values for the
apparent deuterium kinetic isotope effects (KIEapp) were ob-
tained from (νH)exp/(νD)exp ratios, which are equivalent to (kH/
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
47.5
34.0
25.0
19.5
15.2
11.6
4.02
2.78
1.97
1.50
11.8
12.2
12.7
13.0
13.7
13.7
14.3
14.6
25.6
20.5
14.4
10.9
4.13
2.87
2.05
1.48
6.21
7.13
7.01
7.34
7.31
7.41
7.75
7.62
1.11
7.90
1.08
0.852
0.615
0.470
5.78
4.26
2.82
0.778
0.547
0.369
8.88
6.87
a DCV measurements at 291 K. b Substrate concentration 1.0 mM.
c Substrate concentration 2.0 mM. d Each data point obtained from 18
to 60 voltammograms.
Table 2. Kinetic Data for the Reaction between MPA•+ and
•+
MPA-d3 with DPP and LUT in Dichloromethane-Bu4NPF6 (0.2
M)a
DPP (50 mM)b
LUT (2.0 mM)b
R′I νH (V/s)c νD (V/s)c KIEapp νH (V/s)c νD (V/s)c KIEapp
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
36.9
27.2
21.6
16.7
12.1
9.59
7.70
5.53
4.24
2.72
1.92
1.49
8.7
10.0
11.2
11.2
11.1
11.3
11.8
11.8
13.5
9.27
2.54
1.68
1.09
0.816
0.577
0.433
0.293
0.195
5.3
5.5
6.2
6.2
6.7
6.8
7.1
7.3
6.72
5.04
3.85
2.94
2.08
1.43
1.09
0.850
0.653
0.470
a DCV measurements at 291 K. b Substrate concentration 1.0 mM.
c Each data point obtained from 18 to 60 voltammograms.
(12) Reitstøen, B.; Parker, V. D. J. Am. Chem. Soc. 1990, 112, 4968;
Xue, J.-Y.; Parker, V. D. J. Org. Chem. 1994, 59, 6564.
(13) Parker, V. D.; Chao, Y.-T.; Zheng, G. J. Am. Chem. Soc. 1997,
119, 11390.
(14) Chao, Y.-T. Master of Science Thesis, Utah State University: Logan,
VT, 1993.
(15) Parker, V. D. Electroanal. Chem. 1983, 19, 131; Ahlberg, E.; Parker,
V. D. J. Electroanal. Chem. 1981, 121, 57.
(16) Bard, A. J.; Faulkner, L. Electrochemical Methods; Wiley: New
York, 1980.
(17) Reitstøen, B.; Norrsell, F.; Parker, V. D. J. Am. Chem. Soc. 1989,
111, 8463; Parker, V. D.; Reitstøen, B.; Tilset, M. J. Phys. Org. Chem.
1989, 2, 580; Reitstøen, B.; Parker, V. D. J. Am. Chem. Soc. 1991, 113,
6954; Parker, V. D.; Handoo, K. L.; Reitstøen, B. J. Am. Chem. Soc. 1991,
113, 6218; Parker, V. D.; Tilset, M. J. Am. Chem. Soc. 1991, 113, 8778.
(18) Rudolph, M.; Reddy, D. P.; Feldberg, S. W. Anal. Chem. 1994, 66,
589A and references therein.
kD)app values.15 Comparable data for the reactions of MPA•+
and MPA-d3 with LUT and DPP are tabulated in Table 2.
•+
Apparent second-order rate constants (kapp) for all reactions,
calculated from the comparison of experimental DCV data (R′I
) 0.50 and 0.55) with those from digital simulation15 assuming
a simple second-order proton-transfer mechanism, are gathered
in Table 3.
The most interesting feature of the data for all four reactions
is that KIEapp vary with R′I and approach constant values at
about R′I ) 0.50 (at this R′I, the reverse peak on the cyclic
voltammogram is still visible). The expected result for a single-
step second-order proton-transfer reaction is that KIEapp will
be a constant, independent of the extent of reaction (proportional

12722 J. Am. Chem. Soc., Vol. 120, No. 49, 1998
Parker et al.
Table 3. Apparent Second-Order Rate Constantsa for Radical
Cation Proton-Transfer Reactions
eqs 8 and 9 we have kpH/kb ) CH/(1 - CH) and kpD/kb ) CD/
(1 - CD). The value of the real kinetic isotope effects is then
given by eq 10. Thus, in addition to the apparent rate constants,
reactants
kappH (M-1 s-1
)
kappD (M-1 s-1
)
DMA•+/DPP
DMA•+/LUT
MPA•+/DPP
MPA•+/LUT
601
4820
485
44.3
650
44.0
651
KIEreal ) [CH/(1 - CH)]/[CD/(1 - CD)]
(10)
4688
it is necessary only to determine kf in order to evaluate KIEreal
.
a Calculated from R′I ) 0.50 data.
H
The lower limit of kf is (kapp )s.s., since the right-hand side of
eq 8 can approach but cannot exceed unity as kpH/kb becomes
large. The practical upper limit of kf where the reaction rate
Table 4. Apparent and Real Deuterium Kinetic Isotope Effects
under Steady-State Conditionsa
H
still depends on kp /kb, is when this ratio is equal to <0.1. Thus,
kpH/kb
KIEapp/KIEreal
1000
0.011
100
0.0198
10
1.0
0.1
0.01
apparently it is feasible to resolve the rate constants in the
0.10 0.505 0.91 0.99
proton-transfer mechanism only under non-steady-state condi-
tions and where kf falls in the range between (kapp
a For reactions following eq 1, calculated from KIEapp ) KIEreal(kb
+ kpD)/(kb + kpH).
H
)
s.s.
and
H
11(kapp
) .
s.s.
Fitting Experimental DCV Data to Theoretical Data
Obtained by Digital Simlulation. For a simple second-order
proton-transfer reaction between a radical cation and a base,
theoretically the slope of the R′I vs ν-1 response curve is
independent of the second-order rate constant and KIEapp are
independent of R′I. Obviously, the experimental data in Tables
1 and 2 do not conform to this mechanism. On the other hand,
each of the eight sets of data in Tables 1 and 2 can be readily
fit to theoretical data for the complex mechanism (eqs 3-5),
keeping the individual rate constants consistent with the
appropriate kapp (Table 3). However, the experimental to
theoretical data fit is not unique when a single response curve
is considered. Generally, several combinations of rate constants
will give an acceptable data fit.
to R′I), and can be equated to the real kinetic isotope effect
(KIEreal) for the proton transfer.
The data, however, suggest a complex mechanism, such that
kapp values, and hence KIEapp values, approach the steady-state
values as the extent of reaction increases. The obvious hypoth-
esis developed at this stage of our study, supported by indirect
evidence from previous studies,12,13 is that the proton-transfer
reactions between the methylarene radical cations and the
nitrogen-centered bases follow a general mechanism (eq 1) in
which all three rate constants (kf, kb, and kp) contribute to the
overall rate of the reaction. Furthermore, the variation of KIEapp
with decreasing R′I suggests that the reactions do not reach
steady-state until R′I is approaching 0.5. Preliminary digital
simulation of the mechanism (eqs 3-5) verified that with the
•+
The concurrent fitting of data for both ArCH3•+ and ArCD3
ArCH3•+ + B a ArCH3•+/B K3kf/kb
(3)
(4)
(5)
results in a unique set of rate constants that gives the best
correspondence between the experimental response and the
theoretical data.19 To arrive at this unique set of rate constants,
the parameters must be varied over a wide range in a systemati-
cally. Arriving at the unique set of rate constants requires a
ArCH3•+/B f ArCH2 + BH+kp
•
•
ArCH2 + ArCH3•+ + B f products fast
large number of simulations.
H
proper selection of the rate constants, theoretical data could be
generated that closely mimicked the experimental data.
Apparent and Real Deuterium Kinetic Isotope Effects for
Proton-Transfer Reactions. Application of the steady-state
approximation on HA/B (eq 1) results in the rate law (eq 6),
Our data-fitting procedure involves calculating (νexp
-
D
νsimH)2/νsimH, (νexp - νsimD)2/νsimD, and (KIEexp - KIEsim)2/
KIEsim, where νH and νD are the voltage sweep rates necessary
for a particular R′I and the subscripts refer to experimental (exp)
and simulation (sim) data. The calculations are made at each
R′I from 0.85 down to 0.50 and each of these is summed over
the entire range (∑H, ∑D, and ∑KIE). Since the dependence of
KIEapp on R′I is the most important parameter in arriving at a
unique fit, ∑KIE is used directly in the data fit, while ∑H and
∑D are monitored to ensure that good fits to the two response
curves are obtained.
rate ) kapp[HA][B]
(6)
(7)
D
H
KIEapp ) KIEreal(kb + kp )/(kb + kp )
and the apparent rate constant (kapp) is equal to kfkp/(kb + kp).
This gives rise to an expression (eq 7) to relate KIEapp to KIEreal
(equal to kpH/kpD). The data in Table 4 illustrate the effect of
kpH/kb on KIEapp/KIEreal when holding kf equal to kb. An
important consequence of eq 7 is that under steady-state
conditions, that is, those under which the rate law (eq 6) is
applicable, KIEapp are constants independent of the extent of
reaction. However, the data in Table 4 show that KIEapp cannot
necessarily be equated to KIEreal, even under steady-state
conditions; rather, KIEapp/KIEreal varies with kpH/kb from 0.011
The fact that the range of kf over which R′I-dependent KIEapp
H
are expected to be observed is limited [i.e. from kapp to
11(kappH)] suggests what strategy to follow. We typically select
about 10 kf values over the applicable range, starting at
1.1(kappH), and determine what values of the other rate constants,
kpH, kb, and kpD, give the best fit for that particular kf value.
H
This is done by varying kp and adjusting (kb and kpD) to
conform to eqs 8-10. For each kf, a series of ∑KIE as a function
of kpH is obtained. Plots of ∑KIE vs kpH are parabola-like curves
to 0.99 for a 105-fold change in the ratio in the range given.
H
(19) In numerous cases we have used theoretical instead of experimental
data as the known input to challenge our ability to assign unknown rate
constants with kappH, kappD, and R′I-dependent KIEapp as the “experimental”
data, using the concurrent fitting of DCV data for reactions of HA and DA
and the procedure described here. We have, in all cases where kf ranges
from 1.1(kappH) to 11(kappH), been able to precisely identify kf, kb, kpH, and
kpD. On the other hand, when theoretical R′I vs ν-1 data are used to attempt
to assign rate constants based only on a single response curve (reactions of
either HA or DA), several alternative sets of rate constants are observed to
give acceptable fits of the data.
Since the steady-state apparent rate constants, (kapp
)
and
s.s.
D
(kapp
)
(eqs 8 and 9), are available from the kinetic studies,
s.s.
H
H
H
(kapp )s.s./kf ) (kp /kb)/(1 + kp /kb) ) CH
(8)
(9)
D
D
D
(kapp )s.s./kf ) (kp /kb)/(1 + kp /kb) ) CD
let us relate these quantities to the kinetic isotope effects. From

Non-Steady-State Kinetics in Proton-Transfer Reactions
J. Am. Chem. Soc., Vol. 120, No. 49, 1998 12723
different from those of the parent radical cations. The effect of
K3 on E° for charge-transfer reaction 11 is given by eq 12,20,21
where ∆E° is the difference in electrode potential for reduction
of the radical cations as compared with the corresponding
complexes. In all four cases ∆E°, calculated with eq 12, was
<4 mV (Table 5). The fact that the second-order rate constants
for formation of the complexes (discussed in the following
sections and summarized later in Table 7) are relatively small
suggests that the homogeneous electron-transfer reactions (reac-
tion 13) will be of greater importance than diffusion of the
•+
ArCH3•+/B + ArCH3 a ArCH3/B + ArCH3
(13)
(14)
(15)
Figure 2. Three-dimensional least-squares analysis for the reaction
of DMA•+ with LUT. For the sake of clarity, the data shown are only
a fraction of the total used in the analysis. Rate constant assignment:
ArCH3/B f ArCH3 + B
kf ) 6120 M-1 s-1 and kp ) 817 s-1
.
H
log K13 ) F∆E°/2.303RT
ArCH3•+/B f ArCH3•+ + B
(16)
Table 5. Electron-Transfer Rate Constants for Reduction
Reactionsa
b
d
e
complex
K3(M-1
)
∆Eo(mV)c
Ket
ket(M-1 s-1
)
complex to the electrode and charge-transfer reaction 11.
Reaction 13 is expected to be accompanied by the very rapid
dissociation of the neutral complexes (reaction 14). The equi-
librium constants for reactions 13 and 14 are readily calculated
(eq 15). Since ArCH3 is consumed in reaction 13 and regener-
ated in reaction 14, the overall effect of these two reactions is
reaction 16, which is identical to reverse reaction 3. Thus
reactions 13 and 14, although they do not contribute to the rate
determining steps, will affect the DCV data and must be taken
into account in the simulations.
MPA•+/DPP
MPA•+/LUT
DMA•+/DPP
DMA•+/LUT
2.83
74.4
2.10
25.2
3.3
3.7
2.5
2.4
0.88
0.86
0.91
0.92
8 300
14 000
<5 000
<5 000
a See text for description of rate constant assignments. b Evaluated
from kf and kb for complex formation. c Calculated from eq 12: ∆Eo
) Eo(ArCH3•+/ArCH3) - Eo(complex•+/complex). d Equilibrium con-
stant for reaction 13. e Forward rate constant for reaction 13.
with minima that reflect the best fit of experimental to theoretical
data for the kf value. The overall result is a series of the fitting
parameter at the minimum, (∑
A plot of (∑KIE min vs kf is, again, parabola-like with a
•+
The source of radical cation in the reduction of ArCH3 at
the electrode during the cyclic voltammetry reduction scan is
not only reactions 13 and 14 but also equilibrium 3, which is
KIE)min, as a function of kf.
•+
)
drawn to the left by the depletion of ArCH3 in the reaction
minimum that reflects the best value of kf.
layer. The latter has already been accounted for in the simulation
iterations described in the previous section. Reactions 13 and
14 were readily taken into account by treating 13 as an
irreversible reaction followed by a very rapid 14. Further
iterations in the simulations were carried out in which k13 was
varied to give the best fit of experimental to theoretical DCV
data. The last column in Table 5 gives the values of k13 obtained
from the fitting procedure. Improvements in the data fits were
observed when reactions 13 and 14 were included for the
reactions of MPA•+ but little effect was observed for those of
DMA•+. In all cases, including these reactions caused only a
minor perturbation in the simulated DCV response for the
proton-transfer reactions.
The fitting procedure is demonstrated graphically in Figure
2, which plots kp on the x-axis, kf on the y-axis and ∑KIE on the
z-axis. Each of the five parabola like curves projected onto the
x-z plane give the dependence of ∑KIE on kp at the particular
value of kf. The minima of the five curves when projected in
the y-z plane are used to define the best fit of experimental to
theoretical data for kf, which is assigned the value at the
minimum of this curve.
The procedure described to this point is the first iteration for
estimating the rate constants. It is then necessary to refine the
data in the regions of the minima to obtain high resolution of
the data fit.
Since our primary interest is in the proton-transfer reactions
of the radical cations, we have evaluated only the rate constants
for reaction 13 in connection with gaining the best fit of
experimental and theoretical data for the DCV response. No
attempt was made to study the details of the electron-transfer
reactions.
Uncertainty in Rate Constant and KIEreal Assignments.
The experimental error in the determination of R′I values was
observed to be on the order of (0.004, which corresponds <1%
in the range where R′I equals 0.5 to 0.85. However, the
experimental error is not the primary source of the uncertainty
in assignment of microscopic rate constants. The data in Table
6 show that the error in KIEreal has a marked dependence on
kf/kappH. When this ratio is close to unity, even a 1% error in kf
gives rise to a very large error in KIEreal. At the other extreme,
For each kp used at each kf it is necessary to carry out ∼30
simulations (15 for HA and 15 for DA) to generate the R′I vs
ν-1 response curves for HA and DA. The first iteration of fitting
H
usually involves ∼25 kp at 10 kf, which requires ∼7500
simulations to gather the necessary theoretical data. This is
followed by further refinements, including the electron-transfer
reaction discussed in the following section. On the order of
10 000 or more simulations were carried out for each system
studied. Details of the simulation procedure are given in the
Experimental Section.
Homogeneous Electron-Transfer Reactions during the
Reverse Cyclic Voltammetry Scan. The equilibrium constants
(K3) for the radical cation-base reactions (eq 3) studied here
are moderate (Table 5). This implies that the complexes,
ArCH3•+/B, will have reduction potentials (reaction 11) not very
(20) Shifts in reversible electrode potentials have been used to determine
equilibrium constants for a number of reactions following charge transfer
(see ref 21 and literature cited therein).
(21) Parker, V. D. Acta Chem. Scand. 1984, B-38, 125, 189, 741; Peover,
M. E.; Davies, J. D. J. Electroanal. Chem. 1963, 6, 46.
ArCH3•+/B + e- f ArCH3/B
(11)
(12)
∆E° ) (2.303RT/F)log(1 + K3[B])

12724 J. Am. Chem. Soc., Vol. 120, No. 49, 1998
Parker et al.
Table 6. Error in Real Kinetic Isotope Effects as a Function of
kf/kapp
conditions and we did not observe an experimentally significant
minimum under these conditions. Significant minima are
observed with kf > 1.1(kappH), which places kf in the region
between kapp and 1.1(kappH) and justifies our assignment of kf
) 1.05 ((0.05)(kappH). We did not include rate constant
assignments for the DMA•+/DPP reaction in Table 7 since we
were unable to assign a unique set of rate constants in this case
and our assignment of kf ) 1.05((0.05)(kappH) gave rise to very
large errors in the other rate constants and in KIEreal (see Table
6).
Ha
H
kf/kapp
kf (M-1 s-1
)
KIEreal
range ((1% error in kf)
1.02
1.05
1.10
1.20
1.50
2.00
5.00
4916
5061
5390
5784
7230
9640
24100
328
135
61.5
39.4
20.2
13.8
8.99
220-693
113-170
56.8-67.1
37.5-41.4
19.8-20.6
13.7-13.9
8.97-9.01
H
D
a Assumes kapp equal to 4820 and kapp equal to 652.
Since the proton-transfer reactions involve the abstraction of
a proton from -CH3 and -CD3, the KIE are the product of the
primary KIE for cleavage of the C-H(D) bond and the
R-secondary KIE.22 R-Secondary KIE are typically on the order
of 1.15 per deuterium atom.23 This suggests that the primary
KIE for the proton-transfer reactions are on the order of 0.76
times the total KIE.
Protonated pyridines (BH+) probably also take part in
equilibrium 17, at least to some extent, in dichloromethane-
Bu4NPF6 (0.2 M). If the equilibrium constant (K17) is sufficiently
BH+ + B a BHB+
(17)
large, this will affect [B] in the reaction layer. The nitrogen
centers in DPP and LUT are crowded because of the bulky 2,6-
substituents, which makes formation of BHB+ somewhat
unfavorable. Reaction 17, regardless of the magnitude of K17,
is not expected to have any effect on the kinetics of the reactions
studied under pseudo first-order conditions (B ) DPP) but a
large K17 would affect the reactions involving LUT under
second-order conditions. Including reaction 17 with a large K17
in simulations had little effect on ∑KIE but gave significantly
larger values of ∑HA and ∑DA for both reactions involving LUT,
thus indicating a less satisfactory comparison between experi-
mental and theoretical data. Since the reactions of HA and DA
are studied at the same extent of reaction, the relative effect of
equilibrium 17 should be the same in both reactions and have
a minimal effect on the magnitude of KIEappswhich is consistent
with the insensitivity of ∑KIE to the change in conditions.
Digital simulations were carried out to determine whether
incompletely labeled substrate could be responsible for the
observation of extent-of-reaction-dependent KIEapp. The Grig-
nard reaction with methyl-d3-iodide (99.5+%) is expected to
yield 9-methyl-d3-10-phenylanthracene of the same isotopic
purity. Simulations according to a simple proton-transfer mech-
Figure 3. Fitting procedure applied to the reaction between MPA•+
and DPP in dichloromethane-Bu4N+PF6- (0.2 M) at 291 K. [MPA] )
1.0 mM, [DPP] ) 50 mM.
for kf/kappH ) 5, a 1% error in kf gives rise to only a negligible
error in KIEreal. This relation must be kept in mind when
assessing errors.
Rate Constant Assignments for the Reactions of Methyl-
arene Radical Cations with Pyridine Bases. The results of
the fitting procedure used to assign rate constants to the radical
cation-base reactions are summarized in Figure 3 and in the
•+
anism of the reactions of MPA and MPA-d3 (98.5%)/MPA-
Supporting Information (Figures S1-S3). The (∑KIE min vs kf
)
d2•+ (1.5%) with DPP and LUT in which no KIE was assumed
for the reaction of MPA-d2•+ failed to reveal extent-of-reaction-
dependent KIE.
data, obtained from the 3-dimensional analysis presented earlier,
are indicated for kf values ranging from kappH to >5(kappH). The
inserts in the figures are the expanded-scale plots in the region
where the minimum, if observed, occurs. The interpretation for
the analysis for the reactions of MPA•+ with both LUT (Figure
S1) and DPP (Figure 3), as well as that for DMA•+ with LUT
(Figure S2), are straightforward and the rate constants calculated
Discussion
Once steady-state is achieved, the rate law (eq 6) applies to
proton-transfer reactions following general mechanism 1, and
kapp cannot be resolved into the individual contributions of kf,
kb, and kp. On the other hand, during the period before steady-
state is reached, the various rate constants affect the overall
reaction rate to different degrees than those implied by the rate
law (eq 6). Provided that kpH/kb falls in the range from ∼0.1 to
1000, the possibility of resolving the kinetics exists. Under these
at the minima of the (∑KIE min
are summarized in Table 7.
On the other hand, (∑
KIE min for the reaction between DMA•+
)
vs kf plots are assigned. These
)
and DPP (Figure S3) steadily increases as kf increases through-
out the applicable range of kf. We cannot assign unique rate
constants in this case since the required minimum was not
observed. Even in this case, minima were observed in all of
H
D
conditions, the relative effects of kp and kp differ during the
H
the ∑KIE vs kp plots at constant kf (i.e. the dimension
non-steady-state period, and the KIEapp values dependent on the
perpendicular to the (∑KIE min vs kf plane of Figure S3). Our
)
(22) Streitweiser, A.; Jagow, R. H.; Fahey, R. C.; Susuki, S. J. Am. Chem.
Soc. 1958, 80, 2326.
(23) Carrol, F. A. PerspectiVes on Structure and Mechanism in Organic
Chemistry; Brooks/Cole: Pacific Grove, CA, 1998.
conclusion is that in this case kf is too close to kapp to allow
observation of the minimum. We confirmed by simulations for
kf ) 1.05(kappH) that a minimum is not expected under these

Non-Steady-State Kinetics in Proton-Transfer Reactions
J. Am. Chem. Soc., Vol. 120, No. 49, 1998 12725
Table 7. Rate Constants and Kinetic Isotope Effects for Proton-Transfer Reactionsa
R.C. (mM)b
base (mM)c
kf (M-1 s-1
)
kb (s-1
67.6
250
244
)
kpH (s-1
)
KIEapp
KIEreal
MPA•+(1.0)
MPA•+(1.0)
DMA•+(2.0)
LUT (2.0)
DPP (50)
LUT (4.0)
5300
707
6120
518
545
817
5.3-7.3
8.7-11.8
6.2-7.8
47
33
31
a Reactions in dichloromethane-Bu4NPF6 (0.2 M) at 291 K. b Radical cation, substrate concentrations in parentheses. c Base concentrations in
parentheses.
extent of reaction are predicted by theory (simulation). The latter
provide an effective means of comparing experimental with
theoretical data and allows the assignment of unique sets of
rate constants for the proton-transfer reactions. We have
experimentally observed this behavior during kinetic studies of
the proton-transfer reactions of methylarene radical cations with
pyridine bases.
cannot be equated to KIEreal (often called the intrinsic isotope
effect in enzyme reactions).28
The experimental data reported here show the characteristic
feature of KIEapp which vary with extent of reaction, thus
indicating that the data are obtained under non-steady-state
conditions where the rate law (eq 6) is not applicable. The data
presented (Tables 1-3 and Figures 1-3, S1-S3) clearly
demonstrate that the fitting of experimental to theoretical data
for mechanism 1 provides a means to resolve kapp into the
microscopic rate constants kf, kb, and kp.
Kinetic studies carried out under steady-state conditions where
the rate law (eq 6) applies clearly leave considerable uncertainty
in the significance of the observed KIE. One would need to
resolve kapp (eq 6) and evaluate the microscopic rate constants
(kf, kb, kpH, and kpD) to have any confidence that deuterium
KIEapp could be equated to KIEreal. It seems highly likely that
proton-transfer reactions of many different classes of carbon
acids, including organic radical cations other than those studied
here, fall in the dynamic range where KIEapp are only a fraction
of KIEreal. Should this be the case, the KIE reported for these
reactions may only be fractions of KIEreal. Further studies in
which proton-transfer kinetics are resolved are necessary to
answer this question.
Aside from the intrinsic interest in the phenomenon of
tunneling during proton-transfer reactions, there are other
important reasons to know the magnitude of the effect. The
magnitude of the deuterium KIE is often used in the description
of transition states for proton transfer, for example, as in the
discussion of the variation of deuterium KIE as a function of
acid strength of AH and BH+, wherein bell-shaped curve is often
obtained when KIE are plotted vs pKa of BH+ for the proton-
transfer reaction of AH with B. A common explanation of this
phenomenon, attributed to Westheimer,29 is that the maximum
occurs when the proton is bonded equally to A- and B in the
transition state. On the other hand, the explanation of Bell et
al.30 is that maxima are found when HA and BH+ are of equal
strength, under which conditions quantum mechanical tunneling
is expected to be maximal. Regardless of which interpretation
is correct, neither will be valid unless the KIE values used in
the correlation are real rather than apparent values.
The kinetic data for the proton-transfer reactions are sum-
marized in Table 7. The forward rate constants (kf) for the
formation of radical cation-base complexes are 8-10 times
greater when the base is LUT than for DPP. On the other hand,
the limited data are not conclusive with regard to the effect of
structure on the magnitude of the other two rate constants. All
of the KIEreal, taking into account the expected R-secondary
KIE, are clearly out of the range expected for classical KIE
and implicate the occurrence of quantum mechanical
tunneling.24-27 Further studies on all of these reactions are
needed to determine the temperature effect on kpH and kpD and
to gain more knowledge on proton tunneling in these systems.
As was pointed out earlier, the proton-transfer kinetics for
mechanism 1 can be resolved only when kpH/kb falls in a rather
limited range. If kapp for the 2-step mechanism is moderate and
the encounter complexes are formed at diffusion-controlled rates,
the reaction kinetics cannot be resolved. For the proton-transfer
reactions of methylarene radical cations with pyridine bases,
the complexes involved in mechanism 1 cannot be the encounter
complexes that form at every collision. On the other hand,
previous studies12,13,17 have evidence that both proton-transfer
reactions and nucleophile-combination reactions of the radical
cations pass through kinetically significant intermediates, which
have been proposed to be π-complexes between the radical
cations and the nitrogen-centered bases (nucleophiles). The
detailed structure of these complexes is unknown and perhaps
further studies in which the proton-transfer kinetics are resolved
will provide indirect evidence of structure.
The data in Table 4 indicate that for proton-transfer mech-
anism 1 in the dynamic range between limiting rate laws, where
kapp corresponds to neither kf nor kpKeq, KIEapp differ signifi-
cantly from KIEreal. For this mechanism these terms are identical
only under conditions where kapp ) kpKeq, that is, conditions
under which the preequilibrium is rapidly established before
slower proton transfer takes place. The observation of extent-
of-reaction-dependent KIEapp during the stage of the reaction
before steady-state is established thus provides evidence that
the experimental conditions correspond to the dynamic range
between the limiting rate laws. It is well-known that reactions
having multiple rate-limiting steps can give rise to KIEapp that
It is of interest to note that most of the clearly documented
cases31 of large kH/kD values attributable to hydrogen tunneling
are for unimolecular reactions,32 including rearrangement of
sterically hindered aryl radicals,33 enol-ketone transformation
(28) Melander, L.; Saunders, W. H. Reaction Rates of Isotopic Molecules;
John Wiley: New York, 1980; pp 158-162. Cleland, W. W. Bioorg. Chem.
1987, 15, 283. Sinnott, M.; Garner, C. D.; First, E.; Davies, G. Compre-
hensiVe Biological Catalysis, A Mechanistic Reference; Academic Press:
San Diego, 1998; Vol. 4, pp 47-58, and references therein.
(29) Westheimer, F. H. Chem. ReV. 1961, 61, 265.
(30) Bell, R. P.; Sachs, W. H.; Tranter, R. L. Trans. Faraday Soc. 1967,
67, 1995.
(31) The examples of hydrogen tunneling presented are not the most
recent ones and were chosen on the basis of the magnitude of KIE.
(32) Siebrand, W.; Wildman, T. A.; Zgierski, M. Z. J. Am. Chem. Soc.
1984, 106, 4089.
(24) Bell, R. P. The Tunnel Effect in Chemistry; (Chapman & Hall:
London, 1980).
(25) Siebrand, W.; Wildman, T. A.; Zgierski, M. Z. J. Am. Chem. Soc.
1984, 106, 4083.
(26) There have been many more recent advances in the dynamics of
proton transfer and proton tunneling and the introductory27 and subsequent
papers in an issue of the Deutsche Berichte Bunsen-Gesellschaft fur
Physikalische Chemie devoted to “Hydrogen Transfer: Theory and Experi-
ment” provide ready access to this literature.
(27) Limbach, H.-H.; Manz, J. Ber. Bunsen-Ges. Phys. Chem. 1998, 102,
289.
(33) Brunton, G.; Gray, J. A.; Griller, D.; Barclay, L. R. C.; Ingold, K.
U. J. Am. Chem. Soc. 1978, 100, 4197.

12726 J. Am. Chem. Soc., Vol. 120, No. 49, 1998
Parker et al.
of 2-methylacetophenone,34 a sigmatropic hydrogen shift in
hexahydrocarbazole,35 and intramolecular hydrogen migration
between nitrogen atoms of meso-tetraphenylporphine.36 For
unimolecular reactions, KIEapp, which are not necessarily equal
to KIEreal, are not expected to depend on the extent of reaction.
Two examples of large KIE reported for proton-transfer reac-
tions include proton transfer to benzene anion in ethanol (kH/
kD ) 100 at 77 K)37 and proton transfer in naphthol/ammonia
complexes (kH/kD > 200 at 10 K).38
of proton tunneling in these reactions can be much greater than
previously realized. Our successful resolution of the four
pertinent rate constants (kf, kb, kpH, and kpD) for microscopic
reaction steps during the proton transfer reactions of methylarene
radical cations suggests that the approach may be applicable in
other cases and has the potential to greatly enrich the funda-
mental knowledge of proton-transfer reactions.
Experimental Section
Hydrogen tunneling in biochemical systems has recently been
observed in a number of cases and is of intense current interest.39
A pertinent example is the hydrogen atom abstraction from
linoleic acid by soybean lipoxygenase,40,41 a reaction in which
both hydrogen and deuterium tunneling have been found.42 But
it is not clear whether the KIE reported in these studies are
apparent or real values.
The observation of very large kH/kD during proton-transfer
reactions of 4-nitrophenylnitromethane with amine bases43 was
attributed to extensive tunneling and served as strong evidence
for the phenomenon. Several later investigations44 showed that
the KIE were actually normal and that the observation of the
large KIEapp was an artifact that could be attributed to kinetic
deviations caused by the exchange of D in the labeled substrate
with amino H and adventitious water. Could a similar artifact
be responsible for the large KIE observed in the present study?
We do not believe that exchange of D in the labeled radical
cation is feasible. We know of no instances where methylarene
radical cations undergo exchange of methyl H or D. In fact,
the abstractions of H and D from the methyl group are
thermodynamically favorable reactions followed by rapid
electron transfer from the resulting radical to generate the
carbenium ion.
Materials. Dichloromethane was allowed to reflux for several hours
over CaCl2 and, after passing through active neutral alumina, was used
without further purification. Bu4NPF6 (Aldrich) was recrystallized from
dichloromethane-ether before use. DMA (Aldrich) was recrystallized
from 2-propanol before use. MPA was obtained from bromination of
9-phenylanthracene in CCl4, followed by halogen-lithium exchange
with tert-butyllithium under an argon atmosphere and reaction with
methyl iodide in tetrahydrofuran at -78 °C. 9-Methyl-d3-10-pheny-
lanthracene was prepared in the same way, with use of methyl-d3-iodide
(99.5+%) as alkylating agent. 9,10-Dimethyl-d6-anthracene was pre-
pared according to the method of Fieser and Heymann45 with methyl-
d3-magnesium iodide (99.5+%) used as the Grignard reagent. 2,6-
Dimethylpyridine (99+%, Aldrich) was distilled under reduced pressure
before use. DPP from Aldrich was recrystallized from 2-propanol-
petroleum ether. We observed differences between the rate constants
determined with recrystallized DPP and with that used as received.
Instrumentation and Data Handling Procedures. Cyclic voltam-
metry was performed with a Princeton Applied Research (Princeton,
NJ) Model 173 potentiostat/galvanostat driven by a Hewlett-Packard
3314A function generator. After passing through a dual-channel low-
pass filter (Stanford Research Systems, Inc., Model SR640), the data
were recorded on a Nicolet Model 310 digital oscilloscope with 12-bit
resolution. The oscilloscope and function generator were controlled by
a personal computer via an IEEE interface.
The current-potential curves were collected at selected trigger
intervals to reduce periodic noise,46 and 3 or more curves were averaged
before treating with a frequency domain low pass digital filter and
numerical differentiation. The standard deviation in R′I obtained this
way was (0.004.
We also considered the presence of isotopic impurities in the
radical cations as a possible source of the extent-of-reaction-
dependent KIEapp. Simulation results for KIE studies of the
reactions of MPA•+ and MPA-d3•+ contaminated with 1.5% of
MPA-d2•+, in which the latter was assumed to react at the same
rate as MPA•+, failed to predict extent-of-reaction-dependent
KIEapp. Furthermore, if the observation of extent-of-reaction-
dependent KIEapp were due to isotopic impurities, the effect
would be expected to be as much as 2 times greater for the
Cyclic Voltammetry Measurements. A standard 3-electrode 1-com-
partment cell was used for all kinetic measurements. Positive-feedback
IR compensation was used to minimize the effects of uncompensated
solution resistance. Reference electrodes were Ag/AgNO3 (0.01 M) in
acetonitrile constructed as described by Moe.47 The working electrodes,
0.2-0.8 mm Pt, were prepared by sealing wire in glass and polishing
to a planar surface as described previously.48 The working electrodes
were cleaned before each series of measurements with a fine polishing
powder (Struers, OP-Alumina Suspension) and wiped with a soft cloth.
The cell was immersed in a water bath controlled to 18 ( 0.2 °C.
Kinetic Measurements. Rate constants were obtained by comparing
DCV15 data with the theoretical data obtained by digital simulation.16
The reactions were studied both under pseudo-first-order and second-
order conditions and using solutions (CH2Cl2/0.2 M Bu4NPF6) contain-
ing substrate (1.0-2.0 mM) and base (0.5-50 mM) at 291 K. The
experimental R′I data were adjusted to 0.05 intervals in the range 0.85
to 0.50 by linear log-log interpolation. (We have previously observed
that log R′I vs log ν-1 curves are nearly linear in this interval.49) To
avoid interpolation error, we recorded several R′I values very close to
either side of the desired value and averaged them before interpolation.
For example, to determine ν0.5, the voltage sweep rate where R′I is
equal to 0.50, we selected ν values to give R′I of ∼0.51-0.52 and
∼0.48-0.49. Three or more determinations were made in these ranges
and the average values were* then used in the interpolation. The
minimum number of experimental cyclic voltammograms processed
to give a single R′I value was 18; most R′I values were derived from
>30 experimental voltammograms. The resulting ν values, for example,
•+
reactions of DMA-d6 than for those of MPA-d3•+. The
experimental results show the opposite trend: KIEapp for the
reactions of DMA•+ increased by factors of ∼1.23 in the range
R′I ) 0.85-0.50, whereas those for the corresponding increases
for MPA•+ were ∼1.37.
Our results show that KIEapp during proton-transfer reactions
can be far less than KIEreal, which suggests the distinct
possibility that the magnitude of KIEreal and hence the influence
(34) Grellman, K.-H.; Weller, H.; Tauer, E. Chem. Phys. Lett. 1983, 95,
195.
(35) Grellman, K.-H.; Schmitt, U.; Weller, H. Chem. Phys. Lett. 1982,
88, 40.
(36) Limbach, H.-H.; Henning, J.; Gerritzen, D.; Rumpel, H. Faraday
Discuss. Chem. Soc. 1982, 74, 229.
(37) Miyazaki, T.; Shiba, T.; Fueki, K.; Kamiya, Y. J. Phys. Chem. 1991,
95, 9115.
(38) Swinney, T. C.; Kelley, D. F. J. Phys. Chem. 1991, 95, 2430.
(39) Cha, Y.; Murray, C. J.; Klinman, J. P. Science 1989, 243, 1325.
(40) Glickman, M. H.; Wiseman, J.; Klinman, J. P. J. Am. Chem. Soc.
1994, 116, 793.
(41) Hwang, C. C.; Grissom, C. B. J. Am. Chem. Soc. 1994, 116, 795.
(42) Jonsson, T.; Glickman, M. H.; Sun, S.; Klinman, J. P. J. Am. Chem.
Soc. 1996, 118, 10319.
(45) Fieser, L. F.; Heymann, H. J. Am. Chem. Soc. 1942, 64, 372.
(46) Lasson, E.; Parker, V. D. Anal. Chem. 1990, 62, 412.
(47) Moe, N. S. Anal. Chem. 1974, 46, 968.
(43) Caldin, E. F.; Warrick, P.; Mateo, S. J. Am. Chem. Soc. 1981, 103,
202 and earlier work cited therein.
(44) Rogne, O. Acta Chem. Scand. 1978, A-32, 559. Kresge, A. J.; Powell,
M. F. J. Phys. Org. Chem. 1990, 3, 55.
(48) Lines, R.; Parker, V. D. Acta Chem. Scand. 1977, B31, 369.
(49) Ahlberg, E.; Parker, V. D. J. Electroanal. Chem. 1981, 121, 73.

Non-Steady-State Kinetics in Proton-Transfer Reactions
0.85 or ν0.5, were directly proportional to apparent rate constants at the
J. Am. Chem. Soc., Vol. 120, No. 49, 1998 12727
ν
reactions of both HA and DA, differentiates the resulting current-
potential curves, assigns sweep rates necessary for each R′I beginning
at 0.85 and ending at 0.50, stores these data in files for reactions of
both HA and DA, and calculates and stores KIEapp at each value for
R′I. All of the theoretical data for the first iteration of the data-fitting
procedure can be obtained from a single input session. Each simulation
requires from <1 to ∼20 s, depending on the simulation potential step
width, on a 350 MHz personal computer.
We used DIGI to test the reliability of the (FD) simulations. We
compared the FD and DIGI simulations in the voltage sweep rate ranges
where the experimental data were obtained for each of the radical cation
proton-transfer reactions. Each pair of simulations gave R′I results within
0.001 of each other (the experimental error in R′I is around (0.004).
The comparisons are summarized in Table S1 in the Supporting
Information.
extent of reaction corresponding to R′I, that is, R′I ) 0.85 or 0.50 in
the examples given.50
Digital Simulation. Digisim 2.1 (DIGI; BioAnalytical Systems, Inc.,
W. Lafayette, IN)18 has 3 variable model parameters affect the accuracy
of the simulation: (i) the potential step width, (ii) the exponential grid
factor, and (iii) the number of iterations. The 3 parameters were varied
sequentially over a factor of at least 10 while holding all other variables
constant. No changes in R′I were observed over the entire range of
model parameters. The cyclic voltammograms (potential step width, 1
mV) obtained with DIGI were differentiated by the method of Savitzky
and Golay.51
Although DIGI is the state-of-the-art software for carrying out
simulation of cyclic voltammetry, we found it prohibitively time-
consuming for carring out the massive number of simulations necessary
to find the optimum fit between experimental and theoretical DCV data
(see Results). Consequently, we developed our own simulation program
to automate theoretical data collection. The simulation program is based
on Feldberg’s explicit finite difference (FD) method.52 The input for
the program includes experimental values of kappH, kappD, initial
experimental voltage sweep rates, any number of kf (typically 10) and
any number of kpH (typically 25) values at each kf. Changes in simulation
kpH were accompanied by changes in kb and kpD to maintain consistency
with eqs 8-10. The program simulates cyclic voltammograms for the
Acknowledgment. We acknowledge the donors of the
Petroleum Research Fund, administered by the ACS, as well
as National Science Foundation (CHE-970835) for support of
this research.
Supporting Information Available: Table of comparison
of finite difference to digisim simulations and figures of fitting
procedures (5 pages). See any current masthead page for
ordering and Internet access instructions.
(50) Parker, V. D. Acta Chem. Scand. 1981, B35, 233.
(51) Savitzky, A.; Golay, M. Anal. Chem. 1964, 36, 1627.
(52) Feldberg, S. W. Electroanal. Chem. 1969, 3, 199.
JA982682K