General Model for the Nonlinear pH Dynamics in the Oxidation of Sulfur(-II) Species

Article abstract of DOI:10.1021/jp0023314

A general kinetic feature has been observed experimentally for the oxidation of the sulfur(-II) species thiosulfate, thiourea, thiocyanate, and sulfide by chlorite and other multi-equivalent oxidants under appropriate, unbuffered batch conditions. This fingerprint consists of an initial rise in pH followed by an autocatalytic drop in pH or oligo-oscillatory behavior. These systems also exhibit oscillations and other complex dynamical behavior in a continuous-flow stirred tank reactor (CSTR). The previously proposed general models that are oxidant based do not successfully explain the observed pH effects. We propose a simple, general model that is based upon the changing oxidation states of sulfur to explain the general pH features. The scheme qualitatively models autocatalysis and oligo-oscillations in batch and simple and complex oscillations in a CSTR. The general model consists of three separate stages: negative hydrogen ion feedback (S(-II) to S(0)), a transition of S(0) to S(IV), and positive proton feedback from S(IV) to S(VI).

Full text of DOI:10.1021/jp0023314

J. Phys. Chem. A 2000, 104, 11561-11565
11561
General Model for the Nonlinear pH Dynamics in the Oxidation of Sulfur(-II) Species
C. Wayland Rushing and Richard C. Thompson*
Department of Chemistry, UniVersity of Missouri-Columbia, Columbia, Missouri 65211
Qingyu Gao†
Department of Energy Utilization and Chemical Engineering, China UniVersity of Mining and Technology,
Xuzhou, 221008 P. R. China
ReceiVed: June 30, 2000; In Final Form: September 20, 2000
A general kinetic feature has been observed experimentally for the oxidation of the sulfur(-II) species
thiosulfate, thiourea, thiocyanate, and sulfide by chlorite and other multi-equivalent oxidants under appropriate,
unbuffered batch conditions. This “fingerprint” consists of an initial rise in pH followed by an autocatalytic
drop in pH or oligo-oscillatory behavior. These systems also exhibit oscillations and other complex dynamical
behavior in a continuous-flow stirred tank reactor (CSTR). The previously proposed general models that are
oxidant based do not successfully explain the observed pH effects. We propose a simple, general model that
is based upon the changing oxidation states of sulfur to explain the general pH features. The scheme qualitatively
models autocatalysis and oligo-oscillations in batch and simple and complex oscillations in a CSTR. The
general model consists of three separate stages: negative hydrogen ion feedback (S(-II) to S(0)), a transition
of S(0) to S(IV), and positive proton feedback from S(IV) to S(VI).
Introduction
reinvestigated the pH effects observed during the oxidation of
several sulfur(-II) species by chlorite and hydrogen peroxide.
The first of the chlorite-driven oscillators was discovered in
981. Since then, the number of known oscillators based upon
Under appropriate, unbuffered batch conditions all the systems
that were investigated have the same general features as those
previously reported.
1
1
chlorite have grown to include a wide variety of chemical
systems. They require the presence of another halogen-contain-
ing species, with the notable exception of a subfamily in which
Due to the similarities of the various systems we believe that
the chemistry of the sulfur species is a main driving force for
the nonlinear behavior observed in these systems. We therefore
propose a simple, general model based upon changes in the
oxidation state of sulfur in the presence of a generic oxidant.
The proposed model is not meant to explain the experimental
findings of any individual system, as this would require a more
detailed mechanism. We only considered general reactions that
would be common among all the systems. The proposed model
successfully simulates the general pH features that are experi-
mentally observed in batch and predicts pH oscillations in a
continuous-flow stirred tank reactor (CSTR).
2
3
the reductant is a S(-II) species: thiosulfate, sulfide, thio-
4
5
cyanate, and thiourea. These S(-II) systems exhibit some of
the most complex dynamical behavior of any of the chlorite
oscillators.
Although considerable progress has been made in understand-
ing the mechanistic details of chlorite-based oscillators, a
number of issues remain. While some specific systems have
been successfully modeled, the most notable being the chlorite-
iodide system,6 most are still awaiting explanation. R a´ bai and
Orb a´ n have suggested a general model for the chlorite-based
oscillators in an attempt to explain the general nonlinear
-8
9
behavior observed. While their model is in many respects quite
Experimental Section
successful, it does have several limitations. The most notable
is no provision for pH dynamics which prevents it from
accurately describing the nonlinear behavior that is observed
during the oxidation of sulfur(-II) species.
Materials. Reagent grade H2SO4, KSCN, Na2S, Na2S2O3,
H2O2, and NaOH were used without further purification. NaClO2
obtained from Aldrich (=80%) was recrystallized twice from
We noted that the literature provides a number of examples
of nonlinear pH kinetic profiles during the oxidation of several
8
0% ethanol-water in the range 25 to -20 °C resulting in a
20
purity >99% determined by iodometric titration. Thiourea was
used as received or was recrystallized twice. Purified DTPMP
1
0
sulfur(-II) species by not only chlorite but also hydrogen
peroxide,1
1-16
iodate, periodate, and peroxodisulfate. All
17
18
19
(diethylenetriaminepentamethylenephosphonic acid) and EDT-
of these systems have similar pH kinetic profilessan initial rise
in pH followed by an autocatalytic or oligo-oscillatory drop. If
sulfur is a main driving force in the nonlinear behavior then
we would expect to see the same features in other multi-
equivalent oxidant-S(-II) systems. To this end, we have
MP (ethylenediaminetetramethylenephosphonic acid) were ob-
tained from Dow Chemical. Sodium sulfide solutions were
assayed iodometrically every 2 days. H2O2 was standardized
with permanganate every day. All solutions were prepared in
triply distilled water (with the last distillation from basic
permanganate).
*
Corresponding author. Fax: (573) 882-2754. E-mail: thompsonr@
The batch reactions were carried out in a thermostated
Plexiglas reactor with a volume of 42-45 mL. All reactant
missouri.edu.
†
E-mail: gaoyu1@public.xz.js.cn.
1
0.1021/jp0023314 CCC: $19.00 © 2000 American Chemical Society
Published on Web 11/18/2000

11562 J. Phys. Chem. A, Vol. 104, No. 49, 2000
Rushing et al.
Figure 2. Chlorite-thiourea batch reactions when left in contact with
+
-4
air (a) and when protected (b). [H ]
o
) 6.00 × 10 M; [thiourea]
o
)
-
4
-
-3
9
.00 × 10 M; [ClO
2
]
o
) 2.50 × 10 M.
the basic sulfide solution), or biphasic, as in the thiourea system.
The change in pH is accompanied by a very similar change in
potential (Figure 1S). A monophasic or biphasic rise in potential
is followed by an autocatalytic rise. This general feature of pH
and potential change is not unique to the chlorite and hydrogen
peroxide systems, but has also been observed under appropriate
Figure 1. Measured pH vs time curves for the chlorite-sulfide (a
+
and b) and chlorite-thiourea (c and d) systems. Chlorite-sulfide: [H ]
o
-
4
2-
-4
-
-3
-4
)
3.28 × 10 M; [S
]
o
) 7.25 × 10 M; [ClO
2
] ) 6.13 × 10
-3
conditions with a number of additional multi-equivalent
-
3
+
10,13-19
M (a), 6.03 × 10 M (b). Chlorite-thiourea; [H ]
o
) 6.00 × 10
oxidants.
This feature can therefore be thought of as a
-
4
-
M; [thiourea]
o
) 9.00 × 10 M; [ClO ] ) 3.00 × 10 M (c), 2.00
2
“
fingerprint” for nonlinear behavior arising from the sulfur(-
II) chemistry.
Nagyp a´ l and Epstein have reported very poor reproducibility
-
3
×
10 M (d).
solutions were deaerated by passing a nitrogen stream through
the solutions for 10 min. The reactions were initiated by addition
of the S(-II) species. To prevent air interference during the
reaction a small layer of N2 was kept above the solution. The
pH and potential were followed with a glass and platinum
electrode, respectively, utilizing a double junction ISE reference
electrode.
-
-2
21
in their investigation of the ClO2 and S2O3 clock reaction.
They found that for “identical” reaction conditions the reaction
took from a matter of seconds to a couple of hours to reach
completion. We apparently avoided this difficulty by using
significantly lower reactant concentrations; the system showed
very good reproducibility under a wide variety of conditions.
An important feature of these systems is whether the reactions
are exposed to air. Figure 2 illustrates the behavior observed
for the chlorite-thiourea system when the reactions were
performed in the presence of air and under nitrogen. These
systems are very air sensitive and yielded reproducible results
only when protected from air. (All of the experiments in Figure
The bulk of the batch experiments were concerned with
finding the appropriate conditions in which the general finger-
print of the sulfur(-II) systems was present. For the hydrogen
peroxide systems previous studies allowed for a rapid determi-
nation of the necessary conditions. However, previous studies
of the chlorite systems provided little hint of the appropriate
conditions since they generally used buffered or highly acidic
media. Our methodology involved a systematic variation of the
reactant concentrations until the general kinetic feature was
observed. It was found that the fingerprint is present only under
a very narrow range of experimental conditions, but was found
in each S(-II) system examined.
1
were performed nitrogen. Under these conditions the oligo-
oscillatory behavior observed is not present until the oxidant
-
3
concentration is below 2.30 × 10 M. But if exposed to air it
occurs at higher oxidant concentrations as seen in Figure 2.)
Many of the sulfur(-II) systems have previously been shown
1
2-15,19
to be catalyzed by copper ions.
Studies of the oxidation
of sulfur(-II) species are often done with added copper(II) in
order to obtain nonlinear behavior. Doona and Stanbury have
shown that traces of Cu(II) can dramatically catalyze the
Results
2
-
-
-
-
Batch Experiments. Buffers or highly acidic conditions were
used in most of the previous studies of the chlorite-sulfur(-
II) batch reactions. We investigated the pH effects in the chlorite
oxidation of sulfide, thiourea, thiosulfate, and thiocyanate and
the H2O2 oxidation of the thiosulfate and thiocyanate under
unbuffered batch conditions.
oxidation of thiourea by [IrCl6] , ClO2 , IO3 , BrO3 , and
2
2
ClO2. We tested for trace metal ion catalysis in our systems
in several ways. First, results obtained with commercially
available thiourea were compared to those with twice-recrystal-
lized thiourea. Both reactions were repeated with 6 µM DTPMP
and EDTMP (1% of total sulfur concentration); these chelators
have been shown to effectively sequester many trace metal
Figure 1 displays the general kinetic features observed in the
chlorite-sulfide and chlorite-thiourea reactions under appropri-
ate conditions. Similar results were obtained for the chlorite-
thiosulfate, chlorite-thiocyanate, hydrogen peroxide-thiosul-
fate, and hydrogen peroxide-thiocyanate systems. There is an
initial rise in pH followed by an autocatalytic or oligo-oscillatory
drop. The initial rise is either monophasic, as in the sulfide
system (the large, instantaneous rise in pH is due to addition of
2
3
ions. These chelators were also added to the sulfide systems.
No appreciable difference in any of the traces was observed.
Model. We propose the general model summarized in Table
1to simulate the pH dynamics associated with the oxidation of
sulfur(-II) species. To make the model as general as possible,
no acid-base equilibria of the oxidant or reductant are taken
into account. No distinction is made among the chemical species

Nonlinear pH Dynamics in the Oxidation of Sulfur(-II)
J. Phys. Chem. A, Vol. 104, No. 49, 2000 11563
TABLE 1: Reactions of the General Model for the
Sulfur(-II) Systems
no.
reactions
-
(
(
(
(
(
(
(
(
(
1)
2)
3)
4)
5)
6)
7)
8)
9)
OX + S(-II) f S(0) + OH
a
-
S(0) + S(-II) f S(-I) + OH
S(-1) + OX f 2S(0)
-
-
S(-1) + OX + OH f 2S(0) + OH
S(0) + OX f S(II)
-
-
S(0) + OX + OH f S(II) + OH
-
+
S(II) + OX f HSO
3
+ H
2- +
-
HSO
HSO
3
+ OX f SO
4
+ H
2-
-
+
+ 2H+
3
-
+ OX + H f SO
4
2
2-
(
(
(
(
(
10)
11)
12)
13)
14)
SO
HSO
3
+ OX f SO
4
-
2-
+
-
3
-
f SO
3
+ H
2
+
SO
3
+ H f HSO
3
+
Figure 3. Batch simulation of the general model. [H ]
o
) 5.00 ×
+
-
H
2
O f H + OH
-
4
-3
-2
1
2
0
M; [S(-II)]
o
) 4.00 × 10 M; [OX] ) 1.00 × 10 M (a),
o
+
-
H + OH f H
2
O
-
2
.10 × 10 M (b).
a
S(-1) represents a disulfide.
TABLE 2: Rate Equations and Rate Constants for the
General Model
that could predominate at sulfur oxidation states less than +4.
o
rate equations
rate constants at 25
(
The sulfur(-II) species in the model can be related to the
-
1 -1
-
2-
-
R
R
R
R
R
R
R
R
R
R
1
2
3
4
5
6
7
8
9
) k
) k
) k
) k
) k
) k
) k
) k
) k
1
2
3
4
5
6
7
8
9
[OX][S(-II)]
[S(0)][S(-II)]
[S(-I)][OX]
k
k
k
k
k
k
k
k
k
k
1
2
3
4
5
6
7
8
9
) 0.021 M
s
specific species HS , S2O3 , SCN , and (NH2)2CS. The sulfur-
) 50 M s-1
-1
-1) species would likely be S22-, O3S-S-S-SO3 , or
2-
(
(
) 0.01 M s-1
-1
-
SCN)2 while sulfur(0) could be 1/8 S8, HOSH, HOS-SO3 ,
-
) 8.0 × 105 M s-1
-2
[S(-I)][OX][OH ]
-
1 -1
or HOSCN. In our model we propose the S(II) intermediate.
[S(0)][OX]
[S(0)][OX][OH ]
[S(II)][OX]
) 1.0 M
s
-
) 1.2 × 10 M s-1
6
-2
However, in many of the specific systems the S(0) species is
1
1
-1 -1
taken directly to SO32- by 2 equivalents of oxidant. For a more
) 1.0 × 10
M
s
-
-1 -1
[HSO
[HSO
3
][OX]
) 7.0 M
s
detailed explanation of the proposed models for the specific
systems see refs 10, 19, 33, and 34.) The inclusion of specific
species for S(IV) and S(VI) is necessary for the success of the
model. Any nonlinear behavior arising from the chemistry of
the oxidant is omitted from this model to emphasize that arising
from the changing oxidation states of sulfur.
-
+
) 1.48 × 10 M s-1
7
-2
3
][OX][H ]
2-
-1 -1
10 ) k10[SO
3
][OX]
10 ) 0.2 M
s
-
3 -1
R11 ) k [HSO
11
3
-
]
k11 ) 3.0 × 10 s
2
+
12 ) 5.0 × 10 M-1
10
-1
R
R
R
12 ) k12[SO
13 ) k13[H
3
][H ]
k
k
k
s
O] ) 1.0 × 10 M s-1
-
3
2
+
O]
13[H
2
-
11
-1 -1
14 ) k14[H ][OH ]
14 ) 1.0 × 10
M
s
Many of the oscillatory systems that involve the oxidation
of S(-II) can be described as “pH-driven” oscillators. Therefore,
there must be both positive and negative hydrogen ion feedback,
with one of them also being autocatalytic in nature. The model
can be broken down into three constituent parts: negative
hydrogen ion feedback, positive hydrogen feedback, and transi-
tion reactions.
Discussion
The numerical integrations of the rate equations shown in
Table 2 were carried out using a semi-implicit Runge-Kutta
2
6
27
method utilizing the program SIMULATE. To check the
results of the Runge-Kutta method, selected experiments were
2
8
also simulated using the Gear algorithm and the program
GEPASI, which utilizes the LSODE integrator.30 There was
no difference in the results obtained by the different methods.
To test the validity of the proposed model we set out to
simulate several general features of the sulfur systems. As we
are going to show, the model can qualitatively simulate nearly
all of the behavior observed experimentally in the various
systems. The model is capable of simulating the general behavior
seen in batch; a general rise in pH followed by an autocatalytic
drop in pH (Figure 3, trace a). The more complex phenomena
of oligo-oscillations in batch can also be successfully modeled
(Figure 3, trace b). The resulting simulations not only agree
well with the experimental results of this paper but also with
29
Reactions 1 and 2 provide the negative hydrogen ion feedback
and account for the initial rise in pH that is observed under
batch conditions. Reactions 8 and 9, the oxidation of bisulfite
to sulfate, provide positive proton feedback with reaction 9 also
being autocatalytic in hydrogen ion. Reactions 3-7 are transition
reactions allowing for the conversion of S(-1) f S(IV) and
reactions 11-14 are protonation equilibria. Reactions 4 and 6
have previously been proposed in thiosulfate and thiocyanate
reactions and are necessary for oscillations to occur in a
1
4,15,16
CSTR.
Reported rate constants when OX ) H2O2 were used for
reactions 8-10 as a starting point.2 Values for k1 - k7 were
assigned mainly by trial and error while attempting to fit the
general features and oligo-oscillatory behavior that was found
under batch conditions. For successful modeling a number of
constraints were needed. First, there needed to be effective
competition for the sulfur(0) intermediate by the sulfur(-II)
species and the oxidant, with the sulfur(0)-sulfur(-II) reaction
predominating under acidic conditions and the sulfur(0)-oxidant
reaction predominating under basic conditions. Second, the
autocatalytic formation of the sulfur(0) intermediate by reactions
4,25
1
0,13-19
many of those previously reported.
general model gives very similar results to the model developed
Interestingly, the
by Kurin-Cs o¨ rgei et al. for the hydrogen peroxide-thiosulfate
1
4
system.
In several of the S(-II) systems the initial rise in pH is
biphasic (Figure 1, traces c and d). However, the general model
is not capable of simulating this behavior. Shortening the model
by omitting reactions 4 and 6 allows simulation of the two
separate rises in pH (Figure 2S). Reactions 4 and 6 allow for a
too rapid of a conversion of S(-I) to S(II). This would have
the effect of allowing the acid producing reactions, 7-9, to
predominate early and obscure the second rise. Unfortunately,
the shortened model is incapable of producing either oligo-
oscillations in batch or oscillations in a CSTR.
2
-4 is necessary for oligo-oscillatory behavior. Any detailed
mechanism for specific oxidant and S(-II) species would be
far more complex than the simple model presented. It is
important to note that the proposed model is a minimal oscillator
under open conditions.

11564 J. Phys. Chem. A, Vol. 104, No. 49, 2000
Rushing et al.
+
-4
Figure 6. Simulated batch oscillations. [H ]
o
) 5.00 × 10 M; [S(-
-
3
II)]
o
) 4.00 × 10 M; [OX]
o
) 0.02 M.
Figure 4. Simulated bi-stability. Hysteresis between a low-pH and a
+
high-pH steady state as the flow rate is varied in both directions. [H ]
o
-
4
-4
-
2 o
)
5.00 × 10 M.; [S(-II)]
o
) 4.00 × 10 M; [ClO
]
) 0.02 M.
+
Figure 7. Simulated complex oscillations in a CSTR. [H ]
o
) 5.00 ×
) 1.20 ×
-
-
4
3
-3
1
0
M; [S(-II)]
o
) 4.00 × 10 M; [OX]
o
) 0.02 M; k
o
-1
+
10
s .
Figure 5. Simulated simple oscillations in a CSTR. [H ]
o
) 5.00 ×
) 4.00 ×
-
-
4
4
-4
1
0
0
M; [S(-II)]
o
) 4.00 × 10 M; [OX]
o
) 0.02 M; k
o
-1
1
s
.
ena. As the flow rate is increased further, the model no longer
gives oscillations but yields the high pH steady state.
An important feature in nonlinear systems is bi-stability. The
The most exotic and complex behavior observed in the sulfur-
(-II) systems occurs when chlorite is the oxidant.5 The
chlorite-based model proposed by R a´ bai and Orb a´ n successfully
demonstrates that nonlinear behavior can arise from the oxidant
,32
S(-II) systems exhibit bi-stability between two stationary states
associated with low and high pH values. The simulated
bi-stability, i.e., the hysteresis between the two steady states as
the flow is varied, is shown in Figure 4. However, the range of
input concentrations and flow rates where bi-stability occurs is
much narrower than the experimentally derived hysteresis plots.
This is not of great concern as the shape of the hysteresis plot
is very dependent upon the rate constants and would vary
depending on the particular system being investigated.
9
chemistry only. One can envision that in systems that involve
not only a sulfur(-II) species but also chlorite there are two
nonlinear driving forces which could lead to the exotic
phenomena that are observed.
There have been great strides in the development of general
models for chemical oscillators. In most of these models the
systems are thought of as being oxidant driven, i.e., bromate
and chlorite systems. However, these models are not successful
in portraying the complex pH effects that are associated with
the nonlinear oxidation of sulfur(-II) species. This is mainly
due to the complex chemistry that S(-II) species can undergo
and the variety of intermediate oxidation states that can be
formed. The model proposed here is the first that is substrate
based. Comparison of the calculated results and the known
chemical systems suggests to us that our model can serve as a
guide in many of the chemical systems involving oxidation of
S(-II). The model successfully simulates the qualitative be-
havior that is seen in many of the experimental systems. The
general model that is proposed may aid in the systematic design
of new sulfur(-II) oscillators.
Simple pH oscillations calculated with this model under flow
conditions are shown in Figure 5. The shape and other
characteristics of the oscillations are sensitive to the adjustable
rate constants. Interestingly, the model is capable of simulating
oscillations not only in the CSTR but also under batch conditions
as seen in Figure 6. Batch oscillations in potential have
31
previously been observed in S(-II) systems. Since these
experiments were done under buffered conditions no pH
oscillations were observed. However the model does suggest
the possibility that some S(-II) systems might have the
capability for pH oscillations under batch conditions.
Many of the S(-II) systems not only exhibit simple oscil-
lations but also complex oscillations and chaos.5,13-15 At low
flow rate the system exhibits simple oscillations as discussed
earlier. As the flow rate is increased, a second oscillatory peak
begins to emerge after the initial large peak (Figure 3S) resulting
in period 2 oscillations (Figure 7). Unfortunately the model is
incapable of simulating chaos or other more complex phenom-
Acknowledgment. One of us (Q.G.) acknowledges funding
by the National Science Foundation of China (29573109) and
the Science fund of CUMT (A81301).

Nonlinear pH Dynamics in the Oxidation of Sulfur(-II)
J. Phys. Chem. A, Vol. 104, No. 49, 2000 11565
Supporting Information Available: Figure 1S, showing the
pH and mV vs Time. Figure 2S, showing simulated batch pH
vs Time trace for the shortened model. Figure 3S, showing
simulated CSTR pH vs time trace for changing concentrations
of oxidant. This Material is available free of charge via the
Internet at http://pubs.acs.org.
(16) R a´ bai, G.; Orb a´ n, M.; Epstein, I. R. J. Phys. Chem. 1992, 96, 5414.
(17) Beck, M. T.; R a´ bai, G. J. Phys. Chem. 1985, 89, 3907.
(18) R a´ bai, G.; Beck, M. T.; Kustin, K.; Epstein, I. R. J. Phys. Chem.
1
989, 93, 2853.
19) Epstein, I. R.; Orb a´ n, M. J. Am. Chem. Soc. 1989, 111, 2891.
(20) Peintler, G.; Nagypal, I.; Epstein, I. R. J. Phys. Chem. 1990, 94,
954.
21) Nagypal, I.; Epstein, I. R. J. Phys. Chem. 1986, 90, 6285.
(
2
(
(
(
22) Doona, C. J.; Stanbury, D. M. J. Phys. Chem. 1994, 98, 12630.
23) Evans, D. F.; Upton, M. W. J. Chem. Soc., Dalton Trans. 1985,
References and Notes
(
1) De Kepper, P.; Epstein, I. R.; Kustin, K. J. Am. Chem. Soc. 1981,
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1
(24) R a´ bai, G.; Kustin, K.; Epstein, I. R. J. Am. Chem. Soc. 1989, 111,
3870.
(
4
31.
(25) Mader, P. M. J. Am. Chem. Soc. 1958, 80, 2634.
(
(
(
(
(
(
3) Orb a´ n, M. React. Kinet. Catal. Lett. 1990, 42, 343.
(26) Kaps, P.; Rentrop, P. Numer. Math. 1979, 33, 55.
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(27) SIMULATE was provided courtesy of L. Gyorgi, Institute of
Inorganic and Analytical Chemistry, L. Eotvos University, H-1433, Budap-
est, Hungary, and R. J. Field, Department of Chemistry, University of
Montana, Missoula, MT 59812.
(28) Hindmarsh, A. C. Gear: Ordinary Differential Equation Solver;
9
5
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Technical Report No. UCM-30001, Rev. 2; Lawrence Livermore Labora-
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(
9) R a´ bai, G.; Orb a´ n, M. J. Phys. Chem. 1993, 97, 5935.
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