Temperature and ionic strength effects on some reactions involving sulfate radical [SO4-(aq)]

Article abstract of DOI:10.1021/jp9603703

Sulfate radical anion SO₄^- was generated by 248 nm laser flash photolysis of K₂S₂O₈ solutions and monitored by time-resolved multipass absorbance at 454 nm. The 320-520 nm absorption spectrum of SO₄^- was unaffected by up to 2 M added HClO₄, under the experimental conditions. Three reactions were investigated: (a) SO₄^- + SO₄^- → S₂O₈^2-, (b) SO₄^- + H₂O → HSO₄^- + OH, and (c) SO₄^- + S₂O₈^2- → products. Rate constant k_c was too slow to be measured, and only an upper limit was determined: k_c ≤ 10⁴ M^-1 s^-1. Arrhenius parameters were determined at low ionic strength over the range 11.8-74.4°C: 2k_a/∈ = (4.8 ± 2.0) × 10⁵ exp(-1.7 ± 1.1 kJ mol^-1/RT) cm s^-1 and k_b = (4.7 ± 0.1) × 10³ exp(-15.5 ± 0.6 kJ mol^-1/RT) M^-1 s^-1, where ∈ is the SO₄^- absorption coefficient at 454 nm. At 296 K, the values are in good agreement with literature values: 2k_a/∈ = (2.5 ± 0.2) × 10⁵ cm s^-1 and k_b[H₂O] = 440 ± 50 s^-1. Rate constants k_a and k_b were found to increase strongly and nonlinearly with increasing ionic strength (added NaClO₄) or acidity (added HClO₄). Ion-pair formation provides a possible explanation, and a quantitative empirical model is presented for conditions with [Na⁺] ≤ 1.6 M and [H⁺] ≤ 3 M. Using the ion-pair model, estimated ionization equilibrium constants are obtained for the H⁺SO₄^- and the Na⁺SO₄^- radical ion pairs.

Full text of DOI:10.1021/jp9603703

9780
J. Phys. Chem. 1996, 100, 9780-9787
Temperature and Ionic Strength Effects on Some Reactions Involving Sulfate Radical
-
[SO4 (aq)]
Zhen-Chuan Bao† and John R. Barker*,†,‡
Departments of Atmospheric, Oceanic, and Space Sciences and Chemistry, UniVersity of Michigan,
Ann Arbor, Michigan 48109-2143
ReceiVed: February 7, 1996; In Final Form: April 2, 1996X
Sulfate radical anion SO4- was generated by 248 nm laser flash photolysis of K2S2O8 solutions and monitored
-
by time-resolved multipass absorbance at 454 nm. The 320-520 nm absorption spectrum of SO4 was
unaffected by up to 2 M added HClO4, under the experimental conditions. Three reactions were
investigated: (a) SO4 + SO4 f S2O82-, (b) SO4 + H2O f HSO4 + OH, and (c) SO4 + S2O8
f
-
-
-
-
-
2-
products. Rate constant kc was too slow to be measured, and only an upper limit was determined: kc e 104
M-1 s-1. Arrhenius parameters were determined at low ionic strength over the range 11.8-74.4 °C: 2ka /ꢀ
) (4.8 ( 2.0) × 105 exp(-1.7 ( 1.1 kJ mol-1/RT) cm s-1 and kb ) (4.7 ( 0.1) × 103 exp(-15.5 ( 0.6 kJ
mol-1/RT) M-1 s-1, where ꢀ is the SO4- absorption coefficient at 454 nm. At 296 K, the values are in good
agreement with literature values: 2ka/ꢀ ) (2.5 ( 0.2) × 105 cm s-1 and kb[H2O] ) 440 ( 50 s-1. Rate
constants ka and kb were found to increase strongly and nonlinearly with increasing ionic strength (added
NaClO4) or acidity (added HClO4). Ion-pair formation provides a possible explanation, and a quantitative
empirical model is presented for conditions with [Na+] e 1.6 M and [H+] e 3 M. Using the ion-pair model,
estimated ionization equilibrium constants are obtained for the H+SO4 and the Na+SO4 radical ion pairs.
-
-
I. Introduction
Free radical reactions in cloud droplets and aerosols have
received increased attention in recent years and are now included
in some atmospheric chemistry models.1-14 In the troposphere,
cloud droplets and aerosols which are present as liquid or
supercooled liquid can absorb gases and free radicals from the
surrounding air mass. Chemical reactions inside the particles
can affect not only particle chemical and physical properties
but also gas phase chemistry.15 The rapid oxidation of SO2 to
sulfate observed in the troposphere can only occur in the aqueous
phase.16 Condensed phase reactions can also affect ozone in
the troposphere.17 In the stratosphere, aerosols consist largely
of sulfuric acid solutions, which not only have low freezing
points, but which can persist as supercooled liquids for long
periods.18 Low temperature, low pH, and high ionic strength
are common characteristics of some tropospheric and strato-
spheric aerosols. Laboratory experimental studies are needed
under these extreme conditions to investigate potentially
important aqueous phase atmospheric reactions.
Figure 1. Experimental apparatus. For clarity, not all components are
shown.
from 11.8 to 74.4 °C, in the acidity range [H+] from ≈ 0 to 3
M, and at ionic strengths ranging from ∼10-3 to >1.5 M. Not
all of these reactions are important in the atmosphere, but all
are important in laboratory studies of the atmospheric reactions
and therefore must also be understood. Where they can be
compared, the rate constants obtained in the present work are
generally similar to literature values, although they differ in
some details. The measured rate coefficients at high ionic
strength and acidity do not obey the Debye-Huckel theory, or
its usual extensions, but they can be rationalized quantitatively
with an ion-pair model.
Among many potentially important radicals found in cloud
droplets and aerosols, SO4- has a high redox potential and can
even oxidize Cl- .19 Investigations of SO32-(aq) photooxidation
-
implicate the SO4 radical anion as an important chain carrier
in the oxidation of SO2 in acid rain chemistry.21 However,
20
reactions of SO4- with other species can reduce the chain length.
For accurate atmospheric models, it is important to obtain rate
constants of these reactions under atmospheric conditions.
-
Many SO4 reaction rate constants have been measured in
the laboratory,22-37 but few have been measured at low
temperature, low pH, or high ionic strength. The aim of this
-
II. Experimental Section
work is to investigate SO4 reaction kinetics under such
conditions. We have determined rate constants for the reactions
The experimental apparatus consists of excimer laser flash
photolysis and time-resolved detection of transient species by
multipass absorbance (Figure 1). The 248 nm (KrF) excimer
laser (Lumonics HyperEx-400) was typically operated at 1.01
Hz with output energy of ∼80 mJ per pulse, although other
operating conditions were investigated, as described below.
of SO4- with SO4-, S2O82-, and H2O in the temperature range
* Address correspondence to this author (jrbarker@umich.edu).
† Department of Atmospheric Oceanic, and Space Sciences.
‡ Department of Chemistry.
X Abstract published in AdVance ACS Abstracts, May 15, 1996.
S0022-3654(96)00370-X CCC: $12.00 © 1996 American Chemical Society

Reactions Involving Sulfate Radical
J. Phys. Chem., Vol. 100, No. 23, 1996 9781
Laser pulse energies were measured with a calorimetric power
meter (Scientech). The laser beam was passed without focusing
through a rectangular mask, and it illuminated the reaction cell
nearly uniformly. The 2.5 × 2.5 × 4.7 cm (∼29 cm3)
rectangular reaction cell was custom-fabricated from Spectrosil
far-UV silica by Spectrocell Inc. Reaction mixtures were drawn
through the cell by a peristaltic pump (Masterflex Model 7553-
70) situated downstream from the cell; flow rates were varied
from ∼0.4 to 5 cm3/s. In experiments with added NaClO4 or
HClO4, the flow rates were maintained near ∼0.4-0.6 cm3 s-1
,
while in the other experiments, flows were ∼5 cm3 s-1. The
reaction solutions were not recirculated, and they came in
contact only with Pyrex glass and Teflon prior to entering the
reaction cell.
Light from a 200 W Hg-Xe arc lamp (Oriel Model 6291)
was passed through a lens for collimation, then through a liquid
water filter to remove infrared radiation, and finally through a
400 nm cutoff filter (Oriel Filters 51265) to eliminate the UV
fraction. A White cell38 with end mirrors of 15 cm radius of
curvature was used with 12-16 passes to produce a total
absorption pathlength of 60-80 cm. Various apertures and
lenses were used to direct the analytical beam and to minimize
scattered laser light (for clarity, Figure 1 does not show all
details). Light from the White cell was directed to a mono-
chromator (Jarrell-Ash Model NO3) equipped with a photo-
multiplier tube (Hamamatsu 1P28). The photomultiplier anode
current was maintained less than ∼4 µA in order to insure
linearity of response. Two ∼400 nm cutoff filters (Oriel Filters
51265 and 51272) were used at the monochromator entrance
slit to minimize scattered laser light. The photomultiplier output
was terminated by a 1K Ω resistor; the resulting voltage signal
was amplified (Tektronix AM502) and captured with a digital
oscilloscope (LeCroy 9400). Signals were averaged for 150-
400 laser shots by the oscilloscope and stored on a laboratory
computer (Macintosh, Apple Computer) for further analysis. The
oscilloscope was triggered by a pyroelectric detector (Molectron
Model P3-D1) which monitored the laser pulse. By using the
pretrigger feature of the oscilloscope, the transmitted intensity
I0 prior to the pulse was recorded, as well as the time-dependent
intensity I(t) following the pulse.
The temperature of the reaction solution was measured at
both entrance and exit of the cell by calibrated copper-
constantan thermocouples installed in glass thermocouple wells.
The average reading from two thermocouples was used to
establish the solution temperature. Typically, the two temper-
atures differed by e0.5 °C, except at the extreme high and low
temperatures of the experiments, where the differences were
sometimes as large as 2 °C.
All solutions were freshly prepared immediately before use
from the following reagents: K2S2O8 (Fisher), >99.5%, certi-
fied; HClO4 (Fisher), 70%, reagent ACS; NaClO4 (Aldrich),
98%, ACS reagent. The water used for the preparation of the
solutions was purified by a Millipore Milli-Q system, and the
resistivity was >16 M Ω cm.
Figure 2. Time-resolved transmittances (points) at room temperature
(296 K) for the concentrations shown. Nonlinear least-squares fits are
shown as barely visible solid lines through the data.
immediate increase in the absorbance observed at the 454 nm
observation wavelength, followed by a decay, as shown in
-
Figure 2. The SO4 decay kinetics can be described by the
following reactions, according to McElroy and Waygood:24
2-
SO4- + SO4- f S2O8
(2)
(3)
(4)
SO4- + H2O f OH + HSO4
-
SO4- + S2O82- f SO42- + S2O8
-
-
This mechanism predicts that the SO4 decay is described by
mixed first- and second-order kinetics. The time-dependent
transmittance of the cell depends on the absorbance A(t)
according to the Beer-Lambert equation:
I(t)
I0
) exp(-A(t))
(5)
For the mechanism above, the absorbance A(t) can be shown
to depend on the sulfate concentration [SO4-] as follows:24
-1
C2
C2
1
-
A(t) ) ꢀl(SO4 ] ) exp(C1t)
+
-
(6)
{
}
[
]
A0 C1l
C1l
where ꢀ is the molar absorption coefficient (base e) of SO4-, l
is the optical pathlength, t is time, and A0 ) ꢀl[SO4-]0 is the
initial absorbance (which depends on the initial SO4- concentra-
tion). Nonlinear least-squares fits of eq 6 are also presented in
Figure 2, and it is clear that the fits are very good. For the
above mechanism, constant C1 is equal to the pseudo-first-order
rate constant:
III. Results and Discussion
1. Chemical Mechanism and Data Analysis. In the present
work, the sulfate radical (SO4-) was generated by 248 nm laser
flash photolysis of (1-2) × 10-3 M K2S2O8 solutions:28,39
2-
kI ) k3[H2O] + k4[S2O8
]
(7)
where k3 and k4 are the rate constants of (3) and (4), respectively.
Constant C2 is given by C2 ) 2k2/ꢀ, where k2 is the rate constant
for (2). By varying experimental conditions and determining
constants A0, C1, and C2 by nonlinear least-squares fits, it is
possible to determine the three rate constants. By varying
detection wavelength, it is possible to determine the relative
absorption coefficients ꢀ(λ) as a function of wavelength. Note
-
S2O82- + hν f 2SO4
(1)
-
The SO4 absorption spectrum has a broad absorption band
in the visible region with peak absorbance around 450
nm.22,24-26,29,32,40-42 Following the laser flash, there is an

9782 J. Phys. Chem., Vol. 100, No. 23, 1996
Bao and Barker
-
Figure 3. The SO4 absorption spectrum at 296 K.
that k2 cannot be determined independently from the absorption
coefficient ꢀ. As discussed below, (6) can also be used to fit
the data obtained with added NaClO4 and HClO4, but constants
C1 and C2 take different meanings. The nonlinear least-squares
fits were carried out using KaleidaGraph v. 3.0.4 (Abelbeck
Software) for Power Macintosh computers. KaleidaGraph
utilizes the Marquardt-Levenberg algorithm, and we verified
the accuracy of the software by comparison with a well-
established code for the algorithm.43
Figure 4. Plots of 2k2/ꢀ and kI vs [S2O82-] at room temperature (296
K). The solid lines are linear least-squares fits. The small dependences
on [S2O82-] are not significant (see text).
TABLE 1: Experimental Parameters
All of the values for C1 and C2 presented in the figures are
tabulated in Table IS (Supporting Information) along with the
experimental conditions for each experiment.
parameter
range
laser repetition rate
laser pulse energy
number of white cell passes
sample flow rate
probe light intensity (450 nm)
[K2S2O8]
0.11-19 Hz
∼9-130 mJ
1-16
-
2. The SO4 Absorption Spectrum. In the present work,
-
the spectrum of SO4 was measured over the range 320-520
∼0.5-14 cm3 s-1
∼10 times
nm following 248 nm laser flash photolysis of K2S2O8 solutions,
and it is shown in Figure 3. In a series of runs, data for I(t)/I0
were fitted as described above and the fitted values for A0 were
obtained. We assumed that A0 ) ꢀ(λ)l[SO4-]0 and [SO4-]0 )
0.1-2 mM
2.22 In the present work, we have not measured the absolute
value of ꢀ and we will only report values for 2k2/ꢀ, rather than
absolute values of k2.
σ
248φ248[K2S2O8]Q, where σ248 is the absorption cross section
2-
-
of S2O8 at 248 nm, φ248 is the quantum yield for SO4
3. Experimental Parameters. Experimental parameters
such as laser repetition rate, laser pulse energy, analytical light
intensity and sample flow rate may affect the observed reaction
rates, due to accumulation of reaction products in the cell and
multiphoton effects. Signal averaging is used to improve the
signal/noise ratio in the present experiments and thus many laser
pulses are needed. However, if the cell is completely emptied
and refilled between laser shots, large volumes of solution are
consumed. Thus it is particularly important to determine
whether the results depend on laser pulse repetition frequency
and solution flow rate. No such dependence was observed.
Table 1 lists experimental parameters that were investigated;
the experimental results showed no systematic dependence on
any of these parameters within the ranges investigated.
2-
production from S2O8 at 248 nm, and Q is the 248 nm laser
fluence. According to these assumptions, the absorption cross
section is related to measured values of A0, [K2S2O8], and Q
according to (8):
A0
ꢀ(λ)
(8)
[K2S2O8]Q
The relative absorption coefficients agree well with prior
work,22,24-26,29,32,40-42 except that the absorption peak occurs
near 440 nm, rather than 450 nm. When 1, 1.5, and 2 M of
perchloric acid (HClO4) are added to the solution, the observed
absolute cross sections remained constant within (20% (as-
suming σ248 and φ248 to be independent of added HClO4). The
relative absorption coefficients presented in Figure 3 show little
variation as HClO4 is added, in agreement with other investiga-
tions.22,26,36 This lack of variation indicates that the protonated
sulfate radical (HSO4) has the same spectrum as the sulfate
radical anion. Although at first surprising, the SO4- radical is
not unique in having the same spectrum in ionic and protonated
forms: in the carbonate radical system, CO3-, and HCO3 have
virtually identical spectra.44
-
4. Rate Constant Determinations. Three typical SO4
time-resolved transmission profiles observed after 248 nm laser
flash photolysis of aqueous S2O82- solution are shown in Figure
2. Because the pseudo-first-order rate constant kI depends on
[S2O82-] according to (7), we carried out experiments with
[S2O82-] ranging from 1 × 10-4 to 2 × 10-3 M at two
temperatures (296 and 321 K). The results obtained at 296 K
are presented as plots of 2k2/ꢀ and kI vs [S2O82-] in Figure 4;
the results obtained at 321 K are qualitatively similar. The
minor dependences of the rate coefficients on [S2O82-] are
probably due to correlations between C1 and C2 in the nonlinear
least squares analysis and are not physically significant. This
conclusion is consistent with the nonphysical decrease in kI as
[S2O82-] is increased. That neither rate constant varies signifi-
cantly with [S2O82-] is in contrast with the results for kI reported
To obtain k2 from measured values of 2k2/ꢀ, a value for the
absorption coefficient is required. Recent values for ꢀ at 450
nm obtained by other workers are in good agreement.3,22,24 For
example, Tang et al.22 found ꢀφ248 ) 2770 ( 280 M-1 cm-1
and McElroy 23 reported ꢀ ) 1600 ( 100 M-1 s-1 (both values
are base 10). These are in good agreement, based on φ248
≈

Reactions Involving Sulfate Radical
J. Phys. Chem., Vol. 100, No. 23, 1996 9783
TABLE 2: Recent Data on Sulfate Radical Reactions Near 300 K
SO4- sourcea
pH
µ (mM)
kI (s-1
)
2k2/ꢀ (105 cm s-1
5.6b
)
reference
S2O82- LFP (193 nm)
S2O82- LFP (248 nm)
S2O82- LFP (248 nm)
H2SO4 PR
0.3-3
1.5-410
0.75-8.3
concd acid
not reported
0.28-6.1
not considered
360 ( 90
Huie et al. (1989, 1993)
Tang et al. (1988)
McElroy and Waygood (1990)
Jiang et al. (1992)
Herrmann et al. (1995)
this work
1.5-5.4
4.9-5.2
concd acid
5
3.9 ( 1.1 (at µ ) 0)
5.5 ( 0.2 (at µ ) 0)
9.5 (at all [H+])
2.3 ( 0.6b
500 ( 60
500 (assumed)
660 ( 40
S2O82- LFP (248 nm)
S2O82- LFP (248 nm)
4.8-5.8
440 ( 50
2.5 ( 0.2 (µ ≈ 0)
a LFP ) laser flash photolysis; PR ) pulse radiolysis. b Obtained by dividing a reported rate constant by ꢀ/2 ) 1385/2 M-1 cm-1
.
TABLE 3: Differences Due to Fitting Method
fit I(t)/I0 vs time
fit A(t) vs time
kI (s-1
2k2/ꢀ (105 cm s-1
)
440
2.5
442
3.4
)
by McElroy et al.24 and Herrmann et al.,37 but it is in agreement
with the results of Tang et al.22 From our results, we estimate
that the upper limit to the rate of (4) is k4 e 104 M-1 s-1
.
At 296 K, the average values obtained for 2k2/ꢀ and kI are
(2.5 ( 0.2) × 105 cm s-1 and 440 ( 50 s-1, respectively, where
the (1σ errors represent precision only. Our results are
compared with those from other recent investigations in Table
2, where it is apparent that our results for 2k2/ꢀ tend to fall near
the lower end of the range. One possible reason for the range
of reported rate constants is because least squares fitting of I(t)/
I0 instead of A(t) gives systematically different results. In the
experiments, I(t) and I0 were measured directly; transforming
them to A(t) by taking the logarithm of I(t)/I0 requires that the
data be weighted unequally when carrying out the least-squares
analysis. If the change in weighting is neglected in the least-
squares analysis, the accuracy of the results can be affected.
As a test of this effect, we repeated the least-squares analysis
for several runs, but fitted A(t) vs t, rather than I(t)/I0 vs t. The
results are listed in Table 3, and it is clear that fitting A(t)
produces significantly larger values for 2k2/ꢀ, while kI is affected
only slightly. The magnitude of this difference is the same as
the difference between our results for 2k2/ꢀ and those of Tang
et al.22 Since directly fitting I(t)/I0 vs t with equally weighted
data is the proper procedure, all of the rate constants reported
in this work were obtained by that method.
Figure 5. Arrhenius plot of 2k2/ꢀ (points) and nonlinear least-squares
fit (line).
over the range 11.8-74.4 °C, and the results are presented in
Figures 5 and 6. Nonlinear least-squares fits gave the following
Arrhenius expressions:
2k2/ꢀ )
(4.8 ( 2.0) × 105 exp(-1.7 ( 1.1 kJ mol-1/RT) cm s-1
(12)
kI )
(2.6 ( 0.6) × 105 exp(-15.5 ( 0.6 kJ mol-1/RT) s-1 (13)
k3 )
2-
The reduction potential for the SO4-/SO4 couple can be
(4.7 ( 0.1) × 103 exp(-15.5 ( 0.6 kJ mol-1/RT) M-1 s-1
determined from our result for k3 and results obtained by Tang
et al.22 for the reverse reaction, which can be neglected under
our conditions, because [HSO4-] is negligible. From the present
work, k3 ) kI/[H2O] )7.9 M-1 s-1 and from Tang et al.,22 k-3
) 3.5 × 105 M-1 s-1. Thus the equilibrium constant K3 is
(14)
where kI has been identified with k3[H2O] and the uncertainties
are (1σ statistical errors. The rate constant for (2) is nearly
independent of temperature, in agreement with the results
obtained by Huie et al.48 The activation energy for (3) is
consistent with the fact that the reaction is endothermic. The
values obtained in the present work for k3 are in fair agreement
with those reported by Herrmann et al.,37 but both the activation
energy and A factor differ significantly. The reason for this
difference cannot be determined, because no experimental details
or actual experimental results are described in that report.37
6. pH and Ionic Strength Dependence. In concentrated
solutions, the ionic strength is an important parameter, because
each ion is surrounded by an extended solvation shell which
can affect ionic activities and rate constants. The influence of
ionic strength on reaction rate constants k2 and k3 has been
investigated previously for µ e 0.4.22,24 In the present work,
we carried out systematic experiments at ionic strengths up to
>1.5 M by adding NaClO4 to the K2S2O8 solutions. At NaClO4
concentrations greater than ∼2 M, the results were not
reproducible when different batches of reagent were used,
probably due to the presence of impurities. Impurities may also
be responsible for the appearance of tiny suspended particles,
k3
K3 )
) 2.26 × 10-5
(9)
k-3
which is related to the reduction potential difference ∆E3°
between the couples by the expression45
∆E3°/mV ≈ 59.1 log K3
(10)
where
-
∆E3° ) E°(SO4 /SO42-) - E°((H+, OH)/H2O) (11)
If we take E°((H+, OH)/H2O) ) 2.72 V,46 we find E°(SO4
/
-
2-
SO4 ) ) 2.45 V, which is in very good agreement with a
recommended value: 2.43 V.47
5. Temperature Dependence. Although (2) and (3) have
been studied previously in several investigations at room temp-
erature, only one temperature-dependent measurement has been
performed for (2)48 and for (3).37 We carried out experiments

9784 J. Phys. Chem., Vol. 100, No. 23, 1996
Bao and Barker
Figure 6. Arrhenius plot of kI (points) and nonlinear least-squares fit
(line).
Figure 7. Ionic strength dependence of 2k2/ꢀ at 296 K; the solid line
is a least-squares fit of the experimental data to the DHBD equation.
The Debye-Huckel (DH) theory limiting slope is shown for comparison
(broken line). The ionic strength was varied by adding NaClO4.
which scattered the laser light, at high NaClO4 concentrations.
Thus we report only the results up to ∼2 M of NaClO4 (µ ≈
1.6 M) where the results were reproducible.
At relatively low ionic strengths, transition state theory can
be combined with the Debye-Huckel (DH) theory and an
empirical term (-bµ) introduced by Davies to obtain the
Debye-Huckel-Bronsted-Davies (DHBD) equation:49
µ1/2
1 + µ1/2
log k ) log k° + 2ZAZBA
- bµ
(15)
{
}
where k is the observed rate constant, k° is the rate constant at
infinite dilution, A is the Debye-Huckel constant (A ) 0.509
at 298 K), ZA and ZB are charges for species A and B,
respectively, µ is the total ionic strength, and b is an empirical
parameter. Equation 15 predicts an increase of k with increasing
ionic strength for the reaction between two ions of like sign.
For a few reaction systems for which data are available, (15)
with b ≈ 0.2-0.3 is reasonably accurate up to moderate ionic
strengths (µ < 0.5).50 Note that the activity coefficient of an
ion does not monotonically approach zero as the ionic strength
increases, but it can increase again for large µ, because the
amount of solvent available for solvation of ions decreases as
µ increases; decreasing the amount of water tends to reduce
solvation screening and the ions become more active.51
If one of the reactants is a neutral molecule, then ZB ) 0 in
(15) and the rate constant depends on the empirical linear term.52
At higher ion concentrations, higher order terms may become
important, but usually only the linear term is retained:
Figure 8. Ionic strength dependence of kI; the broken line is merely
intended to guide the eye. The ionic strength was varied by adding
NaClO4.
Figure 8 is totally inconsistent with (16), which predicts a
straight line. Higher order empirical terms could be added to
(16) to account for the strong “saturation” behavior, but they
provide no further insight. We find it more satisfactory to seek
a physical explanation for the observed behavior. The ion-pair
mechanism described below provides such an explanation and
gives a self-consistent description of all of our experimental
data.
log k ) log k° - b′µ
(16)
The concept of ion-pair formation was introduced by Bjerrum
in order to explain the strong nonlinear dependence of some
reaction rate constants on ionic strength.54 He proposed that
the Coulombic attraction between ions of opposite charge could,
to some extent, overcome the thermal energy that tends to
separate ions after they have been hydrated in aqueous solutions.
Later workers invoked ion-pair formation to explain the
numerous deviations of reaction rates from the Debye-Huckel
theory in studies of salt effects.55-63 Davies64 pointed out that
ion association can modify the reaction in two ways: firstly,
ion pair formation affects the total ionic strength of the medium;
secondly, one (or more) such ion pairs may be involved in the
where b′ is an empirical constant which depends on the
difference between the activities of the reactants and that of
the transition state: not even its sign can be predicted.53
Plots of some of the data for 2k2/ꢀ and k3 as functions of
µ
1/2/(1 + µ1/2) and µ, respectively, at 296 K are presented in
Figures 7 and 8, where it is clear that both 2k2/ꢀ and k3 increase
with increasing ionic strength. The results for 2k2/ꢀ are in
reasonable agreement with those obtained by Tang et al.22 (aside
from the differences in least-squares fitting discussed above),
and they can be fitted by the DHBD equation (15), as shown in
Figure 7. In sharp contrast, the plot of k3[H2O] vs µ shown in

Reactions Involving Sulfate Radical
J. Phys. Chem., Vol. 100, No. 23, 1996 9785
rate-determining step, thus altering the charge of the activated
complex and reaction rate.
In the present work, large excesses of NaClO4 or HClO4 were
added to K2S2O8 solutions. Neutral ion pairs (e.g. K+SO4
,
-
Na+SO4-, and H+SO4-) can coexist in equilibrium with single
-
ions (e.g. SO4 , K+, Na+, etc.). It is possible that these ion
pairs react even more rapidly than the single ions, because of
the reduction in the charge of the activated complex.64 The
higher their charge and the smaller their radius in solution, the
more efficient the cations become in accelerating the rate of
reaction between ions of like sign.49 Reactions between an ion
and a neutral ion pair can also have a larger rate constant than
that for the reaction between two ions.
When NaClO4 is in great excess, low concentration species
-
-
(e.g. K+, SO4 , S2O82-, K+SO4-, Na+SO4-, and H+SO4
)
contribute little to the total ionic strength, compared with Na+,
ClO4-, and the ion pair Na+ClO4-. The Na+ClO4- association
constant is K ≈ 0.2 M-1 at room temperature,65 but the
dependence of this association constant on ionic strength is not
known. In order to calculate the total ionic strength, we
neglected the minor species and assumed K ) 0.2 M-1 for
Figure 9. Constant C1 vs [Na+] (points) and least-squares fits according
to the ion pair mechanism (lines) at two temperatures. See text for
details.
Na+ClO4 at all µ.
-
A mechanism incorporating sulfate radical ion pairing can
be written by neglecting possible ion complexes containing more
than one cation.52 In the following reaction scheme, Y ) Na
or H:
2-
SO4- + SO4- f S2O8
(2a)
(2b)
(2c)
(3a)
(3b)
SO4- + Y+SO4- f Y+ + S2O8
2-
Y+SO4- + Y+SO4- f 2 Y+ + S2O8
2-
SO4- + H2O f OH + HSO4
-
Figure 10. Constant C2 vs [Na+] (points) and least-squares fits
according to the ion pair mechanism (lines) at two temperatures. See
text for details.
-
Y+SO4- + H2O f OH + Y+ + HSO4
For the Y+SO4- radical ion pair, there is an association constant
KY ) [Y+SO4-]/[Y+][SO4-], and we assume that this equilib-
rium is maintained at all times. The differential equations for
this reaction set can be integrated and the result is the same as
(6), except that constants C1 and C2 take the following meanings:
C1 ) {ꢀe(KY[Y+] + 1)}-1{ꢀik3a + ꢀipk3bKY[Y+]} (17)
C2 ) {ꢀe(KY[Y+] + 1)}-2
×
2
{2ꢀik2a + (ꢀi + ꢀip)k2bKY[Y+] + 2ꢀipk2cKY [Y+]2} (18)
-
where ꢀi and ꢀip are the absorption coefficients for SO4 and
Y+SO4-, respectively, and ꢀe is given by the following expres-
sion:
ꢀipKY[Y+]
Figure 11. Constant C1 vs [H+] (points) and least-squares fit according
to the ion pair mechanism (line) at 296 K. See text for details.
ꢀi
(1 + KY[Y+]) (1 + KY[Y+])
ꢀe )
+
(19)
Figures 9-12, where the data were least-squares fitted to (6)
to obtain values of C1 and C2. In Figures 9 and 11, the solid
lines were obtained from nonlinear least-squares fits of C1 as
functions of [Na+] and [H+], respectively, according to (17) to
obtain values of k3a, k3b, and KY (for Y ) Na, H). We then
used the resulting values of KY in (18) and carried out least-
squares fits of C2 as functions of [Na+] and [H+], in order to
determine the remaining rate coefficients. The least-squares fits
included the ionic strength dependence of k2a, as described
In the present system, these expressions simplify, because the
absorption spectrum does not depend on [H+] and [Na+];
therefore ꢀi ) ꢀip. Rate constant k2a is probably the most
dependent on µ, because it involves two anions; the others
involve an ion and a neutral, or two neutrals. Therefore, we
assumed that k2a obeys (15) with b ) 0.3, but the other
individual rate coefficients are independent of µ.
The experimental results obtained using added NaClO4 and
HClO4 are consistent with this ion pair model, as shown in

9786 J. Phys. Chem., Vol. 100, No. 23, 1996
Bao and Barker
TABLE 4: Parameters for Ion-Pair Mechanism
Y ) Na+ (296 K)
Y ) Na+ (278.5 K)
Y ) H+ (296 K)
KY ) [YSO4]/[Y+][SO4-] (M-1
(2ꢀi/ꢀe2)k2a° (105 cm s-1
((ꢀi + ꢀip)/ꢀe2)k2b (105 cm s-1
(2ꢀip/ꢀe2)k2c (105 cm s-1
(ꢀi/ꢀe)k3a [H2O] (s-1
(ꢀip/ꢀe)k3b [H2O] (s-1
)
0.99 ( 0.25
2.48 ( 0.02
2.5 ( 0.8
2.2 ( 0.8
1.86 ( 0.04
3.5 ( 0.5
0.7 ( 0.5
2.42 ( 0.03
1.9 ( 0.8
)
)
)
12.8 ( 0.8
436 ( 7
6.4 ( 0.3
12.5 ( 0.7
432 ( 7
)
302 ( 20
1030 ( 97
)
2150 ( 215
781 ( 124
much smaller than k2c, however, and this difference produces
the “kinks” in the plots.
The ion-pair model and the rate constants summarized in
Table 4 provide a complete quantitative description of the
reaction system for [Na+] e 1.6 M and [H+] e 3 M. Although
the kinetics evidence indicates an ion-pair mechanism is
operative, this conclusion has not been investigated with other
independent measurements. The assumed reaction model is
consistent with the experimental kinetics data, but these data
may be consistent with other models, as well. The present
mechanism provides a good description of the experimental
results, but it is not necessarily correct. Moreover, the least-
squares fits of these rate coefficients are somewhat statistically
correlated and they depend on the equilibrium constant, which
is itself somewhat uncertain. In light of these considerations,
the numerical rate constants and equilibrium constants are only
applicable within the experimental range of the present inves-
tigation and should not be used for extrapolations outside this
range.
Figure 12. Constant C2 vs [H+] (points) and least-squares fit according
to the ion pair mechanism (line) at 296 K. See text for details.
above. The parameters extracted from this analysis are sum-
marized in Table 4.
First, consider the association equilibrium constants. Davies52
observed that for inorganic salts a decrease of one unit of
negative charge leads to a decrease by about a factor of 10 in
the association equilibrium constant, as long as the ionic size
IV. Atmospheric Implications
Because of the relatively low concentration of SO4- generated
in the atmospheric condensed phase, (2) is probably not
important and need not be included in atmospheric chemistry
models. This eliminates the need to consider many of the
complexities of the ion-pair mechanism in atmospheric models.
In laboratory studies, however, (2) is very important and must
be included in a complete mechanism. Since liquid water is
usually present in the atmospheric condensed phase, (3) can be
important and it is included in a model developed by Jacob et
al.,66 although with an incorrect estimate for the activation
energy, which has been measured in the present work.
An important feature of the present work is use of the ion-
pair model to explain the rate constant dependences on pH and
ionic strength. The present work cannot prove or disprove the
ion-pair model, but if it is correct, it can be a very useful tool
in modeling high-concentration environments, such as occur in
stratospheric aerosols, which are composed principally of 70-
80 wt % sulfuric acid with traces of a variety of other
compounds,67 and in sea salt aerosols, which contain highly
concentrated NaCl solutions. In future work, we will investigate
reactions in such environments. Since rate coefficients depend
strongly and unpredictably on the ionic strength, much further
work remains to be done.
remains constant. The NaSO4 association constant52 at 296
-
K is about K ≈ 5 M-1 and that for the Na+SO4 association
-
was found in the present work to be about 0.99 ( 0.25 M-1
(see Table 4), in reasonable agreement with Davies’ observation.
-
For the HSO4 ion, the association constant is about K ≈
102 M-1 49
,
while that for the H+SO4 ion pair was found in
-
the present work to be 0.7 ( 0.5 M-1 (Table 4). The ratio of
the two association constants is greater than predicted by Davies,
but it is consistent with a suggestion by Jiang et al.36 that this
equilibrium be compared with that of the isoelectronic phosphate
radical and some other structurally similar radicals. All of the
radicals listed by Jiang et al. are much stronger acids than their
parent species, and the association constant ratios are often of
the order of 102, similar to the HSO4-/H+SO4 comparison.
-
It is difficult to determine whether the fitted rate coefficients
for (2a)-(2c) are reasonable. The fitted results for (2a) in the
limit of µ ) 0 are quite consistent with the results obtained
above for µ ≈ 0, but this is expected, since the µ ≈ 0 data are
included in the least-squares fits. For Y ) Na, rate constant
k2b, which is assumed to have no dependence on µ, is in close
agreement with the k2a° result, while rate constant k2c is 3-5
times as large, in general agreement with the conclusion that a
larger rate coefficient is expected when the charge of the
transition state is reduced, as mentioned above. For Y ) H,
however, rate constant k2b is less than k2a°, the reverse of the
expected trend. Considering the large uncertainty in k2b,
however, this inconsistency is not serious.
Acknowledgment. We thank NASA for financial support
under the auspices of the Atmospheric Effects of Aviation/
Subsonic Assessment Program. We also thank Dr. Paul Wine
and Dr. Robert Huie for very helpful conversations and
suggestions regarding experimental details.
Supporting Information Available: Measured parameters
and experimental conditions for all the data presented in the
figures (Table IS) (6 pages). Ordering information is given on
any current masthead page.
The nonmonotonic behavior exhibited by C2 in Figures 10
and 12 comes directly from the nonlinear least-squares fits and
is due to the changing composition of the solutions as Y+
)
Na+, H+ concentrations are varied. At higher [Y+], the ion-
pair concentrations increase and the simple ion concentrations
decrease. These changes cause C2 to depend more on k2b and
k2c than on k2a at high [Y+]. Rate constants k2a and k2b are
References and Notes
(1) Chameides, W. L.; Davis, D. D. J. Geophys. Res., D: Atmos. 1982,
87, 4863.

Reactions Involving Sulfate Radical
J. Phys. Chem., Vol. 100, No. 23, 1996 9787
(2) Graedel, T. E.; Goldberg, K. I. J. Geophys. Res., D: Atmos. 1983,
88, 10865.
(3) Chameides, W. L. J. Geophys. Res., D: Atmos. 1984, 89, 4739.
(4) Jacob, D. J. J. Geophys. Res., D: Atmos. 1986, 91, 9807.
(5) Jacob, D. J.; Gottlieb, E. W.; Prather, M. J. J. Geophys. Res., D:
Atmos. 1989, 94, 12975.
(6) Heikes, B. G.; Thompson, A. M. J. Geophys. Res., D: Atmos. 1983,
88, 10883.
(7) Lelieveld, J.; Crutzen, P. J. Nature 1990, 343, 227.
(8) Dentener, F. J.; Crutzen, P. J. J. Geophys. Res., D: Atmos. 1993,
98, 7149.
(9) Warneck, P. Fresenius J. Anal. Chem. 1991, 340, 585.
(10) Warneck, P. Ber. Bunsenges. Phys. Chem. 1992, 96, 454.
(11) Graedel, T. E.; Weschler, C. J. ReV. Geophys. Space Phys. 1981,
19, 505.
(12) Chameides, W. L.; Stelson, A. W. Ber. Bunsenges. Phys. Chem.
1992, 96, 461.
(13) Colvile, R. N.; Sander, R.; Choularton, T. W.; Bower, K. N.; Inglis,
D. W. F.; Wobrock, W.; Schell, D.; Svenningsson, I. B.; Wiedensohler,
A.; Hansson, H.-C.; Hallberg, A.; Ogren, J. A.; Noone, K. J.; Facchini, M.
C.; Fuzzi, S.; Orsi, G.; Arends, B. G.; Winiwarter, W.; Schneider, T.; Berner,
A. J. Atmos. Chem. 1994, 19, 189.
(14) Sander, R.; Lelieveld, J.; Crutzen, P. J. J. Atmos. Chem. 1995, 20,
89.
(15) Kolb, C. E.; Worsnop, D. R.; Zahniser, M. S.; Davidovits, P.;
Keyser, L. F.; Leu, M.-T.; Molina, M. J.; Hanson, D. R.; Ravishankara, A.
R.; Williams, L. R.; Tolbert, M. A. Problems and Progress in Atmospheric
Chemistry; Barker, J. R., Ed.; World Scientific publishing Co.: Singapore,
1995; p 771.
(16) Warneck, P. Chemistry of the Natural Atmosphere; Academic Press,
Inc.: San Diego, 1988.
(17) Lilieveld, J.; Crutzen, P. J. Nature 1990, 343, 227.
(18) Dye, J. E.; Baumgardner, D.; Gandrud, B. W.; Kawa, S. R.; Kelly,
K. K.; Loewenstein, M.; Ferry, G. V.; Chan, K. R.; Gary, B. L. J. Geophys.
Res. 1992, 97, 8015.
(19) Huie, R. E. Problems and Progress in Atmospheric Chemistry;
Barker, J. R., Ed.; World Scientific publishing Co.: Singapore, 1995; p
374.
(20) Deister, U.; Warneck, P. J. Phys. Chem. 1990, 94, 2191.
(21) Scire J. A.; Venkatram, A. Atmos. EnViron. 1985, 19, 637.
(22) Tang, Y.; Thorn, R. P.; Mauldin, R. L.; Wine, P. H. J. Photochem.
Photobiol. A 1988, 44, 243.
(23) McElroy, W. J. J. Phys. Chem. 1990, 94, 2435.
(24) McElroy, W. J.; Waygood, S. J. J. Chem. Soc., Faraday Trans.
1990, 86 (14), 2557.
(33) Maruthamuthu, P.; Neta, P. J. Phys. Chem. 1978, 82, 710.
(34) Eibenberger, H.; Steenken, S.; Neill, P. O. J. Phys. Chem. 1978,
82, 749.
(35) Huie, R. E.; Neta, P. Atmos. EnViron. 1987, 21, 1743.
(36) Jiang, P. Y.; Katsumura, Y.; Nagaishi, R.; Domae, M.; Ishikawa,
K.; Ishigure, K. J. Chem. Soc., Faraday Trans. 1992, 88 (12), 1653.
(37) Herrmann, H.; Reese, A.; Zellner, R. J. Mol. Struct. 1995, 348,
183.
(38) White, J. U. J. Opt. Soc. Am. 1942, 32, 285.
(39) Hayon, E.; McGarvey, J. J. J. Phys. Chem. 1967, 71, 1472.
(40) hayon, E.; McGarvey, J. J. J. Phys. Chem. 1967, 71, 1472.
(41) Robke, W.; Renz, M.; Henglein, A. Int. J. Radiat. Phys. Chem.
1969, 1, 39.
(42) Lesigue, B.; Ferradini, C.; Pucheault, J. J. Phys. Chem. 1973, 77,
2156.
(43) Bevington, P. R. Data Reduction and Error Analysis for the
Physical Sciences; McGraw-Hill: New York, 1969; Chapter 11.
(44) Chen, S.-N.; Cope, V. W.; Hoffman, M. Z. J. Phys. Chem. 1973,
77 (9), 1111.
(45) Wardman, P. J. Phys. Chem. Ref. Data 1989, 18 (4), 1639.
(46) Stanbury, D. M. AdV. Inorg. Chem. 1989, 33, 75.
(47) Huie, R. E.; Clifton, C. L.; Neta, P. Radiat. Phys. Chem. 1991, 38
(5), 477.
(48) Huie, R. E.; Clifton, C. L. Chem. Phys. Lett. 1993, 205 (2, 3), 163.
(49) Perlmutter-Hayman, B. Progr. React. Kin. 1971, 6, 239.
(50) Pankow, J. F. Aquatic Chemistry Concepts; Lewis Publishers: Boca
Raton, FL, 1991; p 48.
(51) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry;
Plenum Press: New York, 1970, Vol. 1, p 238.
(52) Davies, C. W. Ion Association; Butterworths: North Ryde,
Australia, 1962.
(53) Moore, J. W.; Pearson, R. G. Kinetics and Mechanism; John Wiley
& Sons: New York, 1981; p 272.
(54) Bjerrum, N. K. Dan. Vidensk. Selsk. 1926, 7, 1.
(55) Bronsted, J. N.; Livingston, R. J. Am. Chem. Soc. 1927, 49, 435.
(56) Olson, A. R.; Simonson, T. R. J. Chem. Phys. 1947, 17, 1167.
(57) Bell, R. P.; Prue, J. E. J. Chem. Soc. 1949, 362.
(58) Lamer, V. K. J. Am. Chem. Soc. 1929, 51, 3341, 3678.
(59) King, C. V.; Jacobs, M. B. J. Am. Chem. Soc. 1931, 53, 1704.
(60) Howells, W. J. J. Chem. Soc. 1939, 463.
(61) Indelli, A.; Prue, J. E. J. Chem. Soc. 1959, 107.
(62) Bronsted, J. N.; Delbanco, A. Z. Anorg. Chem. 1925, 144, 248.
(63) Holmberg, B. Z. Phys. Chem. 1913, 81, 339.
(64) Davies, C. W. Progr. React. Kin. 1961, 1, 163.
(65) Ho¨gfeldt, E. Stability Constants of Metal-Ion Complexes. Part A:
Inorganic Ligands, IUPAC Chemical Data Series, No. 21; Pergamon
Press: New York, 1982.
(25) Heckel, V. E.; Henglein, A.; Beck, G. Ber. Bunsenges. Phys. Chem.
1966, 70, 149.
(26) Dogliotti, L.; Hayon, E. J. Phys. Chem. 1967, 71, 2511.
(27) Dogliotti, L.; Hayon, E. J. Phys. Chem. 1967, 71, 3802.
(28) Kraljic, I. Int. J. Radiat. Phys. Chem. 1970, 2, 59.
(29) Hayon, E.; Treinin, A.; Wilf, J. J. Am. Chem. Soc. 1972, 94, 47.
(30) Redpath, J. L.; Willson, R. L. Int. J. Radiat. Biol. 1975, 27, 389.
(31) Chawla, O. P.; Fessenden, R. W. J. Phys. Chem. 1975, 79, 2693.
(32) Kim, K. J.; Hamill, W. H. J. Phys. Chem. 1976, 80, 2320.
(66) Jacob, D. J.; Gottlieb, E. W.; Prather, M. J. J. Geophys. Res. 1989,
94 (10), 12975.
(67) Turco, R. P.; Whitten, R. C.; Toon, O. B. ReV. Geophys. Space
Phys. 1982, 20 (2), 233.
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