Reactive uptake of ozone by oleic acid aerosol particles: Application of single-particle mass spectrometry to heterogeneous reaction kinetics

Article abstract of DOI:10.1021/jp020527t

The technique of single-particle mass spectrometry has been coupled to a reaction flow tube to measure the uptake coefficient, γ, of ozone (O₃) by oleic acid (9-octadecenoic acid) aerosol particles. The reaction was followed by monitoring the decrease of oleic acid in the size-selected particles as a function of O₃ exposure. The reactive uptake coefficient is found to depend on the size of the particle, with γ_meas ranging from (7.3 ± 1.5) × 10^-3 to (0.99 ± 0.09) × 10^-3 for particles ranging in radius from 680 nm to 2.45 μm. It is suggested that the decrease in γ_meas with increasing particle size results from the reaction being limited by the diffusion of oleic acid within the particle, and based on our measurements we estimate the value of γ to be (5.8-9.8) × 10^-3 for particles that are not limited by oleic acid diffusion. A reaction model that includes simultaneous diffusion and reaction of both O₃ and oleic acid is developed and used to fit the observed rates of reaction. Solutions obtained from this model indicate that oleic acid must diffuse within the particle more slowly than is predicted by the measured oleic acid self-diffusion constant.¹ It is proposed that this oleic acid-diffusion-limited uptake is attributable to the ozonolysis reaction products. Furthermore, these experiments demonstrate that it is not always possible to describe heterogeneous uptake by a model that decouples all relevant processes, including reaction and diffusion. Finally, the possible implications that these findings have for the role of particle morphology in the reaction of gas-phase species with atmospheric aerosols are discussed.

Full text of DOI:10.1021/jp020527t

J. Phys. Chem. A 2002, 106, 8085-8095
8085
Reactive Uptake of Ozone by Oleic Acid Aerosol Particles: Application of Single-Particle
Mass Spectrometry to Heterogeneous Reaction Kinetics
Geoffrey D. Smith,† Ephraim Woods, III,‡ Cindy L. DeForest,§ Tomas Baer,* and
Roger E. Miller*
Department of Chemistry, UniVersity of North Carolina, Chapel Hill, North Carolina 27599-3290
ReceiVed: February 26, 2002; In Final Form: June 18, 2002
The technique of single-particle mass spectrometry has been coupled to a reaction flow tube to measure the
uptake coefficient, γ, of ozone (O3) by oleic acid (9-octadecenoic acid) aerosol particles. The reaction was
followed by monitoring the decrease of oleic acid in the size-selected particles as a function of O3 exposure.
The reactive uptake coefficient is found to depend on the size of the particle, with γmeas ranging from (7.3 (
1.5) × 10-3 to (0.99 ( 0.09) × 10-3 for particles ranging in radius from 680 nm to 2.45 µm. It is suggested
that the decrease in γmeas with increasing particle size results from the reaction being limited by the diffusion
of oleic acid within the particle, and based on our measurements we estimate the value of γ to be (5.8-9.8)
× 10-3 for particles that are not limited by oleic acid diffusion. A reaction model that includes simultaneous
diffusion and reaction of both O3 and oleic acid is developed and used to fit the observed rates of reaction.
Solutions obtained from this model indicate that oleic acid must diffuse within the particle more slowly than
is predicted by the measured oleic acid self-diffusion constant.1 It is proposed that this oleic acid-diffusion-
limited uptake is attributable to the ozonolysis reaction products. Furthermore, these experiments demonstrate
that it is not always possible to describe heterogeneous uptake by a model that decouples all relevant processes,
including reaction and diffusion. Finally, the possible implications that these findings have for the role of
particle morphology in the reaction of gas-phase species with atmospheric aerosols are discussed.
Introduction
especially unsaturated hydrocarbons, are known to react with
O3 in both the gas phase and in solution. Reaction rates and
product yields for reactions between O3 and unsaturated
compounds in the gas phase have been measured, but these
reactions are fairly inefficient, with second-order rate constants
Organic aerosols and aerosols containing significant quantities
of organic species are common throughout the troposphere.2-4
Nonurban aerosols contain a variety of organic species of
biogenic origin including alkanes, alkenes, fatty acids, alcohols,
and aromatics.5-8 In urban areas, particles composed of organics
can also be emitted directly into the atmosphere from various
sources including automobile exhaust, natural gas combustion,
wood and cigarette smoke, and cooking exhaust.9,10 Particles
can also be formed or increase in size through the photochemical
oxidation of organic species to create multifunctional organics
which often have vapor pressures low enough that they exist
primarily in the condensed phase. These organic aerosols may
significantly affect the climate by acting as cloud condensation
nuclei, which may consequently affect cloud albedo.7 Organic
species may also play a role in the chemistry occurring within
these aerosols as well as affect the atmospheric budgets of gas-
phase species, including important oxidizing species such as
O3 and HO2.11,12
ranging in value from 10-18 cm3 molecule-1 s-1 to 10-14 cm3
7
molecule-1 s-1
.
On the other hand, it has been shown that
heterogeneous reactions of O3 with unsaturated species in
solution can be efficient13 and may have significant implications
for the loss of O3 in the troposphere and on the identities of the
condensed-phase species in aerosols.
Many previous studies measuring gas-phase uptake by
condensed-phase samples have been conducted by monitoring
the loss of the gas-phase species using Knudsen cell reactors14
and coated-wall flow tubes.15,16 With these techniques, nonre-
active uptake (such as physical adsorption) cannot be distin-
guished from reactive uptake. In addition, when uptake by a
liquid sample is monitored, reaction with vapor above the liquid
cannot be distinguished from reaction in the bulk of the liquid.
A notable exception to these techniques is the recent work of
Worsnop and co-workers, in which they used a thermal
desorption-EI aerosol mass spectrometer to monitor reactive
uptake of a gas-phase species from the perspective of the
particle.17
Ozone is one of the most important oxidants in the tropo-
sphere, and it is the photolytic precursor to the hydroxyl radical
that is primarily responsible for the oxidation of hydrocarbons
in the atmosphere. Ozone is also a significant component of
photochemical smog and can lead to adverse effects on human
respiratory health as well as on vegetation. Organic species,
In the present work, a dual-laser single-particle mass spec-
trometer is used to follow the progress of a reaction by
monitoring the concentration of the condensed-phase species.
The use of separate vaporization and ionization lasers results
in less fragmentation than with other methods, making the
approach particularly well suited to the study of reactions with
organic species.18 Additionally, the dual laser approach makes
* Corresponding authors.
† Present address: University of Georgia, Department of Chemistry,
Chemistry Bldg., Athens, GA 30602.
‡ Present address: Colgate University, Department of Chemistry, 13 Oak
Dr., Hamilton, NY 13346.
§ Present address: Davidson College, Department of Chemistry, Box
7120, Davidson, NC 28035.
10.1021/jp020527t CCC: $22.00 © 2002 American Chemical Society
Published on Web 08/02/2002

8086 J. Phys. Chem. A, Vol. 106, No. 35, 2002
Smith et al.
Figure 1. Schematic of aerosol flow tube. Oleic acid particles are created with the VOAG and enter the flow tube through the moveable aerosol
injector. Ozone passes through the UV absorption cell into the rear of the flow tube. The reaction time between the O3 and the oleic acid particles
is varied by moving the position of the aerosol injector. Particles exit the flow tube and are sampled by the aerosol mass spectrometer.
it is possible to gently probe the identities of the species present
in the condensed phase following reaction and may offer insight
into the mechanism of the reaction. Furthermore, since many
heterogeneous reactions occur near the surface, the uptake
kinetics may be limited by diffusion of the reactants within the
particle to the surface region. Measuring the uptake as a function
of particle size allows us to assess the importance of this
diffusion.
We have coupled an aerosol flow tube to our dual-laser single-
particle mass spectrometer to measure the reactive uptake of
O3 by particles of a particular unsaturated species, oleic acid.
Oleic acid is found in many meats and cooking oils and is, in
fact, the primary unsaturated species in olive oil. Oleic acid
has been measured in tropospheric aerosols at concentrations
long cell is constructed from 1/2 in. o.d. quartz with 1/4 in.
o.d. inlets and outlets for the O3 flow. Light from a pen-ray Hg
lamp (VWR Scientific), selected with a notch filter (Melles
Griot, λ ) 254 nm ( 5 nm), passes through the length of the
cell and is then detected with a calibrated, UV-sensitive
photodiode (Edmund Scientific). The concentration of the O3
flowing through the cell is determined from the measured
absorption, the known path length, and the known cross section
(1.169 × 10-17 cm2 molecule-1 at λ ) 254 nm20). The
concentration of O3 in the flow tube (6 × 10-6 to 1 × 10-4
atm) is calculated from the concentration measured in the cell
and the known gas flows regulated with calibrated mass flow
controllers (MKS Instruments). The concentration of O3 is also
monitored with a residual gas analyzer (RGA) (Stanford
Research Systems), sampling gas through one of three ports
located along the length of the flow tube.
of ∼1 ng m-3 10
,
and in the present experiments it serves as a
suitable representative of many unsaturated species typically
found in these aerosols. In addition, the kinetics of this reaction
have been studied recently by Worsnop and co-workers,17
providing a useful comparison for this novel technique.
The reaction of O3 with pure oleic acid particles is studied
by monitoring the concentration of oleic acid within the particles
as a function of O3 exposure using our single-particle mass
spectrometer. The O3 exposure (PO t) is varied by adjusting the
reaction time with a fixed O3 partial pressure. The residual gas
analyzer is used to monitor the partial pressure of O3 in the
3
Experiment
Aerosol Flow Tube. The apparatus used in these experiments
consists of an aerosol flow tube (Figure 1) and a dual-laser
single-particle mass spectrometer, which is described in detail
in our previous work.18,19 A vibrating orifice aerosol generator
(VOAG) (TSI, Inc.) is used to create a monodisperse stream of
droplets containing oleic acid (Aldrich, CAS# 112-80-1) and
2-propanol. The droplets are directed into an acrylic drying
tower by an argon flow (Air Products, research grade) of 2 STP
L min-1, where the 2-propanol from the droplets evaporates,
thereby generating a monodisperse aerosol of pure oleic acid.
Another 2 STP L min-1 flow of argon sweeps the particles out
of the tower into a one-meter long, 1/4 in. o.d. Pyrex injector
tube, which enters the 1-m long, 1 in. i.d. Pyrex reaction flow
tube from the rear. The drying tower and injector are moveable
along an optical rail, allowing for a variable injection position
of the aerosols along the length of the flow tube.
Ozone is generated by flowing O2 (Air Products, research
grade) through an ozonizer (Pacific Ozone Technology, model
L11), and it is stored in a glass trap containing silica gel and
cooled to 196 K with a 2-propanol/dry ice bath. Ozone is carried
from the trap by a flow of argon (10-60 STP cm3 min-1), passes
through an absorption cell where the absolute concentration is
measured, is diluted by another flow of argon (500 STP cm3
min-1), and then enters the rear of the flow tube. The 10-cm
+
flow tube both with and without particles present. Stable O2
and O3+ signals from the residual gas analyzer indicate that the
partial pressure of O3 is uniform throughout the flow tube and
that the number density of particles is not great enough to deplete
it.
The reaction time is varied by moving the position of the
aerosol injector within the flow tube. A “turbulizer” placed
coaxially with the injector and located 3 cm from the exit end
serves to hold the injector in place and introduce turbulence
downstream of the injector. The flow velocities of the flow tube
flow (1.6 cm s-1) and the injector flow (530 cm s-1) are
sufficiently different so that turbulent flow develops past the
exit of the injector. However, the turbulizer enhances mixing
of the O3 flow with the flow of particles exiting the injector to
ensure that there is no time delay due to diffusive mixing of
the O3 (which would require ∼0.5 s).
The residence time of the particles is calculated as a function
of injector position within the flow tube by measuring the time
delay between the initiation of their creation with the VOAG
and their detection via the light scattering stations in the aerosol
mass spectrometer. This total time represents the time required
for the particles to travel through the drying tower, the moveable
injector, the flow tube, and the aerosol inlet. Then, the time

Reactive Uptake of Ozone by Aerosol Particles
J. Phys. Chem. A, Vol. 106, No. 35, 2002 8087
Figure 2. Mass spectra of oleic acid particles of radius 2.45 µm with
PO ) 1.3 × 10-4 atm and reaction times of 0, 3, and 8 s.
3
required for the particles to travel just through the drying tower
and the moveable injector (measured separately) is subtracted
from the total time, thereby allowing a direct correlation to be
made between the position of the injector within the flow tube
and the particle residence time in the flow tube. The residence
times measured in this manner are found to approximate closely
the times predicted by the bulk flow velocity of the gas, and all
decay data presented here are analyzed with the measured times.
Single-Particle Mass Spectrometer. Aerosol particles from
the flow tube are sampled into the single-particle mass
spectrometer through an 18-cm long, 1/4 in. o.d. stainless steel
inlet terminated by a 100-µm i.d. flow-limiting orifice. After
passing through this orifice the particles enter a 20-cm long
aerodynamic lens,21-23 which focuses them along a well-defined
axis and greatly improves the efficiency with which they are
detected. The focused particles are then accelerated through two
stages of differential pumping, after which they pass through
two 532-nm diode lasers separated by 10 cm. The velocities of
the particles are calculated from the time difference between
the scattering signals, and a digital timing circuit triggers the
pulsed lasers to vaporize and ionize the particles when they reach
the mass spectrometer. The measured velocities are also used
to calculate the aerodynamic diameter of each particle.
A two-laser scheme is employed consisting of a pulsed TEA-
CO2 laser (Lumonics) for vaporization of the particle and a
pulsed vacuum-ultraviolet (VUV) laser for ionization of the
resulting gas plume. The typical CO2 laser energies range from
30 to 65 mJ/pulse, with an estimated focal spot size of
approximately 1 mm2. After a variable delay of 2-30 µs, a
118.5 nm pulse, created by frequency-tripling the 355 nm output
of a Nd:YAG laser (Continuum) in a Xe/Ar gas cell, ionizes
the vapor plume. The resulting ions are analyzed with the time-
of-flight mass spectrometer.
Figure 3. Mass spectra digitized at high resolution for particles of
radius 2.45 µm with PO3 ) 1.3 × 10-4 atm and reaction times of 0, 3,
and 8 s: (a) high-mass peaks including molecular ion of oleic acid
(M+) at m/z ) 282, (b) low-mass peaks showing emergence of peaks
at m/z ) 171 and m/z ) 155.
with spectra digitized at higher resolution. These spectra, with
>1 amu resolution at m/z ) 282, were used in analysis of the
data to ensure that this peak could be distinguished from
neighboring peaks, such as the unsaturated acid impurity at m/z
) 284.
The ozonolysis of alkenes has been studied in both the gas
phase and the condensed phase, and the identities and fates of
some of the products from the reactions have been character-
ized.11,13,24,25 The mechanism of the ozonolysis of oleic acid is
not expected to differ significantly from that of alkenes, as the
double bond in oleic acid is located in the middle of the long
molecule and is therefore relatively isolated from the carboxylic
acid group. It is known that the initial step in ozonolysis
reactions is the addition of the ozone across the double bond of
the alkene forming the primary ozonide (see Figure 4). This
primary ozonide is unstable, and the C-C bond breaks forming
two possible pairs of fragments depending on which O-O bond
breaks (also shown in Figure 4). In both cases, an aldehyde
and a Criegee intermediate are formed. In general, the Criegee
intermediate can decompose, be stabilized, or react with an
aldehyde fragment to form a secondary ozonide. Thus, a number
of primary and secondary products can be formed.
Results
Mass Spectra of Reacted Particles. The soft vaporization
and ionization of the two-laser technique allow the detection
of oleic acid as its parent ion (M+, m/z ) 282), and from this
feature in the mass spectrum (Figure 2) the concentration of
oleic acid within each particle can be measured. The parent ion
at m/z ) 282 (M+) accounts for approximately 15% of the entire
ion signal (see lowest spectrum in Figure 2), indicating much
less fragmentation than with electron impact ionization, which
usually results in less than 1% of the ion signal at m/z ) 282.17
As the O3 exposure time increases (upper spectra in Figure 2),
the peak at m/z ) 282 decreases in intensity and a peak at m/z
) 155 grows proportionally. Figure 3 illustrates these trends
In the spectra shown in Figure 3b, the growth of a feature at
m/z ) 155 can be seen with ozonolysis reaction times of 3 s
and 8 s. Other, smaller peaks that appear upon exposure to O3
are also evident in the spectra. In particular, peaks at m/z )
125, 129, 141, 143, 144, 159, and 171 all grow slightly with
reaction. The mechanism outlined in Figure 4 reveals possible
assignments for these peaks, with the mass-to-charge ratios of
experimentally observed peaks underlined. The largest peak in
the spectrum, at m/z ) 155, and the peak at m/z ) 171 may
represent fragments from one of the ozonolysis products,
9-oxononanoic acid (m/z ) 172). In particular, the peak at m/z
) 171 could be the H-loss fragment from 9-oxononanoic acid,

8088 J. Phys. Chem. A, Vol. 106, No. 35, 2002
Smith et al.
Figure 4. Mechanism for ozonolysis of oleic acid. The mass-to-charge ratios for each species (and associated fragments) are indicated with peaks
identified in the spectra of oleic acid particles underlined.
and that at m/z ) 155 may be the OH-loss fragment from the
same product. Interestingly, these same two peaks represented
the majority of the product features observed in the study of
the reaction of O3 with oleic acid particles conducted by
Ziemann and co-workers using thermal desorption particle beam
mass spectrometry.26 The feature at m/z ) 141 could be
attributed to the H-loss fragment of the nonanal product (m/z
) 142), which has been observed by Rudich and co-workers in
the gas phase in a coated-wall flow tube study of the reaction
of O3 with oleic acid.27 These assignments are not definitive,
however, and the peaks at m/z ) 141, 155 and 171 could also
be attributed to fragments of product species, including second-
ary ozonides. The other peaks are more difficult to assign to
particular ozonolysis products, although their growth is clearly
correlated with the increasing exposure to O3. In particular,
lower-mass fragments (including peaks at m/z ) 29, 43, 57,
70, 82, and 97), which are observed in the spectra of reacted
oleic acid particles (shown in Figure 2), also appear in electron-
impact spectra of the nonanal ozonolysis product.28
pressure of O3 (in atm), cj the mean kinetic speed of O3
molecules in the gas phase (3.6 × 104 cm s-1), R the gas
constant (0.082 atm K-1 M-1), and T the temperature (298 K).
The factor SA/V () 3/a, where a is the particle radius) is the
ratio of the surface area to the volume and normalizes the rate
of O3 uptake to the volume of the particle.
There are many processes that can limit the rate at which a
gas-phase species is taken up by a particle. These processes
include: gas-phase diffusion to the particle, accommodation at
the surface of the particle, reaction on the surface of the particle,
incorporation into the bulk of the particle, and reaction within
the bulk of the particle.29,30 The aggregate of these processes
can be represented by a set of coupled partial differential
equations in the concentrations of the species of interest, with
appropriate boundary conditions at the interfaces. Analytical
solutions to these equations have been obtained for certain
limiting cases.31-33
A convenient simplification that is often made in interpreting
the interaction of gas-phase species with heterogeneous systems
is the use of an electric circuit resistance model.30,34,35 In this
model, each of the processes listed above is considered to be
independent of, or decoupled from, all other processes. Each
process is represented as an individual conductance (Γ) which
is normalized to the gas-particle collision rate and is therefore
unitless. This formulation is useful because the individual
resistances (1/Γ) can be combined in series or in parallel to
represent the overall uptake coefficient (γ) of the gas-phase
species.35,36 Many experimental studies have utilized this and
similar formulations to relate the measured uptake (γmeas) of a
gas-phase species by a bulk sample to the reaction probability
(Γrxn) within the sample.37-39
Reactive Uptake Model. Though reactive uptake by a
condensed phase is typically described in terms of the loss of
a gas-phase species, Worsnop and co-workers have shown that
it is also possible to interpret the uptake in terms of the loss of
a species in the bulk.17 In the reaction of O3 with oleic acid,
the rate of change of the oleic acid concentration, [Oleic], can
be expressed in terms of the reactive O3 uptake coefficient:
PO cj
SA
V
d[Oleic]
3
) -γ
(1)
(
)
dt
4RT
where γ is the probability that an O3 molecule colliding with
the particle will react either at the surface or in the bulk of the
particle. Because γ is a function of [Oleic], the functional form
of γ must be known before eq 1 can be solved for [Oleic] as a
function of time. The term in parentheses represents the O3-
We make use of this resistance model to interpret the reactive
uptake of O3 by oleic acid in the present experiments. The
limiting loss process for O3 is reaction with oleic acid, either
in the bulk or on the surface, and the net uptake of O3 by the
oleic acid particle is the sum of the uptake due to reaction at
particle collision rate (mol cm-2 s-1), with PO the partial
3

Reactive Uptake of Ozone by Aerosol Particles
J. Phys. Chem. A, Vol. 106, No. 35, 2002 8089
the surface (Γsurf) and the uptake due to reaction in the bulk
equation cannot be solved analytically to obtain a solution for
(Γrxn
)
[Oleic], it can be solved numerically with knowledge of PO ,
3
H, D, k2, and a. Alternatively, analytical solutions can be found
for two limiting cases of the diffuso-reactive length, l. Deriva-
tions for the solutions for these cases are given in Appendix II.
Case 1a: Fast Diffusion of O3 Throughout the Particle (l >
a). In this case the rate of reaction will not be limited by O3
diffusion, such that
γ ) Γrxn + Γsurf
(2)
It is possible that O3 could be reacting at both the surface
and in the bulk, and thus the measured rate represents both Γsurf
and Γrxn. However, these processes are not coupled to one
another as long as the concentration of O3 at the surface is not
significantly affected by either process. The following treatment
will utilize analytical expressions for the uptake coefficient to
derive expressions for the oleic acid concentration as a function
of time in the two limits: Case 1, where Γsurf is negligible
compared to Γrxn and therefore γmeas ) Γrxn; and Case 2, where
[Oleic] ) [Oleic]0 exp(-PO Hk2t)
(6)
(7)
3
4HRT a
γ )
k2[Oleic]
cj
3
Γ
rxn is negligible compared to Γsurf and therefore γmeas ) Γsurf.
This case is clearly not valid for the experiments described here,
given that l ∼ 20 nm and a is at least 680 nm.
These expressions can then be used to fit the oleic acid
concentration decays measured in the present experiments.
Case 1: ReactiVe Uptake Dominated by Reaction in the Bulk
(γmeas ) Γrxn). The rate of change of the O3 concentration within
the particle can be described by a differential equation including
diffusion of O3 (in the particle) and reactive loss:
Case 1b: Reaction of O3 Near the Surface of the Particle
(l < a/20). This case is often referred to as the “diffusion-
limited” case because the rate of reaction depends on the rate
of O3 diffusion (and γ therefore depends on D). This case is
applicable for all particle sizes used in the present experiments.
For this case eq 1 can be solved to obtain
∂[O3]
2
) D∇ [O3] - k[O3]
(3)
∂t
3P H Dk
x
O3
[Oleic] ) [Oleic] -
2t
(8)
(9)
x
x
0
Here, D is the diffusion coefficient for O3 in oleic acid (cm2
2a
s-1) and k () k2[Oleic]) is the first-order rate coefficient (s-1
)
4HRT
for the reaction of O3 with oleic acid. This differential equation
can be solved analytically if two common assumptions are
made: (1) that the concentration of oleic acid is uniform
throughout the particle, i.e., oleic acid diffusion is fast, and (2)
[O3] is in steady state, i.e., d[O3]/dt ) 0. The derivation of the
solutions with these assumptions is given in Appendix I. The
consequences of making these assumptions will be discussed
in detail in a later section.
γ )
Dk [Oleic]
2x
x
cj
Case 2: ReactiVe Uptake Dominated by Reaction at the
Surface (γmeas ) Γsurf). The reaction of O3 with oleic acid at
the surface of the particle can be written as a second-order loss
process, similar to the formulation of Worsnop et al.:43
d[Oleic]
surf ) -k2surf[O3]surf[Oleic]surf
With these assumptions, eq 3 can be solved to yield the
steady-state O3 concentration as a function of position within
the particle. From this solution, the flux of O3 into the particle
can be calculated. Normalizing this flux to the O3 particle
collision rate yields the reactive uptake coefficient (derivation
in Appendix I):
dt
) -k2surf(PO Hδ)([Oleic]δ)
3
(10)
where the surface concentration of O3, [O3]surf (mol cm-2), is
approximated as the product of the Henry’s Law equilibrium
fluxsurf
4HRT D
value, H‚PO , and the depth of the surface layer, δ. Likewise,
3
Γrxn
)
)
(coth(a/l ) - l /a)
(4)
the concentration of oleic acid at the surface, [Oleic]surf (mol
PO cj/4RT
cj
l
3
cm-2) is also approximated as the product of [Oleic] and δ.
surf
The second-order rate coefficient, k2
(cm2 mol-1 s-1), is
where H is the Henry’s Law solubility constant of O3 in oleic
acid (M atm-1) and a is the radius of the particle. The parameter
l is often referred to as the “diffuso-reactive length,” and it
represents the characteristic distance that an O3 molecule diffuses
before it reacts, namely:
specific to the surface of the particle and is different from the
bulk rate coefficient, k2.
An expression for the reactive uptake coefficient at the surface
is obtained by normalizing the rate in eq 10 by the gas-particle
collision rate:
D
l )
(5)
d[Oleic]surf
k [Oleic]
x
2
-
dt
4HRT
cj
Γsurf
)
)
δ2k2surf[Oleic]
(11)
In this experiment, D ≈ 1 × 10-5 cm2 s-1 (estimated based on
the diffusion of O2 in a variety of organic solvents),40 k2 ) 1 ×
PO cj/4RT
3
106 M-1 s-1 41
,
and [Oleic]0 ) 3.16 M,42 yielding a value of l
An expression for the rate of change of [Oleic] with time can
be found by substituting eq 11 in eq 1 and solving for [Oleic]:
∼ 20 nm. This distance is much smaller than the radius of even
the smallest particles used in these experiments (680 nm). This
large difference leads to a convenient simplification of eq 4
and is discussed below as Case 1b.
3δ2
[Oleic] ) [Oleic] exp -
PO Hk2surft
(12)
(
)
0
3
a
The above expression for Γrxn (eq 4) can be substituted for γ
in eq 1, resulting in a differential equation in [Oleic]. The
solutions to this equation can be fit to the observed [Oleic]
decays to yield a value for γmeas. Although the differential
Since the functional forms of the solutions for Cases 1a, 1b,
and 2 are different, the shapes of the measured [Oleic] decays
should allow us to identify the operative mechanism. Although

8090 J. Phys. Chem. A, Vol. 106, No. 35, 2002
Smith et al.
Figure 6. Size dependence in the measured reactive uptake coefficients.
No size dependence is predicted by the applicable uptake model (Case
1b).
Figure 5. Oleic acid decay profiles as measured with the single-particle
mass spectrometer for four particle sizes: [ 680 nm, b 1.38 µm, 2
1.86 µm, 9 2.45 µm. Though linear fits can be made to each decay
independently, no single set of parameters (H, D, and k2) can fit all of
the data simultaneously.
The value of the reactive uptake coefficient depends on the
concentration of oleic acid in the particle, and eq 14 therefore
represents the probability of reactive uptake by a pure oleic acid
particle. In this expression for γmeas there is no explicit size
dependence, and the values calculated for each of the four
particle sizes used in these experiments should be identical.
However, a plot of γmeas vs particle radius (Figure 6) demon-
strates a distinct size dependence in the uptake coefficient. In
fact, the observed uptake coefficient decreases linearly with
particle size. Though linear fits can be made independently for
each of the decays in Figure 5, there is no single set of
parameters (H, D, and k2) that will allow all four sets to be fit
simultaneously. Even if the possibility that the data are described
by a surface reaction is considered (Case 2), no size dependence
is predicted from the resulting expression for γ, eq 11. Clearly,
the uptake model does not describe our data adequately.
Nonetheless, the functional form of the oleic acid decay given
by eq 13 is useful for fitting the entire set of data and allows us
to calculate the initial rate of decay and thus the value of γ at
t ) 0.
The most obvious explanation for the size dependence in γmeas
is that oleic acid diffusion within the particle is limiting the
reaction and that the effect of this diffusion is more pronounced
in the larger particles. The observed linear relationship will not
hold in the limit of large particles since γmeas cannot decrease
below zero. Likewise, this size dependence cannot hold for
sufficiently small particles since the rate of reaction will not be
limited by diffusion and γmeas will be constant with particle size.
In fact, the spherical geometry of the particles can result in
values of γmeas that decrease with decreasing particle size when
a/l is small enough that the (coth(a/l ) - l /a) term in eq 4 is
less than unity. Thus, a more general approach to the solution
of simultaneous reaction and diffusion must be taken to describe
uptake over a range of particle sizes and experimental condi-
tions.
Case 1a and Case 2 both yield exponential functions, the solution
for Case 1a does not depend on the particle radius, a, whereas
the solution for Case 2 does. Both Case 1b and Case 2 depend
on the particle size, but they differ in their functional forms
(exponential vs square root).
Measurement of Oleic Acid Reaction Kinetics. The con-
centration of oleic acid in the particles is measured as a function
of O3 exposure (PO t) for four different particle sizes ranging
3
in radius from 680 nm to 2.45 µm. The oleic acid concentration
in the particles is measured as the integrated signal of the oleic
acid parent peak (m/z ) 282) normalized to the signal in the
absence of O3 and is obtained by averaging spectra for 100-
200 particles per reaction time. The decay profiles are then fit
to each of the two possible solutions identified above, namely
the square root decay of Case 1b (l < a/20) and the exponential
decay of Case 2 (surface reaction). Both cases fit the data
approximately, although the square root (Case 1b) is better than
the exponential (Case 2) when the data for all of the particle
sizes are considered. Of course, it is possible that O3 reacts at
both the surface and in the bulk of the particle, but it is difficult
to estimate the relative contributions of the two cases without
knowledge of the values of k2, k2surf, H, and D. Therefore, we
assume that one or the other of these processes dominates the
reactive uptake, and we believe that the data are better
represented by a bulk reaction (Figure 5).
Each of the decay profiles in Figure 5 can be fit to the solution
obtained for Case 1b:
[Oleic]
x
3
) 1 -
(H Dk )(P t) (13)
x
2
O3
[Oleic]
2a [Oleic]
x
x
0
0
where the function is expressed as a fractional decay in the oleic
acid signal. Values for the quantity H Dk are obtained from
x
2
Uptake Model Including Oleic Acid Diffusion. We believe
that the observed decrease in γmeas with particle radius is the
result of oleic acid diffusion limiting the kinetics of the reaction.
The oleic acid is not able to diffuse to the reaction region near
the surface fast enough to maintain a uniform oleic acid
concentration throughout the particle. The larger the particle,
the farther the oleic acid must diffuse, hence the decrease in
reactive uptake as the particle size increases.
the slopes of the linear fits in Figure 5 since the values of a,
PO and [Oleic]0 are known for all of the experiments. This
3
quantity can then be replaced in eq 9 to obtain a value for γmeas
,
the uptake coefficient for pure oleic acid:
4HRT
cj
γmeas ) Γrxn
)
Dk [Oleic]
(14)
x
2x
0
In this way, the value of γmeas is determined from the oleic acid
decay profiles without the need for direct measurement of the
individual quantities, H, D, and k2.
Explicitly including the effect of oleic acid diffusion results
in a set of coupled partial-differential equations in both [O3]
and [Oleic]:

Reactive Uptake of Ozone by Aerosol Particles
J. Phys. Chem. A, Vol. 106, No. 35, 2002 8091
∂[O3]
2
) D∇ [O3] - k2[O3][Oleic]
(15)
(16)
∂t
∂[Oleic]
2
) D′∇ [Oleic] - k2[O3][Oleic]
∂t
where both [O3] and [Oleic] are functions of time and position.
It is not possible to obtain analytical solutions to this set of
partial differential equations, but with values of D, k2, H, PO ,
3
a, and D′ (the oleic acid diffusion constant), solutions for [O3]
and [Oleic] can be obtained numerically. This approach to
solving eqs 15 and 16 does not rely on assuming that [O3]
reaches a steady-state value or that oleic acid diffuses quickly
throughout the particle. In fact, no functional form is assumed
for either [O3] or [Oleic], and both concentrations are allowed
to vary as functions of both position and time. A more detailed
description of this procedure will be given in a forthcoming
paper.44
To investigate the role of oleic acid diffusion in the reactive
uptake of O3, we obtain solutions numerically for [Oleic] using
the differential equation solver in Mathematica (version 4.1,
Wolfram Research). The values of the radius, a, and the O3
Figure 7. Numerical fits to the measured oleic acid decay profiles for
four particle sizes: [ 680 nm, b 1.38 µm, 2 1.86 µm, 9 2.45 µm.
The decays are fit with the diffusion-reaction model using the following
parameters: H ) 0.48 M atm-1, k2 ) 1 × 106 M-1 s-1, D ) 1 × 10-5
cm2 s-1 The value of D′ is allowed to vary for each particle size, with
D′ ranging from 4 × 10-10 cm2 s-1 to 10 × 10-10 cm2 s-1
.
partial pressure, PO , were chosen to match those used in the
3
experiments. The other parameters used are representative of
the O3-oleic acid system: D ) 1 × 10-5 cm2 s-1, H ) 0.48
M atm-1, and k2 ) 1 × 106 M-1 s-1 41
,
where the value for D
is an approximation based on measured diffusion constants for
O2 in organic solvents.40 The value for H is an estimate based
on measured solubility constants for O3 in a variety of organic
solvents45,46 and has been adjusted slightly to yield an uptake
coefficient of γmeas ) 7.3 × 10-3 that is consistent with our
estimate of γmeas ) (5.8-9.8) × 10-3 for small particles for
which oleic acid diffusion is not limiting.
Using the self-diffusion constant of oleic acid, D′ ) 3 × 10-7
cm2 s-1,1 it was found that oleic acid is well-mixed in the
particle and that the rate of reaction is not limited by oleic acid
diffusion. Under these conditions the simple model presented
earlier (Case 1b) is valid. However, as the reaction with O3
proceeds, oleic acid will be replaced with products of the
reaction, and these products may slow diffusion to the reaction
region near the surface. Indeed, the numerical solutions obtained
for [Oleic] can fit only the experimental data (Figure 7) with
D′ slowed to (4-10) × 10-10 cm2 s-1 (depending on particle
size). Since D′ is necessarily large (the self-diffusion constant)
at the beginning of the reaction and then is reduced due to the
formation of products, the constant value of D′ represents an
“effective” diffusion constant, and its inclusion allows the model
to describe the rate of reactive uptake more accurately. By
Figure 8. Rise of peak at m/z ) 155 as a function of O3 exposure
(PO t) for four particle sizes: [ 680 nm, b 1.38 µm, 2 1.86 µm, 9
3
2.45 µm. Lines represent expected signal if every oleic acid molecule
that reacts results in the creation of a fragment at m/z ) 155.
where the reaction is completely quenched after only 30% of
the particle has reacted.47
All of the parameters, including H, PO , k2, a, D, and D′, are
3
coupled in determining the rate of uptake, and no simple
expression for γ can be formulated. For example, D′ becomes
more significant as [O3] increases in the particle (and thus H
and PO ) or as the particle size, a, increases. Additionally, since
3
the rate of diffusion of oleic acid is likely changing as the
reaction products are formed, treating D′ as a constant does not
necessarily describe the effect of diffusion accurately. The most
general approach accounting for these coupled processes and
the changing environment within the particle is the numerical
solution to the coupled diffusion-reaction equations (15) and
(16), and in future work we will address the general relationship
between all of the relevant parameters.44
Product Kinetics. In this study the appearance of one of the
products is also monitored as a function of O3 exposure. The
rise in signal of the feature at m/z ) 155, perhaps a fragment
of the ozonolysis product 9-oxononanoic acid, is shown in
Figure 8 for each of the four particle sizes used in these
experiments. The signals have been normalized to the initial
oleic acid signal (at m/z ) 282), assuming equal detection
efficiency for the peaks at m/z ) 155 and m/z ) 282, and the
corresponding solid curves are calculated from the measured
choosing realistic values for H, D, and k2 (with a and P
O3
reflecting the conditions of each experiment), we can fit all of
our data with reduced values of D′, indicating that oleic acid
diffusion is limiting and that the observed size dependence in
γmeas can be attributed to slow diffusion of oleic acid.
It is possible that the slowing represented by the decrease in
D′ reflects a reduction in the solubility of O3 in the particle as
the reaction proceeds and not a change in the oleic acid
diffusion. This change in solubility could affect the rate of
uptake, but it would not result in an observed size dependence
in γmeas. Thus, we favor the explanation based on the decrease
in the rate of oleic acid diffusion. One explanation for this large
decrease in D′ is that the ozonolysis reaction initiates polym-
erization, which effectively inhibits diffusion of oleic acid.
Indeed, we have observed a similar, but even more drastic, effect
on the rate of reactive uptake in the ozonolysis of 1-octadecene

8092 J. Phys. Chem. A, Vol. 106, No. 35, 2002
Smith et al.
A possible explanation for this discrepancy is that the signal
monitored by Worsnop and co-workers included peaks other
than m/z ) 282. In their mass spectrum, the peak near m/z )
282 appears to shift to a smaller mass-to-charge ratio upon
reaction with O3, and with insufficient resolution or digitization
integration of this signal could lead to an overestimation of the
oleic acid concentration and thus an underestimation in the rate
of uptake and in γ. Nonetheless, the lack of a size dependence
in their values of γ confirms our expectation that the diffusion
of oleic acid does not limit the rate of reaction for sufficiently
small particles.
Recently, Moise, and Rudich27 have also measured the rate
of uptake of O3 by oleic acid in a coated-wall flow tube study.
By monitoring the rate of change of the concentration of O3 in
the gas phase, they were able to calculate a value of γ ) (8.3
( 0.9) × 10-4, also significantly smaller than our value of γ
for particles not limited by oleic acid diffusion. It is likely that
this discrepancy results from the fact that the oleic acid coatings
in their flow tube study are much thicker than our particles.
The thick coatings result in a smaller uptake because oleic acid
diffusion to the region near the surface is more acutely inhibited.
Atmospheric Implications. Individual particles found in the
atmosphere often contain many different species including
alkanes, alkenes, alcohols, and carboxylic acids. The rate at
which these particles react with gases such as O3 will depend
on the composition of the particle as well as the rate of reaction
of O3 with each of the component species. With a value of γ )
7 × 10-3 and [O3] ) 100 ppb, 99% of a 50 nm particle of pure
oleic acid would react in less than two minutes. However, the
oleic acid concentration measured in atmospheric particles10
suggests that oleic acid in aerosols has a much longer lifetime.
This apparent discrepancy could be attributed to the diffusion
of oleic acid. The calculation assumes that oleic acid diffusion
within the particle is fast and therefore does not limit the uptake.
However, we have seen that under conditions in which the oleic
acid diffusion is significantly reduced (compared to self-
diffusion), uptake is much slower. Therefore, particle morphol-
ogy, as well as composition, can play a role in the uptake of a
gas-phase species. If oleic acid exists in a particle with other
species that effectively inhibit diffusion within the particle,
uptake will be reduced. Additionally, the location of the oleic
acid within each particle could affect uptake, since oleic acid
near the surface reacts much more quickly than it would if it
were deep within the particle. Also, a particle with a very porous
structure may inhibit the reaction more than a liquid droplet,
for example, and therefore reduce O3 uptake.
In addition to affecting the composition and properties of
particles in the atmosphere, the reaction of O3 with organic
particle constituents may also affect the lifetime of O3 in the
troposphere.16 The rate at which an aerosol removes a gas-phase
species through reactive uptake can be estimated. Again, using
a value of γ ) 7.3 × 10-3 and a representative organic aerosol
content of 1.5 × 105 particles cm-3 (for an urban setting48) with
a radius of 25 nm and an average oleic acid content of 1%, this
rate is calculated to be 8 × 10-6 s-1, corresponding to an O3
lifetime (with respect to reaction with the aerosol) of 36 h. The
lifetime with respect to photochemical loss, for comparison,
varies from a couple of days to a few hundred days depending
on season and latitude.49 Thus, organic species, such as oleic
acid, residing in particles could have an impact on the
atmospheric lifetime of O3.
Figure 9. Growth of aerodynamic diameter of particles as a function
of the extent of reaction. An estimate of 3.9 × 10-3 is calculated for
γmeas from the size increase (compared to a value of 3.1 × 10-3
calculated from the oleic acid decay data in Figure 5).
oleic acid decay rates. The similarity in the calculated and
measured rises indicates a branching ratio of near unity for the
peak at m/z ) 155. This peak may not represent a fragment of
the 9-oxononanoic acid product because this is only one of two
product pathways expected to be possible (see Figure 8), and
therefore the branching ratio should be less than unity. The peak
may also represent a fragment of a secondary ozonide, in which
case the branching ratio could be close to unity. However, a
more definitive interpretation of these kinetics and the implica-
tions for the rates of product formation is not possible without
more complete information regarding all of the products as well
as the rate of unimolecular dissociation of the primary ozonide.
Particle Growth. As O3 reacts with the oleic acid particles,
it is incorporated into the particle through the creation of
ozonolysis products. This incorporation leads to a net increase
in the mass of the particle as long as the products do not
evaporate. If the density of the particle were to remain constant,
this mass increase would result in an increase in the volume,
and thus diameter. The increase in the aerodynamic diameter
of the particle, which is proportional to the geometric diameter
and the square root of the density, can be measured with laser
velocimetry as the reaction proceeds. Indeed, the aerodynamic
diameters of the particles are seen to increase with exposure to
O3, as shown in Figure 9. Here, pure oleic acid particles 3.72
µm in diameter grow linearly in size to a diameter of 3.86 µm.
By assuming no change in the density of the particle, we can
calculate a value of γmeas ) 3.9 × 10-3 from the measured rate
of mass uptake. This value is in good agreement with the uptake
(γmeas ) 3.1 × 10-3) that we measured independently from the
rate of oleic acid decay for the same particle size. Changes in
particle radius as small as 3 nm are detectable with this method
and provide an alternative diagnostic of the extent of reaction
in the particle.
Discussion
Comparison to Other Studies. Two other groups have
recently studied the uptake of O3 by oleic acid and their
experiments offer useful comparisons to our measurements. The
work of Worsnop and co-workers17 is similar to ours in that
the composition of oleic acid particles reacted with O3 was
monitored with an aerosol mass spectrometer. However, their
particles were smaller than ours, ranging in radius from 100 to
300 nm. The reported uptake coefficient of (1.6 ( 0.2) × 10-3
Conclusion
(for all sizes) is significantly smaller than our value of γmeas
)
The reactive uptake of O3 by oleic acid particles has been
measured using a single-particle mass spectrometer to monitor
(7.3 ( 1.5) × 10-3 for our smallest particle, 680 nm in radius.

Reactive Uptake of Ozone by Aerosol Particles
J. Phys. Chem. A, Vol. 106, No. 35, 2002 8093
the composition of the aerosol as it reacts. The measured uptake
coefficients display a size dependence, with γmeas ranging from
(0.99 ( 0.09) × 10-3 for particles of radius 2.45 µm to (7.3 (
1.5) × 10-3 for particles of radius 680 nm. Numerical solutions
of the coupled partial differential equations describing the
simultaneous diffusion and reaction of both O3 and oleic acid
indicate that slow oleic acid diffusion can account for the
observed size dependence in γmeas. An approach that couples
all processes relevant to uptake must be used to accurately
describe the overall rate of uptake which is represented by γ.
In particular, it is necessary to account for the interdependence
of reaction and diffusion within the particle, not included in
the conventional electric circuit resistance model. Finally, these
experiments indicate that morphology, in addition to composi-
tion, may determine the reactivity of particles in the troposphere.
equal to the Henry’s Law solubility value, HPO (assuming that
O3 from the gas-phase replenishes the surface faster than the
rate at which it reacts in the particle) which results in
3
sinh (r/l )
a
[O3](r) ) HPO
(I.6)
3
r
sinh (a/l )
The flux of O3 into the particle can be obtained from Fick’s
first law by calculating the [O3] gradient as the derivative of eq
I.6
d[O3]
1
|
r)a ) HPO [(a/l )coth(a/l ) - 1]
(I.7)
3
dr
a
Acknowledgment. We gratefully acknowledge support from
AFOSR, Grant F496020-99-1-0064, NSF, Grant CHE 9727788,
DARPA, Grant F496020-98-1-0268, and EPA, Grant R-826767-
01-0. C.L.D. acknowledges support from the EPA STAR
fellowship. We thank Dr. Doug Worsnop of Aerodyne Research,
Inc., Dr. Yinon Rudich of the Weizmann Institute, and Dr. Paul
Ziemann of UC-Riverside for providing us with unpublished
results. We also thank Dr. Worsnop for numerous useful
discussions regarding the mathematics of reactive uptake.
Finally, we thank the two anonymous reviewers for many useful
comments.
d[O3]
D
l
fluxsurf ) -D -
) P H (coth(a/l ) - l /a) (I.8)
(
)
O3
dr r)a
Here, the negative of the [O3] gradient is used since it is the
flux into the particle that is being calculated. An expression for
the uptake coefficient, eq 4, is obtained by normalizing this flux
to the O3-particle collision rate
fluxsurf
4HRT D
Γrxn
)
)
(coth(a/l ) - l /a)
(4)
Appendix I
PO cj/4RT
cj
l
3
Making the common assumptions that: (1) [O3] is in steady
state and (2) [Oleic] is uniform throughout the particle, the
differential equation in [O3] ( eq 3) can be simplified
Appendix II
2
D∇ [O3] ) k[O3]
(I.1)
Substitution of the expression for Γrxn (eq 4) for γ in the
expression relating the rate of change of [Oleic] to γ (eq 1)
yields a differential equation for [Oleic] in time
Since the particles are spherically symmetric, this equation can
be rewritten as
∂[O3]
∂r
D ∂
2
PO cj
D∇ [O3] )
r2
+
SA
(
)
d[Oleic]
3
2
[
∂r
r
) -γ
(1)
(
)
dt
4RT
V
∂[O3]
∂θ
∂2[O3]
1
∂
1
sin θ
+
) k[O3]
(
)
sin 2θ ∂φ2
]
d[Oleic]
D
al
sin θ ∂θ
) -3PO H (coth(a/l ) - l /a)
3
dt
∂[O3]
∂r
D ∂
)
r2
) k[O ]
(I.2)
(
)
3
3P H k D
2 ∂r
x
O3
2
r
) -
[Oleic] ×
x
a
This equation has been simplified by utilizing the fact that [O3]
is spherically symmetric and is only a function of the radial
depth in the particle, i.e., ∂[O3]/∂θ ) ∂2[O3]/∂f 2 ) 0.
Equation I.2 can then be solved by a substitution of
variables: u ) [O3]r to yield
k2[Oleic]
1
D
coth a
-
(II.1)
(
)
[
]
x
D
a
k [Oleic]
x
2
Equation II.1 is the general differential equation for the
concentration of oleic acid including the diffusion of O3 within
the particle and reaction with oleic acid in the bulk of the
particle. A general analytical solution to this equation cannot
be found. However, solutions can be found in certain limiting
cases.
d2u
dr2
u
2
)
(I.3)
(I.4)
l
u(r) ) Acosh(r/l ) + Bsinh (r/l )
u(r)
r
1
r
[O3](r) )
) [Acosh(r/l ) + Bsinh (r/l )] (I.5)
Case 1a: Rapid Diffusion of O3 Throughout the Particle
(l > a). In the limit that l is larger than the radius of the particle
(a), the coth(a/l ) term in eq II.1 can be approximated by a
Taylor expansion. In the following derivation, the substitution
x ) a/l has been made, and the resulting expression, eq 7, is
valid to within 6% when a/l < 1.
where l ) (D/k)1/2 is the diffuso-reactive length.
Since [O3](0), the concentration of O3 at the center of the
particle, must be finite, A ) 0. Additionally, [O3](a), the
concentration of O3 at the surface, is constant in time and is

8094 J. Phys. Chem. A, Vol. 106, No. 35, 2002
Smith et al.
References and Notes
a
l
a
ex + e-x
ex - e-x
1
x
coth
-
)
)
-
(II.2)
(II.3)
( )
(
)
l
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(x - 1) 1 + 2x +
+
+ ‚‚‚ + (x + 1)
(
)
2!
3!
)
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2!
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(II.4)
(II.5)
1
1 a
3 l
= x )
3
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Substituting this approximation for the [coth(a/l ) - l /a] term
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d[Oleic]
D
= -PO H
2
3
dt
l
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(II.6)
3730.
3
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[Oleic] ) [Oleic]0 exp(-PO Hk2t)
(6)
(7)
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4HRT a
γ )
k2[Oleic]
cj
3
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a/20). In the limit that l is much smaller than the radius of the
particle (a), the [coth(a/l ) - a/l ] term in eq II.1 approaches an
asymptotic value of 1. Approximating this term as 1 is valid to
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D
al
= -3PO H
3
dt
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3P H Dk
x
O3
2
) -
[Oleic]
(II.7)
x
a
Solving this differential equation yields
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3P H Dk
x
O3
[Oleic] ) [Oleic] -
2t
(8)
(9)
x
x
0
2a
4HRT
γ )
Dk [Oleic]
2x
x
cj
Case 2: Reaction at the Surface. Substituting Γsurf (eq 11)
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d[Oleic]surf
-
dt
4HRT
cj
Γsurf
)
)
δ2k2surf[Oleic] (11)
PO cj/4RT
3
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for γ in eq 1 yields
d[Oleic]
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3δ2
a
) -
PO Hk2surf[Oleic]
(II.8)
3
dt
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and solving this equation leads to an expression for [Oleic]
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3δ2
[Oleic] ) [Oleic] exp -
PO Hk2surft
(12)
(
)
0
3
a

Reactive Uptake of Ozone by Aerosol Particles
J. Phys. Chem. A, Vol. 106, No. 35, 2002 8095
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