A photochemical box model based on the framework of Capaldo et al.  is used to model regional nucleation events occurring in July 2001 and January 2002. Atmospheric processes included in the model are gas-phase chemistry, condensation, aerosol coagulation, and nucleation. The model is a fixed sectional model with 221 size sections: 186 size sections between 3 nm and 10 μm corresponding to the measured size distribution of particles and 35 additional size sections evenly distributed in logarithmic space by diameter for particles below the detection limit of the field instruments. The concentration of SO2 was measured directly and is an input to the model and the input concentrations of NH3 and OH were calculated from measurement data and are discussed in section 3. The governing equations for the model, giving the change over time of the concentration of gaseous sulfuric acid, and the number concentration of particles in section i, Ni, are:
where Rgas is the rate of change due to gas-phase chemistry, Rnuc is the rate of nucleation, n* is the number of sulfuric acid molecules in the critical nucleus, Rcoag is the coagulation rate, Rcond is the condensation rate, and Rdep is the rate of dry deposition, with dependence on the relative humidity RH, temperature T, pressure P, dry particle size distribution, Nj, and gas-phase concentrations shown. The rate terms are discussed in detail below.
 The rate of nucleation, Rnuc, is calculated using the ternary NH3-H2SO4-H2O parameterization of Napari et al. . The parameterization uses the NH3 gas-phase concentration, H2SO4 gas-phase concentration, temperature, and relative humidity as inputs, and provides a nucleation rate as output. The upper limit of the nucleation rate for which the parameterization is valid is 106 particles cm−3 s−1, so rates higher than this are capped at this value. Napari et al.  also give an approximation for the radius of the initial nuclei as a function of the nucleation rate and temperature. The initial nuclei diameter was calculated to be 0.8 nm for the winter and 1.0 nm during the summer, assuming average temperatures of 275 K and 298 K, respectively. The nuclei diameter was assumed to be constant for each period. The number of sulfuric acid molecules in the critical nucleus, n*, is also calculated using the approximation given in the work of Napari et al. . There were roughly two sulfuric acid molecules in the critical cluster during the winter and four during the summer.
 Figure 2a shows the nucleation rates given by the ternary NH3-H2SO4-H2O parameterization for typical July daytime conditions: 298 K, 60% RH, and H2SO4 from 107 to 3 × 108 molecules/cm3 (0.4 to 4.0 ppt). Ammonia is allowed to vary over the entire range of the parameterization, 0.1 to 100 ppt. The nucleation rates are not high enough for nucleation to occur unless some gas-phase ammonia is present, although ammonia concentrations above 10 ppt and sulfuric concentrations above 2.4 ppt result in nucleation rates of at least 10 particles cm−3 s−1, so even this small amount of gas-phase ammonia can be enough for nucleation to occur on summer days in Pittsburgh.
Figure 2. Nucleation rates (in particles cm−3 s−1) for typical conditions in (a) July and (b) January. The conditions for July are: temperature 298 K, relative humidity 0.6, sulfuric acid 0.4–4 ppt, and gas-phase ammonia 0.1–100 ppt. The conditions for January are: temperature 275 K, relative humidity 0.7, sulfuric acid 0.00375–3.75 ppt, and gas-phase ammonia 0.1–100 ppt.
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 In contrast to July, where the presence of gas-phase ammonia gives the best indication of when nucleation will occur, both the H2SO4 concentrations and gas-phase ammonia can influence whether or not nucleation will occur during the winter. Figure 2b shows the nucleation rates for typical January conditions: 275 K, 0.7 RH, 105–108 molecules/cm3 H2SO4 (0.00375 to 3.75 ppt), and 0.1–100 ppt NH3. At higher sulfuric acid concentrations, nucleation rates can still reach about 1000 particles cm−3 s−1 even with very low gas-phase ammonia concentrations. At lower sulfuric concentrations, nucleation rates can be less than 100 cm−3 s−1 even with high gas-phase ammonia concentrations. Consequently, the amount of H2SO4, which varies more than in the summer months, plays a more important role during the winter.
 The condensation rate, Rcnd, is described using the modified form of the Fuchs-Sutugin equation [Fuchs and Sutugin, 1971; Hegg, 1990; Kreidenweis et al., 1991]. The condensation rate J to a particle of diameter Dp is given by:
where D is the diffusivity of sulfuric acid in air (set to 0.1 cm2 s−1), Kn is the Knudsen number, and F(Kn) is a coefficient correcting for free molecular effects:
A is a coefficient correcting for the interfacial mass transport limitations described by the accommodation coefficient ae:
Here p is the bulk partial pressure of sulfuric acid and p0 is its partial pressure at the particle surface. The value of the accommodation coefficient depends on the composition of the particle, with the presence of organic species in the aerosol likely to result in a lower accommodation coefficient. Jefferson et al.  report values of 0.31 and 0.19 for the accommodation coefficient onto a NaCl aerosol coated with stearic acid with high and low coverage, respectively, compared to values 0.73 ± 0.21 for ammonium sulfate particles and 0.79 ± 0.23 for NaCl. Since the preexisting aerosol mass is likely to be a mixture of inorganic and organic species, the accommodation coefficient is set at 0.2 in this work. Sensitivity analysis is performed to see how changes in this parameter affect the model results.
 H2SO4 is assumed to be the major condensing species. Although it is likely that other species, such as nitrate, ammonium, and organic compounds, also are involved in the growth of the nuclei, their respective roles remain unclear. Measurements taken during PAQS in September 2002 indicate that sulfuric acid is the primary initial species involved in the growth of the nuclei, followed by ammonium and then organics, whose presence lags behind sulfuric acid by .5 hours to up to 2.5 hours in the smallest measured particles (AMS measurements of particles 22–40 nm, estimated physical diameter) [Zhang et al., 2004]. Although contributions of other species cannot be ruled out, condensation of sulfuric acid alone produces growth that is similar to the observations, as will be discussed in more detail below.
 The vapor pressure of sulfuric acid at the surface of the aerosol can be estimated from the data of Bolsaitis and Elliott . For example, at a temperature of 293 K and a relative humidity of 90%, the value of p0 is approximately 10−5 ppt. Since the calculated values of p0 are much smaller than ambient sulfuric acid concentrations, p0 is assumed to be zero.
 For the ith aerosol size section from xi = log10(Di) to xi+1 = log10(Di+1), the sulfuric acid condensation rate is given by
where the sectional mass transfer coefficients Kmti are calculated by
The total change in the gas-phase concentration of sulfuric acid due to condensation is equal to the sum of Ji over all size sections i.
 The diameter of the aerosol particle is adjusted to be in equilibrium with the ambient RH before the calculation. The dry aerosol diameter, Dp,dry, is increased due to the addition of water vapor according to the following parameterization for ammonium bisulfate based on data presented by Seinfeld and Pandis :
The parameterization is valid for RH between 50% and 98%. When RH is below 50%, the particles are assumed to have negligible amounts of water. This parameterization neglects the effect of curvature, which can be significant for particles less than 10 nm in size.
 Finally, the change in the number concentration in section i, Ni, is calculated from the relation:
where Fi is the flux from section i into section i + 1:
MH2SO4 is the molecular weight of sulfuric acid, ρ is the density of the particles, and R is the ideal gas constant.
 The coagulation rate of aerosol particles, Rcoag, is modeled according to Seinfeld and Pandis , using linear interpolation to preserve both mass and particle number concentrations:
where Kk,j is the coagulation coefficient of particles in section k and j, and fk is a correction factor to preserve mass, giving the fraction of the newly formed particle that will go into section k. The generalized coagulation coefficient for the collision of two particles is defined as:
where β is the Fuchs correction factor [Fuchs, 1964]. Linear interpolation is then used to determine the value of fk. For example, if Vp is in between sections k and k + 1, with volumes Vk and Vk+1 respectively, then fk is defined as:
2.5. Dry Deposition
 Dry deposition is modeled using a species and aerosol size-dependent deposition velocity, vdep, such that
where ci is the concentration of gas species or aerosol size section i, and H is the mixing height. Aerosol dry deposition rates range from 0.1 to 0.015 cm s−1 dependent on particle size according to Hummelshoj et al.  while the deposition velocity of H2SO4 is assumed to be 1.0 cm s−1 [Brook et al., 1999].