The observed concentrations of aerosol nitrate in July were relatively low. The average observed concentration was 0.6 μg/m3 and reached a maximum of 6 μg/m3 on the morning of 3 July. The mean observed nitric acid concentration was 3.4 μg/m3 and 73% of the total nitrate was in the gas phase during July. Aerosol nitrate was observed to consistently peak in the late night and early morning hours when the temperatures were lower and relative humidities higher, but evaporated during the day when temperatures increased, and relative humidity dropped (Figure 1).
 Predictions from GFEMN follow the diurnal trend of aerosol nitrate quite well, except during a few daytime periods. During the afternoons of 9, 13, 20, 21, and 27 July, simulations predict that the aerosol nitrate concentrations were rising during the day, when observations indicate that they were close to zero. During these periods, RH was often below 40% and temperatures were generally above 20°C with thermodynamics favoring the existence of nitrate in the gas-phase. In a subsequent section we will test the hypothesis that the ambient aerosols during these periods existed as an external mixture of solid ammonium sulfate particles and wet, acidic particles. The model predictions for this month had an absolute error of 0.41 μg/m3 (Table 1) when the PM during periods with RH below 40% is modeled as an external mixture.
 The observed average aerosol nitrate concentration during January 2002 was around 2.1 μg/m3. As in July, the aerosol nitrate exhibited a diurnal variation with a late night or early morning peak and a minimum in the late afternoon. Figure 2 shows the comparison between predicted and observed aerosol nitrate concentrations for January 2002. In general, there was enough ammonia to neutralize both sulfate and nitrate. During a few periods, for example, 12, 15, 16, 20, and 22 January, the model predicts more aerosol nitrate to be present than observed, though in many of these cases measurement uncertainties may account for the discrepancies. For the period 26 to 28 January, no aerosol nitrate is predicted to form, though observations indicate otherwise. The relative humidity was low during these days, and frequently dropped below 40% during the daytime. The concentrations of ammonia during these periods were such that after neutralizing the sulfate, there was not enough free ammonia remaining to react with the nitric acid vapor to form solid ammonium nitrate particles. Therefore, all of the total nitrate was predicted to remain in the gas phase as nitric acid vapor. Uncertainties and sensitivities of these predictions to measurement error will be discussed in a subsequent section. On average, the model predictions for this month had an absolute error of 0.64 μg/m3 (Table 1).
4.1. Experimental Uncertainty
 Both the predictions and the observations are affected by experimental error. For the predictions, measurement errors in the input variables (TS, TN, TA, T, and RH) affect directly the predicted nitrate. To show how each of these measurement uncertainties can combine to contribute to uncertainties in predictions, a Monte Carlo simulation was performed for two days in July and four days in January. A Latin Hypercube Sampling routine [McKay, 1988] was used considering measurement uncertainties in four of the five input variables (RH, TA, TN, and TS). Normal distributions were chosen for the input distribution of these variables, as uncertainties arising from measurement errors are generally considered to be independent and normally distributed [Ripley and Thompson, 1987] with a mean of zero. A coefficient of variation of 0.15 was assumed for the measured concentrations of TS, TA, and TN, in accordance with our assessment of measurement errors. In addition, a standard deviation of 5% was assumed for RH. To avoid generating artificial correlations among the different input variables and to allow the statistics (e.g., mean and variance) of the output distribution to converge, a sampling size of 200 was selected. Figure 3 shows the predictions with error bars corresponding to the 5th and 95th percentiles of the simulations' output cumulative distribution function. Combined uncertainties in model inputs can result in uncertain aerosol nitrate concentrations. Furthermore, a standard deviation of 0.5 μg/m3 was assumed for uncertainty in aerosol nitrate measurements from the R&P 8400N instrument as reported by Solomon et al. , and 200 random samples for aerosol nitrate concentrations were generated for each period. The shaded band in Figure 3 illustrates the range between the 5th and 95th percentiles of the output cumulative distribution function, though concentrations below zero are not shown. In most cases the predictions are within the uncertainties introduced by measurement errors.
Figure 3. Results from Monte Carlo simulations performed for selected periods in July 2001 and January 2002. Error bars extend to the 5th and 95th percentiles of the cumulative distribution function associated with each prediction. The shaded area bounds the interval between the 5th and 95th percentiles of the observed aerosol nitrate cumulative distribution functions, although concentrations below zero are not shown.
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4.2. Sensitivity to Assumed Physical State (Solid or Liquid) of Aerosol
 In three-dimensional chemical transport models simulating atmospheric aerosol thermodynamics it is often assumed that the aerosol growth follows the deliquescence branch of the hysteresis curve [Nenes et al., 1999]. The error introduced by this assumption can be evaluated by comparing the base-case results (the physical state is an input based on the complementary measurements) to the deliquescence branch predictions.
 In the base-case January simulations we assumed that the particles were solid below 60% and aqueous solutions in all other cases, as was observed experimentally (Khlystov et al., submitted manuscript, 2004). The ammonium nitrate deliquescence relative humidity is approximately 60%, so this choice corresponds approximately to the deliquescence branch of the growth curve of an aerosol containing ammonium nitrate. If deliquescence behavior was assumed for January 2002 with GFEMN calculating the deliquescence RH for each case, particles were predicted to be practically always solid below 66% RH (142 cases), while particles were predicted to be always liquid above 80% RH (79 cases). Out of the 103 remaining cases, the model predicted 57% to be solid, 18% liquid, and 25% a mixture of both. In these instances when the physical state of the particle predicted by deliquescence was different from the base case, the relative difference in mean aerosol nitrate predictions was around 20%. These differences did not have much impact on the overall agreement of predictions and observations of nitrate partitioning, however. Figure 4 shows the model error, defined as the difference between the predicted and observed concentrations of aerosol nitrate, for the 120 cases when the observed and predicted particle states differed: the overall error between the base case and the deliquescence case simulations is not considerably different (Table 1). This indicates that the model's performance using the deliquescence branch of the growth curve is satisfactory during the winter.
Figure 4. Error distribution assuming different states of particles in July 2001 and January 2002. Errors are calculated as predicted minus observed values of aerosol nitrate. Points included are only those for which the predicted solid or liquid state was different from the one determined based on in situ aerosol water measurements (base case).
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 For the base case for July 2001, we assumed that particles were liquid at all relative humidities. If deliquescence behavior was assumed for this month, however, particles were always predicted to be liquid above 80% RH (127 cases), but below this RH, 15% were predicted to be liquid, 72% solid, and 13% a mixture of both (415 cases altogether). In the instances when the physical state of the particle differed between the deliquescence and base case, the relative mean difference in nitrate predictions was almost a factor of two (Table 1). This is consistent with the results of Ansari and Pandis  who suggested that the physical state of the particles affects significantly the aerosol nitrate concentration at low nitrate concentration levels. Figure 4 shows a comparison of the model error for the 353 points when the aerosol is predicted to be solid or a mixture of solid or liquid instead of a pure liquid. The deliquescence case shows a large number of observations have errors close to zero μg/m3 with an overall mean of −0.32 μg/m3, but a consistent underprediction is observed. This underprediction can be explained by the fact that no nitrate was predicted to be in the aerosol phase in 314 out of the 353 simulations for the deliquescence case, while the same was true for only one instance during the same period when particles were simulated as pure liquid. The spread in the error is greater for the base case, but the distribution of these errors is centered about a mean of 0.25 μg/m3.
 The assumed physical state affected the relative difference in predictions of aerosol nitrate concentrations during July more so than during January because the nitrate concentrations were generally lower during the summer. Prior knowledge of the physical state of the aerosol was a lot more important for the summer months than in the winter (Table 1).
4.3. Sensitivity to Mixing State (Internal Versus External) During Periods of Acid PM
 We examine here the possibility that the discrepancies during the low RH acidity periods in July were due to the existence of an external mixture of acid and neutral particles. The crystallization RH for ammonium sulfate is 40% and the crystallization RH for ammonium bisulfate is much lower. As a result the aerosol could consist of solid neutral particles (ammonium sulfate) and liquid acidic particles (ammonium bisulfate) when the RH dropped below 40%. We first examine the impact of this assumption on the nitrate partitioning on the afternoons of 9 and 21 July, when the disagreement between predicted and observed aerosol nitrate is particularly pronounced in the base case. Figure 5 shows that if the particles are modeled as an internally mixed population, the aerosol nitrate concentration is predicted to be 1–2 μg/m3 while observations indicate that it is close to zero in the afternoon. However, assuming that the particles are externally mixed and using the algorithm described in the previous section, the predicted aerosol nitrate concentration drops to practically zero, as observed, when the measured RH drops below 40% between noon and 7 pm on these two days. Because the RH threshold below which we simulate the particles as an external mixture is set at 40%, the selection of these periods is very sensitive to measurement errors in RH. Figure 6 shows that if a similar approach is used for the other periods in July in which the measured RH fell below 40%, the performance of the model improves significantly during these times. The premise of hypothesizing the existence of this specific external mixture during these low RH periods is that in the base case, the aerosol nitrate concentrations are overestimated because the particles are liquid and sufficiently neutralized to accommodate a fraction of the total nitrate present in the system. If we assume that one type of particle is neutral but devoid of water and the other is wet but significantly more acidic, the partial pressure product of ammonia and nitric acid vapor required to form ammonium nitrate becomes significantly higher than in the base case and hence results in lower aerosol nitrate concentrations, as we observe. While it is possible that the particles do not exist in one extreme condition or the other but a mix of the two, comparisons with observations suggest that particles may be closer to the externally mixed state. This simple algorithm is not suitable for running simulations at higher RHs, however, when both neutral and acidic particles are assumed to be liquid. In these cases, the nitric acid and ammonia can be incorporated into both particle types and the algorithm cannot predict the gas-phase vapor pressures that will satisfy equilibrium conditions with the two particle types simultaneously.
Figure 5. Simulations for 9 and 21 July assuming that particles are (1) internally mixed liquid aerosols, and (2) an external mixture of crystallized ammonium sulfate and wet acidic aerosols when the relative humidity is below 40%.
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Figure 6. Errors between predictions and observations for periods with RH < 40% assuming different states of particles in July 2001. Errors are calculated as predicted minus observed values of aerosol nitrate.
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 Particles can exist in an externally mixed state if their sources and histories are different. For instance, neutralized particles could be transported into the Ohio River Valley and western Pennsylvania from the ammonia-rich midwestern United States while acidic particles are generated locally. The different types of particles could be mixed together as the nighttime boundary layer vanishes in the morning, or some particles could be formed as a result of cloud processing, while others could be generated through condensation and nucleation. Suitable thermodynamic models and further experimental evidence may reveal more information on the existence of different mixing states.
4.4. Other Sources of Error
 Though predictions of nitrate partitioning capture diurnal trends reasonably well, errors in some periods can be high (Table 1) and explanations other than measurement error are needed. As noted previously, many assumptions are invoked when using thermodynamic models. The equilibrium assumption may not be valid if some particles are in the upper size range of PM2.5. However, data collected at the PAQS suggest that the geometric mean diameters of PM2.5 during the summer were 0.33 μm for ammonium, 0.36 μm for sulfate, and 0.66 μm for nitrate; and during the winter the geometric mean diameters were 0.30 μm for ammonium, 0.31 μm for sulfate, and 0.34 μm for nitrate [Cabada et al., 2004]. As crustal elements are found in low levels in Pittsburgh, salts of nitrate and crustal elements are not expected to be significant. Also, since measurements at reasonably short timescales were used, it is difficult to explain the observed differences by the averaging of different air masses.
 Two additional sources of error remain unaccounted for. Though previous studies have shown that on average, the expected influence of organics on partitioning of semivolatile inorganics is low [Koo et al., 2003], their impacts on individual cases are still unknown. In addition, the bulk equilibrium approach used in this study can introduce errors, as nitrate does not partition equally into different particles, and modeling them in bulk can introduce errors.
4.5. Sensitivity Analysis
 A perturbation analysis was performed to examine the sensitivity of our model to changes in input variables. Such an analysis can strengthen our understanding of the system's chemistry and further elucidate the potential influence of measurement errors. It can also provide a notion of the sensitivity of atmospheric processes to changes in meteorology and chemistry and thus guide our efforts to make preliminary estimates of PM response to changes in concentrations of precursor species. This section will focus primarily on the first objective, but implications for PM control strategies will also be discussed.
 The response of the predicted aerosol nitrate concentrations to changes in input parameters depends greatly on the point in the domain space that is examined. Ansari and Pandis  proposed a parameter, the gas ratio (GR), that combines three input variables (TA, TS, and TN) to describe different regions of the domain in terms of the amount of free ammonia relative to the amount of total nitrate:
 Table 2 summarizes the characteristics of the different regions of ammonia availability parameterized by GR. Dimensionless sensitivity coefficients can be defined by
where NO3 is the aerosol nitrate concentration and x is an input variable. The sensitivity coefficient can be used to assess the potential impacts of uncertainty in each input variable on the predicted aerosol concentration: Sx is a measure of the relative change in the predicted aerosol nitrate concentration normalized by the relative change in the input variable x when all other input variables are held fixed. Because coarse nitrate is not included in our calculations, we implicitly assume that the small amount of nitrate associated with particles larger than 2.5 μm in the area [Cabada et al., 2004] will not change during these perturbations. Figures 7 and 8show values of Sx when the temperature is perturbed by −3°C, RH is perturbed by −5% (absolute), and TA, TN, and TS are perturbed by −15% (relative) for the months of July 2001 and January 2002, respectively. The response of nitrate is approximately linear over a ±15% range of input concentrations. Overall, the variability in Sx is much larger during the summer, when TA, TN, and TS concentrations spanned a wider range than during the winter.
Figure 7. Sensitivity coefficients (Sx) of aerosol nitrate in July 2001 in response to model inputs of temperature (−3°C), RH (−5% absolute), and TA, TN, and TS (−15% relative). Separate plots are shown for different regimes of the gas ratio (GR). Shaded boxes cover the interquartile range of Sx, while whiskers extend to the 5th and 95th percentiles of Sx. The black line in the box corresponds to the median.
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Figure 8. Sensitivity coefficients (Sx) of aerosol nitrate in January 2002 in response to model inputs of temperature (−3°C), RH (−5% absolute), and TA, TN, and TS (−15% relative). Separate plots are shown for different regimes of the gas ratio (GR). Shaded boxes cover the interquartile range of Sx, while whiskers extend to the 5th and 95th percentiles of Sx. The black line in the box corresponds to the median.
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Table 2. Characterization of GR Regions
|Region Label||Value of GR||Description|
|Low GR||<0||there is insufficient ammonia to neutralize all of the sulfate to (NH4)2SO|
|Moderate GR||≥0 and <1||there is enough ammonia to neutralize all sulfate but not nitrate|
|High GR||≥1||there is a sufficient quantity of ammonia to neutralize all sulfate and nitrate|
 During the summer, the sensitivity of nitrate to temperature is relatively small but distinctly negative (median Sx = −1.01), because decreasing the temperature lowers the dissociation constant of ammonium nitrate and allows more nitric acid to exist in the aerosol phase. During the winter, nitrate is not as sensitive to this parameter (median Sx = −0.03), as most of the nitrate is found in the aerosol phase. The change in nitrate concentrations with respect to RH is on average close to zero (median Sx = −0.26 in July and 0.20 in January), but can be significant during certain periods in July 2001 when most particles are liquid, as illustrated by the large variability of Sx for RH (Figure 7). At or near crystallization points, the change in nitrate with respect to RH can be even more significant. Since crystallization is a function of RH, changes in RH by a few percent can change the picture of nitrate partitioning considerably when the predicted aerosol makes a transition from the aqueous to the solid state.
 The nitrate response to sulfate can be very large for GRs less than 0 (median Sx = −9.77 in July and −4.30 in January), as shown in Figures 7 and 8, because removing sulfate from the system can free up ammonia to react with nitric acid, and adding sulfate can remove ammonia gas and prevent nitrate aerosols from being created. At higher GRs, adding sulfate can actually increase nitrate because sulfates are hygroscopic and increase the water content of aerosols, thus allowing more nitric acid to be pulled into the aerosol phase. Since concentrations of sulfate are generally higher than those of other species, small changes relative to the nominal sulfate can elicit a large response in nitrate, and is reflected in the large values of Sx for this parameter. Sx for TA can be quite large in the case of low and moderate GRs (GR < 0, median Sx = 5.34 in July and 3.44 in January; 0 ≤ GR < 1, median Sx = 1.05 in July and 1.52 in January), as the formation of nitrate aerosols is sensitive to the availability of ammonia and perturbation in TA directly changes this quantity. This ammonia-responsive condition is observed 61% of the time in July 2001 and 45% during January 2002. The magnitude of Sx in response to TA can depend quite largely on the sulfate levels if the particles are liquid, as the equilibrium constant of ammonium nitrate is dependent on the ionic strength of the solution, which in turn is influenced by sulfate concentrations [Ansari and Pandis, 1998]. As a result, predictions of aerosol nitrate are generally sensitive to measured concentrations of ammonia as well as sulfate, more so during the summer than in the winter. Since aerosol nitrate formation is dependent on the amount of TN in the system, measurement errors in TN can also influence aerosol nitrate predictions. When there is an excess of ammonia, as reflected by higher GRs, much of the nitrate can be found in the aerosol phase. When this occurs, e.g., at night or during the winter, the predicted aerosol nitrate concentrations are more directly influenced by the measured TN. However, the actual sensitivity is also subject to ambient temperatures and relative humidities.
 This sensitivity analysis provides insight as to how measurement errors in input variables can affect predictions in nitrate partitioning according to the domain space, but it also suggests implications for PM2.5 control strategies. For instance, sulfate reduction is an obvious choice for reducing PM mass since reducing sulfate reduces a large component of PM2.5, but this strategy can significantly increase aerosol nitrate concentrations when the formation of nitrate particles is limited by ammonia availability. Our sensitivity analysis shows that aerosol nitrate concentrations are responsive to reductions in ammonia concentrations under such ammonia-limited conditions, which were often observed during July 2001. In these cases, reductions in ammonia may be an additionally effective measure in reducing PM mass.