Measurement of prenucleation molecular clusters in the NH 3,H 2 SO 4, H 2 O system

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. D12, 4158, /2001JD001100, 2002 Measurement of prenucleation molecular clusters in the NH 3,H 2 SO 4, H 2 O system D. R. Hanson and F. L. Eisele 1 Atmospheric Chemistry Division, National Center for Atmospheric Research, Boulder, Colorado, USA Received 16 July 2001; revised 30 November 2001; accepted 3 December 2001; published 29 June [1] The molecular cluster ions HSO 4 (H 2 SO 4 ) n 1 (NH 3 ) m corresponding to the neutral species (H 2 SO 4 ) n (NH 3 ) m for n = 2 to 6 and m =0ton 1 have been observed at temperatures up to 285 K. A transverse chemical ionization apparatus was located inside a cooled flow tube where water, sulfuric acid, and ammonia vapors mixed and formed clusters. The complexities of the experimental technique and the interpretation of the results are extensively discussed. Typical NH 3 and H 2 SO 4 concentrations were cm 3, i.e, 100 pptv at atmospheric pressure. For these conditions, cluster concentrations were estimated to be a few times 10 6 cm 3 and the critical, particle-forming cluster likely contained 2 H 2 SO 4 molecules at 275 K. The results are consistent with the species (H 2 SO 4 ) 2 NH 3 playing an important role in the formation of new particles in the atmosphere. INDEX TERMS: 0305 Atmospheric Composition and Structure: Aerosols and particles (0345, 4801); 0317 Atmospheric Composition and Structure: Chemical kinetic and photochemical properties; 0335 Atmospheric Composition and Structure: Ion chemistry of the atmosphere (2419, 2427); 0394 Atmospheric Composition and Structure: Instruments and techniques; KEYWORDS: ammonia, sulfuric acid, nucleation, molecular clusters, particle formation 1. Introduction [2] Atmospheric aerosol particles can have a large influence on radiative forcing of climate and are also believed to pose a health hazard [Intergovernmental Panel on Climate Change (IPCC), 1995; U.S. Environmental Protection Agency, 1997]. For these and other reasons, research on the sources of these particles is very important. Gas-toparticle nucleation involving H 2 SO 4 vapor has long been believed to be a robust source of atmospheric particles. A number of recent studies have shown a strong correlation between H 2 SO 4 and new particles [Weber et al., 1999; Birmili et al., 2000]; however, the process by which these particles are formed is not well understood. Recently, ammonia vapor has been proposed as a potential key player in the formation of atmospheric particles [Coffman and Hegg, 1995; Larsen et al., 1997; Ball et al., 1999; Korhonen et al., 1999]. [3] In a recent paper [Eisele and Hanson, 2000] we described an apparatus designed to measure molecular clusters of sulfuric acid under quiescent (as opposed to free jet expansion) particle-growth conditions. Neutral H 2 SO 4 clusters containing 2 to 8 H 2 SO 4 molecules at 240 K were detected. The cluster measurements were performed for H 2 SO 4 concentrations of cm 3, much greater than would be found in the atmosphere and were conducted at temperatures of 240 K and relative humidities (RH) of 15 to 1 Also at School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia, USA. Copyright 2002 by the American Geophysical Union /02/2001JD %. The critical cluster was found to contain 3 to 4 H 2 SO 4 molecules and its abundance was as high as 10 7 cm 3. [4] This technique provides a window through which particle nucleation can be observed, and in this paper we extend the system to include ammonia vapor. For example, molecular clustering involving H 2 SO 4 and NH 3 can be illustrated as H 2 SO 4 þ H 2 SO 4 þ NH 3 $ðh 2 SO 4 Þ 2 þ NH 3 $ðh 2 SO 4 Þ 2 NH 3 ð1aþ H 2 SO 4 þðh 2 SO 4 Þ 2 NH 3 þðm 0 1ÞNH 3 $ðh 2 SO 4 Þ 3 ðnh 3 Þ m ; ð1bþ... H 2 SO 4 þðh 2 SO 4 Þ n 1 ðnh 3 Þ m * þðm m*þnh 3$ðH 2 SO 4 Þ n ðnh 3 Þ m : ð1cþ For the sake of simplicity, H 2 O molecules are not taken into account here. Note that (H 2 SO 4 ) n (NH 3 ) m can be written in shorthand as (n, m). The monomer reactants H 2 SO 4 and NH 3 are indicated as separate entities, but they may be associated (i.e., as NH 3.H 2 SO 4 ) or they may add stepwise to a cluster. Note that reactions (1a) (1c) are only to illustrate the clustering processes and some equations are composed of multiple steps. [5] Here we report the observation of (H 2 SO 4 ) n (NH 3 ) m clusters for n = 2 to 6 and m < n at number densities up to several 10 6 cm 3. We detected these clusters at temperatures up to 285 K; much warmer than those necessary to form detectable levels of clusters in the H 2 SO 4 -H 2 O system AAC 10-1

2 AAC 10-2 HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS (240 K [Eisele and Hanson, 2000]). A number of tests designed to understand the ionization and detection of these clusters are presented, and several potential problems are identified and thoroughly discussed. Also, a computational fluid dynamics model [Fluent Incorporated, 1999] was employed to gain a better understanding of the flow, temperature profile, and reactant concentrations within the flow reactor. We present evidence that the critical cluster, the cluster accountable for the formation of stable particles, contains three or fewer H 2 SO 4 molecules and two or fewer NH 3 molecules. 2. Experimental Method [6] The measurement of H 2 SO 4 and NH 3 vapors and their molecular clusters were made in a thermostatted flow tube using chemical ionization mass spectrometry (CIMS). The apparatus is described in detail by Eisele and Hanson [2000], and here a brief description is given along with details of significant changes in the experiment. As previously discussed in that paper, H 2 SO 4 and the clusters (H 2 SO 4 ) n (H 2 O) m were monitored by reacting them with NO 3 (HNO 3 ) 1,2 ions [Viggiano et al., 1997]. Here, the reaction proceeds as ðh 2 SO 4 Þ n ðnh 3 Þ m þ NO 3 ðhno 3Þ 1;2! HSO 4 ðh 2SO 4 Þ n 1 ðnh 3 Þ m þ 2; 3ðHNO 3 Þ: ð2þ This equation is written assuming the HNO 3 ligands do not associate with the HSO 4 core ions and that these core ions do not lose H 2 SO 4 or NH 3 ligands upon ionization. The assumptions about HNO 3 and NH 3 are made for conciseness, and their validity does not in general significantly affect our interpretation of the experimental results. Note that H 2 O molecules are not included in reaction (2). As demonstrated by Eisele and Hanson [2000], neutral sulfuric acid clusters that contain H 2 O molecules easily lose them upon ionization and/or sampling, but the H 2 SO 4 content of a cluster is preserved due to the high affinity that H 2 SO 4 molecules have for HSO 4. Thus, despite the loss of H 2 O and, to a lesser extent, NH 3 molecules upon ionization and sampling, much can be learned about prenucleation clusters in the H 2 SO 4 /NH 3 system. It is convenient to introduce a shorthand nomenclature (1&n, m) for the ionic species HSO 4 (H 2 SO 4 ) n (NH 3 ) m, which indicates one bisulfate ion + nh 2 SO 4 and mnh 3 molecules. [7] A flow of N 2 carrier gas was directed over liquid sulfuric acid (98 wt%) and another was humidified to provide separate control of the [H 2 SO 4 ] and of relative humidity. These flows and an additional flow of N 2 (to make a total N 2 flow rate of 15 STP L min 1 ) were mixed in a region held at 300 K. The gas passed through an aluminum heat exchanger and into the vertically aligned flow tube (9.5 cm ID 110 cm in length) the majority of which is surrounded by a jacket through which a thermostatted coolant was circulated. The average flow speed down the tube was 4cms 1. Located 40 cm into the cooled region (and 50 cm below the heat exchanger) were ports for the CIMS ion source and mass spectrometer inlet. These extended into the flow tube and were 3.5 cm apart to provide in situ detection transverse to the gas flow [see Eisele and Hanson, 2000, Figure 1]. The voltage difference between the source and inlet determined the ion-molecule reaction time. A variable ion-molecule reaction time provided a means to distinguish between ionization of neutral H 2 SO 4 clusters and ion-molecule clusters produced by stepwise addition of H 2 SO 4 or NH 3 molecules to HSO 4 ions [Eisele and Hanson, 2000]. H 2 SO 4 and NH 3 concentrations ranged from and cm 3, respectively. Total pressure was 0.8 atm (610 torr.) The flow tube temperature T f was held at 260 to 283 K to induce formation of neutral clusters of H 2 SO 4 and NH 3 (a comparison experiment in the H 2 SO 4 H 2 O system was performed at 243 K). [8] Water vapor was present typically at a partial pressure of 0.23 torr, although a few mass spectrums were taken at ph 2 O up to 3 torr. The ph 2 O of 0.23 torr is equivalent to a relative humidity of 10% at 265 K and 5% at 275 K. For water partial pressures lower than this, H 2 SO 4 coming off the mixing region walls became noticable. For higher water contents the mass spectrum became quite complicated, littered with NO 3 (HNO 3 ) 1,2 (H 2 O) x and other ions. Note the pressure in the collisional dissociation chamber (CDC) was 0.01 torr. The CDC is schematically presented in Figure 1b and in Figure 1 of Eisele and Hanson [2000]. In that study of the H 2 SO 4 H 2 O system the pressure in the CDC was 0.08 torr. A lower pressure here was chosen to minimize stripping of NH 3 from cluster ions. However, that lower pressure also causes the stripping of water molecules from cluster ions to be less efficient. Therefore the water content in the flow reactor was kept relatively low to minimize interferences with the cluster ion signals. As a test of this setup, the flow tube was cooled to 243 K to reinvestigate the clusters formed in the H 2 SO 4 H 2 O system. [9] As in our previous study, the temperature of the gas in the flow tube was not uniform owing to the cooling that the gas must undergo. The gas temperature at the level of the mass spectrometer inlet was found to be [Eisele and Hanson, 2000] 2 4 K warmer than the wall (depending on proximity to the flow tube wall). This measurement was repeated here, and a comparable temperature difference was observed. Also, the fluid dynamics model (details given below) showed the gas temperature was 1 K warmer than the flow tube wall in this region. We report the temperature of the gas in the detection region to be 2 K warmer than the flow tube wall (uncertainty of ±1 K) NH 3 [10] Ammonia vapor entered the flow tube through a separate inlet located 25 cm above the detection region. Low levels of [NH 3 ] were accomplished by passing a dilute NH 3 (27 ppmv) in N 2 mixture through a dilution system [Ball et al., 1999] resulting in a 30 sccm (STP cm 3 min 1 ) flow of N 2 containing 10 to 100 ppbv NH 3. This flow entered the main flow through the inlet, a 1/8 inch (0.32 cm) OD Teflon tube, whose end was a distance of 25 cm upstream of the detection region. The inlet was positioned slightly off the center of the tube based on maximizing the cluster signals. The cluster signals were not extremely sensitive to this position; e.g., they varied 10 to 30% when the injector was moved 1 cm either way off the maximum setting. Assuming NH 3 is not lost on surfaces and it is fully mixed in the main flow, the average NH 3 concentration would be 200 parts per trillion by volume (pptv; one pptv here is equal to molecules cm 3 ) from an initial

3 HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS AAC 10-3 Figure 1. Results from computational fluid dynamics (CFD) simulation of the flow reactor with addition of NH 3. (a) Lower half of the reactor in cross section is shown with the axial distance from the beginning of the cooled portion of the reactor indicated along one side. The ammonia inlet is the small tube 25 cm into the flow reactor; the ionization region is located at 50 cm. [NH 3 ] indicated in mixing ratio contours at 5 pptv intervals. (b) Radial cross section at 50 cm. The ion source and the mass spectrometer inlet are overlaid on this plot. 100 ppbv ammonia in the NH 3 inlet flow. In the flow reactor, however, there is efficient loss due to H 2 SO 4 on surfaces. Also, the time allowed for NH 3 to diffuse into the main flow may not be sufficient for full mixing. [11] Positive ions emanate from the ion source when it is biased at a positive voltage, and they consist of water proton clusters due to the presence of small amounts of water. NH 3 was detected by reacting it with water proton clusters: NH 3 þ H 3 O þ ðh 2 OÞ n! NH þ 4 ðh 2OÞ m þðn mþh 2 O: ð3aþ This reaction is known to be fast (k 2 = cm 3 s 1 ) for n up to 9 [Viggiano et al., 1988]. It is likely to proceed at significant rates for n even larger. Note the products of equation (3a) are difficult to assign, and thus they are ambiguously written as some stripping of water molecules from the core NH 4 + ion may occur during sampling. In our system with a typical electric field of 300 V cm 1 the mean ion energy is essentially the same as the gas temperature and the proton water clusters are dominated by values of n between 4 and 7 [Lau et al., 1982]. Therefore a rate coefficient of cm 3 s 1 was used to calculate the approximate ammonia concentrations from ½NH 3 Š¼lnð1 þ S NH4þ =S H3Oþ Þ=k 3 t; ð3bþ where the ions signals S i are the sum of all i species plus associated water clusters and t is the ion molecule reaction time (varies from to s) and is given by t ¼ l 2 =ðmvþ; where l is the distance and V is the voltage difference between the source and mass spectrometer inlet (3.5 cm and 500 to 2500 V, respectively), and m is the mobility of the reactant ion which is estimated to be 2.1 cm 2 (sv) 1 at 0.8 atm and 275 K for the n = 5 proton cluster (see the appendix). A typical value of the measured NH 3 level from (3) is [NH 3 ]= cm 3 (i.e., 55 pptv at 610 torr and 275 K, n N2 = cm 3 ) for a typical NH 3 addition (50 ppbv in 30 sccm, i.e., a fully mixed, average NH 3 level of 100 pptv). Measured NH 3 levels were typically half of that calculated assuming full mixing, and this is probably due to loss on the flow tube wall and other surfaces. In the appendix, we estimate an uncertainty of +50/ 35% in the reported NH 3 and H 2 SO 4 concentrations Computational Fluid Dynamics [12] The mixing of ammonia into the main flow and NH 3 loss to the flow reactor wall was evaluated with a computational fluid dynamics (CFD) model. Information about the flow and temperature fields is also obtained. A portion of ð4þ

4 AAC 10-4 HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS the cold region (275 K) of the flow reactor 50 cm long by 10 cm inner diameter was modeled. The aluminum heat exchanger was not explicitly modeled, and thus the temperature of the incoming N 2 carrier gas was set at a temperature of 285 K (this is the approximate temperature of the gas just below the heat exchanger measured with a thermocouple). The ammonia inlet was located 23 cm from the main flow inlet, and ammonia was introduced at a level of 16 ppbv in the 30 sccm flow, which would result in an average [NH 3 ] of 33 pptv. NH 3 was assumed to be lost from the gas phase upon each collision with a surface. In the interest of maintaining simplicity, we did not include the mass spectrometer inlet or ion source in the model. Note that these simulations are not expected to yield results that are crucial to the interpretation of our data. They are considered to be illustrative and will provide supporting evidence for that interpretation. [13] The results of this simulation show that after the 25 cm of travel from the NH 3 inlet to the detection region, the ammonia has spread out over a significant portion of the flow reactor radial cross section. Figure 1 shows contour plots of the distribution of ammonia on an axial cross section (Figure 1a) and at the level of the detection region (Figure 1b). In Figure 1a it can be seen that [NH 3 ] takes maximum values of 90, 50, 35, 30, and 25 pptv at distances of 20, 15, 10, 5, and 0 cm above the detection region, respectively. The decreasing [NH 3 ] down the flow reactor reflects the loss of NH 3 to the flow reactor wall. The maximum ammonia concentration in the detection region in Figure 1b is 75% of that calculated assuming full mixing with no losses. This value was very sensitive to the value of the diffusion coefficient selected for NH 3 in N 2 (we used 0.2 cm 2 s 1 ). This, along with the provisions discussed above, lead us to expect only qualitative agreement with the measurements where ammonia was 50% of the calculated average ammonia. [14] The ion source, the mass spectrometer inlet, and the CDC are schematically shown in Figure 1b. Q a is a flow of clean, dry N 2 that is introduced on the high-pressure side of the orifice. It is meant to minimize the amount of water vapor that enters the vacuum system and is typically set such that there is excess flow of the order of 50 to 100% of that which enters the vacuum system (90 sccm). The ions formed in the ionization region traverse this region due to the 100-V potential between the first plate and the orifice. Below we present evidence that the excess Q a flow also interacts with the neutral clusters, drying them out of NH 3 and probably H 2 O molecules Ionization Processes [15] The production of ions from neutral clusters via equation (2) has a linear time dependence. Therefore the ratio of the signals of any two ions produced via direct ionization of neutral clusters is independent of ion reaction time. We use the signal due to H 2 SO 4, the HSO 4 ion, to scale (ratio) the other ion signals because the signal due to H 2 SO 4 cannot be due to ion-molecule association reactions. In addition, scaling an ion signal to HSO 4, (1&0, 0), takes into account the changes in sampling efficiency that occur as the ion reaction time is varied. It is possible that ionization of H 2 SO 4 NH 3 would lead to HSO 4 ; however, its presence is thought to be very low (<10 4 cm 3, see section 4.3 below) for the typical amount of ammonia present in our experiments. Finally, decomposition of HSO 4 H 2 SO 4 producing HSO 4 is negligible, owing to its 42 kcal mol 1 bond strength [Lovejoy and Curtius, 2001]. Therefore plots of the signal ratios (1&n, m)/(1&0, 0) versus ion reaction time will have a slope of zero if (1&n, m) is due to ionization of neutral (H 2 SO 4 ) n+1 (NH 3 ) m clusters Ion-molecule association [16] An ion may be due in part to addition (or loss) of a neutral species to (or from) another ion. This is discussed at length by Eisele and Hanson [2000] for the H 2 SO 4 -H 2 O system; here, we present a short synopsis of the issues, while the appendix has a detailed discussion. The basic difference between forming an ion via ion-molecule association reactions and ionization of a preexisting neutral cluster is their dependence on ion-molecule drift (or reaction) time t. The association reactions have a quadratic (or higher) dependence on t, while the ionization of neutral clusters is linearly dependent on t. [17] The difference in the time dependence between these two ionization processes limits the extent to which an association reaction can contribute to a particular ion signal. The conversion of a (1&n, m) ion to a (1&{n + 1}, m) ion via the process H 2 SO 4 þ HSO 4 ðh 2SO 4 Þ n ðnh 3 Þ m! HSO 4 ðh 2SO 4 Þ nþ1 ðnh 3 Þ m ð5aþ is governed by the ion-molecule association rate coefficient k 5a, the monomer concentration [H 2 SO 4 ], and the ion drift time t. The extent of this conversion in our system is limited to a few percent because we maintained low H 2 SO 4 concentrations and short ion drift times. For example, the ratio of the (1&{n + 1}, m) ion to the (1&n, m) ion due to 5a, if (1&n, m) is due solely to ionization of (H 2 SO 4 ) n+1 (NH 3 ) m neutrals, is given by ½HSO 4 ðh 2SO 4 Þ nþ1 ðnh 3 Þ m Š=½HSO 4 ðh 2SO 4 Þ n ðnh 3 Þ m Š ½H 2 SO 4 Šk 5a t=2: ð5bþ For [H 2 SO 4 ] of cm 3 and a typical value of k 5a t equal to cm 3 s 1, a ratio of 1 2% would be expected for successive ions. Therefore, when the observed ratio of (1&{n + 1}, m) to (1&n, m) is much higher than a few percent, it can be concluded that the (1&{n + 1}, m) ion is due to ionization of the preexisting neutral cluster (n +2,m). If the observed ratio of successive ions is comparable to a few percent, however, then the larger ion could be due to both addition of monomer H 2 SO 4 to a smaller ion and to ionization of the neutral cluster. In this case the variations of the ion signals with reaction time t can be used to estimate the amount of signal to attribute to ion-molecule association reactions and therefore to obtain the signal that pertains to the neutral cluster [Eisele and Hanson, 2000] Ionization rate coefficient and ion decomposition [18] There are additional complications in the detection of (H 2 SO 4 ) n (NH 3 ) m. (1) When m is comparable to n, the composition of the cluster is close to that of ammonium bisulfate, and it is not known how efficiently it can be

5 HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS AAC 10-5 ionized by NO 3 (HNO 3 ) 1,2 reactant ions. The gas-phase acidity of a cluster, and thus possibly the rate of proton transfer to the reactant ion, may decrease significantly when its composition reaches a one-to-one ratio of NH 3 to H 2 SO 4. It is illustrative but not explicatory to point out that the acidity of ammonium bisulfate solutions (NH 4 HSO 4 ) is much less than that of sulfuric acid solutions. We present evidence below that ionization of (H 2 SO 4 ) n (NH 3 ) n clusters may not be efficient. (2) Is there a significant loss of NH 3 and/or H 2 SO 4 molecules upon ionization of a neutral cluster? A recent thermodynamic study (K. Froyd and N. Lovejoy, private communication, 2001) indicates the lifetime at 275 K of the cluster HSO 4 (H 2 SO 4 ) 3 (NH 3 ) m at the high-pressure limit is 5 ms for the process HSO 4 ðh 2SO 4 Þ 3 ðnh 3 Þ m! HSO 4 ðh 2SO 4 Þ 3 ðnh 3 Þ m 1 þ NH 3 : ð6þ This time is comparable to the ion molecule times in our study indicating that loss of NH 3 from HSO 4 (H 2 SO 4 ) n (NH 3 ) m ions could occur. [19] Lovejoy and Curtius [2001] found that the (1&3, m) ions preferentially lose NH 3 molecules. However, HSO 4 (H 2 SO 4 ) 4 NH 3 tends to lose an H 2 SO 4 molecule faster than it does an NH 3 ligand. An estimate for the rate coefficient for loss of H 2 SO 4 from (1&4, 1) can be made from an estimate of the equilibrium constant. The equilibrium (1&4, 1) $ (1&4, 0) + NH 3 was observed at 298 K resulting in a G of 11 kcal mol 1 (K. Froyd and E. R. Lovejoy, private communication, 2001). As that process is slower than (1&4, 1)! (1&3, 1) + H 2 SO 4, a value of 11 kcal mol 1 can be used as an upper limit for G for the reaction (1&4, 1) $ (1&3, 1) + H 2 SO 4. This results in a lower limit of 200 s 1 for the decomposition rate coefficient at 298 K, assuming an association rate coefficient of 10 9 cm 3 s 1. Extrapolation to 275 K assuming an enthalpy of 22 kcal mol 1 yields a decomposition rate of 10 s 1 or a lifetime 100 ms. The loss of H 2 SO 4 from HSO 4 (H 2 SO 4 ) 4 NH 3, 506 amu, could affect the measured H 2 SO 4 content of the (4, 1) and (5, 1) clusters. [20] The H 2 SO 4 molecules are bound more strongly in the smaller clusters. For example, HSO 4 (H 2 SO 4 ) 3 at 391 amu has a bond dissociation energy (loss of H 2 SO 4 ) of 23.5 kcal mol 1 [Lovejoy and Curtius, 2001], which is 5 kcal mol 1 stronger than that for HNO 3 in NO 3 (HNO 3 ) 2. Using the procedure described in the appendix for estimating the lifetime of NO 3 (HNO 3 ) 2 and the thermodynamic information presented by Lovejoy and Curtius [2001], the 391 amu ion has negligible loss rates for H 2 SO 4 (1 and 5 s 1 at 275 and 285 K, respectively.) Note the (1&2, 0) and (1&1, 0) ions are more strongly bound than (1&3, 0) [Lovejoy and Curtius, 2001]. Because an NH 3 molecule will be lost before an H 2 SO 4 molecule and the (1&{n 1}, 0) ions are stable, we conclude that the H 2 SO 4 content of clusters with n 3 is not significantly disturbed in our experiments Mass Discrimination [21] In addition to the biases in the cluster ions due to their decomposition and consequent ion processes discussed above, the sampling, filtering, and detection of an ion are likely to depend on its mass. Throughputs and channel electron multiplier (CEM) efficiencies were evaluated by monitoring a series of masses while varying lens voltages and CEM accelerating potential. We found that for most of the ion lenses, mass discrimination was small, with the exception of the lens at the entrance to the quadrupole mass filter. Variation of this voltage led to variations of approximately a factor of 2 in the ratio of mass 540 to mass 125. This ratio was also dependent on the resolution settings of the mass filter. Also, if the accelerating voltage on the cone of the CEM was greater than 2500 V, we saw little or no variation in the ratio of large (600 amu) to small (125 amu) masses. For a cone voltage of 2200 V, however, the ratio of ions at 655 and 672 amu to that at 293 amu, decreased by 35% with respect to 3000 V on the cone. Ions at 540 and 557 amu saw a decrease of 20 25%, while 408 amu experienced a decrease of 10% or less. Note that data from a newly installed CEM were less sensitive to these parameters; thus the required accelerating voltage is probably a function of the condition of the CEM. [22] We believe that these biases are minimal if the signals of ions with masses within a few hundred amu are compared. However, on the basis of the evaluation of mass discrimination effects discussed above, it is possible that if masses separated across a large fraction of the mass range of the instrument (60 to 900 amu) are compared, mass discrimination biases could be of the order of +100/ 50%. 3. Results and Discussion [23] Figure 2 shows a mass spectrum taken in the presence of 90 pptv H 2 SO 4 and 180 pptv NH 3 at 265 K. A mass spectrum taken without ammonia present was subtracted from the signals from 300 amu and above. This subtraction was done for clarity in the presentation because there are many mass peaks in the spectrum (with intensities of 1 to 5 Hz) that are not associated with NH 3,H 2 SO 4,or their clusters. There were no mass peaks with intensities greater than a few hertz attributable to HSO 4 (H 2 SO 4 ) n 1 for n > 3 in either spectrum; thus this subtraction process did not mask the detection of these clusters. The signals for 195 amu (H 2 SO 4 dimer) and 293 amu (H 2 SO 4 trimer) were obtained from the measured signals by subtracting the estimates of the signal due to ion-molecule clustering reactions such as equation (5a). Note that the ions at 195 and 293 amu are likely due to ionization of neutral clusters that contain NH 3 in addition to the neat (H 2 SO 4 ) 2,3 species. The trimer, tetramer, pentamer, and hexamer of sulfuric acid along with an assortment of NH 3 ligands are readily observed in the data. It is apparent that the ion signals for successive HSO 4 (H 2 SO 4 ) n 1 (NH 3 ) m species are comparable, and thus they are due primarily to ionization of pre-existing neutral clusters. Further evidence for this is the time independence of these signals, which is demonstrated in Figures 4 and 5 below. The solid designator lines represent the mass for the nth mer of H 2 SO 4 along with m =0ton NH 3 molecules. The dashed lines indicate an HNO 3 ligand that may arise during the ionization and transport processes. [24] Note that there are peaks in Figure 2 that correspond to the binding of an HNO 3 molecule to the ions for the trimer plus NH 3 ligands. For the sake of clarity, it is fortunate that the (H 2 SO 4 ) 4,5 clusters do not appear to exhibit this behavior. The trimer data are also complicated by unidentified impurities that result in mass peaks with

6 AAC 10-6 HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS Figure 2. Mass spectrum at 265 K revealing the presence of (H 2 SO 4 ) m (NH 3 ) n clusters. Solid lines indictate (n, m) with the ticks indicating the number of NH 3 molecules (m =0ton). The dashed lines indicate these same clusters but with an HNO 3 ligand attached. Reactant concentrations: 90 pptv H 2 SO 4 and 180 pptv NH 3. intensities of 10 Hz at 308, 312, and 356 amu. These were highly variable and did not depend on the presence of H 2 SO 4 or NH 3. The ion signal at 310 amu, due to (H 2 SO 4 ) 3 NH 3, is strongly affected by these impurities and the subtraction process leads to an oddly shaped peak, as seen in Figure 2, and at times leads to a highly uncertain signal level. This general trend that an ion holds onto HNO 3 ligands more strongly the smaller it gets is opposite to the trend for H 2 O and NH 3. This is consistent with previous studies of the association of HNO 3 with negative ions [Davidson et al., 1977; Viggiano et al., 1982]. [25] The signals for the hexamer and heptamer are in general lower than those for the tetramer and pentamer. Also, the distribution of ammonia ligands is less uniform than the (H 2 SO 4 ) 4,5 clusters. We may be detecting the hexamer and heptamer clusters less efficiently than the smaller clusters if a majority of the hexamer and heptamer neutral clusters are present as (H 2 SO 4 ) n (NH 3 ) m where m n. In fact, the data presented in Figure 2 show no indication of the ions HSO 4 (H 2 SO 4 ) m (NH 3 ) m+1, designated in shorthand as (1&m, m + 1), for all m. We present evidence below that these types of clusters might not be as easily ionized as the case where m < n. [26] Water molecule ligands are not generally observed in the mass spectrums (with the possible exception of the H 2 SO 4 hexamer and heptamer umbrellas.) This was also the case in our previous work at 240 K in the H 2 SO 4 -H 2 O system [Eisele and Hanson, 2000]. The CDC chamber, however, was at a lower pressure in this work, and some water ligands might be expected to survive into the mass filter as the ions experience less collisions. A mass spectrum in the H 2 SO 4 -H 2 O system was taken at 243 K with the current low CDC pressure and is presented in Figure 3. A mass spectrum taken at a lower [H 2 SO 4 ](40% of that for the other experiment) is also shown. H 2 O ligands are bound strongly enough by the pentamer and hexamer cluster ions to be clearly observed. This is not the case for the smaller H 2 SO 4 cluster ions (note the H 2 SO 4 trimer ion signals have been divided by 4.) It is likely that all the neutral clusters contain H 2 O molecules at this RH (60%) because they depend on the presence of H 2 O as was discussed previously [Eisele and Hanson, 2000]. Also, the monomer likely picks up one H 2 O at 10% RH and a second around 50% RH [Hanson and Eisele, 2000]. Apparently, small cluster ions do not hold onto ligands that were associated with the neutral cluster as well as those produced from large clusters. These

7 HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS AAC 10-7 Figure 3. Mass spectrum at 243 K taken in the absence of ammonia showing waters of hydration on the (H 2 SO 4 ) m clusters (110 pptv, [H 2 SO 4 ]= cm 3 ). A second mass spectrum is also shown taken with a much lower [H 2 SO 4 ]( cm 3.) Solid lines indicate the number of H 2 O molecules in the cluster. Note the change in scale for the signals for masses of 380 amu and less. data indicate the low pressure in the CDC will allow for some preservation of the ligands on cluster ions, which is one of the goals of this work. Note that the preservation of H 2 O ligands in this mass spectrum probably reflects the stability of the ion + ligand at this temperature. At the warmer temperatures in the ammonia cluster studies, however, we might not expect H 2 O ligands to remain on the ions Ion Processes [27] Figure 4 shows the ion signals (divided by the monomer signal) versus ion drift time for dimer plus ligands (Figure 4a), trimer plus ligands (Figure 4b), and tetramer plus ligands (Figure 4c). The temperature was 265 K, H 2 SO 4 was at 50 pptv, and NH 3 was at 230 pptv. Not shown are the pentamer + NH 3 ligands, which were comparable in signal levels to those of the tetramer. A plot of this ratio for a particular ion will be flat if it is due to a neutral cluster and if the product ion is stable for times of the order of 0.02 s. The ions HSO 4 (H 2 SO 4 ) m (NH 3 ) n for m =1,2,3 we will denote in shorthand as dimer, trimer, and tetramer of H 2 SO 4 for all n. There are a few ions for n = 1 and 2 that contain an HNO 3 ligand. Note that in Figure 4a the 195 amu ion signal has been divided by 10. Twice the slope of this line divided by the measured monomer concentration of cm 3 yields a value of cm 3 s 1 for the association rate coefficient for H 2 SO 4 + HSO 4! HSO 4 (H 2 SO 4 ) at 265 K. This is close to the value of cm 3 s 1 at 240 K [Eisele and Hanson, 2000] (this previously reported value has been revised according to a detailed treatment of the ion mobility; see the appendix.) Note that the signals at 212 and 275 amu decrease with reaction time, indicating that they may be decomposing. Their contribution to the 195 amu ion is minimal (remember the 195 signal ratio has been divided by 10.) [28] In Figure 4b the trimer plus ligand signals at 310, 327, and 390 amu decrease with time indicating that they are losing NH 3 and HNO 3 as the ion drift time is increased. The slope of the 293 signal divided by estimated [dimer] (obtained from the intercept in Figure 4a) yields a value for the clustering rate coefficient of 10 8 cm 3 s 1. This is an unrealistic value and points to an increase with time of the 293 amu signal as the 310 (and 327 and 390) amu ions lose ligands. Evidence for this is seen in the sum of these ion ratios, also shown in Figure 4b, which is constant with time within the scatter. In Figure 4c the ion signals for the tetramer do not show decomposition behavior. [29] Figure 5 shows the results for clusters up to the pentamer at 275 K with higher [reagent] present: [H 2 SO 4 ]= cm 3 (120 pptv) and [NH 3 ]= cm 3 (280 pptv). In this case, the dimer and trimer ions, HSO 4 H 2 SO 4 and HSO 4 (H 2 SO 4 ) 2, do not have ligands associated with them to a significant extent. Apparently, the decomposition rates are too fast to allow them to be observed if they are sampled at times 3 ms. The association rate constant (e.g., reaction (5a)) obtained from the slope of the 195 amu ion signal versus time from Figure 5a is cm 3 s 1. This is higher than the values found at lower temperatures. A rate coefficient this large at 275 K is consistent with results from other experiments conducted without ammonia and at lower [H 2 SO 4 ] with and without ammonia (values for k 1,2 ranged from cm 3 s 1.) A 40% increase in the ion-molecule association rate coefficient as the temperature is increased from 265 to 275 K is unexpected. Further experimentation is needed to fully understand this. [30] At this temperature the ions for the tetramer and pentamer show decomposition. The HSO 4 (H 2 SO 4 ) 3 (NH 3 ) 3 ion at 442 amu, denoted shorthand as (1&3, 3), decreased by 50% as the ion time increased from 3 to 10 ms. This is consistent with a lifetime of 10 ms. The (1&4, 4) ion at 557 amu decreases by 30% as the ion time increases from 3 to 10 ms resulting in a lifetime of 25 ms. While a decrease in the (1&3, 2) ion at 425 amu and an increase in the (1&3, 1) ion at 408 amu are apparent, the lifetimes of these ions are not easily estimated owing to there concomitant production. See the appendix for an example of measuring decomposition rates of the NO 3 (HNO 3 ) 2 ion with this system. [31] The decomposition times observed here are in the ballpark of the 5 ms estimate from the measurements by K. Froyd and E. R. Lovejoy (private communication, 2001) discussed above. The uncertainty in this estimate of the lifetime is high and is due to the uncertainty in G and in the association rate constant (a value of cm 3 s 1

8 AAC 10-8 HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS Figure 4. Ion signals due to molecular clusters (ratioed to the monomer signal) versus ion drift time at 265 K: (a) H 2 SO 4 dimer + NH 3 and HNO 3 ligands (195 amu signal is shown divided by 10), (b) trimer + ligands, (c) tetramer + ligands. The [H 2 SO 4 ] was cm 3 and [NH 3 ] was cm 3. was assumed for the high-pressure limit.) Note that a ±0.5 kcal mol 1 uncertainty in the measured G translates into a factor of +150/ 60% uncertainty in the lifetime at 298 K. Extrapolation to 275 K introduces an additional /1.33 factor in the uncertainty of the lifetime for a ±2 kcal mol 1 uncertainty in the enthalpy. Thus the 5 ms lifetime has an uncertainty of +200/ 70% if G at 298 K is known to within 0.5 kcal/mol. The uncertainty due to estimating the association rate coefficient is likely to be of the order of a factor of 2. Considering these uncertainties, the 10 to 25 ms lifetimes we extract from the data are consistent with those inferred from previous observations. [32] Finally, the scatter in the data taken at 265 K is quite large, e.g., the 442 amu ion in Figure 4c. This could be due to convectively driven processes in the flow that affect mixing. A nonsteady flow due to convective processes might be expected due to the cooling of the gas. Note the ion signals at 275 K do not have as much scatter, consistent with a more steady flow profile due to less convection. A difference in H 2 SO 4 profiles in the detection region at 265 and 275 K might also arise. This difference may partially explain the anomolous temperature dependence of the ion-molecule association rate coefficient discussed above Effect of Dilution Flow on Molecular Clusters [33] In each of the distributions within a cluster the absence of signal directly attributable to the (H 2 SO 4 ) n (NH 3 ) n

9 HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS AAC 10-9 Figure 5. Ion signals due to molecular clusters (ratioed to the monomer signal) versus ion drift time at 275 K: (a) dimer (divided by 10) and trimer + ligands, (b) tetramer + ligands, (c) pentamer + ligands. [H 2 SO 4 ] was cm 3 and [NH 3 ] was cm 3. cluster is clear. This may be due to loss of NH 3 ligands upon ionization by NO 3 core ions as discussed above. It could also be due in part to less efficient ionization of these types of clusters by NO 3. As alluded to earlier, there is evidence that the neutral clusters that contain as many NH 3 molecules as H 2 SO 4 molecules may not be easily ionized. We discovered this by observing that the cluster ion signals went through a maximum when there was an outflow of dry N 2 from the mass spectrometer inlet into the main flow. As outlined in Figure1b the mass spectrometer inlet has a 0.20-inch ID aperture on the front plate through which the ions pass before entering a 125-mm orifice (mounted on the orifice plate) through which they enter the vacuum system. A flow of dry N 2, labeled Q a in Figure 1b, between these plates limits the amount of water that enters the vacuum system. Excess Q a that does not enter the vacuum system enters the flow reactor. [34] The results of these tests are shown in Figure 6, a plot of the ion signals divided by the signal due to H 2 SO 4 (except for the HSO 4 signal itself which is presented as ln(1 + S HSO4 /S NO3 )) versus the flow rate of gas added at the inlet Q a. The flow entering the mass spectrometer through the 125 mm orifice at 0.8 atm is 90 sccm; any flow in addition to this enters the main flow. An arrow in the figure indicates this point along the X axis. The temperature was 265 K, while NH 3 was 130 pptv and H 2 SO 4 was 70 pptv. For some of the (H 2 SO 4 ) n clusters the ion signals were summed: for the tetramer, the sum of 408, 425, and 442 amu; and for the pentamer, the sum of 540 and 557 amu. An additional experiment with much more ammonia, 780 pptv, present is also included in Figure 6. Also shown as the gray symbols and curve is the ratio of 442 amu/408 amu (divided by 1000). [35] Note the marked increase in the ion signals due to clusters when Q a exceeds the inlet flow. At the maximum effect where Q a is 55 sccm in excess, the cluster ion signals increase by multiplicative factors of 2 3 over the signals at the lowest Q a. Also, at the high [NH 3 ] level, the pentamer signal increases by a factor of 5. The signals due to the monomer and the dimer shown in Figure 6a show little if any effect indicating there is not a large distortion in the gas or reagent ion concentration in the main flow due to Q a.

10 AAC HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS Figure 6. Ion ratios versus aperture flow rate at 265 K: (a) ln(1 + [HSO 4 ]/[NO 3 ]) and dimer/monomer, (b) sum of the trimer + ligands signals, sum of the tetramer + ligands signals, sum of the pentamer with 3 and 4 NH 3 ligands, and the ratio of (4, 3) to (4, 1) divided by 1000 are shown. A set of pentamer signals with much higher NH 3 present is also shown. An experiment at 275 K with [NH 3 ] at 270 pptv and [H 2 SO 4 ] at120 ptv shows a much smaller peak in the cluster ion signals at Q a = 150 sccm. The maximum effect at 275 K for the trimer ion signal was +50%, while the signals due to the tetramer and pentamer (including 506 and 523 amu) both increased by 75%. [36] A possible explanation for these observations is that the excess flow acted to dry out the flow in the detection region, causing neutral clusters to lose NH 3 and thus become amenable to ionization by NO 3. The decrease in the cluster ions as the flow exceeded this critical value is probably due to sufficient dilution of the flow such that the clusters fall apart (i.e., defined here as loss of H 2 SO 4 ) due to inadequate amounts of ammonia. This explanation is consistent with our observations that the effect increases as NH 3 increases and that it decreases as temperature increases. Also shown in Figure 6b is the ratio of 442 to 408 amu ions, the (1&3, 3) to (1&3, 1) ratio, which we assign as the ratio of the 3 ammonia tetramer to the 1 ammonia tetramer. It decreases as Q a increases, indicating a change in the neutral distribution due to this excess flow (note these ions are stable at 265 K on timescales of many tens of milliseconds as shown in Figure 4). [37] Because the excess Q a flow is <1% of the total flow, it likely influenced only the clusters in the region near the ion source and mass spectrometer inlet. Then the neutral clusters (H 2 SO 4 ) n (NH 3 ) m undergo loss of ammonia molecules on the timescale of a second. This suggests that the relatively high [NH 3 ] near the ammonia inlet (see Figure 1a) does not determine the neutral cluster distribution observed many seconds downstream in the ionization region. 4. Cluster Distribution Dependence on Temperature, [H 2 SO 4 ], and [NH 3 ] [38] We have discussed a number of potential biases in the detection of these clusters including ionization efficiency, loss of product ion ligands (preserving sulfate moieties however), and, within the vacuum system, ion transmission, and detection efficiencies. In addition, there are the issues raised in our earlier work on clusters in the H 2 SO 4 -H 2 O system [Eisele and Hanson, 2000]. These include determining whether the neutral clusters had enough time to exhibit a distribution that reflects thermodynamics. In other words, how much do the kinetics of formation influence the measurements? We were able to demonstrate

11 HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS AAC Figure 7. Observed cluster distributions as a function of bath temperature. in that work that it was likely that the cluster distribution was largely influenced by thermodynamics by showing that the cluster signals were strong functions of temperature. [39] Figure 7 shows the ion signals for the H 2 SO 4 cluster plus NH 3 ligands for the trimer (293, 310, and 327 amu), the tetramer (391, 408, 425, and 442 amu), and the pentamer (540 and 557 amu). The data for three different temperatures (circulating fluid temperatures of 263, 273, and 283 K, which are gas temperatures of 265, 275, and 285 K) are shown. Ammonia was present at 160 pptv, and H 2 SO 4 was 85 pptv. The signals are strongly dependent on temperature, indicating that to a large degree the thermodynamics of the neutral clusters influences the H 2 SO 4 content of the measured cluster distribution at temperatures warmer than 265 K. While ammonia ligands on the neutral clusters may be lost once these clusters are ionized, this is not the case for H 2 SO 4 molecules (with the possible exception of (1&4,1).) Note that loss of H 2 SO 4 from (1&4, 1) to produce (1&3,1), if it occurred at a significant rate (see the discussion below for an argument that it does not), would cause a buildup of ions at (1&3,1) and (1&3, 0). Thus a correction to the H 2 SO 4 content of the n = 4 clusters would strengthen their temperature dependence, bolstering our assertion that the thermodynamics of the neutral clusters rather than their kinetics is largely determining their abundance. [40] It is noteworthy that clusters are present at concentrations of several 10 5 cm 3 at temperatures as high as 285 K. In the H 2 SO 4 -H 2 O system for similar [H 2 SO 4 ], cluster concentrations comparable to this appeared at temperatures below 245 K [Eisele and Hanson, 2000]. This comparison highlights the potential importance of NH 3 in new particle formation in the atmosphere. [41] Figure 8 shows cluster distributions at 275 K for three different levels of [H 2 SO 4 ] (2.3, 1.5, and cm 3 ; 104, 70, and 41 pptv, respectively) with ammonia present at 280 pptv. Figure 8a shows the distributions including the NH 3 ligands for the trimer, tetramer, and pentamer while Figure 8b shows the sum over the NH 3 ligands within an H 2 SO 4 cluster (the dimer is also shown here.) For the data presented in Figure 8 a signal level of 10 Hz corresponds to a cluster concentration [(n, m)] of roughly cm 3. This is calculated from an equation that is analogous to (3), [(n, m)] = ln(1 + S (1&{n 1},m) / S NO3 )/kt, where k is assumed to be the same for all clusters. A more accurate determination of the cluster concentrations would require knowledge of the ionization and sampling efficiencies of the neutral clusters. The estimated signals due to ion-molecule clustering have been subtracted from the 195 and 293 amu signals to obtain the dimer and trimer signals (see the appendix). These estimated signals range between 44 and 63% of the 195 amu signals and for the 293 amu signals were 4 12%. [42] The H 2 SO 4 trimer ion has very little ammonia associated with it while the H 2 SO 4 tetramer and pentamer are detected primarily with NH 3 ligands. As discussed above, the neutral H 2 SO 4 trimer likely has NH 3 ligands associated with it that are lost rapidly when it is ionized. The (1&3, 1) ion at 408 amu is the largest among the tetramer clusters. Its dominance is probably due to the loss of ammonia ligands from the (1&3, 2) and (1&3, 3) ions as demonstrated in Figure 5. Also, a contribution to the signal at 408 amu from loss of an H 2 SO 4 molecule from the (1&4, 1) ion at 506 amu cannot be ruled out. It is likely to be small, however, as the sum of the ions at 523, 540, and 557 amu in Figure 5c and the sum of the ions at 391, 408, 425, and 442 amu in Figure 5b are both constant with time (to within ±10%). The former observation indicates the amount of 506 amu produced via loss of NH 3 from 523 amu is small, and the latter observation shows that there is not a buildup of ions at (1&3, m). At short ion-drift times the distribution of NH 3 ligands within the H 2 SO 4 pentamer umbrella indicates a higher concentration of (H 2 SO 4 ) 5 (NH 3 ) m as m increases. Note that a naïve extrapolation of this trend would lead to the conclusion that a significant amount of (H 2 SO 4 ) 5 (NH 3 ) 5 is present H 2 SO 4 and NH 3 Content of the Critical Cluster [43] The critical cluster is the cluster at the saddle point on the free energy surface for formation of the cluster (see Coffman and Hegg [1995] for an example of a Gibbs free energy change surface in the ternary system.) Clusters larger

12 AAC HANSON AND EISELE: PRENUCLEATION MOLECULAR CLUSTERS Figure 8. (a) The 275 K cluster distributions containing 3, 4, and 5 H 2 SO 4 molecules. Three different [H 2 SO 4 ] levels are shown at constant [NH 3 ] of 280 pptv ( cm 3 ). (b) Sum over the NH 3 ligands within each H 2 SO 4 envelope. than the critical one are characterized by growth rates that exceed their decomposition rates. In many cases then the critical cluster lies near a minimum in the number density distribution of clusters. Yet the data from our experiment is complicated by ion-molecule and dilution processes. Specifically, direct identification of the molecular content of the critical cluster is difficult because NH 3 is lost rapidly upon ionization. In addition, the water molecule content was not measurable. [44] To use this approach to obtain an approximate number of H 2 SO 4 molecules in the critical cluster, the data were summed over the NH 3 molecules in each H 2 SO 4 cluster. The summing within an (H 2 SO 4 ) n umbrella is justified if they can be viewed as being in equilibrium (or steady state) with NH 3. This is equivalent to assuming that the most important growth step of the critical cluster is the addition of a sulfuric acid molecule. This approach was implicitly taken for the H 2 SO 4 -H 2 O clusters observed by Eisele and Hanson [2000] in regard to water molecules. The sum over NH 3 ligands within an (H 2 SO 4 ) n umbrella presented in Figure 8b shows that a minimum exists at the H 2 SO 4 trimer at the two highest [H 2 SO 4 ]. This may represent the H 2 SO 4 content of the critical cluster for these conditions. [45] Viewing the clusters as being in equilibrium (or steady state) with the very abundant species H 2 O ([H 2 O] cm 3 ) is reasonable. On the other hand, this may not be the case for the low levels of NH 3 present (<10 10 cm 3 ). A case involving the H 2 SO 4 dimer illustrates the potential problem. Suppose the abundance of (H 2 SO 4 ) 2 NH 3 is greater than that of (H 2 SO 4 ) 2 and that the addition of NH 3 to (H 2 SO 4 ) 2 is the critical step to forming new particles. It is likely that an NH 3 molecule is much less strongly bound to HSO 4 H 2 SO 4 than to (H 2 SO 4 ) 2 such that it is lost very rapidly when NO 3 core ions react with (H 2 SO 4 ) 2 NH 3.In the overall production of (H 2 SO 4 ) 2 NH 3 from H 2 SO 4 + H 2 SO 4 +NH 3,(H 2 SO 4 ) 2 NH 3 is the major contributer to the signal at HSO 4 H 2 SO 4, and a minimum at n =2,m =0 would be obscured. Similarly, if the abundance of H 2 SO 4 NH 3 is large enough to play a role in cluster formation, its presence could be obscured by loss of NH 3 from HSO 4 NH 3. [46] An alternative approach not dependent upon observing a minimum in the distributions involves application of the nucleation theorem [Oxtoby and Kashchiev, 1994]. The power dependency of the nucleation rate on the activity of a species is a linear function of the number of molecules of that species in the critical cluster (see also Ball et al. [1999, and references therein]). Oxtoby and Kashchiev [1994] showed that this concept holds even for the smallest clusters and that a number between 0 and 1 should be subtracted