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Water Solubility of Cloud-Active Aerosols V. K. Saxena Department of Marine, Earth and Atmospheric Sciences, North Carolina State University, Raleigh, NC 27695 and G. F. Fisher Ballistic Missile OBce, Norton Air Force Base, CA 92409 Nineteen common electrolytes, which usually constitute the hygroscopic fraction of cloud-active aerosols (CAA), are examined for their solubility in water. Water droplets nucleated upon CAA have so far in the literature been treated as weak solution droplets. Upon examination of this assumption, it is revealed that sulfates that are so commonly found in urban-industrial and nonurban environments are least water soluble, and large errors are likely to be encountered if their nucleation-growth behavior is simulated based on the assumption of weak solution INTRODUCTION Cloud-active aerosols (CAA) represent that fraction of the total aerosol budget which constitutes one of the necessary ingredients in the formation and persistence of clouds. The CAA may be divided into three categories, namely cloud condensation nuclei (CCN), ice forming nuclei (IFN), and cloud interstitial nuclei (CIN). Very recent investigations (e.g., Radke, 1982) have brought to light the importance of CIN. A large fraction of CCN and CIN consists of sulfate particles, even in regions remotely located from the sources of urban-industrial pollution (Rahn, 1981; Heidam, 1981; Shaw, 1981). Growth of submicroscopic sulfate particles from the dry state to cloud, fog, or haze droplets crucially depends upon their initial dry radii, upon the density and surface tension of solution droplets, and the relative humidity of the environment (Hhel, 1976; 1981a, 1981b). It is demonstrated in this paper that these factors are more crucial to droplets which nucleate upon sulfate par- droplets. A new classification scheme is presented and all nineteen electrolytes are arranged in three groups according to their water solubility. Approximation formulas relating the critical supersaturation to dry state particle radius, critical radius, and radius at 100% relative humidity are examined and magnitudes of errors calculated, if the former are used regardless of the solute concentration in droplets. A method for correcting errors is presented for isothermal haze chambers. ticles than, for example, on chloride particles, primarily because sulfates have a lower degree of solubility in water. This fact has seldom been recognized thus far and, consequently, all droplets are assumed (e.g., Fitzgerald, 1975; Chylek and Ramaswamy, 1982) to consist of weak solutions regardless of their chemical composition. In this paper, we define a weak solution, classify common electrolytes according to this definition, and then examine and analyze approximation formulas customarily used in the study of water droplets. A method to correct errors resulting from this assumption of weak solution is also presented. WEAK SOLUTIONS When electrolytes are dissolved in water, the physical characteristics of the resulting solution are different than those of the pure water. In case of droplets, the surface tension is also dependent upon the radius. As a solution droplet grows by diffusion of water Aerosol Science and Technology 3:335-344 (1984) 0 1984 Elsevier Science Publishing Co.. Inc.
V. K. Saxena and G. F. Fisher vapor, the concentration of the solute decreases so that at sufficiently large sizes the surface tension and density approach the values for pure water and the electrolyte is dissociated into individual ions. The degree and amount of variability of these characteristics are dependent upon the properties of the electrolyte, such as crystalline structure, valence state of the ions, molecular size, and lattice energy of crystals. The effect of such physical characteristics on droplet growth is usually presented by the Kohler equation (See, for example, Pruppacher and Klett, 1980), which represents the solution droplet growth under equilibrium conditions: S = (1 - b)exp(a/r), (1) where S is the saturation ratio, r the droplet radius, and a and b are defined as follows: where o' is the surface tension of droplet, p, the density of water, R, the gas constant for water vapor, T the absolute temperature, M,, and M, the molecular weights, m, and rn, the masses of water and solute, respectively, and i is the van't Hoff factor which represents the degree of ionic dissociation. The most frequent applications of this equation assume that droplets consist of weak solutions, yet the definition of weak is scarcely, if ever, found in the literature. We therefore chose to define a weak solution as one in whch the surface tension, solution density, and van't Hoff factor deviate 1% or less from their values for pure water. Ths introduces an error less than one half of 1% in the Kohler equation. In the following, we have examined nineteen electrolytes studied by Low (1969) to determine the relationships among crystal structure, valence states, lattice energy, and solution concentrations required to qualify as a weak solution according to our definition. TABLE 1. Classification of Electrolytes According to the Concentrations at which They May Be Considered a Weak Solution Weak solution conc. Crystal Ionic charge Lattice energy Electrolyte (molality) structure (of + and - ion) KJ mol-' Group I LiCl N abr NaCl NH, Cl KBr KI KC1 Subgroup 1 NH4N03 NaNO, Group I1 CaCl, MgCl, BaCl, Subgroup 2 KNO, ZnW3 12 Group I11 (NH4)?S04 Na,SO, ZnSO, Mg'SO, cuso, hexagonal tetragonal
Water Solubility of Cloud-Active Aerosols 337 Our results are shown in Table 1. The classification in this table reveals three distinct groups of electrolytes based solely on the concentrations in water at which the solution may be treated as weak. Because of their unusual bonding structures, the nitrates are separated into subgroups. Except for these subgroups, each group is distinct according to its physical and chemical composition. It is clear that the Group I11 electrolytes, consisting entirely of sulfates, are the least soluble and remain a weak solution at concentrations of only up to 0.005 molality. Group 11, the dichlorides, is more soluble, and Group I is the most soluble, remaining weak at concentrations three times as strong as the sulfates. Since most continental aerosols are found to contain sulfates (Hanel, 1981b), the weak solution assumption is invalid for haze and fog droplets. We have conducted extensive calculations to evaluate the errors resulting from the weak solution assumption. We chose sodium chloride (NaC1) and ammonium sulfate ((NH4),S04) as representatives of Group I and Group I11 electrolytes, as they are among the most common salts found in natural CAA. ESTIMATION OF ERRORS The Kohler equation, Eq. (I), may be simplified by first expanding the exponential term in a Taylor's series and eliminating those terms whose contribution is negligible. Assuming a weak solution, we can substitute the surface tension of pure water (IT) for (TI, the dissociation constant (v) for i, and use the definition of density (p) to replace m,v with 4.irp,r3/3. Substituting these values into Eq. (1): where a = IT/^,^ RUT, and (3) P = 3iM,vms/4~pWMs. (4) Since a and /3 are constants, we may now follow the method of Lakitonov (1972), as later expanded by Alofs (1978), and 'apply the optimizing condition to Eq. (2) in order to obtain an approximation for the critical supersaturation (S,): Furthermore, at 100% relative humidity, S = 1, and substituting this into Eq. (2), we obtain r,,, the equilibrium radius at 100% relative humidity. Substituting values for the coefficients in a and p and using algebraic manipulations, we can finally arrive at the following approximation formulas for the critical supersaturation (S,), critical radius (rc), and r1,: Additionally, experimental evidence by Fitzgerald (1974, 1975) shows that the aerosol dry radius (r,) can be found by where k is an empirically determined constant with a value of 1.7 X 10-l' cm3/* for ammonium sulfate and 1.2 xib-" cm3/2 for sodium chloride, respectively, Equations (6)-(9) have been used extensively in the study of haze formation, and considered as unique relationshps (Hoppel and Fitzgerald, 1977; Fitzgerald and Hoppel, 1981; Hudson, 1980). These relationships are not unique (Hanel, 1976; Corradini and Tonna, 1979), as will be demonstrated in the following by the results of our calculations. Using the Kohler equation in the form of Eq. (I), we carried out computations to determine relationships between Sc, rc and r,,,, and compared them with those predicted by Eqs. (6)-(9). The resulting errors are represented in Figure 1, along with errors for Eq. (9). In order to determine the magnitude of errors involved, values of S were determined for various radii at given masses of
V. K. Saxena and G. F. Fisher SODIUM CHLORIDE AMMONIUM SULFATE 1.35. I 1. - 1 1-4: 11s. U 1.10-2, ~ S E L Sc=.IO/o Sc=.O1O/o 1 1 MASS OF SOLUTE (gms) FIGURE 1. Errors resulting from the use of approximation formulas given in Eqs. (6)-(9). Solid lines represent 100% soluble nucleus, dashed lines lo%, and dotted lines 1% soluble nucleus. Use of Eq. (9) for 10% and 1% soluble nucleus leads to exceedingly large errors, which are not shown, plotted on the top pair of diagrams. MASS OF SOLUTE (gms)
Water Solubility of Cloud-Active Aerosols 339 TABLE 2. Concentration of Solution in Droplets at the Equilibrium Radius, rlou Fraction Mass of solute r100 Electrolyte soluble (75) (9) (~m) Molality Sodium Chloride 100 10-l8 0.012 2.35 lo-17 0.036 0.880 10-l6 0.113 0.288 lori5 0.359 0.089 Ammonium Sulfate aassumption of weak solution is valid at these values of molality. (NH,),SO, and NaC1, using Eq. (1). Exact form of the Kohler equation was used, since computations revealed that truncated exponential expansion led to appreciable errors at the smallest radii. The algorithm used in computation selected a radius value, and the solution concentration, rc, was determined as 3ms (10) '= 4r(r3 - r:) where r, is the radius of insoluble portion.
340 V. K. Saxena and G. F. Fisher If E, the soluble fraction of the nucleus, is defined as of these equations. The percentage error was calculated as where mi represents the mass of the insoluble material. The value of ri may be found as Here, the nucleus and the droplets are assumed spherical. In our calculations, quartz (SiO,) with a density of p, = 2.65 g cmp3 has been used for the insoluble fraction of the nucleus material. Using standard data for aqueous solutions (e.g., Low, 1969), the values of a, p, and b in Eqs. (1) and (2) were computed by interpolation, and a value of S determined for each equation. To determine r,,, and rc for a given nucleus, we first determined the interval of radii in whch the r,,, and rc occurred, and, through subsequent halving of the interval, determined that value of the radius where S = 1 and where S = S,,. In this manner, no error occurred in the determination of r,,,, and Sc was determined with an error less than 0.5%. However, the error in the determination of rc is considerably more. We estimated error in rc to be as hgh as 6% in a few cases and less than 2% in majority of cases. The values were calculated for NaCl and (NH4),S04 with masses ranging from 10-18g to 10-12g and for E = 1, 0.1, and 0.01. Greater accuracy in rc is possible by modifying the numerical scheme, but it would require improportionately longer computer time. Our results are summarized in Table 2, showing the equilibrium droplet size (r,,,) of the droplet at 100% relative humidity. To determine the magnitude of error arising from the use of approximation formulas given in Eqs.(6)-(9), we used our calculated values for real solution concentrations to compute the value of the "constant" in each and where ql, q2, q3, and 4, represent the percentage of error arising from the use of Eqs. (6)-(9), respectively. In Figure 1 these errors are plotted along the ordinate on the righthand side. RESULTS AND DISCUSSION In Figure 1 are displayed not only the errors to be expected, but also the deviations from the straight line relationships expressed by the frequently used Eqs. (6)-(9), based on the assumption of weak solution. Indicated in each diagram is the mass at which the solution concentration may be considered as weak in the droplet, following our definition and scheme of classification given in Table 1. Since the nucleation-growth behavior of droplets in the supersaturation range of 0.01-0.1% is of primary importance in studying the haze and fog phenomena, masses of the electrolyte corresponding to this supersaturation range are indicated on each diagram. As is obvious from figure 1, appreciable errors arise from the use of the approximation formulas, especially for concentrated solutions. The largest errors are for (NH4),SO4 in particular and class I11 electrolytes (cf. Table 1) in general. It is worth mentioning that the largest errors are encountered if Eq. (9) is used. Calculations for E = 0.1 and 0.01 and (NH4),S04 droplets cannot even be accommodated on the scale shown in Figure 1, and that is why the corresponding curves are
Water Solubility of Cloud-Active Aerosols 341 missing from the top pair of diagrams in Figure 1. For this approximation formula, errors are less than 5% in only a very narrow range, and only for 100% soluble nuclei. For 10% and 1% solution droplets, the errors exceed 250% and 600% limits, respectively. Since Eq. (9) has been used in the literature (e.g., Fitzgerald, 1974, 1975; Fitzgerald and Hoppel, 1981), previous results should be carefully revised. For Eq. (6), which forms the basis of the isothermal haze chamber introduced by Laktionov (1972), a comparison of errors between the NaCl and (NH,),SO, solution droplets is shown in Figure 2. Evidently, supersaturation spectra using such chambers can be determined more reliably for Group I electrolytes than for those in Group 111, although the present evidence (e.g., Twomey, 1977) indicates that natural aerosols contain a large fraction of Group I11 electrolytes. In isothermal haze chambers, r, is measured and other parameters derived through the approximation formulas. The, growth time I I I l ' I ' ' " 1 \ \, from dry radius to r,, increases exponentially with size (Aleksandrov et al., 1969; Robinson and Scott, 1981). Due to limited residence time within the chamber (Saxena and Fisher, 1982), the larger droplets are only grown to 0.95rloo and measured at this size. In Eq. (7), a linear relationship exists between r,,, and Sc and, therefore, a 5% error in size measurement results in the same 5% error in the derived critical supersaturation. For critical supersaturations rangmg from.0l% to 0.1%, which occur in natural fog and haze, this error occurs toward the lower values of Sc. From Figure 2, we see that the errors resulting from the weak solution assumption are close to 5% at the high supersaturation range for the Group I11 electrolytes. Due to the combination of errors in measurement and formulas, the actual error remains nearly constant at 5% in the range FIGURE 2. Comparison of errors for NaCl and (NH,) SO, droplets. - AMMONIUM SULFATE --- 10PA SOLUBLE 10% SOLUBLE - 10 IIIIIIIUUII \ NaCl \ 1% SOLUBLE MASS OF SOLUTE (GRAMS)
V. K. Saxena and G. F. Fisher.01% to.l% of critical supersaturation. Therefore, when the approximation formulas are used to determine the critical supersaturation, the results can be reduced by 5%, thereby correcting for the errors incurred. The above method was applied to the data published by Alofs (1978) and Hudson (1980). Applying the correction formula smooths out the discontinuity resulting between the different instruments (cf. Figure 3). At supersaturations greater than 0.1%, a discontinuity remains; but, as shown in Figure 2, the error is considerably larger for this range of supersaturations. Since this is within FIGURE 3. An application of corrections applied to approximation formulas used in the determination of the nuclei supersaturation spectra from isothermal haze chambers. The dashed line represents corrected data. : THERMAL DIFFUSION' CHAMBER : ISOTHERMAL HAZE CHAMBER V) : 5% CORRECTION APPLIED 2 v the operating range of thermal diffusion chambers and applies more to cloud droplets than to fog and haze droplets, we did not examine correction methods for critical supersaturations exceeding 0.1 %. Of more importance are the errors resulting in calculations of other cloud physical parameters such as radiative transfer fluxes, liquid water contents, and droplet growth rates. For example, liquid water content (w) of a population of droplets is generally found using the relationship: = $ ~p,,/~~~r~n (r) dr, 'mm where n(r) represents the cloud droplet size distribution and r, and r, are radii of smallest and largest droplets, respectively. Obviously, a 10% error in r would result in a 33% error in the liquid water content.
Water Solubility of Cloud-Active Aerosols 343 TABLE 3. Errors Resulting in the Estimation of Liquid Water Content of Droplets when Computation is Based on the Weak Solution Assumption Electrolyte Fraction of the nucleus Sodium Chloride Ammonium Sulfate mass soluble Mass of solute Error (%) Error (%) Mass of Solute Error (% Error (76) in water (%) (g) (at hoo) (at rd (9) (at ~10)) (at rc) 100 lo-'x 7 1.5 10-18 26 6 lo-'' 2 0.5 10-17 10 1.7 10-l6 1 0.1 10-16 3 0.6 10 - l5 0.2 0 10-15 0.7 0.1 10-1~ 0.1 0 10-14 0.2 0 lo-13 0 0 10-13 0.1 0 10-l8 47 11 10-18 142 29 10-l7 19 4 10-17 56 11 10F16 6 1.2 10-16 17 3.3 lo-'5 2 0.3 10-15 5 0.7 10-1~ 0.6 0.1 10-14 1.2 0.2 lo-13 0.2 0 10-13 0.4 0 lo-'" 302 54 10-18 742 100 lori7 132 25 10-17 306 50 10-l6 10-15 52 18 10 3 10-16 10-15 116 38 19 6 lo-'4 6 1 10-14 11 2 10~'~ 2 0.2 10-13 3 0.5 Liquid water content computations based on the weak solution assumption were compared with the real values of liquid water content. The resulting errors are shown in Table 3. The shape of the error curves closely follow those shown in Figure 2, although, as expected, magnitudes were much larger. It was found that for the range of masses shown in Figure 2, the errors for 100% soluble nuclei were as high as 26% (at r,,,) for ammonium sulfate, but only 7% for sodium chloride. For 10% and 1% soluble nuclei, the errors in the liquid water content estimation for ammonium sulfate amounted to 142% and 74256, respectively, and for sodium chloride corresponding errors were 47% and 302%. Likewise, radiative transfer is dependent upon the solution concentration of droplets in a similar manner. As Hanel (1971) has pointed out, the extinction coefficient is proportional to the square of the radius, and the index of refraction is a function of the solution concentration. A 10% error in radius is likely to yield a 20% error in the extinction coefficient. Finally, the rate of growth of droplets is also a function of the solution concentration in a complex manner. The growth equation (e.g., Pruppacher and Klett, 1980) contains variables such as latent heat, diffusion coefficient, and density that are functions of the droplet solution concentration and will therefore introduce errors if the solution is considered weak. CONCLUSIONS A new classification scheme for nineteen common electrolytes is introduced. This placed the solutes into three distinct groups characterized not only by the concentrations at whch they may be considered weak, but also according to their physical and chemical characteristics. The classification scheme shows that the sulfate group is the least soluble in water if the above criteria are used, and it is suggested that approximation
344 V. K. Saxena and G. F. Fisher formulas may not apply to Group I11 electre lytes. Computations were carried out to determine the actual relationships between r,, r,,,, r, and S,. These relationshps were compared to the weak solution approximation formulas for both Group I and Group I11 electrolytes. It was determined that, as suspected, approximation formulas are invalid for Group 111 solutes. A correction method was developed to correct for errors resulting when approximation formulas are used in analyzing data from isothermal haze chambers. Two samples were chosen and correction applied to demonstrate the effectiveness of the method. This work was supported in part by the National Science Foundation under grants DPP-7922058 and DPP- 8305714. One of us (G.F.F.) thanks the U.S. Air Force for sponsoring his graduate studies. Some of this paper was presented at the Conference on Cloud Physics, Chicago, Nov. 15-17,1982; and is part of Capt. Fisher's MS. thesis. REFERENCES Aleksandrov, E. L., Levin, L. M., and Sedurov, Yu. S. (1969). Tr. IEM, No. 6:16-96. Alofs, D. J. (1978). J. Appl. Meteor., 17:1286-1297. Corradini, C., and Tonna, G. (1979). J. Aerosol Sic. 10:465-469. Chylek, P., and Ramaswamy, V. (1982). J. Atmos. Scr. 381171-177. Fitzgerald, J. W. (1974). J. Atmos. Sci. 31:1358-1367. Fitzgerald, J. W. (1975). J. Appl. Meteor. 14:1044-1049. Fitzgerald, J. W., and Hoppel, W. A. (1981). Paper presented at the International Conference on Condensation and Ice Nuclei, IAMAP Thrd Scientific Assembly, Hamburg, FRG. Hiel, G. (1971). Beitr. Phys. Atmos. 41:137-167. Hhel, G. (1976). In Aduances in Geophysics, vol. 19, (J. Simpson, ed.), Academic Press, New York, 73-188. Hanel, G. (1981a). Atmos. Enuiron. 15:403-406. HBnel, G. (1981b). Beitr. Phys. Atmos. 54:159-172. Heidam, N. Z. (1981). Atmos. Environ. 15:1421-1428. Hoppel, W. A,, and Fitzgerald, J. W. (1977). Proc. Symposium on Radiation in the Atmosphere, Sciencc Press, 62-64. Hudson, J. G. (1980). J. Atmos. Sci. 37:1854-1867. Laktionov, A. G. (1972). IZV. Atmos. Ocean. Phys. 8:672-677. Low, R. D. H. (1969). Rept. ECOM-5249, Army Electronics Command, Ft. Monmouth, NJ, 553 pp. Pruppacher, H. R., and Klett, J. D. (1980). Microphysics of Clouds and Precipitation, D. Reidel, Boston, 714 pp. Radke, L. (1982). Paper No. ICS-4. Presented at the Fourth International Conference on Precipitation Scauenging, Dry Deposition and Resuspension, Santa Monica, CA. Rahn, K. A. (1981). Atmos. Enuiron. 15:1457-1464. Robinson, N. F., and Scott, W. T. (1981). J. Atmos. Sci. 28:1015-1026. Saxena, V. K., and Fisher, G. F. (1982). Preprints, Conference on Cloud Physics, Boston, Am. Meteor. SOC., 286-288. Shaw, G. E. (1981). Atmos. Enuiron. 15:1483-1490. Twomey, S. (1977). Atmospheric Aerosols, Elsevier, New York, 297 pp. Received 1 March 1984; accepted 8 March 1984