135 6 Reversible Chemical Reactions in Aerosols Mark Z. Jacobson CONTENTS Introduction 135 Definitions 136 Equilibrium Equations and Relations 136 Equilibrium Equations 136 Equilibrium Relations and Constants 139 Temperature Dependence of the Equilibrium Coefficient 142 Forms of Equilibrium Coefficient Equations 143 Mean Binary Activity Coefficients 144 Temperature Dependence of Mean Binary Activity Coefficients 146 Mean Mixed Activity Coefficients 147 The Water Equation 148 Method of Solving Equilibrium Equations 151 Solid Formation and Deliquescence Relative Humidity 153 Equilibrium Solver Results 154 Summary 155 References 155 INTRODUCTION Aerosols in the atmosphere affect air quality, meteorology, and climate in several ways. Submicron- sized aerosols (smaller than 1 µ m in diameter) affect human health by directly penetrating to the deepest part of human lungs. Aerosols between 0.2 and 1.0 µ m in diameter that contain sulfate, nitrate, and organic carbon, scatter light efficiently. Aerosols smaller than 1.0 µ m that contain elemental carbon, absorb efficiently. Aerosol absorption and scattering are important because they affect radiative fluxes and, therefore, air temperatures and climate. Aerosols also serve as sites on which chemical reactions take place and as sinks in which some gas-phase species are removed from the atmosphere. The change in size and composition of an aerosol depends on several processes, including nucleation, emissions, coagulation, condensation, dissolution, reversible chemical reactions, irre- versible chemical reactions, sedimentation, dry deposition, and advection. In this chapter, dissolu- tion and reversible chemical reactions are discussed. These processes are important for determining the ionic, solid, and liquid water content of aerosols. L829/frame/ch06 Page 135 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC 136 Aerosol Chemical Processes in the Environment DEFINITIONS Dissolution is a process that occurs when a gas, suspended over a particle surface, adsorbs to and dissolves in liquid on the surface. The liquid in which the gas dissolves is a solvent . A solvent makes up the bulk of a solution, and in atmospheric particles, liquid water is most often the solvent. In some cases, such as when sulfuric acid combines with water to form particles, the concentration of sulfuric acid exceeds the concentration of liquid water, and sulfuric acid may be the solvent. Here, liquid water is assumed to be the solvent in all cases. A species, such as a gas or solid, that dissolves in solution is a solute . Together, solute and solvent make up a solution , which is a homogeneous mixture of substances that can be separated into individual components upon a change of state (e.g., freezing). A solution may contain many solutes. Suspended material (e.g., solids) may also be mixed throughout a solution. Such material is not considered part of a solution. The ability of the gas to dissolve in water depends on the solubility of the gas in water. Solubility is the maximum amount of a gas that can dissolve in a given amount of solvent at a given temperature. Solutions usually contain solute other than the dissolved gas. The solubility of a gas depends strongly on the quantity of the other solutes because such solutes affect the thermodynamic activity of the dissolved gas in solution. Thermodynamic activity is discussed shortly. If water is saturated with a dissolved gas, and if the solubility of the gas changes due to a change in composition of the solution, the dissolved gas can evaporate from the solution to the gas phase. Alternatively, dissociation products of the dissolved gas can combine with other components in solution and precipitate as solids. In solution, dissolved gases can dissociate and react chemically. Dissociation of a dissolved molecule is the process by which the molecule breaks into simpler components, namely ions. This process can be described by reversible chemical reactions , also called chemical equilibrium reac- tions or thermodynamic equilibrium reactions . Such reactions are reversible, and their rates in the forward and backward directions are generally fast. Dissociated ions and undissociated molecules can further react reversibly or irreversibly with other ions or undissociated molecules in solution. Irreversible chemical reactions act only in the forward direction and are described by first-order ordinary differential equations. When they occur in solution, irreversible reactions are called aqueous reactions . EQUILIBRIUM EQUATIONS AND RELATIONS Reversible chemical reactions describe dissolution, dissociation, and precipitation processes. In this section, different types of equilibrium equations are discussed and rate expressions, including temperature dependence, are derived. E QUILIBRIUM E QUATIONS An equilibrium equation describes a reversible chemical reaction. A typical equation has the form (6.1) where D , E , A , and B are species and the ν ’s are dimensionless stoichiometric coefficients or number of moles per species divided by the smallest number of moles of any reactant or product in the reaction. Each reaction must conserve mass. Thus, (6.2) νν νν DE AB DE AB++…⇔ ++… , km ii i i ν ∑ = 0, L829/frame/ch06 Page 136 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC Reversible Chemical Reactions in Aerosols 137 where m i is the molecular weight of each species and k i = +1 for products and –1 for reactants. The reactants and/or products of an equilibrium equation can be solids, liquids, ions, or gases. Reversible dissolution reactions have the form (6.3) where (g) indicates a gas and (aq) indicates that the species is dissolved in solution. In this equation, the gas phase and dissolved (solution) phase of species AB are assumed to be in equilibrium with each other at the gas–solution interface. Thus, the number of molecules of AB transferring from the gas to the solution equals the number of molecules transferring in the reverse direction. In the atmosphere, gas–solution interfaces occur at the air–ocean, air–cloud drop, and air–aerosol inter- faces. Examples of dissolution reactions that occur at these interfaces are (6.4) (6.5) (6.6) (6.7) The reaction (6.8) is also a reversible dissolution reaction. In equilibrium, almost all sulfuric acid is partitioned to the aqueous phase; thus, the relation is rarely used. Instead, sulfuric acid transfer to the aqueous phase is treated as a diffusion-limited condensational growth process. Once dissolved in solution, the species on the right sides of Equations 6.4 to 6.8 often dissociate into ions. Substances that undergo partial or complete dissociation in solution are electrolytes . The degree of dissociation of an electrolyte depends on the acidity of solution, the strength of the electrolyte, the concentrations of other ions in solution, the temperature, and other conditions. The acidity of a solution is a measure of the concentration of hydrogen ions ( protons or H + ions) in solution. Acidity is measured in terms of pH , defined as (6.9) where [H + ] is the molarity of H + (moles H + L –1 solution). The more acidic the solution, the higher the molarity of protons and the lower the pH. Protons in solution are donated by acids that dissolve. Examples of such acids are H 2 CO 3 (aq), HCl(aq), HNO 3 (aq), and H 2 SO 4 (aq). The abilities of acids to dissociate into protons and anions vary. HCl(aq), HNO 3 (aq), and H 2 SO 4 (aq) dissociate readily, while H 2 CO 3 (aq) does not. Thus, the former species are strong acids and the latter species is a weak acid . Because all acids are electrolytes, a strong acid is a strong electrolyte (e.g. , it dissociates significantly) and a weak acid is a weak electrolyte . Hydrochloric acid is a strong acid and strong electrolyte in water because it almost always dissociates completely by the reaction (6.10) AB(g) AB(aq),⇔ HCl(g) HCl(aq)⇔ HNO (g) HNO (aq) 33 ⇔ CO (g) CO (aq) 22 ⇔ NH (g) NH (aq) 33 ⇔ H SO (g) H SO (aq) 24 24 ⇔ pH H=− [] + log , 10 HCl(aq) H Cl⇔+ +− . L829/frame/ch06 Page 137 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC 138 Aerosol Chemical Processes in the Environment Sulfuric acid is also a strong acid and strong electrolyte and dissociates to bisulfate by (6.11) While HCl(aq) dissociates significantly at a pH above –6, H 2 SO 4 (aq) dissociates significantly at a pH above –3. Another strong acid, nitric acid , dissociates significantly at a pH above –1. Nitric acid dissociates to nitrate by (6.12) Bisulfate is also a strong acid and electrolyte because it dissociates significantly at a pH above about +2. Bisulfate dissociation to sulfate is given by (6.13) Carbon dioxide is a weak acid and electrolyte because it dissociates significantly at a pH above only +6. Carbon dioxide converts to carbonic acid and dissociates to bicarbonate by (6.14) Dissociation of bicarbonate occurs at a pH above +10. This reaction is (6.15) While acids provide hydrogen ions, bases provide hydroxide ions (OH – ). Such ions react with hydrogen ions to form neutral water via (6.16) An important base in the atmosphere is ammonia. Ammonia reacts with water to form ammonium and the hydroxide ion by (6.17) Since some strong electrolytes, such as HCl(aq) and HNO 3 (aq), dissociate completely in atmospheric particles, the undissociated forms of these species are sometimes ignored in equilibrium models. Instead, gas-ion equilibrium equations replace the combination of gas-liquid, liquid-ion equations. For example, the equations (6.18) can replace Equations 6.4 and 6.10. Similarly, (6.19) can replace Equations 6.5 and 6.12. H SO (aq) H HSO 24 ⇔+ +− 4 . HNO (aq) H NO 3 ⇔+ +− 3 . HSO H SO 44 2 −+ − ⇔+ . CO (aq) H O(aq) H CO (aq) H HCO 22 23 3 +⇔ ⇔+ + − . HCO H CO 33 2− + − ⇔+ . H O(aq) H OH . 2 – ⇔+ + NH (aq) + H O(aq) NH OH . 32 4 – ⇔+ + HCl(g) H Cl – ⇔+ + HNO (g) H NO 33 – ⇔+ + L829/frame/ch06 Page 138 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC Reversible Chemical Reactions in Aerosols 139 Once in solution, ions can precipitate to form solid electrolytes if conditions are right. Alter- natively, existing solid electrolytes can dissociate into ions if the particle water content increases sufficiently. Examples of solid precipitation/dissociation reactions for ammonium-containing elec- trolytes include (6.20) (6.21) (6.22) Examples of such reactions for sodium-containing electrolytes are (6.23) (6.24) (6.25) If the relative humidity is sufficiently low, a gas can react chemically with another adsorbed gas on a particle surface to form a solid. Such reactions can be simulated with gas-solid equilibrium reactions, such as (6.26) (6.27) In sum, equilibrium relationships usually describe aqueous-ion, ion-ion, ion-solid, gas-solid, or gas-ion reversible reactions. Relationships can be written for other interactions as well. Table 6.1 shows several equilibrium relationships of atmospheric importance. EQUILIBRIUM RELATIONS AND CONSTANTS Species concentrations in a reversible reaction, such as Equation 6.1, are interrelated by (6.28) where K eq (T) is a temperature-dependent equilibrium coefficient and {A}…, etc., are thermody- namic activities. Thermodynamic activities measure the effective concentration or intensity of the substance. The activity of a substance differs, depending on whether the substance is in the gas, undissociated aqueous, ionic, or solid phases. The activity of a gas is its saturation vapor pressure (atm). Thus, (6.29) NH Cl(s) NH Cl 44 – ⇔+ + NH NO (s) NH NO 43 4 3 – ⇔+ + NH SO (s) NH SO 4 2 444 2– () ⇔+ + 2. NaCl(s) Na Cl – ⇔+ + NaNO (s) Na NO 33 – ⇔+ + Na SO (s) Na SO 24 4 2– ⇔+ + 2. NH Cl(s) NH (g) HCl(g) 43 ⇔+ NH NO (s) NH (g) HNO (g). 43 3 3 ⇔+ AB DE KT AB DE eq {} {} {} {} = () νν νν , Ap sA g () {} = , . L829/frame/ch06 Page 139 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC 140 Aerosol Chemical Processes in the Environment The activity of an ion in solution or an undissociated electrolyte is its molality ( m A ) (moles solute kg –1 solvent) multiplied by its activity coefficient ( γ ) (unitless). Thus, (6.30) (6.31) respectively. An activity coefficient accounts for the deviation from ideal behavior of a solution. It is a dimensionless parameter by which the molality of a species in solution is multiplied to give the species’ thermodynamic activity. In an ideal, infinitely dilute solution, the activity coefficient of a species is unity. In a nonideal, concentrated solution, activity coefficients may be greater than TABLE 6.1 Equilibrium Reactions, Coefficients, and Coefficient Units No. Reaction A B C Units Ref. a 1 HNO 3 (g) ⇔ HNO 3 (aq) 2.10 × 10 5 mol kg –1 atm –1 D 2NH 3 (g) ⇔ NH 3 (aq) 5.76 × 10 1 13.79 -5.39 mol kg –1 atm –1 A 3CO 2 (g) ⇔ CO 2 (aq) 3.41 × 10 –2 8.19 -28.93 mol kg –1 atm –1 A 4CO 2 (aq) + H 2 O(aq) ⇔ H + + HCO 3 – 4.30 × 10 –7 –3.08 31.81 mol kg –1 A 5NH 3 (aq) + H 2 O(aq) ⇔ NH 4+ + OH – 1.81 × 10 –5 –1.50 26.92 mol kg –1 A 6 HNO 3 (aq) ⇔ H + + NO 3 – 1.20 × 10 1 29.17 16.83 mol kg –1 N 7 HCl(aq) ⇔ H + + Cl – 1.72 × 10 6 23.15 mol kg –1 O 8H 2 O(aq) ⇔ H + + OH – 1.01 × 10 –14 –22.52 26.92 mol kg –1 A 9H 2 SO 4 (aq) ⇔ H + + HSO 4 – 1.00 × 10 3 mol kg –1 R 10 HSO 4 – ⇔ H + + SO 4 2– 1.02 × 10 –2 8.85 25.14 mol kg –1 A 11 HCO 3 – ⇔ H + + CO 3 2– 4.68 × 10 –11 –5.99 38.84 mol kg –1 A 12 HNO 3 (g) ⇔ H + + NO 3 – 2.51 × 10 6 29.17 16.83 mol 2 kg –2 atm –1 A 13 HCl (g) ⇔ H + + Cl – 1.97 × 10 6 30.19 19.91 mol 2 kg –2 atm –1 A 14 NH 3 (g) + H + ⇔ NH 4 + 1.03 × 10 11 34.81 –5.39 atm –1 A 15 NH 3 (g) + HNO 3 (g) ⇔ NH 4 + + NO 3 – 2.58 × 10 17 64.02 11.44 mol 2 kg –2 atm –2 A 16 NH 3 (g) + HCl(g) ⇔ NH 4 + + Cl – 2.03 × 10 17 65.05 14.51 mol 2 kg –2 atm –2 A 17 NH 4 NO 3 (s) ⇔ NH 4 + + NO 3 – 1.49 × 10 1 –10.40 17.56 mol 2 kg –2 A 18 NH 4 Cl(s) ⇔ NH 4 + + Cl – 1.96 × 10 1 –6.13 16.92 mol 2 kg –2 A 19 NH 4 HSO 4 (s) ⇔ NH 4 + + HSO 4 – 1.38 × 10 2 –2.87 15.83 mol 2 kg –2 A 20 (NH 4 ) 2 SO 4 (s) ⇔ 2 NH 4 + + SO 4 2– 1.82 –2.65 38.57 mol 3 kg –3 A 21 (NH 4 ) 3 H(SO 4 ) 2 (s) ⇔ 3 NH 4 + +HSO 4 +SO 4 2– 2.93 × 10 1 –5.19 54.40 mol 5 kg –5 A 22 NaNO 3 (s) ⇔ Na + + NO 3 – 1.20 × 10 1 –8.22 16.01 mol 2 kg –2 A 23 NaCl(s) ⇔ Na + + Cl – 3.61 × 10 1 –1.61 16.90 mol 2 kg –2 A 24 NaHSO 4 (s) ⇔ Na + + HSO 4 – 2.84 × 10 2 –1.91 14.75 mol 2 kg –2 A 25 Na 2 SO 4 (s) ⇔ 2 Na + + SO 4 2– 4.80 × 10 –1 0.98 39.50 mol 3 kg –3 A Note: The equilibrium coefficient reads, where T 0 = 298.15K and the remaining terms are defined in Equation 6.45. a A: Derived from data in Reference 21; D: From Reference 22; N: Derived from a combination of other rate coefficients in the table; O, R: From Reference 23. With permission. KT T T T T T T eq ()               =−+−+AB C 000 exp ln ,11 A AA + {} = ++ m γ and A AA aq () {} = m γ , L829/frame/ch06 Page 140 Thursday, February 3, 2000 8:39 AM © 2000 by CRC Press LLC Reversible Chemical Reactions in Aerosols 141 or less than unity. Debye and Huckel showed that, in relatively dilute solutions, where ions are far apart, the deviation of molality from thermodynamic activity is caused by Coulombic (electric) forces of attraction and repulsion. At high concentrations, ions are closer together, and ion-ion interactions affect activity coefficients more significantly than do Coulombic forces. The activity of liquid water in an atmospheric particle is the ambient relative humidity (fraction). Thus, (6.32) where a w denotes the activity of water and f r is the relative humidity, expressed as a fraction. Finally, solids are not in solution, and their concentrations do not affect the molalities or activity coefficients of electrolytes in solution. Thus, the activity of any solid is unity; that is, (6.33) Equation 6.28 is derived by minimizing the Gibbs free-energy change of a system. The Gibbs free-energy change per mole (∆G) (J mole –1 ) is a measure of the maximum amount of useful work per mole that may be obtained from a change in enthalpy or entropy in the system. The relationship between the Gibbs free-energy change and the composition of a chemical system is (6.34) where µ i is the chemical potential of the species (J mole –1 ) and k i = +1 for products and –1 for reactants. Chemical potential is a measure of the intensity of a chemical substance and is a function of temperature and pressure. It is really a measure of the change in free energy per change in moles of a substance, or the partial molar free energy. The chemical potential is (6.35) where µ I o is the chemical potential at a reference temperature of 298.15K, and {a i } is the thermo- dynamic activity of species i. The chemical potential can be substituted into Equation 6.34 to give (6.36) Rewriting this equation yields (6.37) where (6.38) HOaq 2 () {} ==af wr , A s () {} = 1. ∆Gk iii i = ∑ νµ, µµ ii i RT a=+ {} o* ln , ∆Gk RTka iii i ii i i = () + {} () ∑∑ νµ ν o* ln . ∆∆GGRT a i k i i i =+ {} ∏ o * ln , ν ∆Gk iii i oo = () ∑ νµ L829/frame/ch06 Page 141 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC 142 Aerosol Chemical Processes in the Environment is the standard molal Gibbs free energy of formation (J mole –1 ) for the reaction. Equilibrium occurs when ∆G = 0 at constant temperature and pressure. Under such conditions, Equation 6.37 becomes (6.39) The left side of Equation 6.39 is the equilibrium coefficient. Thus, (6.40) Substituting Equation 6.40 into Equation 6.39 and expanding the product term gives (6.41) which is the relationship shown in Equation 6.28. TEMPERATURE DEPENDENCE OF THE EQUILIBRIUM COEFFICIENT The temperature dependence of the equilibrium coefficient is calculated by solving the Van ’t Hoff equation, (6.42) where ∆ H T o is the change in total enthalpy (J mole –1 ) of the reaction. The change in enthalpy can be approximated by (6.43) when the standard change in molal heat capacity of the reaction (∆c P o ) (J mole –1 K –1 ) does not depend on temperature. In this equation, is the standard enthalpy change in the reaction (J mole –1 ) at temperature T o = 298.15K. Combining Equations 6.42 and 6.43 and writing the result in integral form gives (6.44) Integrating this equation yields the temperature-dependent equilibrium coefficient expression (6.45) exp . * − () [] = {} ∏ ∆GRT a i i i o ν KT G RT eq () =− () [] exp . * ∆ o KT a AB DE eq i k i ii AB DE () = {} = {} {} {} {} ∏ ν νν νν d d o ln , * KT T H RT eq T () = ∆ 2 ∆∆∆HHcTT TTp ooo o o ≈+ − () ∆H T o o dd o o o oo o ln . * KT HcTT RT T eq T T Tp T T () = +− ()         ∫∫ ∆∆ 2 KT KT H RT T T c R T T T T eq eq o T p () = () −−       −−+                     exp ln ** ∆ ∆ o o o o o oo 11 L829/frame/ch06 Page 142 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC Reversible Chemical Reactions in Aerosols 143 where K eq (T o ) is the equilibrium coefficient at temperature, T o . Values of and ∆c p o are measured experimentally. Table 6.1 shows temperature-dependent parameters for several equilibrium reactions. FORMS OF EQUILIBRIUM COEFFICIENT EQUATIONS Each reaction in Table 6.1 can be written in terms of thermodynamic activities and an equilibrium coefficient. For example, an equilibrium coefficient equation for the reaction (6.46) is (6.47) where is the saturation vapor pressure of nitric acid (atm), is the molality of nitric acid in solution (moles kg –1 ), and is the activity coefficient of dissolved, undissociated nitric acid (unitless). The equilibrium coefficient has units of (moles kg –1 atm –1 ). When the equilibrium coefficient relates the saturation vapor pressure of a gas to the molality (or molarity) of the dissolved gas in a dilute solution, the coefficient is called a Henry’s constant. Henry’s constants (moles kg –1 atm –1 ), like other equilibrium coefficients, are temperature and solvent dependent. Henry’s law states that, for a dilute solution, the pressure exerted by a gas at the gas–liquid interface is proportional to the molality of the dissolved gas in solution. For a dilute solution, = 1, and Equation 6.47 obeys Henry’s law. A dissociation equation has the form (6.48) The equilibrium coefficient expression for this reaction is (6.49) where the equilibrium coefficient has units of (moles kg –1 ). In Equation 6.49, the activity coefficients are determined by considering a mixture of all dissociated and undissociated electrolytes in solution. Thus, the coefficients are termed mixed activity coefficients. More specifically, are single-ion mixed activity coefficients, and is a mean (geometric mean) mixed activity coefficient. When H + , and NO 3 – are alone in solution, are single-ion binary activity coefficients, and is a mean (geometric mean) binary activity coefficient. Activity coefficients for single ions are difficult to measure because single ions cannot be isolated from a solution. Single-ion activity coefficients are easier to estimate ∆H T o o HNO (g) HNO (aq) 33 ⇔ HNO aq HNO g HNO aq HNO aq ,HNO g 3 3 33 3 () {} () {} == () () () () m γ p KT s eq , p s,HNO (g) 3 m HNO (aq) 3 γ HNO (aq) 3 γ HNO (aq) 3 HNO (aq) H NO 3 + 3 – ⇔+ . HNO HNO aq 3 - H H NO NO HNO aq HNO aq HNOHNO HNO aq HNO aq 3 - 3 - 3 - 3 - + () () () () {}{ } () {} == ++ + + 3 2 33 33 mm m mm m γγ γ γ γ , γγ HNO and 3 + γ HNO 3 -+ , γγ HNO and 3 + γ HNO 3 -+ , L829/frame/ch06 Page 143 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC 144 Aerosol Chemical Processes in the Environment mathematically. Mean binary activity coefficients are measured in the laboratory. Mean mixed activity coefficients can be estimated from mean binary activity coefficient data through a mixing rule. A geometric mean activity coefficient is related to a single-ion activity coefficient by (6.50) where γ ± is the mean activity coefficient, γ + and γ – are the activity coefficients of the single cation and anion, respectively, and ν + and ν – are the stoichiometric coefficients of the cation and anion, respectively. In Equation 6.48, ν + = 1 and ν – = 1. Raising both sides of Equation 6.50 to the power ν + + ν − gives (6.51) which is form of the mean activity coefficient used in Equation 6.49. When ν + = 1 and ν – = 1, the electrolyte is univalent. When ν + > 1 or ν – > 1, the electrolyte is multivalent. When ν + = ν – for a dissociated electrolyte, the electrolyte is symmetric; otherwise, it is nonsymmetric. In all cases, a dissociation reaction must satisfy the charge balance requirement (6.52) where z + is the positive charge on the cation and z - is the negative charge on the anion. MEAN BINARY ACTIVITY COEFFICIENTS The mean binary activity coefficient of an electrolyte, which is primarily a function of molality and temperature, can be determined from measurements or estimated from theory. Measurements of binary activity coefficients for several species at 298.15K are available. Parameterizations have also been developed to predict the mean binary activity coefficients. One parameterization is Pitzer’s method, 1,2 which estimates the mean binary activity coefficient of an electrolyte at 298.15K with (6.53) where γ 0 12b is the mean binary activity coefficient of electrolyte 1-2 (cation 1 plus anion 2) at the reference temperature (298.15K), Z 1 and Z 2 are the absolute value of the charges of cation 1 and anion 2, respectively, m 12 is the molality of electrolyte dissolved in solution, and ν 1 and ν 2 are the stoichiometric coefficients of the dissociated ions (assumed positive here). In addition, (6.54) (6.55) γγγ νν νν ± + () = () +− +− +- 1 , γγγ νν νν ± + () +− +− = + – , zz ++ −− +=νν0, ln γ νν νν νν νν γγ γ 12 0 12 12 12 12 12 12 2 12 32 12 1 2 2 2 b fB C=+ + + () + ZZ mm f γ =− + ++ ()       0 392 112 2 12 112 12 12 12 . ln . I I I Be 12 12 1 12 2 212 2 2 4 1122 12 γ β β =+ − +− () [] () () − I II I , L829/frame/ch06 Page 144 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC [...]... iterating all equations many times Suppose a system consists of a single aerosol size bin and 15 equations representing the equilibrium chemistry within that bin At the start, the first equation is iterated When the first equation converges, the updated and other initial concentrations are used as inputs into the second equation This continues until the last equation has converged At that point, the first... Equations 6. 63 and 6. 64 Jacobson et al.4 list B, G, and H values for 10 electrolytes and the range of validity for data Determining the temperature-dependent binary activity coefficients of bisulfate and sulfate is more difficult They can be found by combining equations from the model of Clegg and Brimblecombe6 with Equations 6. 72 and 6. 73 of Stelson et al.7 in a Newton-Raphson iteration Figure 6. 2 shows... E3 m 3 2 + E4 m 2 , (6. 79) where El = Al −  0.5(l − 2)mv  TL  Ul −2 + TC Vl −2  * 1000 R  T0  (6. 80) for each l greater than 2 Equation 6. 79 shows that temperature affects the water-activity polynomial beginning only in the fourth term In Equation 6. 79, temperature affected the solute activity beginning with the second term of the polynomial These equations indicate that the effect of temperature... ions in solution, water vapor condenses to maintain saturation over the solution surface, increasing the liquid water content Liquid water content is a unique function of electrolyte molality and sub-100% relative humidity As the relative humidity increases up to 100%, hydration increases the aerosol liquid water content The liquid water content also increases with increasing solute molality in solution... water In the dilute solution case, the vapor-phase concentration of water is not reduced The mixed and binary molalities in Equations 6. 72 differ from each other because, in a mixture, the quantity and type of ions differ from in a binary solution; thus, a different quantity of water is hydrated in each case Table 6. 3 gives mixed and binary molalities of sucrose and mannitol alone and mixed together in. .. including gas-solution equations, starting with the first size bin Updated gas concentrations from the first bin affect the equilibrium distribution in subsequent bins After the last size bin has been iterated, the sequence is repeated in reverse order (to speed convergence), from the last to first size bin The marches back and forth among size bins continue until gas and aerosol concentrations do not change... concentrations, and is mass- and charge-conserving at all times The only constraints are that the equilibrium equations must be mass- and charge-conserving, and the system must start in charge balance For example, the equation HNO3(aq) = H+ + NO3– conserves mass and charge The charge balance constraint allows initial charges to be distributed among all dissociated ions, but the initial sum, over all species,... which occurs when m = 0 Equations 6. 58, 6. 61, and 6. 61 can be combined to give temperature-dependent, mean binary activity coefficient polynomials of the form ln γ 12 b (T ) = F0 + F1m1 2 + F2 m + F3 m 3 2 + , where F0 = B0 and © 2000 by CRC Press LLC (6. 63) L829/frame/ch 06 Page 147 Monday, January 31, 2000 2:59 PM Reversible Chemical Reactions in Aerosols 147 FIGURE 6. 2 Binary activity coefficients of sulfate... alone in solution Results are valid for 0 to 40 m total H2SO4 (From Reference 18 With permission.) Fj = Bj + G j TL + H j TC (6. 64) for each additional term, beginning with j = 1 In Equation 6. 64, Gj = 0.5( j + 2)U j ( ν1 + ν2 ) R*T0 Hj = 0.5( j + 2)Vj ( ν1 + ν2 ) R* and (6. 65) (6. 66) With sufficient data, many temperature- and molality-dependent mean binary activity coefficients can be written in terms... j ,a  ∑ ∑m i =1 j =1 (6. 81) where binary molalities of species alone in solution (ma) are obtained from Equation 6. 76 at the given relative humidity In this equation, i,j is an electrolyte pair (where the odd/even subscripts used previously are ignored), and c is the hypothetical mole concentration of the pair when mixed in solution with all other components In a model, hypothetical mole concentrations . reactions for ammonium-containing elec- trolytes include (6. 20) (6. 21) (6. 22) Examples of such reactions for sodium-containing electrolytes are (6. 23) (6. 24) (6. 25) If the relative humidity is sufficiently. +mmm L829/frame/ch 06 Page 1 46 Monday, January 31, 2000 2:59 PM © 2000 by CRC Press LLC Reversible Chemical Reactions in Aerosols 147 (6. 64) for each additional term, beginning with j = 1. In Equation 6. 64, (6. 65) (6. 66) With. and Owen 5 as (6. 77) If the water activity at the reference temperature is expressed as (6. 78) then Equations 6. 77, 6. 78, 6. 61, and 6. 62 can be combined to form (6. 79) where (6. 80) for each l