Atmospheric aerosol models for systems including the ions H+, NH4+, Na+, SO42−, NO3, Cl, Br, and H2O



[1] Mole fraction based equations for aqueous phase activities, together with equilibrium constants for the formation of gases and solids, have been combined with a Gibbs free energy minimization algorithm to create equilibrium phase partitioning models of inorganic atmospheric aerosols. The water content, phase state (solid or liquid), and gas/aerosol partitioning are predicted for known ionic composition, relative humidity, and temperature. The models are valid from <200 to 328 K for the subsystems (H+-SO42−-NO3-Cl-Br-H2O) and (H+-NH4+-SO42−-NO3-H2O), and 298.15 K only for (H+-NH4+-Na+-SO42−-NO3-Cl-H2O). The models involve no simplifying assumptions and include all solid phases identified in bulk experiments, including hydrated and double salt forms not treated in most other studies. The Henry's law constant of H2SO4 is derived as a function of temperature, based upon available data, and the model treatment of the solubility of HBr in aqueous H2SO4 is revised. Phase diagrams are calculated for the (NH4)2SO4/H2SO4/H2O system to low temperature. The models are also used to explore the importance of the double salts in urban inorganic aerosols. These Aerosol Inorganics Model (AIM) models can be run on the Web for a variety of problem types at and∼e770/aim.html, and their use is summarized here.

1. Introduction

[2] Atmospheric aerosols are important in many air pollution problems including global climate change, stratospheric ozone depletion, visibility, and smog [Seinfeld and Pandis, 1997]. The effects of aerosols are dependent on the phase state of the particles (solid or liquid), their water content, and the partitioning of volatile components between the aerosol and vapor phases. Models are needed that can quickly and accurately predict the composition and state of the aerosol over the wide ranges of temperature and relative humidity experienced from the lower troposphere to the winter polar stratosphere. Calculations of aerosol composition and phase state often assume thermodynamic equilibrium, which is reasonable if the atmospheric processes that lead to toward equilibrium are fast compared to those that lead away from it [Wexler and Potukuchi, 1998]. It is also important to realize that thermodynamic equilibrium is the end point of all kinetic processes, so that calculating the extent of a reaction often requires a knowledge of the thermodynamics.

[3] The phase state of atmospheric particles can be complex even when equilibrium thermodynamics applies. First, there are a wide range of possible solid phases that can form. Second, the behavior of any liquid phase in the aerosol is likely to be highly nonideal at the high solute concentrations encountered at low relative humidities. This can be a particular difficulty given that many aerosols probably exist in a metastable liquid state under some conditions, supersaturated with respect to dissolved salts [Martin, 2000]. A large fraction of aerosol mass is in many cases made up of inorganic electrolyte compounds that dissociate in water to form ions and a number of models of such systems have been developed. These include EQUIL [Bassett and Seinfeld, 1983], MARS [Saxena et al., 1986], EQUILIB [Pilinis and Seinfeld, 1987], AIM [Wexler and Seinfeld, 1991], SCAPE [e.g., Kim et al., 1993], EQUISOLV [Jacobson et al., 1996], and ISORROPIA [Nenes et al., 1998], each with their own strengths and weaknesses. Models are often judged in terms of their speed and accuracy, which to some extent trade off against each other. See Zhang et al. [2000] for a recent comparison of a number of inorganic aerosol models including one from this study.

[4] Here we describe flexible, highly accurate, phase equilibrium models for inorganic aerosols based upon the work of S. L. Clegg and coworkers [Clegg et al., 1998a, 1998b, and references therein]. The equilibrium state of the gas/aerosol system is calculated using Gibbs free energy minimization as was done in the original AIM (based on the work of Potukuchi and Wexler [1995a, 1995b]), but the formulation, algorithm, and chemistry have all been improved. Our intent is to produce accurate models while applying no simplifying assumptions. The codes are sufficiently fast that they can readily be used for the analysis of experimental data, but remain too slow to be incorporated into large-scale atmospheric models. For computationally intensive problems a faster code can be employed at some sacrifice in accuracy, or specialized algorithms such as neural networks [Potukuchi and Wexler, 1997] can be trained with data sets generated by accurate (though slower) models.

[5] The purpose of this work is as follows: (1) to describe in detail how gas/liquid/solid equilibrium partitioning models based on the results of Clegg and coworkers are constructed and solved; (2) to summarize the capabilities of the models as they are implemented on the Worldwide Web; (3) to present extensions and revisions to the equilibrium thermodynamics of systems treated by the model; and (4) to apply the model to investigate the possible significance of the formation of double salts in ambient aerosols.

[6] We present three models. The first two are intended for applications involving the stratosphere and upper troposphere, and include a limited range of ions. The third model is more extensive, but applies only at 298.15 K. The species and reactions included in our models, and those of other workers, are listed by Zhang et al. [2000]. Although we do not include as many ionic species as, for example, SCAPE2 or EQUISOLV, the present models are notable both for a comprehensive treatment of double salts and their extension to low temperatures.

2. Models

[7] We have developed codes for three chemical systems: (1) H+-SO42−-NO3-Cl-Br-H2O for 180 ≤ T < 328 K [Carslaw et al., 1995; Massucci et al., 1999]; (2) H+-NH4+-SO42−-NO3-H2O for ∼200 ≤ T < 328 K [Clegg et al., 1998a]; and (3) H+-NH4+-Na+-SO42−-NO3-Cl-H2O at T equal to 298.15 K [Clegg et al., 1998b]. The chemical species included in each model are listed in Table 1. The equilibrium constants for reactions between gases, solids, and liquid phase species are given in the references above, together with parameters for the Pitzer, Simonson, and Clegg equations [Pitzer and Simonson, 1986; Clegg et al., 1992], which are used to calculate the activity coefficients of water and ions in the liquid phase.

Table 1. Species Present in the Modelsa
Primary SpeciesbLiquid Phase SpeciesGasesSolids
Acid HydratesSalts
  • a

    The species above are present in all models (1–3), except where the model numbers are given in parentheses. Thus, for example, liquid water is present in all models but the ion NH4+ occurs only in models 2 and 3.

  • b

    Total amounts of ions and water are expressed in terms of the primary species when defining problems, as are the mole balances which must remain constant for each primary species and are used to define linear constraints (section 4).

H+H+HNO3H2SO4 · H2O(1,2)(NH4)3H(SO4)2(2,3)
NH4+(2,3)NH4+(2,3)HCl(1,3)H2SO4 · 2H2O(1,2)NH4HSO4(2,3)
Na+(3)Na+(3)NH3(2,3)H2SO4 · 3H2O(1,2)NH4NO3(2,3)
SO42−SO42−H2SO4H2SO4 · 4H2O(1,2)2NH4NO3 · (NH4)2SO4(2,3)
NO3HSO4HBr(1)H2SO4 · 6.5H2O(1,2)3NH4NO3 · (NH4)2SO4(2,3)
Cl−(1,3)NO3 HNO3 · H2O(1,2)NH4NO3 · NH4HSO4(2,3)
Br−(1)Cl−(1,3) HNO3 · 2H2O(1,2)Na2SO4(3)
 Br−(1) HNO3 · 3H2O(1,2)Na2SO4 · 10H2O(3)
   HCl · 3H2O(1)Na3H(SO4)2(3)
    NaHSO4 · H2O(3)
    NaH3(SO4)2 · H2O(3)
    Na2SO4 ·(NH4)2SO4 · 4H2O(3)
    NaNO3 · Na2SO4 · H2O(3)

[8] The models described here do not include the Kelvin effect, which influences the partitioning between gases and aerosols for particle diameters less than about 0.1 μm, nor do they treat aerosols that are externally mixed. Thus the position of chemical equilibrium is the same as that for a single bulk condensed phase in the same system volume. The models can be run for essentially two different types of calculation: (1) where the total number of moles of water in the system is specified (and this partitions between the gas and condensed phases); (2) the system is equilibrated to a fixed relative humidity. In both cases the equilibration of trace gases between the vapor and condensed phases can be enabled or disabled as required, as can the formation of solids which allows the properties of liquid aerosols supersaturated with respect to solid phases to be investigated. Where the total water in the system is specified, and the ionic content of the system is zero, then the equilibrium between liquid water, water vapor, and ice will be determined.

[9] Aerosol models are generally complex, and can be difficult both to access and use. In order to address these problems we have made our models available on the Worldwide Web. The sites have simple input forms covering a variety of common types of calculation. Output is flexible and comprehensive, and includes graph plotting. Further details, including the Universal Resource Locators (URLs), are given in Appendix A.

3. Theory

[10] In this section the thermodynamic basis of the models is described, including the calculation of relative humidity at low temperature and the equilibrium between liquid acid sulphate aerosols and gas phase sulphuric acid. A revision of the treatment of HBr solubility in aqueous H2SO4 at low temperature is summarized in Appendix C.

3.1. Equilibrium Constants

[11] Definitions of these quantities, for the formation of solids and equilibria of aqueous species with trace gases, are given by Carslaw et al. [1995] and Clegg et al. [1998a] and are briefly summarized here. For a solid Mv+Xv · zH2O(cr) the dissolution reaction is

equation image
equation image
equation image

where xKS (dimensionless) is the thermodynamic solubility product of the solid, prefix a indicates activity, and x indicates mole fraction as defined by Clegg et al. [1998a, equation (1)]. The quantity fi* is the mole fraction activity coefficient of ion i for a reference state of infinite dilution in water (thus fi* → 1 as xH2O →1). In our models the mole fraction activity coefficient of water (equation image) is relative to a pure liquid reference state, thus equation image → 1 as xH2O → 1 and the activity of pure water is unity. The activity of the solid does not appear as a denominator in equation (2) as the activity of any pure solid phase is equal to unity.

[12] We also define a saturation ratio S, which is equal to the actual value of the activity product from equation (2) in a solution divided by the thermodynamic value xKS. In a solution saturated with respect to the solid, the value of S will be equal to 1.0.

[13] For an acid gas HX the Henry's law equilibrium is given by

equation image
equation image
equation image

where xKH (atm−1) is the mole fraction based Henry's law constant and pHX (atm) is the equilibrium partial pressure of HX. For equilibria involving gas phase NH3 the present models are only valid for acid systems, and the treatment of the Henry's law equilibrium and acid dissociation is described by Clegg et al. [1998a] (see also section 4.1). The equation for the association of H+(aq) and SO42−(aq) to form HSO4(aq) is given by Clegg and Brimblecombe [1995].

[14] The studies describing the thermodynamic basis of the present models, and referred to above, do not list equilibrium constants for gas-solid reactions. However, these are readily calculated from the values for gas-liquid (Henry's law) and solid-liquid (xKS) equilibria.

3.2. Relative Humidity

[15] Equilibrium water vapor pressures (pH2O) above aqueous aerosols are related to the water activity (aH2O) of the aerosols and the vapor pressure over pure liquid water (po(H2O(l))) at the same temperature by

equation image

where the water activity is also equivalent to the equilibrium relative humidity (RH), expressed as a fraction. Aerosols containing dissolved acids or salts can remain in a liquid state to as low as 200 K without becoming saturated with respect to solid phases. Values of po(H2O(l)) at these temperatures are thus essentially extrapolations, although freezing point depression measurements together with vapor pressures of water over ice allow some checks for self consistency [Clegg and Brimblecombe, 1995].

[16] Here vapor pressures of pure water above 275.15 K are taken from the Goff-Gratch equation [McDonald, 1965], and at lower temperatures from the modification of this equation presented by Clegg and Brimblecombe [1995]. Values for a range of temperatures are listed in Table 2.

Table 2. Vapor Pressures of Pure Liquid Water Used in the Modelsa
  • a

    Units: T, K; po(H2O(l)), atm.

183.152.047 × 10−7233.151.865 × 10−4283.151.211 × 10−2
188.154.869 × 10−7238.153.095 × 10−4288.151.682 × 10−2
193.151.101 × 10−6243.155.018 × 10−4293.152.307 × 10−2
198.152.378 × 10−6248.157.960 × 10−4298.153.125 × 10−2
203.154.920 × 10−6253.151.237 × 10−3303.154.187 × 10−2
208.159.790 × 10−6258.151.886 × 10−3308.155.550 × 10−2
213.151.879 × 10−5263.152.825 × 10−3313.157.281 × 10−2
218.153.486 × 10−5268.154.159 × 10−3318.159.460 × 10−2
223.156.272 × 10−5273.156.027 × 10−3323.151.218 × 10−1
228.151.096 × 10−4278.158.604 × 10−3328.151.554 × 10−1

3.3. Gas Phase H2SO4

[17] Equilibrium partial pressures of H2SO4 above H2SO4/H2O mixtures for compositions approaching the pure acid have been measured by Roedel [1979] for ∼96 mass percent and ∼82.5 mass percent acid, by Ayers et al. [1980] for 98.01 mass percent acid, and most recently by Richardson et al. [1986] for the azeotropic mixture close to room temperature. Following the work of Clegg et al. [1998a], we use the results of Ayers et al. [1980] to estimate a Henry's law constant xKH for H2SO4 as a function of temperature. Ayers et al. present the following equation for the equilibrium partial pressure of H2SO4 (in atm) in their experiments from 338 to 445 K:

equation image

The Henry's law reaction and the equation for the equilibrium constant are

equation image
equation image

The equilibrium partial pressure pH2SO4 is also related to the vapor pressure of the pure acid by

equation image

where aH2SO4′ is the activity of H2SO4, but for a reference state of unity for the pure acid (for which aH2SO4′ = 1). Combining equations (6b) and (7), we obtain

equation image

The second term in equation (8), the ratio of the activities, is simply the reference state correction for the activity coefficients, which varies with temperature (but not solution composition).

[18] The following expression for po(H2SO4(l)) as a function of temperature was determined using activities of 98.01 mass percent acid from Giauque et al. [1960] and equations (5) and (7):

equation image

The value of the reference state correction in equation (8) was obtained first at 298.15 K by comparing mole fraction based activities obtained from the critical review of Clegg et al. [1994] for the infinite dilution reference state with values of aH2SO4′ from Giauque et al. [1960], yielding ln((aH+)2aSO42−/aH2SO4′) equal to 9.67197. The variation of this quantity from 180 to 323.15 K was then calculated using the model of Carslaw et al. [1995] (infinite dilution reference state) and results of Giauque et al. [1960] to yield

equation image

where Tr is the reference temperature of 298.15 K. This reference state correction was then combined with equation (9), giving ln(xKH(298.15 K)) = 27.475, ΔrHo(298.15 K) = −180.383 × 103 J mol−1, and ΔrCp = −1877.57 + 4.8685T (J mol−1 K−1) for the Henry's law reaction.

[19] What are the likely uncertainties in the Henry's law constant? Bolsaitis and Elliott [1990] have evaluated the vapor pressure of pure liquid H2SO4 in a modeling study of the thermodynamics of the H2SO4/H2O system over the entire concentration range. They note that water activities for compositions approaching the pure acid, evaluated by Giauque et al. [1960], do not appear to be consistent with pH2SO4 measured by Roedel [1979] and Ayers et al. [1980], and compositions of azeotropic mixtures [Kunzler, 1953] extrapolated to room temperature. In their correlation, Bolsaitis and Elliott [1990] therefore adjust the H2O and H2SO4 activities for high H2SO4 mole fractions. They compare predicted equilibrium pH2SO4 with the measurements of Ayers et al. [1980] for 98.01% acid in their Figure 10. At 65°C, the lower temperature limit of the Ayers et al. data, predicted partial pressures exceed the measured values by a factor of almost 3, and are also greater than estimates of Vermeulen et al. [1982]. The equilibrium pH2SO4 for H2SO4/H2O particles of azeotropic composition determined by Richardson et al. [1986] also exceed the partial pressures of Ayers et al. [1980] for 98 mass percent acid by about 50% from 320 to 330 K and almost a factor of 2 at 298.15 K (extrapolated). Partial pressures of H2SO4 determined by Roedel [1979] for (82 to 83) mass percent and (95 to 97) mass percent acid, when adjusted to 99%, yield a mean pH2SO4 of 3.29(−1.1, +1.7) × 10−8 atm at 296.15 K, which is greater than that extrapolated from the equation of Ayers et al. [1980] by about a factor of 2.

[20] These comparisons suggest that the equilibrium pH2SO4 determined by Ayers et al. [1980] may be too low, though this is by no means certain. The implications of this for the derived Henry's law constant (that it would be too large) are complicated by the uncertainties in the activities of H2SO4 in highly concentrated solutions as these affect the term aH2SO4′ in equation (10). This needs to be evaluated, preferably including new measurements of equilibrium water vapor pressures above highly concentrated H2SO4 solutions. For the present, we note that we have previously used a value of xKH(H2SO4) based on the measurements of Ayers et al. [1980] to compare measured and predicted pH2SO4 for (NH4)2SO4/H2SO4/H2O mixtures at 303.15 K [Clegg et al., 1998a]. There was reasonable agreement, even though the concentrations of the solutions were at or beyond the limits of the present model. We therefore retain the data of Ayers et al. [1980], and activities of Giauque et al. [1960], as the basis for Henry's law constants in the model. From the comparisons carried out here it appears that H2SO4 partial pressures calculated using the model are more likely to be too low than too high and that these partial pressures are probably uncertain to within a factor of 2–4 at room temperature, and more at lower temperatures.

3.4. Gibbs Energies

[21] The Gibbs energies of formation (ΔfGo/J mol−1) of each species at the temperature of interest are calculated from the equilibrium constants using the base set of ΔfGo, enthalpies (ΔfHo/J mol−1), and heat capacities at constant pressure (Cp/J mol−1 K−1) listed in Table 3 for the eight primary liquid phase species. We note that the accuracy of the model is not dependent on these values, as all that is required is that the complete set of ΔfGo are consistent with the equilibrium constants for all reactions. In this section we summarize the thermodynamic relationships that enable expressions for ΔfGo as functions of temperature to be obtained for all solids and gases, and also the HSO4(aq) (bisulphate) ion which is a reaction product of H+(aq) and SO42−(aq).

Table 3. Thermodynamic Properties of the Liquid Phase Primary Speciesa
SpeciesΔfGo (298.15 K)ΔfHo (298.15 K)Cp
SO42−−744530c−909270c−5569.8294 + 31.76549T – 0.04710391T2e
NO3−111250c−207360c−568.03908 + 2.2990619T – 2.13091 × 10−3T2f
Cl−131228c−167159c−514.195 + 1.3Tg
H2O−237129c−285830c295.1612 – 1.540498T + 2.7023 × 10−3T2g

[22] Each equilibrium constant K(T) at temperature T is related to the Gibbs energies of formation of the reactant species j and product species k by the equation

equation image

where R (8.3144 J mol−1 K−1) is the gas constant, and vj and vk are the stoichiometric numbers of each reactant and product in the reaction equation. As an example, consider the equilibrium between the solid (NH4)2SO4(cr) and an aqueous solution at temperature T:

equation image
equation image

For this reaction, equation (11) yields

equation image

which enables ΔfGo((NH4)2SO4(cr),T) to be calculated from the ΔfGo values of the primary species listed in Table 3 and xKS from Clegg et al. [1998a, equation (16)].

[23] The Gibbs energy of formation of a chemical species at temperature T can be expressed as the following function of T:

equation image

where ΔfGo(Tr) and ΔfHo(Tr) are the Gibbs energy and enthalpies of formation of the species at reference temperature Tr, and symbols a to d are parameters for the equation for its heat capacity Cp:

equation image

We note that the use of mole fraction equilibrium constants in equation (11) yields values of ΔfGo for the product species k on the same basis. For comparison with literature values of ΔfGo, which are consistent with equilibrium constants on a molality basis, the quantity ln(1000/M(H2O)) · equation image (where M(H2O) is the molar mass of water in g) should be added to the mole fraction ΔfGo calculated using equation (11).

[24] The relationships between the enthalpy and heat capacity changes for a reaction, and the equilibrium constant K, are given by

equation image
equation image
equation image
equation image
equation image

We further differentiate ΔrCp with respect to T in order to completely describe the variation of the equilibrium constants with temperature. Coefficients a to d for equations (14) and (15) for each reactant and product species are related to these differentials as follows:

equation image
equation image
equation image
equation image

We note that the d coefficient in the heat capacity equation is needed only in order to represent the variation of ΔfGo with temperature for water vapor and for ice. Constants for equation (14) are given for both species in Table 4.

Table 4. Constants for the Gibbs Energy Equation for Water Vapor and Icea
QuantityWater VaporbWater VaporcIce
  • a

    The switching temperature for the water vapor values is set at 275.15 K because at this temperature the calculated ln(po(H2O(l))) from the two sets of constants agree to seven digits.

  • b

    For temperatures below 275.15 K.

  • c

    For temperatures above 275.15 K.

ΔfGo(Tr)−229.685737145 × 103−228537.98−241.294330406 × 103
ΔfHo(Tr)−242.64644281 × 103−241.77239 × 103−293.73615 × 103
b1.13261 × 10−3−0.439385−0.57943577
c−1.099821 × 10−5−9.3701 × 10−42.879618 × 10−3
d3.573575 × 10−84.27758 × 10−6−3.8444421 × 10−6

4. Implementation

[25] A system is in chemical and phase equilibrium when its Gibbs free energy (G) is minimized. Equilibrium composition has previously been calculated in atmospheric applications by minimizing G directly [Bassett and Seinfeld, 1983; Wexler and Seinfeld, 1991], or indirectly by solving equilibrium relations for each phase and chemical reaction [e.g., Pilinis and Seinfeld, 1987; Kim et al., 1993]. In this work we adopt the former approach and cast the problem as the minimization of a nonlinear objective function (the total G for the system), subject to linear constraints that express the mole balances for each primary species. The method implicitly treats all possible reactions between the species included in the expression for G.

[26] The total Gibbs free energy of a system at constant total pressure P and temperature T containing liquid water, dissolved ions (i), solids (s), and gases (g) is given by

equation image

where n is the number of moles of the indicated species. In equation (23), fugacities are assumed to be equal to partial pressures (p), which are calculated for each vapor phase species g by the equation

equation image

The total partial pressure of the vapor phase water and trace gases in the final summation in equation (23) is likely to contribute only a small amount to the total pressure P. The major constituent of the gas phase would be air for atmospheric calculations, or some inert carrier gas in an experiment. In order to account for this, an amount of ideal nonreacting gas equal to that contained in a volume V at the system T and P is included as a gas phase species. As the partitioning of this species is not itself of interest, its Gibbs energy of formation ΔfGo is set to zero at all T.

[27] It should be noted that the total volume of the vapor phase after the partitioning of water vapor and trace gases will exceed V by a small amount. The condensed phase also contributes to the total volume of the system. However, for calculations involving aerosols the effect is negligible due to the small amounts of material involved, and for almost all practical calculations V can be considered the total system volume.

[28] The equilibration of an aerosol/gas system to a fixed relative humidity (RH) is a common requirement in atmospheric modeling. This is implemented in the model following the approach of Shvarov [1978]. Consider a system consisting of a gas phase at a known relative humidity, and an aerosol phase with (initially) zero water content. When equilibrium is reached, some fraction of the gas phase water may exist in the condensed phase either as liquid water and/or water of hydration in one or more solids. The contribution of the gas phase water to the total Gibbs energy of the system is unaltered as the relative humidity is constant, and it may therefore be ignored. However, the change in the total Gibbs energy of the system due to the movement of water vapor into the condensed phase (as liquid water and/or water of hydration in a solid) must be taken into account. This change is equal to −ΔnH2O(g)fGo(H2O(g),T) + RT · ln(pH2O)) where ΔnH2O(g) is the total amount of water existing in the condensed phase at equilibrium. When pH2O in this expression is related to the relative humidity by the equation for the vapor pressure over pure water, and ΔnH2O(g) is divided into contributions for the liquid phase and individual solids, we obtain the terms below.

[29] For calculations at fixed relative humidity the expression for the liquid water contribution to G in equation (23) is replaced by

equation image

In addition, the summation of all solids in equation (23) is modified to take account of the numbers of water molecules of hydration, if any, associated with each solid. The summation in equation (22) becomes

equation image

where equation image is the number of water molecules of hydration for salt s (e.g., zero for NaCl(cr) and 10 for Na2SO4 · 10H2O(cr)). The number of moles of gas phase water is not a variable in calculations at a fixed relative humidity. However, it must be taken into account when calculating the partial pressures of the other gases. At a total system pressure P the sum of the partial pressures of all gases excluding water vapor is equal to (P − RH · po(H2O(l))). Thus the partial pressure of the background gas, and each trace gas, is calculated from

equation image

where the summation of course excludes gas phase water.

[30] For the linear constraints involving each primary species P we have

equation image

where the summation j is over all species in all phases, and vP,j is the stoichiometric number of primary P in species j. Thus, for example, vH,j and equation image are both equal to unity where j is the gas HNO3, and for the solid (NH4)3H(SO4)2(cr) we have vH,j equal to 1, equation image equal to 3, and equation image equal to 2.

4.1. Minimization

[31] The equilibrium state of the gas/aerosol system is reached when the value of G, given by equation (23), is at a minimum. In our work this is determined using the commercial routine E04UCF [Numerical Algorithms Group, 1986], which is designed to minimize an arbitrary smooth function subject to bounds on the variables, and both linear and nonlinear constraints, using a sequential quadratic programming method. (Note: the routine E04UCF as used in the present codes is dated 7 July 1989, but was apparently revised at MK 15B of the NAG Fortran Library in 1991.) The routine is based upon the work of Gill et al. [1986], and is essentially identical to the routine NPSOL that they describe.

[32] The variables in the present problem are the numbers of moles of each species in each phase. As noted in the previous section, linear constraints are applied to ensure that the total amount of each primary species in the system remains constant. The fact that the H2O(l) = H+(aq) + OH(aq) equilibrium is not yet included in the model restricts predictions of the partitioning of NH3 and the acid gases H2SO4, HNO3, HCl, and HBr. For systems in which an aqueous phase exists, the model is valid for H+(aq) molalities above about 10−5 mol kg−1 at 298.15 K, for which the water equilibrium has a negligible effect. This limit will vary with temperature as it is related to the equilibrium constant for the dissociation of liquid water.

[33] The treatment of gas phase NH3 in the models also requires special mention. The linear constraint for ammonia is expressed in terms of the total NH4+ present in the system rather than total NH3. Equilibria with the aqueous phase and, for example, solid phase NH4NO3(cr) are described by [Clegg et al., 1998a]

equation image
equation image

In both reactions the formation of one molecule of gas phase NH3 is accompanied by a corresponding increase in the total H+ in the system, in the first case as H+(aq) and in the second as a H atom in the HNO3 molecule. Consequently, in the calculation of the linear constraint for primary H+ in the system the number of moles of gas phase NH3 is subtracted to take this into account. Thus equation image is equal to −1.

[34] Partial differentials of equation (23) with respect to each variable nj are needed to determine the minimum of G. These are given below for each phase for the general case in which RH is not fixed:

equation image
equation image
equation image
equation image

In calculations for which the relative humidity is fixed, the partial differential with respect to the moles of liquid water becomes

equation image

The differentials for the other liquid phase species and the trace gases are unchanged. For solids which include water of hydration, equation (31c) is replaced by

equation image

In many calculations it will not be known initially whether the system, in its equilibrium state, includes a liquid phase. Thus both liquid water and ions must be included as variables even though they may be zero in the final result. However, both mole fractions (xH2O, xi) and activity coefficients equation image, become undefined where the amounts nH2O(l) and/or ni are zero. The lower bounds on these variables must therefore be set to small positive values in the minimization calculation, and a decision must be made at the end of the computation as to whether a liquid phase indeed exists. Similarly, positive lower bounds must be assigned for water vapor and trace gases that are being equilibrated, as the logarithms of the partial pressures appear in equation (23).

[35] We note that the setting of these positive lower bounds is complicated somewhat by the requirements of the minimization software. Further details are given in Appendix B.

[36] In systems where soluble gases such as NH3 and particularly H2SO4 are being equilibrated, the very low equilibrium partial pressures of these species (perhaps <10−20 atm for H2SO4(g)) means that the contribution to the total Gibbs energy in equation (23) is very small. It is also likely to be many orders of magnitude less than that of the most abundant species. There are a number of ways of ensuring the correct result for such a problem. For example, the Henry's law equilibrium between the gas and liquid aerosol could be specified as a nonlinear constraint. However, it is not necessarily known at the start of a calculation whether a liquid phase will exist in the system and therefore whether the constraint is valid. The same objection would apply if the equilibrium between the gas and a solid phase were specified. Alternatively, where the gas phase species is expected to be present in extremely small amounts compared to the quantity in other phases then the amount can simply be calculated from equilibrium with the reacting species present in the other phases. This would have a negligible effect on the overall mass balance in most cases. In order to reduce the need for such approximations all computations with the models are carried out in extended (quadruple) precision, which enables G to be calculated to a precision of greater than 30 digits (see section B2 of Appendix B).

[37] At the endpoint of each calculation nonzero amounts of liquid water and all ions will always be returned, due to the positive lower bounds on these quantities. Although the lower bounds on the solids are generally set to zero, a result may also include very small amounts (either positive or negative), rather than exactly zero, when the solids should not exist. This is because all results are correct only to some finite precision, and constraints and bounds are obeyed only to within the feasibility tolerance of the minimization software (see Appendix B). The existence of liquid and solid phases is therefore determined as follows. First, the fraction of the total amount of each primary species present in the liquid phase, and in each solid, is determined. For the liquid phase species, if any of these fractions exceed a set tolerance (1 × 10−4), then the liquid phase is deemed to exist, and all ion and water amounts are therefore accepted. For each solid a similar test is first applied (tolerance 1 × 10−5). If this condition is met but the maximum fraction of any primary species present in the solid is also less than a second lower tolerance (1 × 10−3), then the problem is run again, generally from the previous endpoint. If no liquid phase is found to exist (from the above test), then the calculation is also repeated, this time with all liquid phase species excluded from the variable set (which now contains only ns and ng). No tests are needed to determine the existence of any trace gas that is equilibrated, as the equilibrium amount will always be greater than zero.

4.2. Validating Results

[38] Each result is subjected to a number of tests after the output of the Gibbs energy minimization has been processed and the presence of liquid and solid phases has been determined. First, the result must conform to Gibbs' phase rule. This states that the number of degrees of freedom in a chemical system is equal to the number of chemical components (H2O, and the acids and salts) minus the number of phases present, plus two. An error is flagged for results in which the number of degrees of freedom, taking into account that both T and P are fixed, appears to be less than zero. A system can be invariant, and have zero degrees of freedom, although this is rare. Here a warning is given.

[39] A series of further tests are then carried out to check for equilibrium between the various liquid, solid, and gas phase species. The tests are formulated in terms of the equilibrium constants from which the ΔfGo values used in the minimization calculation were derived, and are summarized in Table 5. Tolerances for these tests are generally set to five parts in a thousand for saturation ratios (solids) and fractional differences between actual partial pressures and those calculated from liquid phase activities or from solids (for trace gas partial pressures or partial pressure products).

Table 5. Tests of the Validity of a Result
Species PresentTest
RHWater ActivityHenry's Law, Pressure ProductSaturation
  • a

    The water activity of the liquid phase is compared to the value of RH (aH2O = RH).

  • b

    If a liquid phase exists, then the partial pressure of water is compared with the water activity of the liquid phase (pH2O/po(H2O(l)) = aH2O).

  • c

    If a liquid phase exists, then the partial pressure of the trace gas is compared with the equilibrium value calculated from the ion activities and Henry's law constant (e.g., pHX = aH+aX/xKH, and for ammonia: pNH3 = aNH4+/(aH+xKH). See section 3.1 and Clegg et al. [1998a] for definitions of the Henry's law expressions. If one or more solids exist, but no liquid phase, it may be possible to check the product of the partial pressures of two trace gases (Kp) against the equilibrium value calculated for a solid. For example: Kp(NH4NO3(cr)) = pNH3pHNO3; Kp(H2SO4 · 4H2O(cr)) = pH2SO4pH2O4; Kp(NH4HSO4(cr)) = pH2SO4pNH3.

  • d

    The calculated equilibrium vapor pressure of water above ice (po(H2O(cr)) is compared to the value of RH (po(H2O(cr))/po(H2O(l)) = RH).

  • e

    If a liquid phase exists, then the saturation ratio S(H2O(cr)) is checked (S(H2O(cr)) should equal 1.0).

  • f

    If trace gases have been equilibrated, then it may be possible to check their partial pressure products against the value for the solid; see the second part of note c above.

  • g

    If a liquid phase exists, then the saturation ratio S of the solid is checked (S should equal 1.0).

Liquid phasexa   
Water vapor xb  
Trace gases  xc 
Icexd  xe
Other solids  xfxg

[40] The results of the tests are used in two ways. First, to determine whether the calculation should be repeated, up to a maximum of four times or until no errors or warnings are encountered. Second, any failures of the tests are indicated in the output of results to the user.

5. Practical Calculations

[41] The unique features of the present models are the inclusion of double salt and hydrate formation (see the last column of Table 1), and their validity to the very low temperatures encountered in the upper troposphere and stratosphere. In the sections below we show example calculations of phase equilibrium calculations at low temperature, and examine the possible significance of double salt formation in ambient aerosols in the Los Angeles basin.

5.1. Phase Equilibrium in the (NH4)2SO4/H2SO4/H2O System

[42] In a previous study of water vapor pressures and freezing properties of NH4HSO4 solutions [Yao et al., 1999] we presented a calculated phase diagram for the system NH4HSO4/H2SO4/H2O, and those model predictions have been shown to be consistent both with other measured water vapor pressures [Chelf and Martin, 1999] and melting temperatures of solid NH4HSO4/H2O [Koop et al., 1999]. In Figure 1 we show a calculated phase diagram for (NH4)2SO4/H2SO4/H2O as a function of relative humidity and the salt fraction equation image (equal to n(NH4)2SO4/(n(NH4)2SO4 + nH2SO4)) for temperatures up to 323.15 K. For compositions close to pure aqueous (NH4)2SO4 liquid solutions exist only for relative humidities greater than about 80%. As acid is added, thus moving from high to low equation image values on Figure 1, the liquid range greatly increases. Thus, for example, solutions close to pure aqueous H2SO4 in equilibrium with a relative humidity of about 55% remain as liquids to about 210 K.

Figure 1.

Calculated phase diagram of aqueous (NH4)2SO4-H2SO4-H2O, as a function of relative humidity (RH) and the salt fraction of (NH4)2SO4 (equation image, equal to n(NH4)2SO4/(n(NH4)2SO4 + nH2SO4)). The plot shows the temperatures (T/K) at which the solutions become saturated with respect to one or more of the following solids: (1) ice; (2) (NH4)2SO4(cr); (3) (NH4)3H(SO4)2(cr); (4) NH4HSO4(cr); (5) H2SO4 · 4H2O(cr); (6) H2SO4 · 6.5H2O(cr).

[43] Supersaturation of aqueous atmospheric aerosols with respect to solid phases is thought to be common, and the models have been parameterized using laboratory data for such systems [Clegg et al., 1998a, 1998b] where possible. For this reason, the codes on the Worldwide Web allow the formation of solid phases to be disabled in order to investigate the properties of supersaturated aerosols.

[44] The results plotted in Figure 1 are also shown as contours in Figure 2. This enables the temperatures at which saturation with respect to each of the six solid phases occurs to be read directly for any combination of relative humidity and salt fraction.

Figure 2.

Calculated phase diagram of aqueous (NH4)2SO4-H2SO4-H2O, as a function of relative humidity (RH) and the salt fraction equation image, shown as a contour plot. The temperature (°C) of each contour line is indicated on the plot, and the numbered solids are the same as those listed in the caption to Figure 1.

5.2. Importance of Double Salts

[45] Aerosols containing ammonium and both nitrate and sulphate have been observed often in the western United States [John et al., 1990] and western Europe [ten Brink et al., 1996], where they affect health and visibility. These particles are likely to become prevalent in more locations as the sulphur content in fuels continues to be reduced. This is because the lower concentrations of aerosol sulphate, which is formed as H2SO4, can be more completely neutralized by ammonia and at the higher pHs there will be less tendency for nitrate to be volatilized as HNO3. In most locations with ammonium nitrate/sulphate aerosols, the number of moles of sulphate is comparable to or exceeds that of nitrate. Thermodynamic data for the double salts, in this case (NH4)2SO4 · 2NH4NO3(cr) and (NH4)2SO4 · 3NH4NO3(cr), and their equilibria with liquid solutions are sparse, and their formation has not always been treated satisfactorily in models (e.g., MARS, EQUILIB, GFEMN, and ISORROPIA do not account for double salts [Saxena et al., 1986; Pilinis and Seinfeld, 1987; Ansari and Pandis, 1999; Nenes et al., 1998, 1999]).

[46] What is the practical effect of including ammonium sulphate/nitrate double salt formation in equilibrium aerosol models? Consider a particle at 290 K with an ammonium nitrate salt fraction equation image fraction of 0.5, and assume that crystallization occurs at the deliquescence point which is at about 76% relative humidity at that temperature (point a on Figure 3). If the relative humidity drops below this value, (NH4)2SO4(cr) will form thus increasing the salt fraction of NH4NO3 in solution and the composition of the aqueous phase will follow the 290 K line to the right from points a to b. At about 70% relative humidity (point b on Figure 3) the salt (NH4)2SO4 · 2NH4NO3(cr) will form, and any further reduction in relative humidity will cause the remaining water to evaporate leaving a particle composed of the two solids (NH4)2SO4(cr) and (NH4)2SO4 · 2NH4NO3(cr). In model calculations that do not include double salt formation the aqueous phase would be predicted to persist until a relative humidity of about 63% (point c on Figure 3). At that point, NH4NO3(cr) would form, and the particle become completely solid as mixture of (NH4)2SO4(cr) and NH4NO3(cr).

Figure 3.

Relative humidities (RH) at which aqueous NH4NO3/(NH4)2SO4 mixtures become saturated with respect to salts, as a function of the salt fraction of ammonium nitrate (equation image, equal to nNH4NO3/(nNH4NO3 + n(NH4)2SO4)). The equilibrium solid phases that form at each temperature as the relative humidity above the aqueous mixture is reduced are as follows: (1) NH4NO3(cr); (2) (NH4)2SO4(cr); (3) (NH4)2SO4 2NH4NO3(cr); (4) (NH4)2SO4 3NH4NO3(cr). The extensions of the curves for salts 1 and 2 below those for the two double salts indicate the predicted behavior when double salt formation is neglected.

[47] The formation of the double salt in atmospheric aerosols has two main implications. First, if it is neglected then the predicted mass of the particle will be incorrect. This would affect light scattering predictions, for example. One mole of NH4NO3 and one mole of (NH4)2SO4 have a total mass of 212 g. At 70% relative humidity and 290 K there is about 90 g of water associated with these electrolytes. If the double salts are permitted to form, this drops to zero just below 70%, whereas if they are not, solidification does not occur until about 63% relative humidity where there would be 50 g of aerosol water.

[48] Between 63 and 70% relative humidity the presence of water predicted in the absence of double salt formation would lead to a 20 to 30% overprediction in particle mass. At 270 K the overprediction occurs over a narrow relative humidity range of 75 to 77%, but since the relative humidity is higher at this temperature the error in mass is larger (over 30%). At 310 K the overprediction spans the range 61 to 53% relative humidity and leads to errors in particle mass of the order of 15 to 20% if double salt formation is neglected.

[49] Second, neglecting the formation of double salts also has implications for the predicted vapor-particle partitioning of NH3 and HNO3. The vapor pressure product of ammonium nitrate (for the reaction NH4NO3(cr) = NH3(g) + HNO3(g)) varies strongly with temperature. Thus we do not expect much particulate NH4NO3 at 310 K, where the product is over 500 ppb2 (7.4 × 10−16 atm2), under typical atmospheric conditions. Conversely, at 270 K, more ammonium and nitrate are likely to be in the particulate phase since the vapor pressure product is only 0.015 ppb2 (1.9 × 10−20 atm2) at this temperature. We have used the model to compare the equilibrium vapor pressure products of NH3 and HNO3 over ammonium sulphate/nitrate aerosols by calculating values of (pNH3 · pHNO3) at points corresponding to points b and c in Figure 3 for the temperatures 270 and 310 K. The results show that neglecting double salt formation can lead to overpredicting the vapor pressure product by up to 25 to 35%. This in turn would cause the predicted partitioning of ammonium nitrate to the particle phase to be too low in model calculations for gas/aerosol systems.

[50] How often are double salts likely to occur in real atmospheric systems? Sun and Wexler [1998a, 1998b] have simulated size distributions of particulate matter in the South Coast Air Basin of California (SoCAB) that encompasses Los Angeles. This region has the highest particulate matter loadings in the United States. The runs were performed for an episode in June 1987 where extensive measurements were available from the Southern California Air Quality Study (SCAQS). Aerosol models that do not include double salt formation will incorrectly predict the particulate mass if the following conditions are met: first, the aerosol composition is such that the ammonium sulphate/nitrate double salts can form, and second, the ambient relative humidity is sufficiently low that the aqueous aerosols become saturated with respect to one or both of the double salts but not so low that the particle would be solid anyway. (This corresponds to the relative humidity range between points b and c on Figure 3 for a temperature of 290 K.) All 2340 surface cells were scanned for modeled cases that fulfilled these criteria for 24 hours on 24 June 1987 for a total of 56,160 cell hours. Of these, only about 2500 cell hours had conditions where ammonium sulphate/nitrate double salts would be likely to form. The original runs segregated the particle sizes into eight sections. On average, two of the eight sections had suitable compositions.

[51] Sodium sulphate double salts can also occur under typical atmospheric conditions, and may be more important than those discussed above. Two sodium double salts are relevant to atmospheric aerosols: sodium ammonium sulphate tetrahydrate (Na2SO4 ·(NH4)2SO4 ·4H2O) and sodium sulphate nitrate dihydrate (Na2SO4 ·NaNO3 ·2H2O). At 298.15 K the sodium salts occupy more than half the phase plane in the NH4+-Na+-SO42−-NO3-H2O system (compared to about 5% for the ammonium nitrate/sulphate double salts), and more than equation image of the plane in NH4+-Na+-SO42−-Cl-H2O [see Potukuchi and Wexler, 1995a, Figures 1e and 2]. Note that in Los Angeles and other polluted coast cities, sulphuric and nitric acids displace chloride from sea salt particles as HCl(g). Consequently, sea salt particles, the source of most particulate sodium, often contain substantial sulphate and nitrate after a few hours of atmospheric processing [e.g., John et al., 1990]. The particles are also often near neutral pH because the volatile HCl readily evaporates, thus acting as a buffer.

[52] In order to examine the possible occurrence of sodium sulphate double salts, the same SoCAB runs were searched for cases where the mass of sodium not associated with sodium chloride was more than 10% of the total mass, and sulphate was also more than 10% of the mass. Of the 56,160 cell hours examined, 15,443 had at least one size bin that satisfied the criteria with on average over three of the eight sections satisfying the criteria. More than one third of these sections were between 2.5 and 10 μm in diameter because the sodium is found almost completely in the coarser modes, and two thirds of the sections were between 0.6 and 10 μm. Accumulation mode sections are important because they contain substantial quantities of sulphate in addition to the tail of the sea salt size distribution.

[53] What does the formation of the sodium sulphate double salts imply for predictions of atmospheric particulate matter? It is likely that vapor-particle partitioning will not be significantly altered by their presence or absence, as compounds in the double salt are essentially nonvolatile. However, the effect on the calculated water content of the particles may be important, because the double salts will cause the particle to be solid at higher relative humidities than would otherwise be the case. For instance, a particle containing equimolar amounts of Na+, NH4+, and SO42− is predicted to be aqueous to about 72% relative humidity at 298.15 K if the double salt is ignored, compared to only 83% if Na2SO4 ·(NH4)2SO4 ·4H2O(cr) forms. The corresponding relative humidity range for the salt NaNO3 ·Na2SO4 ·2H2O(cr) is about 73 to 81%.

6. Summary

[54] We have presented three comprehensive phase equilibrium models for atmospheric aerosols that are solved by minimization of the total Gibbs energy of the gas/aerosol system. The models are based on fundamental thermodynamic principles, and a large body of experimental data at subsaturated, saturated, and supersaturated conditions. We have also derived an expression for the Henry's law constant of sulphuric acid, based upon available data. The models, which are available for use on the Worldwide Web, contain no simplifying assumptions and incorporate all the known solid phases in the three systems treated. They are highly flexible and permit equilibrations to fixed relative humidities and the inclusion or exclusion of trace gases or individual solids as required. The latter capability enables the models to be used to calculate the behavior of the liquid aerosols, supersaturated with respect to solid salts, that are thought to exist at low relative humidities.

[55] One of the principal features of the present models, compared to other available codes, is their validity to very low temperatures for the systems H+-SO42−-NO3-Cl-Br-H2O and H+-NH4+-SO42−-NO3-H2O. We present the calculated equilibrium phase diagram for the system (NH4)2SO4/H2SO4/H2O for relative humidities as low as 10% and temperatures to 210 K.

[56] We have explored the importance of the double salts, which have been neglected in most thermodynamic models of inorganic atmospheric aerosols. The ammonium sulphate/nitrate salts (NH4)2SO4 · 2NH4NO3(cr) and (NH4)2SO4 · 3NH4NO3(cr) are likely to occur in locations with relatively low sulphur emissions where resulting particulate sulphuric acid can be largely neutralized by ambient ammonia. If the double salts form, they may have two influences on atmospheric aerosol predictions. First, they alter the vapor-particle partitioning of ammonia and nitric acid, and second, they reduce the aerosol water content relative to the case where only the individual (single) salts form. Because these effects occur over a limited range of relative humidity and composition, the inclusion of these double salts in equilibrium thermodynamic models of aerosols affects predictions in a relatively small number of cases during a typical simulation for Los Angeles.

[57] The sodium sulphate double salts (Na2SO4 · (NH4)2SO4 · 4H2O(cr) and NaNO3 · Na2SO4 · 2H2O(cr)) are likely to be more important. They occupy a much larger fraction of the phase diagram than the ammonium sulphate/nitrate salts and so form over a much wider range of compositions. The sodium sulphate double salts do not contain volatile components and therefore do not directly influence particle-vapor partitioning. However, their occurrence affects the predicted aerosol water content. The primary source of urban particulate sodium is sea salt, so the influence of these double salts is confined to coastal cities, such as Los Angeles, which experience onshore breezes.

Appendix A.

[58] We have installed our codes on the Worldwide Web (WWW). The URLs are and∼e770/aim.html. These sites accept aerosol composition data interactively from the user and calculate the phase composition (of both aerosol and gas phase), water content, and other thermodynamic properties of the aerosol system on-line. Problems can be solved for one or more individual cases, or for a range of values of a selected variable such as temperature or relative humidity. The results of such multiple calculations can be output in column form or graphed. The following types of calculation are available:

  1. Simple: A single electrolyte (i.e., a salt or acid), or a mixture of electrolytes, is equilibrated to a specified temperature and relative humidity. Chemical composition is entered as the number of moles of each ion. At the option of the user the formation of individual solids can be disabled. The model output consists of the solution composition at equilibrium, including the amount of liquid water present, and the amounts of any solids (salts and their hydrates) that have formed. Equilibrium partial pressures of volatile components (for example, NH3 and acids such as HNO3 and HCl, depending on the model) that would exist over the condensed phase are also calculated. A single problem is entered at a time.
  2. Comprehensive: This is similar to the “simple” case above, but with more facilities and options available to the user. Here the electrolyte/water system is considered to exist in 1 m3 of atmosphere. Again, it can be equilibrated to a fixed relative humidity, or to a specified total amount of water. In the latter case the equilibrium distribution of water between the vapor and condensed phases will be determined; thus the relative humidity is a quantity calculated by the model. Partitioning of NH3 and the acid gases into the vapor phase is also calculated, at the option of the user. As for the “simple” calculation, the formation of solids can be switched off individually, in order to investigate the properties of supersaturated solutions.
  3. Aqueous Solution: Here the model determines the equilibrium thermodynamic properties of an aqueous solution whose composition is specified in moles of each ion per kilogram of water. Ion and water activities are calculated, together with any solids formation. As for the other types of input, the formation of individual solids can be switched off if required. Equilibrium partial pressures of volatile components are also determined.
  4. Batch is the most flexible mode of use of the model, in which one or more problems can entered (one per line) in an html text box. All of the above types of problem, “simple,” “comprehensive,” and “aqueous solution,” can be entered. The options controlling each calculation are given as numerical codes. Results can be output in normal or column form, or as graphs.
  5. Parametric: This type of calculation allows the effect of the variation of a single quantity, such as temperature or some element of the ionic composition, for example, to be investigated. The flexibility of the input is similar to the “comprehensive” type. The user specifies the variation of the chosen model quantity as the maximum and minimum values of the range, and the number of points to be calculated. For example, if relative humidity were being varied with a range of 0.8 to 0.9, with three points to be calculated, then the model would determine the equilibrium properties of the system at relative humidities of 0.8, 0.85, and 0.9. The output of model results can be in normal or column form, or as graphs, at the option of the user. A “lapse rate” parametric option allows the effect of the adiabatic movement of an air parcel, with its consequent changes in temperature and relative humidity, to be examined.

Appendix B.: Bounds Setting

B 1. Bounds Setting

[59] In this section the setting of upper and lower bounds for the variables is described. The minimization routine E04UCF requires a “feasibility tolerance” that defines the maximum acceptable absolute violations of the bounds and linear constraints. We typically set the lower bounds on the amounts of ions, water, and trace gases to factors of 2× to 5× the feasibility tolerance, which is assigned a value of about 10× the machine precision.

[60] The fact that the liquid and gas phase species all have positive lower bounds reduces the amounts of each primary species that can be present in the product compounds containing it. For example, consider a system containing a number of ions and 1 mole of water. The water can occur in the liquid and gas phase, and as water of hydration in one or more solids. Let the lower bounds for liquid and gas phase water both be 1 × 10−10 moles, and the lower bounds for all solids be zero. The amount of free water available is equal to the total moles input less the sum of the amounts held by species on their lower bounds, that is (1.0 – 2 × 10−10) moles. The amount of water that can be held in any one phase is therefore equal to the amount held on the lower bound in that phase plus the amount of free water available. Generalizing, the amount of primary species j (nj(free)) that is available to be distributed among the species that contain it is given by

equation image

where nj(total) is the total moles of j present in the system, vj,k is the number of molecules or ions of primary species j in species k, and nk(bl) is the number of moles of species k on its lower bound (bl).

[61] The maximum amount of any species that can exist is limited by the amounts of the constituent primary species available, and the stoichiometry. For example, consider a system containing 1 mole of free H+, 1 mole of free NH4+, 1 mole of free SO42−, and 20 moles of free H2O. The maximum amount of solid H2SO4 · 4H2O(cr) that can be formed if all the other products involving these primary species are on their lower bounds is equal to the lower bound set for the solid itself plus 0.5 moles. This is because the limiting quantity is the free H+ which has a stoichiometric factor of 2 for the solid. More generally, for any species k,

equation image

where nk(bu) and nk(bl) are the upper and lower bounds, respectively, of species k. Quantities nj1(free), nj2(free), etc. are the amounts of each primary species j present in k, and vj,k are the stoichiometric factors as before. We note that H+(aq) is not limiting to the formation of gas phase NH3 as it is produced by the dissociation of primary species NH4+ (equation (29)).

[62] Although the lower bounds for aqueous H2O and the ions are set greater than zero by an amount that exceeds the feasibility tolerance, the minimization software occasionally assigns an amount (nH2O(l), ni) at or below zero. This is accommodated by using a Taylor expansion in ln(n) about a value z (which is close to zero), and replacing all n for which n < z by the exponent of the expansion of ln(n) about z. This exponent is always positive, of course. The critical value z is assigned a value of 1.1 × feasibility tolerance, and is thus less than the lower bounds on the liquid water and ions by a factor of about 2. The expansion is given by

equation image
equation image

B.2. Program Notes

[63] Running the model in extended precision requires recompilation of the NAG library routines, and changes to the following subroutines which contain machine- and precision-dependent code: X02ajft.f, X02bjft.f, and X02aayt.f. The following settings of optional parameters for E04UCF are used: feasibility tolerance = 1 × 10−28, function precision = 1 × 10−28, optimality tolerance = 1 × 10−28, crash tolerance = 0.0, linesearch tolerance = 0.5, step limit = 0.02, big bound = 1000.

Appendix C.

[64] The solubility of HBr in aqueous H2SO4 at stratospheric temperatures has been studied by several groups [see Massucci et al., 1999, Table 6] including, most recently, Kleffmann et al. [2000]. It has been pointed out by L. Williams (personal communication, 2001) that although the measured effective Henry's law constants (H*) of HBr are fitted well by the model, the predicted trend in H* with HBr molality in the liquid phase is the opposite of that observed for some H2SO4 concentrations. This can be significant when estimating values of H* for the trace concentrations of HBr that apply in atmospheric problems.

[65] We have therefore systematically reevaluated the available data for HBr solubility in aqueous H2SO4 at low temperature and refitted the model in order to correct the error. There are changes to three activity coefficient model parameters and the variation of the Henry's law constant of HBr with temperature. The new values are given below:

equation image
equation image
equation image
equation image

Both the atmospheric relevance of the HBr-H2SO4-H2O system, and the available data, are confined to temperatures ≤298.15 K. When the model is used for calculations at higher temperatures, the term in (T − 298.15) for equation image in equation (C4) should be divided by 2, to give a factor of −0.244375.


[66] This work was supported by NATO Collaborative Research grant CRG 960160, and grants from the IBM Environmental Research Program (to A.S.W.) and the Natural Environment Research Council of the United Kingdom (Advanced Fellowship GT5/93/AAPS/2 for S.L.C.).