9.2.1 Size Distribution of Atmospheric Particles

The dynamics of particles covers by definition the processes that affect directly the size distribution of a population of aerosols, i.e., a population of particles in suspension in a gas. Therefore, emission and deposition processes are not included in aerosol dynamics; they are addressed separately in Chapters 2 and 11, respectively.

The size distribution of a population of particles may be represented in several ways. For example, the concentration of particles may be represented in terms of their number, N_p, their surface, S_p, their volume, V_p, or their mass, M_p. These different variables are interrelated via the Stokes diameter and the particle density. For example, for a population of particles with a same size, d_p,St (monodispersed population), and density, ρ_p:

Mp=Vp ρp=Np πdp,St36ρp(9.2)

However, atmospheric particles cover a large range of sizes. Therefore, the particle concentration must be represented by a concentration distribution as a function of size. Such distributions are typically presented as a function of the particle volume or diameter, and the particles are assumed to be spherical. It is a reasonable assumption for aged particles, which have undergone condensation of semi-volatile chemical species. It is, however, a poor assumption for freshly emitted primary particles, which may have fractal shapes. For the sake of simplicity, one uses here d_p rather than d_p,St to represent the particle diameter. The two particle size distributions that are the most commonly used are the number distribution and the mass distribution. The mass distribution is equivalent to the volume distribution if the particle density is uniform for all particles (then, ρ_p is the factor converting volume to mass). Let n_p(v_p) be the distribution of the number concentration of particles as a function of their volume, v_p, and n_p,d(d_p) be the distribution of the number concentration of particles as a function of their diameter d_p. The main equations for those number concentration distributions are as follows:

Np=∫0∞np(vp) dvp=∫0∞np,d(dp) ddp;  np,d(dp)=π2 dp2np(vp) Vp=∫0∞np(vp) vp dvp=∫0∞np,d(dp)π6dp3ddp Mp=∫0∞np(vp) vp ρp(vp) dvp=∫0∞np,d(dp)π6dp3 ρp(dp) ddp (9.3)

Here, N_p, V_p, and M_p represent the total concentration in terms of number, volume, and mass, respectively, of the polydispersed particle population. Typically, one distinguishes three modes in the particle size distribution (Whitby, 1978):

1. 
   

   – A nucleation mode

   
   
2. 
   

   – An accumulation mode

   
   
3. 
   

   – A coarse mode

The nucleation mode corresponds to particles that have been formed from gaseous molecules and have later grown via the condensation of other gaseous molecules and coagulation with other nucleated particles. This mode is located within the ultrafine fraction of PM.

The accumulation mode results from the emissions of fine particles and from dynamic processes such as condensation and coagulation. It is called the accumulation mode because these dynamic processes lead to the accumulation of particles in that size range (see Sections 9.2.3 and 9.2.4).

The coarse mode consists mostly of particles emitted via mechanical processes (abrasion, wind erosion, etc.). Condensation and coagulation have little effect on these particles, because their number concentration is low (although their mass concentration can be significant) and particle dynamics favors fine particles, which are typically present in greater numbers and have greater available surface area for condensation and coagulation processes to occur.

Figure 9.1 shows idealized volume and number concentration distributions of a typical urban particle population. Lognormal distributions are used here for the volume and number concentrations (see Section 9.7 for a description of the lognormal representation of particle concentration distributions). It appears that the volume (and, therefore, the mass) of the particles is present mostly in the accumulation and coarse modes and that the number concentration is dominated by the ultrafine particles of the nucleation mode.

Figure 9.1. Schematic representation of the size distribution of the volume concentration (top) and number concentration (bottom) of atmospheric particles typical of a polluted urban area.

Source of the data: Whitby (1978).

One should note that this classification into three modes is too simplistic. For example, several accumulation modes can be found in the atmosphere, because an accumulation mode resulting from the condensation of gaseous pollutants will have a different size distribution than an accumulation mode resulting from the evaporation of cloud and fog droplets where aqueous-phase chemical reactions have taken place (Hering and Friedlander, 1982).

Three processes are considered to govern particle dynamics and affect the size distribution of a population of particles:

1. 
   

   – Nucleation

   
   
2. 
   

   – Condensation (and the reverse process, which is evaporation)

   
   
3. 
   

   – Coagulation

The mathematical representation of these three processes is called the general dynamic equation, GDE (Friedlander, 2000). The different terms of this equation are presented in detail below.

9.2.2 Nucleation

Nucleation is the formation of a new particle from gaseous molecules. This nucleation process may involve two or three different chemical species. Nucleation involving two species (for example, sulfuric acid and water) is referred to as binary nucleation. Nucleation involving three species (for example, sulfuric acid, ammonia, and water) is referred to as ternary nucleation. Nucleation may occur spontaneously in the gas phase; then, one refers to homogeneous nucleation. In some cases, nucleation may be favored by the interaction of gaseous chemical species with a surface; then, one refers to heterogeneous nucleation. Nucleation rates are difficult to estimate theoretically and uncertainties may be of several orders of magnitude. The most robust algorithms have been developed for the homogeneous binary nucleation of sulfuric acid and water. A detailed review of nucleation algorithms has been conducted by Zhang et al. (2010), who recommended the algorithm of Kuang et al. (2008) for the nucleation rate, J_n (particles cm⁻³ s⁻¹), of sulfuric acid particles:

Jn=1.6×10−14 (NH2SO4)2 (9.4)

where N_H2SO4 is the number of gaseous sulfuric acid molecules (or other molecules containing sulfate, such as ammonium bisulfate) (molec cm⁻³).

The nucleation process is actually in competition with condensation (described in Section 9.2.3), which transfers gaseous molecules toward existing particles. If the particle concentration is important, then condensation tends to dominate. Therefore, nucleation is mostly observed in the atmosphere when the concentration of gaseous molecules of a chemical species with a low saturation vapor pressure is high and when the concentration of existing particles is low or moderate. Thus, one observes nucleation of new particles near emission sources (e.g., near vehicle exhaust) or under atmospheric conditions where the formation of a chemical species with a low saturation vapor pressure is important. This is the case, for example, in forested areas where the atmosphere is pristine (i.e., with low particle concentrations) and where emissions of biogenic VOC are important. These biogenic VOC can be oxidized rapidly to form compounds with low volatility, which may then nucleate. In addition, episodes of sulfate particle nucleation have been observed under conditions where the oxidation of sulfur dioxide (SO₂) is rapid and important.

Nucleation increases the number of particles, as well as the total mass of particles. However, the particle mass created by nucleation is typically very low compared to the mass of existing fine and coarse particles.

9.2.3 Condensation and Evaporation

Condensation consists of the transfer of gaseous molecules toward an existing particle. This particle can be liquid or solid and the new condensed mass can also be liquid or solid. The reverse process is evaporation; it consists of the transfer of molecules from the particle toward the gas phase. Condensation (or evaporation) occurs when the gas phase and the condensed phase are not in thermodynamic equilibrium and transfer between these phases must occur to establish this thermodynamic equilibrium. The two phases may be out of equilibrium, for example, because of a change of concentration in the gas phase (formation of a semi-volatile or non-volatile species) or because of a change of temperature, pressure or relative humidity.

The evolution of the distribution of the particle number concentration as a function of volume, n_p(v_p), where v_p is the particle volume, is given by the following equation:

∂np(vp)∂t=−∂Ivnp(vp)∂vpIv=dvpdt(9.5)

where I_v is called the growth law. It represents the rate of growth by condensation of a particle of volume v_p. This growth law corresponds to the flux of molecules from the gas phase to the particle and depends on (1) the mass transfer flux by diffusion of the molecules from the gas phase toward the particle surface and (2) the particle surface area available for condensation of these gaseous molecules. For spherical coarse particles, continuum mechanics implies that the mass transfer flux is proportional to the molecular diffusion coefficient and inversely proportional to a characteristic distance, which is the particle radius. Thus, the growth law is given as the product of three terms, which are the particle surface area, the mass transfer coefficient (diffusion coefficient/particle radius), and the condensing species concentration difference between the bulk gas phase and the particle surface (expressed in terms of molecular volume per unit volume of air):

dvpdt=π dp2 2Dmdp vm(Cg−Cg,e)=2πdp Dm vm(Cg−Cg,e)(9.6)

where v_p (cm³) is the particle volume, d_p is the particle diameter (cm), D_m is the diffusion coefficient of the condensing molecules in the air (cm² s⁻¹), v_m is the molecular volume of the condensing molecule (cm³ molec⁻¹), C_g is the gas-phase concentration of the condensing molecules (expressed as number of molecules per unit volume of air; here, molec cm⁻³), and C_g,e is the gas-phase concentration of the condensing molecules in thermodynamic equilibrium at the particle surface (molec cm⁻³). The molecular volume is obtained by dividing the molecular mass of the condensing species by its density in the condensed phase. Since the particle surface area is proportional to the square of the particle diameter and the mass transfer flux is inversely proportional to the particle diameter, the growth law is proportional to the particle diameter.

For particles that have a diameter significantly less than the mean free path in the air (which is about 70 nm at an atmospheric pressure of 1 atm), particle dynamics can no longer be represented by continuum mechanics, but must be treated as a free molecular flow. As a result, the mass flux of molecules condensing on a particle is governed by the kinetic theory of gases and does not depend on the particle size. Therefore, the particle diameter only appears in the particle surface area available for condensation, which is proportional to the square of the particle diameter:

dvpdt=π4 dp2(8kB Tπ mg )12 vm(Cg−Cg,e)(9.7)

where m_g is the mass of the condensing gaseous molecule (g), k_B is the Boltzmann constant (1.381 × 10⁻²³ J K⁻¹ or, here, 1.381 × 10⁻¹⁶ erg K⁻¹) and T is the temperature (K). The term in parentheses to the power 1/2 is the mean thermal velocity of the condensing molecule, ct¯ .

The Fuchs-Sutugin equation (Fuchs and Sutugin, 1971) is commonly used to calculate the growth law over the full range of particle sizes (there are other similar formulas to represent the growth law):

dvpdt=2π dp Dm1+(1.33Kn+0.711+Kn)Knvm(Cg−Cg,eexp(4σpvmdp kB T))(9.8)

where Kn is the Knudsen number, which is equal to (2 λ_m,a/d_p), λ_m,a is the mean free path of a molecule in the air (cm), and σ_p is the surface tension of the particle (erg cm⁻²). The term in the denominator (which is a function of Kn) accounts for the transition between the molecular regime for condensation on ultrafine particles and the continuum regime for condensation on coarse particles. Condensation on fine particles (i.e., d_p between about 0.1 and 2.5 μm) occurs in this intermediate regime. For ultrafine particles, Kn becomes important and the growth law becomes proportional to the square of the particle diameter. For coarse particles, Kn tends toward 0 and the growth law becomes proportional to the particle diameter.

The exponential term represents the Kelvin effect, which results in the possible evaporation of molecules from some ultrafine particles because of their strong curvature. For a sphere, the concentration above its surface will be greater than the thermodynamic equilibrium concentration obtained over a flat surface. Conceptually, this phenomenon is due to the fact that the curvature of the spherical surface implies that a molecule in the particle will be farther statistically from molecules in the gas phase, thereby allowing more space for molecules in the gas phase near the particle surface. This phenomenon depends on the surface tension of the particle. It can be significant: for example, it leads to a 10 % increase in the saturation vapor pressure of water over a 10 nm water droplet. It implies that condensation may not take place on the smallest ultrafine particles and that instead these particles may undergo evaporation of some species present in solution.

Figure 9.2 shows the growth law for the condensation of sulfuric acid, H₂SO₄, immediately neutralized into ammonium sulfate, (NH₄)₂SO₄. One notes that the growth law increases with the particle diameter, but that the slope decreases as the diameter increases, since the growth law is proportional to d_p² for ultrafine particles and proportional to d_p for coarse particles. However, in air pollution, one is interested in the overall condensation rate, i.e., the growth of a particle population due to condensation of gaseous molecules on all those particles. Figure 9.2 also shows the overall condensation rate, i.e., the growth law for a single particle of a given size multiplied by the number of particles of that size, for a particle population typical of a polluted urban area (see Figure 9.1). This condensation rate is maximum here for particles with diameters ranging from about 0.03 to 0.3 μm, which corresponds mostly to the accumulation mode. We will see in Section 9.2.4 that coagulation leads to a similar result, which explains why atmospheric particles accumulate in this size range.

Figure 9.2. Condensation. Top figure: growth law due to gas/particle conversion by condensation of 1 ppb of sulfuric acid (immediately neutralized as ammonium sulfate) as a function of the particle diameter; bottom figure: distribution of the condensation rate of 1 ppb of sulfuric acid (immediately neutralized as ammonium sulfate) as a function of particle size for a particle population typical of a polluted urban area (corresponding here to that of Figure 9.1).

Figure 9.3 depicts the Kelvin effect for an organic compound, nonadecane (an alkane with 19 carbon atoms), which is present in diesel engine exhaust. The saturation vapor pressure of nonadecane above a flat surface is 6.1 × 10⁻⁹ atm, i.e., 6.1 ppb at P = 1 atm. One investigates here the case where its gas-phase concentration is twice its saturation vapor pressure, i.e., 12.2 ppb. One notes that the Kelvin effect increases the saturation vapor pressure (or concentration) significantly below 0.025 μm. Thus, evaporation of nonadecane occurs for particles with a diameter less than about 0.025 μm. These particles shrink, whereas particles with a diameter greater than about 0.025 μm grow by nonadecane condensation. This process may occur in the exhaust of a diesel vehicle, where alkanes are initially very concentrated in the gas phase and condense on soot particles until the exhaust dilution leads to lower gas-phase concentrations, which may then become less than the saturation vapor pressure for the ultrafine particles. The decrease of the diameter of these ultrafine particles will favor their transfer toward the accumulation mode via coagulation (see Section 9.2.4).

Figure 9.3. Kelvin effect. Comparison of the gas-phase concentration and saturation vapor concentration of nonadecane as a function of particle diameter. The gas-phase concentration was taken equal to twice the saturation vapor concentration over a flat surface.

Source: Adapted from Devilliers et al. (2013); reprinted with authorization, © 2012 Elsevier Ltd.

Condensation increases the particle mass, but does not change the number of particles. Conversely, evaporation decreases the particle mass, but does not change the number of particles.

9.2.4 Coagulation

Coagulation corresponds to the collision of two particles and the formation of a single particle from the two original particles. One typically assumes that the collision of two particles results in the formation of a single particle and that, therefore, every collision leads to coagulation. Although this is likely to be the case for liquid or liquid-coated particles, it may not always be the case for solid particles. For the sake of simplicity, one will assume here that every collision leads to coagulation. The coagulation rate is minimum for two particles of a same size and increases as the difference between the sizes of the two coagulating particles increases. Therefore, freshly nucleated ultrafine particles will coagulate more readily with a fine or coarse particle than with an ultrafine particle. However, the number concentration of particles must be taken into account as well and, if the number of ultrafine particles is very high compared to those of fine and coarse particles, then coagulation among ultrafine particles becomes important. The change as a function of time of the distribution of the particle number concentration as a function of the particle volume, n_p(v_p), consists of two terms: one represents the increase of the number of particles of volume v_p due to coagulation of particles of smaller volumes (i.e., a particle of volume v_p’ coagulating with a particle of volume (v_p – v_p’) to create a particle of volume v_p) and the other represents the decrease of the number of particles of volume v_p due to coagulation of those particles with other particles. This overall change is given by the following equation, where the terms β(v_p, v_p’) are the coagulation coefficients between a particle of volume v_p and a particle of volume v_p’:

dnp(vp)dt=12∫0vpβ(vp‘,vp−vp‘)np(vp‘)np(vp−vp‘)dvp‘−∫0∞β(vp,vp‘)np(vp)np(vp‘)dvp‘(9.9)

As it was done for the growth law representing the transfer of gaseous molecules toward a particle, the collision between two particles can be addressed by considering the extreme cases of the continuous regime for coarse particles and the kinetic theory of gases for ultrafine particles in the free molecular regime. Between these two extremes, an intermediate regime must be parameterized for the size range that corresponds approximately to fine particles. For coarse particles, the coefficient of coagulation between two particles, β(v_p,i,v_p,j) (expressed here in units of cm³ per particle per second), is given by the following equation:

where, for particle i, v_p,i is the particle volume (cm³), D_p,i is the brownian diffusion coefficient in the air (cm² s⁻¹), and d_p,i is the particle diameter (cm). The coagulation coefficient, β(v_p,i,v_p,j), is minimum when d_p,i = d_p,j and increases when the difference in size of the two particles increases.

For particles of diameter less than the mean free path in the air, the kinetic theory of gases applies:

β(vp,i,vp,j)=π4(c¯t,i2+c¯t,j2)1/2(dp,i+dp,j)2 (9.11)

In the intermediate regime (which concerns fine particles), the Fuchs equation is commonly used (Fuchs, 1964):

β(vp,i,vp,j)=2π (Dp,i+Dp,j)(dp,i+dp,j)[dp,i+dp,jdp,i+dp,j+2(gi2+gj2)1/2+8(Dp,i+Dp,j)(c¯t,i2+c¯t,j2)1/2(dp,i+dp,j)]Dp,i=kB T cc3πμv,a dp,i;cc=1+2 λm,adp,i(1.257+0.400 exp(−0.55 dp,iλm,a))c¯t,i=1.6(kBTmp,i)12;gi=0.47dp,i λm,i[(dp,i+λm,i)3−(dp,i2+λm,i2)32]−dp,i;λm,i=8 Dp,iπ c¯t,i(9.12)

where c_c is the Cunningham coefficient and, for particle i, v_p,i is the volume (cm³), D_p,i is the brownian diffusion coefficient in the air (cm² s⁻¹), d_p,i is the diameter (cm), c¯t,i is the mean thermal velocity (cm s⁻¹), m_p,i is the particle mass, and g_i is a coefficient (cm) that is a function of the particle diameter i and its mean free path in the air λ_m,i. This formula tends toward the solution of the continuous regime when one of the diameters becomes large and toward the kinetic theory of gases when both diameters become less than the mean free path. This brownian coagulation coefficient is shown in Figure 9.4 for particles of diameters 0.001, 0.01, 0.1, 1, 10, and 100 μm coagulating with particles with diameters ranging from 0.001 to 100 μm in diameter. For a particle of a given diameter, the coagulation coefficient is minimum for coagulation with a particle of the same diameter.

Figure 9.4. Brownian coagulation. Top figure: brownian coagulation coefficient between particles of diameters d_p,i and d_p,j; bottom figure: distribution of the coagulation rate of particle j coagulating with particles i of greater size using a particle population typical of a polluted urban area corresponding to that of Figure 9.1.

This coagulation coefficient includes the assumption that all collisions lead to coagulation of the two particles to form a single particle. If that is not the case, one must introduce a correction term to account for the fact that the coagulation probability is less than 1. In addition, this equation corresponds to the brownian motion of particles. For ultrafine particles, the van der Waals forces are no longer negligible and increase the coagulation rate. For electrically charged particles, one must account for the Coulomb forces. Equations are available to take into account these processes if they are relevant (Friedlander, 2000).

The coagulation rate between particles depends not only on the value of the coagulation coefficient, but also on their number concentrations. Therefore, coagulation is more important for ultrafine particles since they are typically present in greater number than fine and coarse particles (see Figure 9.1). For fine particles, coagulation will be important only near sources, because in the ambient background, their number concentrations are typically too low to lead to significant coagulation rates and coagulation can then be neglected. Figure 9.4 shows the distribution of the coagulation rate of a particle of a given diameter (0.001, 0.01, 0.1, 1, and 10 μm) coagulating with particles of a greater diameter using a particle population typical of a polluted urban area. This coagulation rate is equivalent to a pseudo-first-order coagulation rate coefficient, which is defined as the product of the coagulation coefficient and number concentration of the other particles involved in the coagulation process. The particle number concentrations presented in Figure 9.1 were used here. One notes that those coagulation rates are greater for ultrafine particles and that those particles coagulate preferentially with fine particles of diameters ranging from about 0.05 to 0.5 μm. The coagulation rates become negligible for coagulation with particles of diameter greater than 2.5 μm because their number concentrations are very low. Consequently, coagulation increases particulate mass in the accumulation mode (fine particles) and has almost no effect on coarse particles. As for condensation, particles tend to accumulate by coagulation in the size range corresponding to the accumulation mode. We will see in Chapter 11 that atmospheric deposition is less efficient for particles in that size range. Therefore, fine particles, i.e., those in the accumulation mode, have an atmospheric lifetime that is longer than those of ultrafine and coarse particles.

Coagulation decreases the number of particles, but does not affect their total mass.

9.3.1 Saturation Vapor Pressure

The saturation vapor pressure, P_s,i, is the pressure that a gas cannot exceed. If the partial pressure, P_i, of a gaseous species exceeds its saturation vapor pressure, then nucleation and/or condensation occur to decrease the partial pressure and bring the species back to thermodynamic equilibrium. One must note, however, that condensation on liquid particles may occur in cases where the partial pressure is less than the saturation vapor pressure (see Sections 9.3.2 and 9.3.3).

9.3.2 Henry’s Law

Henry’s law applies to dilute aqueous solutions. It relates the concentration of a chemical species in the gas phase (represented by its partial pressure, P_i) to its activity in the aqueous phase as follows:

where C_i is the concentration in the particle in moles per liter (M), γ_i is the activity coefficient of the species in the aqueous phase (γ_i C_i is the activity of the species), and H_i is the Henry’s law constant (which depends on temperature). The activity coefficient reflects the fact that the solution is not ideal and that interactions among species in solution (molecules and ions) affect their ability to be present in solution. By definition, the activity coefficients are equal to 1 in an ideal solution. If interactions with other species present in solution are repulsive, the species will be less present in solution than if the solution were ideal (i.e., without any interactions with other species); on the other hand, if the interactions are attractions, then the species will be more present in solution than if the solution were ideal. An activity coefficient greater than 1 corresponds to the case where the interactions are repulsive and an activity coefficient less than 1 corresponds to the case where interactions correspond to attractions. A dilute solution tends toward ideality; then, the activity coefficient tends toward 1, and the activity tends toward the concentration (Henry’s law was originally formulated for an ideal solution). The partial pressure is generally expressed in atm; then, the Henry’s law constant is expressed in M atm⁻¹.

9.3.3 Raoult’s Law

Raoult’s law applies to concentrated solutions. It relates the concentration of a chemical species in the gas phase (represented by its partial pressure, P_i) to its activity in the liquid phase as follows:

where x_i is the molar fraction of the chemical species in solution. If the solution is ideal, then the molar fraction of the species in solution is equal to the ratio of its partial pressure and its saturation vapor pressure. For a pure solution of the species, its molar fraction is equal to 1 and one obtains the definition of the saturation vapor pressure as being the maximum value of the partial pressure.

9.4.1 Sulfate and Ammonium

Sulfuric acid has a very low saturation vapor pressure and is not likely to remain in the gas phase. Therefore, it undergoes nucleation (either binary with water or ternary with water and ammonia) or condensation on existing particles. In the aqueous phase, sulfuric acid dissociates to form bisulfate (HSO₄⁻) and sulfate (SO₄^2-) ions:

H2SO4 ↔ H++HSO4−(R9.1)

HSO4− ↔ H++SO42−(R9.2)

H₂SO₄ is a strong acid and the equilibria are strongly displaced toward the formation of the sulfate ions. H₂SO₄ and its ions are partially or totally neutralized by ammonia (a base) present in the atmosphere. Depending on the ammonia and sulfate concentrations present in the atmosphere, there may be formation of ammonium bisulfate (NH₄HSO₄), ammonium sulfate ((NH₄)₂SO₄) or letovicite:

Ammonium bisulfate is formed when there are as many moles of ammonia as there are moles of sulfate. Ammonium sulfate is formed when there are at least two moles of ammonia available for each mole of sulfate (sulfuric acid being a diacid, it takes two moles of ammonia to neutralize one mole of sulfuric acid totally). Letovicite corresponds to the intermediate case when there are 1.5 moles of ammonia available per mole of sulfate and its chemical formula is (NH₄)₃H(SO₄)₂.

9.4.2 Nitrate and Ammonium

Nitric acid has a rather high saturation vapor pressure and, therefore, it remains preferentially in the gas phase. However, it can react with ammonia to form a semi-volatile species: ammonium nitrate (NH₄NO₃):

This species is subject to the following thermodynamic equilibrium:

Keq=[HNO3(g)][NH3(g)][NH4NO3(p)](9.15)

where the notations (g) and (p) indicate a gas-phase and particulate-phase species, respectively. In the case of a solid phase, the ammonium nitrate concentration is taken to be 1 by definition and we have the following equilibrium relationship:

The first comparison of this theoretical equilibrium relationship with experimental data in the Los Angeles basin, California, showed satisfactory results (Stelson et al., 1979). In the case of an aqueous phase, ammonium nitrate dissociates into an ammonium cation and a nitrate anion:

NH4NO3 ↔ NH4++NO3−(R9.6)

Then, the activities of the nitrate and ammonium ions must be taken into account and the equilibrium relationship is as follows:

Keq=[HNO3(g)][NH3(g)]γNH4+ [NH4+(aq)] γNO3−[NO3−(aq)] (9.17)

where the notation (aq) indicates an aqueous phase (this notation is generally ignored for ions, because they are typically present in the aqueous phase in the lower atmosphere). If the product of the concentrations of nitric acid and ammonia is low, then there is no formation of solid particulate ammonium nitrate (at low humidity) and negligible formation of ammonium nitrate in solution (at high humidity). A small fraction of nitric acid may dissolve in aqueous particles (e.g., in sulfate particles), but the liquid water content of particles is very low (<1 mg m⁻³) and the fraction of nitrate formed via dissolution in particles is negligible.

The equilibrium constant of ammonium nitrate decreases when the temperature decreases and when relative humidity increases (e.g., Stelson and Seinfeld, 1982, Mozurkewich, 1993). Therefore, cold and humid atmospheric conditions (for example, in wintertime) are conducive to the formation of particles containing ammonium nitrate. However, solar radiation is limited during winter. Therefore, the formation of HNO₃, which requires OH radicals or ozone, both of which are produced by photolysis, is low then. Thus, it is typically during springtime or in fall that the concentrations of ammonium nitrate will be highest, because there is sufficient production of HNO₃ and a moderate temperature favors the formation of particulate ammonium nitrate.

Ammonia will preferentially neutralize sulfuric acid before reacting with nitric acid. Therefore, the formation of ammonium nitrate will occur only if (1) there is sufficient ammonia to neutralize sulfuric acid first and (2) the product of the nitric acid and remaining ammonia concentrations is sufficiently high for ammonium nitrate formation.

Nitrate may occasionally be found in coarse particles, because nitric acid can react with sea salt and alkaline soil particles. The thermodynamics is favorable to the formation of nitrate salts and the following reactions may occur:

Sea salt:HNO3(g) + NaCl(p) → NaNO3(p) + HCl(g)(R9.7)

Soil particles:2 HNO3(g)+CaCO3(p) → Ca(NO3)2(p)+CO2(g)+H2O(g)(R9.8)

Similar reactions are possible with gaseous sulfuric acid. However, given its very low volatility, sulfuric acid tends to be in the particulate phase directly via nucleation or condensation, rather than via chemical reactions. Note that sea salt contains a fraction of primary sulfate. Sea salt and soil particle emissions are addressed in Chapter 11.

9.4.3 Deliquescence

When humidity increases, a solid salt (ammonium sulfate, ammonium nitrate …) will become liquid at a given humidity. This humidity is called the humidity of deliquescence. It depends on temperature. At 25 °C for example, it is 40 % for ammonium bisulfate, 62 % for ammonium nitrate, and 80 % for ammonium sulfate.

When the relative humidity decreases, a salt present in solution will become solid at a given humidity. This humidity of crystallization is less than the humidity of deliquescence and the solution is metastable between these two values of the relative humidity, being in a state of supersaturation for this salt. This phenomenon is called hysteresis.

Thermodynamic data pertaining to the inorganic equilibria of atmospheric aerosols are available in several books on atmospheric chemistry (e.g., Finlayson-Pitts and Pitts, 2000; Jacobson, 2005a; Seinfeld and Pandis, 2016).