10.2.1 Mass Transfer of Chemical Species between the Gas Phase and the Liquid Phase

The mass transfer of a chemical species in the gas phase toward the liquid phase can be seen as a sequence of three steps:

1. 
   

   – The mass transfer of the gas-phase molecule toward the droplet surface by molecular diffusion

   
   
2. 
   

   – The thermodynamic equilibrium between the gas-phase and aqueous-phase concentrations at the droplet surface

   
   
3. 
   

   – The mass transfer of the dissolved molecule in the aqueous phase from the droplet surface toward the inner droplet by molecular diffusion

These processes occur in the reverse direction in the case of a chemical species that volatilizes from the aqueous phase.

In most models of cloud chemistry, the system is assumed to be at equilibrium and the two mass transfer steps are neglected. Nevertheless, it is important to take them into account for the absorption of pollutants by falling raindrops (precipitation scavenging) as well as for heterogeneous reactions, which occur at the droplet surface.

10.2.2 Henry’s Law

Thermodynamic equilibrium at the surface of a droplet is governed by Henry’s law. Henry’s law applies to dilute aqueous solutions. It relates the concentration of a chemical species in the gas phase (characterized by its partial pressure, P_i) to its activity in the aqueous phase as follows:

where C_i is the concentration in the droplet in M (moles per liter), γ_i is the activity coefficient of the chemical species in the aqueous phase, and H_i is the Henry’s law constant, which depends on temperature. Partial pressure is generally expressed in atm and the Henry’s law constant is, therefore, expressed in M atm⁻¹.

If the solution is very dilute, the solution can be assumed to be ideal and the activity coefficients become 1 (i.e., the chemical species concentrations are equal to their activities). As a matter of fact, the initial formulation of Henry’s law used the species concentration, rather than its activity. This assumption is generally appropriate for clouds that have a large liquid water content. However, it may not be applicable to fogs, because they may contain high aqueous-phase concentrations of pollutants, particularly during their formation and evaporation.

The Henry’s law constants of selected atmospheric chemical species are provided in Table 10.1.

10.2.3 Ionic Dissociations

In the aqueous phase, some chemical species dissociate into cations (with a positive charge) and anions (with a negative charge). This ionic dissociation leads to a displacement of the gas/droplet equilibrium if one considers the total aqueous-phase concentration of the chemical species, i.e., the sum of the concentrations of the non-dissociated species and corresponding ions. For example, for a diacid, such as sulfuric acid, H₂SO₄, or sulfur dioxide (which hydrolyzes into H₂SO₃):

H2A(g) ↔ H2A(aq)H(R10.1)

H2A(aq) ↔ HA−+H+K1(R10.2)

HA− ↔ A2−+H+K2(R10.3)

Assuming here an ideal solution, the concentration of the non-dissociated species is obtained according to Henry’s law:

where H is the Henry’s law constant in M atm⁻¹, the aqueous-phase concentration of H₂A is expressed in moles per liter, i.e., M, and the gas-phase concentration of H₂A is expressed in atmosphere (atm).

The dissociation equilibria lead to the formation of HA⁻ and A^2- and their concentrations are related to that of H₂A as follows:

K1=[HA−] [H+][H2A(aq)] (10.3)

(10.4)K2=[A2−] [H+][HA−]

The ensemble of the concentrations of H₂A in solution, including the ionic species HA⁻ and A^2-, [H₂A(aq)]_t, can be defined as follows:

Thus:

[H2A(aq)]t=[H2A(aq)](1+K1[H+]+K1K2[H+]2)(10.6)

For example, in the case of sulfur dioxide, H₂SO₃, A is SO₃, which leads to:

SO2(g) (+H2O(l))↔ H2SO3(aq)HSO2=1.23 M atm−1(R10.4)

H2SO3(aq)↔ HSO3−+H+K1=1.3×10−2 M(R10.5)

HSO3−↔ SO32−+H+K2=6.6×10−8 M(R10.6)

where the notation (l) indicates liquid water. The values of the Henry’s law constant and dissociation equilibrium constants are given at 25 °C. These constants depend on temperature, according to the van’t Hoff relationship, as follows:

H(T)=H(Tref)exp(ΔHAR(1Tref−1T)) K(T)=K(Tref)exp(ΔHRR(1Tref−1T))(10.7)

where ΔH_A is the enthalpy of dissolution and ΔH_R is the enthalpy of reaction. The enthalpy of dissolution is negative for most major atmospheric chemical species (ΔH_A = – 6.25 kcal mole⁻¹ at 25 °C for SO₂). Therefore, if T < T_ref, the Henry’s law constant will be greater than its reference value (generally given at 25 °C). In other words, a lower temperature favors dissolution. The enthalpy of reaction can be positive (for example, for the dissolved species of CO₂, H₂CO₃) or negative (for example, for the dissolved species of SO₂, H₂SO₃). If it is positive, a lower temperature will favor the non-dissociated species (for example, H₂CO₃, rather than HCO₃⁻ or CO₃²⁻). If it is negative, a lower temperature will favor the ionic species (for example, SO₃²⁻ and HSO₃⁻, rather than H₂SO₃).

The ensemble of species corresponding to SO₂(aq) is generally noted as S(IV), because sulfur is in oxidation state IV. The ensemble of species corresponding to sulfuric acid is noted S(VI), because sulfur is in oxidation state VI. For S(IV):

[S(IV)(aq)]=[H2SO3(aq)]+[HSO3−]+ [SO32−](10.8)

[S(IV)(aq)]=[H2SO3(aq)](1+K1[H+]+K1 K2[H+]2)(10.9)

Therefore, the equilibrium between gas-phase and aqueous-phase SO₂ is expressed as follows:

[S(IV)(aq)][SO2(g)]=[H2SO3(aq)][SO2(g)](1+K1[H+]+K1 K2[H+]2)=HSO2(1+K1[H+]+K1 K2[H+]2)(10.10)

The relative fractions of the three species of sulfur dioxide in solution can be calculated as a function of pH as follows.

Equation 10.9 provides the fraction of H₂SO₃(aq):

[H2SO3(aq)][S(IV)(aq)]=(1+K1[H+]+K1 K2[H+]2)−1(10.11)

The HSO₃⁻ and SO₃²⁻ fractions are obtained from the ionic equilibrium relationships:

[HSO3−(aq)][S(IV)(aq)]=(1+[H+]K1+K2[H+])−1(10.12)

[SO32−(aq)][S(IV)(aq)]=(1+[H+]2K1K2+[H+]K2)−1(10.13)

These fractions are illustrated as a function of pH in Figure 10.1. When the solution is acidic, the non-ionic S(IV) fraction dominates. On the other hand, when the solution is basic, the ionic equilibria are displaced toward the formation of H⁺ ions and the sulfite ion, SO₃²⁻, dominates. At pH values typical of clouds (i.e., between about 4 and 5.6), the bisulfite ion, HSO₃⁻, dominates.

Figure 10.1. Dissolution of SO₂ in droplets. Effective Henry’s law constant (top figure) and aqueous-phase chemical composition (bottom figure) of sulfur in oxidation state IV, S(IV), as a function of pH at 5 °C.

10.2.4 Effective Henry’s Law Constant

An effective Henry’s law constant, H_eff, includes all forms of a chemical species in solution. For example, for a species with two ionic dissociations:

Heff=H (1+K1[H+]+K1 K2[H+]2)(10.14)

This effective Henry’s law constant is proportional to the standard Henry’s law constant, but it takes into account the dissociation of the molecular species into ionic species (via the dissociation constants, K₁ and K₂) and it depends on the pH of the solution (via [H⁺]). The effective Henry’s law constant is greater than the standard Henry’s law constant, which reflects the fact that the dissociation of a molecular species in solution increases its overall solubility. This effective Henry’s law constant is illustrated as a function of pH for sulfur dioxide in Figure 10.1. As indicated by Equation 10.14, it increases with pH, because a basic solution favors the dissolution of an acid.

For a monoacid HA, such as nitric acid, HNO₃:

Heff=H (1+K1[H+])(10.15)

For a base, such as ammonia, NH₃ and NH₄OH:

Heff=H (1+K1[OH−])=H (1+K1 [H+]KH2O)(10.16)

This latter equation includes the dissociation of water into protons (H⁺) and hydroxide ions (OH⁻):

At 25 °C, the equilibrium constant for water dissociation is as follows:

K‘H2O=[H+][OH−][H2O(l)]=1.81×10−16 M(10.17)

The density of water at 25 °C and 1 atm is 0.997 g cm⁻³, i.e., 997 g liter⁻¹; the molar mass of water is 18 g mole⁻¹, thus:

[H2O(l)]=99718=55.4 M(10.18)

Therefore:

KH2O=[H+][OH−]=10−14 M2(10.19)

10.2.5 pH and Electroneutrality

The pH function is defined as the negative value of the base 10 logarithm of the proton activity:

pH=−log(γH+[H+])(10.20)

where γ_H+ is the activity coefficient of H⁺. In the case of a dilute solution, the solution may be assumed to be ideal. Then, the activity coefficient is unity and one obtains the following relationship, which is most commonly used:

A neutral solution (for example, distilled water) has a pH of 7 at 25 °C, since there are as many H⁺ ions as there are OH⁻ ions:

An acidic solution will contain more H⁺ ions than OH⁻ ions and its pH will be less than 7. A basic solution will contain more OH⁻ ions than H⁺ ions and its pH will be greater than 7. A very concentrated acidic solution (such as polluted fog droplets) may have a negative pH.

Electroneutrality means that the cation positive charges are balanced by the anion negative charges. For example, for S(IV) in solution:

[H+]=[OH−]+[HSO3−]+2 [SO32−](10.23)

The factor of 2 used with the sulfite ion concentration, [SO₃²⁻], accounts for the fact that each sulfite ion has two negative charges.

The pH of a non-polluted rain can be calculated based on electroneutrality and the thermodynamic equilibria of carbon dioxide, CO₂, which has an atmospheric concentration of about 400 ppm. CO₂ is hydrolyzed following its dissolution in the aqueous phase (values of equilibrium constants are given at 25 °C):

CO2(g) (+H2O(l))↔ H2CO3(aq)H=3.4×10−2 M atm−1(R10.8)

Carbonic acid, H₂CO₃, is a weak diacid (i.e., it does not dissociate much), which leads to bicarbonate ions, HCO₃⁻, and carbonate ions, CO₃²⁻:

H2CO3(aq)↔ HCO3−+H+K1=4.3×10−7 M(R10.9)

HCO3−↔ CO32−+H+K2=4.7×10−11 M(R10.10)

The electroneutrality equation is written as follows:

[H+]=[OH−]+[HCO3−]+2 [CO32−](10.24)

The ion concentrations can be expressed as a function of [H⁺] (which is the variable that must be calculated) and of the H₂CO₃ concentration (which can be calculated from the concentration of CO₂(g) and its Henry’s law constant). The fraction of CO₂ present in the aqueous phase is assumed to be negligible compared to the amount present in the gas phase (this assumption will be verified at the end of the calculation). According to Henry’s law, assuming an atmospheric pressure of 1 atm:

Equilibrium R10.9 leads to  : [HCO3−]=[H2CO3(aq)]×4.3×10−7/ [H+]

Equilibrium R10.10 leads to: [CO32−]=[HCO3−]×4.7×10−11/ [H+]

By combining these two equations:

[CO32−]=[H2CO3(aq)]×4.3×10−7× 4.7×10−11/ [H+]2

Given that: [OH⁻] = 10⁻¹⁴ / [H⁺]; the electroneutrality relationship may be written as a function of [H₂CO₃(aq)] and [H⁺] by expressing [OH⁻], [HCO₃⁻], and [CO₃²⁻] as functions of these two concentrations:

[H+]=(10−14/ [H+])+([H2CO3(aq)]×4.3×10−7/ [H+])+(4.04×10−17[H2CO3(aq)]2/ [H+]2)

The last term may be neglected (this assumption will be verified at the end of the calculation) and a quadratic equation is obtained for [H⁺]:

Replacing [H₂CO₃(aq)] by its value, one obtains: [H⁺] = 2.42 × 10⁻⁶ M[H+]=2.42×10−6 M. Therefore, the pH of a non-polluted rain is slightly acidic due to the solubility of CO₂.

The second assumption can be verified as follows (the concentration of CO₃²⁻ is negligible compared to that of HCO₃⁻):

[HCO3−]=2.42×10−6 M; [CO32−]=4.69×10−11 M

The first assumption (CO₂ is mostly present in the gas phase) can also be verified as follows, assuming for example that the liquid water content is 1 g m⁻³, i.e., about 10⁻³ liter m⁻³:

[CO2(aq)]t= 1.36×10−5+ 2.42×10−6+ 4.69×10−11= 1.60×10−5 M= 1.60×10−5× 10−3mole per m3of air=1.60×10−8/ 40.9 atm= 3.92×10−10 atm=3.92×10−4ppm ≈ 0.0001 % of gas-phase CO2

Therefore, both assumptions have been verified.

10.2.6 Chemical Composition of the Aqueous Phase

A cloud or fog droplet is formed from a hygroscopic particle and, consequently, the soluble fraction of this particle will be present in the aqueous phase. Furthermore, the gas-phase species that are water-soluble will be partially transferred to the aqueous phase according to Henry’s law and the corresponding ionic equilibria, if the chemical species is subject to ionic dissociation.

Therefore, one must first define the initial chemical composition of the droplets, which include the particles that acted as condensation nuclei or have collided with a droplet and the gases that are partially dissolved in those droplets. The main chemical species will include water and its ions (H₂O, H⁺, and OH⁻), sulfuric acid and its bisulfate and sulfate ions (H₂SO₄, HSO₄⁻, and SO₄²⁻) originating from sulfate-containing particles, nitric acid and the nitrate ion (HNO₃ and NO₃⁻) originating from nitrate-containing particles and the dissolution of gaseous nitric acid, ammonia and the ammonium ion (NH₄OH and NH₄⁺) originating from ammonium-containing particles and the dissolution of gaseous ammonia, carbon dioxide and its bicarbonate and carbonate ions (H₂CO₃, HCO₃⁻, and CO₃²⁻), sulfur dioxide and its bisulfite and sulfite ions (H₂SO₃, HSO₃⁻, and SO₃²⁻), hydrogen peroxide (H₂O₂), ozone (O₃), oxygen (O₂), and some metals, which may act as catalysts for some reactions (iron and manganese, for example). Aqueous-phase chemistry may also take into account water-soluble organic species, such as aldehydes, and radicals (OH, NO₃, HO₂ …).

The fraction of a chemical species that is present in the aqueous phase depends on its effective Henry’s law constant and the atmospheric liquid water content. This mass balance is calculated as follows. Let A be a chemical species present in the gas phase with its concentration [A(g)] expressed in atm and present in the aqueous phase with a total concentration (i.e., including the molecular species and its ionized species, if applicable) [A(aq)] expressed in mole liter⁻¹ (M). To calculate the fraction present in the aqueous phase, one must express the concentrations in terms of the same air volume, for example in mole m⁻³. The gas-phase concentration is converted as follows:

Cg=[A(g)]R T(10.25)

where C_g is the gas-phase concentration of A expressed in mole per m³ of air, T is the temperature in K and R is the ideal gas law constant (8.206 × 10⁻⁵ atm m³ mole⁻¹ K⁻¹). The concentration present in the aqueous phase is calculated as follows:

where C_aq is the aqueous-phase concentration of A expressed in mole per m³ of air, L is the liquid water content in g per m³ of air and the factor of 10⁻³ is used to convert grams of liquid water into liters. Therefore, the fraction of A present in the aqueous phase, f_aq, is equal to the ratio of C_aq and the sum of C_g and C_aq. One accounts for the fact that the effective Henry’s law constant, H_eff, is equal to the ratio of [A(aq)] and [A(g)]:

faq=CaqCg+Caq=10−3 [A(aq)] L[A(g)]RT+10−3 [A(aq)] L=(1+103L Heff RT)−1 (10.27)

The fraction present in the aqueous phase increases with the Henry’s law constant and the liquid water content, but not in a proportional manner. Table 10.1 provides the dissolved fractions of some typical atmospheric chemical species for a liquid water content of 1 g m⁻³ (typical of a cumulus cloud). Some species such as NO, NO₂, O₃, and CO₂ are not very soluble, whereas others such as H₂O₂, NH₃, and HNO₃ are mostly present in the aqueous phase.

10.3.1 Nitric Acid

Nitric acid (HNO₃) can be formed in the gas phase from the following reactions (see Chapter 8). During daytime, the reaction of nitrogen dioxide with the hydroxyl radical is fairly rapid (the NO₂ lifetime is on the order of one day):

NO₂ may also be oxidized by ozone (O₃):

During daytime, NO₃ is rapidly photolyzed (leading to NO and NO₂ according to two photolysis pathways). During nighttime, it can be converted by reaction with NO₂ into N₂O₅ (nitrogen pentoxide):

which can dissociate back into its precursors; therefore, an equilibrium among these three species is established. N₂O₅ may also be hydrolyzed by water vapor to form nitric acid:

This reaction is slow in the gas phase and does not contribute significantly to the formation of nitric acid, compared to the oxidation of NO₂ by OH. However, this reaction can also occur with liquid water and if a N₂O₅ molecule gets into contact with a cloud or fog droplet, its hydrolysis rate is fast (e.g., Finlayson-Pitts and Pitts, 2000). Then, a heterogeneous reaction involving a gas-phase molecule and a liquid molecule takes place:

On the other hand, NO₃ may also react rapidly when getting into contact with a liquid water molecule:

NO3(g)+H2O(l) → NO3−+H++OH(aq)(R10.16)

In the presence of clouds, the kinetics of nitric acid formation by NO₃ and N₂O₅ is commensurate with the gas-phase oxidation of NO₂ by OH.

10.3.2 Sulfuric Acid

Sulfuric acid is formed in the gas phase via a set of reactions that also involve the OH radical (Stockwell and Calvert, 1983):

SO2+OH+M → HOSO2+M(where M is O2 or N2)(R10.17)

The last two reactions are very fast and the kinetics of the first reaction governs the overall kinetics of sulfuric acid formation. The rate constant expressed as a bimolecular reaction between SO₂ and OH is 9.6 × 10⁻¹³ cm³ molec⁻¹ s⁻¹ at 1 atm and 25 °C. This kinetics is, therefore, ten times slower than that of the oxidation of NO₂ by OH. Thus, this chemical pathway does not lead to rapid formation of sulfuric acid. One notes that this set of reactions leads to the formation of a hydroperoxyl radical, which can subsequently react with NO to regenerate the OH radical and produce NO₂.

In the aqueous phase, SO₂ is rapidly oxidized into sulfuric acid, bisulfate or sulfate ions. Hereafter, the term sulfate is used to refer to all three chemical species and the rates are given in terms of [H₂SO₄] representing all forms of sulfate. Three oxidants may convert SO₂ into sulfate in the aqueous phase: hydrogen peroxide (H₂O₂), ozone (O₃), and oxygen (O₂). The latter reaction must be catalyzed by metal ions (such as iron and manganese) to occur rapidly.

The reaction with H₂O₂ is very fast and occurs within a few minutes in a cloud. Thus, it is completed when one of the two reactants (SO₂ or H₂O₂) is entirely consumed. This type of reaction is called a titration reaction. Therefore, the amount of sulfate formed is the initial amount (in mole or ppb) of the reactant present in the smaller amount. For example, if 2 ppb of SO₂ react with 1 ppb of H₂O₂, 1 ppb of sulfate will be formed; 1 ppb of SO₂ will remain and all H₂O₂ will have reacted. This aqueous-phase kinetics can be represented by the following expression (based on Lind et al., 1987):

d[H2SO4]dt=7.2×107 [H+] [HSO3−] [H2O2](10.28)

where the rate constant is expressed in M⁻² s⁻¹ and the concentrations are expressed in M. This expression results from experiments conducted at 18 °C and it applies to pH values between 4 and 5.2; no activation energy data are available for this rate constant.

The reaction with O₃ is slower than that with H₂O₂, but it may nevertheless be faster than that with OH in the gas phase. The O₃ reaction is important in winter when H₂O₂ concentrations are low. The following rate expression (M s⁻¹) may be used (Hoffmann, 1986):

d[H2SO4]dt=(2.4×104 [H2SO3]+3.7×105 [HSO3−]+1.5×109 [SO32−]) [O3](10.29)

where the rate constants are given at 25 °C in M⁻¹ s⁻¹ or M⁻² s⁻¹, and the concentrations are those of the aqueous-phase species (M). No activation energy is provided for the first rate constant; those of the second and third rate constants are 46 kJ mole⁻¹ and 43.9 kJ mole⁻¹, respectively.

The O₂ reaction can also be significant, but its kinetics is more uncertain than those of the two previous reactions, because metal concentrations in cloud and fog droplets are highly uncertain. The following rate expression accounts for the catalytic effect of iron and manganese, including their synergistic effect (Martin and Good, 1991):

d[H2SO4]dt=(2.6×103 [Fe(III)]+7.5×102 [Mn(II)]+1010 [Fe(III)][Mn(II)]) [S(IV)](10.30)

where the rate constants are given at 25 °C in M⁻¹ s⁻¹ or M⁻² s⁻¹, and the concentrations are those of the aqueous-phase species (M). This expression applies to pH values between 3 and 5. No activation energy data are provided for these rate constants.

A comparison of the rates of formation of sulfate, S(VI), from these different oxidation reactions of SO₂ in the aqueous phase is illustrated in Figure 10.2. For the selected oxidant concentrations, the oxidation by H₂O₂ dominates up to a pH of 5, when the kinetics of the oxidation by O₂ catalyzed by iron and manganese becomes more important. The oxidation by O₃ becomes commensurate with that by H₂O₂ for pH values above 6, which are rare in the atmosphere. These results depend of course on the oxidant concentrations and the relative importance of these oxidation mechanisms varies depending on those concentrations.

Figure 10.2. Kinetics of the oxidation of SO₂ into sulfate by reaction with H₂O₂, O₃, and O₂ (catalyzed by Fe and Mn) in the aqueous phase as a function of pH and by OH in the gas phase. Conditions for the cloud aqueous phase are as follows: the cloud is at about 3 km altitude, P = 0.69 atm, T = 5 °C, L = 0.5 g m⁻³. Conditions for the gas phase are as follows: P = 1 atm, T = 25 °C, clear sky. Concentrations are as follows: [SO₂(g)] = 2 ppb, [H₂O₂(g)] = 2 ppb, [O₃(g)] = 40 ppb, [Fe(aq)] = 10 μM, [Mn(aq)] = 0.5 μM, [OH(g)] = 10⁶ cm⁻³. See text for the pH range to which the rate expressions actually apply.

Example: Kinetics of formation of nitric acid and sulfuric acid in the atmosphere

One is interested in calculating the half-life of NO₂ and SO₂ in the gas phase in the presence of OH radicals at a concentration of 10⁶ cm⁻³. The following kinetic constants are given: 13600 ppm⁻¹ min⁻¹ for the oxidation of NO₂ and 1400 ppm⁻¹ min⁻¹ for the oxidation of SO₂. The initial concentrations are 1 ppb of NO₂ and 1 ppb of SO₂. Calculate how much (in ppb) of HNO₃ and H₂SO₄ will be formed after eight hours at a temperature of 25 °C and a pressure of 1 atm. Next, calculate the half-life of SO₂ for its aqueous-phase oxidation by H₂O₂, present at a concentration of 1 ppb, in a low-level cloud, which has a temperature of 5 °C, an atmospheric pressure of about 1 atm, a pH of 5 and a liquid water content of 0.1 g m⁻³. The aqueous-phase kinetics of the reaction between S(IV) and H₂O₂(aq) is given by Equation 10.28. The Henry’s law constants are given in Table 10.1. The ionic equilibrium constants of SO₂ are as follows: K₁ = 1.7 × 10⁻² M, ΔH_R1 = −4.16 kcal mole⁻¹, K₂ = 6 × 10⁻⁸ M, ΔH_R2 = −2.22 kcal mole⁻¹. The ideal gas law constant is R = 1.986 cal mole⁻¹ K⁻¹. Compare the formation rate of HNO₃ by oxidation of NO₂ in the gas phase and that of H₂SO₄ by oxidation of SO₂ in the gas and aqueous phases.

The half-life, t_½, corresponds to the time required for half of the initial concentration of NO₂ to react; it is provided by Equation 8.4:

t½(NO2(g))=ln(2)kNO2 [OH]

One must convert [OH] into ppm; according to Equations 3.9, 7.1, and 7.2: [OH] = 10⁶ cm⁻³ ≈ 4 × 10⁻⁸ ppm.

Therefore, for NO₂: t_½(NO₂(g)) = 21 hours 27 mint½(NO2(g))=21 hours 27 min

In less than one day, half of NO₂ initially present will have reacted. After eight hours, the amount of NO₂ remaining after gas-phase oxidation is given by Equation 8.2:

[NO2]=[NO2]0 exp(− kNO2 [OH] t) = 0.77 ppb

Therefore, the amount of HNO₃ formed is: [HNO₃] = 1 – 0.77 = 0.23 ppb

About one quarter of the initial amount of NO₂ has been transformed into nitric acid in eight hours.

The half-life of SO₂ is calculated similarly: t_½(SO₂(g)) = 8 days 16 hours

Similarly, the amount of SO₂ remaining in the gas phase following eight hours of oxidation is:

[SO₂] = 0.97 ppb; [H₂SO₄] = 1 − 0.97 = 0.03 ppb

[SO2]=0.97 ppb; [H2SO4]=1–0.97=0.03 ppb

Only 3 % of the initial amount of SO₂ have been transformed into sulfuric acid after eight hours.

To address the aqueous-phase oxidation of SO₂ (i.e., S(IV)), one must first calculate the Henry’s law constants and the ionization constants at 5 °C (278 K) using the law of van’t Hoff (Equation 10.7). For SO₂:

HSO2(5 °C)=1.23 exp(−6.25×1031.986(1298−1278))=2.63 M atm−1

In order to calculate the effective Henry’s law constant, one must calculate the values of the equilibrium constants, K₁ and K₂, corresponding to the formation of HSO₃⁻ and SO₃²⁻, respectively, at T = 278 K based on their values at T_ref = 298 K.

K1(5 °C)=K1(Tref)exp(ΔHR1R(1Tref−1T))=1.7×10−2exp(−4.16×1031.986(1298−1278))=2.82×10−2 MK2(5 °C)=K2(Tref)exp(ΔHR2R(1Tref−1T))=6×10−8exp(−2.22×1031.986(1298−1278))=7.85×10−8 M

Therefore, the effective Henry’s law constant of SO₂ is, based on Equation 10.14:

HSO2,eff(5 °C) = 7.48×103  M atm−1

The ionic dissociation of SO₂ in the aqueous phase increases its solubility by a factor of about 6,000. For H₂O₂:

HH2O2(5 °C)=7.45×104 exp(−14.5×1031.986(1298−1278)) = 4.34×105 M atm−1

The fraction of SO₂, which is dissolved in cloud droplets, is calculated according to Equation 10.27:

faq,SO2=(1+103LHeff RT)−1=(1+1030.1×7.48×103×8.2×10−5×278 )−1=1.7 %

The fraction of H₂O₂, which is dissolved in cloud droplets, is calculated similarly:

faq,H2O2=49 %

Therefore, the gas-phase concentration of SO₂ is little affected by its dissolution in the cloud, but that of H₂O₂ decreases by about half. The gas-phase concentrations of SO₂ and H₂O₂ are: [SO₂(g)] = 0.98 ppb and [H₂O₂(g)] = 0.51 ppb.

One seeks the half-life of the oxidation of SO₂ by H₂O₂ in cloud droplets. One is interested in the half-life of total SO₂ (i.e., the sum of gas-phase and aqueous-phase SO₂). First, one calculates the fraction of aqueous-phase SO₂ that is present as HSO₃⁻, since it is the species that reacts with H₂O₂. Based on Equation 10.12, the values of K₁ and K₂ calculated previously, and a pH of 5:

[HSO3−(aq)][S(IV)(aq)]=(1+[H+]K1+K2[H+])−1=0.992

Therefore, S(IV) is almost entirely present in the aqueous phase as HSO₃⁻, as illustrated in Figure 10.1. HSO₃⁻ corresponds to 1.7 % of total (gaseous + aqueous) SO₂, since only 1.7 % of SO₂ is present in cloud droplets. Before calculating the oxidation rate of HSO₃⁻ by H₂O₂, one must calculate the aqueous-phase concentration of H₂O₂, based on its gas-phase concentration (calculated previously) and its Henry’s law constant:

[H2O2(aq)]=HH2O2 [H2O2(g)]; [H2O2(aq)]=2.21×10−4 M

Using the kinetics of the oxidation of HSO₃⁻ by H₂O₂ given by Equation 10.28, the half-life of HSO₃⁻ in the aqueous phase is calculated as follows:

t1/2(HSO3−)=ln(2)kHSO3− [H2O2(aq)] [H+]=4.4 s

Therefore, HSO₃⁻ (and, therefore, S(IV) since SO₂ is present almost entirely as bisulfite ions in the cloud droplets) is oxidized very rapidly by H₂O₂. However, HSO₃⁻ represents only 1.7 % of the total amount of SO₂. To calculate the half-life of SO₂, one must take into account the fact that only a small fraction of SO₂ (1.7 %) is present in cloud droplets:

t1/2(SO2(total))=4.4 s0.017=4 min 19 s

Therefore, the oxidation of SO₂ by H₂O₂ in a cloud is very fast due to the kinetics of the aqueous-phase reaction and the high solubility of H₂O₂. This half-life is commensurate with the lifetime of a cloud droplet, which is on the order of a few minutes. Therefore, aqueous-phase oxidation will dominate the transformation of SO₂ into sulfuric acid, since the half-life of the gas-phase oxidation was calculated to be about eight days. In the presence of clouds, the oxidation of SO₂ (via its aqueous-phase reaction) is faster than the gas-phase oxidation of NO₂.

10.3.3 Organic Compounds

Organic chemistry is of interest because the oxidation of some water-soluble organic compounds (for example, aldehydes) may lead to compounds of lower volatility, which will be more conducive to the formation of secondary organic aerosols (SOA) if the cloud of fog evaporates. However, studies conducted so far suggest that the contribution of cloud chemistry to average SOA concentrations is small (<10 %) compared to the gas-phase oxidation (see Chapter 9 for a more detailed discussion of the aqueous-phase reactions of organic compounds); nevertheless, it may be important for some specific locations and periods.

10.4.1 Non-linearity of the Oxidation of SO₂ into Sulfate

The gas-phase formation of sulfate is proportional to the SO₂ concentration and, therefore, to its emissions. However, this oxidation pathway is generally minor compared to the aqueous-phase oxidation pathways.

The aqueous-phase oxidation of SO₂ by H₂O₂ is fast and may be considered to be a titration reaction. If the SO₂ concentration is greater than that of H₂O₂, a reduction of the SO₂ concentration will not have any effect, because the amount of sulfate formed is limited by the amount of H₂O₂ available. Then, the system is non-linear, because a reduction of the SO₂ concentration will not result in a proportional reduction in sulfate.

The aqueous oxidation of SO₂ by O₃ and O₂ is also slightly non-linear, because these reactions depend on pH: the kinetics of the reaction decreases when the pH decreases. Therefore, these two reactions are self-limiting because the pH decreases as sulfate is being formed (since sulfate is an acid) and the kinetics slows down. Furthermore, if the initial concentration of SO₂ is reduced, the amount of sulfate formed will be less, which implies a greater pH and, consequently, a faster kinetics. Therefore, the reduction in sulfate will be less than the reduction in the SO₂ concentration and the system is non-linear.

10.4.2 Non-linearity of the Oxidation of NO₂ into Nitrate

The decrease of NO_x concentrations may have different effects on the concentrations of the oxidants involved in the oxidation of NO₂ into nitric acid (i.e., OH and O₃). In a high-NO_x regime, a decrease of NO_x emissions generally leads to an increase of oxidant concentrations and, therefore, the decrease of the amount of nitric acid being formed will be less than that of the NO_x emissions (in some cases, there may even be a slight increase of the nitric acid concentration if the increase of the oxidant concentration exceeds the decrease of the NO₂ concentration). On the other hand, in a low-NO_x regime, a decrease of NO_x emissions leads to a decrease of oxidant concentrations and the decrease of the nitric acid concentration will be more important than that of the NO_x emissions. Therefore, the system is non-linear and can be antagonistic (in a high-NO_x regime) or synergistic (in a low-NO_x regime) in terms of nitric acid reduction.

10.4.3 Reduction of Acid Rain in the United States

The Clean Air Act imposed SO₂ and NO_x emission controls for coal-fired power plants in order to reduce acid rain in the United States (see Chapter 15). The reduction of sulfate deposition was initially less than proportional compared to the reduction of SO₂ emissions, for the reasons mentioned in Section 10.4.1 on the non-linearity of atmospheric chemistry. Nevertheless, these emission controls became efficient in the long term to reduce atmospheric acid deposition.

Indeed, SO₂ emission controls are not efficient to reduce sulfate levels when the SO₂ levels are greater than those H₂O₂; however, they become efficient once the SO₂ levels become less than those of H₂O₂. Therefore, beyond a certain amount of SO₂ emission controls, the decrease in sulfate concentrations becomes nearly proportional to that of SO₂ emissions (there is a slight non-linearity due to the effect of pH on the oxidation of SO₂ by O₃ and O₂). It is possible that in some regions of China, the current conditions are such that [SO₂] > [H₂O₂].

NO_x emission controls have led mostly to decreases in oxidant concentrations, because the power plants are located in rural areas, i.e., areas that are in a low-NO_x regime (see Chapter 8). Therefore, it is likely that the NO_x emission controls led to decreases in both NO₂ and oxidant levels and, consequently, a decrease in nitrate levels greater than that of the NO_x emissions.