(5.88)
where the steric factor P accounts for processes that are not included in the simple collision theory. This last equation provides a justification for the empirical Arrhenius equation (Figure 5.12)
(5.89)
Here, is again the gas constant equal to 8.314 J mol–1 K–1 and A is the so-called pre-exponential factor. We see from (5.88) that A varies with the square root of the temperature T, but this weak dependence is often ignored. If the enthalpy ∆H of the reaction (difference between the enthalpy of formation of the products C and D and of the reactants A and B) is negative, the reaction is said to be exothermic and may proceed at a rapid rate. Otherwise the reaction is said to be endothermic and much less likely to occur at a rapid rate.
Figure 5.12 Energy transfer along a reaction path. The activation energy is the minimum amount of energy needed for colliding species to react. The heat of reaction is the potential energy difference between the reactants and products. The reaction is said to be exothermic if heat is released by the reaction. Otherwise, it is said to be endothermic, and heat must be absorbed from the environment.
A three-body reaction describes the combination of two reactants A and B to form a single product AB, and requires an inert third body (M) to stabilize the product. The reactants collide to form a product AB* that is internally excited due to conversion of the kinetic energy of the colliding molecules:
(k1) ; A + B → AB*
(5.90)
The excited product AB* either decomposes
(k2) ; AB* → A + B
(5.91)
or is thermally stabilized by collision with M (in the atmosphere, M is usually N2 or O2):
(k3) ; AB* + M → AB + M*
(5.92)
where the asterisk describes the addition of internal energy to M. This internal energy is eventually dissipated as heat. Thus, the overall reaction is written
A + B + M → AB + M
(5.93)
Although M has no net stoichiometric effect in the overall reaction, it is important to include it in the notation of the reaction because it can play a kinetic role. The rate of the overall reaction is given by
(5.94)
where [M] is effectively the air density [molecules cm–3]. AB* has a very short lifetime and is therefore lost as rapidly as it is produced; this defines a quasi-steady state (as opposed to true steady state, since the concentration of AB* is still changing with time). The corresponding steady-state expression is
k1[A][B] = (k2 + k3[M])[AB*]
(5.95)
Replacing into (5.94), we obtain
(5.96)
with
(5.97)
which is the Lindemann–Hinshelwood rate expression for a three-body reaction. In the low-pressure limit [M] ≪ k2/k3, we have k → (k1k3/k2)[M] so that the rate depends linearly on the air density. In the high-pressure limit [M] ≫ k2/k3, we have k → k1; the rate is then independent of the air density, as [M] is sufficiently high to ensure that AB* stabilizes by reaction (5.92) rather than decomposes by reaction (5.91).
A standard formulation for the rate expression of a three-body reaction is
(5.98)
Here, ko = k1k3/k2 is the low-pressure rate constant, k∞ = k1 is the high-pressure rate constant, and F is a correction factor for the transition regime between low-pressure and high-pressure limits. The Lindemann–Hinshelwood rate expression has F = 1. More accurate is the Troe expression:
(5.99)
where FC is the broadening factor. Kinetic data for three-body reactions are commonly reported as FC, ko, and k∞, with temperature dependences for ko and k∞.
5.4 Chemical Mechanisms
As we saw in Chapter 4, the chemical operator of an atmospheric model solves the chemical evolution equation
(5.100)
for an ensemble of species. Here ni is the number density [molecules cm–3] of the ith species, Pi is the production rate [molecules cm3 s–1] representing the sum of contributions from all reactions producing i, and ℓini [molecules cm–3 s–1] is the loss rate representing the sum of contributions from all reactions consuming i. If Pi and ℓi are constants, then the solution to (5.100) is a simple exponential approach to the steady state Pi/ℓi:
(5.101)
The problem is more complicated if Pi and ℓi depend on the concentrations of other species that are themselves coupled to species i. This is frequently the case in atmospheric chemistry because of catalytic cycles, reaction chains, and common dependences on oxidant concentrations. One must then solve (5.100) for the ensemble of coupled species as a system of coupled ordinary differential equations (ODEs). Computational methods for this purpose are described in Chapter 6.
Here, we discuss the general task of defining the ensemble of coupled species and reactions that need to be taken into account in a chemical transport model to address a particular problem. This collection of reactions represents the chemical mechanism of the model. It includes not only the species of direct interest to the problem but also the precursors and reactants for these species, which themselves may have precursors and reactants. The mechanism must represent a closed system where the concentrations of all species can be computed. Box 5.2 gives an example of a simple mechanism.
Box 5.2 A Simple Mechanism for Tropospheric Ozone Formation
We describe here an simple mechanism for production of ozone from oxidation of hydrocarbons in the presence of NOx (see Chapter 3). The mechanism includes just nine coupled reacting species: O3, OH, HO2, RO2, RH, HCHO, CO, NO, NO2. Emissions of NOx and RH complete the closure. No closure is needed for species that are only products (such as H2, ROOH, H2O2, HNO3). Although this mechanism is considerably oversimplified relative to mechanisms used in research models, it serves to illustrate some of the ideas presented in the text. Comments on individual reactions (1–3 and 7) are listed following the mechanism.
1. This reaction convolves four elementary reactions: (1a) O3 + hν → O(1D) + O2, (1b) O(1D) + M → O(3P) + M, (1c) O(3P) + O2 + M → O3 + M, (1d) O(1D) + H2O → 2 OH. O(3P) and O(1D) have lifetimes of much less than a second and can be assumed to be at steady state through the above reactions. Thus the overall rate of reaction (1) is computed in the mechanism as –d[O3]/dt = k1[O3] = (k1ak1d[O3][H2O])/(k1b[M] + k1d[H2O]).
2. RH in this reaction represents a lumped hydrocarbon accounting for the overall reactivity of hydrocarbons RHi with elementary rate coefficients ki for oxidation by OH. Thus [RH] = Σ [RHi] and k2 = (Σ ki[RHi])/Σ[RHi]. The RH + OH reaction produces the R radical, which immediately adds O2 to produce the lumped organic peroxy radical RO2. Thus, O2 is involved in the stoichiometry of the reaction although it does not control the rate; customary practice is then to put it on top of the reaction bar.
3. This reaction is not stoichiometrically balanced and does not conserve carbon. The reaction RO2 + NO actually produces RO and NO2, but we assume that RO immediately adds O2 to produce HCHO and HO2. This is based on analogy with the fate of CH3O2 and CH3O produced from methane oxidation. It is obviously a very crude treatment, as higher RO radicals may react by various pathways to produce a range of oxygenated organic compounds. However, we may not have the information needed for the RO species of interest, and/or accounting for the full suite of compounds would greatly increase the number of species in the mechanism. An important attribute of the formulation of reaction (3) is that it conserves radicals through the formation of HO2.
7. This reaction represents the sum of two branches for HCHO photolysis, with an assumed 50:50 branching ratio: (7a) HCHO + hν → H + CHO and (7b) HCHO + hν → CO + H2. H and CHO both rapidly add O2 to yield HO2 and CO.
The list of possible reactions involving atmospheric species is exceedingly large, but only a small fraction is sufficiently fast to need to be taken into account. Placing limits on the size of the chemical mechanism is essential for computational tractability of the model. The computational cost is mainly driven by the number of coupled species and the stiffness of the system. If a
species plays a significant role in the chemical mechanism, but is not coupled to the others, it should be separated from the coupled system and its chemical evolution calculated independently.
A number of compilations of reactions of atmospheric relevance are available in the literature. Many of these reactions have large uncertainties in their rate coefficients and products, reflecting the difficulty of laboratory measurements of reaction kinetics. A particular challenge is the chemistry of organic species, which involves a very large number of species and a cascade of oxidation products including radicals with varying functionalities and volatility. Most of the reaction rate constants have never been actually measured except for the smallest organic compounds, and must instead be inferred by analogy with reactions of similar species. To limit the size of the mechanism as well as to reflect the limitations in our chemical knowledge, large organic compounds are typically lumped into classes of species with the same functionality or volatility, and the evolution of a particular class is represented by a single surrogate or lumped species.
The choice of species to be included in the coupled chemical mechanism must balance chemical completeness and computational feasibility. Although experience and chemical intuition are important in the construction of a chemical mechanism, one can also use objective considerations. Consider for example the construction of a chemical mechanism to compute OH radical concentrations in the troposphere. Tropospheric OH has a concentration ~106 molecules cm–3 and a lifetime ~1 s, so that the important reactions controlling OH concentrations must have rates ~104–106 molecules cm–3 s–1. Species for which total production rates (Pi) and loss rates (ℓini) are orders of magnitude lower under the conditions of interest will not play a significant role in OH chemistry, either directly or indirectly, and can thus be decoupled or eliminated from the mechanism. Starting from a large ensemble of candidate species, it is thus possible to construct objectively a reduced mechanism. Calculations of Pi and ℓini can be made locally in a chemical transport model so that reduced mechanisms adapted to the local conditions can be selected (Santillana et al., 2010).
5.5 Multiphase and Heterogeneous Chemistry
So far we have discussed reactions involving the gas phase. Liquid and solid phases in the forms of aerosols and clouds enable a different type of chemistry. Chemical species partition between the gas and the condensed phase, and reactions can take place at the surface or in the bulk of the condensed phase (Figure 5.13). Standard usage is to refer to this chemistry as multiphase or heterogeneous. Some authors make a distinction between multiphase chemistry as involving reactions in the bulk condensed phase, and heterogeneous chemistry as involving surface reactions, but atmospheric chemistry literature tends to use these two terms interchangeably.
Figure 5.13 General schematic for uptake of a chemical species A by an aqueous aerosol particle with subsequent reaction to produce non-volatile species B. Chemical reactions are indicated by solid arrows and molecular diffusion by dashed arrows. The s subscript indicates surface species with properties possibly different from the bulk.
We see from Figure 5.13 that heterogeneous chemistry involves a combination of molecular diffusion, interfacial equilibrium, and chemical reaction. We examine here how these processes determine the rate of the overall reaction.
5.5.1 Gas–Particle Equilibrium
Chemical species in the atmosphere partition between the gas and particle phases in a manner determined by their free energies in each phase. Equilibrium partitioning always holds at the gas–particle interface, and extends to the bulk gas and particle phases in the absence of mass transfer limitations (Figure 5.13). Mass transfer limitations will be discussed in Section 5.5.2. For liquid particles, the timescale to achieve gas–particle bulk equilibrium is typically less than a few minutes.
Gas–particle equilibrium for a species X is described by
X(g) ⇄ X(a)
(5.102)
where X(g) and X(a) denote the species in the gas and aerosol phases, respectively. The general form of the equilibrium constant is
(5.103)
where px is the partial pressure of X and [X(a)] is the concentration in the particle phase. Different measures of concentration are used depending on the type of particle phase.
Aqueous particles
When the particle phase is an aqueous solution (aqueous aerosol or cloud), the equilibrium expression (5.103) is called Henry’s law and the equilibrium constant K is called the Henry’s law constant. [X(a)] (commonly written [X(aq)]) is the molar concentration (or molarity) of the species in solution. In the atmospheric chemistry literature, K is commonly given in units of [M atm–1] where M denotes moles per liter of solution. The Henry’s law constant varies with temperature T [K] according to the van’t Hoff law
(5.104)
where To is the reference temperature commonly taken as 298 K, ΔH is the enthalpy of dissolution [J mol–1] at that reference temperature, and = 8.314 J K–1 mol–1 is the ideal gas constant. ΔH is always negative for a gas-to-aqueous transition so that K increases with decreasing temperature. Table 5.2 lists the Henry’s law constants for selected species. Dependences on temperature are strong; a typical value ΔH/ = –5900 K implies a doubling of K for every 10 K temperature decrease.
Table 5.2 Henry’s law constants K for selected speciesa
Species K(298 K)
[M atm–1] ΔH/
[K] K*
[M atm–1]
O3 1.1 (–2) –2400 1.8(–2)
CH3OOH 3.1 (2) –5200 9.5(2)
SO2 1.2(0) –3150 1.6(3)
CH2O 1.7 (0) –3200 1.4(4)
HCOOH 8.9 (3) –6100 2.2(5)
H2O2 8.3 (4) –7400 4.1(5)
NH3 7.4(1) –3400 9.2(8)
HNO3 2.1 (5) –8700 4.3(11)
a Read 1.1(–2) as 1.1 × 10–2. Effective Henry’s law constants K* are calculated at T = 280 K and pH = 4.5, including complexation with water for dissolved CH2O, acid dissociation for dissolved SO2, HCOOH, and HNO3, and protonation for NH3. Data are from the Jacob (1986, 2000) compilations.
Fast dissociation or complexation of the species in the aqueous phase can increase the actual solubility beyond the physical solubility specified by Henry’s law. Consider for example the dissolution and dissociation of an acid HA:
HA(g) ⇄ HA(aq)
(5.105)
HA(aq) ⇄ H+ + A–
(5.106)
with acid dissociation constant Ka [M]:
(5.107)
To account for the dissociation of HA in the aqueous phase, we define the effective Henry’s law constant K* [M atm–1] as
(5.108)
Similar expressions can be derived for other dissociation and complexation processes.
The dimensionless partitioning coefficient f of an atmospheric species X between the aqueous phase and the gas phase can be defined by the concentration ratio in the two phases, referenced in both cases to the volume of air. By making use of the ideal gas law we obtain:
(5.109)
Here, nX(aq) and nX(g) are the concentrations of X in the aqueous and gas phases, respectively, both in units of molecules per cm3 of air. L is the atmospheric liquid water content [cm3 liquid water per cm3 of air]. K in (5.109) should be replaced by K* if X(aq) dissociates or complexes. With K in units of [M atm–1], the ideal gas constant is = 0.08205 atm M–1 K–1. Liquid water contents are typically in the range 10–9–10–11 for aqueous aerosol under non-cloud conditions, so that f ≪ 1 for all species in Table 5.2 except for NH3 (if aerosol pH is low) and HNO3 (if aerosol pH is high). Gas–aerosol equilibrium of NH3 and HNO3 is discussed in Section 3.9. By contrast, application of (5.109) to a precipitating cloud with liquid water content L ~ 1 × 10–6 and typical pH range 4–5 yields f ≫ 1 for HNO3, NH3, H2O2, and HCOOH, and f ~ 1 for CH2O. We conclude that HNO3, NH3, H2O2, and HCOOH are efficiently scavenged by rain, CH2O is partly scavenged, and SO2, CH3OOH, and O3 are not efficiently scavenged. SO2 can be efficiently scavenged only if it oxidizes rapidly to sul
fate in the aqueous phase (see Section 3.8). In-cloud partitioning is discussed further in Section 8.8 as a driver of wet deposition.
Organic particles
Gas–particle equilibrium involving non-aqueous solutions is harder to quantify because of uncertainty in the composition of the particle phase. Formation of organic aerosol is thought to involve at least in part an equilibrium partitioning between semi-volatile organic vapors and the organic phase of the aerosol:
(5.110)
where [X(g)] and [X(a)] are the gas- and particle-phase concentrations of X, and CO is the concentration of pre-existing organic aerosol, all in units of mass per volume of air. Nonlinearity in gas–particle partitioning arises because condensation of X contributes to CO. An alternative way to express the same equilibrium is with respect to the volatility of the species, CX* = 1/K in units of mass per volume of air. The partitioning coefficient between the particle and the gas phases is then given by f = CO/CX*. As in the case of aqueous-phase partitioning, f can range over many orders of magnitude depending on the species; most species will be in a limiting regime of near-total fractionation in the gas phase (f ≪ 1) or in the particle phase (f ≫ 1), with some species switching between regimes depending on CO. Because of the very large number and poor characterization of the species contributing to organic aerosol formation, an effective modeling approach can be to partition the ensemble of organic species into order-of-magnitude volatility classes Ci* that are transported independently in the model as their total (gas + particle) concentration Ci. Gas–particle partitioning is then diagnosed locally by
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