Aqueous-Phase Diffusion and Reaction

In Section 11.2.4 we showed that after time Tja = / r Daq the concentration profile inside a cloud droplet becomes uniform. The characteristic time for aqueous-phase reaction was found to be equal to = [A]// aq. If the characteristic aqueous-phase diffusion time is much less than the characteristic reaction time, then aqueous-phase diffusion will be able to maintain a uniform concentration profile inside the droplet. In this case, there will be no concentration gradients inside the drop and therefore no aqueous-phase mass transport lim- 

FIGURE 11.7 Gas-phase and aqueous-phase mass transport limitation for a species with = 0.1 cm s , Daq = 10 - cm s . The lines represent onset (10%) of mass transport limitation for the indicated values of drop diameter. Diagonal sections represent mass transp ort limitation vertical sections represent aqueous-phase limitation (Schwartz, 1986). 

3 Interfacial Mass lYansport and Aqueous-Phase Reactions 

Despite the fact that for typical cloud droplets gas-phase mass transport is in the continuum regime, mass transport across the air-water interface is, ultimately, a process involving individual molecules. Therefore the kinetic theory of gases sets an upper limit to the flux of a gas to the air-water interface. This rate is given by (11.25) and depends on the value of the accommodation coefficient. 

Following our approach in Section 11.3.1 the molar flux per droplet volume through the interface, J v, will be 

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Reaction of dissolved gases in clouds occurs by the sequence gas-phase diffusion, interfacial mass transport, and concurrent aqueous-phase diffusion and reaction. Information required for evaluation of rates of such reactions includes fundamental data such as equilibrium constants, gas solubilities, kinetic rate laws, including dependence on pH and catalysts or inhibitors, diffusion coefficients, and mass-accommodation coefficients, and situational data such as pH and concentrations of reagents and other species influencing reaction rates, liquid-water content, drop size distribution, insolation, temperature, etc. Rate evaluations indicate that aqueous-phase oxidation of S(IV) by H2O2 and O3 can be important for representative conditions. No important aqueous-phase reactions of nitrogen species have been identified. Examination of microscale mass-transport rates indicates that mass transport only rarely limits the rate of in-cloud reaction for representative conditions. Field measurements and studies of reaction kinetics in authentic precipitation samples are consistent with rate evaluations. 

Solution of the coupled mass-transport and reaction problem for arbitrary chemical kinetic rate laws is possible only by numerical methods. The problem is greatly simplified by decoupling the time dependence of mass-transport from that of chemical kinetics the mass-transport solutions rapidly relax to a pseudo steady state in view of the small dimensions of the system (19). The gas-phase diffusion problem may be solved parametrically in terms of the net flux into the drop. In the case of first-order or pseudo-first-order chemical kinetics an analytical solution to the problem of coupled aqueous-phase diffusion and reaction is available (19). These solutions, together with the interfacial boundary condition, specify the concentration profile of the reagent gas. In turn the extent of departure of the reaction rate from that corresponding to saturation may be determined. Finally criteria have been developed (17,19) by which it may be ascertained whether or not there is appreciable (e.g., 10%) limitation to the rate of reaction as a consequence of the finite rate of mass transport. These criteria are listed in Table 1. 

To analyze laboratory uptake data it is necessary to determine how y depends on the other parameters of the system. Let us assume that potentially any or all of gas-phase diffusion, interfacial transport, and aqueous-phase diffusion may be influential. We assume steady-state conditions and that species A is consumed by a first-order aqueous phase reaction (12.101). At steady state the rate of transfer of species A across the gas-liquid interface, given by (12.1 IS), must be equal to that as a result of simultaneous aqueous-phase diffusion and reaction, (12.112) ... 

Reactions taking place on the surface of solid or liquid particles and inside liquid droplets play an important role in the middle atmosphere, especially in the lower stratosphere where sulfate aerosol particles and polar stratospheric clouds (PSCs) are observed. The nature, properties and chemical composition of these particles are described in Chapters 5 and 6. Several parameters are commonly used to describe the uptake of gas-phase molecules into these particles (1) the sticking coefficient s which is the fraction of collisions of a gaseous molecule with a solid or liquid particle that results in the uptake of this molecule on the surface of the particle (2) the accommodation coefficient a which is the fraction of collisions that leads to incorporation into the bulk condensed phase, and (3) the reaction probability 7 (also called the reactive uptake coefficient) which is the fraction of collisions that results in reactive loss of the molecule (chemical reaction). Thus, the accommodation coefficient a represents the probability of reversible physical uptake of a gaseous species colliding with a surface, while the reaction probability 7 accounts for reactive (irreversible) uptake of trace gas species on condensed surfaces. This latter coefficient represents the transfer of a gas into the condensed phase and takes into account processes such as liquid phase solubility, interfacial transport or aqueous phase diffusion, chemical reaction on the surface or inside the condensed phase, etc. 

Aqueous-Phase Mass Transport Limitation The concentration of a species A, specifically, C(r, t), undergoing aqueous-phase diffusion and irreversible reaction inside a cloud droplet, is governed by... 

Diffusion into the bulk. This is determined by the diffusion coefficient in the liquid (D,). Diffusion within the bulk aqueous phase is much slower than gas-phase diffusion and can be rate-limiting under conditions of high reactant concentrations where the rate of the chemical reaction is high. This appears to have been a problem in some experimental studies of some aqueous-phase reactions relevant to the atmosphere where either bulk solutions or large droplets and reactant concentrations higher than atmospheric were used (Freiberg and Schwartz, 1981). 

A commonly used mass transfer reaction model is presented in Figure 8.1a, where the reaction occurs in the bulk aqueous phase [35, 47]. It is assumed that the substrate dissolved in the organic phase diffuses into the aqueous phase, reaching equilibrium. In the absence of reaction, once equilibrium is achieved, apparent mass transfer ceases. Given the presence of active enzyme, depletion of substrate in the aqueous phase occurs, and the system moves into a new equilibrium. Thus, the overall reaction rate depends both on reaction and mass transfer. 

Because the cellulose ether alkoxide is present entirely in the aqueous phase, the rate-limiting step may be the partitioning (phase transport) of the hydrophobic electrophile across the interface from the organic to aqueous phase. If the reaction rate is controlled by diffusion of the electrophile across the interface, then one would expect a correlation between water solubility of the hydrophobe and its alkylation efficiency. The fact that the actual alkylation reaction is probably occurring in the aqueous phase (or at the interface) yet the electrophile itself is principally soluble in the organic phase has important mechanistic ramifications. This type of synthetic problem, in which one reactant is water soluble and the other organic soluble, should be amenable to the techniques of phase transfer catalysis (PTC) to yield significant improvements in the alkylation efficiency. 

Based upon a detailed analysis of reaction transients, a mechanism was proposed for chlorophyll a-photosensitized transmembrane oxidation-reduction of aqueous phase donors and acceptors that included electron transfer between juxtaposed Chi a+ r-cations and Chi a molecules as the transmembrane charge-transfer step [112]. The maximum apparent first-order rate constant for this step was 10 s , which seems large for thermal electron transfer between chlorophyll molecules located at the opposite membrane interfaces, even considering that nuclear activation barriers may be relatively small for this reaction. Transverse flip-flop diffusion of Chi b across the membrane is 10 -fold slower than transmembrane redox under these conditions, so this alternative mechanism is almost certainly unimportant. Kinetic mapping studies have shown that some of the Chi a becomes localized within the membrane at sites that are inaccessible to aqueous phase electron acceptors, presumably within the membrane interior [114]. This suggests the possibility of a transverse hopping mechanism involving electron transfer over relatively short distances from buried Chi a to interfacial Chi a+, followed by electron transfer from Chi a at the opposite interface to the buried Chi a" ". 

If all six of these assumptions are satisfied, then (12.45) is valid and the calculations are simplified considerably. Our goal now will be to first quantify the rates of the five necessary process steps (gas-phase diffusion, interfacial transport, ionization, aqueous-phase diffusion, reaction) calculating appropriate timescales. Then, we will compare these rates to those of aqueous-phase chemical reactions. Finally, we will integrate our conclusions, developing overall reaction rate expressions that take into consideration, when necessary, the effects of the mass transport limitations. 

The equations above have been the basis of most atmospheric aqueous-phase chemistry models that include mass transport limitations [e.g., Pandis and Seinfeld (1989)]. These equations simply state that the partial pressure of a species in the cloud interstitial air changes due to mass transport to and from the cloud droplets (incorporating both gas and interfacial mass transport limitations). The aqueous-phase concentrations are changing also due to aqueous-phase reactions that may be limited by aqueous-phase diffusion included in the factor Q. 

MASS TRANSFER ASPECTS OF ATMOSPHERIC CHEMISTRY 11.3.1 Gas-Phase Diffusion and Aqueous-Phase Reactions... 

At the beginning of an emulsion polymerization performed above the CMC, the free radicals generated in the aqueous phase promote the nucleation of particles by the homogeneous and micellar mechanisms explained previously. The fact that the surface area of all monomer droplets is by far much smaller than that of all the other colloidal species makes it unlikely that the radicals existing in the aqueous phase enter and polymerize into monomer droplets. Thus, the droplets play the role of monomer reservoirs. The diffusion of this component through the aqueous phase provides the monomer needed to replace that consumed by reaction and to swell the polymer produced in the particles. 

Figure 19.10 shows a typical plot of the intraparticle effectiveness factor versus dimensionless time for different values of the Thiele modulus. It should be noted that the reversibility of the aqueous phase ion-exchange reaction leads to lower effectiveness factors than those for irreversible reactions. However, the figure also shows that as the reaction becomes increasingly diffusion controlled, reversibility can actually produce a favorable effect. Note that, in addition to the two effects mentioned, simulation can also give the concentrations of the various components as functions of the radial position, time, and the Thiele modulus. 

If ksoi kdes it follows that y = a Fsoi denotes the process of the solvation and/ or dissolution of adsorbed gas molecule. Comparing Eq. (4.269) with (4.265) we see that Faq = r o/in the absence of chemical reactions (Eq. 4.287). In general, the aqueous phase conductance Faq can include dissolution (limited by the slow diffusion into the droplet and by saturation), aqueous phase chemical reaction and heterogeneous surface reactions. The coefficient F oi is also a pure number but its value (in contrast to a and y) is not restricted to numbers less than one. The solubility limited uptake coefficient F oi varies with time t. The gas is exposed to the droplet and is given by the relationship (Fogg 2003) Daq with the aqueous phase diffusion coefficient ot the dissolved substance ... 

In this modeling approach, the interfacial reactions are also considered to be prominently affecting the overall transport profile of species. This is possible when the rate of chemical reaction at the interfaces is of the same order as the diffusion of species across the boundary layers and membrane phase. The net resistance for transport can be composed of resistances due to aqueous boundary layers, membrane phase diffusion, and the chemical reaction taking place at the interfaces. This can be mathematically written as follows ... 

The model was developed in the framework of the two-film model [9]. It was therefore assiuned that the species diffuse in a quiescent layer in the aqueous phase with chemical reactions described by steps 2, 3 and 4. 

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