the characteristic time to achieve interfacial phase equilibrium in cloud drops

Atmospheric Enuironmnt Vol. 23, No. 10, pp. 2299-2304, 1989.

Printed in Great Britain.

Oco&6981/89 53.cQ+o.O0 0 1989 Pcrgamon Press plc

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THE CHARACTERISTIC TIME TO ACHIEVE INTERFACIAL PHASE EQUILIBRIUM IN CLOUD DROPS

SUDARSHAN KUMAR

Environmental Science Department, General Motors Research Laboratories, Warren, MI 48090-9055, U.S.A.

(First received 20 October 1988 and injnalform 21 March 1989)

Abstract-The problem of gas absorption in a cloud drop for the case of a single isolated drop as well as for the case of a drop within a cloud is considered. Expressions for the characteristic times for achieving phase equilibrium between the gas-phase and liquid-phase pollutant concentrations at the air-water interface for the two cases are derived. According to the work reported herein, for highly soluble gases, the characteristic time for the case of an isolated drop as well as for the case of a drop within a cloud is directly proportional to the drop radius and inversely proportional to the accommodation coefficient of a gas. Furthermore, in the first case, the characteristic time is directly proportional to the solubility of the gas, while in the second case, it ‘Is independent of the solubility but inversely proportional to the liquid water content of the cloud.

This problem has been considered previously, however, the expression in the literature is based on the concentration at the interface of a semi-infinite liquid body and is not valid for a cloud drop. The findings in this paper disagree with those obtained from the expressions derived in the literature. In addition, for highly soluble gases, the expression in the literature predicts unrealistically large characteristic times for establishment of interfacial equilibrium.

The reaction characteristic times for reactions of SO,, H,O, and 0, leading to the formation of SO:- within cloud drops are calculated and compared to the characteristic times, predicted by the expressions derived here, for establishment of interfacial equilibrium for these gases. The comparison shows that the establishment of interfacial equilibrium is a rapid process for all cases considered, and does not limit the chemical reactions leading to the formation of SOi- within cloud drops.

Key word index: Interfacial equilibrium, characteristic time, mass-transfer limitation, cloud modeling, phase equilibrium, time constant.

NOMENCLATURE 7 the characteristic time to achieve interfacial phase equilibrium

Liquid-phase concentration within the drop the characteristic time for a chemical reaction liquid-phase concentration within the drop at equilib- 5 net flux of A into the drop (4’ - +-). rium with the initial gas-phase concentration p. concentration of A at the drop surface concentration of A in the drop at steady state

INTRODUCTlON

diffusivity of the pollutant species in water Henry’s law constant for solubility of the gaseous species in water Boltzmann constant a function of a, H and M as defined in (9) liter liquid water content of the cloud molecular weight a function of K, R and D as defined in (11) the number concentration of molecules of A corre- sponding to the partial pressure p. in gas-phase partial pressure of the pollutant species initial partial pressure of the pollutant species in the gas phase partial pressure of A in equilibrium with the drop surface concentration C, radial distance from the center of the drop radius of the drop universal gas-law constant time initial time absolute temperature average molecular velocity in the gas-phase the accommodation coefficient the nth positive root of (11) the nth positive root of (21)

The chemical composition of cloudwater is greatly influenced by the absorption of various pollutant gases into the cloud drops and subsequent chemical reactions among various species within the drops. The process of absorption and aqueous-phase reaction within the drops involves the following steps.

(a) Diffusion of the gaseous species from the gas phase to the air-water interface at the drop surface.

(b) Establishment of equilibrium between the gas- phase concentration and the aqueous-phase concentration at the interface.

(c) Possible ionization of the pollutant species in the aqueous phase.

(d) Diffusion of the ionic and/or non-ionic species within the drop.

(e) Possible chemical reactions involving various ionic or non-ionic species.

These steps are necessary during the production of acidity and sulfate within cloud drops. Each of these

2299

2300 S~DARSHAN KUMAR

steps is characterized by a time constant or a characteristic time. The formation of the products of chemical reactions in the cloud drops is controlled by the slowest step; the one with the largest characteristic time. It is important to have an accurate sense of the characteristic time for each of these steps for sound formulation of mathematical models describing cloudwater chemical composition.

In this work, the characteristic time for the second step, i.e. the establishment of interfacial phase equilibrium for cloud drops is considered in detail. Schwartz and Freiberg (198 1) and Seinfeld (1986) have considered the mass transport aspects for cloud drops and have derived expressions for the characteristic’ times for each of the five steps mentioned above. However, they have based the derivation of the characteristic time for the interfacial phase equilibrium on Danck- werts’ (1970) solution for the concentration at the interface of a semi-infinite liquid body. In the following section, the equations governing the concentration within a single isolated spherical drop are formulated, and a new expression for the characteristic time for establishment of the interfacial equilibrium is derived. In the next section, establishment of interfacial phase equilibrium for a drop within a cloud is considered. The characteristic times for the limiting cases of gases with very low solubility and gases with very high solubility are obtained. These results are then compared to the characteristic time expression derived by Schwartz and Freiberg (1981) and Seinfeld (1986).

INTERFACIAL PHASE EQUILIBRIUM FOR A SINGLE

ISOLATED DROP

Formulation of the problem

Consider a spherical drop of pure water with radius R immersed in air containing a gaseous pollutant species A at partial pressure pO. The pollutant molecules diffuse through the air and impinge on the drop surface. A fraction of the molecules impin~ng on the surface are transferred across the air-water interface resulting in the absorption of the pollutant in the drop. The characteristic times z,, and z1 for the diffusion of a pollutant in the gas phase and liquid phase, respectively, are given by (Schwartz, 1984)

and T~=R*/~D,, (1)

7, = R2/n2D. (2)

Here D, and D are the diffusivities of the pollutant in the gas phase and the liquid phase, respectively. For a typical cloud drop of radius R = 10 pm, and typical molecular diffusivities of D, =O.l cm2 s- ’ and D = 10m5 cm2 s-l, r,,=3.33~10-~s and r,=10P2s. Therefore, diffusion in the gas phase is much faster than that in the liquid phase. Thus, we can assume that the gas-phase concentration is at steady state and the characteristic time for interfacial equilibrium can be

derived from the equation for diffusion in the liquid phase. For the case of a single isolated drop, the mass of pollutant absorbed by the drop is very small and consequently, the pollutant concentration in the gas phase can be assumed to stay constant. The concentration C(r, t) of pollutant A within the drop is gover- ned by the di~usion equation

g=D[Ex+;TE]

in which C is the pollutant concentration within the drop and D is the diffusion coefficient of A in water. To solve this equation, we formulate appropriate initial and boundary conditions.

Initial condition: C(r, tO)=O, r< R. (4)

First boundary condition:

dC/c?r=O at r=O for tar,. (5)

The second boundary condition involves tlie rate at which the molecules of A impinge at the air-water interface. In order to derive the second boundary condition, we follow the treatment in Danckwerts (1970). The net flux to the drop is given by

dp,--PS)

#=~~~~~~~)’ (6)

where t( is the accommodation coefficient, and p, = C,/H, the partial pressure of A at equilibrium with the drop surface ~on~ntration C,. We now define Cs =poH, the liquid-phase concentration that would be in equilibrium with the ambient partial pressure pO. The net flux to the liquid drop, in terms of liquid-phase concentrations, is therefore

h=__f(cx-C,)

HJ(27MT) (7)

Thus, the boundary condition at the drop surface is:

DE= -K(C;-C,) at r=R 03)

where K =a/(H,/2aM%‘T). (9)

characteristic time for i~ter~ciai equilibrium

The diffusion Equation (3) with initial condition (4), and boundary conditions (5) and (8) can be solved analytically with the solution in an infinite series form. It is possible to obtain the analytic solution either by using the Laplace transforms or by the separation of variables technique. This equation was solved by these two methods and the solution obtained by either method is

x exp (-8.2WR’) I (10)

Interfacial phase equilibrium characteristic time 2301

where @. (n= 1, 2, . . . , co) is the nth positive root of

/?cot/!?+N=O (11) and

N=KRJD-1. (1 la)

The equations describing the problem posed here are analogous to the problem of a sphere at zero temperature and subject to radiation from a medium at a constant temperature (Carslaw and Jaeger, 1959). The characteristic time for interfacial phase equilibrium can now be evaluated by considering the concentration at the interface, which is given by

C,(t)=C(R,t)=C:

x exp( -fl,,fDt/R2) 1 (12) The roots of (11) are all real and simple and lie one in each of the intervals (0, n), (n, 2a), . . . ; there are no repeated roots, and the first six roots of (11) for various values of N are readily available (Carslaw and Jaeger (1959)). These roots can also be calculated iteratively for any given value of N.

The infinite series in (12) converges very rapidly, and for values of Dt/R2 applicable to cloud and fog drops, only the first term in the infinite sum in (12) is of importance. It will be shown later that for the particu- lar gases of interest in the acidification of cloudwater, the second term in this series is no more than 5% of the first term and is often considerably smaller. The subsequent terms are even smaller and therefore, the concentration can be calculated accurately by employing only the first term in the infinite series. Thus,

2KR D(N+N2+82)exp(-8:DtlK*) .

1 1 (13)

From (13), it is clear that the surface concentration approaches C,*. The rapidity of approach depends on b: D/R2 and the inverse of this quantity is the characteristic time for interfacial phase equilibrium. The characteristic time 7 is, thus, given by

T=R’/~?:D (14)

where /Jr is the first positive root of (11). It should also be noted that at steady state, the concentration of A is uniform within the drop and is equal to that at the surface. This steady state concentration is given by

c,,=c,*. (15)

Limiting values of 7 for high and low solubility gases

The characteristic time 7 depends on the solubility constant H through its dependence on fl,. Since H varies widely, it is of interest to find the limiting values of 7 for very high and very low values of H. For high values of H, KR/D is small and approaches 0. Thus, from (1 la), N = - 1. Using a simple series expansion

for the cotangent function in (11) (Abramowitz and Stegun, 1972), it can be easily proven that

for large H, N+ - 1, and j?,+,/(3KR/D).

Substituting for N and fii into (14), shows that for highly soluble gases,

z=R/3K=(HR/3&/(27rMRT), (16)

i.e. the characteristic time for interfacial phase equilibrium is directly proportional to the water solubility of the pollutant and inversely proportional to its accommodation coefficient. On the other hand, for gases with low solubility, N is very large, and fir +A. There- fore, for this case (14) becomes

7 = R2/n2D. (17)

Thus, for gases with low solubility, 7 approaches a lower limit of R2/n2D and is independent of the solubility and the accommodation coefficient.

CHARACTERISTIC TIME FOR INTERFACIAL EQUILIBRIUM FOR A DROP WITHIN A CLOUD

In the previous section, the characteristic time for interfacial equilibrium was calculated when a single drop of water is present. In the presence of a single water drop, the gas-phase concentration for a pollutant species does not change even for the most highly soluble gases. Thus, the assumption of a constant gas- phase concentration is well justified. However, in the presence of a large number of drops (as in a cloud or fog), the gas-phase concentration cannot be assumed to remain constant with time. The gas-phase concentration will decrease because of the absorption of the pollutant species by water drops. This decrease in the gas-phase concentration is influenced directly by the liquid-water content of the cloud or fog and the solubility of the gas.

In the derivation presented in this section, it is assumed that the cloud is monodisperse (drop radius R), and has a liquid water content L. Furthermore, the liquid-phase concentration of pollutant is uniform within the drops and is denoted by C. A mass balance for the pollutant A is given by

po/BT=p/9T+ LC(t).

The time-dependent gas-phase concentration of A is, therefore, given by

p(t)=p,-LHgTp,. (18)

The net flux to a drop within the cloud is then obtained by replacing pO by p in Equation (6). The interfacial boundary condition for this case, after some algebraic manipulation, is

Dg= -K(C,*-C(l+LHWT)) at r=R. (19)

The solution for the concentration C(r, t) within the drop can be obtained by methods outlined in the

2302 SUDARSHAN KUMAR

previous section. In fact the concentration C(r, t) is given by an equation very similar to (10X namety

c,* C(r* [)=(I +LH$lT)

2KR2 sin (@j,r/R)

-=I ~D(~+~z+~~2)sin~~

xexp( -jl;2Dt/R2) 1 . (20) In this case /I’s are the positive roots of

~~ot~+~=o, (21)

where N’ = KR( I+ LH5W)/D - 1. The characteristic time for interfacial phase equilibrium is then given by

~=R"lfi;'D (22)

where #I‘, is the first positive root of (21). It is clear that for the case of a single drop (i.e. L-+0), I?‘= N and the two solutions are identical. One notes that the steady state concentration within the drop is reduced by a factor of (1 + LHWT) and is given by C$/( 1 + LHSBT). The concentration within a drop in the two cases is identical provided LHgT<tl. For low solubility gases, under typical cloud conditions (i.e. L = 5 x to-’ and T=288 I(), LHWT<< 1, and the concentradon profile within the drop evolves in accordance with (lo), and the characteristic time is given by R2/n2D. On the other hand, for a highly sduble gas in a cloud situation, LHBTx 1, and

iV'=(~RL/D)&tT/2nhf)- 1. (23)

For typical values of R, L, D and T, N’ approaches - 1; therefore,

and z=( R/3aL)J(2nM/WT). (24)

Thus, the characteristic time for a highly soluble gas is directly proportional to the drop radius R, inversely proportional to the liquid water content L and the accommodation coefficient ~1, and most interestingly, independent of the solubility, provided LHBT>> 1.

COMPARI~N WITH RRSULTS IN THE ~TRRATURE

The results derived in this report indicate that the time constant for interfacial equilibrium for a single drop depends on the solubility, accommodation coefficient, and diffusivity of the gaseous species in water. It has been shown that for highly soluble gases (NH,, H20z and HNO,, for example), the time constant T is directly proportional to H/a. On the other hand, for a drop within a cloud, the time constant for interfacial equilibrium for a highly soluble gas is independent of the solubiiity H, but inversely dependent on the liquid water content L, and the a~rn~ation ca&kient a. However, in both cases, the time constant t for gases

with low solubility approaches a constant value independent of H or E(, and is given by R2/n2D.

These results are significantly different from the results in the literature (e.g. Schwartz, 1984; Seinfeld, 1986), based on the expression

T= 2~~~TDH2~~2 125)

which predicts that 5 is proportional to Hz/al, and independent of the drop radius R for all gases. The Expression (25) for t has been derived (Seinfeld, 1986) by treating the drop as a semi-infinite liquid body, whereas in this work the equations for finite spherical drops of radius R have been employed.

The accommodation coefficient is a fundamental parameter affecting the rate of transfer of a gaseous species to cloud and fog drops. However, measurements of accommodation coefficients for only a few gases on water surfaces are available. The accommodation coefficients for 0, (Lee and Tang, 1985) and NO, (Lee and Tang, 1988) have recentl’y become available. In addition, for SO,, measurements of only the lower limits of accommodation coefficient values (Tang and Lee, 1987; Gardner et al., 1987) have been possible. Therefore, Heikes and Thompson (1983) and Chameides (1984) have treated 01 as a variable in investigating mass transfer to aqueous drops. Chamei- des (1984) has shown that for cloud drops, mass transfer of a gaseous species across the air-water interface may become the rate-determining step for chemical reaction within the drops when the accommodation coefficients for the gaseous reactants are low. Therefore, in this section, we will compute the characteristic times for interfacial phase equilibrium for a range of CI values.

The t values for various species that influence the chemical composition of cloudwater have been calculated from (22) for various values of the accommodation coefficient and are given in Table 1. In these calculations, the cloud drops are assumed to have a radius of 10pm and a pH of 5. For purposes of comparison, Table 1 also gives I values calculated from (14) and (25) for same conditions. It can be seen that the characteristic time for a drop to attain interfacial equilibrium in the presence of a cloud (Equation (22)) is < I s in all cases. The gaseous species are listed in order of increasing solubihty in this table, and we see that the characteristic time r increases as the solubihty of the gas under consider- ation increases. Even for HNO,, which has a very high solubility, the T values are < 1 s. On the other hand, Expression (25) would indicate that the attainment of interfacial phase equilibrium for highly soluble gases such as HNO, and NH, wilt take a much longer time. It is clear that the Expression (22) derived in this work predicts much lower T values than those predicted by (25) for almost all cases. The table also shows the ratio, evaluated at t = z, of the second term to the first term in the infinite sum in (20). As mentions earlier, this ratio is never >0.05. Furthermore, this ratio decreases

Interfacial phase equilibrium characteristic time

Table 1. Comparison of characteristic times for establishment of phase equilibrium as calculated by three different expressions (cloud drop radius = 10 pm, pH = 5)

2303

Characteristic time (s) In a Single Literature

Gaseous cloud drop expression species a: (Equation 22) (Equation 14) (Equation 25)

0, 1.00 5.1 X 10-a 5.1 x 1o-3 3.9 x lo- I4 0.10 5.1 X 1o-3 5.1 x 10-3 3.9 x lo-l2 0.01 5.1 x 1o-3 5.1 x 1o-3 3.9 x lo-‘0

SO, 1.00 8.4 x 1O-3 8.6 x 10-S 3.6x 1O-3 0.10 4.6 x 1O-2 4.8 x lo-’ 3.6 x lo- ’ 0.01 4.3 x 10-l 4.5 x lo- 1 3.6 x lO+ ’

HK’, 1.00 4.5 x 1o-2 1.2 x 10-l 2.6 0.10 4.2x 10-l 1.2 2.6 x lo+’

NH, 1.00 4.8 x lo-’ 1.8 5.8 x lo+’ 0.10 4.4 x lo- ’ 1.8 x lo+’ 5.8 x lo+“

HNO, 1.00 9.0x lo-* 9.2 x lo+’ 1.5 x lo+14 0.10 8.7 x 10-l 9.2 x 1O+6 1.5 x 10+16

*The numbers denoted by 0.00 are all less than 10m50.

Ratio in Equation (20)

5.0 x 10-Z 5.0 x lo-* 5.0 x 10-Z

1.2 x 1o-2 1.6 x lo-”

0.00*

4.2 x lo-“’ 0.00

0.00 0.00

0.00 0.00

exponentially with time, so that the concentration C(R, t) and the characteristic time t can be estimated quite accurately by employing only the first term in the infinite sum.

Figure 1 shows the concentration of dissolved SO, within a cloud drop of radius 10 pm immersed in air containing SO, as a function of radial position at various times. The concentration gradient within the drop is quite small at t/r = 1, decreases as t/7 increases, and becomes practically negligible at t/t = 3.

DISCUSSION +2.2x lo”[so:-][O,]. (26)

Two major reaction paths by which cloudwater derives its acidity are the aqueous-phase oxidation of S(N) (i.e. SOl(aq), SO:-, HSO;) by 0, and H,O,. It is instructive to calculate the reaction characteristic

u- I I

Fig. I. Approach of the interfacial concentration to its equilibrium value as a function of time for a drop of

radius 10 pm in air containing SO,.

times for S(N), H,O, and O,, and compare them to the characteristic times for establishment of interfacial phase equilibrium to determine if a limitation to the chemical reaction is present. The kinetics of these reactions are well known (e.g. Martin, 1984), and the rates of formation of sulfate (in mol d- ’ s- I) as a result of these reactions are given by (Erickson et al., 1977; Martin and Damschen, 1981)

d[SO;-]

dt =3.1 x lO’[HSO;][O,]

d[SO:-]

dt =5.2x lO’[H+] [HzO,] [HSO;]. (27)

The reaction characteristic time for a species A in- volved in a chemical reaction is given by (Seinfeld, 1986)

CA1 7’=dCAl/dt. (28)

The reaction characteristic times for Hz02, 0, and SO2 were calculated for the case of absorption and chemical reaction in cloud drops for typical atmospheric concentrations of H,O, = 2 ppb, 0, = 50 ppb, SO, = 15 ppb, L= 5 x lo-’ and various values of cloudwater pH. The characteristic time for establishment for phase equilibrium for a drop within a cloud based on Expression (22) is shown in Table 2 (assuming a=O.l) along with the characteristic time for chemical reactions for cloud drops. The characteristic time for phase equilibrium in a cloud drop depends on the pH for SO, but not for 0, and H,O,; of these three gases, only the solubility of SO, is a function of pH. In addition, the reaction characteristic times for SO, and O3 depend on drop pH because the net rate of sulfate formation due to oxidation of S(W) by 0, depends on cloud drop pH (Martin, 1984).

2304 SUDARSHAN KUMAR

Table 2. Comparison of characteristic times for interfacial phase equilibrium and chemical reaction for three gaseous species responsible for the production of sulfate in cloud drops (assuming a=O.l for all gases, L= 5 x lo-‘, and initial

drop pH as indicated in the table)

Characteristic time (s)

Species

SO*

0,

H@,

Reaction pH (Equation 28)

5.0 12.0 4.5 4.3 4.0 1.4

5.0 2.8 x 10-J 4.5 2.7 x 1O-2 4.0 2.4x 10-l

5.0 78.0 4.5 78.0 4.0 78.0

Phase equilibrium in a cloud drop (Equation 22)

4.6 x lo- 2 1.4 x 1o-2 7.4 x 10-X

5.1 X 10-3 5.1 x 1o-3 5.1 X 10-J

0.42 0.42 0.42

However, the reaction characteristic time for W,O, does not vary with pH because the net rate of SO:- formation as a result of S(N) oxidation by H,Oz is inde~ndent of cloud drop PH.

From Table 2, it is clear that the characteristic time for the establishment of interfacial phase equilibrium is much smaller than that for chemical reaction for almost all cases. The establishment of interfacial phase equilibrium is, therefore, not the rate determining step in the formation of SOi- in cloudwater. Thus, in the formulation of modefs simulating the chemical composition of cloudwater, one need not explicitly take into account the process of establishment of phase equilibrium at the air-water interface. All models fo~uIated to date (e.g. Jacob and Ho~mann, 1983; Schwartz, 1984) implicitiy assume ‘instantaneous’ establishment of equilibrium at the interface; and based on the results of this work, this is a valid assumption. However, if the characteristic times calculated from the often used Expression (25) are compared to the reaction characteristic times, one would reach the erroneous conclusion, for example, that formation of SOi- through oxidation of S(W) by H,O, in cloud drops is mass-transfer limited for t( values close to 0.1.

REFERENCES

Abramowitz M. and Stegun I. A. (1972) Handbook I$ Marh- ematical Functions. Dover, New York.

Adamson A. W. (1973) A Textbook of Physical Chemistry. Academic Press, New York.

Carslaw H. S. and Jaeger H. C. (1959) Conduction offfeat in Solids, 2nd ed. Clarendon Press, Oxford.

Chameides W. L. (1984) The photochemistry of a remote marine stratiform cloud. i. geophys. Res. 89, 47394755.

Danckwerts P. V. (1970) Gas-Liquid Reactions. McGraw- Hill, New York.

Erickson R. E., Yates L. M., Clark R. L. and McEwen D. (1977) The reaction of sulfur dioxide with ozone in water and its possible atmospheric significance. ~~~~s~~e~~~ En~~~ro~rne~t 12, 227-230.

Gardner J. A., Watson L. R., Adewuyi Y. G., Davidovits P., Zahniser M. S., Worsnop D. R. and Kolb C. E. (1987) Measurement of the mass accommodation coefficient of SO, on water droplets. J. geophys. Res. 92, 10,887-10,895.

Heikes B. G. and Thompson A. M. (1983) Effects of heteroge- neous processes on NO,, HONO and HNO, chemistry in the troposphere. i. geopkys. Res. 8% Cl& 10,883-10,895,

Jacob D. J. and Hoffmann M. R. (1983) A dynamic model for the production of H+, NO, and SOi- in urban fog. J. geophys. Res. 88 Cll, 6611-6621.

Lee J. H. and Tang I. N. (1985) Accommodation coefficient measurements of ozone on water surfaces. In Proc. Symp. Heterugerteous Processes in Source-Dominated Atmos- pheres (edited by Novakov T.).

Lee J. H. and Tang I. N. (1988) Accommodation coefficients of gaseous NO, on water surfaces. Atmospheric Environ- ment 22,1147-1151.

Martin L. R. (1984) Kinetic studies of sulfite oxidation in aqueous solution. In SO,, NO and NO, Oxidation: At- rnosp~er~c Considerations (edited by J. G. Calvert). Butter- worth, Boston, MA.

Martin L. R. and Damschen D. E. (1981) Aqueous oxidation of sulfur dioxide by hydrogen peroxide at low pH. At- mospheric Environment 15, 1615-1621.

Schwartz S. E. (1984) Mass-transport considerations per- tinent to aqueous-phase reactions of gases in liquid-water clouds. In Chemistry of ~ultip~ase Atmospheric Systems (edited by W. Jaeschke). Springer, Heidelberg.

Schwartz S. E. and Freiberg J. E. (1981) Mass-transport limitation to the rate of reaction of gases in liquid droplets: application to oxidation of SO, in aqueous solutions. &mospheric Enoironmenr 15, 1129-l 144.

Seinfeld J. H. (1986) Atmospheric Chemistry and Physicx ofAir P~~~~~tion. John Wiley, New York.

Tang I. N. and Lee J. H. (1987) Accommodation coefficients of ozone and SO,: implications for SO, oxidation in cloud water, In 7%e Chemistry of Acid Rain (edited by Johnson R. W. and Gordon G. E.). ACS Symposium Series.

the characteristic time to achieve interfacial phase equilibrium in cloud drops

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