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CHAPTER 2Presentation of Solubility Data:Units and ApplicationsStephen SchwartzAtmosphetic Sciences Division, Brookhaven National Laboratoty, Upton, NY, USA

The solubility of gases in water and other aqueous media such as seawater andmore concentrated solutions is central to the description of the uptake and reac-tions of these gases in aerosols, precipitation, surface water and other aqueousmedia such as the intracellular fluids of plants and animals. It is also pertinentto sampling of soluble atmospheric gases in aqueous medium for analytical pur-poses. This book presents evaluated summaries of data pertinent to the solubilityof gases in aqueous media. This chapter introduces the terminology by which thissolubility is described and the pertinent units and presents examples of applica-tions pertinent to atmospheric chemistry. As is seen below, a variety of units havebeen and continue to be employed for gas solubility data, so some attention mustbe given to this subject. As this is an IUPAC publication, every effort is made toemploy units that are consistent with the International System of Units (SystemeInternational, SI) [l]. However, in IUPAC publications of solubility data it isusual to publish data in the original units in addition to SI units. The consistencyof SI makes this system of units convenient for application in atmospheric chem-istry and related disciplines [2]. However, as elaborated below, there are somedepartures from strict SI that persist in chemical thermodynamics that requirespecial consideration.The solubihty of weakly soluble gases in liquids is one of the oldest branchesof physical chemistry. The origins of this study go back to William Henry(1775- 1836), whose presentation before the Royal Society and subsequent pub-lication [3] gave rise to what has come to be known as Henrys law. That paperdescribes a series of experiments with several gases which remain of concernto present-day atmospheric scientists, CO2, Hz& N20, 02, CI-L N2, H2, COand PHs. The gas and liquid were contacted and allowed to reach solubilityequilibrium at various values of applied pressure, with the amount of absorbedChemicals in the Ahnosphere - SoiubiliQ, Sources and Reacrivio. Edited by P. G. T. Fogg and .I. Sangster0 2003 IUPAC ISBN 0-471-98651-8

__________By acceptance of this article, the publisher and/or recipient acknowledges the U.S. Government's right to retain a

nonexclusive, royalty-free license in and to any copyright covering this paper.

Research by BNL investigators was performed under the auspices of the U.S. Department of Energy under ContractNo. DE-AC02-98CH10886.

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20gas determined from the decrease inthe water. The generalization reachedas follows:

Chemicals in the Atmospheregaseous volume after equilibration withby Henry is stated in the original paper

The results of a series of at least fifty experiments on [the several gases] establish thefollowing general law: that under equal circumstances of temperature, water takesup, in all cases, the same volume of condensed gas [at higher applied pressure] asof gas under ordinary pressure. But as the spaces [volumes] occupied by every gasare inversely as the compressing force [pressure], it follows, that water takes up,of gas condensed by one, two, or more additional atmospheres, a quantity which,ordinarily compressed, would be equal to twice, thrice &c. the volume absorbedunder the common pressure of the atmosphere. [italics in original]

In other words, the amount of gas absorbed, as measured by volume at ordinarypressure, is proportional to the applied pressure.It is also clear in the presentation that Henry reaIized that the amount ofdissolved material (extensive property) scaled linearly with the amount of water;in fact, Henry normalized his results to a standard amount of water, so that it isclear that he recognized that it is the concentration (intensive property) that isproportional to the pressure of the gas.In a subsequent appendix, Henry [4] noted that the residuum of undissolvedgas was not always entirely pure and exhibited an increasing proportion of lesssoluble contaminant, necessitating correction of his originally presented data. Hestates in this cormection:For, the theory which h4r. DALTON hm suggested to me on this subject, and whichappears to be confirmed by my experiments, is, that the absorption of gases by wateris a purely a mechanical effect, and that its amount is exactly proportional to thedensity of the gas, considered abstractedly from any other gas with which it mayaccidentally be mixed. Conformably to this theory, if the residuary gas contain 1/2,MO, or any other proportion, of foreign gas, the quantity absorbed by water willbe IL?, l/IO, &c. short of the maximum.

In this elaboration, Henry makes use of Daltons newly enunciated law of partialpressures to correct his previously reported data.This brief summary of these experiments permits us to restate Henrys lawwith complete fidelity to his original formulation, changing only the terminol-ogy, as follows: the equilibrium concentration of a dissolved gaseous species isproportional to the partial pressure of the gas, the proportionality constant beinga property of the gas and a function of temperature, or[X(aq)] = kHx(T)pX (2.1)

where [X(aq)] is the aqueous concentration, px is the gas-phase partial pressureand kux (T) is the Henrys law coefficient of the gas X in water at temperature T.The law is readily generalized to other solvents, the Henrys law coefficientdepending on both the solvent and the solute, to mixed solvents, and to solvents

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presentation of Solubility Data 21cont~~ng nonvolatile solutes. In the te~nology of modem physicaI chemistry,Henrys law is a limiting law:

the reason being that the properties of the solvent change with increasing con-centrations of solute. This situation can be accounted for by expressing Henryslaw in terms of the activities in the pertinent phases:

We thus observe that the Henrys law coefficient is an equilibrium constant forthe reactionX(g) = X(aq)

Recognition of this leads to an intrinsic connection to chemical ~e~odyn~cs,and in particular to the fact that the Henrys law coefhcient is reIated in thecustomary way to the free energy change of reaction ask H xxp(-AG/I?T)

where in this case AGo is the free energy of dissolution:AGo = AfG(X& - AfGO(Xs)

the Af G ' seing the standard Free energies of formation of the aqueous andgaseous species. In practice, it is the usual situation that knowledge of the Henryslaw coefficient allows one to determine the unknown standard free energy of for-mation of one of the two species from the known vahre for the other, rather thanthe Henrys law coefficient being calculated from two known standard free ener-gies of formation. However, there are important practical exceptions in instancesof highly reactive gases whose reactivity precludes establishment of the solubilityequilibtium at measurable concentration and partial pressure.The reason for focusing here on Hemys law and for quoting from Henrysoriginal paper is in part to provide justification for the form of Henrys lawemployed in the present work, and advocated for universal use, namely that itis the aqueous-phase concen~tion (or, alte~atively, molahty) that is viewed asproportional to the gas-phase partial pressure, rather than the other way around,as has been a customary expression of Henrys law by many subsequent investi-gators in the nearly 2OOyears since Henrys initial report. From the perspectiveof the atmospheric scientist, this approach has a certain further justification aswell: it is usually the gas-phase partial pressure that can be considered the inde-pendent or controlling variable, with the aqueous-phase concentration dependenton the gas-phase partial pressure. Henrys law is the basis for any considera-tion of exchange of material between the gas phase and natural waters, and theHenrys law coefficient is central to expressions that describe the extent and rate

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22 Chemicals in the Atmosphereof such exchange, even for species whose equilibrium solubility is not adequatelydescribed by Hemys law.It must be stressed that expressing Henrys law according to Equation (2.1)is by no means universal either in the physical chemistry literature. or in theenvironmental sciences. Frequently it is customary to express Henrys law theother way around (gas-phase partial pressure proportional to aqueous-phase con-centration); this can even be sometimes more convenient, as in considerations ofthe evasion of dissolved gases from solution, as from a lake to the atmosphere.In any event, it is inevitable to encounter Henrys law expressed in this reversesense. Also, a variety of modes of expression have been employed for describingthe concentration of dissolved gas, such as the volume that it would occupy at astandard pressure, or mole fraction relative to solvent, that have led to yet furthervariants of Henrys law and to tabulations of Henrys law coefficients in a mul-tiplicity of tmits. It is thus imperative, at the present time and for the foreseeablefuture, to pay attention to the sense and units of Henrys law as employed bydifferent investigators and to convert Henrys law coefficients from one set ofunits (and sense of the law) to another. Consequently, we address these issuesin this chapter and present algorithms for and examples of such conversions. Asnoted above, there is a further desire to employ SI units, and the consequentnecessity for making the use of these units more familiar and for dealing withspecial problems associated with the fact that the units conventions of chemicalthermodynamics are not entirely consistent with SI.For a detailed review of modeling interactions between gaseous species andliquid water in the atmosphere, see Sander [5].1 UNITSAs a measure of aqueous-phase abundance we employ molality, the mixing ratioof the solute in the solution, having unit mole of solute per kg of solvent,mol kg-l. Use of this measure and unit is consistent with current practice inchemical thermodynamics and is preferred to concentration, which, because ofdensity change associated with temperature change, is not constant with tem-perature for a solution of a given composition. Nonetheless, concentration iscommonly employed, with the most common unit being molar (moles per litreof solution, mall- or M). This unit suffers from the further disadvantage thatthe litre is not an SI unit, the SI unit of volume being the cubic meter. In practice,because the mass of 11 of water is approximately equal to 1 kg, the molality ofa dilute aqueous solution is approximately equal numerically to the molar con-centration. More precisely, for a single component solution, the concentration ofa solute X, Cx, is related to the molality of the solute rnx as

where ~~~1~~~~enotes the density of the solvent in the solution:A o l v e n t+ ~ x m x

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presentation of Solubility Data 23where &,tUtiO,, enotes the solution density and MWx denotes the molar mass(molecular weight) of species X, in units of kg mol- . More generally, for amulticomponent solutions

(2.3)where the summation is taken over all solutes. For dilute aqueous solutions thesolvent density is nearly equal to I kg molwl, so the concentration and molalityare numerically essentially equal, i.e.

where the quantity in braces can be omitted for numerical work but must beretained for units calculus. For more concentrated solutions of importance inatmospheric chemistry, such as aqueous clear-air aerosol, including sea salt, orfor stratospheric sulfmic acid aerosol, the volume fraction of solute and theresulting departure from unit solvent density can be appreciable.

Where sufficient pressure dependence data are available to report the Henryslaw coefficient as a limiting law, then that value is reported, i.e.knx (mol kg- brt-l) z lim mx(mol kgsorVent-PZ+0 px (bar)

When such data are not available (which is generally the si~ation), the Henryslaw coefficient is reported as the quotient of molality by partial pressure, withthe partial pressure (range) of the measurements specified:

,& (mol kg- bar-) = mx(mol kgsOIVent-)px (bar)For application in atmospheric chemistry, the lack of such pressure-dependentdata should make little practical difference provided that there are good reasonsto assume that the measurements were made within a pressure range in whichHenrys law is applicable. This means that linear extrapolation to zero pressurewould correspond to zero concen~ation.Solubility data are reported at the temperature(s) of measurement. Where suffi-cient temperature-dependence data are available, the enthalpy change of solutionis reported as AH0 dln(kn.Jl mol kg-l bar-)-=-

4 dU,r)In principle, this quantity can also be obtained from the limiting-law solubilitydata, but again in practice such data are not generally available, again a situationof little practical consequence.The unit mol kg-l bar- for the Henrys law coefficient is practical for manyapplications, generally, md specifically in atmospheric chemistry, with the obvi-ous proviso that the partial pressure of the gas be expressed in units of bar.

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24 Chemicals in the AtmosphereWe adopt the symbol kH for this practical quantity, and likewise employ thesymbol px to denote the partial pressure in bar of the species X. However,as these units are not products of powers of base SI units, their use leads tocomplications, especially in expressions in which they are combined with SIquantities. It is necessary to employ a Henrys law coefficient in SI units in suchexpressions, which we distinguish by a caret, &r, where &r = 10-5kn. Likewiseto avoid ambiguity, when we employ partial pressure in pascals, we use thesymbol fix.Increasingly the mixing ratio xx is gaining favour in atmospheric chemistryas the quantity for expressing local abundance; units are mol/mol (air), e.g.mnol/mol. Since Hemys law solubility depends on the partial pressure of thegas, rather than mixing ratio, it is necessary to convert from mixing ratio topartial pressure. This is readily achieved as

px(bar) = p(bar)xx(mol/mol)so that

mixn01ol"ent -) = knx (mol kggbar-)p(bar)xx(mol/mol)In many applications, such as the evaluation of chemical kinetics and masstransport rates, the concentration is required, rather than the molahty, and it istherefore often convenient to employ a Henrys law coefficient that yields theconcentration directly. If we denote this concentration Henrys law coefficientHx (units M bar-l), then within the approximation that the solvent density isequal to 1 kg mold,

In many considerations of mass transport it is useful to employ a dimensionlessHenrys law coefficient, the ratio of concentration in solution (aqueous) phase tothat in the gas phase (or vice versa; extreme caution is suggested in using dimen-sionless Henrys law coefficients because the sense of the equilibrium cannot beinferred from the units of the quantity). We take this dimensionless Henrys lawcoefficient also in the sense aqueous/gaseous:

where the dimensionless property is denoted by the tilde over the symbol. Withinthe accuracy of the ideal gas law (adequate for all practical purposes in atmo-spheric chemistry, at least on earth), the concentration (molme3) of a gaseousspecies X having partial pressure j3x (Pa) is given as

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Presentation of Soiubility Data 25w h e r eg s h en i v e ra so n sn ds h eb se mrE q u a t i2 . 2 )h eq u e o u s -o n c e n tf h ei s sa sm W s

B o t hh eq u e on da s e oo n c e n tc ai ta rr ehr a t i of h e s eu a n t i th ei m e n s ie n ra wo e fs f oi n d e p e n df a r t ir e s s

F o rl lu ~ t i tn h ei g h t -i d ef q u a2 .n t rI n hq u a n t i. r . ~ ~ ~ t ~ ~ ~si m e n s ir oh ix pxi t si m p l eo r ms i r e co n s e qfm p lo n sn pi c a l l yh ee n r ya wo e f f in n i tf o lg s Oa -o h od i l u t eq u e oo l u th eo l ve n sk g ~ , ~ ~ * ~a sa fa p p r o x i m a t3g ~ ~ ~ ~ ~ r n- .F o rh ee n r ya wo e f f ii v en n i tf o lg s O , Vov e r s i oa c t ou s te p p l i

F o ro l v e ne n s ix p r e sn n i tfg s O, ~ ~ th ie c

w h e r eh ei n a lp p r o x iq u a lo l di t hh ep p r of h ov e n te n s i tq u a lo g - .h eo n v ea cr ei ro no f s efn c o n s i sn i t so rh i se a ss ef t r3 n io rh el a wo e f f i c im o lg s O t Va - s r e fh eh iu sc o m b i ni t ht h e rnl g e bx p r e s

2 Q U E O U S - PO L U B IQ U I LB e f o r er o c e e d ie h o u lo t eum p ou a l io e an a m e l yh a tt p p l iop a r io l ua s er ,o rr e co a ha r eh y s i c ai s s o lu to tn d e rh e me an o lo oa h e m i c ai f f e rp e c ie r eh y sb s on co lud o e so tn c l ue a c to o r mi s th e mp eu cs y

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26 Chemicals in the AtmosphereFor example, the dissolution of acetaldehyde can be represented by the followingsequence of reactions:

CHsCHO(g) -----+ CHsCHO(aq)CHsCHO(aq) + Hz0 ---+ CHsCH(OH}z(aq)

The first of these is physical ~ssolu~on, i,e. the Henrys law equi~b~um; thesecond is a chemical reaction. The pertinent equilibrium expressions are:

and

The total dissolved acetaldehyde concentration is evaluated as the sum of the twodissolved species:I?X-hCHO(aq, tot)1 = K2HdI HOGt qN + [CHSH(OH&(aq)l

k H~~~~~~PcH~cHo~~ + b136~~JIn practice, whether or not to include the hydrated species as a part of thedissolved component depends on the experiment or application and whether thedissolution process is rapid or slow compared with the hydration reaction orother possible reaction, such as reaction of &CO with dissolved S(IV). What isimportant to note for the present purpose is that the equ~ib~um concen~ation ofthe hydrated species is itself proportional to that of the unhydrated species, so thatthe total is proportional to the concentration of the unhydrated species and theoverall linearity between gas-phase partial pressure and aqueous concentration isretained. One can take advantage of this linearity to define an effective Henryslaw coefficient. in this instance

such that one has for the total dissolved acetaldehyde concentration an expressionthat is formally identical with Henrys law, viz.

It is of course crucial in any potentially ambiguous situation to specify whetherthe Henrys law coefficient refers to the total or only to the unhydrated species.This is important also in correlations of Henrys law coefficients with chemicalstructure. Physical solubility depends on properties such as pol~zability, whichdepend on molecular structure and number of electrons; correlations based onsuch properties can obviously be disrupted if there is a structural change associ-ated with the hydration.

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pt-esentation of Solubility Data 27Much important in this context is the exclusion from the rubric of Henrys

law any chemical reaction that results in a solubility that is not linear in thepartial pressure of the gaseous species. Consider the solubility equilibrium forthe dissolution of the strong acid HCl:HWd -

for which the equilibrium relation isk, =

H+ (aq) -I- Cl-(aqj

WIW-1PHCl

For both H+ and Cl- deriving entirely from the dissolved hydrochloric acid,which may be considered entirely dissociated, thenpllcl = kcq-1[Acid]2

where [Acid] denotes the aqueous concentration of the dissolved acid, a nonlinearrelation, and ipsofucto, not Hemys law (here we have considered the aqueousacid concentration to be the independent variable, consistent with the fact thatvirtually all the material would be present in solution, rendering this concen~ationthe more readily measured variable). For a variety of reasons such as descriptionof mass transport rates it may be useful to consider the actual Henrys lawconstant of HCl. Formally this is related to the overall equilibrium constant kqthrough the sequence of reactionsHCl(g) --+ HCl(aq)

HCl(aq) --+ H+(aq) + Cl-(aq)which sum to give the overall reaction

HCl(g) = H+(aq) + Cl-(aq)Correspondingly, the equilibrium expressions are

and keq

= lB+lKl-1p H C1

where kHHcis the Hemys law coefficient for HCl, k3 is the acid dissociation con-stant of aqueous HCl and k,.q is the equilibrium constant for the overall reaction.If the overall equilibrium constant keq can be determined, by measurement ofpartial pressure and aqueous activities of the two ionic species, and if, generally

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28 Chemicals in the Atmospherein a separate experiment, the acid dissociation constant Ka can be determined,then the Henrys law coefficient (equilibrium constant of physical solubility) ofHCl can be evaluated as

Where data are available to permit the determination of both the Henrys lawcoefficient and the equilibrium constant(s) of the pertinent aqueous-phase reac-tion(s), then all these data are presented. Frequently only the eq~ib~um constantfor the overall reaction is known or, because of uncertainties in deten-nination ofthe individual component equilibrium constants, it is known with greater accu-racy. In that case the overall solubility equilibrium constant is presented, togetherwith the enthalpy change associated with the overall reaction.

3 EFFECTIVE HENRYS LAW CCEFFICIENTSIn the case of a weakly acidic gas dissolving in an aqueous solution that maybe considered well buffered relative to the incremental acidity resulting fromdissolution of the gas, it may be possible to assume that the acid concentration isconstant, in which case the total amount of dissolved gas is linear in the gas-phasepartial pressure. Consider the ~ssolution of SOz:

For this system it is possible and convenient to define a pH-dependent effectiveHenrys law coefficient for total dissolved sulfur(IV):

ks% = W2@41PSO2SO2 + Hz0 + H+ + HSOj-

k W+lW%-1d = [SO2(aq)]HS03- + H-+ + SOzz-

kti = ~~+lW32-lWSO3-1

such that under the assumption that the aqueous solution is well buffered oneobtains a Henrys law-like expression for total dissolved sulfur(IV):

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Presentation of Soiubility Data 29

H202

H2C0

OHCH&HOPAN

03NO2NO02

Figure 2.1 pH dependence of the effective Henrys law constant for gases whichundergo rapid acid-base dissociation reactions in dilute aqueous solution, as afunction of solution pH. The buffer capacity of the solution is assumed to exceedgreatly the incremental concentration from the uptake of the indicated gas. Alsoindicated at the right of the figure are Henrys law constants for nondissociativegases. T z 300 K (modified from Schwartz [6]; for references, see Schwartz v])

Expressions such as this have found much application in consideration of scav-enging of gases by cloudwater and in considerations also of the coupled masstransport and chemical reaction in cloudwater. Figure 2.1 shows the pH depen-dence of. the effective Henrys law coefficient for gases which undergo rapidacid-base dissociation reactions in dilute aqueous solution as a function of solu-tion pH. Also shown for reference are the Henrys law coefficients of severalnondissociative gases of atmospheric interest. Note the great increase in effectiveHenrys law coefficient resulting from the greater solubility of the ionic speciesthan the parent neutral species.

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30 Chemicals in the Atmosphere4 PARTITI ON EQUILIBRIA IN LIQUID-WATER CLOUDSThe key initial characterization of the distribution of a volatile substance betweengas and liquid phases in a cloud is the equilibrium distribution. This distri-bution depends in the first instance on the liquid water content of the cloud.Values of cloud liquid water content L are typically of the order of 1 gmp3, i.e.1 x 10e3 kgmv3. (The commonly used measure of liquid water content, the liq-uid water volume fraction, is related to L as L/pew, where pcW he density ofcloudwater is ca 1 kgme3; the liquid water volme fraction is thus of the orderof 1 x 10p6.) For a volatile substance dissolving according to Henrys law, theamount of material per cubic meter in the aqueous phase is

The amount of material per cubic meter in the gas phase iswgl = @xU - ~/P~~~I~~~c zxI&T for lpcn7 < 1

The partition ratio, the ratio of material in the liquid to gas phase, is thus

This result is readily extended to more complicated solution equilibria, especiallythrough the use of the effective Henrys law coefficient.Similar considerations pertain to the evaluation of the equilibrium distributionbetween air and an aqueous scrubbing solution employed in extracting a solublegas from the atmosphere; here the liquid water volume fraction would be replacedby the ratio of flow-rates of solution to air.Note that we have employed the strict SI version of the Henrys law coefficienthaving units, mol kg- Pa-. Unfortunately, if we use the Henrys law coefficientin practical units, mol kg- bar-, we must use a units conversion factor of 1 xlo- barPa_:

The partition ratio can be readily assessed from the value of the quantity L& Rg T.For a given situation of liquid water content and temperature it is useful to considerthe value of the Henrys law coefficient ,& such that 50 % of the substance is ineach phase: &+, = (LRgT)- or km0 = 105(LRgT)-

For scoping purposes, we note that for T rz 300 K, RgT sz 2.5 x 103 m3 Pa/(molkg-), so that for L FG1 x 10W3 gmv3, & RZ0.4molkg- Pa-l andknjO =4 x 104 mol kg bar-. Note that for most gases of atmospheric interest kH

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Presentation of Solubility Data 31so that the equilibrium distribution consists of virtually all of the substance presentin the gas phase; a key exception is HzOz. However, the effective Henrys lawcoefficient approaches or in some instances subst~ti~ly exceeds knzO, or exampleHNOs, which at equilibrium is present virtually entirely in solution phase.The fraction of the total volatile materiaI that is present in the aqueous phase is

For a weakly soluble gas such that kHxRgT CK 1 (i.e. 10-5Z,ku~l?a~ .+< ),

As noted above, the recurrence of the factor 105 is due to the standard stateof gaseous substances being taken as 1 bar (rather than 1 Pa), which leads tothe Henrys law coefficient kHx having the units mol kg-l bar-, rather thanmol kg-* Pa-, which would be consistent with SI units.

5 EXPRESSIONS FOR ABUNDANCE OF MATERIALSIN MULTl-PHASE SYSTEMS

Frequently it is useful to compare on an equivalent basis the amounts of tracespecies present in a volume of air in different phases (gas, aqueous, particulate).A variety of quantities and units have been employed to express such abundances,as summarized by Schwartz and Warneck [2}, Use of mole based units is advo-cated whenever possible (known com~sition and molecular weight) becausethese units display chemically meaningful relationships. Because mixing ratio byvolume is approp~ate only for gas-phase species, this quantity is not suitable.Suitable quantities and units are concentration (mol me3), mixing ratio (mole permole air) and equivaIent partial pressure, the partial pressure that the specieswould exhibit if an ideal gas, fix* (Pa) or px* (bar). Of the several quantities,mixing ratio seems most general and suitable.Consider a species X present in cloudwater (liquid water content L kgmT3)with molality mx mol kgei. The several equivalent measures of abundance arepresented in Table 2.1.

Table 2.1 Expressions for molar abundance of cloudwater dissolved species. Cloudliquid water volume fraction, L: modality of dissolved species, mx mol kg-Quantity Unit Symbol ExpressionConcentration mol m-3 Cx LmxPartial pressure Pa bX* L&,TmxPartial pressure bar PX* 3 O-5Li?gTmxMixing ratio mol/mol (air) xx 10~5LR~7hx/~~~r(b~r)

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Chemicals in the Atmosphere26 MASS ACCOMMODATION AND INTERFACIAL

MASS-TRANSPORT RATESMuch attention has been paid in recent years to the measurement of mass accom-modation coefficients of gaseous substances on the air-water interface knd to theexamination of the implication of such mass accommodation coefficients on therate of uptake and reaction of gases in cloudwater. The mass accommodationcoefficient is the fraction of collisions from the gas phase on to the solutioninterface that result in transfer of molecules from the gas phase to the liquidphase. Here we review the pertinent derivations and implications. The Hemyslaw coefficient is central to both.Consider the flux of a species X from the gas phase into a solution. Thecolhsion rate (mol mV2s-l) is evaluated from the kinetic theory of gases asox + = ~~x~xcx~s)(o+) (2.7)

where 3x = (8Rs~/~~ Wx) 4 is the mean molecuhtr speed of the species X,ox is the mass accommodation coefficient and Cxcs)(Oi-) is the concentrationof the species at the interface, on the gas side, which may differ appreciablyfrom the bulk concentration because of a diffusive gradient in the vicinity of theinterface. This fiux can be expressed equivalently in terms of the hypotheticalaqueous-phase concentration CX(~~)*(O+) hat would exist in equilibrium withthe gas-phase concentration Cx(sj (O+), asfJX + = +~5Xcx(q)*(o+l/@X

Here it is most convenient to use the dimensionless (conc~n~atio~concen~ation)Henrys law coefficient. The flux given by Equation (2.7) is a gross flux, whichmay be offset by a return flux from the solution phase into the gas phase. Thisreturn flux is crx- = &$x@&J-)/&where Cx(aQj(O-) denotes the concentration of the species at the interface, on thewater side. This is so because, by microscopic reversibility, ox- must be equalto crx+ at phase equilib~um [i.e. when Cx(aqj(O-) = &Cx(sj(O+)] and becausethe flux scales linearly with the concentration. The net flux into solution is

+crxG0x -ffx-== $x~xk{sj(O+) - ~x@qp~/~xl @-QEquation (2.8) sets an upper limit to the mass transfer rate that must be consideredin any treatment of interphase mass transfer; dependmg on the value of ox, thismay or may not present an appreciable citation to the overall rate.7 COUPLED MASS TRANSFER AND AQUEOUS-PHASE REACTIONThis discussion (based on Schwartz [8]) develops expressions describing therate of gas and aqueous-phase mass transport and aqueous-phase reaction. The

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P r e s e n t af c d u b ia t a 3exchange flux of a gaseous species between the gas and aqueous phases maybe de&bed phenomenolo~~a~y (e.g. Danckwerts [9]) as the product of anoverall mass-transfer coefficient Ka times the difference between the gas-phaseconcentration Cx(s) and the aqueous-phase concentration Cxcaq) ivided by thedimensionless Henrys law coefficient fix describing the equilibrium solubilityof the gas in the aqueous medium:

crx = &]CX(a) - ~XQq~/&l (2.9)where Cx(sj and Cx~~~j efer to concentrations in the bulk of the phase, thatis, at distances sufficiently far from the interface to be outside the region ofstrong gradient in the vicinity of the interface. For concennations in units ofmol mV3, the flux has units mol mW2 -r. Note that here we use the diiensionlessHenrys law coefficient fix. This expression is fairly general, being applicableto a variety of situations of interest in atmospheric chemistry, for example to theuptake of a gas into cloudwater by uptake followed by aqueous-phase reactionor to dry deposition of a gas to surface water. All of the physics (and chemistry)of the mass-transfer process is embodied in the mass-trausfer coefficient Ka; thesubscript g denotes that this is the gas-side mass transfer coefficient. This quantityhas dimension of length/time, or velocity.It is evident from Equation (2.9) that in order for there to be a net flux, theaqueous-phase concen~ation Cxcaq)must not be in equilibrium with the gas-phaseconcentration Cx(a), i.e. that Cx~~sj# &&x~a); more specifically, for there to bea net uptake in the aqueous medium Cxcaq) must be less than tixCx(aj. Thelack of equilibrium may be due to aqueous-phase chemical reaction depleting theconcentration the dissolved gas or, alternatively, in the case of dry deposition tosurface water, simply to the fact that the water is undersaturated relative to theatmospheric concen~ation of the gas, for example because of transfer from themixed layer of the ocean to the deep ocean.If the .aqueous-phase concentration of the depositing substance Cx~~j is 0[more precisely, if Cxcaqj < &$x(s)], then ox = KsCx(a~. However, the con-dition Cxtaqj < &&x(a) does not necessarily imply that the flux is equal to itsrn~~urn possible value, as governed by atmospheric mass transport only, sincethere may be a return flux (aqueous phase to gas phase) resulting from a near-interface aqueous-phase concentration that is substantial compared to &.$x(s)even when CX(~~) K BxCxcs).The choice of the gas-phase concentration, rather than the aqueous-phaseconcentration, as that to which to refer the flux is arbitrary, reflecting the gas-phase o~entation of atmospheric chemists. The flux might entirely equiv~endybe referred to the aqueous-phase ~oncen~ation, viz.

where Ki is referred to as the liquid-side mass transfer coefficient; it is seen thatKl = I$/&.

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34 Chemicals in the AtmosphereConsiderable progress is made in understanding and describing the overall

mass-transfer process by considering it to be a sequence of processes, from thebulk gas phase to the interface, across the interface, and from the interface to thebulk aqueous phase. These fluxes are described, respectively, as

where Cx(sj(O+) denotes the concentration of the species at the interface, onthe gas side, which may differ appreciably from the bulk gas-phase concentra-tion because of a diffusive gradient in the vicinity of the interface; likewise,CX(~~J(O-) denotes the concentration of the species at the interface, on the waterside. The gas-phase and liquid-phase mass-transfer coefficients kg and kl describephysicul mass transfer (turbulent diffusion plus molecular diffusion through thelaminar boundary layer immediately adjacent to the interface); the magnitudesof these coefficients depend on the degree of physical agitation characterizingthe system. The coefficient ,L3,o be discussed later, represents enhancement ofthe aqueous-phase mass-transfer flux due to removal of the material by chemicalreaction; when there is no enhancement, #I = 1.Under steady-state conditions the flux is constant and equal in both media andacross the interface. By equating the several expressions for the flux [Equations(2.8)-(2.11)], one obtains the overall mass-transfer coefficient as the inverse sumof the mass-transfer coefficients in the two media and at the interface:

1-= L+KG k

(2.12)In the absence of any aqueous-phase chemical reaction of the dissolved gas thatwould enhance the rate of uptake, the enhancement coefficient p is equal to unity.This corresponds to the two-film expression commonly employed in evaluatingair-sea fluxes of nonreactive gases (e.g. [lo]). Under the assumption that theinterfacial resistance is negligible, then there is a critical solubility @tit = kg/k1for which the gas- and liquid-phase resistances are equal. For g &fir, gas-phase mass transport is controlling and the uptake rate isindependent of I?. For reactive gases the effect of the chemical removal processis exhibited in the enhancement coefficient p, and it is #Il? that is to be comparedwith I&fit.

To examine reactive enhancement of uptake, it is necessary to model theconcurrent mass-transfer and reactive processes, and a number of models havebeen introduced over the years, largely in the chemical engineering literature.The most familiar such model is the so-called diffusive-film model, which positsan unstirred laminar film at the interface having thickness Dxq/ kl, where Dxaqis the molecular diffusion coefficient of the dissolved gas in aqueous solution.

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P r e s e n tf o l u b ia t a 5T h eeactive gas must diffuse through this film to the bulk convectively mixedliquid. Reaction occurs in the film and/or in the bulk hquid depending on therelative rates of reaction and mass transport. Although criticized as unrealistic,this model has received widespread application in the geochemical literature.Alternative models posit systematic or stochastic transport of stagnant, near-surface parcels of liquid into the bulk, again with reaction occurring to greater orlesser extent in these unstirred parcels depending on the relative rates of reactionand mass transport. In all cases the ~q~d-phase mass transport is characterizedby the single parameter kt as well as by I&P.

Rates of reactive uptake for the three models were examined and comparedby Danckwerts and Kermedy [1 l]. For the diffusive film model the expressionfor 6 for a reversible first-order reaction of the dissolved gas can be written as(2.13)

where n is the ratio at equilibrium of the total concentration of dissolved materialto the concen~ation given by Henrys law ~ssolution alone, n = &*/fix, andK is a dimensionless rate coefficient for reaction,(2.14)

where I& U) is the effective first-order rate coefficient for aqueous-phase reactionof the dissolved gas.A few words should be said about the factors n/(n - 1) and L$/&J employedin the definition of K. At n >> 1, which corresponds to large equilibrium enhance-ment of the soIubili~ and which is often the si~ation of mterest for reactiveuptake, n/(n - 1) zs 1 and this factor can thus be neglected in evaluation of K.At low values of n (recall that n 2 1) the factor n/(n - 1) gives rise to substan-tial enhancement of K ssentially because the reaction does not need to proceedvery far to reach eq~ilib~um. The quantity &,+rG kt,Dxq has ~mension ofinverse time; comparison of /+(I) with kCfitpermits the importance of reactivee~~cement to the rate of uptake to be i~ediately assessed.Despite somewhat different and more realistic assumptions of the surfacerenewal models, these modeIs were found to yield expressions for p that differfrom Equation (2.13) by no more than a few percent [ll]. These models havegamed substantial support in laboratory studies (e.g. [9]). Expressions for thekinetic enhancement equivalent to Equation (2.13) were later obtained in thecontext of atmosphere-surface water exchange by Bolin [12] and by Hooverand Berkshire [13] and are sometimes is associated with those investigators.Although all that remains to evaluate p (and ultimately I$) is knowledge.of the pertinent parameters, it is worthwhile to examine the dependence of theenhancement of uptake on the equilibrium constant and reaction rate constant.

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36 Chemicals in the AtmosphereAt low values of the argument (~0.5), i.e. for low reaction rate coefficients, thehyperbolic tangent function tanh closely approximates the argument itself, so thattanh K 1 /K i approaches umty and in turn fi approaches unity. Thus approximately(within IO %)

/3=1 for K 5 0.3 (12.13a)In this limiting situation there is essentially no enhancement of the rate of uptakeover that given by physical dissolution alone. For somewhat greater values ofK, where the enhancement becomes appreciable, p is approximated by seriesexpansion of the tanh function (again within 10 %) as

for K 5 2.1 (2.13b)

At the other extreme, for values of the argument of tanh greater than about 1.5(i.e. at high reaction rates), tanh approaches unity, and to good approximationL

K21 for K 2 2.4 (2.13~)

K2 -1

For values of K such that (~4 - 1) < n, this expression simplifies toL

#?=K2 for 2.4 5 K 5 (1 + 0.1~)~ (2.13d)Finally, if K; is sufficiently great, the first term in the denominator of Equation(2.13) predominates and

fi%rI for K 2 2.4 and K 2 lOO(q - 1)2 (2.13e)In this limit the rate of uptake is equal to that for a nonreactive gas whosesoiubility is enhanced, owing to instantaneous chemical equilibrium, by the factorq. This limit justifies the use of an effective Henrys law coefficient &* =nknX for rapidly established equilibria and specifies the range of applicability ofthis treatment.The dependence of j3 on the effective first-order rate coefficient for reactionI&,(~) s illustrated in Figure 2.2 for a range of values of g. The plateauing of bfor large values of kaqC1) nd intermediate values of n corresponds to the limit inEquation (2.13e). The linear region of the graphs (slope = 0.5 on the logarithmicplot) corresponds to the limit in Equation (2.13d). This is equivalent to the wellknown [9] expression for diffusion controlled reaction for n > 1:

for 2.4 5 K 5 (1 +O.lr~)~ (2.15)

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Presentation of Solubility Data 3?

Equilibrium enhancement

Figure 2.2 Dependence of the kinetic enhancement factor p of the aqueous-phasemass-transfer coefficient /Q_as a function of the effective first-order rate constantfor reaction, @, normalized to kL2,/&.+ where DW is the aqueous-phase moleculardiffusion coefficient, for indicated values of the equilibrium solubility enhancementfactor q (adapted from Schwartz [8])8 RATES OF MULTI-PHASE REACTIONS IN LIQUID

WATER CLOUDSIt is desired to evaluate the rate of aqueous-phase reactions in cloudwater underthe assumption of solution equilibrium of the dissolved reacting gas and to expressthis reaction rate in units such that it can be compared with rates of gas-phasereactions. We take the cloud liquid water content L such that the liquid watervolume fraction L/pCw

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38 Chemicals in the AtmosphereFor a gaseous reagent species predominantly present in the gas phase, @x >> ix*and we obtain the result

1YX.Y = CPX + PX*).= LRgTkHxkq (I = lO-5L RgTkH, k$)x

This important quantity may be readily evaluated for known aqueous-phase first-order reaction rate coefficient kqcl). yx,aq scales linearly with liquid water content,Henrys law coefficient, and aqueous-phase rate constant It may be readily andimmediately compared with the rate coefficient for gas-phase reaction of speciesX and thus provides the means of comparing rates of gas- and aqueous-phasereactions of a given species.For other than first-order reaction, kaq~1 is replaced by the effective first-orderreaction rate coefficient, evaluated, for example, as the product of second-orderreaction rate coefficient and concentration of other reacting species.9 DEPARTURE FROM PHASE EQUILIBRIUM

AT AIR-WATER INTERFACEFor a given flux into the aqueous phase, corresponding to an assumed reactionrate, the fractional departure from equilibrium at the interface can be evaluatedto examine for mass-transport limitation to the rate of reaction as a function ofmeasured or specified mass accommodation coefficient and reaction rate [ 131.This fractional departure is

fiXCX&) O+) - ~X(~lJ)O-) CX CXkXCX(g) co+) = ~xZXCX(~)(O+)/~ = ax~xjx&)+)/4&T (2.17)Note that here the partial pressure of the gas is in units of Pa. Particular attentionmust be paid here to consistency of units or else errors can result (e.g. Ref. 9p. 69); a major strength of SI is that such consistency is intrinsic.Following Schwartz [14], we may use Equation (2.17) to test whether the rateof reaction in a spherical drop is appreciably limited by the rate of interfacialmass transport. For departure from phase equilibrium not to exceed a criterion,

arbitrarily k&en as 10 %,

For a spherical drop of radius a, the flux ox corresponds to an uptake rate 41ru~ox ora mean volumetric reaction rate xx~~~ = (4ru2~x)/(4~u3/3) = 3ox/u, whencethe criterion that must be satisfied for mass transport 1imitaGon not to exceed 10 % is

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Presentation of Solubility Data 39For the aqueous-phase reaction rate given by Equation (2.16) this criterion becomes

Note that here, as in general where the Henrys law coefficient appears inexpressions, we must use the Henrys law coefficient in strict SI units, i.e.mol kg- Pa-. In the second form of the criterion this Henrys law coefficienthas been replaced by that in mol kg-l bar- units.A similar expression has been given by Schwartz [ 141 as a criterion for absenceof li~tation to the rate of multiphase reaction due to the finite rate of gas-phasediffusion, viz.

IO DRY DEPOSITION OF GASES TO SURFACE WATERAs presented above, the flux of dry deposition of atmospheric gases to surfacewater, or more generally the exchange flux between the atmosphere and surfacewater, may be evaluated for known values of the pertinent mass-transfer coef-ficients in the two phases and the Hemys law coefficient, taking into accountany reactive enhancement. Dry deposition is generally expressed as the productof the atmospheric concentration times a deposition velocity ?,d:

UX = udcX(g)whence ?.+j Ks provided that cx~~j < &Cxtgj.Here we examine the dependence of deposition velocity on the several param-eters. An approximate value for kg is 0.13 % of the wind speed, for a referenceheight of 10 m; the exact proportionality coefficient depends somewhat on the ref-erence height and on the atmospheric stability [151.Thus, for typical wind speedsof 3 - 15 m SC*,values of kg range from 4 to 20 mm s-l. An expression for lot givenby Liss and Merlivat [16], which exhibits three linear regions of dependence onwind speed (increasing slope with increasing wind speed), has gained substa.n-tial support (e.g. [17]). That expression gives kl = 1.4 x 10e6, 1.3 x 10F5 and1.1 x 10e4 m s-l for wind speeds of 3, 5 and 10 m s-l, respectively; the actualvalues depend somewhat on the identity of the transported gas and on tempera-ture. We take these values as representative of the range that must be consideredfor the present purpose of identifying factors governing dry deposition of gasesto surface waters. However, it must be stressed that although these values arerepresentative, the problem of specifying the precise values pertinent to a givenenvironmental situation is by no means solved.Figure 2.3 shows the overall mass-transfer coefficient Ks in the absence ofreactive enhancement as a function of Henrys law coefficient for representativevalues of k, and kg. Also shown are values of Henrys law and effective Henryslaw coefficients for several gases of atmospheric interest. The points marked byl represent the equilibrium solubility after.the reaction (acid-base dissociation,

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4 0 C h e mn h et m

F i g u r e. 3v e r a la s s - to e f fG ( i nh eb sf n tm a s s - t ri m i t ao ro n r ea s ep ) s u nf oi t yo ra l u e sf G n dL n d i cn h es y mf h eu rh ec at hr i g h te r m i t so m p a rf n t e rn dv e ro n do rn dao f h ea s s - a co e f f. l s oh o wr ee na wo ekH ( o p e ni r c l en df f e c te n r ya wo e fH *f i li rf uo f t m o s p ha s e s ;h en t e rb s c ic a li v eH n dH* i n n fm o lg - / ( 15 a )a d a p tr o mc h w a8 ] )p He p e n d e n t ,rl d e h y d ey d r o l y s i sa so n eoo m p l e th ea to hp h y s i c a lo l u b i l i t yi n d i c a t e dy )i v eh eq u i l i b rn h a n ca c.T h eh r u s tf h ei g u r es h a ti t h o u te a c t i vn h a n c e mh er e at h a tr eu f f i c i e n t l yo l u b l ena t e rh a to re p r e s e n t aa l uf u n rt h e i re p o s i t i o ne l o c i t yo u l deo n t r o l l eyt m o s p h ea sr a n shf i g u r ee m o n s t r a t e sh em p o r t a n tn f l u e n cfe a c t in h a n c en tA u r t h e re a t u r ef h ei g u r es h ee v i ct h ep pi g hh ii s pt h en t e r f a c i a lo n d u c t a n c e ,v a l u a t e ds1 / 4 ) %o rn d i c aa l uf h aa c c o m m o d a t i o no e f f i c i e n t. e r eh ee a no l e c up e eh ie po no l e c u l a re i g h t ,a se e na k e ns 0 2- ln t e r f ae s i som a s sr a n s p o r th o u l dea k e nn t oo n s i d e r a t ifn t e r f ao n d usc o m p a r a b l eog e v a l u a t e di t hn l yh ea sn di q u i d -e s i sI n t e r f a c i a le s i s t a n c es o ti m i t i n go h ea t ef r ye p o s ioa lfc x 0 e 4u t ,e p e n d i n gn h ea l u efg e v a l u a ti tn lh a nl i q u i d - p h a s ee s i s t a n c e s ,i g h te c o m ei m i t i no ro w ea l uf z1 1O N C L U S I O N SE x c h a n g efo l a t i l ep e c i e se t w e e nh ea sh a sf h et m o s pni qw a t e rs e yr o c e s sn h ev o l u t i ofa t e r i an h ea r tn v i r

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Presentation of Solubility Data 41system. The driving force for this exchange is the Hem-ys law solubility equili-brium together with rapid equilibria in aqueous solution. The Henrys law con-stant, or the effective Henrys law constant that takes these rapid equilibria intoaccount, is thus a fundamental property of volatile atmospheric species in theearths environment that must be well known in order to describe the extent andrate of these phenomena under circumstances of interest. The Henrys law con-stant (or effective Henrys law constant) ranges fairly widely for substances ofinterest to the earths atmospheric environment, by some 18 orders of magnitude.Because of this wide range of solubilities, the rates and extents of various pro-cesses, such as the distribution between gas and liquid phase in clouds, likewisevary substantially. In many situations it is sufficient to identify limiting casesapplicable to sparingly soluble or highly soluble gases. For example, the zeroth-order question is where the bulk of the material resides, and often knowledge ofthis is sufficient for the task at hand, for which an order of magnitnde estimateof the solubility is often sufficient. Clearly insoluble gases such as nitrogen andoxygen are present essentially entirely in the gas phase. Very soluble material,such as nitric acid, would be expected to be present essentially entirely in the liq-uid phase, at least for clouds of sufficient liquid water content. In some instancesit is necessary to know the amount of material present in the lesser compart-ment, for example in the calculation of the rate of aqueous phase reactions incloudwater. These calculations require rather precise knowledge of the Henryslaw solubility.

This chapter has presented formalism to describe several applications of Henryslaw solubility to atmospheric chemistry. It is hoped that it provides a sense of thepertinence and use of solubility data in atmospheric chemistry and will stimulateinterest in the evaluated Henrys law data presented in this volume.

REFERENCES1. Mills, I.; Cvita$ T.; Homann, K.; Kallay, N.; Kuchitsu, K., Quantities, Units and

Symbols in Physical Chemistv (The Green Book), Blackwell Scientific Publications,Oxford, for International Union of Pure and Applied Chemistry, 1993.2. Schwartz, S. E.; Wameck, P., Units for use in atmospheric chemism, Pure Appl.Chew., 1995, 67, 1377-1406.3. Henry, W., Experiments on the quantity of gases absorbed by water, at differenttemperatures, and under different pressures, Philos. Trans. R. Sot. bndon, 1803, 93,29-43.

4, Henry, W., Appendix to Mr. William Henrys paper, on the quantity of gases absorbedby water, at different temperatures, and under different pressures, Philos. Trans. R.Sot. London, 1803, 93, 274-6.

5. Sander, R., Modeling atmospheric chemistry: interactions between gas-phase speciesand liquid cloud/aerosol particles, Surv. Geophys., 1999, 20, 1-31.6. Schwartz, S. E., in The Chemistry of Acid Rain: Sources and Atmospheric Processes,Johnson, R. W.; Gordon, G. E. (eds), American Chemical Society, Washington, DC,1987, pp. 93-108.7. Schwartz, S. E., Mass-transport considerations pertinent to aqueous-phase reactionsof gases in liquid-water clouds, in Chemistry of Mdtiphase Amspheric Systems,Jaeschke, W. (ed.), Springer, Heidelberg, 1986, pp. 415-71.

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42 Chemicals in the Atmosphere8. Schwartz, S. E., Factors governing dry deposition of gases to surface water, in

Precipitation Scavenging and Atmosphere-Surface Exchange, Schwartz, S. E.; Slinn,W. G. N. (coordinators) Hemisphere, Washington, DC, 1992.9. Danckwerts, P. V., Gas-Liquid Reactions, McGraw-Hill , New York, 1979.10. Liss, P. S.; Slater, P. G., Flux of gases across the air-sea interface, Nature, 1974,247, 181-4.11. Danckwerts, P. V.; Kennedy, B. E., Kinetics of liquid-film process in gas absorption.Part I: models of the absorption process, Trans. Znst. Chern. Eng., 1954,32, S49-S52.

12. Bolin, B., On the exchange of carbon dioxide between the atmosphere and the sea,Tellus, 1960, 12, 274-81.13. Hoover, T. E.; Berkshire, D. C., Effects of hydration on carbon dioxide exchangeacross an air-water interface, J. Geophys. Res. 1969, 74, 456-64.

14. Schwartz, S. E., Chemical conversions in clouds, in Aerosols: Research, Risk Assess-ment and ControZ Strategies, Lee, S. D.; Schneider, T.; Grant, L. D.; Verkerk, P. J.(eds), Lewis, Chelsea, MI, 1986, pp. 349-75.15. Liss, P. S.; Me&vat, L., Ai-sea gas exchange rates: introduction and synthesis, inThe Role ofAir-Sea Exchange in Geochemical Cycling, Buat-Menard, P. (ed), Reidel,Dordrecht, 1986, pp. 113-27.16. Hicks, B. B.; Liss, P. S., Transfer of SO2 and other reactive gases across the ai-seainterface, Tellus, 1976, 28, 348-54.17. Watson, A. J.; Upstill-Goddard, R. C.; Liss, P. S., Air-sea gas exchange in roughand stormy seas measured by a dual-tracer technique, Nature, 1991, 349, 145-47.