8. Compartmental Modeling of Two Phase Systems: Reactions in Gas and Liquid

Transcription

1 ChE 505 Chapter 8N Updated 0/0/07 8. Compartmental Modelin of Two Phase Systems: Reactions in Gas and Liquid In many situations, includin the atmosphere, reactions occur in more than one phase e.. as/air) and liquid (water). We outline here the eneral approach to modelin such two-phase systems. We consider closed and open systems, equilibrated and nonequilibrated. Later on we extend this to multiphase systems. In this chapter first, we present the eneral methodoloy for treatin well mixed two phase systems. Then we show its application to addressin the sulfur and acid rain problem in the atmosphere. ChE 505 Chapter 8 Addition A Guide throuh Chapter 8 Let us reflect first on what we should have mastered in previous chapters 4 throuh 7. Upon introducin the concept of reaction mechanisms and illustratin how to derive rate forms from them, we applied that knowlede, coupled with the assumption of a well-mixed system, to study ozone depletion in the stratosphere and ozone dynamics in polluted urban atmosphere. Considerations of a sinle phase, as phase, were sufficient in these situations. We proceeded then in Chapter 5 to formalize the methodoloy of obtainin rate data from mechanisms (PSSA and RCSA methods were covered) and we covered various catalytic and noncatalytic reactions includin enzyme promoted ones. In Chapter 6 and 7 we focused on the theory of elementary reactions, which are the basic buildin blocks of any mechanism. We reviewed the kinetic theory of ases, and its use in collision theory, and introduced the classical transition state theory (CTST). Elementary reactions in ases were considered from the viewpoint of these two theories. In Chapter 7 we extended the CTST to elementary reactions in liquids, and illustrated the effect of dielectric constant, ionic strenth and chares in the ions on the prediction of the rate constant. We concluded by explainin the oriin and consequences of diffusion-limited reaction on the rate constant of an elementary reaction. The purpose of Chapter 8 is to introduce the extension of a well-mixed compartmental model to systems (closed and open) that contain two phases as and liquid. The eneral mass balance is presented first for a eneral species that can exist in both phases and that can react in each phase. The use of the developed approach is then illustrated in the problem of acid rain. Sulfur dioxide history in the as and liquid phase is monitored and the dynamics of S(VI) creation and ph of the liquid phase is followed in time. Section 8.1 introduces the nomenclature to be used. Note by the form of the mass transfer flux N l that the flux from as to liquid is defined as positive and that the drivin force for this flux is the difference in partial pressure of in the as phase p, and the partial pressure of that would be in equilibrium with the liquid phase at concentration C L. Hence, once equilibrium between as and liquid is reacted the drivin force oes to zero 1

2 and the net transfer of between as and liquid becomes zero. The Henry s constant` H includes the effect of ionization of the dissolved as (see Chapter ), and if there is no ionization is reduced to the ordinary Henry s constant value H. One should note that in settin the system for use in the sulfur dioxide oxidation problem we introduce the ratio of the volumes of the liquid and as phase L V L /V. Only when we start nelectin L compared to unity do our equations lose enerality and become applicable only to the problems atmospheric chemistry where typically < L < Equations (4) and (4a) present the balance for the as phase and equations (5) and (5a) for the liquid phase. Both are for a closed system. Equation (7) represents the dynamics due to reactions of the equilibrated closed system. Method of findin the rates and procedure to be taken with insoluble or nonvolatile species is outlined. Equations (14a), (14b), 15) are restated as the eneral ones to be solved for a non-equilibrated closed as-liquid system. At that point a procedure is introduced (see additional Hints sub-section) that relies on conservation of the atomic species (that cannot be created or consumed by reaction). This procedure is employed in solvin the sulfur dioxide oxidation problem and the conservation of sulfur equation for the closed system is presented as eq (18). The as phase sulfur content is all in oxidation state four form S(IV), i.e. SO, as indicated by eq (19). In the liquid sulfur exists in S(IV) form (e.. various undissociated and dissociated forms of H SO 4. The rate of chane of S(VI) is iven by equation () where the oxidation rates are contributed by various species. As these rate forms depend on hydroen ion concentration the electroneutrality equation (4) must be solved simultaneously to eq (). Equation () representin the overall sulfur conservation as atomic species must be used also simultaneously to update the SO as phase concentration. This procedure outlines the alorithm that must be used to solve the dynamics of sulfur dioxide oxidation in a closed system. The eneral equations for an open system are shown in eqs. (5a), (5b) (5c). We consider here only the limitin case when only the as phase is constantly refreshed while the liquid phase does not have a continuous input and output. This is often used as an extreme model in atmospheric chemistry. The closed system model represents water droplets that travel with the same mass of air and are exposed to the

3 same air all the time. This limitin open system model assumes that air is abundantly and constantly replenished but the droplets of water are the same. If the replenishment rate of as is very lare then the partial pressure of all components in the as phase is constant and the only equations to be solved are for the reactions in the liquid. Now this becomes a sinle-phase system (liquid) that is fed by constant composition of the surroundin as phase. This is the other extreme model that we will solve for the SO problem. The equations to be solved in this case are eqs (), (a) with eqs (1) and (). Section 8. outlines the story of the sulfur cycle in the atmosphere. The main conclusion to be derived from Section 8..1 on atmospheric chemistry of SO is that as phase reactions are slow (see Table and ). Section 8..1 introduces the water phase reactions of SO oxidation. Table 6 provides the Henry s constants for soluble ases and Table 7 displays the pertinent dissociation constants. The concepts of electroneutrality and modified Henry s constant are reviewed. The effect of partial pressure of SO, of temperature and ph on solubility of al S(IV) is shown in Fiure. Fiure displays the mole fraction of various dissociated and undissociated forms of H SO. Abundance of water in different type of atmosphere is reviewed and its effect on maximum concentration of soluble ases is revealed. Conversion of units and comparison of as and liquid reactions is shown. The section on Reaction Kinetics of Aqueous Sulfur provides the kinetic rates and their constants for sulfur dioxide oxidation by oxyen, ozone, hydroen peroxide, etc. and illustrates the mechanism and rates catalyzed by iron, etc. Table 8 lists the rate expressions and rate constants for aqueous oxidation of S(IV). Table 9 lists the important reactions and dissociations of nitroen oxides and ives the dissociation constant for nitric acid (its Henry s constant is reported in Table 6). The relative manitude of various oxidation rates is shown in Fiure 4. The section on Dynamic Behavior sets up all the equations for the development of a computer alorithm needed to solve the sulfur dioxide oxidation problem. Since we are considerin oxidation of sulfur peroxide by ozone and peroxide and the effect of presence of nitric acid we must write the conservation equations for all these species as shown in the rest of Chapter 8. Now you are armed with the tools to solve HW 4 which is also listed in the chapter. 8.1 Two Phase Systems: Introduction

4 To set up the compartment type model for a two phase system of al mass M and volume V, schematically presented below, we need to make assumptions listed below and keep track of the quantities listed below. Needed assumptions: - Each phase is perfectly mixed - Rate of transport between phases is proportional to a volumetric mass transfer coefficient (mass transfer coefficient times interfacial area per unit volume) and the difference in species concentration in the two phases. We illustrate this below for a as liquid system. 1. Closed System Gas in system of mass M & volume V interfacial area A l V V V L M M M const L Liquid of system of mass M L & volume V L MUST DIFFERENTIATE MOLAR CONCENTRATION AND REACTIONS IN DIFFERENT PHASES C C R R L L mol molar concentration of in as phase m as mol concentration of in liquid phase m liquid mol as phase reaction rate of m as mol liquid phase reacti on rate of m liquid ( ) as liquid interfacial area ( ) A m A a V a l l l m l m specific interfacial areas (per unit volume of al system) 4

5 m as ε as holdupin the system m al K p C H ( atm) mol molar concentration of m liquid m mol liquid x atm ( ε V V ) ( ε 1 ε V V ) m liquid ε L liquid holdupin the system m al L mol N l K ( p C L H ) mass transfer flux of s L mol overallas liquid mass transfer coefficient for atm m s partial pressure of in as phase in liquid phase Henry's constant (equilibrium as L fromas to liquid liquid partition coefficient for. Open System Q in C o GAS V, M Q out C Assume ideal as Q L C ol p C RT p o o C RT A l LIQUID V, M L L Q L C L Note that due to the assumption of perfect mixin the exit system concentration or partial pressure are representative of the conditions in the whole system. Now we develop the model equations for closed, then for open system. Two Phase Systems: Model Development The model is based on the mass balance for species in each compartment (as and liquid) which in eneral can be written as: Rate of Rate of Rate of Rate of accumulation of input of output of eneral of In a closed system the only (rate of input) (rate of output) is by the exchane mass trade flux of from as to liquid. We can open system flow terms in and out also contribution. The rate of eneration is the sum of the reaction rates that occur on that compartment and that produce (consume ) multiplied by appropriate stoichiometric coefficients. 5

6 1. Closed System Let us consider air-water system like in the atmosphere. Then the al volume of the system V V V L (as volume) (liquid volume) (1) Let V L /V G L, and L << 1 such as is the case in the atmosphere. Then V V (1 L) V. (1a) In addition, a species may be found in the as phase at concentration C and in the liquid at concentration C L. When equilibrium is established between the two phases and species does not cause upon dissolution. C L H H C RT p H p () where p (atm) is the partial pressure of in the as phase, R is the ideal as constant while H (mol/l atm) and H are two forms of the Henry's constant (based on ideal system linear equilibrium assumption). For a liquid species that underoes ionization in the liquid phase C L H C H RT p H p (a) and the modified Henry's constant, H, must be used which is a function of ph of the solution and can be readily obtained if the dissociation reactions and their equilibrium constants are known. Based on the introduced notation the rate of transport between the as and liquid phase for species is where N& l mol Ls ( K a )( p C H ) l L / () a dm l - is the interfacial area of the as-liquid interface per unit volume of the system dm K mol - is the overall mass transfer coefficient for dm s atm ( p C H )( atm) L - is the drivin force for transport which oes to zero at equilibrium. The balance on species in the as phase is iven by 6

7 dc V R V N & ls V; t 0, C C o (4) dc R N Ý l (1 L) (4a) R is the reaction rate of production of in the as phase. The balance for species in the liquid phase yields V L dc L L dc L R V Ý L L N l V ; t 0, C L C Lo (5) R L L Ý N l (1 L) (5a) The electroneutrality equation f([h ]) 0 must be solved in conunction with the above. a) Equilibrated System If we assume that equilibrium is always present between as and liquid then & 0 (6) N l And H C d, so that addin eq (4a) and (5a) we et [( 1 H L )C ] R LR L t 0, C C o (7) Now one needs to sort out the reactions in the mechanism proposed for the as phase: S S' υ m i 1 A 0 ; i 1,,...R ' m Where v i is the stoichiometric coefficient of in reaction I occurrin in the as phase; and we have independent as reactions and for the liquid phase (8a) ' R S S' υ m il 1 A 0 ; i 1,,...R L ' (8b) 7

8 m Where v i is the stoichiometric coefficient of in reaction I occurrin in the liquid phase; and we have independent liquid phase reactions ' R L The rate of reaction of is then found by summin the rate of each reaction multiplied by the stoichiometric coefficient of in that reaction, as shown by equation (9a) and (9b) for the as and liquid phase, respectively. R ' R υ m r i (9a) i i 1 and R L R L υ m il r il (9b) i1 where r i, r il is the rate of the i-th reaction in the as and liquid phase, respectively. If we have listed complete mechanisms then each reaction is elementary and its rate law can be written from the law of mass action. Aain, exact and approximate solutions are possible, dependin whether we use PSSA or solve the full system of equations. For the exact solution one must solve equations repeated below as equations (10): d [( 1 H L )C ] R LR L, t 0, C C o (10) for all 1,,,...S, S 1,...S S' for S' intermediates as well as S stable species (reactants or proceeds). For the approximate solution via PSSA one solves S' nonlinear equations for the concentrations of active intermediates R ' R m i i i 1 R'L υ m il i 1 ' υ r 0 ; i 1,... R (11a) ' R L r il 0 ; i 1,,...R L (11b) Then one solves eq (7) for the stable species 1,,...S. Nota bene: Numerical solutions tend to become inaccurate when one attempts to divide by zero or multiply by infinity. Hence, if the aseous species is practically insoluble. H L << 1, i.e H RTL << 1 8

9 then for those species only the as phase reactions should be considered. dc R (1) for 1,..Ns where NS indicates the al number of nonsoluble species. One should pay attention in particular if a hihly insoluble species is created by the liquid phase reaction that one does not attempt to calculate its as phase concentration by C C L / H as this involves division by a very small number. If on the other hand the species is hihly soluble, i.e practically nonvolatile, H L >> 1, i.eh RTL >> 1 then only the liquid phase reactions should be accounted for. dc L R L (1) for all -s of nonvolatile species ( 1, NU, whre NU is the al number of nonvolatile species. For nonvolatile species one should use only liquid phase concentrations C L and not attempt to convert them to as phase concentration C C L / H. b) Non-equilibrated System This results in a lare and stiff set of differential equations since transport terms must be included in the calculation. The set of equations to be solved are: 1 dp R RT ( K a l ) p ( C L / H )(1 L) Q C V in p RT (14a) dc L R L ( K a l ) ( p C L / H ) 1 L L (14b) t 0 p p o, C L C Lo (15) c) Additional Hints It is sometimes useful in solution of the above problems to work in terms of al concentration of atomic species, say a i. 9

10 In a closed system the al concentration of an atomic species is constant. () a i (1 L) () a i La ( i ) L (16) The atomic species concentration in each phase is found by summin the product of the number of atoms of species i present in species, α i and the concentration of species, over all chemical species present in that phase ( 1,...S). S (a i ) α i C (17a) 1 S (a i ) L α i C L 1 (17b) For example for the sulfur dioxide pollution problem the al sulfur concentration must be constant in a closed system. [] S (1 L) [] S LS [] L (1 L) p SO, o RT (18) The constant sulfur concentration is readily obtained from the initial partial pressure of SO assumin no sulfur is initially present in the liquid. Now in the as phase SO is the only form in which S(IV) exists so that [] S [ S(IV )] p So RT (19) In the liquid phase various S(IV) forms exist: SO H O, HSO, SO and S(VI) forms SO H O, HSO 4, SO 4 so that [] S L ( S(IV) ) L SVI ( ) [ ] L (0) If we deal with the equilibrated model then [ S(IV) ] L H S(IV) p SO (1) So that eq (18) can be written as ( 1 L) p SO,o RT (1 L) p SO RT H S( IV) Lp SO L[(S(IV))] L () 10

11 The rate of chane of S(VI) by oxidation in the liquid phase can be written as [ ] ( dsvi N ( ) R ox,s(vi), i () i1 t 0 [ S(VI) ] 0 (a) Here, R ox,s(vi), i is the i-th oxidation rate form (i.e with oxyen, with ozone, with peroxide, catalyzed by iron, etc.) and N forms can be included. In each oxidation rate the liquid phase concentrations can be expressed as C L H p in terms of their current partial pressures, p, except for the nonvolatile species for which C L is used directly. The electroneutrality equation (4) f ([ H ]) 0 (4) must be used and solved at each step to calculate [H ] and evaluate all modified Henry s constants H -s which depend on it. Equation () can be used at each interation step, as [S(VI)] builds up and [ S( IV )] evaluate and update p SO. declines, to For other species that can be found in as and liquid phase an equation of the type of eq ( can be written to update the partial pressure p of such species. Or equation (7) is solved for all such species.. Open System In an open two phase system inputs and outputs of both phases may be present and must be accounted for. The most eneral set of equations is obtained by addin such input and output terms to equations (0a, b) to obtain: 1 dp RT R ( K a l ) p ( C L / H )(1 L) Q ( ) 1 L C V in p RT (5a) ( ) (5b) L dc L LR L (K a l ) p C L / H L Q L C in C L V L t 0 p p o, C L C Lo (5c) For an equilibrated open system we have aain (when Q L 0) and C L H p (6) 11

12 d 1 RT LH p R LR L V RT p in p t 0, p p o Q ( ) (7) (7a) In addition, in the air pollution literature it is often assumed that the dominant term in eq (5a) is the last one, i.e Q /V is hue so that C in p in RT p RT (8) This implies that for all as phase species p p in const p o (9) The only equations to be solved now are for the species in the liquid phase dc (0) L R L t 0 C L C Lo (0a) In this particular case the al concentration of atomic species cannot be assumed constant since the system is open. But the partial pressures of all aseous species are taken as constant. This means that in our SO oxidation example [] S const but eq (, a) are still valid N ds(vi) [ ] R ox,s(vi),i () i 1 t 0 [S(VI)] 0 (a) In the rate forms above, however, one uses now the assined p o values and only updates in time the liquid phase concentration of all species C L and the electroneutrality equation f( [ H ]) O (1) so that the proper solubilities can be calculated for aseous species 1

13 C L H p () Eventually as ph falls as [H ] increases, H phase. drops off and additional SO is hindered from enterin the water 8. THE SULFUR PROBLEM The maor sulfur components in the environment are - carbonyl sulfide, C O S - carbondisulfide, CS - dimethyl sulfide, ( CH) S - hydroen sulfide, H S - sulfur dioxide, S O - sulfate ion, SO 4 - elemental sulfur, S 8 The lobal sulfur cycle is depicted in Fiure 1. Table 1 presents estimates from various sources (clearly there is reat uncertainty). Nevertheless the anthropoenic emissions of S O are an appreciable fraction of the al flux to the atmosphere. Most abundant are COS, CS, ( CH) S and H S. In the troposphere [ COS] 500 ± 50 ppt indicatin a lon life time. [ CS ] 15 to 0 ppt in surface nonurban air and 100 to 00 ppt in polluted urban air. [ CS ] decays rapidly with altitude indicatin short life time. ( CH ) S is most abundant in seawater about 1 x 10-7 (/L). It is produced by alae and bacteria. H S sources are anearobic processes in sulfur rich soils. Concentration levels are not well known. TABLE 1: Estimates of annual fluxes (T yr -1 ) of environmental sulfur a 1

14 14

15 FIGURE 1: The lobal sulfur cycle, showin the maor reservoirs, pathways, and forms of occurrence of sulfur. Fiures enclosed in circles refer to the individual fluxes and correspond to fiures in column, Table 1.. Flux between marine plants and dead oranic matter estimated as 00 T yr -1. (Moss, 1978). 15

16 ChE 505 Chapter 8N Updated 0/08/ Atmospheric Chemistry of S O Thermodynamically the tendency to react is hih SO O SO (1) [ SO ] so that [ SO ] eq 8 x in air at 1 atm and 5 o C. 4 ( ) SO H O H SO aq () Reaction () is in humid atmosphere instantaneous for all practical purposes. However, reaction (1) cannot proceed at atmospheric conditions without a catalyst. Such catalyst may be present in aerosols but the rates at 5 o C are still very slow. SO hν SO () The above photoexcitation of S O is effective only at λ < 18 mm and hence does not occur to a reat extent. Moreover quenchin with N, O and H O occurs 45.7%, 41.7% and 1.% of the time makin this reaction () a neliible source of S O in the atmosphere. Other possible reactions include those with - electronically excited O, reactive molecules and free radicals (e.. O, N O, O H, H O ) - other peroxyalkyl ( CHC ( O) O ) and acyl peroxy radicals. Possible reactions are summarized in Table and their relative importance in Table. By far the most important as phase reaction is the one with hydroxyl radical. OH SO HOSO (4) M The HOSO ultimately leads to HSO. 4 Recent evidence suests that the [O H ] concentration in the photo-oxidizin mixtures of HNO, NO, NO is insensitive to even lare addition of S O. Hence, reaction (4) is slow compared to reactions in that cycle and is the rate determinin step for HSO 4 formation. For a value of the rate constant reported in Table with [O H ] 10 7 (molecules/cm ) we et an estimate that S O would be converted at a rate of.96%/hr or 95%/day. However, usin an improved estimate of the daily averae [O H ] 1.7 x 10 6 (molecules/cm ) we et 0.67%/hr conversion rate or 16.1%/day. Clearly, the main uncertainty is in estimatin the (OH ) concentration. The reactions of reduced sulfur species are reported in Table 4. Here a lower estimate of [O H ] 10 6 molecules/cm is used as these processes tend to occur in a cleaner atmosphere. 16

17 ChE 505 Chapter 8N Updated 0/08/05 TABLE : Rate constants for potentially important reactions of round state SO and SO molecules in the lower atmosphere a TABLE : Estimated relative contributions to SO oxidation by as-phase reactions k SO X 17

18 ChE 505 Chapter 8N Updated 0/08/05 TABLE 4: Reactions of reduced sulfur species: RSH alkane thiols, RSR sulfides, RSSR disulfides. 8. HETEROGENEOUS REACTIONS IN THE ATMOSPHERE The atmosphere in addition to ases contains liquid (aqueous) droplets and particulate matter (aerosols). In addition to the homoeneous as phase reactions discussed thus far, heteroeneous reactions can occur in the droplets or on the surfaces of droplets and of particulate matter. These types of reactions will cause us to examine a few additional concepts such as - phase equilibria - interphase transport - intraphase diffusion and reaction. The particles in the atmosphere vary in size from less than 1 μ m (aerosols), μ m water droplets in fo and clouds, and hydrometeors of 0.1 to mm diameter. Precipitation results from hydrometeors reachin the earth's surface. Let us consider now reactions in such aqueous droplets Aqueous-Atmospheric Reactions - SO Oxidation These reactions are oxidative in nature and involve neutral free radicals, free radical ions, ions, nonradical nonionic species such as H O and O. The best known set of atmospheric aqueous reactions is related to SO oxidation that results in acid rain. SO rises in power plant plumes and reacts in as phase via OH or other radicals, to form H SO 4 which instantly -1 acquires water and condenses. Gas phase reactions can contribute up to 1% hr SO conversion rate. Much hiher observed conversion rates (see Table 5) indicate the importance of heteroeneous reactions. The solubility of definition of phase equilibrium, Henry s constant and modified Henry s constant, see chapter N. Temperature dependence of some Henry s constants is in Table 7. 18

19 ChE 505 Chapter 8N Updated 0/08/05 TABLE 5: Field measurements on the rates of SO oxidation in plumes a Diurnal averae rate 19

20 ChE 505 Chapter 8N Updated 0/08/05 TABLE 6: Henry s Law coefficients of atmospheric ases dissolvin in liquid water a Water Ionization The ionization or recombination reaction for water is for all practical purposes infinitely fast HO H OH so that equilibrium is always established [ ] ' 16 K w H OH / HO 1.8 x10 M at 98K Since molar concentration of water is constant, we et 14 Kw H OH 1.0 x10 M at 98 K 0

21 ChE 505 Chapter 8N Updated 0/08/05 For pure water [H ] [O H - ] M. Recall that p H - lo [H ] so that p H 7 for pure water at 98K. TABLE 7: Thermodynamic data for calculatin temperature dependence of aqueous equilibrium constants Sulfur Dioxide/Water Equilibrium The scenario is similar as in the case of C O absorption SO H O SO H O H ( ) SO SO H O H HSO K S1 HSO H SO K S The concentration of dissolved species are [ SO H O] H SO p SO HSO K s1 p SO HSO H H K K p SO H SO S1 S SO 1

22 ChE 505 Chapter 8N Updated 0/08/05 Satisfyin the electroneutrality relation requires ( ) w SO S1 SO SO S1 S SO H K H K p H H K K p 0 The al dissolved sulfur in oxidation state 4 is S(IV) and its concentration is iven by K K K S IV H p H p H H S1 S1 ( ) SO 1 SO S S ( IV ) SO where H is the modified Henry's constant which we can express as S( IV) H H 1 10 K 10 K K ph ph S( IV) SO S1 S1 S Clearly the modified constant H is dependent on ph. The larer the ph, i.e the more alkaline the solution, the larer the equilibrium content of S (IV ). This is illustrated in Fiure for p SO 0. ppb and 00 ppb. FIGURE : Equilibrium dissolved S(IV) as a function of ph, as-phase partial pressure of SO and temperature

23 ChE 505 Chapter 8N Updated 0/08/05 Let us express now mole fractions of various dissolved sulfur species as function of ph (really these are mole ratios, i.e moles of a particular sulfur species divided by al moles of S (IV ) or mole fractions on water free basis since water is the dominant component. [ SO HO] ph ph 1 α0 XSO HO 1 KS110 KS1K 10 S S( IV) HSO 1 1 ph ph α1 X 1 KS110 KS10 HSO S( IV) SO ph ph α X 1 KS10 ( KS1KS) 10 SO S( IV) Common notation for these are α0, α1, α in aquatic chemistry. 9 Fiure illustrates the three mole fractions for p SO 10 atm (1 ppb). Since these species have different reactivities ph will affect reaction rates. At low ph SO H O dominates, at hih ph all S (IV) is in the form of SO. Intermediate ph contains mainly HSO. FIGURE : Mole fractions and concentrations of the three dissolved S(IV) species, SO H O, HSO, and SO, as a function of ph at T 98 K, and 9 p SO 10 atm (1ppb).

24 ChE 505 Chapter 8N Updated 0/08/05 Abundance of Liquid Water in the Atmosphere Liquid water content is expressed in ( water/m air) or (m water/m air). Let L be the second dimensionless measure. Then clouds L 10-7 to 10-6 fo L 10-8 to 5 x 10-7 aerosols L to The distribution of a species A between as and aqueous phases, in say a cloud, can be expressed as moles of A in solution per m of air moles of A in air per m of air H P L A A PA H RTL RT H A R T L << 1 species A is mainly in the as H A R T L >> 1 species A mainly in the liquid L 10, RTL 1 4 x 10 4 M atm For ( ) A Hence for H A << 4 x 10 4 M atm -1 H A >> 4 x 10 4 M atm -1 species A in the as species A in the liquid Table 6 indicates that, with exception of HC l, HCHO and especially H O and HNO, other species will remain mainly confined to the as phase. 1 For S O at ph 4, S( IV) 10 H S( IV) RTL and most of the (almost all of) the S (IV) is in the as phase. In contrast for H N O the modified Henry's constant is H and all of nitric acid is in solution. Maximum Solubility H and at L 10-6 ( ) For example, the equilibrium dissolved concentration of A is iven by [ A ] H P e A A but the al amount of species present is N C V C V A A AL L V L PA V CA C A V L CA L L V RT N P P C C L V RT RT A A o A A A o L 4

25 ChE 505 Chapter 8N Updated 0/08/05 Since the followin relationships hold C H P AL A A PA o 1 PA PA HAL HARTL RT RT RT PA P o A 1 H RT L C AL A HAPA o 1 H RT L A [ 1 ] Then, when H RT L>> 1 A the maximum concentration of dissolved A is: C AL max P Ao RT L Clearly, then, for a very hihly soluble as we must take precautions in calculatin the equilibrium composition not to violate the mass balance, (i.e "not to dissolve more as than there is available"). The maximum concentration in the liquid reachable by the hihy soluble as is as shown above and repeated here with different notation P RTL A [ A] max and this cannot be exceeded. Say P A 10-9 atm (1 ppb) and L 10-6, [A ] max 4 x 10-5 M. Reardless of the value of H, say H and [A] eq 10 M, clearly [A ] max cannot be exceeded. [A ] max can only be exceeded if the droplets are brouht into contact with much larer volume of air not ust the volume containin L amount of water. Let us now develop a methodoloy for comparison of as and liquid phase reactions. Consider the aqueous reaction ( ) k A S IV products with the rate 1 ( ) [ ( )] ( ) R Ms k Aaq S IV The same rate can be written in terms of equilibrium partial pressures R kh H PP A S( IV) A SO 5

26 ChE 505 Chapter 8N Updated 0/08/05 Note that R is the rate of depletion of A (or S (IV) ) per unit volume of water and per unit time. To convert to the basis of unit volume of air we must multiply by L R a L R (mol/m air s) Further conversion of units is often practiced. For example Ra RT x 9 10 x 600 Pa yields the rate in (ppb h -1 ) based on volume of air. R% a LR [ A] LRT R P A 1 ( s ) is the fractional rate of conversion. Characteristic reaction time is 1 τ R % R a A nomoram for rate conversion is presented in Fiure. FIGURE : Nomoram relatin aqueous reaction rates, in M sec 1, to equivalent as phase reaction rates, in ppb hr -1, at T 98 K, p 1 atm, and iven liquid water content L. The diaonal lines are constant aqueous reaction rates (National Center for Atmospheric Research, 198). 6

27 ChE 505 Chapter 8N Updated 0/08/05 Reaction Kinetics of Aqueous Sulfur Dissolved SO enerates three aqueous S (IV) species: SO H O, HSO and SO. Various oxidants converts S (IV) to S (VI). These oxidants are dissolved oxyen, ozone and hydroen peroxide. Due to their lare Henry's constants the last two are particularly abundant. The reactions can be catalyzed by dissolved metal ions. 1. Uncatalyzed oxidation of S (IV) by dissolved oxyen has been proven to be neliibly slow mainly due to low reactivity of dissolved oxyen and small concentration of dissolved oxyen 1 4 H 1. x10 M atm which yields O H O x10 M. O [ ]. Oxidation of S (IV ) by dissolved ozone proceeds by nucleophilic attack on ozone by SO H O, HOSO and SO with an increasin reactivity. The rate of reaction of S(IV)can be written as ( ) [ ] R ( k X k X k X ) S IV O S( IV) 0 SO HO 1 HSO SO At room temperature k0.4 x10 M s, k1.7 x10 M s, and k 1.5 x10 M s where X is the mole fraction of. [ ] O O O O H P where H 9.4 x10 M atm 1. O.84 x10 to5.6 x10 M. Since At normal tropospheric ozone concentrations of 0-60 ppb, [ ] the rate depends on [S (IV) ], which decreases rapidly at low ph, the rate of reaction slows down at low ph.. Oxidation of S(IV) by H O The prevailin hypothesis is that this oxidation involves mainly the bisulfide ion in formation of peroxymonosulfurous acid as reactive intermediate by the followin mechanism eq k1 f k1b HSO H O SO OOH H O SO OOH H H SO k 4 ' Use of PSSA and definin k k [ H O] yields the followin rate of sulfuric acid formation 1b 1b 7

28 ChE 505 Chapter 8N Updated 0/08/05 k1f k H HSO [ HO] RHSO 4 ' k1 b k H which can be expressed as the rate of S (IV) loss where [ ][ ( )] HSO kh H O S IV X RS( IV) 1 K H k k k k 7.47 x 10 M s and K 1 M at 98K. For ph < 1 rate decreases with ph 1f ' ' k1 b k1 b but above it is independent of ph. 4. Oxidation of S (IV) catalyzed by iron Several modes of interaction with iron and its compounds are possible. - physical adsorption occurs on iron oxide - chemisorption occurs with subsequent catalytic reaction on the particle surface - aerosol particles containin iron oxide serve as catalysts - dissolved iron catalyzes the reaction. Detailed kinetic expression is only available for bulk aqueous phase Fe (III) catalyzed reaction with dissolved oxyen. The reaction rate depends on [S (IV)], [Fe (III)], ph, ionic strenth and temperature. It is very sensitive to the presence of certain anions (e.. SO ) and cations (e.. Mn ). The equilibria involved are: Fe H O FeOH H FeOH H O Fe( OH ) H Fe( OH ) H O Fe( OH ) ( s) H 4 FeOH Fe( OH ) Total soluble iron is: [ Fe ] Fe FeOH Fe ( OH ) Fe ( OH ) 4 sol Total iron present is: [ Fe] [ F ] Fe( OH ) sol 4 8

29 ChE 505 Chapter 8N Updated 0/08/05 The relative amounts of soluble iron species are a stron function of ph. The overall equilibrium between solid Fe( OH ) and Fe is Fe( OH ) H Fe H O with (at 98 o K) K 10 Fe [ H ] At ph 4.5 this yields 11 Fe x 10 M. The followin rate is observed [ ][ ] SO RS( IV) k Fe( III) S( IV) X with k 1. x 10 6 M --1 s -1 at 9K. At ph 4.5 solubility of iron decreases sinificantly and the rate chanes to (at ph 5) [ ] R x Fe S IV. 5 S( IV) 5 10 ( ) When Fe (II) is added to the mixture there is an induction period needed for Fe (II) oxidation before reaction proceeds. Available rate expressions for - oxidation promoted by Mananese - oxidation catalyzed by Mananese and Iron - oxidation by dissolved NO - oxidation by dissolved HNO - oxidation by carbon are summarized in the enclosed Table 8. Needed data for nitroen oxides is in Table 9. Comparison of various rates at an assumed set of conditions is illustrated in Fiure 4. At all ph < 5, H O oxidation rate is the fastest and relatively independent of ph. Catalyzed rates become sinificant at ph > 5. The effect of temperature is as follows. Increased T lowers solubility but increases the rate constant. Except for Mn and Fe the solubility effect dominates, i.e the rate increases with decreasin temperature. 9

30 ChE 505 Chapter 8N Updated 0/08/05 TABLE 8: Rate expressions for sulfate formation in aqueous solution TABLE 9: Equilibrium constants for aqueous-phase nitroen oxide reactions a 0

31 ChE 505 Chapter 8N Updated 0/08/05 FIGURE 4: Comparison of aqueous-phase SO oxidation paths. The rate of conversion of S(IV) to S(VI) as a function of ph. Conditions assumed are: [SO ()]5ppb; [HNO (] ppb; [H O ()] 1 ppb; [NO ()] 1 ppb; [O ()] 50 ppb; [Fe (aq)] x10-7 M; [Mn (aq)] x10-8 M. Dynamic Behavior If instead of the instantaneous values of the rates of conversion at a iven set of conditions we are interested in the variation of species concentration in time, i.e the dynamics of the system, we must set up appropriate species balances based on our control volume and basic conservation equation. The control volume is the larest arbitrarily selected volume of the system within which, at a iven instant of time, there are no spatial radients in composition. Now, to this control volume we apply for the mass of each species of interest our fundamental conservation law ( ) () ( ) ( ) ( ) ( ) ( ) ( ) Rate of Input Rate of Output Rate of Generation Rate of Accumulation 1 4 If our system (control volume) is closed, i.e no material streams cross its boundaries by convection or diffusion, then terms (1) and () are identically zero. Rate of eneration (which is neative if species is depleted) is equal to the rate of accumulation, which is the time rate of chane of the observed quantity. If the system is open then all terms are present, un1less the system is at steady state in which case term (4) is identically zero. If more than one phase exists in the system then the above balance has to be applied to each phase and the input-output terms contain the interphase transfer term (what is input to one phase had to come from the other and hence is the output for the balance on the other phase). Moreover, knowlede of phase 1

32 ChE 505 Chapter 8N Updated 0/08/05 holdup, i.e volume functions occupied by each phase, are needed (e.. knowlede of L in our air droplets system). Consider now a closed system (air droplets) containin P HO const (no reaction). If P HO is the partial pressure of H O, and H O( aq) is the concentration of the dissolved peroxide, then at equilibrium (which is very rapidly established) we have But n n n HO HO ( ) HO ( aq) P HO V V air HO aqv RT HO [ ] ( ) V V P HO HO aq L RT where L Vair V and L HO [ ] ( ) ( ) HO aq H P HO HO L The concentration of the dissolved species is ( aq) [ HO] H RT[ HO] HO 1 L 1 H H RT HO HO HO and the remainin equilibrium vapor pressure is: RT L P HO [ ] RT H O 1 H RT L HO The fraction of al peroxide that resides in the liquid phase is: ( ) [ ] HO aql HHO LRT HO H LRT 1 HO with H HO 7.1 x 10 4 M atm -1 at 98K. Similarly for ozone ( ) [ ] O aq L HO L RT O 1 H O L RT 1 with HO 9.4 x10 M atm.

33 ChE 505 Chapter 8N Updated 0/08/05 This results in the followin comparison: L aqueous fraction HO aqueous fraction O. x 10. x 10. x Fiure 4 indicates that aqueous peroxide concentration rapidly approaches equilibrium value for L<<10-7. FIGURE 4: Aqueous-phase H O as a function of liquid water content L for 1 and 5 ppb H O The same approach can be used for dissociatin species such as HNO PHNO [ HNO] ( HNO( aq) NO ) L RT HNO ( aq) H HNO P HNO Kn 1HNO( aq) HHNO K n1 P HNO NO H H NO H HNO K n1 PHNO / H Total concentration of aqueous species ( ) ( 1 10 ph ) HNO aq NO H K P H P HNO n1 HNO HNO HNO Hence HNO aq NO H RT HNO HNO ( ) [ ] 1 HHNO RT L

34 ChE 505 Chapter 8N Updated 0/08/05 and the al fraction of dissolved species is ( ( ) ) [ ] HNO aq NO L H RT L HNO H RT L 1 Dynamics of Reactions in a Droplet HNO HNO Consider now a droplet immersed at t0 in air containin SO, NH, H O, O and HNO. We write the mechanism, i.e the set of plausible physical and chemical steps involved: SO H O SO H O K SO H O H HSO KS1 HSO H SO K NH HO NH HO H NH H O NH OH K HO HO HO aq H O HO O aq H HNO H O HNO aq H HNO aq H NO Kn ( SO ) ( ) ( ) S ( NH ) ( ) 4 a1 ( ) ( HO ) ( ) ( O ) ( ) ( HNO ) ( ) ( ) and finally HO H OH ( Kw ) All the above absorption and dissociation steps are fast and can be assumed in equilibrium. The appropriate equilibrium constants are iven in parenthesis. Now the rate limitin step is the oxidation of S (IV) to S (VI). If we assume that no metal ions are present then RS( IV) ( k0 [ SO HO] k 1 )[ ] o o HSO k o SO O k p H [ HO][ S( IV) ] X HSO 1 K p H 6 x10 S( IV) NO [ ][ ] where above all [ ] concentrations are for the aqueous species, subscript o refers to ozone and p to peroxide. 4

35 ChE 505 Chapter 8N Updated 0/08/05 Now, since S (VI) is formed, we consider its rapid dissociation also HSO aq H HSO K ( ) ( ) 4 4 S HSO H SO K ( ) 4 4 S 4 In addition the electroneutrality principle holds: H NH 4 OH HSO SO SO 4 HSO 4 NO Clearly, if metal ions are present (especially Mn, Fe, etc.) additional rate forms need to be considered. One also needs to examine the possibility of ammonia oxidation then. We now want to understand the dynamics of a closed system with droplets loadin L. This would be the situation if droplets and air travel toether and do not receive additional input of any species. We would make a balance on as phase. [ ] d SO R SO but the rate of as phase reaction is neliible and this equation can be nelected. The balance in the liquid phase yields where [ ( )] d[ S( VI) ] d S IV RS( IV) HSO RT[ SO ] t 0 [ S( IV) ] [ S( IV) ] o 1 H RT L SO o H H 1 10 K 10 K ph ph SO SO s1 s K K K H SO 1 H H s1 s1 s We reconize that we can express all of the ionic concentrations in terms of the appropriate equilibrium constants, hydroen ion concentrations and as phase concentrations: HSO H K SO s1 H P SO 5

36 ChE 505 Chapter 8N Updated 0/08/05 H SO K s K 1 s SO P SO H HNH K a1 NH 4 P NH H K w OH Kw / H HHNO K n P 1 HNO NO H [ SVI ( )] HSO 4 [ H ] Ks4 1 Ks H [ SVI ( )] SO 4 H H 1 K K K In a closed system s S4 s4 ( ) [ ] [ ] SO HSO SO SO H O L P SO S( IV ) L RT [ ] [ ] SO P SO RT But SO [ SIV ( )] H P [ SO] H RT SO SO 1 HSO RT L and P SO [ ] RT SO 1 H RT L SO It follows then that H RT 1 HSO RT L o H RT 1 H RT L SO [ SIV ( )] [ SIV ( )] [ SO] o SO SO [ SO ] o In the rate form we need to substitute all concentrations now in terms of [ ] SO or other known quantities 6

37 ChE 505 Chapter 8N Updated 0/08/05 SO [ SO H O] H P [ SO ] SO SO 1 HSO RT L [ SO ] SO s1 H 1 HSO RT L H RT H K RT HSO H SO K [ ] s K 1 s RT SO SO H 1 H SO RT L HO RT [ O] [ O] 1 H RT L H O RT HO [ HO] [ HO] 1 HHORT L H 1 H SO [ SIV ( )] [ SO] X HSO SO RT RT L HSO H K [ SIV ( )] H H HNO RT NO 1 H RT L SO S1 SO [ NO ] [ ] In addition NO K K K H H H H H S1 S1 S SO S ( IV ) SO 1 Finally the needed [H ]is obtained from the electroneutrality equation. H K RT H NH H NH a1 Kw H RTL ( ) [ ] 1 NH K HSO K w ( ) [ ] s RT 1 SO H H 1 HSO RT L HSO K [ ] s K 1 s RT SO H 1 HSO RT L SIV ( ) HSO RT/ 1 H o SO RTL SO [ ] ( ) [ ] H H 1 K K K s s4 s4 7

38 ChE 505 Chapter 8N Updated 0/08/05 [ ] ( SIV ( ) H ) [ ] SO RT/ 1 H o SO RTL SO H Ks 4 1 Ks H HNO K [ ] n RT NO 1 H ( 1 HNO RTL) Now we write the mass balance for other soluble species that participate in the S( IV ) oxidation reaction such as ozone, hydroen peroxide, and nitroen dioxide. Appropriate differential equations have to be set for all of those and with [ ] d O ( k0 [ SO HO] k 10 HSO k 0 SO )[ O] O [ ] 1 O [ ] d O dp d O L RT 1 RT dp O R O where - R is the ozone decomposition rate in the as phase (due to OH or NO reactions). Similarly O and [ ] K [ ][ ( )] P H H O S IV X HSO d H O 1 K P H [ ] 1 HO [ ] d H O dp d H O L RT with 1 HO RT dp R HO where - R HO is the peroxide decomposition rate in the as phase. Finally for [NO ] 8

39 ChE 505 Chapter 8N Updated 0/08/05 d[ NO ] 1 dpno d[ NO ] ( aq) L RT where [ ] d NO with 6 [ ][ ] 4 x 10 SIV ( ) NO dp 1 NO RT R NO where - R NO is the rate of disappearance of NO via as phase reactions. We can interate now the differential equations for S( VI),[ O],[ HO] and [ ] NO assumin instantaneous equilibration between the as and liquid phase. The initial conditions are: At t0 [ S( VI )] [ O ] ( H O RTL 1)( P ) O [ H O ] ( H RTL 1) P RT H O H OO0 [ N ] ( H RTL 1) P RT O 0 NO 0 NOo RT The electroneutrality balance equation must also be solved at t0 to evaluate the initial values of the concentrations of all ionized species and to obtain the initial value of all rates. After each interation step, considerin that this is a closed system, the al concentration of S ( IV) can be updated usin the requirement that al sulfur concentration must be constant. H LRT P S( VI) L S( IV) 1 H LRT RT S( IV) S IV ( ) SO o Since S( VI ) is obtained at next time step by interation, S( IV ) can be obtained from the balance equation shown. The electroneutrality balance is rerun due to presence of S( VI ) species and the new partial pressure of SO can be evaluated ( ) S( IV) SO S VI H P toether with all other S( IV ) species. 9

40 ChE 505 Chapter 8N Updated 0/08/05 We need to interate all of these differential equations simultaneously and at each interation step calculate [H ] from the electroneutrality balance. In usin the balance one should not foret that H are also functions of [H ]. Consider now the system with the followin initial conditions at t 0 [ ] [ NO ] [ ] [ O ] [ H O ] SO 5 ppb ph ppb NH 5 ppb L 10 5 ppb 1 ppb 6 and compute [S (IV) ] in μ M as a function of time in minutes in a closed system assumin absence of as phase reactions. Now consider an open system, i.e a system in which droplets are constantly exposed to the air of the same partial pressures as corresponds to the above initial al concentrations. This means that partial pressures are fixed at P SO P NH 9 PHNO 8.54 x 10 ppb P O P HO.0 ppb 1.87 ppb 5 ppb 0.65 ppb Now the differential equation to be solved is [ ( )] d S IV RS( IV) where all concentrations are eliminated in terms of [H ] and partial pressures P, P, P, PO, P. The [H ] concentration is still obtained from the electroneutrality balance. SO NH HNO HO Comparison of closed and open system is shown in Fiures 1 and 1. Less sulfate is produced in the closed system due to depletion of H O and O. ph increases less in the open system due to increased neutralization of more sulfate formed with NH. The low ph in the closed system keeps SO out i.e reduces dissolution and depresses sulfate formation via O. Total HNO decreases in the open system due to decrease in ph. 40

41 ChE 505 Chapter 8N Updated 0/08/05 FIGURE 5: Aqueous sulfate concentration as a function of time for both open and closed systems. The conditions for the simulation are: [S(IV)]al 5 ppb; [NH ]al 5 ppb; [HNO ]al 1 ppb; [O ]al 5 ppb; [H O ]al 1 ppb; L 10-6 ; ph FIGURE 6: ph as a function of time for both open and closed systems as Fiure 1. 41