JOURAL OF CHEMICAL PHYSICS VOLUME 108, UMBER 16 22 APRIL 1998 Binry nuletion of sulfuri id-ter: Monte Crlo simultion I. Kusk, Z.-G. Wng, nd J. H. Seinfeld Deprtment of Chemil Engineering, Cliforni Institute of Tehnology, Psden, Cliforni 91125 Reeived 11 ovember 1997; epted 22 Jnury 1998 We hve developed lssil mehnil model for the H 2 SO 4 /H 2 O binry system. Monte Crlo simultion s performed in mixed ensemble, in hih the number of sulfuri id moleules is fixed hile tht of ter moleules is lloed to flutute. Simultion in this ensemble is omputtionlly effiient ompred to onventionl nonil simultion, both in smpling very different onfigurtions of lusters relevnt in nuletion nd in evluting the free energy of luster formtion. The simultion yields moleulr level informtion, suh s the shpe of the lusters nd the dissoition behvior of the id moleule in the luster. Our results indite tht the lusters re highly nonspheril s result of the nisotropi intermoleulr intertions nd tht luster ith given number of id moleules hs severl very different onformtions, hih re lose in free energy nd hene eqully relevnt in nuletion. The dissoition behvior of H 2 SO 4 in luster differs mrkedly from tht in bulk solution nd depends sensitively on the ssumed vlue of the free energy f hb of the dissoition retion H 2 SO 4 H 2 O HSO 4 H O. In smll luster, no dissoition is observed. As the luster size beomes lrger, the probbility of hving n HSO 4 H O ion pir inreses. Hoever, in lusters relevnt in nuletion, the resulting ion pirs remin in ontt; bout 240 ter moleules re required to observe behvior tht resembles tht in bulk solution. If lrger vlue of f hb is ssumed to reflet its unertinty, the probbility of dissoition beomes negligible. A reversible ork surfe obtined for ondition typil of vpor to liquid nuletion suggests tht the rte-limiting step of ne prtile formtion is binry ollision of to hydrted sulfuri id moleules. The ion pirs formed by dissoition ply key role in stbilizing the resulting luster. The reversible ork surfe is sensitive to the ssumed vlue of f hb, thus pointing to the need for n urte estimte of the quntity either by b initio lultions or experiments. 1998 Amerin Institute of Physis. S0021-96069851716- I. ITRODUCTIO The theory of binry nuletion dtes bk to pper by Reiss. 1 Despite suessive modifition ounting for trnsient behviors nd pths in the viinity of the sddle point, 2 7 the theory is n extension of the lssil nuletion theory for single omponent systems. While the lssil theory 8 10 is presently the only prtil pproh for prediting nuletion rtes, limittions of the theory, rising from its mrosopi nture, re ell knon. Thus, there hs been gret interest in estblishing moleulr level pprohes to nuletion. One of the most importnt binry nuleting systems is H 2 SO 4 /H 2 O, to hih numerous ppers re devoted on both theoretil 11 26 nd experimentl 27 2 fronts. Comprisons of the lssil preditions ith experimentl dt for H 2 SO 4 /H 2 O nuletion, hile not extensive, yield onfliting results s to the vlidity of the lssil theory. 19,28,29,1,2 Thus, it is of gret interest on both fundmentl nd prtil grounds, to seek moleulr level desription of binry H 2 SO 4 /H 2 O nuletion. There re to mjor trends in developing moleulr level theory of nuletion. One is moleulr level simultion, 41 hih n be pplied regrdless of the omplexity of the intermoleulr intertion. Hoever, the free energy of luster, the quntity of entrl importne in nuletion theory, is usully evluted by integrting its internl energy obtined t different tempertures from seprte simultions. This spet renders the pproh omputtionlly demnding nd, s result, virtully ll of the simultions re limited to single omponent system. The lterntive pproh is to use sttistil mehnil density funtionl theory, 42 first pplied to homogeneous nuletion by Oxtoby nd Evns. 4 In this pproh, the grnd potentil of the system is ritten s funtionl of order prmeters. Then, the sttionrity ondition of the grnd potentil determines the order prmeters for the ritil nuleus nd the orresponding grnd potentil follos from the funtionl. When luster possesses high degree of symmetry nd the intermoleulr potentil is reltively simple, this pproh is omputtionlly fr less demnding, lloing one to investigte muh ider rnge of the prmeter spe. Thus, one usully employs model potentils tht pture the essentil fetures of the moleules under onsidertion. When the results re ompred ginst preditions from the lssil theory, hih uses bulk thermodynmi quntities obtined from the sme theoretil frmeork, one n isolte the devitions from the lssil preditions rising from the moleulr level detils. Although the pproh is more pproximte ompred to moleulr simultion, the theory is extremely poerful in ddressing the devitions in semiquntittive mnner nd hs been employed to investigte severl interesting systems 44 52 for hih no simultion hs 0021-9606/98/108(16)/6829/20/$15.00 6829 1998 Amerin Institute of Physis
680 J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld been performed. The theory s reently pplied to ternry system. 5 In the present se of the H 2 SO 4 /H 2 O binry system, hoever, density funtionl theory does not offer ny dvntge over simultion, for the intermoleulr potentils tht n fithfully represent the system re very omplited nd s result of the strongly nisotropi intermoleulr intertions, smll luster is expeted to be highly non-spheril. Consequently, e resort to method of omputer simultion, in prtiulr Monte Crlo simultion, sine e re primrily onerned ith equilibrium properties of the lusters. To our knoledge, the present ork is the first exmple in hih n extensive evlution is mde of the free energy of luster formtion in binry system. The lk of simultion in binry system is result of the extensive omputtion involved in the free energy lultion. Thus, one of our gols in the present ork is to estblish simultion tehnique tht onsiderbly redues the omputtionl effort. The intermoleulr intertion potentil is the fundmentl informtion prerequisite in pplying moleulr theory. To reprodue this potentil in simple y, one usully represents moleule s set of intertion sites rigidly held together in its representtive geometry. The intertion prmeters re then optimized to reprodue b initio results for geometries nd energies of smll lusters or ertin bulk thermodynmi properties. The min onern in this pproh is the qulity of the potentil thus obtined. For exmple, the neessity of inorporting moleulr polrizbilities or three-body potentils in urtely reproduing the energy nd the geometry of luster is often stressed. 54,55 In the se of luster of highly ssoitive moleules, hoever, simultion must be performed for very long period of time to smple fithfully the relevnt prts of the phse spe. For ter luster, for exmple, this is neessry beuse the strong intermoleulr intertion rising from n extensive hydrogen bond netork hinders the evolution of the luster from one struture to nother, even though very different strutures hve to be tken into ount in evluting the thermodynmi properties of the luster. In the present se of the H 2 SO 4 /H 2 O system, even stronger hydrogen bonds re expeted to be involved, 56 s is suggested by the reltively high boiling point of the pure id (0 C. Thus, in order to perform fesible lultion to obtin the free energy of the lusters, e must be ontent ith simple model potentils. onetheless, simultion provides signifint moleulr level insight tht is otherise unttinble. Sine the size of typil luster, ontining fe id moleules nd severl tens of ter moleules, is smller thn the Bjerrum length, the distne t hih the Coulombi intertion of pir of ions beomes omprble ith the therml energy, the dissoition behvior of sulfuri id moleule in the luster n be quite different from tht in bulk solution. In the lssil nuletion theory, hoever, the reversible ork of luster formtion is expressed in terms of bulk thermodynmi quntities. It follos tht the reversible ork estimted by the lssil theory my not reflet the true dissoition behvior in the luster, hose effet n be ddressed only by moleulr pproh. Another oneptully, if not quntittively, importnt problem reltes to the very foundtion of the thermodynmis of interfes. Unless luster is spheril on verge, the formlism loses its vlidity s the thikness of the interfil region beomes non-negligible ompred to the size of the luster itself. 57 As mentioned bove, hoever, smll luster is expeted to be highly nonspheril even fter therml verge is tken. Given the limittions on the qulity of the intermoleulr potentils, sensible y to ddress the effet of the moleulr level detils is to ompre the results from simultion ith the lssil preditions obtined by using bulk thermodynmi properties for the system ith the sme model potentils used in simultion. Computtion involved in determining these properties from simultion ith the required ury is demnding nd beyond the sope of this ork. Insted, e shll diretly ompre the reversible ork surfe from simultion ith the lssil preditions for rel H 2 SO 4 /H 2 O system. This pper is orgnized s follos. The model potentil is developed in Se. II. In Se. III, e derive the expression for the reversible ork of luster formtion. In this setion e employ mixed ensemble, in hih the number of ter moleules is lloed to flutute hile tht of id moleules is fixed. As result, luster is hrterized solely by the number of id moleules inside it. In Se. IV, e hrterize luster by the numbers of moleules of both ter nd id nd derive the expressions for the reversible ork nd the equilibrium distribution for this luster. The resulting expressions n be redily evluted from the results of the mixed ensemble simultion. Setion V desribes ertin detils of the simultion. Results of simultion re presented in Se. VI, here omprison is mde beteen the reversible ork from moleulr simultion nd tht from the lssil theory. The pper onludes in Se. VII ith brief disussion on the implitions of our ork. II. MODEL Given the possibility of dissoition nd protontion of the moleules involved, oneptully the simplest pproh is to represent the H 2 SO 4 /H 2 O system s mixture of retive speies of ter, sulfte ion, nd proton. In the initil stge of this ork, e investigted this possibility 58 nd enountered serious diffiulties. Briefly, to tke dvntge of the existing models for ter 59 nd sulfte ion, 60 proton s desribed s unit hrge e, here e is the hrge of n eletron, ith Lennrd-Jones prmeters tht reprodue O-H bond energies of hydronium ion nd sulfuri id. Sine both bond energies re of the order of 10 2 kl/mol, hoever, protons rrely hnge their positions during simultion. Consequently, the system beomes loked into lol minimum ditted minly by the initil onfigurtion of protons. This behvior is in serious ontrdition ith the experimentl vlue of the tivtion energy for proton trnsfer in ter, hih is estimted to be bout fe kl/mol. 61,62 An lterntive model s proposed by Hle nd Kthmnn, 6 in hih prtil hrge less thn e is ssigned to proton. Although the model nnot desribe either dissoition of id or protontion of ter, nd hene is inpplible for the present purpose of ddressing the disso-
J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld 681 TABLE I. Intertion prmeters for ter nd hydronium ion. Wter Hydronium ion A 2 O kl Å 12 /mol] 0.629510 6 0.89110 6 C 2 O kl Å 6 /mol 0.6254510 0.652810 z O 0.8476e 0.2480e z H 0.428e 0.4160e ro-h Å 1.000 0.97 H-O-H 109.47 111.6 ition behvior in luster, it n expliitly inorporte internl rottions nd vibrtions of O-H bonds in sulfuri id moleule. Suh internl degrees of freedom n be importnt. In H 2 SO 4 H 2 O dimer in vuum, for exmple, the potentil energy brrier for internl rottion of the O-H bonds of the id moleule is estimted to be 5.2 kl/mol by b initio lultion t the MP2/-21g** level of theory, 64 suggesting possibility of more thn one onformtion of the dimer. Some of these onformtions beome unvilble to the dimer s it is trnsferred from the vpor phse to the bulk liquid solution simply beuse of the dissoition of the id moleule. This indites tht the ontribution to the dimer free energy from ertin onformtions is not inluded in the lssil desription hih relies on bulk thermodynmi properties. Thus, the loss of n vilble onformtion n ontribute to n error in the lssil free energy predition. Attempts to develop dissoitive model pose serious problems. ote tht proton in dissoitive model must rry unit hrge of e. In relity, hoever, its effetive hrge ertinly is less thn e, sine proton exists by forming hemil bond to n oxygen from ter or sulfte ion. Moreover, hen one of the O-H bonds is broken nd nother is formed, the entire eletroni struture of the moleules involved hnges. This is inherently quntum mehnil effet, hih, hen reprodued in the relm of lssil mehnis, requires firly detiled intertion potentils inluding expliit polrizbilities nd three-body potentils, s illustrted by dissoitive model for pure ter. 65,66 To develop dissoitive model for the present system, substntil mount of dt on energies nd strutures of sulfuri id-ter omplexes is required. In this ontext, e note tht the three-body potentil n be quite sensitive to the ext environment. Thus, the dissoitive model for pure ter 65,66 is unlikely to remin vlid in the present system. Hoever, vilble literture deling ith suh omplexes is very limited. Apprently, only one suh pper exists, 67 hih provides intertion energies ith prtilly optimized geometries of the neutrl omplex H 2 SO 4 H 2 O nd the ioni omplex HSO 4 H O. The lk of neessry informtion preludes n effort to develop detiled intertion potentil. An b initio moleulr dynmis 68 71 pproh, in hih tomi nulei re treted lssil mehnilly hile the eletroni degrees of freedom re treted expliitly by quntum mehnil density funtionl theory, 72 ppers to be n interesting lterntive. In this pproh, no model potentil needs to be speified, rther it is lulted on-the-fly during the ourse of simultion. Hoever, its pplition is urrently limited to investigting dynmis tht our on the time sle of the order of pioseonds nd is not yet prtil in evluting free energy. The pproh dopted here is to tret the dissoited nd undissoited sttes s distint. Sine proton exists primrily s prt of either sulfuri id or hydronium ion rther thn s free ion, nd the seond dissoition of sulfuri id is negligible ompred to the first, one n introdue ter, hydronium ion, sulfuri id, nd bisulfte ion s the onstituent moleules. A moleule is modeled s set of intertion sites rigidly held together t representtive geometry. It is ell knon tht nondditive intertions, rising from moleulr polrizbilities nd three-body intertions, ply n importnt role in both the energetis nd the strutures of lusters. 54,55 onetheless, the enormous omputtionl effort involved in free energy lultion nd the lk of either experimentl or quntum mehnil dt indite one should dopt the simplest possible potentil, nmely pirise dditive potentil. Even then, in this four omponent system, there re s mny s ten distint intertion pirs, hile the vilble dt over only four of them. For exmple, in ddition to the dt on the to omplexes mentioned bove, vrious ter-ter intertion potentils re vilble. Protonted ter lusters hve been fous of extensive studies on both experimentl 7 76 nd theoretil 77 81 fronts. To estimte the reminder of the intertions, hoever, one needs to invoke some kind of mixing rule, hih in turn requires tht ll of the pir intertions be desribed by the sme type of funtion. In this ork, the intertion potentil u beteen moleules nd is ssumed to be the sum of Coulombi nd Lennrd-Jones intertions, u i on j on z iz j A ia j r ij r C ic j 12 6, 1 ij r ij here r ij is the distne beteen sites i nd j nd the summtion is tken over ll intermoleulr intertion sites. The model prmeters re given in Tbles I III. The nottions for the intertion sites for sulfuri id nd bisulfte ion re defined in Fig. 1. The geometries for these speies re given in Tble IV nd disussed belo. Prmeters for ter orrespond to the SPC/E potentil, 59 hih is knon to reprodue ertin bulk properties of pure ter inluding the oexistene densities 82 nd the surfe tension. 8,84 Prmeters for the other moleules require explntion. As mentioned bove, protonted ter lusters hve been studied extensively nd vrious intertion potentils hve been proposed. 85 88 Hoever, none ssumes form s simple s Eq. 1. Thus, it is neessry to develop potentil s follos. First, the geometry of the hydronium ion is tken TABLE II. Lennrd-Jones prmeters of sulfuri id moleule nd bisulfte ion. Sulfur site Oxygen sites A 2 i kl Å 12 /mol 0.400610 7 0.765110 6 C 2 i kl Å 6 /mol 0.2001610 4 0.7815410
682 J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld TABLE III. Prtil hrges on H 2 SO 4 nd HSO 4. Intertion site H 2 SO 4 HSO 4 S 2.8528 e 2.8272 e O 1 1.025 e 1.2942 e O 1 1.025 e 1.1482 e O 2 0.9582 e 0.9615 e O 2 0.9582 e 0.9615 e H 1 0.564 e H 1 0.564 e 0.582 e from the urte theoretil determintion by Rodell nd Rdom. 89 Eh tomi site bers prtil hrge hile the Lennrd-Jones prmeters re ssigned only for the oxygen sites. Then, these prmeters re tuned to reprodue the experimentlly determined enthlpy of hydrtion of the ion hydrted up to six ter moleules. 7 The differene beteen enthlpy nd internl energy is ignored for simpliity. In prmetrizing the prtil hrge on oxygen, either negtive or positive hrge n be ssigned, yielding similr overll greement to the experimentl enthlpy of hydrtion. Hoever, the ssignment of negtive hrge resulted in intertion prmeters tht re loser to the vlues for SPC/E ter nd is preferred on the bsis tht the protontion of moleule should not signifintly hnge its intertion prmeters. The resulting hydrtion energies re ompred ith experimentl vlues in Tble. V. Exept for the first fe hydrtes, the greement is fir. The experimentlly determined geometry is dopted for sulfuri id 56 s shon in Fig. 1 nd Tble IV. A prtilly optimized geometry for bisulfte ion s obtined by Kurdi nd Kohnski 67 using b initio lultion. In their lultion, S-O 1 -H 1 114 s obtined. Hoever, the vlue is probbly n overestimte insofr s the orresponding vlue for sulfuri id s overestimted by 11.5. In the present ork, e simply use the sme vlue s for sulfuri id. We lso ignore the slight (0.5 ) devition beteen the C 2 xes of the ngle O 1 -S-O 1 nd O 2 vsvo 2. In other ords, TABLE IV. Sulfuri id moleule geometry dopted in this ork. r(o 1 -H 1 ) 0.97 Å r(s-o 1 ) 1.574 Å r(so 2 ) 1.422 Å (H 1 -O 1 -S 108.5 (O 1 -S-O 1 101. (O 2 SO 2 ) 12. (H 1 O 1 SO 2 20.8 (P 1 P 2 b 88.4 Looking don the O 1 -S bond. The H 1 O 1 projetion must be rotted lokise by 20.8 to be is to the O 2 -S bond. b Angle beteen O 1 SO 1 nd O 2 SO 2 plnes. The smll devition from extly perpendiulr plnes brings O 2 nd O 2 loser to O 1 nd O 1, respetively. bisulfte ion s obtined from sulfuri id by removing the proton H 1 nd shortening the bond length r(s-o 1 ) to 1.48 Å. In determining intertion prmeters for these moleules, b initio results 67 on the potentil energies nd geometries of H 2 SO 4 H 2 O nd HSO 4 H O ere used. The experimentlly obtined dipole moment of sulfuri id 56 provided dditionl informtion. The informtion, hoever, is simply not suffiiently extensive to determine ll of the prmeters. To further filitte the prmetriztion, the Lennrd-Jones prmeters for these moleules ere ssumed to be the sme s those of sulfte ion 60 nd only the prtil hrges ere djusted. In ddition, prtil hrges on O 2 nd O 2 sites of bisulfte ion re ssumed to be equl, though these to sites re not equivlent. The resulting intertion energies nd geometries re ompred ith the b initio results in Tbles VI nd VII for sulfuri id nd bisulfte, respetively. In Tble VI, other sets of b initio results re shon. Judged from the level of the theory, the results by Morokum nd Mugurum 90 re perhps the most urte. onetheless, e employ the results by Kurdi nd Kohnski 67 sine the orresponding dt for HSO 4 H O re vilble only in their pper. Beuse of some model prmeters left undjusted, the model is expeted to be suffiiently flexible to inorporte dditionl dt s they beome vilble. An itertive method similr to tht suggested by Hlley et l. 66 is used to find the optimum sets of prmeters. First, simultion s rried out ith some resonble vlues of the prmeters, for hih the differene beteen the lulted nd ext energies nd strutures of the omplexes ere evluted. e sets of prmeters ere obtined by TABLE V. Enthlpy of hydrtion from simultion nd experimentl vlues. n1,n U n1,n kl/mol,b H n1,n kl/mol, 0,1 25.5 1.6 1,2 2.74 19.5 2, 21.98 17.9,4 11.42 12.7 4,5 10.92 11.6 5,6 10.4 10.7 FIG. 1. Model of sulfuri id moleule shoing nottions nd C 2 xis. U n1,n U n U n1, likeise for H n1,n. b This ork. Simultion performed t 298.15 K. Experimentl dt of Lu et l. Ref. 7.
J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld 68 TABLE VI. Energy nd geometry of H 2 SO 4 H 2 O neutrl omplex. Comprison beteen simultion nd ext vlues. This ork Experiment b Kurdi Morokum d Ly e Energy kl/mol 15.76 15.769 10.00 25.7 O-O 1 distne Å f 2.662 2.656 2.696 2.55 Dipole moment of H 2 SO 4 debye 2.718 2.72515 2.2989 Simultion s rried out t 0.1 K. b Kuzkoski et l. Ref. 56. Ab initio SCF-MO-LCGO lultion by Kurdi nd Kohnski Ref. 67. d Ab initio lultion by Morokum nd Mugurum Ref. 90 t the fourth-order MP4SDQ level ith zero-point orretion. The vlues ited here re estimted from their figures. e Ab initio lultion by Ly Ref. 64 t the MP2/-21g** level. f O is on ter hile O 1 is on sulfuri id. rndomly perturbing the old prmeters, hih in turn ere used to generte the next tril prmeters only if they resulted in derese of the devition. The proess s repeted until no further derese of the devition is hieved. III. REVERSIBLE WORK OF CLUSTER FORMATIO TABLE VII. Energy nd geometry of HSO 4 H O ioni omplex. Comprison beteen simultion nd b initio results. This ork Ext vlue b Energy kl/mol 16.9 16.9 O-O 1 distne Å 2.80 2.26 O 1 -H 1 -O 174.6 180 O 1 -H-O 168.9 180 Simultion s rried out t 0.1 K. b Ab initio SCF-MO-LCGO lultion by Kurdi nd Kohnski Ref. 67. H nd O re the tomi sites on hydronium ion, O 1 is on sulfuri id, nd H 1 refers to the position of the proton if the S-O 1 bond is shortened to 1.48 Å ithout removing H 1 from H 2 SO 4. In this setion, e derive sttistil mehnil expression for the reversible ork of luster formtion from H 2 SO 4 /H 2 O binry vpor. Sine the vpor phse serves s referene stte in lulting the reversible ork, its preise nture hs to be speified first. Most of the id moleules in the vpor exist s hydrtes so tht the number of the sulfuri id monomers is signifintly smller thn the totl number of id moleules. 14,15,18,19,22,2,91 In deriving the sttistil mehnil expression for the reversible ork of luster formtion, it is most onvenient to tke referene stte in hih moleules exist s monomers forming n idel gs mixture. One the reversible ork is obtined s funtion of monomer onentrtion of id moleules, it is n esy tsk, if so desired, to re-express it s funtion of the number density of ll sulfuri id moleules lulted regrdless of hydrtion stte. 14,18,22 The sme pplies to the totl number of ter moleules if hydrte formtion leds to serious depletion of ter. 22,92 Inside the vpor phse, e tke system of volume V hih stisfies the folloing to onditions. 9 On one hnd, V is suffiiently mrosopi in the sense tht its oupling ith the surrounding vpor is suffiiently ek. Then the sttistil properties of the system re determined by the grnd nonil ensemble. 94 On the other hnd, V is smll enough tht the probbility of finding more thn one unorrelted density flutution tht prtiiptes in the nuletion proess t ny instnt is negligible, hih implies tht there is t most one luster in the system. Then, the reversible ork W rev to form luster inside V from the referene stte is given by W rev log, 2 r here (k B T) 1 ith k B nd T being the Boltzmnn onstnt nd the bsolute temperture, respetively. r is the prtition funtion of the system onstrined to be in the referene stte, hile is evluted under the onstrint tht the system ontins luster. Assuming the idel gs behvior in the referene stte, r T,V,, s 0 1! 0 1! q e V q se s V e n n s V, s here the subsripts nd s refer to ter nd sulfuri id, respetively. The moleulr prtition funtions of the moleule re given by q nd, the former rising from the internl degrees of freedom of the moleule nd the ltter from the kineti energy of trnsltion nd rottion. is the hemil potentil of the omponent in the referene stte nd rises from the integrtion over the orienttionl oordintes of moleule. Here, the symmetry number of moleule is bsorbed in q. Finlly, n is the number density of the omponent in the referene stte, here e hve mde use of the ft tht n q e. 4 In single omponent system, luster simultion is ommonly relized by onfining fixed number of moleules, sy i, in spheril ontiner of volume v onentri ith the enter of mss of i moleules. To the extent tht these moleules tully form luster nd the thermodynmi properties of the luster re nerly independent of v over ide rnge of v, Lee et l. 4 hrterized the luster by its size i lone. The exlusive use of nonil ensemble in luster simultion stems from the ft tht lusters re
684 J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld unstble ith respet to hnge in their size. In the H 2 SO 4 /H 2 O binry system, there re s mny s 00 lusters ssuming tht the mximum numbers of H 2 SO 4 nd H 2 O moleules in luster re nd 100, respetively. If nonil ensemble is employed, expensive thermodynmi integrtion hs to be rried out to evlute the free energy of eh luster. One n signifintly redue the omputtionl effort by devising simultion tht preferentilly smples lusters relevnt in nuletion, i.e. those found long the vlley pssing through the sddle point of the reversible ork surfe. Under onditions typil of sulfuri id-ter binry nuletion, n s n. Moreover, hen the reltive humidity is less thn 100%, the sulfuri id hydrte nnot gro indefinitely ithout quiring more id moleules. Thus, during the time period required for luster to either quire or lose n id moleule, the luster rehes stble prtil equilibrium ith respet to its internl degrees of freedom nd the exhnge of ter moleules. 12,14,15 These lusters in stble prtil equilibrium re, in ft, those found long the vlley of the free energy surfe. Thus if e employ the mixed ensemble in hih the number of ter moleules is lloed to flutute hile tht of sulfuri id moleules is fixed, simultion ill preferentilly generte the lusters relevnt in nuletion. In this pproh, luster is hrterized only by the number of sulfuri id moleules in it. For brevity, e refer to this luster s the luster. Our gol here is to express in terms of the prtition funtion of the luster in the mixed ensemble. As e shll disuss in Se. IV, the free energy of the luster hrterized by both nd, the ltter being the number of ter moleules in it, n be esily obtined from simultion on the luster. Reently, e developed ne pproh to luster simultion in single omponent system using grnd nonil ensemble. 95 The method is free of ny rbitrriness involved in the definition of luster. Insted, it preferentilly genertes the physil lusters, defined s the density flututions tht prtiipte in nuletion, 5 40 nd diretly determines their equilibrium distribution ithout the omputtionlly demnding free energy evlution. In the present se of H 2 SO 4 /H 2 O binry system, hoever, Monte Crlo moves to rete or nnihilte n id moleule ill rrely be epted sine hydrogen bond netork ill be seriously disturbed in the proess. The present pproh of using the mixed ensemble n be regrded s n pplition of the grnd nonil ensemble pproh to heterogeneous nuletion, in hih fixed number of id moleules in the luster, s hole, re regrded s heterogeneous nuletion site. We first obtin the prtition funtion for the mixed ensemble under the onstrint tht the system ontins n luster. One is obtined, follos immeditely. Sine n s is mny orders of mgnitude smller thn the orresponding vlue in the luster, e n divide the id moleules in the system to belonging to the luster nd v regrded s prt of the vpor. Although no dissoition is lloed in the vpor in ord ith the hoie of the referene stte, possible dissoition in the luster is n essentil feture of the system. We denote the number of resulting bisulfte ions nd hydronium ions by i. In our model representtion of H 2 SO 4 /H 2 O binry system using the four distint moleulr speies, e then hve,v,, v, 1 v q sv! s q s v s i 1 i 0 i! q b 1 i! b i z i h i! z de 0! U, 5 here the subsripts b nd h refer to bisulfte ion nd hydronium ion, respetively. We hve defined the fugities z of ter nd z h of the hydronium ion by z q e nd z h q he, 6 respetively nd ssumed tht the v sulfuri id vpor moleules n be treted s n idel gs to integrte out their ontribution in the onfigurtionl integrl. As result, stnds for i, hose trnsltionl nd orienttionl degrees of freedom re olletively denoted by in the onfigurtionl integrl. ote tht in the definition of z h, Eq. 6, the hemil potentil of ter is used sine hydronium ions re formed from ter, to hih the system is open. To lrify this point, Eq. 5 is derived in Appendix A strting from dissoitive model. ote tht the onfigurtionl integrl is tken over ll onfigurtions onsistent ith the luster. This onstrint is ell-defined nd presents no diffiulty in evluting the integrl. In ft, s e ill see belo, the prtition funtion to be evluted by simultion involves only id moleules. Moreover, the boiling point of pure sulfuri id is muh higher (0 C thn the temperture of interest, here id moleules re bound together regrdless of the vlue of V. This mens tht no expliit onsidertion is neessry to impose the onstrint tht the id moleules form luster. Finlly, multiplying by e s ( v ) nd summing over v, e n onvert to :,V,, s ; h e s v,v,, v, v 0 V e q se n s i s s i 0 z de 0! U, here is defined by z i i! i! i! q bq h s q s q b. h Clerly, k B T log is the free energy of the retion 7 8
J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld 685 H 2 SO 4 H 2 O HSO 4 H O, here eh moleule is onsidered to be isolted in vuum. If the summtion ith respet to i in Eq. 7 is to be evluted diretly in single simultion, it must be possible to reple sulfuri id-ter neutrl pir ith bisulftehydronium ion pir vi tril move in Monte Crlo simultion. This tril move, hoever, ill lmost ertinly be rejeted sine it ill seriously disturb the hydrogen bond netork in the luster, osting very high energy. Insted, eh term in the summtion over i must be lulted seprtely. In prtiulr, the simultion fouses on evluting the expression z,v,,, i de 0! U 10 for eh vlue of i, the number of the ion pirs in the system. Exept for the nlytilly trtble ftors, is the prtition funtion in the mixed ensemble of n luster hose dissoition stte is speified by i. Eqution 10 nnot be evluted in single simultion. Insted, it must be evluted by mens of thermodynmi integrtion. An expression for (,V,,, i ) onvenient for this purpose is derived in Appendix B. Using Eq. B6 in Eq. 7, e obtin,v,, s ; e n s n V n s V i 0 n i i! i! d1 e u hb 1 i exp 0 i u hb d log n0 U n0 log n n Vd log n, 11 hih, hen substituted into Eq. 2 long ith Eq., yields e Wrev,V,, s ; n s V here e define,v,,, i log n i i 0 i! i! if hb 0 i u hb d log n0 9 e,v,,,i, U n0 log n n Vd log n, 12 1 here 0 is hosen to be suffiiently smll tht the system n be regrded s n idel gs omposed of ter, sulfuri id, nd bisulfte-hydronium ion pirs. n 0 is the smllest of the number density of ter moleules in vpor tht e re interested in nd u hb is the intermoleulr potentil beteen HSO 4 nd H O. The therml verge x is evluted t fixed vlue of x, hih remins onstnt long the integrtion pth. The therml verge u hb is lulted in the nonil ensemble of single HSO 4 H O ion pir. f hb () is the free energy required to mke the ion pir in the system from sulfuri id nd ter forming n idel gs onfined in the unit volume t nd is defined through the reltion e f hb d1e u hb 1. 14 When n is expressed in Å, the distne in the integrl of Eq. 14 hs to be mesured in Å so tht Eq. 1 is dimensionlly onsistent. IV. REVERSIBLE WORK SURFACE AD THE CLUSTER SIZE DISTRIBUTIO In the previous setion, luster is hrterized by lone. In the lssil nuletion theory, hoever, it is ustomry to hrterize luster by both nd. To obtin the expression for the reversible ork to form this (, ) luster, e must first speify ho to define for prtiulr onfigurtion of moleules in the system. Let us onsider n exess quntity defined by ex n V, 15 hih is zero for uniform vpor. Hoever, during simultion in the mixed ensemble,, nd hene ex, flututes. For mrosopi V, this flutution rises primrily from tht due to the vpor moleules. Flututions of this kind hve very little to do ith the nuletion proess nd should not be ounted s prt of luster. In the mixed ensemble, hoever, their effet on ex n be mde negligible by deresing the volume until it stisfies n V1. 16 In this limit, the system ontins, on verge, no vpor moleule. In ft, the probbility of finding t lest one vpor moleule of ter in the system is, ssuming the idel gs behvior of the vpor phse, given by 1e n V n V, hih is negligible s result of Eq. 16. Thus, one n ttribute the non-zero vlue of ex to the presene of luster, hih suggests tht one my define ex i i n V, 17 here e inlude hydronium ions in the definition sine the ions re formed from ter. In vie of Eq. 16, emy redefine by i, 18 nmely, ll the ter moleules, hether or not protonted, in the system n be regrded s prt of the luster. The (, ) luster thus defined is physil luster in the sense tht it represents density flututions relevnt in nuletion. 95 We note tht the lusters generted in simultion re onsistent ith n intuitive definition of lusters. To see this,
686 J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld note tht, exept hen 0, is lys lrger thn n V(1), the verge number of ter moleules in V hen filled ith the uniform vpor. Thus, on verge, ny ttempted Monte Crlo move to rete moleule in the system ill be epted ith higher probbility if the nely reted moleule interts more fvorbly ith the rest of the moleules, hile s soon s moleule evportes from the luster, it is more likely to be removed from the system upon its tril nnihiltion. Some omments on V re in order. Clerly, V hs to be lrger thn the sptil extent of luster in it. Tht the system is mirosopi does not ffet the pplibility of the sttistil mehnil desription. It is suffiient to ssume ek oupling beteen the system nd its surroundings. 94 Both onditions re trivilly stisfied in the present se of vpor to liquid nuletion, here the molr volume in the vpor phse is onsiderbly lrger thn physil dimension of the luster nd the intertion beteen the vpor moleules nd luster n be ignored. The expression for the reversible ork to form n (, ) luster n be esily obtined. In ft, the grnd nonil prtition funtion of the (, ) luster is obtined from Eq. 7 by keeping only the term i in the sum over. Thus,,V, V, s ;, e q se n s s s i z i z i 0 i! i! i!! de U V e q se n s s s i z i i 0 i! i! i!,v,,, i p,v,,, i, i, 19 here p is defined by p,v,,, i, z! d{}e U z 0 de! U 20 hih is the normlized probbility of finding ter moleules in the system ontining id moleules, i of hih re ions, nd is diretly obtined from single simultion. Sine Eq. 19 differs from Eq. 7 only by ftor of p, one n obtin the desired expression from Eq. 12: e Wrev,V,, s ;, n s V i 0 e,v,,,i p,v,,, i, i. 21 The equilibrium luster size distribution (,, s,, ) n be obtined s follos. Suppose, tht the entire vpor phse of volume V tot is divided into smll ells of volume V. Beuse of Eq. 16 nd n s n, most of the ells ontin no moleules t ll nd those ontining luster or monomer re on verge sptilly distnt. Thus, one n ssume tht ll of the ells re sttistilly independent. Then, the verge totl number of the (, ) lusters in V tot is given by V tot,v,, s ;,, 22 V,V,, s here is the grnd nonil prtition funtion of the system of volume V tken in the vpor. In lulting, ll the possible mirosttes onsistent ith the metstble stte hve to be ounted for. Hoever, sine the system ontins no moleules t ll for most of the time, it n be pproximted s idel gs:,v,, s r,v,, s. 2 When divided by V tot, Eq. 22 beomes,, s,, ew rev,v,, s ;,, 24 V here e used Eq. 2. AsEq.2indites, e Wrev is the probbility of finding the (, ) luster in the system reltive to the referene stte. Sine the luster n be found nyhere in the system nd the event of finding it t one ple or nother is mutully exlusive, e Wrev is proportionl to V, inditing tht is independent of volume s required. Finlly, e ddress onsisteny issue. Stritly speking, neither the 0,1 luster nor the 1,0 luster is vpor monomer of ter or id moleule, respetively, sine these lusters exlude ter vpor from the system of volume V beuse of the definition Eq. 18, hile the monomers do not. In ft, one n redily sho tht 1,0n s e n V, 25 here the term for i 1 is ignored in Eq. 19, nd tht 0,1n e n V. 26 Hoever, this distintion is ompletely insignifint sine e n V 1. Alterntively, one n onsistently reover the monomer densities by setting V0 in Eqs. 25 nd 26. V. DETAILS OF THE SIMULATIO First, e briefly desribe some of the detils of the simultion. The system is defined s spheril vity of rdius 50 Å. We studied the lusters of 1,2,. As pointed out in Se. III, for given vlue of, there re 1 lusters to be simulted seprtely orresponding to the different dissoition sttes defined by i. Thus, there re nine lusters in totl. For eh of the lusters, the initil onfigurtion of the moleules is reted s follos. The sulfur site of sulfuri id moleule or bisulfte ion is pled t the enter of the vity nd the rest of the moleules, inluding ertin number of ter moleules, re pled rndomly inside the system. After suffiient equilibrtion t T0.15 K nd n 0.104810 6 Å, orresponding to reltive humidity
J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld 687 TABLE VIII. Conditions of the simultion. T298.15 K, nd n s tot 0.1 10 1 Å. o. n Å n V Reltive humidity % S1 0.104810 6 0.548710 1 1.65 S2 0.210 6 0.1047 26.06 S 0.10 6 0.1571 9.08 S4 0.8910 6 0.2010 50.01 S5 0.510 6 0.2618 65.1 S6 0.610 6 0.142 78.17 Sturtion pressure of ter is ssumed to be 2.71 mmhg Ref. 96. of bout 10% if the tul vpor pressure of ter is used, 96 the onfigurtion is used s the initil onfigurtion for the simultions t the nerby vlues of T nd n. The proess is repeted to obtin the initil onfigurtion of the hole rnge of T nd n studied. After equilibrtion, hih typilly tkes 10 5 10 6 Monte Crlo steps, nd U re smpled for bout 10 7 10 9 Monte Crlo steps. Smpling is mde every 10 steps for short runs nd every 10 4 steps for long runs. Here, one Monte Crlo move onsists of i tril rndom trnsltion nd rottion of moleules in the system nd ii one tril grnd nonil move, nmely tril retion or nnihiltion of ter moleule. Moleules to be moved re hosen rndomly so tht eh moleule is piked up one per Monte Crlo step on verge. The mximum displement for tril trnsltion nd the mximum ngle for tril rottion re tuned during the simultion so tht the eptne rtio, defined s the rtio beteen the number of epted tril moves nd the totl number of the tril moves, stys round 50%. As disussed in Appendix C, moleule t the enter of the system undergoes rottion only. To evlute the temperture integrtion in Eq. 1, simultion is performed t 298.15 K nd higher tempertures, the highest of hih is hosen so tht the integrnd U n0 i u hb is negligible t this temperture nd depends on the vlues of both nd i. For exmple, it is 500 K for (, i )(1,0), hile 2400 K for (, i ) (,). Also, n 0 0.104810 6 Å. In evluting the seond integrl of Eq. 1, simultion is rried out t n 0.104810 6, 0.210 6, 0.10 6, 0.8910 6, 0.510 6, nd 0.610 6 Å s summrized in Tble VIII. FIG.. A luster ith 2, i 0tT298.15 K nd n 0.510 6 Å. Further tehnil detils of the simultion is disussed in Appendies B D. In Appendix B, riterion is disussed on the dequy of the number of intermedite points used to evlute the thermodynmi integrtions. In Appendix C, riterion in hoosing the system volume V is disussed from point of vie someht different from tht of Lee et l. 4 Finlly, under resonble pproximtion, one n dispense ith the integrtion ith respet to n indited in Eq. 1, hih results in the improvement of the omputtionl effiieny by ftor of severl. The method is disussed in Appendix D. VI. RESULTS AD DISCUSSIO A. Shpe of the lusters Snpshots 97 of the lusters re shon in Figs. 2 7, here sulfur sites, oxygen sites, nd hydrogen sites re olored red, drk blue, nd hite, respetively. Figure 2 shos hydrte of sulfuri id moleule, in hih loop of hydrogen bonds is formed from the H 1 site to the O 2 site of the id ith to ter moleules, similrly for H 1 to O 2. In hydrtes ith feer ter moleules, the hydrogen bonds re formed preferentilly from the H 1 nd H 1 protons to oxygens of ter moleules. One end of the hydrogen from ter tthed to the H 1 site, for exmple, forms distnt hydrogen bond to the O 2 site. FIG. 2. A luster ith 1, i 0tT298.15 K nd n 0.610 6 Å. FIG. 4. A luster ith 2, i 0tT298.15 K nd n 0.610 6 Å.
688 J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld FIG. 5. A luster ith 2, i 0tT298.15 K nd n 0.610 6 Å. Figures, 4, nd 5 sho lusters ith to sulfuri id moleules. ote tht the id moleules diretly form to hydrogen bonds in Fig., hile one of them is medited by ter moleule in Fig. 4. Around room temperture nd t ll vlues of n investigted in this ork, these to onfigurtions re representtive of the onnetivity of the id moleules. Sine interonversion of these to onformtions requires one or to hydrogen bonds to be broken first, the Monte Crlo lifetimes of the onformtions re firly long, being of the order of 10 7 steps. As the vlue of n is inresed, onfigurtions similr to Fig. 4 beome more probble thn those similr to Fig.. The onfigurtion in Fig. 5 n be regrded s n intermedite beteen those to. ote tht 5 in both Figs. nd 4. Hoever, it is unlikely tht interonversion beteen these to onformtions ours in nonil simultion sine it ould involve diffusion of ter moleule over severl ngstroms on the luster surfe here no fvorble intertion site exists. This shos ler dvntge of the mixed ensemble simultion over onventionl one in nonil ensemble. This effetiveness in smpling very different relevnt onfigurtions hs been stressed in simultion ork tht determines the solvtion shell struture of protein nd nulei id, 98 for exmple. Similr onnetivity is observed in Fig. 6 for lusters ith three sulfuri id moleules. An dditionl omplition rises in this se, hoever, sine the id moleules n no form ring Fig. 7. As is seen from Figs. 6 nd 7, id FIG. 7. A luster ith, i 0tT298.15 K nd n 0.610 6 Å shoing ring onformtion. moleules in the ring onformtion tend to form hydrogen bonds preferentilly ith ter moleules thn mong themselves in omprison to those in the liner onformtion, leding to lrger lusters. Sine both onformtions re observed in the simultion, the orresponding free energies re expeted to be lose, the differene being the order of k B T. To understnd its implition, e monitored the quntity WU log n V, 27! hih plys the sme role s U in nonil ensemble in tht the sttistil eight of given mirostte is proportionl to e W. The dt from simultion of 710 8 Monte Crlo steps ere divided into smll bloks, eh of hih orresponding to 10 7 Monte Crlo steps, nd the verge of W s lulted for eh blok. Figure 8 shos the vrition of this blok verge of W for three vlues of n. Lrger, i.e., less negtive, vlues of W orrespond to the liner onformtion hile the smller vlues of W orrespond to the ring onformtion. Despite the lrge differene in W, both liner nd ring onformtions re observed. This mens tht the ring onformtion is entropilly unfvorble sine it ttrts more ter moleules, onfining them to fr smller volume thn V. As these ter mol- FIG. 6. A luster ith, i 0tT298.15 K nd n 0.610 6 Å shoing liner onformtion. FIG. 8. The hnge in the blok verge of W. The size of eh blok is 10 7 MC steps. The onditions of simultion for S1, S4, nd S6 re given in Tble VIII.
J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld 689 TABLE IX. Het of formtion. Moleule H 2 O b H 2 SO 4 H O HSO 4 b H f kl/mol 57.80 175.671.912 1421 227.774.06 Prigogine nd Defy Ref. 105. b Lis et l. Ref. 106. eules evporte, the luster returns to the liner onformtion. From Fig. 8, it is ler tht dditionl Monte Crlo steps re required to hieve good sttistis. onetheless, qulittive trends re lredy present. For exmple, the ring onformtion ppers more frequently s n is inresed. one of the lusters shon here possesses spheril or n xil symmetry. Given the firly long lifetime of eh onformtion of the lusters, the sme is expeted to be true even fter the therml verge is tken. Thus, the thermodynmi desription of the lusters is no longer menble to Gibbs presription. 57 In prtiulr, the thermodynmi quntities nnot be lssified into extensive nd intensive ones, hih is prerequisite in deriving the fmilir Gibbs Duhem reltion, for exmple. It seems hrdly profitble to try to extend Gibbs interfil thermodynmis to inlude suh ses, sine the quntities introdued into suh theory re unlikely to be subjet to experimentl mesurement. This in turn highlights the importne of the moleulr level pproh. B. Estimte of the vlue of f hb To evlute the reversible ork to form n luster or n (, ) luster, from Eqs. 12 or 21, the vlue of f hb defined by Eq. 14 is required. Using the het of formtion of the moleules given in Tble IX nd ignoring the entropi effet, e obtin k B T log 147.706.98 kl/mol. 28 To evlute the remining ftor in Eq. 14, e introdue the pproximtion 1 d1e u hb 1 v f f euhb, 29 here v f nd f respetively denote the volume nd the solid ngle over hih H O n flutute ith respet to HSO 4. We pproximte their vlues s the ubi of the mximum displement nd ngle used for the tril Monte Crlo move. Then v f 0.25 Å, f 0.. 0 From simultion, u hb 15.040.01 kl/mol t T 298.15 K. Sine Å in v f nels out Å of n in Eq. 1, e n omit the unit of volume here to obtin k B T log 1 d1e u hb 1 127.850.01 kl/mol. 1 Combining Eqs. 28 nd 1, e obtin FIG. 9. A luster ith 2, i 1tT298.15 K nd n 0.610 6 Å. f hb 19.856.99 kl/mol. 2 In ht follos, e dopt 20 kl/mol s the vlue of f hb. At T298.15 K, the indited unertinty in this quntity is s lrge s 12k B T. Effets of this unertinty ill be ddressed subsequently. C. Dissoition of H 2 SO 4 in luster Snpshots of lusters ontining HSO 4 H O ion pir re shon in Figs. 9 nd 10, in hih tomi sites re olored s before ith the exeptions for the O 1 site of the bisulfte ion nd the oxygen site of the hydronium ion, hih re, respetively, olored green nd light blue. In Fig. 9, hydrogen from H O forms hydrogen bond diretly to the O 1 site of HSO 4, hile in Fig. 10, H O forms hydrogen bonds to oxygen sites of H 2 O nd H 2 SO 4. In both lusters, hoever, there is no intervening moleule beteen the ion pir. The sme holds true for ion pirs buried in 97 ter moleules s shon in Fig. 11. The ter moleules strt to intervene in the ion pirs s the number of ter moleules exeeds bout 240 s shon in Fig. 12. Thus our simultion FIG. 10. A luster ith, i 1 t T298.15 K nd n 0.89 10 6 Å.
6840 J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld FIG. 1. Free energy of the luster s funtion of the dissoition stte defined by i. nd T298.15 K. f hb 20 kl/mol. The onditions of simultion for S1, S, nd S5 re given in Tble VIII. FIG. 11. Ion pirs forming ore inside the luster ith 97., i tt298.15 K nd n 0.610 6 Å. Wter moleules re not shon. suggests tht, s fr s the dissoition behvior of H 2 SO 4 is onerned, t lest bout 240 ter moleules re required to ttin behvior tht resembles the bulk solution. The ft tht there is no seprtion of ion pirs in smll luster does not imply tht dissoition is not importnt in the luster. As n H 2 SO 4 H 2 O neutrl dimer dissoites to n HSO 4 H O ion pir, its net dipole moment inreses to ttrt more ter moleules. If these dimers re pled in vuum, the dipole moment hnges from.28 D to 12.0 D t 298.15 K upon dissoition. Consequently, the potentil energy of the luster dereses. Whether or not dissoition ours in given luster is determined by the free energy (, i ), defined by Eq. 1, of the luster s funtion of i. The quntity is shon in Fig. 1 for the se of tt298.15 K for three vlues of n. Clerly, dissoition ours for ll ses. The inrese in is observed s i hnges from 0 to 1 hen n 0.104810 6 Å, refleting the ft tht the ion pir nnot ttrt suffiient ter moleules hen pled in vpor ith suh lo ter onentrtion. If 27 kl/mol is ssumed for the vlue of f hb, hoever, e find tht dissoition no longer ours for this luster Fig. 14. This hnge t the qulittive level points to the neessity of more urte estimte of the vlue f hb. D. Effet of hydrtion In Eqs. 12 nd 21, n s represents the number density of unhydrted sulfuri id moleules in ordne ith the hoie of our referene stte in lulting the reversible ork. In relity, hoever, most id moleules exist s hydrtes nd one n speify only the totl number density n s tot. Thus, e must express n s in terms of n s tot. The derivtion given here is nlogous to the orresponding one in the lssil theory. 14,18,22 Sine given id moleule is either hydrted or unhydrted regrdless of its dissoition stte, e hve n tot s 1,. From Eqs. 21 nd 24, e obtin FIG. 12. Ion pirs forming ore inside the luster ith 24., i tt298.15 K nd n 0.610 6 Å. Wter moleules re not shon. FIG. 14. Free energy of the luster s funtion of the dissoition stte defined by i. nd T298.15 K. f hb 27 kl/mol. The onditions of simultion for S1, S, nd S5 re given in Tble VIII.
J. Chem. Phys., Vol. 108, o. 16, 22 April 1998 Kusk, Wng, nd Seinfeld 6841 n tot s 1 V n s V 1 i 0 e,i, 4 here e hve mde use of the normliztion ondition p,v,,, i, 1. 0 5 Assuming tht the hydrtes re dominted by those ontining only one id moleule ( 1), e find tht p 0 n 1 s tot n s i 0 1 e 1, i. 6 In the folloing, e ssume tht n s tot 0.110 1 Å, hih is typil of experiments in H 2 SO 4 /H 2 O binry nuletion. 2 E. Reversible ork of luster formtion nd omprison ith the lssil preditions Rell tht the luster size distribution is given in the lssil nuletion theory by, n expw lssil,, 7 here W lssil is the reversible ork of luster formtion evluted by the lssil theory. Thus, reriting Eq. 24 s, n expw rev, logn V, 8 e find tht sensible omprison is mde beteen our moleulr theory nd the lssil by ompring the quntity defined by REV, W rev, logn V 9 ginst W lssil (, ). Similr quntities n be defined for n luster by summtion ith respet to of Eqs. 7 nd 8. Thus, e ompre W lssil ( ) defined by expw lssil expw lssil, 40 0 ith REV ( ) defined by exp REV exp REV,. 0 41 The upper limit () of the summtion is only forml sine the summnd deys quikly s is inresed hen the reltive humidity is less thn 100%. Figure 15 ompres the reversible ork REV of the luster formtion obtined by ssuming the vlue of 20 kl/mol for f hb ith W lssil obtined by the lssil theory. 99 While the lssil theory predits tht -luster is still subritil, our simultion predits tht 2-luster is the ritil nuleus for onditions S1 to S4 nd tht REV dereses monotonilly ith for onditions S5 nd S6. See Tble VIII. This mens tht the rte-limiting step of FIG. 15. Reversible ork of -luster formtion t T298.15 K. Clulted ith f hb 20 kl/mol. The onditions of simultion for S1, S, nd S5 re given in Tble VIII. C1 shos the lssil predition under the ondition S1. Similrly for C nd C5. ne prtile formtion is the binry ollision of sulfuri id moleules. ote tht the lssil theory predits tht W lssil inreses s n is inresed. This trend reflets the ft tht higher vlue of n results in more signifint depletion of the id moleules nd reverses s beomes lrge enough so tht the luster is ritil nuleus. When 27 kl/mol is employed for the vlue of f hb, the results of our simultion beome more or less omprble ith the lssil predition s shon in Fig. 16. This indites tht the lrge disrepny beteen the moleulr theory nd the lssil s observed in Fig. 15 rises from the differene in the behvior of luster involving the HSO 4 H O ion pirs. To obtin more detiled informtion regrding the lusters ith ion pirs, e lulted REV of the (, )-luster formtion for the ondition S. The results re shon in Figs. 17 nd 18 for the se of f hb 20 kl/ mol nd 27 kl/mol, respetively. Figures 17 nd 18 lso ompre the simultion ginst the lssil predition. For smll vlues of the greement is surprisingly good. Figure 17 shos double minimum in the reversible ork. The FIG. 16. Reversible ork of -luster formtion t T298.15 K. Clulted ith f hb 27 kl/mol. The onditions of simultion for S1, S, nd S5 re given in Tble VIII. C1 shos the lssil predition under the ondition S1. This is similr for C nd C5.