Oppgavesett kap. 6 (1 av..)
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1 Oppgvstt kp. 6 (1 v..) hns.brnn@go.uio.no Problm 1 () Wht is homognous nucltion? Why dos Figur 6.2 in th book show tht w won't gt homognous nucltion in th tmosphr? ˆ Homognous nucltion crts cloud droplts without condnstion nuclus. Figur 6.2 illustrts th rltiv humidity (RH) ruird to kp droplt in quilibrium s function of rdius. For droplt with rdius 0.01 µm RH of bout 112 % is rquird. Th dpndnc on rdius mns tht smllr droplts nd vn lrgr RH to mintin quilibrium. Homognous nucltion hppns whn wtr molculs collid nd stick togthr whil othr molculs continu to condnc onto this rst nuclus, but sinc vry smll droplts rquir vry high suprsturtion ths homognous nucli will not surviv long nough to grow in norml tmosphric conditions whr suprsturtion rchs mximum of fw pr cnt. Homognous nucltion dos not hppn in th tmosphr. (b) Wht hppns to surfc tnsion nd Gibbs fr nrgy rspctivly whn gs condnsts to liquid. Rlt this to th qution blow nd xplin th trms. E = Aσ nv (µ v µ l ) ˆ (Th qution corrsponds to qution 6.1 in th txtbook.) To form droplt, work must b don to form th surfc. This is rprsntd by Aσ in th qution whr σ is th mount of work ndd to crt unit of surfc btwn gs nd liquid nd A is th r of th droplt. Th Gibbs fr nrgy for liquids µ l is smllr thn for vpor µ v which mns th Gibbs fr nrgy will dcrs. E is th nt incrs in th nrgy of th systm du to th formtion of th droplt. 1
2 (c) 1. Wht is illustrtd in Figur 6.1 in th book? 2. Givn suprsturtion, will th droplt grow spontniously or vport whn R < r? Explin why. ˆ Figur 6.1 illustrts th chng in nrgy cusd by forming droplt of rdius R givn by th xprssion in qution (6.4) for th two css whr < s (blu) whr w do not hv suprsturtion nd > s (rd) whr w hv suprsturtion. A systm will lwys ttmpt to pproch quilibrium by rducing it's nrgy, this mns tht th growth of droplts will only b fvord if droplt growth rducs th nrgy of th systm. In th cs whr < s growing droplt will lwys dd mor nrgy nd droplt formtion nd growth is not fvord. Any smll nucli formd by collisions will vport quickly. Whn w hv suprsturtion > s thr is mximum E t criticl rdius r. Abov this rdius, E dcrss nd ny droplt bov this rdius will continu to grow spontnously. (d) Hvordn dnns dråpr fr vnndmp i tmosfærn drsom dtt ikk kn skj gjnnom homogn nuklsjon? ˆ Droplts form through htrognous nucltion whr wtr vpor condnss on condnstion nuclus. This rducs th surfc tnsion rquird to sustin droplt of rdius R, mking droplt formtion possibl t lowr rtiv humiditis. Problm 2 () Wht do w cll th curvs w gt whn w plot th right sid of q 6.8 ginst r? (b) Wht hppns t th mximum of ch curv? (c) Wht do w sy th droplt is bfor nd ftr this mximum, rspctivly? (d) Wht do w cll droplts bfor thy crossd this mximum? A mor common nm thn th prvious qustion. ˆ Th curvs r clld Köhlr curvs. At th mxim, droplts r ctivtd, which mns tht from hr thy will continu to grow spontnously givn nough vilbl humidity, bfor th mximum th droplt nd nrgy to grow. Bfor th mximum droplts r clld hz whil ftr th mximum thy r clld cloud droplts or fog. 2
3 Problm Eksmn GF Klvins qution is s follows: r = 2σ ( ) (1) nkt ln s () Explin th symbols in th qution. Giv physicl intrprttion of th qution. ˆ S sction in th txtbook. Th qution shows how th rdius of droplt in unstbl quilibrium dpnds on th suprsturtion. (b) Bsd on th qution(1), drw grph of th rltionship btwn rltiv humidity nd th droplt rdius for droplt in quilibrium. How wll dos th qution bov xplin th formtion of droplts in ntur? ˆ This qution dos not rlisticlly dscrib droplt formtion on ntur du to th vry lrg suprsturtions rquird to form smll droplts. (c) Givn th qution s = xp ( 2σ ) n kt r imm 1 + w M s( 4 πr ϱ m) How dos th curvtur ct nd th solution ct rspctivly ct th formtion of cloud droplts How dos this dpnd on th rdius of th droplt? ˆ Th solution ct mns tht lowr suprsturtions r rquird to form droplt. This is bcus slt molculs occupy som of th spcs on th surfc othrwis occupid by wtr molculs. Th ct is lrgst for smll droplts. Th curvtur ct mks th sturtion prssur ovr th droplt dcrs s th rdius incrss (2) Problm 4 CCN () Wht r CCN nd wht rol do thy ply in th formtion of droplts? CCN (Cloud Condnstion Nucli) r th smll numbr of rosols prsnt in th tmopshr which cn function s nucli for droplt formtion. Sinc htrognous nucltion is virtully impossibl in th tmosphr ll droplts w obsrv hv bn formd by htrognous nucltion on CCNs.
4 (b) Wht r typicl CCN concntrtions ovr lnd nd ocn, rspctivly? Lnd: 00 cm. Ocn: 100 cm. Th point is tht, in gnrl, th concntrtions r highr ovr lnd thn ovr th ocn. (c) Givnn th sm suprsturtion in mrin nd continntl irmsss, whr do you xpct ctivtion of droplts to hppn fstst? Th ctivtion will hppn fstst ovr th ocn. Th rson is tht th comptition btwn th CCNs for th vilbl wtr vpor is lss "intns" hr du to th lowr concntrtion. In typicl continntl ir mss th sm mount of vtr vpor will b shrd by mor CCNs nd mk mor smllr droplts rthr thn fw lrg ons. (d) Imgin thr dirnt prticls, ll with rdius 0.1µm. On is hydrophobic, th othr wttbl nd th third is soloubl. Discuss th prticls bility to ct s CCN. ˆ Th hydrophobic prticl will not ttrct wtr nd cn not ct s CCN. ˆ Th wttbl prticl cn ct s condnstion nuclus but will not hlp th droplt grow through th soultion ct ˆ Th solubl prticl will b n cint CCN sinc it will not only ct s nuclus, but lso mk th droplt grow mor sily through th solution ct. From problm 6.8 in th book: () (b) ˆ Smll droplts will vport vn in sturtd tmosphr du to th curvtur ct, mking th tmosphr undr-sturtd rltiv to th highly curvd surfc of th droplt. ˆ Th slt will ttrct th wtr mking it condns on it nd kping th cupbord dry. 4
5 () ˆ Continntl ir msss usully contins mor nd smllr CCNs, spcilly clos to th surfc. This dpndnc on hight, which is not prsnt in mrin ir, indicts sourcs ovr lnd. Problm 6.10 from th book W us Klvin's formul r = 2σ ( ) () nkt ln s nd sinc w will nd th rltiv humidity / s, w solv for this vlu ( ) 2σ = xp s rnkt Insrting σ = 0.076Jm 2, numbr dnsity of molculs n = m, nd rdius r = m, using T = 27K nd k = J molc 1 K 1, w hv ( ) 2σ = xp (5) s rnkt = = 100.6% (4) Problm 6.11 from th book. From Figur 6. this prticl is givn by curv. W loct th suprsturtion nd go right until w hit th curv, nd thn downwrds whr th rdius is givn: r = µm. b. Hr w hv th rdius of prticl. This prticl corrsponds to lin 5, so w go from th rdius upwrds until w hit th curv nd thn to th lft to nd th rltiv humidity: RH = 9%. c. Th hz stt is givn by dimtrs lss thn 1µm. This prticl corrsponds to lin 5, for which th criticl rdius (top of th curv) is lss thn th 0.5µm (hz limit). Whn th prticl rch this siz, it will continu to grow by itslf (nd thus byond th hz stt), so th mximum suprsturtion rquird is givn t th top of this curv: S = 0.45%. 5
6 Problm 6.12 from th book Th qution is ( ) [ = xp 1 + b ] 1 (6) s r r For not too smll r, /r 1 nd b/r 1 w cn xpnd th functions to sris: ( ) xp 1 + (7) r r nd [ 1 + b ] 1 1 b (8) r r so w cn writ s = ( 1 + ) ( 1 b ) r r = 1 + r b r b r 4 (9) 1 + r b r (10) whr w ssum b/r 4 much smllr thn th othr trms. Th -trm is clld th curvtur ct, nd is du to th fct tht droplt hs curving surfc, not pln surfc. Th b-trm is clld solut/solution ct, n ct cusd by ddd solut. To nd th criticl rdius, w nd th top point of th curv ( ) = r s r + b 2 r = 0 4 = 1 ( ) b r 2 r 2 b r c = (11) (12) 6
7 Insrting into th qution givs (r c ) = 1 + b ( ) s b b = 1 + b b (1) = 1 + b (14) = 1 + (15) = (16) 7
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