ESCI 341 Atmospheric Thermodynamics Lesson 14 Curved Droplets and Solutions Dr. DeCaria

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1 ESCI 41 Atmophric hrmodynamic Lon 14 Curd Dropt and Soution Dr. DCaria Rfrnc: hrmodynamic and an Introduction to hrmotatitic, Can Phyica Chmitry, Lin A hort Cour in Coud Phyic, Rogr and Yau hrmodynamic of Coud, Dufour and Dfay SURFACE ENSION At th urfac btwn a iquid and a apor, thr i a thin ayr known a th urfac ayr. h urfac ayr i ony about mocu thick, but ha om important proprti. h pcific intrna nrgy of th mocu in th urfac ayr i gratr than that for mocu dpr within th fuid. hi i bcau th mocu in th urfac ayr don t ha intrmocuar forc puing on thm from a id. If you want to incra th urfac ara of th fuid, you ha to bring mocu from th intrior of th fuid to th urfac. hi rquir work (bcau you ar incraing thir nrgy). If you incra th urfac ara by an amount, da, thn th numbr of mocu that mut b raid to th urfac i proportiona to da. hrfor, th work rquird i ao proportiona to da dw da da. (1) h proportionaity contant,, ha unit of nrgy pr ara, or forc pr ngth. It i cad th urfac tnion. For watr (at mtoroogica tmpratur) N/m. Bcau of urfac tnion, a dropt of watr ha a prur gratr than that of it urrounding. h diffrnc in prur acro a phricay curd intrfac i gin by 2 p p p0, (2) r

2 whr r i th radiu of curatur, p i th prur inid th drop, and p 0 i th prur outid th drop. 1 For a dropt with radiu of 1.5 m, thi man a prur diffrnc of 1000 mb ( about 1 atm) btwn th inid and outid of th dropt. HE CURVAURE EFFEC In th at on w drid th Poynting Equation, p, () which how th ffct of prur on th aturation apor prur. An incra in prur incra th aturation apor prur. For mot appication th incra in aturation apor prur du to an incra in prur i ma. hat i why w don t uuay concrn our with modifying th Cauiu-Capyron quation to account for air prur. Howr, in a curd dropt th prur i graty incrad. hu, w xpct that th aturation apor prur or a curd drop wi b gratr than or a fat urfac. W tart by uing th chain ru to writ p r p r From th Poynting quation w ha. (4) and o (4) bcom p, (5) p r r. (6) Sinc th apor i an ida ga w can ubtitut R to gt 1 A driation of thi can b found in Dufour and Dfay, pp. -6 2

3 From (2) w ha n p r R r. (7) o that (7) bcom p r 2 2 r (8) Intgrating (9) yid n r R r. (9) Whn r thn C n ( ), o (10) bcom 2 1 n ( r) C. (10) R r or ( r) 2 1 n ( ) R r 2 1 ( r) ( )xp. (11) R r Dfining 2 a (12) R w gt a ( r) ( )xp r. (1) A dcra in drop iz incra th aturation apor prur. HOMOGENEOUS NUCLEAION Condnation occurring in th abnc of condnation nuci i known a homognou nucation. In ordr to gt a tab watr dropt to form, th ambint apor prur mut b qua to th aturation apor prur of th dropt, = (r).

4 Sinc (r) > () (du to th curatur ffct), it i impoib to gt a tab watr dropt to form un th oum i upraturatd, maning ( ), whr ( ) i th aturation apor prur or a fat urfac of pur watr. h amount of upraturation i gin by th aturation ratio, S ( ). Saturation Ratio (14) A aturation ratio of unity man th rati humidity i 100%. A aturation ratio gratr than unity impi upraturation. An mbryonic watr dropt form du to chanc coiion of watr apor mocu. Sinc mbryonic watr dropt ar o ma (a dropt coniting of 20 watr mocu woud ha a radiu of about m), thy ar untab un aturation ratio xcd 10. En if a dropt houd grow by chanc coiion to a iz of nar 100 watr mocu, it woud ti rquir a aturation ratio of nary 4 (rati humidity of 400%) in ordr to b tab, and not aporat. In th atmophr, aturation ratio ar rary argr than 1.01 (101% rati humidity). hrfor, homognou nucation cannot rponib for th formation of coud dropt. HE SOLUE EFFEC For an ida oution (a ry diut oution), th pcific Gibb fr nrgy of iquid watr containing a out i whr g g R n, (15) g i th pcific Gibb fr nrgy of pur watr, and fraction of watr, dfind a n n n i th mo, (16) (n i th numbr of mo of watr, and n i th numbr of mo of out). h uprcript rfr to pur, iquid watr. 4

5 Equation (15) how that dioing an impurity in watr owr th Gibb fr nrgy of th oution (th mar bcom, th mor ngati th at trm in Eq. (15) bcom, dcraing th pcific Gibb fr nrgy). It i not traightforward to undrtand why adding a out owr th Gibb fr nrgy. A quaitati argumnt i: h pcific Gibb fr nrgy i g h 5. (17) Adding a out wi incra th ntropy of th ytm (ntropy of mixing), and thrfor dcra th pcific Gibb fr nrgy. h out mocu wi ao affct th intrmocuar forc, and coud rut in a owring of th intrna nrgy (and hnc, nthapy) of th ytm. Sinc at quiibrium g g, and inc adding an impurity to th watr cau g to dcra, thn g mut dcra if quiibrium i to b maintaind. For th apor w ha dg d d. (18) From Eq. (18) w that in ordr to dcra g th aturation apor prur mut dcra (at contant tmpratur). Adding an impurity to watr wi owr th aturation apor prur. hi i cad th out ffct. o dri quantitatiy what ffct th impurity ha on apor prur, w tart with dg dg. (19) o find an xprion for dg w diffrntiat Eq. (15) to gt W know that o Eq. (20) bcom dg dg R n x d R d n. (20) dg d d, dg d d R n x d R d n. (21)

6 Stting Eq. (18) and Eq. (21) qua to ach othr [a pr Eq. (19)] rut in d d R n d R d n d d. (22) Our aim i to find out how ari with, at contant. hrfor, w can ignor th d trm in Eq. (22) and gt or d R d n d, d Rd n. (2) Sinc a a thi bcom d R d n, and from th ida ga aw, R =, o w ar ft with n n Intgrating thi from = 1 (pur watr) w gt d d. (24) which i ao n n, Raout Law (25) hi rut i known a Raout aw. For ry diut oution w can approximat a 1 n n. (26) h numbr of mo of out and iquid ar gin by n im / M out n m / M whr m i ma, M i mocuar wight, and i i th ion factor (th numbr of ion that on mocu of ubtanc diociat into). hrfor, th mo fraction of iquid watr i h ma of th iquid in th dropt i m M 1 i. (27) m M 6

7 o that th Eq. (27) i 4 m Lr im M 1 Uing Eq. (28) in Raout Law [Eq. (25)] rut in 1 1 br. (28) 4L M r 1 b r (29) whr im M b. (0) 4 M In raity, watr dropt in th atmophr ar not ida oution, bcau thy ar not ry diut. Howr, w oftn approximat thm a though thy wr ida, and u Raout aw, undrtanding that it i ony an approximation. L COMBINING HE CURVAURE AND SOLUE EFFECS If w ha a curd urfac of pur watr, th curatur ffct t u that r ( r) ( )xp a r (1) If th dropt i impur, w nd to ao inok th out ffct to gt Rca that b 1 xp r a r. (2) ( ) i found from th Cauiu-Capyron quation, w gt th fu xprion for th aturation apor prur or an impur, curd dropt, b L xp a r xp. () r R 0 SABILIY OF DROPLES In ordr for a dropt to b tab, th ambint apor prur mut b qua to th aturation apor prur, =. hi man th aturation ratio, S, mut b [from Eq. (2)] b S a r r 1 xp 7 (4)

8 A pot of Eq. (4) for a dropt containing thr diffrnt amount of out (ach cur diffr in out ma by a factor of 10) i hown bow. h tat cur i for th at amount of out. A pot ik that hown abo i omtim rfrrd to a a Kohr cur. Making u of th approximation that for ma x, x 1 + x, w can writ a b S 1. (5) r r h radiu at which th Kohr cur i a maximum can b found by taking S /r and tting it qua to zro. hi radiu i cad th critica radiu, r, and th aturation ratio at thi point i cad th critica aturation ratio, S. hy ha au of r b S 1 a 4a 27b (6) Not that in (6) th do not rfr to pur, iquid watr. h critica radiu i of fundamnta importanc for coud dropt growth. At radii bow th critica radiu (r < r) th dropt ar in tab quiibrium. If S incra th dropt wi grow to a argr iz and thn top. If S dcra th dropt wi hrink to a mar iz and thn top. Dropt at radii bow th critica radiu ar cad haz partic. At radii abo th critica radiu (r > r) th quiibrium i untab, and th dropt wi pontanouy grow argr, n though S i not incraing. 8

9 Dropt who radiu qua th critica radiu (r = r) ar aid to b actiatd. Not that tab dropt can form at aturation ratio w bow unity. 9