10/06/08 1. Aside: The following is an on-line analytical system that portrays the thermodynamic properties of water vapor and many other gases.

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1 10/06/ Th watr-air htrognous systm Asid: Th following is an on-lin analytical systm that portrays th thrmodynamic proprtis of watr vapor and many othr gass Prliminaris a) Rviw on quilibrium: thrmodynamic quilibrium -- a systm is in thrmal quilibrium (no tmpratur diffrnc) mchanical quilibrium -- a systm in which no prssur diffrnc xists btwn it and its nvironmnt chmical quilibrium -- condition in which two phass coxist (watr in this cas) without any mass xchang btwn thm b) For a homognous systm, 2 intnsiv proprtis dscrib th thrmodynamic stat (considr th q. of stat pα = RT); this is not th cas for a htrognous systm consisting of dry air, watr vapor and condnsd watr. c) Considr an xtnsiv proprty Z of th watr-air systm and dfin it as Z = Z g(p,t,n d,n v) + Z c(p,t,n c) Th al diffrntial of this proprty is th following: dz g, g, g, g, g = dt dp dn, d T + p + n + p n T n d n,, T, p, n v T, p, n dn v dz c, c, g, c = dt dp, T + p + n p, n T, n c T, p, n Exampl of chmical quilibrium A closd containr initially filld partly with watr in a vacuum will b in chmical quilibrium whn p = s (th saturation vapor prssur). At this point, th numbr of molculs passing from liquid to vapor quals th numbr passing from vapor to liquid. dn c Vapor in quilibrium (vacuum initially) p quil s watr 5.2 Chmical potntial If a singl molcul is rmovd from a matrial in a crtain phas, with th tmpratur and prssur rmaining constant, th rsulting chang in th Gibbs fr nrgy is calld th chmical potntial of that phas. In othr words, th chmical potntial can b dfind as th Gibbs fr nrgy pr molcul. (Rcall, dg = -sdt + αdp for a rvrsibl transformation.)

2 10/06/08 2 Th chmical potntial of an idal gas is dfind by µ = µ o + RTlnp, [dµ = RTdlnp = RTdp/p = αdp] whr µ o is th chmical potntial for p = 1 atm. For an idal gas, g = µ 2 - µ 1 = RTln(p 2 /p 1 ) [= -w max ] Exampl (Takn from Wallac and Hobbs, pp ): Driv an xprssion for th diffrnc in chmical potntial btwn watr vapor and liquid watr, µ v - µ l, in trms of th bulk thrmodynamic proprtis: (th partial prssur of watr vapor) and T. For a (vapor) prssur chang d, w hav (from dg = -sdt + αdp), for vapor and liquid, rspctivly, dµ v = α v d, dµ l = α l d, whr α v is th spcific volum of a watr vapor molcul and α l is th spcific volum of a liquid watr molcul. Thn it follows, by combining th quations abov, that d(µ v - µ l ) = (α v - α l )d α v d (5.1) sinc α v >> α l. Th quation of stat applid to just on watr vapor molcul is α v = kt, (5.2) whr k, Boltzmann's constant whos valu is x J K -1 molcul -1, is usd in plac of th gas constant to rprsnt on molcul. Upon solving (5.2) for α v, w can writ (5.1) as d(µ v - µ l ) = kt d. (5.3) Sinc µ v = µ l at = s (saturation, which is quilibrium) w can intgrat (5.3) as follows: µ µ v l d(µv - µ l ) = 0 ktd/, s or µ v - µ l = kt ln (for on molcul). (5.4)* s

3 10/06/08 3 This rsult will b usd latr in dvlopmnt of th thory of nuclation of watr droplts from th vapor phas. [Nuclation is thrfor closly rlatd to thrmodynamics.] 5.3 Changs in stat W will now considr various quilibria for watr in th atmosphr Gibbs phas rul For a systm of C indpndnt spcis and P phass, th Gibbs phas rul stats that th numbr of dgrs of frdom F (i.., th numbr of stat proprtis that may b arbitrarily slctd) is F = C - P + 2 (5.5) If w dal with a singl componnt systm (C=1) such as watr thn w can form th tabl blow: P (numbr of phass) F (dgrs of frdom) Thus, for a singl xisting phas (such as watr vapor) w may arbitrarily choos p () and T (i.., th quation of stat). For two phass, th choic of ithr p or T dtrmins th othr, and for thr phass (i.., th tripl point) thr is no choic, i.., this point has spcific valus for p and T. Th phas diagram for watr is dpictd in Figs. 5.1 and 5.2. Along th two coxisting phas lins, on dgr of frdom is possibl, within ach phas two dgrs of frdom xists, and with thr coxisting phass (th so-calld tripl point) thr ar non. W will now considr th phas lins.

4 10/06/08 4 Fig Phas transition quilibria for watr. P t is th tripl point, and P c is th critical point, abov which no distinction btwn th liquid and vapor phass xists. From Iribarn and Godson (1973). Fig PT phas diagram for watr. Stabl phas boundaris ar shown by solid lins; mtastabl phas boundaris ar indicatd by dashd lins. Not that th ordinat in xponntial in prssur, in contrast to th linar p dpndnc in Fig From Young (1993), p Fig. 5.3 Thrmodynamic surfac for watr. Takn from Iribarn and Godson (1973).

5 10/06/ Thrmodynamic charactristics of watr Ky points on a p-t diagram for watr (Figs. 5.1, 5.2 and 5.3): a) tripl point: p t = Pa = mb; T t = K = 0.01 C; α it = x 10-3 m 3 kg -1 ; α wt =1.000 x 10-3 m 3 kg -1 ; α vt = 206 m 3 kg -1 a) critical point: T c = 647 K; p c = atm; α c = 3.07x10-3 m 3 kg Equilibrium btwn solid and liquid - Th Clapyron Eq. Th lins drawn in Figs ar rprsnt th quilibrium condition btwn two phass of watr. Ths ar important in our study of thrmodynamics, and th quilibrium lin btwn th liquid and vapor phas is particularly valuabl. Ths quilibrium conditions can b xprssd analytically, and th basis for th functions drivd in th following is th First Law, from th th Gibbs fr nrgy is drivd. At quilibrium w hav dg=0, so thrfor g(solid) = g(liquid). -s liquid dt + α liquid dp = -s solid dt + α solid dp Dfin th following: Thn s = s solid - s liquid α = α solid - α liquid sdt = αdp. W solv this as follows, and utiliz th dfinition of ntropy chang for a phas transition (Ch. 4): dp/dt = s/ α = - H fusion /(T α) Clapyron Equation (5.6) whr th lattr quality is basd on th ntropy chang associatd with a phas chang, dfind in Chap. 4: s = - H fusion /T. This xprssion is th Clapyron Equation. Sinc H > 0 (hat is libratd upon frzing) and α > 0 (frzing watr xpands), th slop dp/dt < 0 as shown in Figs. 5.1 and Liquid/vapor quilibrium - th Clausius-Clapyron Eq. In this cas α = α liquid - α vapor -α vapor. Thn, from th quation of stat, w can writ -α v = -R v T/ s ( s is usd in plac of p sinc w ar concrnd with quilibrium at saturation)

6 10/06/08 6 Thn d s /dt = - H vap /(T α) = H vap s /(R v T 2 ) or dln s /dt = L vl /(R v T 2 ). (5.7) Rcall that L vl is not a constant, but varis by ~10% ovr th tmpratur intrval (0, 100 C), and by about 6% ovr th intrval (-30, 30 C). (S Tabl 5.1) Howvr, to a first approximation, w can assum that L vl is constant and intgrat Eq. (5.7): s d ln so s = L R lv T v T o dt T 2 which is solvd as ln s so = Lvl 1 1 R T T v o whr T o = K and so = 6.11 mb (from lab masurmnts). Rwriting this xprssion and grouping th constants into nw constants A and B yilds th simpl, approximat xprssion: s (T) = A -B/T, (5.8) whr A = 2.53 x 10 8 kpa and B = 5.42 x 10 3 K. A mor lgant drivation of th Clausius-Clapyron quation givn in Rogrs and Yau, pp W will also considr this. In ordr to gt a mor accurat rprsntation of s (T), th tmpratur dpndnc can b incorporatd using a linar corrction trm involving T in an xprssion for L vl, and thn substitut in Eq (4.10) and intgratd. A mor accurat form is (s HW problm) log10 s = /T log 10 T ( s in mb and T in K) Bolton (1980) dtrmind th following mpirical rlation by fitting a function to tabular valus of s (T). This form is accurat to within 0.1% ovr th tmpratur intrval (-35 C, +35 C): 17.67T s ( T ) = xp (5.9) T

7 10/06/08 7 Tabl 5.1. Saturation vapor prssurs ovr watr and ic, and latnt hats of condnsation and sublimation. T ( C) s (Pa) i (Pa) L lv (10 3 J kg -1 ) L iv (10 3 J kg -1 ) Moistur paramtrs Common variabls usd for masurmnt of moistur: a) mixing ratio (r v ) r v = m v /m d = ε / [p-] ε/p (5.10) b) spcific humidity (q v ) q v = m v /(m v +m d ) = r v /r = r v / (r d +r v ) = ε / [p-(1-ε)] (using th quation of stat for r v and r d ) ε/p Thus, q v r v. c) vapor prssur () (connction to Clausius-Clapyron Eq.) -- Don t confus with s d) rlativ humidity (f) f = r v / r vs (T,p) / s (T) = r vs (T d )/r vs (T) (5.11) B carful - f can b xprssd in % or in fraction ) virtual tmpratur (T v ) (dfind prviously; not rally usd to masur watr vapor) T v T( r v ) f) dwpoint tmpratur (T d ): tmpratur at which saturation occurs; dfind as a procss in Chap. 6. Which of ths ar absolut and rlativ? Which ar consrvd for a substaturatd parcl that is not xprincing mixing with its surroundings?

8 10/06/08 8 Problms 1. Prov that pur watr vapor can rach saturation in th following procsss: a) isothrmal comprssion b) adiabatic xpansion 2. Using th rlation L(T) = L 0 (c-c pv )(T-T 0 ) for th dpndnc of th latnt hat of vaporization on tmpratur, intgrat Eq. (5.7) for obtaining th xprssion for s (T). This rsult is somtims call th Magnus quation or Kifr s formula. Ovr th rang 30 C T +30 C, compar s (T) from this quation with th simplr approximation (5.8), with Bolton s mpirical formula s 17.67T ( T ) = xp T whr s is in mb and T is in C. Also compar with data in Tabl A sampl of moist air has a tmpratur of 5 C, a prssur of 80 kpa, and a rlativ humidity of 65%. Solv for th following proprtis of th sampl by calculations, using approximations only whn ncssary. Confirm your answrs with a skw-t diagram whrvr possibl. a) Potntial tmpratur b) Mixing ratio c) Dw point d) Saturation point tmpratur (tmp. of th LCL) ) Equivalnt potntial tmpratur f) Virtual tmpratur g) Dnsity 4. Dtrmin th watr vapor prssur and mixing ratio for a dw point tmpratur of 90 F and a prssur of 1 atm. (1 atm = 1013 mb). ( F = 9/5 C + 32) 5. From th quation log 10 s = /T log 10 T for th saturation vapor prssur (in mb), driv th latnt hat of vaporization at 10 C.

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