ESCI 341 Atmospheric Thermodynamics Lesson 13 Phase Changes Dr. DeCaria

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1 ESCI 341 Atmopherc Thermodynamc Leon 13 Phae Change Dr. DeCara Reference: Thermodynamc and an Introducton to Thermotattc, Caen Phyca Chemtry, Lene GENERAL A phae change a change between od, qud, or apor (ga). A ytem may een hae more than one od or qud phae Carbon ha eera od phae, ncudng graphte and damond The phae of a ytem n equbrum can be repreented on a phae dagram. A typca phae dagram hown beow The ne eparatng the phae repreent pont where the two phae can coext. The trpe pont repreent the one and ony pont on the dagram where a three phae can coext. The crtca pont occur at the end of the apor-to-qud tranton ne. At temperature aboe the crtca temperature, or preure aboe the crtca preure there no dtncton between the qud and apor phae.

If you go from the apor to qud phae at temperature or preure exceedng the crtca aue, condenaton doe not occur. Intead, there a gradua ncreae n denty and a contnuou tranton from apor to qud.

2 If you go from the apor to qud phae at temperature or preure exceedng the crtca aue, condenaton doe not occur. Intead, there a gradua ncreae n denty and a contnuou tranton from apor to qud. The ope of the od-to-qud tranton ne how how the metng pont change wth preure. The dagram aboe typca for ubtance uch a CO2. In th cae, an ncreae n preure w ncreae the metng pont. If you are near the od-to-qud tranton ne n th cae, queezng the ubtance w caue t to odfy. The trpe-pont temperature and preure for CO2 are 57C and 5.1 atm. Therefore, at room temperature dry ce a ga, and f t cooed beow 57C t w depot nto the od phae, wthout eer gong through the qud phae. Th why od CO2 referred to a dry ce. PHASE DIAGRAM FOR H2O Water one of the few ubtance that can ext n a three phae at the temperature and preure found n the Earth atmophere. The phae dagram for water 2

3 The trpe pont of water at 0.01C and atm (611 Pa). A unque feature of water that the od-to-qud tranton ne ope upward to the eft, ntead of to the rght. GIBBS FREE ENERGY AND MATERIAL EQUILIBRIUM We hae preouy hown the foowng nequate for coed ytem du TdS pdv dh TdS Vdp df SdT pdv dg SdT Vdp (1) Thee nequate are ad for a procee n a coed ytem that n mechanca and therma equbrum, but not matera equbrum 1. For a mut-component ytem the Gbb free energy a functon of T, p, and the mae of each component, m. The dfferenta of G therefore G G G dg dt dp dm T p m (2) p, m T, m T, p We ao know that for a coed ytem wth fxed compoton ( dm 0 ) that ao n equbrum, that dg SdT Vdp, (3) and comparng Eq. (2) and (3) how that G T G p pm, Tm, S V o Eq. (2) can be wrtten a G dg SdT Vdp dm m. (4) T, p We ao know that the tota Gbb free energy of a ubtance equa to the pecfc Gbb free energy of each conttuent mutped by t ma, 1 See Lene, Chapter 4 (partcuary Secton 4.4) f more deta are dered. 3

4 Dfferentatng Eq. (5) how that 2 G g m. (5) G m T, p g. Ung th n Eq. (4) yed Gong back to Eq. (1) we know that Combnng Eq. (6) and (7) ge whch reduce to dg SdT Vdp g dm. (6) dg SdT Vdp. (7) SdT Vdp g dm SdT Vdp gdm 0. (8) Equaton (8) te u that f a mut-component ytem not n matera equbrum, that t w then pontaneouy eoe unt Equaton (9) the condton for matera equbrum. g dm 0. (9) Note that een though the equbrum condton of Eq. (9) n term of the pecfc Gbb free energy, t not mted to contant preure, contant temperature procee. It the matera equbrum condton wthout any retrcton. MATERIAL EQUILIBRIUM IN A TWO COMPONENT SYSTEM 2 Note that the pecfc Gbb free energy, g, of a conttuent doe not depend on the ma of tef or any g m other conttuent, o that 0 for any aue of or j. Howeer, th not true for the other j u G ntene tate arabe,.e., 0. So, athough g, t not neceary true that m m U m u. j 4

5 In a ytem wth two component, Equaton (8) become dm g dt dm g. (10) dt 0 Any ma ot from component 1 mut be ganed by component 2, o we hae and Eq. (10) become dm dt dm dt dm. (11) dt 1 g g 0 Let examne what happen f the two component are not n matera equbrum, o that g g. There are two pobe cenaro: Cae 1: g g Th cae requre dm1 dt 0 o the ma of component 1 w decreae whe the ma of component 2 w ncreae. Cae 1: g1 g2 Th cae requre dm1 dt 0 o the ma of component 1 w ncreae whe the ma of component 2 w decreae. In ether cae, the reut that the component that ha the owet pecfc Gbb free energy w ncreae at the expene of the component wth the hgher Gbb free energy. Another way to tate th that equbrum faor the component wth the owet pecfc Gbb free energy. The ony way the two component can be n matera equbrum f they hae the ame pecfc Gbb free energy, g1 g2. THE CLAPEYRON EQUATION If we hae two phae preent, the condton for equbrum from Eq. (9) g1 g 2. (12) The equbrum condton ay that at equbrum the pecfc Gbb free energe of the two phae w be equa. Dfferentatng Eq. (12) ge 5

6 From the Gbb equaton we hae o we can wrte Eq. (13) a whch can be rearranged to ge dg1 dg 2 dg dt dp. dt dp dt dp From the frt aw of thermodynamc we hae. (13) dp 2 1 dt. (14) dh Td dp. whch, f the temperature and preure are contant, ntegrate to h T L, where L the atent heat of the phae change (unt of energy per ma). Equaton (14) then become dp dt whch known a the Capeyron Equaton. L, (15) T The Capeyron equaton competey genera. It appe to any equbrum phae change. LIQUID VAPOR EQUILIBRIUM If we appy the Capeyron equaton to the phae change between qud and apor, then the preure jut the aturaton apor preure (p = e) and we get de dt. (16) L T Snce the pecfc oume of a apor o much greater than the pecfc oume of a qud, we can approxmate the dfference between the two a. Th make Eq. (16) If the apor an dea ga, then de dt L. (17) T 6

7 and Eq. (17) become 1 de e dt RT e whch known a the Cauu-Capeyron equaton., L, (18) 2 RT Integratng the Cauu-Capeyron equaton w ge an equaton for the nterface of the od-qud phae change on the phae dagram. Integratng between a known reference apor preure and temperature, e0 and T0 (aumng that the atent heat ndependent of temperature and preure) ge e L 1 1 e 0 exp. (19) R T0 T A uazaton a to why aturaton apor preure depend on temperature can be found at th nk: SOLID VAPOR EQUILIBRIUM For the phae change from od to apor we ony need to repace the atent heat of aporzaton wth the atent heat of ubmaton, L n Eq. (19). Thu, oer an ce urface the aturaton apor preure e L 1 1 e 0 exp. (20) R T0 T SOLID LIQUID EQUILIBRIUM For od qud equbrum we mut return to the Capeyron equaton [Eq. (15)] dp dt L. T For mot ubtance the od more dene than the qud, o that >. Thu, for mot ubtance, the od-qud ne on the phae dagram w tt upward to the rght. 7

8 Water unque n that t od phae (ce) e dene than the qud phae, o that <. Th why the od-qud ne on the water phae dagram tt upward to the eft. THE EFFECT OF AIR PRESSURE ON THE SATURATION VAPOR PRESSURE Up to know we e aumed that the ony ubtance we had n our ytem wa water. What happen f we ao hae an nert ga, uch a ar preent? The equbrum condton, Eq. (9), t requre that g dm 0, but now we hae three conttuent: qud water (m), water apor (m), and dry ar (md). The equbrum condton therefore g dm g dm g dm 0. Snce the ar nert, dmd = 0 and dm = dm. So a before, the equbrum condton d d dg dg. (21) At frt gance th ook ke we are gong to get an exacty equaent anwer to what we had wthout ar preent. Howeer, there a dfference. Athough the ar doe not hae any effect on the Gbb free energy of the water apor (nce both are dea gae, and therefore don t nfuence each other), the ar doe ncreae the preure on the qud. So, we hae dt dp dt de (note that f ar were not preent, then p = e and we woud jut get the Capeyron equaton a before). We want to know what affect changng p ha on e, o f we hod the temperature contant we get whch can be wrtten a dp de f temperature hed contant e p T. (22) 8

9 Equaton (22) caed the Poyntng Equaton. The Poyntng Equaton ay that an ncreae n preure w ncreae the aturaton apor preure. Th mpe that t eaer for water moecue to ecape from the qud to the apor phae f the qud under preure. The phyca expanaton for th phenomena not traght-forward, and there are ome popuar though ncorrect expanaton hang to do wth the number of adjacent moecue hodng a water moecue n the qud phae. One eemngy reaonabe phyca expanaton that ometme ued that a the qud put under preure and the moecue are queezed coer together, the attracte force between moecue become weaker and repue force are enhanced. Th make t eaer for a water moecue to ecape or pop out of the qud urface. Une preure are ery hgh or temperature are ery ow, >>. Therefore, for mot atmopherc appcaton e p T 0, and we can gnore the effect of ambent ar preure on the aturaton apor preure. Howeer, we w ee n the next eon that n a cured dropet the nteror preure ery greaty enhanced, and the effect of the Poyntng Equaton cannot be dmed n th tuaton. It mportant to note that the preence of ar doe not gnfcanty ater the aturaton apor preure. We often peak of the ar hodng the water apor, and of warm ar omehow beng abe to hod more water apor. Th noton actuay pretty y, nce the ar dong nothng. There woud be the ame amount of water apor preent regarde of whether or not ar wa preent. THE MYTH OF THE ICE SKATER The anomaou ope of the od-qud equbrum ne for water ha ed many to attrbute the abty to ce kate on preure metng, whereby a a kater 9

10 pae oer the ce, the ncreaed preure on the ce hft the equbrum from the od phae to the qud phae. It turn out that, athough the expanaton ound paube, t ncorrect. The true nature of the pperne of ce much more compex, and noe a proce caed urface metng. The deta are beyond the cope of th coure, but an oerew of the ubject can be found n Roenburg (2005) 3. 3 Roenberg, R, 2005: Why ce ppery?, Phyc Today, December ue, pp

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