Clausius-Clapeyron Equation

Transcription

1 ausius-apyron Equation p (mb) Liquid Soid Vapor (º) oud drops first form whn th aporization quiibrium point is rachd (i.., th air parc bcoms saturatd) Hr w dop an quation that dscribs how th aporization/condnsation quiibrium point changs as a function of prssur and tmpratur

2 Who ar ths pop? ausius-apyron Equation Rudof ausius Grman Mathmatician / Physicist Discord th Scond Law Introducd th concpt of ntropy Bnoit Pau Emi apyron Frnch Enginr / Physicist Expandd on arnot s work

3 ausius-apyron Equation Basic Ida: Proids th mathmatica rationship (i.., th quation) that dscribs any quiibrium stat of watr as a function of tmpratur and prssur. p (mb) Liquid Accounts for phas changs at ach quiibrium stat (ach tmpratur) 03 Soid 6. P (mb) Vapor Vapor (º) Liquid Liquid and Vapor V Sctions of th P-V and P- diagrams for which th ausius-apyron quation is drid in th foowing sids

4 Mathmatica Driation: ausius-apyron Equation Assumption: Our systm consists of iquid watr in quiibrium with watr apor (at saturation) W wi rturn to th arnot yc Isothrma procss Saturation apor prssur 2 B A Adiabatic procss D 2 Saturation apor prssur 2 A, D B, Voum 2 mpratur

5 Mathmatica Driation: Rca for th arnot yc: ausius-apyron Equation W Q NE Q Q 2 Q Q 2 whr: Q > 0 and Q 2 < 0 2 Saturation apor prssur 2 B A Isothrma procss Adiabatic procss Q W NE Q 2 D 2 If w r-arrang and substitut: Voum Q W NE - 2

6 Mathmatica Driation: ausius-apyron Equation Rca: During phas changs, Q = L Sinc w ar spcificay working with aporization in this xamp, Q W NE - 2 Aso, t: Q L 2 d Saturation apor prssur 2 B A Isothrma procss Adiabatic procss Q W NE Q 2 D 2 Voum

7 Mathmatica Driation: ausius-apyron Equation Rca: h nt work is quiant to th ara ncosd by th cyc: W NE dv dp Q W NE - 2 h chang in prssur is: Isothrma procss d h chang in oum of our systm at ach tmpratur ( and 2 ) is: dv α α w 2 dm Saturation apor prssur 2 B A Adiabatic procss Q W NE Q 2 D 2 whr: α = spcific oum of apor α w = spcific oum of iquid dm = tota mass conrtd from apor to iquid Voum

8 ausius-apyron Equation Mathmatica Driation: W thn mak a th substitutions into our arnot yc quation: Q WNE L α αw dmd - d 2 W can r-arrang and us th dfinition of spcific atnt hat of aporization ( = L /dm) to obtain: d d α ausius-apyron Equation for th quiibrium apor prssur with rspct to iquid watr α w Saturation apor prssur 2 B, A, D 2 mpratur

9 Gnra Form: ausius-apyron Equation Rats th quiibrium prssur btwn two phass to th tmpratur of th htrognous systm Equiibrium Stats for Watr (function of tmpratur and prssur) p (mb) dp s d Δ Liquid whr: = mpratur of th systm = Latnt hat for gin phas chang dp s = hang in systm prssur at saturation d = hang in systm tmpratur Δα = hang in spcific oums btwn th two phass Soid Vapor (º)

10 ausius-apyron Equation Appication: Saturation apor prssur for a gin tmpratur Starting with: d d α α w Assum: and using: α α w α R [aid in th atmosphr] [Ida gas aw for th watr apor] W gt: d R d 2 If w intgrat this from som rfrnc point (.g. th trip point: s0, 0 ) to som arbitrary point (, ) aong th cur assuming is constant: s0 d R 2 0 d

11 ausius-apyron Equation Appication: Saturation apor prssur for a gin tmpratur s0 d R 2 0 d Aftr intgration w obtain: n s0 R 0 Aftr som agbra and substitution for s0 = 6. mb and 0 = K w gt: (mb) 6. xp R (K)

12 ausius-apyron Equation Appication: Saturation apor prssur for a gin tmpratur (mb) 6. xp R (K) A mor accurat form of th abo quation can b obtaind whn w do not assum is constant (rca is a function of tmpratur). S your book for th driation of this mor accurat form: (mb) xp n ( K) ( K)

13 ausius-apyron Equation Appication: Saturation apor prssur for a gin tmpratur (mb) xp n ( K) ( K) What is th saturation apor prssur with rspct to watr at 25º? = K = 32 mb What is th saturation apor prssur with rspct to watr at 00º? = K Boiing point = 005 mb

14 ausius-apyron Equation Appication: Boiing Point of Watr d d α α w At typica atmosphric conditions nar th boiing point: = 00º = 373 K = J kg - α =.673 m 3 kg - α w = m 3 kg - d d 36.2 mb K his quation dscribs th chang in boiing point tmpratur () as a function of atmosphric prssur whn th saturatd with rspct to watr ( )

15 ausius-apyron Equation Appication: Boiing Point of Watr What woud th boiing point tmpratur b on th top of Mount Mitch if th air prssur was 750mb? From th prious sid w know th boiing point at ~005 mb is 00º Lt this b our rfrnc point: rf = 00º = K -rf = 005 mb Lt and rprsnt th aus on Mt. Mitch: = 750 mb d d 36.2 rf rf rf 36.2 mb K 36.2 rf mb K = 366. K = 93º (boiing point tmpratur on Mt. Mitch)

16 ausius-apyron Equation Equiibrium with rspct to Ic: W wi know xamin th quiibrium apor prssur for a htrognous systm containing apor and ic p (mb) Liquid P (mb) Soid 03 Liquid 6. Vapor Vapor Soid (º) si B A V

17 ausius-apyron Equation Equiibrium with rspct to Ic: Rturn to our gnra form of th ausius-apyron quation p (mb) d s d Soid Liquid Mak th appropriat substitution for th two phass (apor and ic) d si d α s α i Vapor (º) ausius-apyron Equation for th quiibrium apor prssur with rspct to ic

18 ausius-apyron Equation Appication: Saturation apor prssur of ic for a gin tmpratur Foowing th sam ogic as bfor, w can dri th foowing quation for saturation with rspct to ic si (mb) 6. xp R s (K) A mor accurat form of th abo quation can b obtaind whn w do not assum s is constant (rca s is a function of tmpratur). S your book for th driation of this mor accurat form: si (mb) xp n ( K) ( K)

19 ausius-apyron Equation Appication: Mting Point of Watr Rturn to th gnra form of th ausius-apyron quation and mak th appropriat substitutions for our two phass (iquid watr and ic) dp wi d α w f α i At typica atmosphric conditions nar th mting point: = 0º = 273 K s = J kg - α w = m 3 kg - α i = m 3 kg - dp d 35,038 mb wi K his quation dscribs th chang in mting point tmpratur () as a function of prssur whn iquid watr is saturatd with rspct to ic (p wi )