ThermalPhysics oftheAtmosphere
SecondEdition
MaartenH.P.Ambaum DepartmentofMeteorology UniversityofReading Reading,UnitedKingdom
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5.3 Humidityvariables................................101
5.4 Dewpointtemperature..............................102
5.5 Wet-bulbtemperature...............................104
5.6 Moiststaticenergy .................................106
5.7 ThePenmanequation. .............................109 Problems........................................112
CHAPTER6Verticalstructureofthemoistatmosphere 115
6.1 Adiabaticlapserateformoistair......................115
6.2 Entropyofmoistair................................118
6.3 Finiteamplitudeinstabilities .........................124
6.4 Verticalstructureinthermodynamicdiagrams. ...........125
6.5 Convectiveavailablepotentialenergy ...................130 Problems........................................132
CHAPTER7Clouddrops .................................... 133
7.1 Homogeneousnucleation:theKelvineffect... ...........133
7.2 Heterogeneousnucleation:theRaoulteffect... ...........138
7.3 Köhlertheory ....................................140
7.4 Charge-enhancednucleation .........................144
7.5 Dropgrowthbydiffusion............................148
7.6 Dropgrowthbycollisionandcoalescence ...............156 Problems........................................158
CHAPTER8Mixturesandsolutions 161
8.1 Chemicalpotentials................................161
8.2 Idealgasmixturesandidealsolutions ..................164
8.3 Raoult’slawrevisited...............................166
8.4 Boilingandfreezingofsolutions.. ....................168
8.5 Affinityandchemicalequilibrium. ....................170 Problems........................................174
CHAPTER9Thermalradiation .............................. 177
9.1 ThermalradiationandKirchhoff’slaw ..................177
9.2 TheStefan–BoltzmannandWiendisplacementlaws .......180
9.3 Globalenergybudgetandthegreenhouseeffect ...........182
9.4 Climatefeedbacksandthehydrologicalcycle. ...........186
9.5 Thermodynamicsofaphotongas.. ....................189
9.6 DerivationofthePlancklaw .........................193
9.7 Energyflux,andtheStefan–Boltzmannintegral ...........198 Problems........................................202
CHAPTER10Radiativetransfer 205
10.1 Radiativeintensity................................205
10.2 Radiativetransfer.................................207
10.3 Zenithangles ....................................211
10.4 Radiative–convectiveequilibrium. ....................214
Preface
Shouldweviewclassicalphysicsasatooltounderstandphenomenainatmospheric science,orshouldweviewatmosphericscienceasanappliedbranchofclassical physics?Ofcoursebothviewpointsaremostlyaccurate,relevant,andevenoverlapping.However,byfocussingonthefirstviewpointwemaymissoutontheprofound senseofuniversality,oforganisationthatclassicalphysicsbringstoourunderstandingoftheworld.Herewewillfocusonthesecondviewpoint.Thisbookisanattempt topresentatmosphericscienceasoneofthegreatmodernapplicationsofclassical physics,inparticularofthermodynamics.
Thepresentsecondeditionof ThermalPhysicsoftheAtmosphere hasbeenrevisedandexpandedthroughoutcomparedtothefirstedition.Therevisionfollows fromyearsofteachingthismaterialtopostgraduatestudents.Manystudentshave toldmeoftheirsenseofachievementandsatisfaction—andofrelief!—onfinishing amasterslevelphysicstopic,oftentheirfirstexpositiontoadvancedphysicsmaterial.Istillexperiencethesamesenseofwonder,discoveringandexploringthemyriad waysinwhichwecanusethermodynamicstounderstandandpredictsomanydifferentphenomenaintheatmosphere.Ihopethereaderwillsharethissenseofwonder.
Reading,UnitedKingdom
October2020
MaartenAmbaum
Idealgases
Inthischapterweintroducetheconceptofanidealgas,agasofnon-interacting molecules.Anidealgasisanaccuratemodelofdilutegasessuchastheatmosphere.
Wefurtherintroducethenotionofmacroscopicvariables,amongstthemsuch familiaronesastemperatureandpressure.Thesemacroscopicvariablesmustberelatedtosomepropertyofthemicroscopicstateofthemoleculesthatmakeupthe substance.Forexample,forthesystemsweconsiderhere,temperatureisrelatedto themeankineticenergyofthemolecules.Thelinkingofthemacroscopicandmicroscopicworldsisthesubjectofstatisticalmechanics.Inthischapterwegivean elementaryapplicationofittoidealgases.
1.1 Thermodynamicvariables
Consideravolumeofgas.Ausefulmentalpictureisthatofagasinaclosedcylinder withapiston,similartothedrivingcylinderofasteamengine,see Fig. 1.1.Inthis waywecancontrolcertainpropertiesofthegas,suchasitsvolumeortemperature, andperformexperimentsonit.Suchexperimentsarenormallythoughtexperiments, althoughinprincipletheycanbeperformedinthelaboratory.
Atthemacroscopiclevel,thegashassomefamiliarproperties:
•volume V (units:m3 )
•mass M (units:kg)
•density ρ = M/V (units:kgm 3 )
•temperature T (units:K,Kelvin)
•pressure p (units:Pa = Nm 2 ,Pascal).
Thegasismadeupofmoleculeswithindividualmass M1 ,sothetotalmassofgasis M = NM1 , (1.1)
with N thenumberofmolecules.Thenumberofmolecules N isoftenexpressedas amultipleoftheAvogadronumber NA , NA = 6.022 × 1023 . (1.2)
TheAvogadronumberwasuntilveryrecentlydefinedasthenumberofmoleculesin 12gofcarbonisotope 12C,butisnowdefinedastheexactinteger6 02214076 × 1023
ThermalPhysicsoftheAtmosphere. https://doi.org/10.1016/B978-0-12-824498-2.00008-2 Copyright©2021RoyalMeteorologicalSociety.PublishedbyElsevierInc.incooperationwithTheRoyalMeteorologicalSociety.Allrights reserved.
FIGURE1.1
Gasinacylinderwithpiston.
Thenumberofmoleculesisthendefinedasamultiple n of NA
N = nNA , (1.3)
where n isthenumberof moles.Withthisdefinitionofthemol,themassofthegas canbewrittenas
M = nμ (1.4)
with μ = NA M1 the molarmass.Sothemolarmassof 12Cis12gmol 1 .(Although the‘mol’isnotstrictlyspeakingaphysicalunit,itisdefinedbywritingtheAvogadro numberas NA = 6 02214076 × 1023 mol 1 .)
Thetemperaturecanbedefinedas‘thatpropertywhichcanbemeasuredwitha thermometer’.Thisdefinitionsoundscircularbutitcanbeshowntobeaperfectly validdefinition.TheSIunit1 fortemperatureistheKelvin(K).Temperatureisoften denotedindegreesCelsius, ◦ Cwith T/◦ C = T/K 273 15,orindegreesFahrenheit, ◦ Fwith T/◦ F = 1 8 T/◦ C + 32,see Fig. 1.2.Temperaturecanneverbelowerthan0K, orabsolutezero;thetemperatureinKelvinisalsocalledthe absolutetemperature
FIGURE1.2
NomogramforCelsius–Fahrenheitconversion.
Figure 1.3 illustratesthetypicalmeantemperaturesencounteredthroughthedepth oftheEarth’satmosphere.Thisfigureusesthelogarithmofpressureasaverticalcoordinatebecausethisisapproximatelyproportionaltothealtitudeintheatmosphere.
1 SIstandsfor SystèmeInternationald’Unités,theinternationallyagreedsystemofunitsforphysical quantities.
Temperature,in ◦ C,asafunctionofheight.Tropicalannualmean(thickline),extratropical wintermean(mediumsolidline)andextratropicalsummermean(mediumdashedline). ThetropicsherecorrespondtothelatitudesbetweenthetropicsofCancerandCapricorn; theextratropicsherecorrespondtothelatitudesbeyond45◦ ineitherhemisphereforthe correspondingseason.BasedondatafromRandel,W.etal.(2004) JournalofClimate 17, 986–1003.
Goingupinaltitude,thetemperaturefirstdecreases(troposphere),increases (stratosphere),andthendecreasesagain(mesosphere).Themesosphereendsatabout 90 kmaltitude,abovewhichthetemperaturestartstoincreaseagain(thermosphere).These atmosphericlayersareseparatedbythetropopause,stratopause,andmesopause, respectively.
Thetemperatureincreaseinthestratosphereisduetothephoto-dissociationofozone,O3 , whichabsorbsthesolarenergyintheUVpartofthespectrum(wavelengthsshorterthan about 320 nm).Indeed,theozoneitselfisformedbyphoto-dissociationofmolecular oxygen,O2 ,whichoccursatwavelengthsshorterthan 240 nm.Themaximumozone concentration(‘theozone-layer’)isatabout 25 kmaltitude.
Thetemperatureinthetropospherehasamaximuminthetropics,whileinthe stratosphereithasamaximuminthesummerhemisphereandaminimuminthewinter hemisphere.Thislatitudinaltemperaturegradientisreversedinthemesosphere.Notealso thatthetropopauseiscoldestandhighestinthetropics.
Thethermosphere(outsidethisplot)isheatedbyabsorptionofUVradiationand subsequentionizationofthemolecularconstituents,thusformingtheionosphere.Atthese altitudesthedensityissolowthatenergydoesnotgetthermalizedeffectivelyandlocal thermodynamicequilibriumisnotfullyattained.Thethermospheregiveswaytospacein theexosphere.
FIGURE1.3
Pressureistheforceagasexertsonitsboundingwallsperunitarea.Thisdoes notmeanthatgasonlyhasapressuredefinedattheboundingwalls:theinternalpressureofagascaninprinciplebemeasuredbyinsertingsomeprobeandmeasuring theforceperunitareaontheprobe.Thereareseveralunitsofpressureinuse,each withitsspecificareaofapplication.TheSIunitforpressureisthePascal(Pa)which isequivalenttooneNewtonpersquaremetre.InatmosphericapplicationswenormallyusethehectoPascal(hPa;bydefinition,1hPa = 100Pa)ormillibar(mbar;with 1mbar = 1hPa).
Apressureof1013.25hPaisalsocalled oneatmosphere,thenotionalmeanvalue ofmeansealevelpressureonEarth.Sowehave1atm = 101325Pa.Thispressure unitisalmostexclusivelyusedinhighpressureengineeringapplicationsand,despite itsname,doesnotusuallyfindapplicationinatmosphericscience.Arelatedpressure unitisthe bar with1bar = 105 Pa,fromwhich,ofcourse,themillibarisderived.
Pressureandtemperaturedonotcorrespondtoapropertyofindividualmolecules. Theyarebulkpropertiesthatcanonlybedefinedasastatisticalpropertyofalarge numberofmolecules.Thiswillbediscussedinthenextsection.
Thereareseveralothermacroscopicvariablesthatcanbeusedtodescribethe stateofasimplegas;theseareknownas thermodynamicvariables.Ifweknowall therelevantthermodynamicvariables,weknowthefullthermodynamicstateofthe gas.Allthesevariablesareinterrelatedanditturnsoutthatforasimplesubstance(a substancewithafixedcomposition,suchasdryair)weonlyneedtwothermodynamic variablestodescribeitsthermodynamicstate.2
Formorecomplexsystemsweneedmorevariables.Forexample,inamixtureof varyingcompositionweneedtoknowtheconcentrationsoftheconstituents.Moist airissuchamixture.Thenumberofwatermoleculesintheairishighlyvariable andthesevariationsneedtobetakenintoaccount.Forseawaterweneedtoknow thesalinity—theamountofdissolvedsalts—becauseithasimportantconsequences forthedensity.Finally,forclouddropsweneedtoknowthesurfaceareaaswellas theamountofdissolvedsolute,bothofwhichhaveprofoundconsequencesforthe thermodynamicsofthedrops.
Thermodynamicvariablesareeither:
• extensive,proportionaltothemassofthesystem,or
• intensive,independentofthemassofthesystem.
Volumeandmassareextensivevariables,temperatureanddensityareintensivevariables.Formostvariablesitisobviouswhethertheyareextensiveorintensive.
2 Thenumber N ofthermodynamicvariablesrequiredtodefinethestateofanysystemisgivenbythe Gibbs’phaserule, N = 2 + C P,
with C thenumberofindependentconstituentsand P thenumberofcoexistingphases(gas,liquid,solid) inthesystem.
Extensivevariablescanbedividedbythemassofthesystemtobecomeintensive; suchnewvariablesarethencalled specific variables.Specificandextensivevariables willbedenotedbythesameletter,butwiththespecificvariablewritteninlowercase anditsextensiveequivalentinupper.Forexample,thevolume V ofasystemdivided bythemass M ofthesystembecomesthespecificvolume v with v = V/M .Note that
where ρ isthedensity.Specificvolumeisoften,confusingly,denotedbyaGreek letter α ,anapparentlyarbitrarynotationwhichwewillnotfollowhere.Laterwe willcomeacrossotherextensivevariables.Forexample,theentropy S ofasystem isanextensivevariable,sowecandefine s = S/M asthespecificentropy.Although temperature T isanintensivevariableitisnormallydenotedbyanuppercaseletter, aconventionweadopthereaswell.
Wehaveignoredinternalvariationsinthevolumeofgasormaterialunderconsideration.Forexample,weassumethereisnointernalmacroscopicmotionofthe gas,whichwouldbeassociatedwithpressurevariationsandinternalkineticenergy. Clearlythisisnotthecasefortheatmosphereasawhole.Thepressureanddensity varyenormouslythroughtheatmosphere,usuallymostdramaticallyinthevertical: at10kmheightthepressureisaboutaquarterofitssurfacevalue.
Weassumethatwecandefinetheintensivethermodynamicvariableslocallyand thattheyhavetheirusualequilibriumthermodynamicrelations.Wethensaythatthe gasisin localthermodynamicequilibrium.Localthermodynamicequilibriumisvalid ifthereisalargeseparationbetweenthespatialandtemporalscalesofmacroscopic variationsandthoseofmicroscopicvariations.Thespatialscaleofmacroscopicvariationsneedstobemuchlargerthanthe meanfreepath ofmolecules,themean distanceamoleculetravelsbetweencollisionswithothermolecules.Thetemporal scaleofmacroscopicvariationsneedstobemuchlargerthanthemeantimebetween molecularcollisions.NeartheEarth’ssurfacethemeanfreepathintheatmosphereis about0 1µm(about30timestheaveragemoleculardistance)withtypicalmolecular velocitiesofseveralhundredsofmetrespersecond,solocalthermodynamicequilibriumissatisfied.Itturnsoutthataboveabout100kmheight,localthermodynamic equilibriumbreaksdown:thedensityandcollisionrateissolowthatthermalequilibriumcannotbeachievedonshortenoughtimescales.
Asmallvolumeofgasintheatmosphere,forwhichtheinternalmotioncanbe ignoredandwhichhaswell-defineddensity,temperature,andsoon,iscalledan air parcel.Becauseanairparcelis,bydefinition,inlocalthermodynamicequilibrium, itsthermodynamicvariablessatisfyalltherelationshipsthatarefoundinequilibriumsystems.Atthelevelofanairparcelweneednotworryaboutnon-equilibrium effects.
1.2 Microscopicviewpoint
Fromthemicroscopicviewpoint,temperatureisdefinedastheaveragekineticenergy ofthemolecules,
with (U , V , W ) thethree-dimensionalvectorvelocityofthemolecule.Thebrackets denotetheaverage,atimeaverageforasinglemolecule,theaverageoverall molecules,ortheaverageoveranensembleofgasesinthesamemacroscopicstate. Akeyassumptionofstatisticalmechanicsisthatalltheseaveragesleadtothesame result.Theconstant kB istheBoltzmannconstant,
Instatisticalphysicsaswellasmacroscopicthermodynamics,energyisthefundamentalquantity.Temperatureisaderivedquantitywhichhasbeengivenitsown unitsbecauseitismeasuredwithathermometer.TheBoltzmannconstantismerelya proportionalityconstantbetweenenergyandabsolutetemperature.Thefundamental pointisthatstatisticalmechanicscanbeformulatedsuchthatthemicroscopicdefinitionoftemperatureintermsofthemeankineticenergyofthemoleculescorresponds tothethermodynamicdefinitionoftemperature.
BeforeMay2019theKelvinwasdefinedas exactly 1/273.16ofthetemperatureat thetriplepointofwater.Thestandardoftemperaturewasatriplepointcell,aclosed vesselofglasswhichcontainsonlypurewater,keptatthetemperaturewherethe watercoexistsinitsthreephases.Underthatdefinition,theBoltzmannconstantwas determinedbymeasuringhowmuchenergyamoleculegainsforagiventemperature change.FromMay2019theBoltzmannconstanthaschangedfromameasuredquantitytoadefinedfixedconstant exactly equalto1.380649 × 10 23 whenexpressedin theunitJK 1 .ThisdefinitionthentiesthevalueoftheKelvintothevalueofthe Joule,whichisdefinedindependentlyintermsofotherfundamentalphysicalconstants.Thetemperatureofwateratitstriplepointisnowameasuredquantity.Of courseitisstillmeasuredtobe273.16Kbutnowwithanuncertaintyof0.0001K.
Thefactor3/2inthemicroscopicdefinitionoftemperaturereflectsaclassicresult inthemechanicsofsystemswithmanycomponents,namelythateach degreeoffreedom contains,onaverage,thesameenergy.Adegreeoffreedomisanindependent variableinwhichthesystemcanvary.Asinglemoleculecarriesthreetranslational degreesoffreedom:motioninthe x , y ,and z-directions.Therecanalsobeinternal degreesoffreedomcorrespondingtorotationsandvibrationsofthemolecule.The equipartitiontheorem statesthateachaccessibledegreeoffreedom3 carriesonaveragethesameenergy,andthisenergyequals kB T/2.Addingtheaveragekinetic
3 Notallavailabledegreesarenecessarilyaccessible.Quantizationofenergylevelsimpliesthatthereisa minimumenergyrequiredtoexcitehigherenergylevelsinanydegreeoffreedom.
FIGURE1.4
Transferofmomentumbyamoleculecollidingwiththewall.Thetotalmomentumtransfer istwicethemomentuminthe x -directionofthemolecule.
energiesinthethreespatialdirectionsthengivestheresultofEq. (1.6).Theproofof theequipartitiontheoremisgiveninSection 4.7.
Pressureistheresultofmanycollisionsofindividualmoleculesagainstthewalls ofavesseloraprobe.Ifamoleculeapproachesthewallwithavelocity U and elasticallycollideswiththewall,thenthemolecule’smomentuminthedirectionof thewallchangesby2M1 U ,from M1 U to M1 U .Thismomentumistransferredto thewall.ByNewton’slaws,theamountofmomentumtransferredperunittimeisthe forceonthewall,see Fig. 1.4.Foraninteriorpointwecandefinethelocalpressure asthemomentumfluxdensitythroughsomeimaginarysurfaceintheinteriorofthe fluid.
Sohowmanymoleculescollidewiththewall?Letthenumberdensityof molecules,thatisthenumberofmoleculesperunitvolume,bedenotedwith ˜ n.We cannowwritethenumberdensityofmoleculeswith x -velocitiesbetween U and U + dU as nU ,whichisrelatedtothetotalnumberdensity n by
Overatime δt ,thosemoleculeswithpositivevelocitybetween U and U + dU that arelocatedwithinadistance U δt ofthewallwillcollidewiththewall.Therefore, thenumberofsuchmoleculesthathavecollidedwiththewallwillbe ˜ nU U δtA, with A theareaofthewall.Togetthemomentumtransferperunittime,simply multiplythisnumberbythemomentumtransferpermolecule,2M1 U ,anddivide bythetimetaken, δt .Thisistheforce FU exertedonthewallbymoleculeswith positivevelocitiesbetween U and U + dU ,
Tofindthepressureweneedtodivideby A andintegratetheforceoverallpositive velocities, U > 0,becausemoleculeswithnegativevelocitieswillnotcollidewith
thewallandthuswillnotcontributetothepressure,
Bysymmetry,therewillbeanequalnumberofmoleculeswithpositiveandnegative U .Wecanthereforeintegrateoverallvelocities U ,positiveandnegative,anddivide theresultbytwo.Theexpressionforthepressurethenbecomes
with ˜ n thetotalnumberdensity.Theequipartitiontheoremstatesthat M1 U 2 = kB T sothatthepressuresatisfies
= NkB T, (1.12)
wherewehavesubstituted ˜ n = N/V .Thisisthe idealgaslaw. Bywritingthetotalnumberofmolecules N as nNA ,theidealgaslawcanbe written
where R iscalledthe universalgasconstant,
Beforethemicroscopicdefinitionsoftemperatureandpressurewereknown,itwas alreadyhypothesizedbyAvogadro,andlaterconfirmedtobetrue,thattheconstant R isthesameforalltypesofgases,andthereforeindeeduniversal.
Theidealgaslawimpliesthatforaparticularnumberofmolecules nNA held insomevolume V atsometemperature T ,thepressureisindependentofthetype ofmolecules.Thiscanbeunderstoodfromtheobservationthatboththekineticenergy(andthereforetemperature)andthemomentum(andthereforepressure)ofa moleculescalewiththemassofthemolecule.
Anotherformoftheidealgaslawfollowsbydividingbythemass M = nμ ofthe gastofind pv = RT, or p = ρRT, (1.15) where R istheso-called specificgasconstant, R = R /μ. (1.16)
Thisistheformoftheidealgaslawthatisnormallyusedinatmosphericscience. Confusingly,theconventionistouseacapital R forthespecificgasconstanteven thoughitisaspecificquantity.Notealsothatinmostphysicsandchemistryliterature theletter R standsfortheuniversalgasconstant;itshouldbeclearfromthecontext
whichismeant.Thisisoneofthoseinstanceswheretheconventionusedinatmosphericscienceliteratureisnotparticularlyhelpful.Althoughtheidealgaslawinthe formofEq. (1.13) ismoregeneral,thebigadvantageoftheformofEq. (1.15) isthat itisformulatedintermsofspecificquantities:wedonotneedtodefinethesizeof thesystemwearedescribing.
Theidealgaslawencompasses:
• Boyle’slaw: atconstanttemperature,theproductofpressureandvolumeisconstant
• Gay-Lussac’slaw: atconstantvolume,thepressureofagasisproportionaltoits temperature.
Figure 1.5 illustratestheselawsindiagrams.Theselawswereoriginallydetermined experimentally.Theyareonlystrictlyvalidforidealgases.
TheleftpanelillustratesBoyle’slawandthemiddlepanelGay-Lussac’slaw.Therightpanel illustratesthat,foranidealgasatfixedpressure,thevolumeofagasisproportionaltoits temperature;thisissometimesknownas Charles’slaw
Inderivingtheidealgaslaw,wehavenotconsideredsubtletiessuchasinelastic collisions,whereenergytransferbetweenthegasandthewalloccurs,ortheconsiderationthatthewallisnotamathematicalflatplanebutmadeupofmolecules.These complicationsdonotalterthebasicresult.
Wehavealsonotconsideredinteractionsbetweenthemoleculesandinteractions atadistancebetweenthemoleculesandthewall.This does makeadifferenceand itdefinesthedifferencebetweenrealgasesandidealgases.Idealgasesaremadeup ofnon-interactingmolecules,vanishinglysmallmoleculesthatareunaffectedbythe presenceofanyothermolecules.
Weassumethatmoleculesinanidealgasdonotinteractwitheachotherandalso thatthemoleculesareinthermalequilibrium.Strictlyspeakingtheseassumptionsare inconsistent,asagascanonlyachievethermalequilibriumthroughmanycollisions betweenthemolecules.Thecollidingmoleculesdistributetheenergyamongstallthe accessibledegreesoffreedomandthusachieveequipartition.Thisprocessofenergy distributioniscalled thermalization.Agasisinlocalthermodynamicequilibrium ifalltheavailableenergyisthermalized.Ifcollisionsarerare,energycannotbe thermalizedeffectivelyandthegascannotachievelocalthermodynamicequilibrium.
FIGURE1.5
Thisoccursathighaltitudesintheatmosphere(higherthanabout100km)wherethe energyinputfromradiationisnotthermalizedduetothelownumberofcollisions.
Theidealgaslawisanexampleofan equationofstate.Realgasesarenotideal andwillthereforehaveadifferentrelationshipbetweenpressure,densityandtemperature.Forexample,theequationofstateforrealgasesismoreaccuratelydescribed by vanderWaals’equation,
with a and b constantsthatdependontheparticulargas.Theterm nb represents thereductioninavailablevolumeduetothefiniteeffectivesizeofthemolecules.
Theterm a(n/V)2 isrelatedtotheaverageinteractionenergybetweenmolecules (whichiswhythetermisquadraticinthenumberdensity)anditcontributesasan effectivepressure;therelationbetweenpressureandenergydensitywillbeexplained inSection 2.2,andaderivationofvanderWaals’equationwillbepresentedinSection 3.8.VanderWaals’equationismoreaccurateforgasesathighdensities,and approximatelydescribessuchimportantprocessesasphasetransitions.
DensitiesofgasesintheatmospherearesuchthattheidealgaslawgivesessentiallythesameresultsasvanderWaals’equation,sowesticktothemuchsimpler idealgaslaw.Indeed,forlowdensitiesvanderWaals’equationreducestotheideal gaslaw,
Conversely,forlargepressures,
Sothehighpressure,incompressibleliquidisalsoalimitingcaseofvanderWaals’ equation.
Ingeneral,anequationofstateissomerelationshipbetweenthevariablesofthe system,
where a1 ,a2 representanyothervariablesthatinfluencethestateofthesystem, suchashumidityinair,orsalinityinwater.Theequationofstateisdependenton theprecisenatureofthesystemandresultsderivedusinganequationofstateare thereforeonlyvalidforthatparticularsystem.Wewillseethatmanyoftheresultsin thisbookarederivedwithoutreferencetoanequationofstateandwillthereforebe validforanysubstance.
Eq. (1.17) → pV = nR T if n/V → 0 (1.18)
Eq. (1.17) → V = nb if p →∞ (1.19)
1.3 Idealgasmixtures
Idealgasesaredefinedasgaseswherethemoleculesthemselveshavenegligiblevolumeandhavenegligibleinteractionswitheachother.Soifwemixseveralidealgases atthesametemperatureinasinglevolumetheydonot‘feel’eachother’spresence. Thismeansthateachconstituentgascontributesindependentlytothepressure.The contributionofeachconstituenttothetotalpressureiscalledthe partialpressure.So ni molsofconstituent i willhaveapartialpressure pi equalto
Thefactthatthesepartialpressuresindependentlymakeupthetotalpressureofthe mixtureiscalled Dalton’slaw:
Dalton’slawisonlystrictlytrueforidealgases.Fornon-idealgases,partialpressures cannotbeeasilydefinedingeneral.
AnotherwayofexpressingDalton’slawis
wherethemolarfraction yi ofconstituent i isdefinedby
with n = i ni thetotalnumberofmolesinthemixture. FromDalton’slawitfollowsthat
with Ri thespecificgasconstantforconstituent i and wi themassfractionofconstituent i —thatis,thefractionconstituent i contributestothetotalmassofthe mixture(seealsoProblem 1.6):
Thespecificgasconstantofthemixtureisrelatedtoaneffectivemolarmassofthe mixture μ whichisdefinedby
Table 1.1 liststhemainconstituentsofair.Thedryairconstituentsarewellmixed andlong-livedwhichmeansthatthebulkcompositionofdryairisfixedthroughout
FIGURE1.6
Massfraction(leftaxis,inpartspermillion)andmolarmixingratio(rightaxis,inpartsper million)ofCO2 asmeasuredatMaunaLoaObservatory,Hawaii.Inset:meanannualcycle ofCO2 ,withscalesasinthemaingraph.Thisannualcycleisdominatedbyvegetation growthandresultingCO2 captureintheNorthernHemispheresummer.DatafromDr. PieterTans,NOAA/ESRL(www.esrl.noaa.gov/gmd/ccgg/trends/)andDr.RalphKeeling, ScrippsInstitutionofOceanography(scrippsco2.ucsd.edu/).SeealsoKeeling,C.D.etal. (1976) Tellus 28,538–551.
theatmosphereuptoveryhighaltitudes.Thereisalong-termupwardtrendinCO2 concentrationovertimeduetohumanactivitybutthishasonlyaminoreffecton theidealgaspropertiesofdryair,see Fig. 1.6.However,theCO2 trenddoeshave aprofoundeffectontheradiativepropertiesoftheairanditisthemainagentof human-inducedclimatechange.
Usingthenumbersin Table 1.1 itisstraightforwardtoverifythattheeffective molarmass μd fordryairis
andthespecificgasconstant R fordryairis
Table1.1 Mainconstituentsofair.
FIGURE1.7
Nomogramexpressingtheidealgaslawfordryair, p = ρRT (with R = 287 Jkg 1 K 1 ).
Thenomogramworksbylayingaruleracrossthenomogramtodefineanindexline;forany straightindexlinecrossingthethreegraduatedscales,thecorrespondingvaluesfor ρ , p , and T willsatisfytheidealgaslawfordryair.
Thisvalueof R canbeusedintheidealgaslaw,Eq. (1.15),whichthenrelatesthe pressure,density,andtemperaturefordryair. Figure 1.7 isanomogramexpressing thisrelationgraphically.
Nowwecanalsocalculatetheeffectofwatervapourontheidealgaslaw.The amountofwatervapourintheaircanbequantifiedbyitsmassfraction.Forwater vapourthismassfractioniscalledthe specifichumidity,usuallydenoted q .This meansthatthemassfractionofdryairmustbe1 q .FromEq. (1.26) itthenfollows
thatthespecificgasconstantforthemoistairis
(1 q) R μd + q R μv = R 1 + μd μv 1 q , (1.31)
with μd and μv themolarmassesfordryairandwater,respectively,and R thespecificgasconstantfordryair,Eq. (1.30).Itiscommonusagetoabsorbthefactorwith thedependencyonspecifichumidityinaredefinedtemperature,the virtualtemperature, Tv .Sotheidealgaslawformoistairis
pv = RTv or p = ρRTv , (1.32)
wherethevirtualtemperature Tv isdefinedas
μd /μv beingequalto1.61.Soitturnsoutthatmoistunsaturatedaircanbetreated asdryairaslongaswereplacethetemperatureintheequationofstatebythevirtual temperature.
Thevirtualtemperatureincreaseswithspecifichumidity.Overhumidtropical areasthespecifichumiditycanbeabout30gkg 1 ,leadingtoavirtualtemperature thatisabout2%abovethetemperature(about6◦ C).Overcolderareasthespecific humidityisalwaysmuchless,soherethevirtualtemperatureandthetemperatureare alwayslessthan1◦ Capart.
Problems
1.1 Howmanymoleculesarethereinalitreofairatstandardpressureandtemperature?Whatisthevolume,inlitres,ofonemoleofidealgasatstandardpressure andtemperature?
1.2 Usethemicroscopicdefinitionoftemperature,Eq. (1.6),tofindatypicalrootmean-squaremolecularvelocityforair.
1.3 Considerthetrajectoryofamoleculeofmass m inasphericalvesselofradius r .Themoleculemoveswithspeed c andcollideselasticallywiththewallsof thevessel.Themoleculestrikesthevesselwallatanangle θ withtheradial direction,asin Fig. 1.8
(i)Showthatthemoleculeateverysubsequentcollisionstrikesthevesselwall atthesameangle θ withtheradialdirection.
(ii)Showthatthemomentumexchangedoutwardduringasinglecollisionis 2mc cos θ .
(iii)Showthatthemomentumexchangedoutwardperunittimeforthismolecule equals mc 2 /r
FIGURE1.8
Trajectoryofmoleculeinsphericalvessel.
(iv)Combinethisresultwiththemicroscopicdefinitionoftemperature, Eq. (1.6),toshowthatanidealgasinthesphericalvesselmustobeytheideal gaslaw,Eq. (1.12).
1.4 (i)Calculatethedensityofdryairatatemperatureof15◦ Candapressureof 1000hPa(orusethenomogramin Fig. 1.7).
(ii)Calculatethedensityifwenowassumethatthisairhasaspecifichumidity of3gkg 1 .
(iii)Calculatethedensityofairatavirtualtemperatureof15◦ Candapressure of1000hPa.
1.5 ThemassfractionsofthemajorconstituentsoftheatmosphereonMarsaregiven inthefollowingtable.Calculatetheeffectivemolarmassandthespecificgas constantfortheMartianatmosphere.
1.6 Usetherelationbetweenmassandmolenumber, Mi = μi ni ,tofindanexpressionrelatingmassfraction wi tomolarfraction yi .
1.7 Saturn’slargestmoon,Titan,hasanatmospheremainlymadeupofnitrogen (N2 ).ThetemperatureatTitan’ssurfaceisabout94Kandthepressureisabout 1460hPa.Whatisthedensityatthesurface?Themainvaryingconstituenton Titanismethane(whichhasamolarmassof16gmol 1 ).Whatisthevirtualtemperature,withrespecttomethane,atthesurfaceifthemolarfractionofmethane is y = 1.5%?SeealsoProblem 5.9.
