Physics of planetary atmospheres Lecture 6: Atmospheric...
Transcript of Physics of planetary atmospheres Lecture 6: Atmospheric...
Lecture6:Atmospheric(andionospheric)chemistry
ErikVigren
HT2018
(MainlybasedonHeng,chapters6and7)
Physicsofplanetaryatmospheres
Introduction intogas-phasechemistryandchemicalmodelling
First:environmentscharacterizedbylowtemperatureandlowpressure
Caseexample1:simpleionosphericmodelofMars
Second:Effectofhigherpressureandtemperature
EquilibriumconstantsKeq,Keq’andK’
Caseexample2:ThehydrogenatmosphereconsideredinSection7.2ofbook
Caseexample3:EquilibriumC-H-Ochemistryincluding inH2 dominatedatmosphere(Section7.3-7.4ofbook)
Outline
Itishardtoformoneparticleoutoftwo(conservationofenergyandmomentum isrequired)
Acollisioncomplextendstofallbackintothereactants:H+Hà H2*à H+H
Multibody reactionsarepossibleunderhighenoughpressure.Maybevisualizedasatwo-stepprocess.
ForexampleH+H+Mà H2 +M (Misanymolecule)
1) H+Hà H2*, (*meansexcited)2) H2*+Mà H2 +M*.
Thethirdspeciescomesin,collidingwiththeunstableH2*.MomentumandenergyissharedwithM.
Three-bodyreactionsaretypicallyneglectedwhenmodellingchemistryinlowpressureenvironmentssuchasplanetaryionospheres, cometarycomae,interstellarclouds…
Introduction
TobreakH2 intoH+Hrequires~4.52eV(NISTchemistry webbook)
IfwedefinetheHeatofformationofH2 as0eV,thatis∆Hf (H2)=0eVthenwehave
∆Hf(H2)+4.52eV≈∆Hf (H)+∆Hf (H)à ∆Hf(H)=2.26eV=218kJ/mol.
Tocheckwhetherachemicalreactionisenergeticallyfeasibleatverylowtemperatureonecancomparetheheatofformationofthereactantswiththoseoftheproducts,e.g.,
A+Bà C+D
If∆Hf (C)+∆Hf (D)< ∆Hf (A)+∆Hf (B)theforwardreactionis(ormaybe)energeticallypossible,exoergic reaction
If∆Hf (C)+∆Hf (D)> ∆ Hf (A)+∆Hf (B)theforwardreactionisnotenergeticallypossible,endoergicreaction
Introduction
ChangeinnumberdensityofspeciesX isgivenby
dnX/dt =PX – LX
PX production rateofX
LX lossrateofX
AtchemicalequilibriumPX =LX
Introduction intogas-phasechemicalmodelling(weshallneglecttransport throughout thislecture)
𝑃" = $𝑍"(𝑝)𝑘*+𝑛--(*)*
𝐿" =$𝑌"(𝑝′)𝑘*1 + 𝑛--(*1)*1
Sumupthecontribution fromallprocessespthatleadstoproductionofX
Sumupthecontribution fromallprocessesp’thatleadstolossofX
ZX,p (YX,p’) isthenumberofXspeciesformed (lost)inthereactionp (p’) [dimensionless]kp (andkp’)denotesthereactionratecoefficientforprocessp (p’) [unitdependonnumberofreactants]r(p)denotethereactantsinreactionp,and nr isthenumberdensityofreactantr[m-3]
𝑃" =$𝑍"(𝑝)𝑘*+𝑛--(*)*
𝐿" =$𝑌"(𝑝′)𝑘*1 + 𝑛--(*1)*1
Ozone formation in Earth’s atmosphere exclusively via
p1: O + O2 + M à O3 + M
ZO3(p1) = 1, right? 𝑃23 = 𝑘*4𝑛2𝑛25𝑛6
Ozonelossmainlyviaphotodissociation andreactionwithO…(andpollutantsneglectedhere)
p’1: O3 + hvà O2 + O p’2’: O3 + O à O2 + O2
𝐿23 = 𝑘*´4𝑛23 + 𝑘*´5𝑛23𝑛2
Clarifying(?)example
Wheretofindratecoefficients?
Photodissociation andphotoionization [s-1]
e.g.,Huebner&Mukherjee(2015),PSS,(http://phidrates.space.swri.edu) Providecalculatedfrequenciesat1AUformanyatomsandmoleculesandforvariousradiation fields(includingactiveandquitesun).Foratmosphericmodelling Beer-Lambertlawneedtobeused(centralinputs:impingingstellarspectra,atmosphericdensityprofiles,cross-sections forphoto-processes)
Two-bodyreactions[cm3 s-1]
UMISTdatabase forastrochemistry(http://udfa.ajmarkwick.net)Kineticdatabase forastrochemistry(http://kida.obs.u-bordeaux1.fr)
Provideratecoefficientsparticularly forbinaryreactionswithreferencestooriginal work.
Three-bodyreactions[cm6 s-1],n-bodyreactions[cm2(n-1) s-1]
Notawareofanygooddatabase.Equilibriumchemistrycanstillbetackledasweshallseelater.
Caseexample1
SimpleionosphericmodelofMars
PredictelectronnumberdensityanddominantionsintheMartianionospherefrominformationofabundances ofneutralsnearanaltitudeof~200km
MarsatmosphereisCO2 dominated.Around~200-250kmatomic oxygenbecomes thedominant species.
Wewilluseasimple chemicalmodeltopredictthedaysideelectrondensitynear200-210km.Weshallalsousethemodeltopredictwhichionspeciesthatmaybemostabundant.
Simplifyingassumption:concentrationofCO2andOstable:n(CO2)≈n(O)≈5x107 cm-3
MAVEN/NGIMSdatafromMartiandayside
SimpleionosphericmodelofMars
SimpleMarsionosphericmodel(sorry forbusyslide)
Photoionization ofCO2 andOleadsprimarily to“CO2+ +electron”and“O+ +electron”,respectively.
P(CO2+)≈k1 xn(CO2) k1 ≈1.2x10-6/1.5242s-1 (see Huebner &Mukherjee,P&SS2015)
P(O+)≈k2 xn(O) k2 ≈4.5x10-7/1.5242s-1
HowisCO2+ lost?Checkonlinereactiondatabase(e.g., UMIST).
CO2+ isnotreactivewithCO2.ReactivewithOtoproduceO2
+.CO2+ isalsolostthroughelectronrecombination.
L(CO2+)=n(CO2
+)x[kin1 xn(O)+kDR1 xn(e-)] kin1=1.64x10-10 cm3s-1 kDR1=3.80x10-7 (Te/300)-0.5cm3s-1
HowisO+ lost?Checkonlinereactiondatabase (e.g.,UMIST).O+ isnotreactivewithO.ReactivewithCO2 toproduceO2
+.LossofO+ throughelectronrecombinationnegligible.
WenotethatP(O2+)= n(CO2
+)xkin1 xn(O)+ n(O+)xkin2 xn(CO2)
L(O+)=n(O+)xkin2 xn(CO2) kin2=9.40x10-10 cm3s-1
O2+ isnotreactivewithCO2 orO,soonlylosttroughelectronrecombination.
L(O2+)=n(O2
+)xkDR2 xn(e-) kDR2=1.95x10-7 (Te/300)-0.7cm3s-1
Lastbutnotleast:n(e-)=n(CO2+)+n(O+)+n(O2
+)
CO2 O
CO2+ O+
O2+
Electron
Electron
CO2O
k1 k2
k1ink2ink1DR
k2DR
clearallcloseall
nCO2=5e+7;%cm-3nO=5e+7; %cm-3Te=300;%Kd_AU=1.524;%AU
k1=1.2e-6*(1/d_AU)^2;%s-1k2=4.5e-7*(1/d_AU)^2;%s-1kin1=1.64e-10;%cm3s-1kin2=9.4e-10;%cm3s-1kDR1=3.8e-7*(Te/300)^-0.5;%cm3s-1kDR2=1.95e-7*(Te/300)^-0.7%cm3s-1
dt=0.1;%snCO2p(1)=0;nOp(1)=0;nO2p(1)=0;ne(1)=0;Time(1)=0;
for i=2:1:50000nCO2p(i)=nCO2p(i-1)+…+ nCO2*k1*dt- nCO2p(i-1)*(nO*kin1+ne(i-1)*kDR1)*dt;nOp(i)=nOp(i-1)+nO*k2*dt-nOp(i-1)*nCO2*kin2*dt;nO2p(i)=nO2p(i-1)+ nCO2p(i-1)*nO*kin1*dt+...+ nOp(i-1)*nCO2*kin2*dt-nO2p(i-1)*ne(i-1)*kDR2*dt;ne(i)=nCO2p(i)+nOp(i)+nO2p(i);Time(i)=(i-1)*dt;end
figure (1)semilogy(Time,nCO2p,'k-')hold onsemilogy(Time,nOp,'r-')semilogy(Time,nO2p,'b-')semilogy(Time,ne,'m-')grid onxlabel('Time (s)')ylabel('Number density (cm^-^3)')legend('CO2+','O+','O2+','e-')
SimpleMarsionospheric model…Matlabcode
CO2 O
CO2+ O+
O2+
Electron
Electron
CO2O
k1 k2
k1ink2ink1DR
k2DR
SimpleMarsionospheric model…results
Steadystatereachedratherquickly…tenminutes
ne ≈104cm-3
O2+ dominant~104 cm-3
CO2+ ~103 cm-3
O+ fewhundredcm-3
Predictionsnearz=200km
ne ≈104cm-3
O2+ dominant~104 cm-3
CO2+ ~103 cm-3
O+ fewhundredcm-3
MAVEN/NGIMSobservations
SofarwehavelookedatlowP and lowT environments.Athighpressuremulti-bodyreactionscanbeefficient.Athighenough temperatures…
Moleculesinternallyexcitedtohigherdegree.Moreimportant:Maxwellian speeddistribution. Collisionenergiescanbehigherthanenergybarriers.
WhileA+Bà C+Dmaybetheonlypossibilityreactiondirectionatlowtemperature…athigh temperatureA+B<-->C+D
Forwardreactionà ratecoefficientkfReversereactionß ratecoefficientkr
Equilibriumratecoefficient(possiblydimensional).BalancekfnAnB withkrnCnDKeq’= kf/kr=nCnD/nAnB (inthisexampleKeq´ isdimensionless but that does notapply ingeneral)
Ok,thatwasKeq’,whataboutKeq andK’discussedinthebook?
Keq’ ,K’ andKeq
Keq isdimensionless and relatedtotheGibbsfreeenergyvia(Eq.6.14inbook)
(R=8.314JK-1 mol-1 isuniversalgasconstant,T istemperatureinK)
Δ𝐺;< valuesatP0=1baranddifferentT aretabulatedate.g.,https://janaf.nist.gov
Keq’maybedimensional andisdefinedastheratiokf/kr.ItisrelatedtoKeq via
Keq’=Keq(kBT/P0)x =Keqn0-x (seeEq.6.17inbook)
wherex is“thenumberofreactantsminusthenumberofproducts”involvedintheforwardreaction.
K’isjustanormalizedequilibriumconstant(uptouswhattonormalizeagainst). Forinstance,inalatercaseexamplewerelateittoKeq’ viaK’=Keq’/nH22.
𝐾>? = exp −Δ𝐺->DEFGHI<
𝑅𝑇
Δ𝐺->DEFGHI< =$Δ𝐺;<(𝑝)*
−$Δ𝐺;<(𝑟)-
TheGibbsfreeenergy(Δ𝐺0 = Δ𝐻0 − 𝑇Δ𝑆0)isthemaximumamountofnon-expansionworkthatcanbeextractedfromathermodynamicallyclosedsystem(wikipedia).Foranintroduction intotheGibbs freeenergyassociatedwithchemicalreactions,seeforinstancehttp://chem1.com/acad/webtext/thermeq/TE4.html,
Keq =exp(-∆Greac,0/RT) Keq’=Keq(kBT/P0)x =Keqn0-x K’=Keq’/nH22
Considerthenetreaction
CH4 +H2O<->CO+3H2
Notethatx=-2(differencebetweenthenumberofreactantsandproductsintheforwarddirection)
WegetK’= Keq’/nH22 =Keqn02/nH22 =Keq(P0/P)2=(P0/P)2 exp(-∆Greac,0/RT)
(inagreementwith7.31inbook)
Clarifying(?)example
Δ𝐺->DEFGHI< =$Δ𝐺;<(𝑝)*
−$Δ𝐺;<(𝑟)-
FindΔ𝐺;< varywith𝑇 fortheinvolvedproducts(CO,H,H,H)andreactants(CH4,H2O)(https://janaf.nist.gov)
Caseexample2
ThehydrogenatmosphereconsideredinSection7.2ofthebook
Foragivenpressure,how will therelativeabundances of atomic (H)andmolecular (H2)hydrogenvary with thetemperature?
ThehydrogenatmosphereconsideredinSection7.2ofthebook
2H+M<->H2 +M
dnH/dt =-2kfnH2nM +2krnH2nM@equilibrium Keq´=kf/kr =nH2/nH2
dnH2/dt =-krnH2nM +kfnH2nM
Bookkeeping:ntotal (totalnumberdensityofhydrogenatoms,aconstant)
ntotal =nH +2nH2à nH +2Keq´nH2 – ntotal =0(quadratic equation)
𝑛Y =−1+ 1 + 8𝐾>?′𝑛FHFD\
4𝐾>?′
Reactiondirectionatlowtemperature
à
𝑛Y =−1+ 1 + 8𝐾>?′𝑛FHFD\
4𝐾>?′
Checklimits… 2H+M<->H2 +M @equilibrium Keq´=kf/kr =nH2/nH2
lim`abc →e
−1 + 1 + 8𝐾>?′𝑛FHFD\
4𝐾>?′= 0
limabc →<
−1 + 1 + 8𝐾>?´𝑛FHFD\4𝐾>?′
= limabc →<
12 1 + 8𝐾>?´𝑛FHFD\
gh i⁄8𝑛FHFD\
4 = 𝑛FHFD\
àReactiondirectionatlowtemperature
𝑛FHFD\ = 𝑛Y + 2𝑛Yi → 1 = 𝑛Y𝑛FHFD\
+ 2𝑛Yi𝑛FHFD\
= 𝑛Yk + 2𝑛Yik
Recall
𝑛Y𝑛FHFD\
= 𝑛Yk =−1 + 1 + 8𝐾>?´𝑛FHFD\
4𝐾>?1 𝑛FHFD\=−1 + 1 + 8𝐾′
4𝐾′
𝑛Y =−1 + 1 + 8𝐾>?′𝑛FHFD\
4𝐾>?′
𝑛Yik =1 − 𝑛Yk2
Generating toppanelofFig.7.1isstraightforward.Generating thebottompanelisabitmoretricky.
Keq’ isdefinedastheratiokf/kr.ItisrelatedtoKeq via
Keq’=Keq(kBT/P0)x =Keqn0-x (seeEq.6.17inbook)
wherex is“thenumberofreactantsminusthenumberofproducts”involvedintheforwardreaction.
2H+M<->H2 +Mgivesx =1 andsoK’=Keq’ntotal =Keqn0-1ntotal ≈(P/P0) exp(-∆Greac,0/RT)Weasume;n0 ≈ntotal..Inworstcase n0 = ntotal./2
Results(appearsasifthe“n0≈ntotal assumption” isusedinthebook,butpossibleerrorisquitesmall)
Foragivenpressure,how will therelativeabundances of atomic (H)andmolecular (H2)hydrogenvary with thetemperature?
Whenthetemperatureisbelow~3000K(expected tobethecase inmost exoplanetatmospheres)itisfairtoassume that thehydrogenisnearly exclusively inmolecular form.
Caseexample3
C-H-OchemistryinH2 dominatedatmosphere(consideredinSections7.3&7.4ofthebook)
Givenaseriesof net reactions,elemental abundances of CandO,how will themolecular composition vary with thetemperature?
C-H-OchemistryinH2 dominatedatmosphere (consideredinSection7.3ofthebook)
CH4 + H2O ßà CO + 3H2 ß (1)
2CH4 ßà C2H2 + 3H2 ß (2)
CO2 + H2 ßà CO + H2O ß (3)
Reactiondirectionatlowtemperature
𝑘hl𝑛mn𝑛Yio + 2𝑘il𝑛miYi𝑛Yio = 𝑘hp𝑛mYq𝑛Yin + 2𝑘ip𝑛mYqi CH4
𝑘hl𝑛mn𝑛Yio + 𝑘op𝑛mni𝑛Yi = 𝑘hp𝑛mYq𝑛Yin + 𝑘ol𝑛mn𝑛Yin H2O
𝑘hp𝑛mYq𝑛Yin + 𝑘op𝑛mni𝑛Yi = 𝑘hl𝑛mn𝑛Yio + 𝑘ol𝑛mn𝑛Yin CO
𝑘ip𝑛mYqi = 𝑘il𝑛miYi𝑛Yio C2H2
𝑘ol𝑛mn𝑛Yin = 𝑘op𝑛mni𝑛Yi CO2
Given:Relevantnetreactions(1,2,3)Pressure,Temperaturen(H2)=p/kBTelementalabundance ofOC/OratioGibbsfreeenergiesofformation
𝐾>?,h′ =𝑁mn𝑛Yii
𝑁mYq𝑁Yin
𝐾>?,i1 =𝑁miYi𝑛Yii
𝑁mYqi
𝐾>?,o1 =𝑁mn𝑁Yin𝑁mni
HerewedenotenX/nH2 byNx
𝐾>?,h′ =𝑁mn𝑛Yii
𝑁mYq𝑁Yin
𝐾>?,i1 =𝑁miYi𝑛Yii
𝑁mYqi
𝐾>?,o1 =𝑁mn𝑁Yin𝑁mni
𝐾>?,h′𝑛Yii
=𝑁mn
𝑁mYq𝑁Yin= 𝐾h1
𝐾>?,i′𝑛Yii
=𝑁miYi𝑁mYqi = 𝐾i1
𝐾>?,o1 =𝑁mn𝑁Yin𝑁mni
= 𝐾o1
𝑧𝑦𝑥 = 𝑎
𝑡𝑦i = 𝑏
𝑧𝑥𝑠 = 𝑐
𝑁mYq + 𝑁mn +𝑁mni + 2𝑁miYi =𝑛m∗
𝑛Yi≈ 2𝐹m
𝑁Yin + 𝑁mn + 2𝑁mni =𝑛n∗
𝑛Yi≈ 2𝐹n
𝑦 + 𝑧 + 𝑠 + 2𝑡 = 2𝑑
𝑥 + 𝑧 + 2𝑠 = 2𝑒
Recall:wedenotenX/nH2 byNx
nc* isnumberdensityofcarbonatoms,“howmanycarbonatomsarethereintotalinagivenvolume”
Fc andFO aretheelementalabundances ofcarbon(solarvalues~3x10-4,~6x10-4,respectively)
𝑧𝑦𝑥 = 𝑎
𝑡𝑦i = 𝑏
𝑧𝑥𝑠 = 𝑐
𝑦 + 𝑧 + 𝑠 + 2𝑡 = 2𝑑
𝑥 + 𝑧 + 2𝑠 = 2𝑒
∑ 𝐴G𝑦G = 0�G�<
with
𝐴� = 16𝑏i
𝐴q = (16𝑏 − 2𝑎𝑏𝑐)
𝐴o = 16𝑏 𝑒 − 2𝑑 − 𝑎𝑐 + 4
𝐴i = 2𝑏𝑐𝑎 + 2𝑎𝑐𝑑 − 16𝑑 − 2𝑎𝑐𝑒 + 8𝑒
𝐴h = 16𝑑i − 16𝑑𝑒 +𝑐𝑎 + 4𝑒i + 2𝑐𝑒
𝐴< = −2𝑐𝑑𝑎
Notidentical(andnotequivalent)withequation inbook.(Idonotruleoutmistakesfrommyside)
C/O=0.1
InsightsfrommodellingofH2 dominatedatmospheres(seedistributedprint-outsofFig.7.2andFig.7.3ofHeng)
• Inacarbon-poor atmosphere atmosphere,water isexpected tobethemain carrier of oxygen.
• When C/Otakes onthesolarvalue orhigher CObecomes themain carrier of oxygenathigh temperatures.
• CH4 isthemain carrier of carbon atlow temperature,with increasing temperature COgradually takes over.
• When theelemental abundance of carbon islow,carbon dioxide islessabundant than carbon monoxide.
• Acetelyne (C2H2)istypically sub-dominant incarbon-poor atmospheres.Maybecome dominantcarrier ofcarbon if theatmosphere iscarbon rich andhot.
• When thetemperature is”low”(~800K)thetrendsexhibited bythemixing ratios versus theC/Oratio arerelativily simple:constant mixing ratio of water while thecarbon carrying molecules showsimplescalings.
• Athotter temperatures thetrendsare more complex.When carbon-poor theatmospheres are methane-poorandwater rich.When carbon rich,they are methane rich andwater poor. Transition around C/O≈1.
Someconcludingremarks
• Theatmosphericchemistrydependsinpartonthetemperature…thetemperaturedependsinpartonthechemistry
• Wehavecompletelyover-lookedcondensation (resultinginhazeorcloudformation)
• Wehavenotconsideredhowtransportmayinfluencethepicture(wehavenotcomparedtimescalesforchemicalreactionswithtransporttimescales)
• Whilewecanpredictrelativeabundances ofdifferentspeciesatchemicalequilibrium, wehavenotassessedhowlong timeittakestoreachsuchastate(isitamatterofminutes?days?years?theageof theUniverse?)
• Itdoesnotrequireachemisttodochemicalmodelling.
Homework:
Problem7.6.4(onnitrogenchemistry)inHeng’s book.