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.