Ifthere is only one observed energy level. instead ofEq. (49) the fbllowing fbrmula should be used: E,: Ei - Rtr lh * C)2. (A3) where C is an adjustable parameter.
Thermodynamic Properties High-Temperature Gomponents ofMars-Atmosphere D.Giordar0, L. Marraffa, M.Capitelli, G.Col0nna, D.Pagano, L.Pietanza, P,Minelli, A.Casavola, na F.Taccog
Reprinted from
Journalof Spacecraftand Rockets Volume 42.Number 6. Pages 980-9Bg
gl.4!+4-A publication of the andAstronautics. Inc. American lnstitute ol Aeronautics 1801Alexander BellDrive, Suite 500 -4344 Reston, VA20191
eNoRocrrrs JounNaror Sp,qcecn.qn' 2005 Vol.42.No. 6. November-December
High-TemperatureThermodynamicProperties of Mars-AtmosphereComponents M. Capitelli-and G. Colonnai Isrirurodi MetodologieInoerganichee Plasnti (CNR), 70126Bari,ltal-,D. Giordano+and L. Marraffas EuropeanSpaceReseat'chand Technologt'Center,2200 AG Noordu'ijk, The Netherlands and A. Casavola,$ P.Minelli,sD. Pagano,T L. Pietanza,s and F. Taccognas ttiversird 26 Bari U de,eliStudidi Bari, 70I , Italv Methods of calculation of high-temperature thermodynamic properties for some selected Mars-atmosphere components in the temperature range from 2fi) to 50.fiX) K and results are discussed and compared with previous works. Aspects such as quasi-bound rotational states, cutoff criteria, and autoionizing states are considered,
R R-r' Rr-
Nomenclature aij
L int.f
C
D,
F, G G o ( u r .. . .9, Hi h 1 1 ,1 6 , 1 1 J kg 111 i tl Pi
p, Qr.u.
Q,,
= = =
Qri
=
Q, Qin.,
stoichiometriccoellicient lbr the formation reactionof the i th species intemal specificheat total specificheat speedof light dissociationenergy ionizationenergy internalenergy ener_ey of the nth quantumlevel rotationalenergy(polyatomicmolecules) Gibbs fiee energy vibrationalenergyof the state(ur. r':. . . . . f,,) wcightof the /rth quantumlevel statistical m o l r re n t h a l p o v l ' t h er r h s p e c i c \ PIanckconstant momentsof inertia rotationalquantumnumber Boltzmannconstant massof the ith species principalquantumnumber statistical of thc r th electronicstatcs "veight (polyatomic molccules ) wlc i - s hot 1 ' t h es t a t e( r ' r .r ' : . . . . . L ' , , , ) statistica partition function fbr the harmonicoscillator (polyatomicmolcculcs ) totalpartitiontunctionof thc ith spccies internalpartitiontunctionof the ith species partition tunction fbr the rigid rotator (polyatomicmolccules) partitionlunctionof the i th species translational
s T
T,i' l./ U
AEi LH,., lL=
@il
= =
universalgasconstant Rydbergconstant modilied Rydbergconstant equilibriumdistance enrropy temperature excitationenergyof the lth electronicstates (polyatomicmolecules) volume vibrationalquantumnumber lowering of the ionizationpotential variationof the enthalphyfbr the formation of the i th species reducedmass symmetryt-actor frequencyofthe nth vibrationalmode (polyatomicmolecules) I.
Introduction
f,r UTURE missionsto Mars fbresechypersonicentriesinto and .l-' ae'rohrrkingmaneu\ers through the atmosphereof that planet. Heatshielddesignwill requirethe high-speedaerodynamics analysis of spacecraliconli-uurations. In this regard.the availabilityof accuratethermodynamicpropertiesof thechemicalconstituentsthat may cxist in the flow fieldduring hypersonicflight is mandatory.The litcratureol1ersbasicallya few main sources:the JANAF tables.r ' tabulationedited the tableseriesliorn NASA.: and the impressive by Gurvich et al.+The JANAF tablesarelimited to 6000 K; Gurvich et al.'stablesand the recenttablesfrom NASA reach20.000K. Conall tabulations can be considered in cerningmolecularcomponents, satistactoryagrcementup to 6000 K. However,atomic components and their ions are not dealt with the same accuracybecausethe numberof energylevelsincludedin the calculationof electronic partition functions is not sufficient.Even the old tables produced by Browncs qualif,vfbr better accuracywith regardto this aspect. althoughthis authorincludedonly the observedenergylevelsas publishedby Moore.6 Moreover.the problemsconnectedwith the divergenceof internalpartition functionsand relatedto thc completion of the energy-lcvel sct wcre not dealtwith adequatcly. In this paper.we presenta newcalculationofthe thermodvnamic properties of IVtars-atmosphere componcntsfiom 200 to 50.000 K. carriedout with the purposeof dealingwith the aforementioned problem in an accurateas possiblemanner.In particular.the problem of the energy-lcvclcompletionwill be confiontedby avoiding the hydrogenicapproximationrecentlyusedby Wang rnd Rhodes' fbr the calculationof air-component thermodynamicproperties.ln addition. the thermodynamicpropertiesproducedin this work are
Presentedas Paper2004-2378at the AIAA 37th ThermophysicsConl'erence.Ponland. OR. 28 June-l .1uI1200'1:received28 July 200.1:revision received2 I October2004:acceptedtbr publication2 I October200:1.Copyright O 200;t b1' the American Instituteof Aeronauticsand Astronautics. Inc. All rights resened. Copiesol this paper may be made lbr personalor internal use. on condition that the copier pay the S10.00per-copy f'eeto the Copyright ClearanceCenter.Inc.. 222 RoservoodDrive. Danvers,MA 0 1 9 2 3 :i n c l u d et h e c o d e0 0 l l - 1 6 5 0 / 0 5 S 1 0 . 0 0i n c o r r e s p o n d e n cr vei t h t h e CCC. *Prof'essor.Chemistry Department.Sezione Temtoriale di Bari. Via O r a b o n a . l :c s c o m c O 5 @ a r e a . b a . c nFre. il tl .o u 'A I A A . iResearcher. Sezionc Tembriale cti Bari. Via Orabona .l: cscpgcO8@ N{emberAIAA. area.ba.cnr.it. iResearch Hngineer. Aerothermodynamics Section: I)ornenico. G i o r d a n o @ e s a . i nMt .e m b e rA I A A . tResearchEngineer.Aerothermodynamics Section. 'll Fellowship Researcher. ChemistryDepafiment. Via Orabona.1.
980
981
CAPITELLI ET AL.
meant to constitutethe platfbrm to support the developmentof a gas-mixturethermodynamicmodelof Mars atmohigh-temperature spheresuitablefbr implementationin hypersonicflowfield solvers.
il.
The Helmholtz potential.the intemal contributionE1n,to the thermodynamicenergy,the entropy.and the intemal constant-pressure specilicheat Cin,r, are obtained,respectively,from F - H(0) :
Method of Calculation
A. Partition Function gas mixtures The thermodynamicpropertiesof high-temperature can be obtainedif the partition tunctionsof singlecomponentsare known. In general,the partition tunction can be t-actorizedas the productof the translationalQo.1and the internal Qin,.;contributions Qi :
Qo.'Qin.i
Eint: Rr:('!*!) 1 ,r/
(l)
(3)
B. InternalPartitionFunctionof AtomicComponents For an isolatedatom. an infinite number of bound statesexists below the ionization limit. Then, the internal partition llnction is the sum over the infinite electroniclevels \/r\
(4)
However,the sumin Eq. (4) divergesbecausethe statisticalweight is not bound when the principal quantumnumber increases(rr + cc) indefinitely'.For example,g, = lnr for atomic hydrogen.Thus. the summationin Eq. (4) must be adequatelytruncated.Unlbrtunately. thereis no universalcutofTcriterion.Reviews'nof variouscutofT methodscan be found in the literature.The existingcutolTcriteria yicld partition functions and their derivativesthat depend on either the electrondensity or the sas pressure.This meansthat the thermodynamicpropertiesof the componentsdependnot only on the temperaturebut also on the pressure.In particularwe ret-erto Griem'scriterion.eaccordingto which the ionizationenergyof an atom in the presenceof other componentsis lowered by a factor that.in general.dcpendson the numberofchargedparticlcs.All the levelsthe encrgiesof which are lower than the correctedionization potentialare accountedfor in the sum (4); that is. we sum up to thc last energylevel that arisesliom the limitation E,, < E, - LE,
(8)
s: Rr,, o. +('+\ u \"t
(e)
(5)
ln our calculationsAE; is set as a parameter.and resultsare reprobported for various AEl values.In addition to the diver,qence lem, there is also the issueof includingonly experimentallyobservedenergylevclsor alsoaddingthc theoreticallyprcdictedlevels. and statistical Tabulations('i(' are availablethat provide cner,eies supobservedIcvcls.Nonetheless. weightstbr manyexperimentally lawsis a usefulway plementingthatinformationwith semiempirical calto obtain the missingenergyle'"'els.Exact quantum-rnechanical culationsfor high-lyinglevels.that is. near the continuumlimit, althoughpossible.arestill a prohibitivetask.
Cn:Cint''+fR
(11)
T h e e n t h a l p li s g i v e nb 1 ..;Rf
H - Ht\t:8",
(1 2 )
The Gibbs tiee energy is calculatedfrom the entropyandthe enthalpyas
G _ H ( 0 ): _ R T | ^ Q + R T
(l -l)
The Helmholtz potential.the internal contributionto the thermospecilic heat dynamic energy,and the intemal constant-pressure strongly dependon the assumedsetsof energylevelsand adopted cut-off criterion. Finally, we calculatethe formation enthalpy of the component underconsidcrationfrom L H 1 . 1: H i _
lai,H,
( t4)
where Hi. Hi are given in Eq. (12) and the sum runs on all the fbrming reactants.For example.fbr N lbrmation
1N:-N
(1 s )
lheformationenthalpyis givcnby A H r r ( I ) : D 1 2 + l H I t - H ( 0 ) ] \- \ [ U l t - H ( 0 ) ] \ ' . ( 1 6 ) D. InternalPartitionFunctionofllolecular Components The startingpoint tbr calculatingthe internal partition function equationof a is to solvcthe Schrodinger of molecularcomponents representativcmoleculeto obtain the energy levcls corresponding to the independentmoleculardegreeof tieedom. The Schrodinger which approximation. equationis solvedin the Bom-Oppenheimer separatesthc motion of the electronstiom that of the nuclei. The solutionof the Schrodingerequationfor the elcctronsyields the approximatewavetunctionso1-theinfiniteelectronicstatesas well The energycorrespondingto thenth electronic astheir degeneracies. stateis expressedas a function of the internucleardistancel and constitutesthe potentialenergyI',,(r')seenby the internalrnotion of the nuclei. governedby the nuclearSchrodingercquation. L
C. ThermodynamicProperties From thepartitionfunctionsandtheir first and seconddcrivatives. we can calculateall the thermodynamicpropertiesof the components accordingto the tormulasof statisticalthcrmodynamics.rr
I (ro)
/tJ
The total specific heat is obtained by adding the translational contribution
Equation (3) assumesdifierent explicit fbrms lbr atoms and for molecules.
0 . , : I c , , . * p (- ^ * L )
U - H ( 0 ): g , "+, l R r
L
/r\
e t n . i : ) - s , . * p ( - +' - B / ) / \ 7
,t, /,
*r'(Ip) cn,,:nlzr(L+) ;if: ;tT l, \ \
(2)
The internal partitionfunction can be written in generalterms as
(6)
/
The translationalpartition function is availablein analyticalform
q o , : ( 2 r m i k r rf n | ) i t '
-RT |*Q
DiatomicMolecules The treatmentof the diatomic moleculeslbllows the methoddcct al.rr ln this velopedby Drellishaket al.rr'rrand by Stupochenko methodthe cnergyof a particularstatein the moleculeE,,7, is split theelectronicexcitationenergytel(/1).lhe into threecontributions:
982
CAPITELLI ET AI-,
3 . 51 0 5
4.0lo'" E
:
3.0105
E
t
?5r05
a
J
E z.oto' E 3
t . st o '
7 o
1 . 0l 0 '
= E E
0.0 | 0" . 2 . 01 o - " - 4 . 0l o ' " . 6 . 0l o ' " - 9 . 0l 0 ' "
.a
50 r0'
- 1 . 0l 0 ' ' " - 1 . 21 0 - ' "
0.0 t0'
0.8
L2
1.6
2
1.6 2.0 2.4 | .2 Internuclear distance.cm x 108
ft
Intemucleardistance,cm x I0'
a) I'ig. I
2 . 0l 0 ' "
b)
2.8
N2 ground state for different centrifugal distorsions (numbers indicate the "/s values): a) potential curves and b) their derivative.
(rr. r,) andthe rotationalenergyE,n,(n . u. "/ ). vibrationalenergyE,.15 Thus E , , 1 .:
E . . r ( n*) E . , 6 { / t . lt *
E - o , t r tt .. - / t
(17)
The vibrational energy associatcdwith the uth vibrational level of the irth electronicstateof a diatomic moleculeis exoressedin analytical form asrr
'-;:.,"(". *1)' l)-,","(, +,"," (,. l) . ,"-" (,* i)'
(1 8 )
wherecr;".a(,,tt,,ael e, andar: , arespectroscopic constantstbr each electronicstate.Expression(18) can bc rewrittcnwith the energv ret'erenced to the first vibrationallevel. which reads E,;6(ir.0) : (1 l2)o" I (l ll)a,,,r,.1 ( l/ti)rr.r,,'r',, + (l ll6)ot,,:,, (19) (18) is valid tor all vibrationalstatesup Assumingthat expression we can determinethe maximumpermissible to dissociation. "'alue u.,,, of the vibrationalquantumnumberfor eachrotationless( J : 0) molecularstateby comparingthe vibrationalenergvwith the dissociationenergyofthe nth electronicstateret-erenced to the energl' of the hrst vibrationallevel D,,{n). The rotationalenergy fbr a nonrigid rotator associatedwith thc t,th vibrationallevel of the rrth electronicstatereads
f:l+:!
: B , ' t ( ' t +l ) - D , ' / r ( /+ 1 ) l
(20)
where
B,:8,,-a,,(u+i)
/')l\
D,:D,-1,,(u+*)
r/l') \
The maximun-rpermissiblevaluc ./.", of the rotationalquantum numberl-oreachvibrationalquantumnumberis deterniinedcomparingthevibrational-rotational energvwith thedissocirtirrn energ-r relativeto the electroniclevelwe areconsidering. This represents a simplilicationin which the moleculeis considered as a nonrotating systcm. For a diatomicmoleculethe potentialcurvc is given hr l'
L I a : D , U - e x p [ - l ( r .- , ' , , ) ] ] '
( 2 3)
whercfl is a constant(in cm r) whosei'aluewill be derivedfiom
f : lzr',yf D,,ho,'
(21)
When we consider.on the basis of classical mechanics.a rotating molecule,we must introducean additionalterm in Eq. (23): a centrifugalpotential.'' Thus. when the angularmomentumis "/, the e f l e c t i r ep o t e n t i ael n e r g y( i n c m 1 )b e c o m e s Lllrrt:
L l t * ( h l 8 t : t 1 t r : t .rl . l - l t
/1 5)
A seriesof potentialcurvesfor consecutive./s can be constructed. These potential curves show a maximum at a larger distancerespect to the equilibrium distancelbr 0 < "/ < J*u^. For ./ :0 no centritugaldistortionoccursand there is some maximum value of "/ beyondwhich the potentialcurve no longer displaysa minimum. This meansthat tbr ./ :0 all vibrational stateswith energy lower than the dissociationlimit are present.whereasthcre are no stable stateswith -/ largerthan -/.,,. SeeFig. l. The centrifu-ealdistortionof the potentialenergydeterminesthe existenceof "quasi-bound"statesabovethe dissociationlimit of the molecule. To calculatethe number oi rotational statesabove the dissociationlimit 1br each vibrational state. we dillerentiate Eq. (2-5)respectto r. settingthis derivativeequal to zero and solve the resultingequationlbr the value of r at the hump (calledr,,,) as a function of -/ for any electronicstateof the molecule: dU
^
tJl'
: 2 D , f l e x p l - f J 0-' r , , ) l- e x p [ - 2 p ( r- r ")J] - 2. h. / ( J+ l ) : 0 r) 6n-cU
(26)
Thenthevalueof thepotcntialat r,,,is calculated f'oreach./: t t r l r ' , ) - D . { 1 - e x p [ -l ] ( r , , - r , ) l l 2+ (nf an'1cp'))t(./+ l)
(21) This potentialis comparedwith the energyas calculatedfiom the coupled vibrational-rotationalenergyexpressionfbr any assumed u: ./ is varieduntil thesetwo cncrg)'valuesare equal.When this point is reached.onehasa compatibler-:../ combination. Once the maximumnumberof vibrationallevelsfor eachelectronic stateand the maximum numberof rotationalstatesfor each vibrational statehavebeendetermined.the internal partition tunction can be calculated by the followingexpression: r
o,,,,:lf
"? ,.'I,r, 7
-1,.\1
-
0 , , . '*lo-l r ' 1 r / ? r I f
t,r )?
+ tr"*p[-E L
r t l. I- l . * 'oLl - L ' r '( ^ nn r
I
r . rrr'' / r - l rB1
(28)
I
where the symmetry factor o cquals one or two for. respectively, heteronuclear and homonucle ar diatomicmolecules. ln Sec.llI, we will presentresultsfbr selected diatomicmolecules andcomparethemwith datafrom othercompilations.Hcre,we want
983
CAPITELLI ET AL
5.5 CO
(_)
,ir
50
l
\.\
4.-5 :-
.,
10 -t.5 l0
10000
20000
30000
,10000
2.5
50000
20000
10000
i0000
,10000
50000
l.K 60
)J
CN 5.5
5.0
'
5.0 -.
\
\'.
E;
\
--
\..
+)
'1.0
.J
(b) (ct (d)
\. \\
+(,
3.5 .t.5 30
1.0
l)
0
10000
20000
30000
10000
l)
50000
0
f.K Fig. 2 Nondirnensional specific heat as a function of temperature for quasi-bound states. CO and NO without (---) and with (-)
to discussthe influenceof quasi-boundstatesand of electronically excited stateson the thermodynamicpropertiesof molecularcomponents. Figure 2 illustratesthe clfect ofquasi-boundstateson nondimensionalspecificheatofCO and NO. Considerationofthe quasi-bound statescan introducedillerencesof up to l07c of the maximumspecihc heat. Note also that the nondimensionalspeciftcheat at constantpressurestartsfiom the value 3.5 (activationof translational and rotational degreesof tieedom) and rapidly reachesthe value ,1.5.which includes the vibrational contribution lrom the ground electronic state.The maximum is due to the contribution of the electronicallyexcitedstates. excited In any case.dependingon the numberof electronically statesconsideredin the partition function and on the energy values. the nondimensionalspecific heat slowly convergesalier the contribution(3/2R* R). This behavmaximumto the translational ior is typical for systemscontaininga finite number of rotational. vibrational.and electronicstates. The last observationis betterunderstoodby inspectionof Fig. 3. excitedstatesin fbrmingand which showsthe role of electronically enhancingthe specificheatof O1 and CN. Curves a representthe specific heat of the ground stateof the molecules.which rapidlypassesliom 3.-5to 4.5 lbr the reasonsexplainedearlier.Then, thereis the prescnceof a small maximum, afier which the specilic heat smoothlv declinestoward the translainclusionof electronicallyextional contribution.The successive citedstatesb. c. and d producesa wcll-definedmaximum.the absolute valueof which dependson the specificmoleculc.Caseb refers to the inclusionof the groundstateandthe frrstexcitedstate.Casec refersto the inclusionof the secondexcitedstatealso.Finally,in cased. all ofthe excitedlevelsas siven bv Herzberqrrareincluded in thc calculation. 2. PoLratomic Molecules The intemal partition tunction of polyatomicmoleculesby analogy with diatomic moleculescan bc written in the generalcasein
I 0000
20000 10000 T.K
40000
s0000
Fig. 3 Contribution of electronically excited states to the nondimensional specific heat as a function of temperature for 02 and CN.
tne l o t l o w l n g w a y - ' : Q,n,
: rf r'^n(-Sr;')
TI . I
S-
I
J
|
llL'
^trj
-;--;u,j ) p, cxpl -l^at
k:
.
{ r ' 1 r. ' ;
,r, ,]
r l . /+ l , e * p [ - r r r .( r l
(29)
.l
In this expression. the first sum is takenover the electronicstates (in ,eeneralthe valenceones)of the molecule.characterizedby the ener-uyof excitation {]" and the statisticalweight p1. The electronic statesof nonlinearpolyatomic moleculesare nondegenerate and therefbrethe statisticalweight of these statesis determined only by multiplicity pi :25 -| I . ln the caseof linearpolyatomic molecules,the statisticalweightsarethe sameas for the corresponding statesof diatomic moleculcs.Sum over r, is carried out lbr all the rr normalvibrationsof the molecule.[f a molecule.consisting of N atoms.hasr degeneratenormal vibrationswith degeneracyr/,,, thennr :3N - 5 - r tbr linearmoleculesand nr:3N-o-)-(rt,,-tt 7t tbr nonlinearmolecules.The statisticalweight p, of the state ( r r . u : . . . . . u , , ,i )s e q u atlo
r,:fIffi#
(30)
The energyof the vibrational levcls has beencalculatedin the approximationof harmonicoscillator: G g ( r ' r t. ' : . . . . . u . ) =
1) I.",(,,* t t = l
\
-
/
(3t;
CAPITELLI ET AL,
An additional simplilication that can be included consistsin tactorizing the vibrational panition function. Then it can be written as rltcl
I- -I- l t a t l I r , . * p l - f = o ; j ' r r 1r '.1. , , ,I) II
1l
:
where ,t
n =
L
Lrl
/
I
\
r.^
r..r J
(40)
/J
e , ,: ' o ht:Bt !'T^ (3 2 )
,[ {iz"t'r'-o[-frol "'] (f"+d" -'l):
(33)
* l)l L',,'.(d,,
(4r)
for linear molecules.and
weightof uthlevelin therrthvibrational wherep,,(u) is thestatistical mode P,\u):
f
Q n ' , :^ I1J1 -l ' - . * p (- ir;' a, ,r ) l
(42)
Q,,:tloJ@lA,&,cr)(hryn'
lbr nonlinearmolecules. According to expression(39), the thermodynamicfunctionscan be representedas the sum of the correspondingcomponents:
andGfl)(u) is theenergyof the uth levelin therrthvibrationalmode t4'lrt',t
-,,, w i l \ t. ti tl
(34)
t , l s n / -l )' t |
Summation over -I in Eq. (29) has been carried out for the value u :0. The energy of the rotationallevelsof the polyatomic moleculesis expresseddifferently,dependingon the symmetryof the molecule. For linearpolyatomicmoleculesand moleculesof the type of a sphericaltop, such as diatomic molecules,the energy of the rotational levelsis describedby the expression F'(J ) : B''l ('l + l)
(35;
For moleculesof the type of a symmetricaltop (36)
whereA assumes valuestiom -J to +J . For asymmetrictop molecules 1c.t.l (J + l) + (A, * B,)tt
(37)
(35-37),the quantitiesA,. B,, and C, are the rotaIn expressions tional constantsrelatedto the principal momentsof inertia of the moleculc.In particular Aa: tr/8r)cl ^.
B r : h/ 8 r t c l ,
C,t :
h
/8tt2t'lc
(38) where 1r, 16, and 16 correspondto somc avcragestructurcof the moleculein the groundvibrationalstate. It is important to note that difl'erentelectronicexcited statesof thus the samepolyatomicmoleculecan havedillerentgeometries. ailbcting the symmetry numbcr and also detcrmining a dittcrent rotationalbchavior.Then to includetheseexcitedstatesin our calto takeinto accountthisgeometrvrariation. culationsit is necessary With increasingtemperature. the use ol'the direct summation of theabsence of datafbr methodbecomesimpossiblebothbecause the high vibrational-rotationalenergylevels includin,sthe ,sround electronicstateof almostall polyatomicmolecules.as well as thc knorvledse ol'thedependence ol'the absence of sulhcicntlyaccurate energyof theselcvelson the quantumnumbersr',,and ./. Then caltunctionsof polvatomicmolecules culationsof the thermodynamic oscillator"approximation. Destill usc the "rigid rotator-harmonic of excitedelectronicstates. I'iationstrom this model.the presence and othereftectsaretakeninto accountin the firrm of corrcctions. This approximationassumesthe vibrationsof the moleculein its electronicgroundstateas harmonic.and then the upperlimits fbr r,,,and J in the partition function and its derivativesapproach infinity.Then.the intcrnalpartitionfunctionlbr the moleculein the electronicgroundstateis givenb5r Q i n t: P x 0 l l ' n
"
"
e^ x. .P_(,-.r.t\'t1 r[ 11 - e x P ( - r ' ) l '
(43) n l r o r r r 'll n " L- r'-- L\ - r i , , r ; e x p ( - 4 , ) [ 1 - e x p ( - r l , ) ] '
n
(u)
wherer,, : thc1k sT 1'u,,. and
l H o ( r ) - H ' r ( o )-1 ,l,c i , l ' ] , , - ,
(4s)
RTR fbr linearmolecules. and
F , . ( J . k ) : B , , J ( J+ 1 ) + ( A , - B , ) t :
F,(J):1{a,
l H o ( ? . ) - H o ( o ) 1 h ,_, $ , Ld'u Rr
: px Q::' Q'hx,,'
(39)
lHo(r) - H')(o)1., rf=c-::r r r l RT
r
r_16r
,
fbr nonlinearmolecules. If we assumethat the vibrational-rotationalpartition function of the excitedstatesis equalto that ofthe ground state,the components of the excited electronicstatescan be calculatedas the corrections to the componentsof thc ground electronicstate
l H r r r - j 1 " ( o r :l .).- & + r , , , , , * r ( _ '\ fk + LpyksT RT e lr , , \ / //
jl., )l fL , - f a.-o(\ Aa1 /)
r.,7r
7P,
[ c , . r .r. : I r R
P , . * p_(! r , , r
l4tt'
^Bt
)/ * , ]t
,r+l+, l'.-o(-Lr, \I ultylLsT
I
\
^Br
/l
\l' I'-)-11.^o/-'', I ?r,""\ ^,.' il
,,1 ))'I tL,*#r,;"..n(-fr {'.I #*o(-^-Lrl)}'
('18)
whereI,i" and pi are the energyand the statisticalweight ofthe i th electronicstaterespectivelyand p.y is the statisticalweight of the ground state.
985
CAPITELLI ET AL
m.
Resultsand Discussion
25
A. MonoatomicComponents In this sectionwe comparethe presentresultswith similar ones from the literature.In particularwe will comparethe presentresults (called new) with the results obtaincd by our group some vears agorTrs (called old), the recent calculationsperformed by Wang and RhodesT(called Wang), the old and new calculationsof the NASA group,2and the old calculationsperformedby Browne.s The main dif-ferencesamong the calculationsare in the set of levelsinsertedin the panition function and its derivatives. The presentcalculationscompletethe observedlevels reported in the tables by Moore6 and the National Institute of Standards and Technology(NIST)'u with Rydbergand Ritz extrapolationIaws fbr principal quantumnumber lower than 20. using the hydrogenlike tbrmulation for n > 20 (see Appendix). The calculationsare then presentedwith the lowering of the ionization potential as a parameterto controlthe cutofT.The sameprocedurewasusedin our previouscalculationslTlEeventhoughin this casewe considereda lesscompleteset of energylevels. Wang and Rhodes?have recentlypresentednew calculationsof thermodynamicpropertiesfor very high-temperatureair componentsby includinga completesetof hydrogcn-likelevels.Moreover. the resultspresentedby those authorsare self-consistentbecause they calculatedthe lowering of the ionization potential provided by the Debye-Htickcl theory in their equilibrium code. In principle, their resultscannotbe directly comparedwith ours. However, a rough estimateof the lowering of the ionization potentialfor atmosphericair plasmasin the temperaturerange 20,000-35,000K should be in the range of 600--500 cm r tbr neutral atomic components,reproportionally scaling with the charge of the consideredion [seeEq. (7)]. This point can be understoodby looking at Fig. 4, in which we havereportedthe lowering of the ionizrtion potential for equilibrium oxygenplasmasas a function of temperature for difTerentpressures. The sameloweringshouldbe approximately appliedto air plasmas. Gordon and McBridc.r on the other hand. usedthe f'ew-levelapproximationto calcuiatethe partitiontunctionof atomiccomponents up to 6000 K. They extendedthcir resultsabovef > 6000 K essentially by usingthewell-knownGurvich'stables.The lasttabulations also accounttbr a very limited numberof excited levels. Finally. the Browne's tablcssarc buscdon the observedlevelsas In general.we shouldexpecta good reportedin theMoore'stables.6 agreementabout the thermodynamicpropertiesof atomic (neutral and ionized) componentsfor temperaturesbelow 10,000K when only the low-lying excitedstatescontributeto the partitionfunction. Strong differencesshould appearabove 10.000 K when the highlying excitedstatesbegin to contribute.The etfectbecomesevidcnt when complete sets of energy levels are insertedin the partition function.lt shouldbe alsonotedthatonly our results(newand old) dependon the cutotT. These qualitative observationscan be drawn from the results shownin Figs.5 and 6. wherea comparisonof the nondimensional specificheats(Cn/R) lbr the components N and N is illustratedas
20
+Nerv -- E old _1- -Wang - -a- Broune
l5
l0
I 0000
20000
30000
40000
50000
T.K
a) ll
20
i\ , l \
l5
l0
\-- \- _ --\ --.---\ \ ---H
..'
0
| 0000
20000
s0000
i0()00 t.K
b)
Fig. 5 Comparison of nondimensional specific heat as a funclion of temperature for \ for a) AE = 5fi) cm I a n d b ) A E = 1 0 0 0c m - ' .
25 +New ----E -(Jtd _ r- Wang --a--Brownc
20
\
l5
t;
,.
l0
\ :/----r0000 a)
20000 i0000 T-K
40000
i0000
25 2000 1600
+o { {- ^
u . 0 l a t ml ' p = 0 .l a t m ti a lr^\ ' p=latm l -p=lflatm
,-!
20
-.-.r
1 .,
t /
I /
80t)
/ / '100 0 0
,'/
l5
1200
--*
\-^
;t
---#-
'\*
r\
I
-:- - :,.'2_{_-:---
F'ig. 4 Lowering ofthe ionization potential (AE) ofneutral speciesin a oxygen plasma as a function of temperature at different pressures.
I
,tlf
l0
5 0 0 0 r0 0 0 0 | 5 0 0 0 2 0 0 0 0 2 5 0 0 0 3 0 0 0 0 3 5 0 0 0 T.K
/'-\.
v 0 b)
r0000
20000 i0000 T-K
.10000
50000
Fig. 6 Comparison of nondimensional specific heat as a function of -' temperature for N* for a) AE = 500 cm ' and b) AE = 1000 cm .
986
CAPITELLI ET AL,
a function of temperaturefor different cutofTvaluesin the temperaturerange0-50.000 K. Wangand Rhodes'srresultsat high tempcr.;iurcarein satisfactory agreementwith the prcscntresults.This meansthat the hydrogenlike approximationfbr higherexcitedstatesis a good one.On the otherhand.rvecanobservestrongdiffbrencesin thelow-temperature regimc.Apparentlv.Wangand Rhodeshavecompletelydisregarded the presenceof low-l1inglo'els of the ditferentcomponents. Thus, their spccitic heatspresenta flat behavior (that is. only the transIationalcontributionC /,lR:2.5) below 10.000K and a dramaric increasewhen high-lyingexcitedstatesarepopulated. Gordonand \lcBride's specificheatscloselylbllow our results below 10.000 K. but are lar-eelyunderestimatedabove 10.000 K becauseof the insufficientnumberof levelsconsidered by theseauthors.The samebchavioris observedin relationto Browne'scalculations.The insufficientnumberof levelsconsidered by Gordonand McBride is determinedbv the cutoff criterionadoptedin theircalculations.This cutofTcriterion is temperaturedependent:the panition tunction includes only those levels below rhe ionization porential lowered by A6f . then as temperatureincreases.f'ewerenergy levels are used.which resultsin an underestimationof thermodynamic propertiesat hish temperature. It is also interestingto compare the presentresults with those obtainedby our group someyearsago. A satisfactoryagreementis observedand the minor dilferencesare due to the improvementof the energy-levelsetswe haveused" Similar considerations can be made tiom Fig. 7 in which we have plotted the nondimensionalspecifichearsof C and C-. ln this case.we limit our comparisonto Gordonand McBride'sand Browne'sdata.As in the previouscase.the agreement is excellent below 10.000K but becomespoor abovethat temperature.namely when high-lying excited statesbegin to contributesubstantiallyto the partitionfunction. Before concludingthis section.we want to emphasizethe probIem relatedto the inclusionof the so-calledautoionization states. Thesestatesderiveby consideringothcr seriesin the completionof
+
\ E - 2 5 0c m ' ' =
f E = 5 o oc m ' '
o
l E : l o o oc m ' '
a
flordon & N{cBride
* l0 0
I
/.
25 C
A.
20 l5
f ,, .\,.,
li 1::
l0
/r
0
10000
20000 30000 .T. K
\\\ ' . " '.-\
10000
50000
F'ig. 7 Comparison of nondimensional specific heat as a function of temperature for C and C* for different cutoff.
20 -eonlr lst('S) -l'],'i's) l:ipr'P"r
,^t \
t5 I
F v, -
1
!0
5
0 20000
,10000
60000
80000
l 00000
80000
I 00000
T.K 20 +oni\'2s(rs)
/
-
l5
q
w
/,
..- t(l
5
0
0
20000
tor,,,, 60000 ,. K
Fig.8 Nondimensional specificheat for C+ and C2* as a function of temperaturecalculatedwith and without autoionizinglevels(up to n =20). energylevels,which convergeto ionization potentialsgreaterthan the first ionization potential of the consideredcomponent.These levels are usually eliminatedin the partition-functioncalculations by the cutoff procedureeven thoush in many casesthose are observedexperimentally.To understandtheir influence.we presentin Fig. t3 the nondimensionalspecific heat fbr the C- and Cr- calculatedwith and withoutthe inclusionof autoionizinglevels.For both series.we considerlevelsup to n : 20. The figure showsthat the autoionizationafl-ects the resultsabove20.000K and.therefore. originatesa further problemlbr the determinationof the electronic partition function. The extensionof the resultsup to 100,000K is reportedonly tbr a better appreciationof the ditl'erencesin the two calculations. Indeed,C- and Cr- ionswill disappearfbr f > 50.000K in realistic pressure ranges. B. DiatomicSpecies In this section.we will comparcthe presentresultswith those givenin the JANAF tables.rby McBride er al..-rin ESA tables.17.1E and by JafTe.r() The presentresultsare calculatedb.v-' usin-espectroscopic data from Huber and Hcrzberg.lr The I'ariouscalculations dif-ferbecauseof thc numberof electronicallyexcitedstatcsi ncluded in the partitionlunctionand, in somecases.becauseol the way to accountfor the rotationaland vibrational lcvels. Resultsfor N2. O1.NO. and CN areshownin Fig. 9. In the lowtcmperaturerange( f < 6000 K) our resultsarein perfectagreemcnt with all othercalculations. Smalldifftrcncesemer-qe at highcrtemperatures when the contributionof clectronicallyexcitedstatesbecomesimportant.Deviationsfor NO arepresentonly with the recent calculations of Gordonand McBride. who havelittcd the Gurvich data:with polynomialcurves. It shouldbe notedagainthat diatomicmoleculesdo not survive for temperatureshigherthan 10.000K in realisticpressureranges. The extensionto highertemperaturesis interestingto emphasizethe declineof the specificheat and its convergenceto the translational specificheat.
987
CAPITELLIET AL,
\2 -
CO
ESA 1994
-Jaffe -"-o " JAN.AF ---€- 'this work
l0 A
U
30000 20000 T.K
t0000 5.5
10000
r0000
o
I 0000
5.0
20000
'I"
i0000
'10000
50000
K
'1.5 ( 10,using last method.we obtain A : - 1.0288and B : -0.664'79. Ifthere is only oneobservedenergylevel.insteadofEq. (49) the fbllowing fbrmula shouldbe used:
Estimation of Energy Levels for Neutral Carbon
The availablcdatabasefor observedatomicelectronicenergylevels are\'loore'stables6andthe morecompleteNIST's table.eHowever.manvpredictedelectroniclevelsaremissing,especiallyfbr the higherquantumnumbers,so thc set of electroniclevelsshouldbe completedusingsemi-empirical methods. To cxtendthe electroniclevelsof the ground-state configuration we usethe iso-electronic sequence method.Let us considercarbon a t o mC ( l ) a s a n e x a m p l e . For carbon.the groundstateconfi-uuration we shallconsideris 2 s : 2 p r ( 1 tSt ., ' O r , 2 s r p - r 1 : r5S' .' . 3 P ' . r P 'r. D ' . 1 D ) : 2 p r ( r Sr, P . r D ) . Ifone ofthesetermshasnot yet beenobserved.it can be dcrived throughextrapolationfrom energylevelsol'isoelectronicspecies. Practically.fbr carbonC(t). one missingterm is 2pr [1S]. so. tbr example.we considerthe energiesol N(tl). O(lll). and F(lV). corresponding to the sameterm.andreportin,c themasa functionolthe atomicnumberZ. we perfbrman extrapolation to obtainthe missing energycorresponding to the C(l) (Z:6). The methodworks well: as a matterof lact extrapolating isoelectronic sequences correspondine to 2srpi15S) term we obtaina valuever,vcloseto the observedone. Theexcitedstatescomefrom theexcitationofone electrontoward highervalues of theprincipalquantumnumbers(rr > 2)l suchseries for C(l) are 2sr2p(rP')nx 2s2p:1rP;nx 2s2or(rD)nx' 2s2or(rS)nx" 2s2p:1:P;n' "' 2 p r 1 r S; n x / t Thcse scries arisc fiom thc intcractionof the atomic corc 12sr2p{rP ) . 2 s 2 p r ( r P )... . ) w i t h t h e e x c i t e de l e c t r o n( n x . n x ' . . . . ) withx:s.p,d..... Let us lbcus our attentionon the nrst excited series.In the case o f x : s , t h e r ea r et w o s p c c t r o s c o pticcr m s :r P ' 1 L : l . S : l ) . a n d ' P ' ( L : l . S : 0 ) . N I S T ' st a b l e sr c p o r to b s e r v c lde v e l su p t o r r : l 0 l b r t h e l b r m e ra n d r r : l - 1( m i s s i n gv a l u e sr i : l l a n d n : 1 2 )f o r the latter.To extrapolateto higher principal quantum numbers.the
R.r''
F - F.-
Conclusions
In this paperwe havedescribedand discusseda recentetTortto calculatethe high-temperature thermodynamicpropertiesof Marsatmosphere Associatedproblems.suchas the effects componcnts. of quasi-boundvibrorotationalstates.cutoff criteria,energy-level completionol'the monatomiccomponents. and autoionizinsstates havebeenconsidered and haveshownto be nonnegligible, particularly at hi-ehtemperatures. lndeed.we find agreementwith datafrom other publishcdtablesat low tcmperaturesand marked differences at high temperatures. In particular.lbr monatomiccomponents, the discrepancies arcmainlydueto theinsufficientnumberofelectronic levelsusualll accountedfbr in the othertabulations.
Appendix:
following Ritz-Rydberg seriescan be used:
E,:
E i - R t rl h * C ) 2
(A3)
where C is an adjustableparameter. For thc spcctroscopic series2s:3pt:P tnx. all thc :pcctroscopic terms correspondingto x: g (n > 5) have not been observed.In this caseto obtainthe energylevel of the termscorrespondingto 59. we report in graph the cnergy levels (5s. 5p, 5d. 50 as a function of the azimuthal quantumnumber 1. and extrapolateto / :,1. The energylevelsol'5s. -5p.5d, 5f are the avcragevaluesover all the spectroscopicterms: fbr example.for 5s we have _tD
F-
-
q{'P )* E.f +gtrP rx E.P g(rP')+g(rP')
(A4)
whereg is statistical weights. The total statisticalweight associated to -5genergylevcl.obtained as prcviously.is the sum over the statisticalweights of all the correspondingprcdictedspectroscopic terms. From this last energy value. we extrapolateto n > 5 using the lbrmula(A2). T h i s m e t h o dh a s b e e na p p l i e do n n : 2 0 a n d 1 : 1 9 . w i t h t h e exceptionof thoseserieswhere no observedenergiesare present. For n > 20 energylevels.we assumean hydrogen-like behavior and usethe tbllowingf ormulas: Ei -
E,:
R t ' -l n :
(A.5)
It is possibleto demonstrate that.at flxedrr and fbr eachscries.thc predicted sum over the statisticalweightsof all thc corresponding spectroscopicterms for dillbrent 1 is cqual to 8r
=
l,li-
.,?corc
(A6)
so filr rr > 20 we haveappliedthis lastequationto detcrminethe weight. totalstatistical References rstull. D. R..and Prophet.H.. "JANAF ThermochemicalTables."National StandardRef'erenceData Series.National Bureau of Standards.NSRDSN B S - - 1 7U . . S . D e p t .o f C o m m e r c eW . a s h i n g t o nD, C . J u n e 1 9 7 1 . -Gordon. S.. and McBride. B. J.. "ThermodynamicData to 20000 K for M o n o a t o m i cG a s e s . N " ASA TP-1999-208523 Ju . n e1 9 9 9 .
CAPITELLI ET AI-.
jMcBride. B. J.. Zehe.M. J., and Gordon. S.. "NASA Glenn Coefficients for CalculatingThermodynamicPropeniesof Individual Species."NASA T P - 2 0 0 2 - 2 1 5 5 6 .S e o t .2 0 0 2 . aGurvich,L. V, Veyts.L V. and Alco ck.C.B.,ThermodtnamicProperties Hemisphere.Neu, York. I989. ol lndiv'idualSubslances. 5Browne,W. G.."ThermodynamicPropertiesof SomeAtoms and Atomic Ions," Ceneral Electnc,EngineeringPhysicsTM 2, Philadelphia.1962. 6Moore,C.. 'Atomic EnergyLevelsas Derived fiom the Analysesof OpData Senes.National Bureauof tical Spectra,"NationalStandardRef'erence Standards,NSRDS-NBS-35,Vol. I. U.S. [)ept. of Commerce.Washington. D C . J a n .1 9 7 1 . 7Wang.T. S., and Rhodes.R.. "ThermophysicsCharacterizationol Multiply lonized Air PlasmaAbsorption of Laser Radiation,' .burnal oJ Therm o p h r s t c sa n d H e a t T r a n s J e rV. o l . 1 7 .N o . 2 . 2 0 0 3 , p p . 2 . 1 7 - 2 2 4 . EMcChesney,M., "Equilibrium Shock-WaveCalculationsin Inert Cas. Multiply lonized Debye-Hiickel Plasmas.Canadtan Jtturnal of Pftr'sics. Vol. 42. No. 12. 1964.pp.2413-2494. eGriem.H. R.. "High-DensityCorrectionsin PlasmaSpectroscopy." Phrsi c a l R e t ' i e wV. o l . 1 2 8 .N o . 3 , 1 9 6 2 .p p . 9 9 7 - 1 0 0 3 . I0NIST Atomic Spectra Database. URL: http://physics.nist.gov/ PhysRefData/ASDIlcited l5 March 200'{1. rlHerzberg. G,. Molecular SpectraanclMolet'ular Structure,L Spe
