Particle densities of high temperature gases: hydrogen, nitrogen, oxygen p = 1, 2, 5, 10, 20, 30 kg/cm2, T = 1000 to 50000 degrees K

Particle densities of high temperature gases

Citation for published version (APA):

Pflanz, H. M., & ter Horst, D. T. J. (1966). Particle densities of high temperature gases: hydrogen, nitrogen, oxygen p = 1, 2, 5, 10, 20, 30 kg/cm2, T = 1000 to 50000 degrees K. Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1966 Document Version:

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•

PARTICLE DENSITIES OF HIGH TEMPERATURE GASES. (Hydrogen, Nitrogen, Oxrgen

I

p

=

1, 2, 5, 10, 20, 30 kg/oa T = 1000 to 50000°K.)

H.M. Pflanz and D.Th.J. ter Horst. EH • 66 - R6 •

,

•

1.

SUMMARY:

For a gaseous mixture in physical-chemical equilibrium the formalism is given leading to the computation of particle densities of the general components A

2, A2+, A, A+, A++, A+++ and electrons. Approximate cut-off criteria for the partial sum over rotational energy levels are given and a limiting vibrational quantum number is introduced for the molecular sum over states. Following Griem's work the Debye-Ruckel theory is applied to limit the sum over states of atomic and ionized species and to correct the respective ionization energies in the Saha equations.

The computer routine to obtain sum over states and particle densities is outlined. The influence of the iteration accuracy, of the

limiting vibrational quantum number and the dominating Debye-Huckel correction on sum over states and particle densities is discussed.

Computed data of sum over states. Saha equations and particle densities

2

of Hydrogen, Nitrogen and Oxygen for pressures 1 to 30 kg/em and 0

•

[1]

[2]

[3)

Pa.r.t:Lc.le densi ti.~_s_of high temperature gases, (Hydrogen₁ Nitroilen, Oxygen ;p::::: 1 t 2, 5,

·w,

20, 30 kg/cm2 T = 1000 to 50000°K.)

by H.M. Pflanz and D.Th.J. ter Horst.

Introduction •

!n general a high temperature gas :is a mixt~re of neutral parti.cles

(atoms molecules), excited neutral particles, ions and electrone. For sufficiently small spatial and time variations the equilibrium

eLi e:tribution of energy (or velocity) of these .Part s follows the

!•laxwe 11-Boltznann distribution. I f radiation ie negligible and the

:=unong the ners th>:? rn.ixtur~? collision

dominated thermal equilibrium can. b-:> expected, provided no strong external forces are acting.

To a system a single tempe rat :,re may be assigned, which is then r·roport:lonal to

the entire system.

roost probable kinetic energy of a particle or

This is true for most plasmas ( xc near boundaries such as electr()des

and walls) [1]. The particle dietr~bution such a mixture and other

thermodynamic properties can be calculated from total densities and

t~m:peratures making use also of' the chemical and/or electronic

reactions tak.i ng place in the go.sp Tn essence the .formalism dev~loped

for gases applies with suitabl modifications to the individual

reaction partners.

The eqll';!;_tion of mass-actio.!:..

The thermodynamic \l is def the equation [2,

3J

( '1a) G ~· ~~ - 3T ,.. pV

H. Edels - t:ish Electrical ani ied Industries Research Assoc ..

Tolman - The A of S~ is~ical Me s, Oxford Univ~roity

•

G may be expressed also i.n terms of the partition function Z

which gives the sum over the relative numbers of particles n

in different energy levels referred to a ground level

[2,

3].

(1) where: E ::: internal energy p =pressure V =volume S =entropy G

=

nkT ( 1 _lln Z) n T = temperature (absolute) n =number of particles.

l. em said to be in thermodynamic E-quilibrium if at a given

pressure and temperature the variation of G is zero.

(2)

In addition chemical equilibrium in a reacting gas mixture exists

when

a.A + bB + ... ~ cC + dD + ... ..

i.e. when under given pressure and temperature conditions a mols of component A react with b mole of B to give e and d mole of C

and D, respectively.

Combinjng equ. (1) and (3) with condition (2) a mixture of perfect

gas components taken at their partial pressures is then in physical-chew~cal equilibrium when [2)

In statisti.ea.l mechanics the partition function is defined [2]

..

s:::: number of different eigen-states of a particle, n::::number of particles of a kind.

The total energy E of a state of a particle can be taken as the

s

sum of translational (t) and internal (i) energies,

(6)

The partition function may also be written as the product of the

sum over the translational states Zt and the sum over the internal states Z •• Thus

l.

( 7)

I'he expression for the sum over the translational states

(:!

is removed by the Stirling approximation)

3

zt

=

eVf

(2lrmkT)lz nh

(8)

is well known from ~uanturn Mechanics

[2].

The usual notations are:

e base of natural log-system, m mass of particle,

k and h Boltzmann's and Planck's constants respectively,

g recognizes that twice as many solutions of Schrodinger's equation are obtained wher. the particle in question has a spin. g ia thus

2 for particles with spin and 1 for particles without spin.

Combining equations (7) and (8) the thermodynamic potential equ. (1)

becomes after substitution of pV

=

nkT

•

For the general chemical reaction of interest in our case,

( 10) AB~A + B

the equation of mass aqtion is obtained from condition (4)

(11a) ( m

m...J/2

2'1F

:A:!

5/2 = gAgB •.p • (kT) gAB h~ n

or expressed in terms of particle densities: N = V

( 11)

z

A

z

B

1 i

~ AB

L.i

Indices A, B, AB refer to particles of kind A, B and AB.

Sum over internal energy states of a molecule~

Any internal energy state of a molecule is determined in first approximation by the sum of the electronic (e), the vibrational (v)

and the rotational (r) contribution. Thus relative to the sam of the energies of the separated atoms we have

{ 12) _{E i}

=

_{E e} + E _v + E _r

This is based on weak interaction between the electronic, vibra.tional

.>Jn.d rotational energy.

Substituting equ. (12) into the sum over internal states equ. (7) and allowing for g-fold degeneracy of the electronic and rotational terms there results (

4, 5].

(We express E in cm-1, c is the

velocity of light.)

[1+]

J.D. Fast- Philips :?es. Rep.

f.,

382.

5.

{ 13)

The molecular model which gives a reasonable approximation is that of the anharmonic oscillator and the nonrigid-vibrating rotator. Referred to the ground level the vibrational energy levels of the anharmonic oscillator are given by the equation

[6]

( 14) E

=

CJ v - x fJ ( v2 + v) + • • ..

~m

-1]

v e e e •Nhere:

v"" vibrational auantum number and

w ,

x (,J , • • .. • are characteristic

• e e e ·

factors resulting when solutions of a third ord~r potential function or a Morse pot.ential function are fitted to observed spectral levels .. The rotational energy terms are given to a good approximation

(6]

by

the equation

(15)

where

Due to vibration the internuclee.r distance undergoes variations. TI1e resulting change of the moment of inertia accounted for by ~e. The quantity Be is inversely proportjonal to the molecular moment of i.nert ::md depends on the electronic state.

The term Dv in equ. (15) also depends on the vibrational quantum number v and is a correction for the influence of the centrifugal force on the rotational energy.

The degeneracy g , called also the statist.ical ~eight of the

r

rotational levels, expressed in terms of the rotational quantum number J is

6 ..

(6]

G. Herzberg - Molecular Spectra of ~:,iatomic Mole

Inc. New York,

1955.

("'"') \ I ( g

=

2J + 1

r

For sufficiently high temperatures fully excited rotational energy levels may oe assumed. Then the rotational portion of the sum over states can be replaced by the integral over states. Thus holding in equ. (13) Ee and Ev' the sum over the rotational states becomes with equ. (15) and (17)

(18)

J

(J _!!s_ [J(J+1)B Z J ( e ' v) •

~

( 2J + 1 ) e k'l' 0 2 2 ] - J (J+1) D v dJ

For homonuclear diatomic molecules because of symmetry only even

or odd rotational energy terms are solutions of Schrodinger's equation ..

The symmetry number normally 6" = 1 .ts hence (t = 2 for homonuclear diatomic molecules.

Due to th~ approximate character of equ. (15) integral (18) could diverge. Therefore an upper limit J

=fJ

must be chosen in agreement with the convergence criterium£

>

J(J+1). Since in general B>) Dv high values of~ result. Developing in equ. (18) the second exponential into a series it is found upon integration that the contribution of the upper limit is negligible. Thus we obtain

(19)

or expressed more simply

(20) n (e,v) __

1

~

)

(2j)l (

~ ~)

j

u J _. f' B he

J'

j l B2 he

(

...

)

At high0r temper.'1.tures above simpll fication, however, affect a Z J

seriously, but then d:i ssociation J e complete a.nd hence its influence

on the particle densities or the predominant gaseous components goa s unnoticed.

In actual computations it is found that the numerical values are such that often

(-1i'

f!)

is very much less than 1. Then sum (20)

B c th

CAn be broken off after the contribution of the j term has

become negligible. There are, however. cases where this is not so. Notably the rotational constants of the hydrogen molecule are exceptions, the effect of which is felt already at a few thousand degrees. In this case it is within the approximate character of equ. ( 15) to drop J:)v entirely i.e. to let j = 0 in equ. (20), or more reasonably to break off summation before successive terms begin to increase. Thus from equ. (20} with

(21a) :::; (2j)!

uj j!

and the recurrency requirement

(21b) uj+1 = 2(2j+1)( D;

~)

<1

uj B he

the following cut-off criterion is obtained,which is to be obeyed by all terms included in sum (20)

(21)

.. 2

j

_<

+

B he 1

'+

D1tT -

2

v

In our computations it is observed that this criterion applies when the vibrational quantum number v is held at a high level while the summation over j is carried out. The reason for this is found in the modifying influence of v on the rotational constants B and to a lesser degr~e on D • At low temperatures and low v's sum (20)

v

is generally self-limiting in a practical sense.

8 ..

Taking the energies absolute the previously assigned reference level must be dropped and the s.m over states multiplied with the exponential

•

results finally for the molecular sum o~er iaternal states

.. ')2 ) (.,_ a where ED he

*

kT zi = L;i e

1

~

(

2J ) 1 ( D v kl' ) j X B j ' i · B he

B and E are given by equ. (16) and (14), resp. and the values of

v

. l , _e B , D _{e v} 9 W , _e x _{e e}

w

and E _e may be found in the tables of ref ..

(6].

The factor

1\.e

accounts for

A-

doubling and is two for all electronic states other than

L

states, for which it is one [

6].

I"or sufficiently high temperatures also the partial sum over v of equ. (22) ma;:r appear to be divergent unless we recall that there is only a limited number of discrete vibrational energy levels and recognize the validity range of equ. (14).

In earlier computations the sum over states

[4, 5)

was carried to relatively low temperatures and terminated when the contribution exp. ( - (E + E )hc/kT] became insignificant, i.e. when for a

• v

I

'

kT particular state E e + Ev >) /he.

In practical cases the sum over states is generally self-terminating

0

in the sense of this inequali.ty for temperatures O< T< 5000 K.

For higher temperatures the summation must be cut off with the last •librational quantum number below the respective dissociation limit .. \v'hile the dissociation limits in some cases are known quite accurately, the corresponding limiting vibrational quantum numbers vD may be found approximately from low lying ;ribrational levels by the Birge - Sponer extrapolation (

6, 7].

However, errors up to 25%, depending on the number of experimentally observed levels, must be expected.

An ef:'ti.Mate of the last vibrational quantum number vL just below the d.issc ciation lirrd t is also possible from equ. ( 14) if we let Ev equal the dissociation limit

Ev

We rrust, however, realize that

generally the constants in equ. (14) are derived from low vibrational levels near the potential minimum of the electronic state in question.

[~.A.G. Gaydon- Dissociation Energiec and !5pect:ra of Diatomic Holecules,

"

..

2

I f we cut off equ. (14) after the term x fJ (v +v) and solve for

e e

vD, the limiting vibrational quantum number becomes \'lith v

1

=

vD -1 ( ) w -w x e e e 2c..)x + e e

(

c...~-wx)

2 e e e 2w x e e

~

U X • 1 e e

In '~qu. (23) the (-) sign only is significant since the maximum of

equ. (14) corresponds to the dissociation limit. As before the error involved here is considerable. For example the limiting

vibrational quantum number of the ground state of oxygen is 41 [

7],

while the value computed from equ. (23) is v

1 ~ 35 (EI.

=

41280

cm-1 ,

t..J ::: _e 15Ro, fJ _{e e} x = 12). Greater differences _· must be exPected for other

-states and different molecules. Indeed the use of such values in

any computations appears rather questionable. However, because we are riealing here w.ith high energy levels wh:ich enter the summation via negative exponentials their contributions to the sum over states at low temperatu.res in general should be negligtble. At high

tempere.tures dissociation is nearly complete and the number of molecules is small against other species, hence also the influence of vT _.... on the particle densi.ties should be wiped out. Later in this text th<;; entire temperature range is investigated by comparing the> sum over states and the particle densities of U:t•;! nitrogen species computed with vL and 0 .. 75 "'L. From t.his further deducUons of the accuracy of the appended data can bP made.

Internal sum over states of neutral atoms and ions.

Using tabuleted spect:roscopic a

[8]

sum c.ver internal states readily computed

(2]

wlth the equation

(2~)

E;. he

l.m

kT

[s]c.E.

l'1oore - il.tomic :~nergy Levels,

u.s.

Dept. of Commerce National

Bureau of Standards, 1949.

..

whr;re g

=

(2 J + 1) the AtatisU.cal weight associated with the m m.

energy level E. referred to thf' appropriate ionization level an.d

l.M

J is the total angular momentum. As far as possibl~ missing terms

m

in th~ tabulation of ref.

[8]

must of course be estimated and included in the summation.

For low temperatures when E. hc/kT ))1 it is generally satisfactor1

J.m

to consider a few tt:!rms onl7. At higher temperatures, however, terms

11.

up to the ionization li.mit become significant in the sum. While the energy of these high levels remains essentially constant the

statistical weights are not bound as the ionization limit is

approached. The sum over states th~refore diverges, 11nless appropriate cut-off criteria are applied:

The Unsold cut-off is applicable only to high density plasmas i.e. to electron densit s Ne [ 9, 10) satisfying

{26)

This cut-off [ 11] as well as the lowering of the ionization potential are derived by considering the interaction of an atom and of a

neighboring perturbing ion. The potential barrier set up by the·ir Coulomb fields is regarded as the limit for bound electron motion and is set equal to the possible • highest excited level. Rela.tj,ng

the probable distance (probability e-1) between the two species to

the Bohr radius of this level, Unsold obtained a limiting principal

•

quantum number n with which the sum over states is to be terminated. Burhorn and Wienecke

[12]

applied the Unsold lowering of the

ionization potential to o

2, N2, NO,ai:r, H2 and H2o. Comparing their

results with inequality (26) it is found that their electron densities {or electron pressures) are below the applicable range of the Unsold theory. Moreover, since they do not give an indication of the last term used in their sum over states the error involved in their Z 's a.nd in the computations of partial pressures as well as the thermodynamic properties, however small it may be. CEtnnot be estimated. As will be seen later application of the Debye-Huckel limiting theor·y would have been appropriate •

[9)r1. HcChesney- AIAJI.-Journal Vol.1, No.?t P.P• 1666-1668 (1963). [1o]H.R. Griem- Phys~ .~tev. 128, 3, pp. 997-1003 (1 ). See also

H.R. Griern - Plasma 3pectroscopy, Me Graw-Hill Book Co. New York, 1964.

[1~ A. Unsold - Zeitschr .. Astroph. 24, pp. 355-362 ( 948).

~2)F. Burhorn & Wienecke -

z.

Phys. Chemie 213, 37-43 (1960)

12 ..

rteference to other more elaborate procedures for the termination of the sum over states of hi.gh temperature high density gasee beyond the range of our interest here may be found in the literature (10, 13]. The Debye-Ruckel limiting theory applies to medium density, medium temperature plasmas. A rigorous derivation is given for example in ref.

[3

page 229]. Therefore the following brief discourse is sufficient. From Poisson's equation, applied to a system consisting of a test charge surrounded by a spherically symmetric cloud of

electrons against a uniform back ground of positive ions, the electric potential of the test charge due to all other charged particles is obtained. Fron: this solution i t is apparent that the electron cloud rr.odifies or screens the Coulomb field of the test charge in a

stance known as the Debye length. Contrary to the Unsold theory referring to the nearest neighbor the Debye-Hickel derivation considers long range int<?ractions. The underlying assumptions are

that the electrostatic energy of the electrons is small against the thermal energy and that they are in thermal equilibrium.

From the Debye potential the correction F of the free energy due to Coulomb interactions is derived

[10. 13]

~o

(27)

with

(28) F ::: +F

c

where Z = 0,1, 2 ... for atoms, single, double etc. ionized atoms respectively; Ne is the electron density and Nz are the particle densities of species of charge number

z.

F is the total free energy and F the unperturbed free energy given

0

by the equation F =E-ST.

0

The eq,uation of mass-action fc·r ionization usually referred to as

the Saha equation is obtained from condition {lt) and the free

ent>rgies of the reaction partners analog to the previously discussed case ~>f dissociation.

[13]D.P. ~uclos and A*B. Cambel- Progress in International Research

on ~hermodynamics and Transport Properties, edited by J.F. }~si

a

(li:T)

It

(29)

The chemical reaction (10), however, is to be repla-.4 by tlle ioaisation reaction

(30) A i=!J.

1 + e-I

Z Z+ B

Az and AZ+₁ are the reaction partners, the latter of which is stripped of an electron relatiYe to the first and Iz ia the ionization energy of species of charge namber z.

Included in equ. (29) is the reduction ~Iz of the ioaizatioa enerJ1 which represents the Debre-Huckel correction for tbe

Coulomb disturbance of the free energy~ It is 4eriYed by Griea (10)

(31)

The lowered ionization energy presents itself as a natural •pper level i.e. a cut-off of the sum oYer states. Acoordingl7 ia tlle sum over internal states (equ. 25) all levels

(32)

should be included.

For completeness it should be mentioned that the error iahereat in the above cut-off proeedure as well as the error introduced ia the sum over states by using unperturbed energy levels ap to the cut-off point are in geaeral negligible [10].

It remains to define the range of validity of the Debre-Biekel

theory. This subject is extensively discussed by Duclos aad Caabel

[13].

A plausible upper density boundary as a function of tell))erature

is the Fowler - Be~lin - Montroll - limit. Griem

[10]

specifies the Debre domain for multiply ionized plasmas by the iaequalitr

(33)

In the following section the foregoing formalism is applied to the computation of sum over states, Saba equations and particle densities.

S;y;stem of equations for the determination of Rarticle dtntitit! of an assembly in ph;y;eical - chemical eguil1briua.

Specifically we are interested in plasmas of the diatomic gases H₂,

o

₂, N₂• At high temperatures these gases are in part dissociated and/or ionized. It is assumed that such a gaseous mixture consists of the following generalized perfect coaponents.: (where applicable)

+ + ++ +++

A

2t A2 , A, A , A , A and electrons.

The total particle density N is made up of the sum of the partial densities of the above components.

(34)

With the assumption of neutrality and recognizing that on each successive stage of ionization an additional electron comes free the electron density N is related to the densities of the respective

e

gas components by the equation (35)

The equations of n1ass-action yield expresaioas for the ratios of partial densities of successive components of the gaseous mixture. Applying equations (11) and (29) there results the following syste• of equations:

(36)

(37) OS/2 A + A2+ N _{NA +} E1 he

• i

(2'11" m kT) 2Z 2. kT

•

i

=

SA2+ NA ;: h3 Z A2 e .2 _i (38) 3/2 A+ 2Z A+ :&1 he N eN .A.+ (21tm

•

kT) i kT SA+ NA = h3 Z A e = i (39)

'3/

A++ Ne NA ++ (21t m kT) 2 A++ E₁ he

'

2z1 kT SA++ = e = NA+ h'& _zi A+ (40) 3/2 A+++ N _{e NA+++} (21T m kT) _2Zi J.+++ E1 he kT e

_s

₊₊₊ NA++

=

h3 A++ e

=

zi A A+ A++ A+++ [

-1]

where ED and Ei , E₁ , Ei em are the dissociation aad ionization energies respectively.

E

1 DC Iz -A Iz reduced ionization energy with A Iz per equ.. (31),.

The following simplifications were used:

etc .. Comiutational procedure.

For the computation of the particle densities with temperature as the independent variable use of an IBM 1620 digital computer was made. Since the system of equations (34 to 40) is nonlinear and is further complicated because the sum over internal states for the atomic and ionized species depends on the particle densities an iteratiTe solution was attempted.

First all molecular sum over states were computed according •••· (22) using the tables of ref.

(6]

which were fill~d in by reasonable

estimates. However, for the diss<·oiation energy of .Nitl:'ocen Gaydon. 'e

value of 9.764 eV

[?,

14] was used. Limiting vibrational quantua nwmbers were obtained from equ. (23) or from the Birge-Sponer extrapolation.

Summing over j while e and v were held fixed either one of the following two first occurring cut-off criteria was applied: A practical limit is

(41)

where uj and u

0 are specified by equ. (21a), while the eecoa4 limit is given by inequality (21).

Because of computer limitations only those combinations of electronic and vibrational energy levels satisfying the inequality

(42)

were included in the molecular sum over states. No further limits, modifications or corrections were deemed necessary.

Next the preliminary sum over states for atoms and ionized species were computed according equ.

(25).

The energy levels of ref.

[8]

extended by estimates particularly in the case of hydrogen were utilized. In these initial computations all levels with their associated statistical weight subject to the practical liait

were included ..

With these conditions and the following initial simplifications, where applicable,

T '- 15000° neglect N /:.. ++ and N A+++ for Ni t:rogen and Oxygen

16.

15000( T' 50000° neglect NA and NA + for Nitrogen, Oxygen and Hy4rosen

2 2

the system of equations (34 to (40) was aolved for preliminary particle densities .. Theu the~ Iz t s were comp·ul;ed per oqu... (31). Ap;t.l;}'ing

condition (32) "first round" sum over s·te:tes an•l true partL.;:...e

densities were obtained. In the same manner 11_{1!1e'Cond round" :!.eta}

were computed with revised di

3 's next. All associated speciee of

the first and second round particle densities N and N

1 reapectiYely

..

were now subjected to the following print-out or further iteration order:

(44) stop iteration, print-out

z

₁ •s and N 's

q+1 q+1

continue iteration by recomputiag Iz '•• new

z

1 's etc. until (44) is satisfied bT

two successive rouuds.

Using the procedure outlined above sum over states, solutions of the indivi4ual Saba equations and particle densities for Hydrogen,

2 Nitrogen and Oxygen at pressures of 1, 2,

5,

10, 20 and 30 kg/em were obtained over a temperature range from 1000 to 5Q000°K. These data are appended. Figures 1, 2 and 3 show summary graphs of the particle densities of the above gases as a fuaation of

2 temperature at a pressure of 1 kg/em •

Discussion of namerical results.

Up to ?000°K. generally no more than 2 rounds of computations were

required, while for higher temperatures up to 4 iterations were observed~

Since the N's of all species of the gaseous assembly of a roua4 were checked against the print-out condition, the error in the sum over states and the particle densities is determined by that particle density which converges slowest. The particle densities of one

series of iterations at a typical temperature are comparei in table 1.

I t is seen that the number error shifts two to three dieits per

iteration and one may conclude ths,t the computational for theN's and zi•s at the most, but frequ•ntly does

2 Table 1. Nitrogen, T = 20000 p = 2 kg/em ..

~~-,~-error is 10-1

%

-2o~

not exceed 10 ~·

~~-.. iteration N X 101? N X _{1010 N} X 1011 _{N Ax 1016} NA+ X 10'i? 11t

Az A2+ NA++ x 10 NA+++ X 10

e ~-"1st _{round 3 .. 41510 1 .. 50410 1 39118 2 .. 7~441 3 409?9 2 .. 65639 2.053h5} 2nd round 3.4g442 1 .. 2_1954 1 .. g_1715 2.§p206 .:; .. 4.1830 2·05899 2 .. 2,15?.it 3rd round 3.4244.2, 1.31 1.21§.92 2.60182 3 .. 41B31 3·053,60 2 .. 91114 6

..

In a test case the print out or iteration criterion was reduced from 0.01 to 0.001. It resulted in a minor improvement of the

results. The gain was insaffieient to j~stify the extra eoMp,ter tiae.

The influence of the limiting vibrational quantum number v₁ upon the molec~lar sum over states and the particle densities was investigated in a few sample computations. The variatioas of

N _{N2+ N2} Nz+

z

₁ 2 and Zi computed with 0.75 v₁ and related to

z

1 reap.

z

1 obtained by summing to v₁ are shown on fig. 4. It can be seen that the reduction to 0.75 v

1 is of no practical consequence at the lower to intermediate temperatures, while at the high temperature end the variation is nearly 100%. As stated earlier this variation of

z

1 is

lost In the relative abundance of molecules~ atoms, ionized speciee and electrons of the gaseous assembly as the temperature increases. This is demonstrated by fig. 5 where the Tariation of the particle densities due to changing v

1 to 0.75 v1, taken relatiTe to the total particle density is plotted ags.inst temperature. Clearly the influcnlce

of modifying v

1 is insignificant and the relative variation barely

exceeds 10-6 times the total particle density. It so happens that a

number of points on fig. 5 even represent the numerical limit of the computer ..

It was further deemed necessary to shed some light on the uncertainty of the correction of the ionization potential and its effect on the particle densities. Thus the particle densities of Nitrogen were

recomputed for a number of temperatures with 2 xAiz and 10

xAiz ..

The deviations from the N' a, obtained with 1 x

4Iz

were then related to the total partie density at the respective temperature• and

plotted in figurea 6a and 6b. It is somewhat surprising that the relative deviation of the 10

xAiz-

data is not larger .. As is to be expected Ne is affected most. In the temperature range from appr.

20000 to 40000°K t.he error due to ,Or,. is highest but the difference ll

between the 10

x4Iz

and 2 x~Iz- data is sl~Ulllest. Towards lower and higher temperatures the difference increases. At still lower temperatures the error as a result ot

A

Iz should disappear as follows from equ. ) and also from (31) In the first ease the contributions of the higher energy levela become less sigt1.:tficant

to the point where terms, Iz - Aiz• artrt no longer accounted for .. Since the densiti.ee of charged parth~les deo.rease faster than the tempera.ture, diz should eventually disappear in the second case ..

Sim.i.lar reasoning suggests that the influence of Aiz should becou more important at still higher temperatures. Finally it should be remarked that the general pattern of the deviations is in agreement with the successive rise and decay of dissociated and eucoeaaively ionized particles as the temperature increases.

For hydrogen and oxygen the preceding discussion in essence applies as well. But for hydrogen greater deviations with

Aiz

DRtst be expected because of the high statistical weights of the higher levela.

Generally it ean be stated that variation of~Iz and hence the cut-off level in the computation of the sum over states has by far the largest influence on the particle densities.

The authors are indebted to Messrs. F.C. Bussemaker and A.J. Geurts of the computer centre at the 11_{Technische Hogeschool EindhoYen",}

fig.1

fig.2 fig.3 fig.4

Particle densities of B:rdrogen (p = 1 kg/crl)

Particle densities of Nitrogen (p :1: 1 q./Cil2)

Particle densities of O:'Q'gen (p ₌ 1 kg/c11112>

Relati~e Yariation of sum over states of Nitrosea particles Ito

N

2 and 112 due to changing v1 to nearellt iatepr of 0.75 TL

(p

=

1 ks/cm2)

fig.5 RelatiTe variation ef particle densities of Nitrosen when

N2 If~

z

₁ and Zi are computed to the aea:rest integer of 0.,75 v₁

(p = 1 kg/cm2 )

fig.6a Relative variation of particle densities of Nitrosen (N

8, N2, N~, N) due to modification of 6lz•

fig.6b Relative variation of particle densities

or

Nitrogen

14

10

a~

6

4 ,,

_, ___ ,_,_

!

Njc

! "

-I

HYDROGEN

.

RESSURE

1 kgjc m

2] I _~

)

~

- - "

"'

_""

-~

\~

_1'-- -l-·

\

~~---

r---

_r----

H

total

---~---)\

.-

-

_---

---

!---r---

_1----

H+

I

\

--

1-

--I

\

- - -

__

,

___

1-••c--·-••~ _,

_______

-I

_\----~-

-1/

\

I

\

--I

\

-I

1\

I

I

\

I I

\

\

____

, \

~

1---

I

_,_

""'

f-,, _ _ _ _ ,

--r-+--

~-~ , __ -...,.

r----\

'H---

,

___

..

\

1,, ______ ., --- _{- - - -}

---\

·-···· I - "•"" 1---,--.

--- - - - - -

_,

____

- 1 - - -

1\

_{····!-·-·-} - -~-·-

---"--\H2

l

----\

_I

T

~

5

10

15

20

25

30

35

40

45

50.c10

3

----

.. --.,.---,---.---.----.---. PRESSURE I 2 - - 1 kg/em - · - · - 10 kg/cm2

-~

- - 30kgtcm2 1 1 1

-T

..

5

10

15

20

25

30

35

40

45

17

to

15

10

2

13

10

·----~ ·

--\I

N/cm

3 - ----:---~---:---

~--_\

NITROGEN

.[PRESSURE

1

kgjcm

2]

\

-~--- ~

--!---~--- -·--' - - - \ ~-i---

\(

~ ~

~~~

---~--I\

\

---

r----_

_N

_tot«ll

---

--)

,.,..--

-...__

-.:;;:::::__,

r

-~

~--

Ne

--- \--~ --~--

I

l

\

_I)?-·

-I

\

---

~

---~---~·

Jl

\

N+~

-- ----

--- v

I

\

N+//\

---~-·--

\

----

I\---

~f-·-

-=

L

--·-···--~---\~/

I

I

7

I

-""

I

·---v

--~----

--/\

---·- ---+--- "" """"""

I \

-- I

1\

--- --~- "·---

I

\

I

_I

_\

---" ---~- - -

I

\

I

, ____

\

n~->

I

\

I

\

!--- r---- _._

__

1

\

I

N+\

N2

/

~--~--"·---- ·-- I ---!

\

I

--- i···--·

--~---=t

...

----" " " " " ----

L

f--- " " " "

~v

.

·---, . . .

\

I

---I"

\

" __

/.

____________ \ --

1---\

;---·-

1\

I .•• ---- ·--·----···~ --··

~

---,--- " " " "

-N;

_\\

_\

T

I

5

10

15

20

25

30

35

40

46

50x10

3

·-· - ... -·1-- t··- ---+--···---!

OXYGEN . [PRESSURE 1 kgjcm

2] . -·--··1--·-- - -·--+---1 -·-j~- ·-·- 1

-1

014 ---+---+-+---+-+- ·- ---·-···· ....

c---8•

\

-6

~-

--···

\J ----.

--

.lf-4•

~-~

- -

~

IL

02

1

:

13 ..__1--...,L_f---_---

--~+----~--~1\_-_

---

--_.L _ _ _ .._ _ _ _ _ _ _ _ L....,_·_ --_ _ ---_--j_____;_ _ _ _

·,=~\=_· ----_...J._·---~~==T==

3

5

10

15

20

25

30

35

40

45

50x10

f

I

N(

y") -

N<

0.75

Vd

I

_ 6

NrorA

L

10

-10

10

1012

-14

10

1016

-18

10

-20

10

N

N+

Ni

+

Ne,N

__ L_

I

N

N++

N

2 1(J2 2 ,__ _ _ ____,_ ____ , ______ ... _____ , j _ . _ _ _ _ _ _ _ _,_ _ _ _

~----1.-....-0

10

20

30

40

0

50 x10

3

K

,.__,.,.

! ' ..

-2

10

-6

10

I

-· r--· I Na

-

_I

7~

I I /

///

I

,v

I ···~ 1---+ I

--N2

I

i

I-N2

_i

f

I

10

20

30

-4

10

-6

10

I

(

2 X

Alz)j

I

N(1.6lz)-N

10K

alz

NTOTAL

1012

-~----~--l---­

-2xAiz _ 14 --10K .o.lz

10

-16

1 0

1 . . . . - . . . - - - - ! . . . - - - - '

0

10

1

I

NC ,.

Ab)

-N(,~:

:i:)

I

NroTAL

1 0

°

1 - - - - . - - - · · · - . - - - - . . - - - -.. - - - , -2

1 0

1---+--6

10

N++•

1()8 I • -10

10

____ J

N

+-+I

N+

1012

2)CAb: -14 --10xA1Z

10

·~-N++

I

-16

10

50x10

3

0

10

20

30

₄₀

_OK

SUM OVER INTERNAL STATES OF NITROGEN A Z N2 ~ Z N2+ T{°K) _{1 1} .500 8.70418 E+01 1.81362 E+02 1000 1.80422 E+02 3-79234 E+02 2000 4.30462 E+02 9.22901 E+02 3000 7. 8.54o.5 E+02 1.70.527 E+03 4ooo 1 .. 2.51.52 E+03 2.74120 E+03 .5000 1.832.59 E+03 4.04307 E+03 6000 2.5323.5 E+03 .5.62694 E+03 7000 3·3.5.526 E+03 ?.5143.5 E+03 8000 4.30773 E+03 9.73261 E+03 9000 .5.40019 E+03 1.23146 E+Olt '10000 6.649.58 E+03 1 • .5298.5 E+04 12000 9·73431 E+03 2. 26449 E+04 14000 1.39096 E+04 3.213.59 E+04 16000 1.97229 E+04 4.41}67 E+04 18000 2.79062 E+04 5.891.52 E+o4 20000 3·93280 E+04 7.67406 E+04 22000 5.49348 E+04 9.77635 E+04 24000 · ? • 56446 E+04 1.21978 E+05 26000 1.02394 E+05 1.49331 E+05 28000 1 .. 36016 E+05 1.79954 E+05 30000 1.?7247 E+05 2.13201 E+05 32000 2.26369 E+05 2.49609 E+05 34000 2.83792 E+05 2.88990 E+05 36000 3.49475 E+05 3.30914 E+05 38ooo 4.23978 E+05 3.7.5289 E+05

40000 5.07902 E+05 4.21535 E+05 42000 6.01063 E+05 4.69810 E+05

44ooO 7.03070 E+05 5.19582 E+05

46000 8.11464 E+0.5 5.72288 E+05 48000. _{9.29659 E+0.5} 6.266?4- E+05 50000 1.05743 E+06 6. 808o8 E+05

~

..

'D

=

1

;;w ....

'n - 2 uilul' ·~

!(0][)

z•

_{z ••} If++ z 1f+++ zll z~"' lf++ If+++

i i zi i 1 i zi zi :.:._

SUM OVER IftERII.AL STATES C6 KITROGEN

500 4.00000 7 .. 03139 4 .. 42089 1 .. 00000 4 .. 00000 7.03139 4.42089 1.00000 1000 4.00000 7·93463 5.11184 1.00000 4.00000 7-93463 5.11184 1.00000 2000 4.00000 8.44527 5.52808 1.00000 4.00000 8.41+527 5.52808 1.00000 3000 4.00099 8.62824 5 .. 67885 1 .. 00000 4.00099 8 .. 62824 5.67885 1.00000 4000 4.01010 8.73707 5-75663 1.00000 4.01010 8 .. 73707 5-75663 1.00000 5000 4 .. 04104 8 .. 83353 5.8o409 1 .. 00000 4.041o4 8.83353 5 .. 80409 1.00000 6000 4.1054o 8.93737 5.83608 1.00000 4.10540 8.93737 5.83608 1.00000 7000 4.(!0815 9-05291 5-85916 1.00000 4.20815 9 .. 05291 5.85916 1.00000 8ooo 4.3484-8 9.17892 5-87681 1.00005 4-.34848 9.17892 5.87681 1.00005 9000 4.52220 9-31281 5.89121 1.00019 4.52220 9-31281 5.89121 1.00019 10000 4.72381 9.45201 5.90398 1.00056 4.72381 9.45201 5·90398 1.00056 12000 5.18927 9·73882 5-92971 1.00282 5.18927 9-73882 5.92971 1.00282 14000 5-71279 1.00291 E+01 5.96251 1.00894- 5-71250 1.00291 E+01 5.96251 1.00894 16000 6 .. 28552 1 .. 0318lt E+01 6.00838 1.02125 6.28290 1.03184 E+01 6.00838 1.02125 18000 6.92431 1 • 06054 E+01 6.07100 1.04166 6.91593 1.06054 E+01 6.07100 1.04166 20000 7.67130 1 • 08908 E+01 6.15214 1 .. 07146 7.65386 1 • 08908 E+01 6.15214 1 .. 07146 22000 8.57199 1.11760 E+01 6 .. 25219 1.11121 8.~5 1.11760 E+01 6.25219 1.11121 24000 9.68336 1 .. 14632 E+01 6.37064 1.16095 9 .. 6575.5 1.14632 E+01 6.37064 1.16095

26000 1.10575 E+01 1.17551 E+01 6 .. 50648 1.22033 1. 10228 E+01 1.17551 E+01 6 .. 50648 1.22033 28000 1.27390 E+01 1 .. 20547 E+01 6.65840 1.28875 1.26840 E+01 1.20547 E+01 6.6584o 1.28875

30000 1 .. 4762} E+01 1.23655 E+01 6.82502 1.36552 1 • 46803 E+01 1.23655 E+01 6.82502 1 .. 36552

32000 1.71507 E+01 1.26911 E+01 ? .. 00498 1.44990 1 • 7034.5 E+01 1.26911 E+01 7.00498 1 .. 1M990

34ooo 1. 99172 E+01 1.303.54 E+01 7-19699 1.54116 1 • 97.58! E+01 1.30354- E+01 7-19699 1 .. 54116

}6000 2.30650 E+01 1.34027 E+01 7·39992 1.63863 2. 28.568 E+01 1 .. 34027 E+01 7-39992 1 .. 6}863 38000 2.65~96 E+01 1.37968 E+01 7 .. 61276 1.74167 2.64688 E+01 1.37968 E+01 7.61276 1.74167 4ooOo 3. o4&o2 :a.o1 1.4-2220 :1+01 7.83468 1.84972 3.~2 E+01 1.42220 E+01 7-83468 1. 84972 42000 3.47209 E+01 1.4-6820 E+01 8.06504 1. 96229 3·47209 E+01 1.46820 E+01

8.06504

1.96229

"000

3.92922 E+01 1.51803 E+01 8.30332 2.07890 ,. 92922 E+01 1.518o} E+01 8.30332 2.07890

lt60oo 4-.41723 E+01 1.57203 E+01 8.54-918 2.19917 4. 41723 E+01 1.57203 E+01 8.54918 2.19917

48ooo 4.93379 E+01 1.63048 B+01 8.80243 2.32273 4.9»?9 E+01 1.6}o48 B+01 8 .. 80243 2.}2213

50000 5. 47650 :1+01 1.69363 E+01 9.06301 2.44927

s ..

At?';o

:z..o1

1.693,3 :1+01 9-06301 2.44,27

SUM ORR DfTERliAL S'.rAftS OJ' lU~ROGEII <t "0 P • 5 Jq:tcll" •

•

p = 10 kC/OiP 1'(01()

z.

zlf+ I++

•

•••

z

Jf z ••

z -...

z .... 1 i zi zi i i i i 500 4.00000 7-03139 4-.42o89 1.00000 4.00000 7-0.3139 4.42089 1.00000 1000 4-.00000 7-93463 5.1118't 1.00000 4.00000 7.93463 5.11184 1.00000 2000 4.00000 8 .. 44527 5.528o8 1.00000 4.00000 8.44527 5.528o8 1.00000 :5000 4.00099 8.62824 5 .. 67885 1.00000 4.00099 8.62824 5.6?885 1.00000 4000 4.01010 8.73707 5-75663 1.00000 4-.01010 8.73'70'7 5-75663 1 .. 00000 5000 4-.. o41o4 8.83353 5.804<>9 1.00000 4.o41o4 8.83353 5.80409 1.00000 6000 4.10540 8.93737 5.8,3608 1.00000 4.1054o 8.93737 5.83008 1.00000 7000 4 .. 20815 9-05291 5.85916 1.00000 4 .. 20815 9-05291 5.85916 1.00000 8000 4.34848 9.17892 . 5.87681 1.00005 4.34848 9.17892 5.87681 1.00005 9000 4.52220 9-31281 5 .. 89121 1.00019 4.52220 9.31281 5.89121 1.00019 10000 4.?2381 9.45201 5.90398 1.00056 4 .. 72381 9.45201 5-90398 1.00056 12000 5.18924 9.?3882 5·929?1 1.00282 5.18918 9.73882 5.92971 1.00282 14000 5 .. 71161 1 • 00291 E+01 5.96251 1.00894 5.7110? 1.00291 E+01 5.96251 1.008~ 1&000 6.2?820 1.031~ 1+01 6.00838 1.02125 6.2?575 1.03184 .1+01 6.00838 1.02125

111Deo 6.9()094 1 • 06054 E+01 6.07100 1.04166 6.89315 1.06054 E+01 6.0?100 1.04166

21000 ?.60716 1 • 08908 E+01 6.15214 1.07146 7.58752 1 • 08908 E+01 6.15214 1.07146

22000 8 •. 46449 1.11760 E+01 6.25219 1~11121 8.39303 1.11760 E+01 6.25219 1.11121

24ooo 9.50176 1.14632 E+01 6.37064 1.16095 9.36"+23 1.14632 E+01 6.37064 1.10095

26000 1.07470 E+01 1.17551 E+01 6.5()648 1.22033 1.0586o E+01 1.17551 E+01 6.50648 1.22033

28000 1.22933 E+01 1.2054? E+01 6.6534o 1.288?5 1.20401 E+01 1 • 20547 E+01 6.6584o 1.28875

30000 1.40984 E+01 1.23655 E+01 6.82502 1.36552 1.37216 E+01 1.23655 E+01 6.82502 1.36552

32000 1.62098 E+01 1.26911 E+01 7.0049~ 1.44990 1 • 56764 E+01 1.26911 E+01 7.00498 1.44990 3.1t000 1.86371 E+01 1.30354 E+01 7.19699 1.54116 1.79123 E+01 1.30354 E+01 7·19699 1.54116 36000 2.17675 E+01 1.34<>27 E+01 7-39991 1.63863 2.04303 E+01 1.34o2? E+01 7.39991 1.63863 38000 2.49322 E+01 1.37968 E+01 7.61275 1.?416? 2.32254 E+01 1.37968 E+01 7.61274 1.74167

4oooo 2.85669 E+01 1.42220 E+01 ?.83467 1.84972 2.62881 E+01 1.42220 E+01 7-83467 1.84972

42000 3.26090 E+01 1.46820 E+01 8.06504 1.96229 2.99457 E+01 1 • 46820 E+01 8.06500 1. 96229

~ 3.68943 E+01 1. 51803 E+01 8.30332 2 .. 07890 3.44736 E+01 1 .. 51803 E+01 8 .. 30325 2.07890 lt6000 4.16583 E+01 1.57203 E+01 8.54918 2.19917 3.848?3 E+01 1.57203 E+01 8.54907 2.19917 48000 4.64119 E+01 1 .. 6 3048 E+01 8.80243 2.32273 4 .. 28463 E+01 1.63048 E+01 8.8o229 . 2.32273 ~ 5.21033 E+01 1.69363 E+01 9-06301 2.44927 4 .. 79411 E+01 1.69363 E+01 9.06279 2.1t492?

SUM OVER INTERNAL STATES OF NITROGEN

• • _{'D • 20 k£/ca- »}

=

-3o

u;u~-!(OK) z N N+ z Jl++ ft+++ Z N z.N• Z N++ z I{+++ i

zi

i zi i l. i 1 500 4.00000 7-03139 4.42089 1.00000 4 .. 00000 7-03139 4.42089 1.00000 1000 4.00000 7-93463 5.11184 1.00000 4.00000 7-93463 5.11184 1.00000 2000 4.00000 8.44527 5.52808 1.00000 4.00000 8.44527 5.52808 1.00000 3000 4.00099 8 .. 62824 5.67885 1.00000 4.00099 8 .. 62824 5.67885 1.00000 4000 4.01010 8.73707 5·75663 1.00000 4.01010 8.73707 5.75663 1.00000 5000 4.04104 8.83353 5.80409 1.00000 4.04104 8.83353 5.80409 1.00000 6ooo

I

4 .. 1o54o 8 .. 93737 5-83608 1.00000 4 .. 10540 8.93737 5.83608 1 .. 00000 1000

l

4.20815 9-05291 5·85916 1.00000 4.20815 9-05291 5.85916 1.00000

8oOO

4.34848 9-17892 5.87681 1.00005 4.34848 9 .. 17892 5-87681 1.00005 9000 4.52220 9·31281 5-89121 1.00019 4.52220 9.31281 5 .. 89121 1 .. 00019 10000 4 .. 72381 1 9.45201 5.90398 1.00056 4.72381 9.45201 5-90398 1.00056 12000 5-18910 1 9.73882 5-929?1 1 .. 00282 15-18908 9-73882 5-92971 1 .. 00282

14ooO 5 .. 71051 1.00291 E+01 5 .. 96251 1.00894 1 5 .. 71051 1 .. 00291 E+01 5.96251 1 .. 00894

16000 6.27336 1 .. 03184 E+01 6.00838 1.02125 I 6.27317 1.03184 E+01 6.00838 1 .. 02125 18000 6.88501 1.06054 E+01 6 .. 07100 1 .. 04166 6 .. 87482 1.06054 E+01 6 .. 07100 1 .. 04166 20000 7.56711 I _{1 • 08908 E+01 6.15214 1.0?146 7 .. 53846} 1.08908 E+01 6.15214 1.07146 I 22000 i ~.34974 1.11760 E+01 1 6.25219 1.11121 8 .. 29156 1.11760 E+01 6.25219 1.11121 2-\000 9.26547 1.14632 E+01 6.37064 1.16095 9.22947 1.14632 E+01 6 .. 37064 1.16095

26000 1.03549 E+01 1.17551 E+01 6.50648 1.22033 1.03445 E+01 1 .. 17551 E+01 6 .. 5o648 1~22033 280oo 1.17804 E+01 1.20547 E+01 6.65840 1. 28875 1.16120 E+01 1.20547 E+01 6.6584o 1.28875 .30000 1.34080 E+01 1.23655 E+01 6.82502 1.36552 1 • 30862 E+01 1.23655 E+01 6.82502 1. 36552

32000 1.52329 E+01 1.26911 E+01 7.00498 1.44990 1.4?787 E+01 1 .. 26911 E+01 7.00498 1 .. 44990 34000 1.73104 E+01 1.30354 E+01 7.19699 1. 54116 1.66946 E+01 1.30354 E+01 7.19699 1 .. 54116

36000 1.96405 E+01 1.34027 E+01 7.39990 1 .. 6;863 1.88336 E+01 1.34027 E+01 7·39990 1 .. 63863 38000 2.22183 E+01 1 • 3?968 E+01 7.61272 1.?4167 2.11905 E+01 1 .. 3?968 E+01 7.61272 1 .. 74167

40000 2.50348 E+01 1.42220 E+01 7.~3462 1.84972 2.37892 E+01 1.42220 E+01 7 .. 83462 1.84972 42000 2. 80779 E+01 1.46820 E+01 8.o6491 1.96229 2.65611 E+01 1.46820 E+01 8 .. 06491 1 .. 96229 4k100 3.13334 E+01 1.51803 E+01 8.30309 2 .. 0?890 2.96125 E+01 1.518o3 E+01 8 .. 30309 2.0?890 46000 3-47858 E+01 1.57203 E+01 8.54884 2 .. 19917 3.363o8 E+01 1.57203 E+01 8.5488o 2.19917 ltiOoo ;.84187 E+01 1.63048 E+01 8.00205 2.32273 3•78474 E+01 1 .. 6}048 E+01 8.80180 2.32.273

50000 4.28641 E+01 1.69363 E+01 9 .. 06241 2.44927 4.22155 E+01 1.b9363 E+01 9.06202 2.lf4927

I

"

I

Pa.rti.cle concentrati.ona/ca of Ni.t.rogen at 1 kg/em • ~

,

T(°K) . . total. BN liN

J(N + BN

~·

•a++

liN+++

e 2 2

1000 7. 1o600 E+ 1 8 7.1o600 E+18

2000 3-55300 E+18 3.55300 E+18

3000 2.36866 E+18 2.36863 E+18 3-30062 E+13

4000 1.77650 E+18 1.07110 E+10 1.?7333 E+18 1.03445 E+10 3.16530 E+15

5000 1.42120 E+18 1.18189 E+12 1.3?453 E+18 8.63390 E+11 4.66654 E+16 3. 18508 E+ 11 6000 1.18433 E+18 3.34629 E+13 9.32443 E+17 1.13538 E+13 2.51889 E+17 2.21091 E+13 7000 1.01514 E+18 3-77457 E+14 3.88113 E+17 3.94197 E+13 6.26274 E+17 3-3803? E+14 8000 8.88250 E+17 2 .. 02244 E+15 8.26547 E+16 4.87631 E+13 8.01550 E+17 1.9?368 E+15 9000 ?.89555 E+17 6.80390 E+15 1.49904 E+16 3.90101 E+13 7.60957 E+17 6.?6489 E+15 10000 7.10600 E+17 1.73983 E+16 3.20257 E+15 2.87155 E+13 6.72600 E+17 1.73696 E+16 12000 5.92166 E+17 6+52934 E+16 2.10247 E+14 1.35944 E+13 4.61369 E+17 6.52798 E+16

14000 5.07571 E+1? 1.40844 E+17 1.22295 E+13 3.94866 E+12 2.25869 E+17 1.40840 E+17 1.21295 E+11 16000 4.44125 E+1? 1.84367 E+17 4.?4786 E+11 6.9641? E+11 7.53921 E+16 1.84361 E+17 3-0912? E+12 18000 3.94777 E+17 1. 86556 E+17 1.74975 E+10 1.00226 E+11 2.17034 E+16 1.86479 E+1? 3-85356 E+13 20000 3.55300 E+17 1.?4167 E+1? 1.839Bo E+10 ?.23115 E+15 1. 73636 E+17 2.65243 E+14

22000 3.23000 E+17 1.60880 E+17 2.66343 E+15 1.58032 E+17 1.42395 E+15

24000 2.96083 E+1? 1.50274 E+17 1.11665 E+15 1.39108 E+17 5.58313 E+15 1.32616 E+10

26000 2.73307 E+17 1.44350 E+17 4.98947 E+14 1.12566 E+1? 1.58912 E+16 2.66270 E+11

28o00 2.53785 E+17 1.42853 E+17 2.17890 E+14 7.85768 E+16 3.21:;42 E+16 2.84749 '1+12

JOOOO 2.36866 E+17 1.42364 E+17 8.75356 E+13 4.64838 E+1o 4.79129 E+16 1.8115? E+13

}2000 2.22062 E+17 1-39918 E+17 3.26210 E+13 2.1.t:;841 E+16 5-76477 E+16 7. 958o8 :1+13

34000

2.09000 E+1? 1.35364 E+17 1.19061 E+13 1 .. 21539 :&+16 6.11970 B+16 2.?2208 :&+1.1t

}6000 _{1.97388 E+17} 1.29831 E+17 4.1.t62?2 E+12 6.05430 E+15 6 .. 07186 E+16 7-79991 &+1At

}8ooo 1.87000 E+17 1.24287 E+1? 1.75030 1+12 3.07457 E+15 5-76962 E+16 1.9't007 E+15

~ 1.?7650 E+17 1.19314 E+17 7.14405 E+11 1.58533 E+15 5.25188 E+16 4.2}054 E+15

42000 1 .. 69190 E+17 1.15202 E+17 2.97236 E+11 8.14405 E+1'+ 4.51327 E+16 8.040?8 '1+15

lt4000 1.61500 E+17 1.11931 E+17 1.22739 E+11 4.070?6 E+14 3·59597 Z+16 1.32016 E+16

ltOOOO 1.54478 1+17 1.091?4 E+1? 4.92634 E+10 1.94894 E+14 2.6}450 E+16 1.8?63} &+16

4lSOOO 1.48o41 E+1? 1.06509 E+17 1.91281 E+10 8.94637 E+13 1.79068 :1+16 2.35355 &+16

Particle concentrationa/oa of Nitrogen at 2 kg/em •

.

,

T(°K) ~ NN NN •• + NN NN+

••••

NN+++ totaa.l

•

2 2 1000 1.42120 E+19 1.42119 E+19 2000 7 .. 10600 E+ 18 7.10599 E+81

3000 4.73733 E+18 4.73728 E+18 4.66779 E+13

4oOO

3·55300 E+18 1.50755 E+10 3-54852 E+18 1.47071 E+10 4.47758 E+15

5000 2.84240 E+18 1.61118 E+12 2.77608 E+18 1.27914 E+12 6.63182 E+16 3.32042 E+11

6ooo 2.36866 E+18 4.35694 E+13 1.99978 E+18 1.87019 E+13 3.68884 E+17 2 .. 48675 E+13

I

7000 2.03028 E+18 4.95372 E+14 1 .. 01600 E+18 7.86297 E+13 1.01328 E+18 4.16743 E+14 8ooo 1.77650 E+18 2.78253 E+15 2.84317 E+17 1 .. 21917 E+14 1.48661 E+18 2.66061 E+15 9000 1.57911 E+18 9.58428 E+15 5.83695 E+16 1.07832 E+14 1.50157 E+18 9.47645 E+15

_I

10000 1.42120 E+18 2 .. 4?483 E+16 1 .. 30674 E+16 8.23?02 E+13 1.35863 E+18 2.46659 E+16 12000 1.18433 E+1(5 I 9.5?467 E+16 9.71719 E+14 4.28466 E+13 9.91868 E+1? 9.57038 E+16

1'+000 1.01514 E+18 I

2 .. 25052 E+17 7.65204 E+13 1.54622 E+13 ,5.64960 E+17 2.25037 E+1? 1.26356 E+11 16000 8.8B250 E+17

I

3.27833 E+17 4.52244 E+12

I

3.73058 E+12 2.32585 E+17 3 .. 27827 E+17 3 .. 24455 E+12

18ooO. _{7.89555 E+17 3.56191 E+17} 2 .. 21999 E+11 6 .. 66014 E+11 7.72132 E+16 3.56110 E+17 4.03852 E+13

20000 7.1o600 ~+17 3. 4241•3 E+17 1.31929 E+10 1.21692 E+11 2.60182 E+16 3.41831 E+17 3.05960 E+11t 22000 6 .. 46000 E+17 _I 3.18853 E:+17 2 .. 63037 E+10 9.90938 E+15 3.15621 E+17 1.61566 :&+15

24o0o

l

5.92166 E+17 2 .. 97137 E+17 4 .. 27905 E+15 2.84362 E+17 6.38752 E+15

26000 5 .. 46615 E+17 2.81925 E+17 2 .. 00883 E+15 2.43437 E+17 1.92439 E+16 1. 88823 E+"

28000 5.07571 E+17 ·2.75012 E+17 9.59916 E+14 1.88188 E+17 4.34085 E+16 2. 26079 E+'

30000 4.73733 E+17 2 .. 73360 E+17 4.37673 E+14 1.26526 E+17 7.33928 E+16 1. 62333 E+'

I

32000 4.44125 E+17 2.71076 E+17 1.85513 E+14 7 .. 47288 E+16 9.80567 E+16 7. 79083 E+'

}4000 4.18ooo E+17 2.65151 E+17 7.48816 E+13 4.06779 E+16 1.11815 E+17 2 .. 80?76 E+'

36000 3 .. 94777 E+1? 2.56274 E+17 3 .. 00645 E+13 2.15002 E+16 1 .. 16144 E+17 8.28655 E+'

38000 3-74000 E+17 2.46226 E+17 1.24515 E+13 1.14o53 E+16 1.14247 E+17 2.10878 E+'

40000 3.55300 E+17 2.36175 E+17 5.76930 E+12 6.'+6785 E+15 1 .. 08246 E+17 4.4o5o6 &+'

42000 3.38380 E+17 2.27381 E+1? 2.52940 E+12 3.51125 E+15 9.85868 E+16 8.89881 ;;...·

"<>oo 3.23000 E+17 2.20004 E+17 1.12331 E+12 1.89546 E+15 8 .. 51875 E+16 1 .. 59113 lt+'

46000 3 .. 08956 E+17 2.13985 E+17 4.95141 E+11 9·99402 E+14 6.89253 E+16 2 .. 50453 E+'

48000 2.96083 E+1? 2.08824 E+17 2.13252 E+11 5 .. 08717 E+14 5.19348 E+16 3.48153 E+'

!50000 2.84240 E+17 2.03879 E+1? 8.93859 E+10 2.50114 E+14 3.66998 E+16 4.34100 E+'

2

Particle concentrations/em . ot Nitr ;~:tan at C) k~:t/em _•

_•

T(°K) NN _total NN _e NN ₂ NN + ₂ NN l'fN+

....

NN+++

1000 3-55300 E+19 3.55299 E+19

2000 1.77650 E+19 1.77649 E+19

1.18433 E+19 1.18432 E+19

.

7.38o44 E+13

3000

4ooo 8.88250 E+18 2.37347 E+10 8.<>7541 E+18 2 .. 33646 E+10 7 .. 08131 E+15

5000 7.10600 E+18 2.45902 E+12 7.00068 E+18 2 .. 11353 E+12 1.05314 E+17 3.45486 E+11 6000 5.92166 E+18 6.27285 E+13 5.32000 E+18 3.45567 E+13 6.01664 E+17 2.81718 E+13 7000 5.07571 E+18 7.03362 E+14 3.2594o E+18 1.77656 E+14 1.81490 E+18 5.25705 E+14 8ooo 4.44125 E+18 4.15102 E-t15 1 .. 27939 E+18 3.67749 E+14 3.15355 E+18 3.78328 E+15

9000 3.94777 E+18 1.49247 E+16 3.32747 E+17 3.94758 E+14 3 .. 58518 E+18 1.45299 E+16

10000 3.55300 E+1~ 3.92077 E+16 8.15028 E+16 3.24285 E+14 3·39308 E+18 3 .. 88~34 E+16

I

12000 2.96083 E+18 1 .. 64257 E+17 6 .. 80871 E+1? 1.75001 E+14 2.62551 E+18 1.64o82 E+17

14000 2.53785 E+18 4 .. 00981 E+17 7 .. 22041 E+14 8.18875 E+13 1.73517 E+18 4.00899 E+17 1.34-895 E+11 160001 2.22062 E+18 6.59148 E+17 6.81696 E+13 2.79682 E+13 9.02331 E+17 6 .. 59141 E+1? 3.51775 E+12 18ooo 1.97388 E+18 8.00437 E+17 5.20480 E+12 6.94850 E+12 3·73057 E+17 8.00349 E+17 4.39440 E+13 20000

l

1 .. 77650 E+18 8 .. 17454 E+17 3.97381 E+11 1.53551 E+12 1.41922 E+17 8.16792 E+17 3-30804 E+14 22ooo

I

1.61500 E+18 7-79824 E+17 3.75591 E+10 3.64139 E+11 5 .. ?0869 E+16 7 .. 76351 E+17 1.73690 E+15 24oOO' 1 .. 48041 E+18 7.30978 E+17 9 .. 82341 E+10 2.53890 E+16 7 .. 17119 E+1? 6.92968 E+15

26000 ·1 .. 36653 E+18 6.88040 E+17 2.93384 E+10 1 .. 23504 E+16 6.44254 E+17 2.18926 E+16 9.45797 .1+1 28000 1 .. 26892 E+18 6 .. 58804 E+17 6.36661 E+15 5.48712 E+17 5 .. 50440 E+16 1 .. 27254 E+1

30000 I 1.1ts433 E+18 6 .. 44999 E+17 3.2t16C$2 E+15 4.27104 E+17 1.08931 E+17 1.07904 E+1

I

}2000 1.11031 E+18 6. 3999·8 E+17 1.63007 E+15 2.9?429 E+17 1.71193 E+17 6.07166 E+1

34ooo 1.04500 E+18 6 .. 33795 E+17 7.64402 E+14 1 .. 87329 E+17 2.22863 E+17 2 .. 46355 E+1

360001

9.86944 E+17 6.21185 E+17 3.50670 E+14 1.10416 E+17 2.54206 E+17 7.85626 E+1

:;aooo,

9-35000 E+17 6.02856 E+17 1.56771 E+14 6.32212 E+16 2.66661 E+17 2.10438 E+'

4oooo 8.88250 E+17 5.81737 E+17 7 .. 14125 E+13 3.60832 E+16 2.65418 E+17 4.93944 E+'

42000 8.45952 E+17 5.60492 E+17 3.33651 E+13 2.07512 E+16 2 .. 54285 E+17 1. 03901 E+'

44ooo 8.07500 E+17 5.40903 E+17 1.58782 E+13 1.20053 E+16 2.34828 E+17 1.97468 E+'

46000 7.72391 E+17 5 .. 23911 E+17 7.66990 E+12 6.92015 E+15 2.07663 E+17 3.38884 E+•

48000 7.40208 E+17 5 .. 09614 E+17 3.66660 E+12 3.92543 E+15 1.?4303 E+17 5.2.}606 :g..•

Particle concentrations/em~ of Nitrog&n at 10 kao1_{cm •} I

NN NN NN N NN lflf+ lflf++ NN+++

T(°K) total e 2 N2+

1000 7.10600 E+19 7.10600 E+19

2000 3.55300 E+19 3-55299 E+19 1.00633 E+10

3000 2.36866 E+19 2.36865 E+19 1.04375 E+14

4000 1. 77650 E+19 3.34931 E+10 1.77549 E+19 3.31222 E+10 1. 00156 E+16

5000 1.42120 E+19 3.41234 E+12 1.40627 E+19 3.05948 E+12 1.49262 E+17 3.52861 E+11

6000 1.18433 E+19 8.37387 E+13 1.09790 E+19 5.34222 E+13 8.64331 E+17 3.03165 E+13 ?000 1.01514 E+19 9.17497 E+14 7.41261 E+18 3.09734 E+14 ·2 .. 73697 E+18 6.07762 E+14

8000 8.88250 E+18 5.53056 E+15 3.58935 E+18 7.74370 E+14 5.28208 E+18 4.75619 E+15

9000 7.89555 E+18 2.06264 E+16 1.16009 E+18

l

_{9.95839 E+14} 6.69421 E+18 1.96306 E+16 10000 7.10600 E+18 5.52312 E+16 I 3.15860 E+17 8.92147 E+14 6.6?967 E+18 5.43391 E+16

12000 5.92166 E+18 2.38594 E+1? 2.89686 E+16 5.12585 E+14 5.41550 E+18 2.38082 E+17

14000 5.07571 E+18 6 .. 07073 E+17 3.57011 E+15 2.67436 E+14 3 .. 85799 E+18 6.0680b E+17 1.43230 E+11 16000

I

4 .. 44125 E+18 1 .. 07198 E+18 4.42211 E+14 1.11557 E+14 2.29729 E+18 1 .. 07197 E+18 3 .. 79356 E+12

18000 3.94777 E+18 1 .. 41524 E+18 4.67945 E+13 3·53327 E+13 .1 .. 11732 E+18 1.41515 E+18 4.7?732 E+13 20000 ,3 .. 55300 E+18 1 .. 53648 E+18 4.57665 E+12 9.40873 E+12 4.80396 E+17 1. 53576 E+18. 3.588fl7 E+14 22000 3.23000 E+18 1.51299 E+18 4.96873 E+11 2.48288 E+12 2.05882 E+17 1 .. 50925 E+18 1.87152 E+15 24ooO 2.96083 E+18 1.43688 E+18 6.61571 E+10 7.12875 E+11 9 .. 45038 E+1b 1 .. 42199 E+18 '?.44532 E+15

26000 2.73307 E+18 1 .. 35479 E+18 I 1.0<S284 E+10 2 .. 25296 E+11 4.73061 E+16 1.30?16 E+1~ 2.3CS122 E+16 5.60598 E+1

28000 2 .. 53785 E+18 1.28756 E+18 i _{7.56037 E+10 2.52292 E+16} 1.16256 E+18 6.249?9 E+16 7 .. 92428 E+1 •30000 2 .. 36866 E+18 1.24457 E+18 2.51521 E+10 1.38238 E+16 9.75957 E+17 1.34299 E+17 7.32366 E+1

32000 2.22062 E+18 1.22422 E+18 7.50148 E+15 7.53608 E+1? 2.35240 E+17 4 .. 6o86o E+1

340o0 2 .. 09000 E+18 1 .. 21392 E+18 3.91645 E+15 5 ... 30610 E+17 3.41343 E+17 2.07639 E+1

36000 1 1.97388 E+18 1.19970 E+18 1.95536 E+1Cj 3 .. 45471 E+17 4 .. 26040 E+17 7 .. 17359 E+1

38000 1.8?000 E+18 1.17544 E+18 9.1t7391 E+14 2.13794 E+17 4 .. 7778o E+17 2.03044 E+1

40000 1.77650 E+18 1.14~58 E+18 '+ .. 55395 E+14 1.29276 E+17 4.99226 E+17 4.9528? E+1

42000 1.69190 E+18 1.10542 E+18 2 .. 23476 E+14 7.78431 E+16 4 .. 97650 E+17 1.07609 E+1

44000 1.61500 E+18 1. 06797 E+18 1.1.3326 E+14 4.70601 E+16 4. 78646 E+17 2.12o65 E+1

46000 1.54478 E+18 1.03303 E+18 5.69278 E+13 2.85386 E+16 4.44964 E+17 3.81885 E+1

48ooo 1.48o41 E+18 1.00214 E+18 2.89121 E+13 1-72418 E+16 3-98115•E+17 6.28893 E+1

50000 1 .. 42120 E+18 9.75561 E+17 1.48116 E+13 1.02846 E+16 3 .. 40740 E+17 9.45983 E+1

..

' _{Particle concentratione/cm of •Nitro_g_e:n at 20 U/ca •} • I T(°K) N _Ntotal NH HN N HN NN+ liN++ NN+++ e 2 N2+ 1000 1.42120 E+20 1 .. 42119 E+20 2000 7.10600 E+19 7 .. 10599 E+19 ~ 1.42316 E+10

3000 4.73733 E+19 4 .. 73731 E+19 1.47609 E+14

4ooo 3.55300 E+19 4 .. 72934 E+10 3-55158 E+19 4.69219 E+10 1.41654 E+16

5000 2.84240 E+19 4.?5922 E+12 2 .. 82125 E+19 4.40087 E+12 2 .. 11416 E+17 3.58350 E+11 6000 2.36866 E+19 1 .. 13040 E+14 2.245o6 E+19 8.09250 E+13 1.23598 E+18 3.21149 E+13

7000 2 .. 03028 E+19 1.20353 E+15 1 .. 62482 E+19 5 .. 17573 E+14 4.05218 E+18

I

6. 85959 E+14

Booo

1 .. 77650 E+19 _{7.30163 E+15} 9.26436 E+18 _{1.51389 E+15} 8 .. 48603 E+18 _{5.78773 E+15}

9000 1.57911 E+19 2 .. 81452 E+16 3-73050 E+18 2.34684 E+15 1.20043 E+19 2.57983 E+16 10000 1.42120 E+19 7.72754 E+16 1.17487 E+18 2.37178 E+15 1.28825 E+19 7.49036 E+16 12000 1.18433 E+19 3.45316 E+17 1.20228 E+17 1.46990 E+15 1.10324 E+19 3· 43846 E+17

14000 L01514 E+19 9.07811 E+17 1.66037 E+16 8 .. 31746 E+14 8.31920 E+18 9.06980 E+17 1 .. 53783 E+11 16000 8.88250 E+18 1.69545 E+18 2.52886 E+15 4.03362 E+14 5 .. 49159 E+18 1 .. 69544 E+18 4.15099 E+12 18000 I 7. 89555 E+18 2 .. 41028 E+18 3.55273 E+14 1.57510 E+14 3 .. 07503 E+1B 2.41017 E+18 5.294o9 E+13

20000

I

7.10600 E+18 2.79877 E+18 4 .. 53917 E+13 5.12293 E+13 1 .. 508B4 E+18 2.79798 E+18 3.98572 E+14

i

l

22000

I

6.46000 E+18 2 .. 87782 E+18 5.91051 E+12 1.55278 E+13 7.06421 E+17 2.87369 E+18 2 .. 06574 E+15

I

24000 5.92166 E+18 2 .. 79478 E+18 8 .. 76006 E+11 4 .. 85309 E+12 3 .. 40259 E+17 2.77846 E+18 8 .. 15827 E+15 26000 ,5 .. 46615 E+18 2 .. 65893 E+1o

I

1.53768 E+11 1.63012 E+12 1.74375 E+17 2.60674 E+18 2 .. 60996 E+16 3 .. 54509 E+' 28000 5-07571 E+18 2.52471 E+18 3.14157 E+10 5 .. 84396 E+11 9.61034 E+16 2.38507 E+18 6.98165 E+16 4.98063 E+' 30000

I

4. 73733 E+18 2.41996 E+1b 2.14890 E+11 5.50555 E+16 2 .. 10465 E+18 1.57646 E+17 4 .. 81418 E+' 32000 4.44125 E+18 2.35452 E+18 7.71256 E+10 3.18422 E+16 1.75526 E+18 2 .. 99578 E+17 3 .. 29072 E+' 34000 4.18000 E+1t'S 2.32095 E+18 2.57570 E+10 1.81349 E+16 1.36101 E+18 4 .. 79727 E+17 1 .. 63653 E+'

36000 3.94777 E+18 2 .. 29896 E+18 9.98632 E+15 9·79304 E+17 6.58902 E+17 6.18922 E+'

38000 3.74000 E+18 2.26954 E+18 5.30828 E+15 6.62632 E+17 8.00636 E+17 1.87960 E+'

400oo 3.55300 E+18 2.22489 E+18 2.75567 E+15 4.30624 E+17 8.89901 E+17 4.82298 E+'

42000 3.38380 E+18 2.16711 E+18 1.41968 E+15 2.74319 E+17 9.30082 E+17 1.08761 E+'

.ltJt.ooo 3.23000 E+18 2.10230 E+18 ?.35139 E+14 1.73757 E+17 9·31072 E+17 2 .. 21331 E+'

46000 3.08956 E+18 2 .. 03649 E+18 3.84~89 E+14 1.10165 E+17 9.01245 E+17 4.12779 E+·

48000 2.96083 E+18 1 .. 97413 E+18 2.03871 E+14 6.99156 E+16 8 .. 45518 E+17 7 .. 10607 E+

50000 2 .. 8424o E+18 1.91783 E+18 1.10318 E+14 4 .. 42169 E+16 7.67085 E+17 1.13150 E+·