Survey of Methods of Calculating High-Temperature Thermodynamic

Survey of Methods of Calculating High-Temperature Thermodynamic Properties of Air Species. M. Capitelli, G. Colonna, C. Gorse Centro di Studio per la Chimica dei Plasmi del CNR Dip. Chimica, Universit´a di Bari Via Orabona 4, 70126 Bari, Italy D. Giordano European Space Research & Technology Center P.O. Box 299, 2200 AG Noordwijk, The Netherlands December 12, 1993 Revised December 13, 1994

Abstract

The purpose of this report is to survey critically the efforts made in the past years towards the understanding of the high temperature thermodynamic properties of air species and to describe the basic methods for their calculation. The report also contains the theoretical background and the required input data upon which a new calculation of the thermodynamic properties for high temperature air species, namely + − + N, N+ , N2+ , N3+ , N4+ , O, O+ , O− , O2+ , O3+ , O4+ , N2 , N+ 2 , O2 , O2 , O2 , NO, NO , has been carried out in the temperature range from 50 K to 100000 K. The results from this new calculation are presented in the form of tables in an associated report (ESA STR-237).

iii

Contents 1 INTRODUCTION

1

2 THE PARTITION FUNCTION

2

3 THE INTERNAL PARTITION FUNCTION OF ATOMIC SPECIES

3

3.1

Divergence of the internal partition function

. . . . . . . . . . . . . . . . . . . .

3

3.2

Cut-off criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

3.3

Energy levels, degeneracies and internal partition functions . . . . . . . . . . . .

5

3.4

Thermodynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

4 THE INTERNAL PARTITION FUNCTION OF MOLECULAR SPECIES

9

4.1

Diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.2

Thermodynamic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

5 CONCLUSIONS

15

6 REFERENCES AND NOTES

17

7 TABLES

19

8 FIGURES

28

iv

1

INTRODUCTION

In the last three decades, many papers have been published on the thermodynamic properties of high temperature gases, with particular reference to air species because of their importance in the aerothermodynamic analysis of the hypersonic flow surrounding a space vehicle during its reentry into the Earth’s atmosphere. Recently, new data relating to the thermodynamic properties of some air molecular species have been presented. 1–3 These data have been obtained from sophisticated approaches to the calculation of the rotovibrational energies of the diatomic molecules. In general, these new data are in satisfactory agreement with those provided by older methods based on analytical expressions of the rotovibrational energy. However, at sufficiently high temperatures, an unexpected effect is found: the internal constant pressure specific heat of the diatomic molecules vanishes 1 instead of converging to the expected value of two times the universal gas constant given in standard textbooks. This peculiarity, which is due to the use of a finite number of rotovibrational energy levels in the calculation of the internal partition function and its derivatives, was already found 4 many years ago for H2 . On the other hand, the new methods are important in that they make it easier to understand the problem of separation of the different contributions (electronic, rotational, vibrational) to the internal partition function, a problem which has to be solved if one is to construct thermodynamic models in thermal non-equilibrium. In a situation of thermal disequilibrium, different temperatures characterise the independent degrees of freedom of the molecule and it becomes necessary to separate the associated contributions to the basic thermodynamic properties. This problem was dealt with in the past by introducing simple models for the different energy modes of a molecule but it becomes more complicated when a rigorous quantum-mechanical approach is followed; strictly speaking, a rigorous quantum-mechanical approach works against the separation of the electronic, rotational and vibrational contributions. The separation between translational and electronic degrees of freedom raises no problem for the atomic species and one may believe that the calculation of the thermodynamic properties is straightforward. This impression is, however, illusory. In fact, the problems 5, 6 met many years ago with respect to the calculation of the electronic partition functions of monoatomic species have not yet been solved: the electronic partition function of an isolated monoatomic species is not bounded and a suitable cut-off criterion must be used to truncate it. Unfortunately, there is no universal cut-off criterion; the existing cut-off criteria yield partition functions, and their derivatives, that depend on either the electron density or the gas pressure. This means that the thermodynamic properties of single species depend not only on the temperature but also on the pressure. Additional problems arise from the determination of the electronic energy levels of the atomic species: Moore’s well-known tables 7 help to some extent because they provide energies and statistical weights for many observed levels. Nonetheless, supplementing these tables with semiempirical laws is still the only way to obtain the missing energy levels. Exact quantummechanical calculations for high-lying levels, i.e. near the continuum limit, are still a prohibitive task. The purpose of this report is to survey critically the efforts made in the past years towards understanding the high-temperature thermodynamic properties of air species and to describe the basic methods for their calculation. A subsequent report 8 will present tables of thermodynamic properties for high-temperature air species, namely N, N+ , N2+ , N3+ , N4+ , O, O+ , O− , O2+ , + − + O3+ , O4+ , N2 , N+ 2 , O2 , O2 , O2 , NO, NO , in the temperature range from 50 K to 100000 K. 1

2

THE PARTITION FUNCTION

The partition function Q is the basic construct 9 of statistical thermodynamics required to calculate the thermodynamic properties of single species. In general, the partition function can be factorised in the form of a product of the translational Qtr and the internal Qint contributions Q = Qtr Qint The translational partition function Qtr is given in analytical form as


Qtr =

2πmkT h2

3/2

V

where m is the mass of the molecule, k is the Boltzmann constant, T is the temperature, h is the Planck constant, and V is the volume. The internal partition function can be defined at this stage as Qint =

 n

gn exp(−

En ) kT

where gn and En represent the statistical weight and the energy of the nth internal quantum level of the molecule under consideration, the sum running over all the accessible levels. The calculation of Qint is straightforward when the characteristics gn and En are known; these can be determined, in line of principle, by quantum mechanical methods. Despite the apparent simplicity, however, the existing values of Qint can differ by orders of magnitude, especially for the atomic species.

2

3

THE INTERNAL PARTITION FUNCTION OF ATOMIC SPECIES

In the case of atomic species, the internal partition function is the sum over the infinite electronic levels Qint =

∞  n

gn exp(−

En ) kT

(1)

In principle, the electronic energies En and the corresponding statistical weights gn can be calculated by quantum mechanics. 3.1

Divergence of the internal partition function

For the sake of clarity, let us consider the simple system of the hydrogen-like atom, i.e. a system composed of one electron in the field of a nucleus of atomic charge Z. In this case, the Schr¨ odinger equation Hψ = Eψ (where H is the Hamiltonian operator, ψ the wave function and E the total energy) can be solved exactly, yielding at the same time the eigenfunctions ψn and the eigenvalues En of the hydrogen-like atom in analytical form. It turns out that the energies depend only on the principal quantum number n En = −

1 Z 2 e2 1 2 a0 n2

(2)

where e represents the electronic charge, and a0 is the Bohr radius. If all the energies are referred to that of the ground state (n = 1), then (2) can be rearranged to the form


En − E1 =

1 Z 2 e2 1 1− 2 2 a0 n



(3)

The expression (3) indicates that the last bound level (n = ∞) has an energy referred to the ground state equal to E∞ − E1 =

1 Z 2 e2 = Ei 2 a0

and such a quantity Ei is, by definition, the ionisation potential of the hydrogen-like atom. The statistical weights can be calculated by the degeneracy of the states, i.e. by the number of eigenfunctions describing all the sublevels corresponding to a given principal quantum number n. In the case of the hydrogen-like atom, it turns out that gn is given by gn = 2n2

(4) 3

The factor two on the right-hand side of (4) arises from the spin wave functions. We are now in a position to calculate the internal partition function of an hydrogen-like atom from (1), (3) and (4). It is easily recognised that, when (3) and (4) are substituted into (1), the corresponding internal partition function diverges; in fact, with increasing n the exponential term in (1) converges to exp(−Ei /kT ) but the quadratic term 2n2 diverges. This is clear evidence of the necessity to introduce a cut-off criterion in order to eliminate the divergence. 3.2

Cut-off criteria

The cut-off criterion must be the consequence of physical arguments. In the case of the hydrogenlike atom, a very simple criterion is based on the idea that a level is considered as bound when the mean distance of the electron from the nucleus is less than the average distance between atoms: only in such a case can the electron be considered as belonging to the atom. Keeping in mind that the average distance for an electron in the hydrogen atom is of the order of a0 n2 and that the average distance between two atoms is of the order of (1/N )1/3 , N being the particle number density (p = kN T ), we obtain a0 n2 ≤ (N )−1/3 a restriction which gives us the maximum permissible principal quantum number nmax . It should be stressed that this cut-off criterion is dependent on pressure because of the relation existing between N and p. Similar equations have been developed to account for the presence of electrons and ions around an atom; in this case the cut-off criterion has been derived by equalising the classical semimajor axis of the electron in the considered atom to the Debye shielding distance. The corresponding principal quantum number at which the partition function is to be truncated becomes  2 nmax = 36.11 · 103 Zef fT/

 j

1/4

nj Zj2 

where Zef f is the effective charge of the atom, i.e. the net charge seen by an excited electron, and nj is the particle number density of the j th species with charge Zj . This cut-off criterion was developed by Margenau and Lewis 10 ; it was used by Drellishak 11 and by Drellishak et al. 12–13 to calculate the internal partition functions of Ar, N and O plasmas. It should be noticed that in this case the partition function depends not only on the temperature but also on the number densities of the ionised species. Another class of cut-off criteria is based on the fact that the ionisation energy of an atom in the presence of other species is lowered by a factor which, in general, depends on the number densities of the charged particles, including the electrons. Different theories have been proposed to calculate this lowering; the most popular one is that proposed by Griem 14 , which yields the expression  1/2  ∆Ei = 2(Zj + 1)e3 (π/kT )1/2  nj Zi2 

(5)

j

4

All the levels whose energies are lower than the corrected ionisation potential are accounted for in the sum (1), i.e. we sum up to the last energy level which arises from the limitation En ≤ Ei − ∆Ei

(6)

The use of the criteria proposed by Margenau and Lewis and by Griem require that the number densities of the charged particles be known. These variables are easily obtained in the case of chemical equilibrium: we can write a set of Saha equations for the different ionisation equilibria together with the relevant conservation laws (mass and charge). An iterative procedure is usually performed because of the interdependence between equilibrium composition and partition functions 5, 6. As an example, the internal partition function of H versus temperature for various pressure values is shown in Fig. 1. The data have been obtained from (1), (3), (4), (5) and (6) in combination with the relevant chemical equilibrium equations 15 . Inspection of Fig. 1 shows that the increase of the gas pressure, and, therefore, of the densities of charged particles, yields higher ionisation potential lowerings and decreases dramatically the internal partition function. 3.3

Energy levels, degeneracies and internal partition functions

A major concern associated with the atomic species is the determination of their internal energy levels and corresponding degeneracies, because the Schr¨ odinger equation cannot be exactly solved for atoms with many electrons 9 . Moreover, the available approximate quantum mechanical methods become more and more difficult to use in the case of open shell systems, particularly for high-lying electronic states. To our knowledge, no quantum mechanical data exist for the energies and the corresponding wave functions of high-lying levels of atomic air species. Consequently, the calculation of the internal partition function necessarily requires the use of semiempirical sets of energy levels. We refer in particular to the semiempirical sets of energy levels used in the past by Drellishak and coworkers 11, 13 and by Capitelli and coworkers 5, 6, 16, 17 . These sets of levels were determined by completing the information on the electronic states collected in Moore’s tables 7 as well as in the available literature. Let us consider the N atom as an example. The excited states come from the excitation of one electron towards higher values of the principal quantum number (n > 2): the most important series arises from the interaction of the atomic core 1s2 2s2 2p2 (3 P ) with the excited electron nx (x = s, p, d, . . .). For x = s, p, d enough levels are available in Moore’s tables to construct Ritz-Rydberg series of the type En = Ei − R∗ /(n + A + B/n2 )2

(7)

where Ei is the ionisation potential corresponding to the selected sequence and R∗ is the Rydberg constant. The constants A, B can be determined when at least two levels are available. The same procedure has been adopted for x = f by using the levels given by Eriksson and Johansson 18. For x = g, the level 5g is obtained by extrapolation of the curve E = E(s, p, d, f )n=5 (see Fig. 2). From this extrapolated level and the expression En = Ei − R∗ /(n + C)2

(8)

the n(> 5)g levels can be estimated. The same procedure is adopted to estimate the level 6h and the n(> 6)h levels, and so on. Fig. 2 shows that from n > 10 it becomes justified to group the 5

levels with the same n but different x into a single level with the appropriate multiplicity. For n > 10 this would be the case for levels higher than 10h. Let us now consider the configuration 1s2 2s2 2p2 (1 D) with the excited electron nx . For x = s there is only one observed level. In this case use can be made of (8); Ei represents now the ionisation potential of the principal series augmented with the excitation energy of the 1 D state of N+ . The configurations np (2 P ◦ , 2 D◦ , 2 F ◦ ) can be completed in a similar way. The configurations nd require the use of isoelectronic sequences for their completion. All terms, except the 3d 2 D term, are found to lie above the series limit Ei of the principal series and are therefore disregarded. The degeneracy of the different levels has been calculated according to the L-S coupling scheme 9 . A similar approach was followed also by Drellishak and coworkers who, however, completed the relevant series by extrapolation rather than by using (7) and (8). All the air atomic species were treated in a manner analogous to that described for N. It should be noticed that the ions taken into consideration in the tables by Drellishak et al. 12, 13 and Capitelli et al. 16, 17 do not exceed the third stage of ionisation. In this report we consider temperatures as high as 100000 K and we must include at least the ions N4+ , O4+ , and possibly the ions N5+ , O5+ . Moore’s tables list only a few levels for such ions. However, in the nitrogen system the different levels can be more and more considered as hydrogen-like (N4+ ) and heliumlike (N5+ ); for these two ions the energies, in cm−1 , and the degeneracies can be written, respectively, as En = 789533 − 2743232/n2 gn = 2n2

n≥3

En = 4452800 − 3935840/n2 gn = 4n2

n≥2

The corresponding partition functions, including the low-lying excited states, are then written as Qint (N 4+ ) = 2 + 2 exp(−1.4388 · 80465/T ) + 4 exp(−1.4388 · 80723/T ) + +

n max n=3

Qint (N

5+

) = 1+

2n2 exp(−1.4388 · En /T )

n max n=2

4n2 exp(−1.4388 · En /T )

In the oxygen system, the missing energy levels have been obtained from the empirical formulae (7) and (8) up to n = 10 in the case of O4+ and up to n = 8 in the case of O5+ ; higher levels for these ions have been estimated respectively from En = 918702 − 2745917.5/(n − 0.115)2 gn = 4n2

n ≥ 11

En = 1113999.5 − 3954121.2/n2 gn = 2n2

n≥9 6

Concerning O− , we have allowed only the fine splitting 19 of the ground state according to Qint (O− ) = 4 + 2 exp(−1.4388 · 177.08/T ) Autoionising levels 20 with energies higher than the electron affinity were neglected. Let us now compare the internal partition function for N and O calculated by Drellishak et al. and by Capitelli et al., whose set of energy levels have also been used for the tables 8 calculated in the present study. The comparison is shown in Fig. 3; the data refer to N and O plasmas in equilibrium at p = 1 atm. One can notice that in both cases Drellishak’s values exceed those of Capitelli up to a factor two. The discrepancies are due to the use of different sets of energy levels as well as of different cut-off criteria. It should be noticed, however, that the differences about Qint shown in Fig. 3 affect the concentrations of N and O in the corresponding plasmas only when they are minor species. Similar considerations apply for the partition functions of the corresponding ions. The previous conclusions apply only to equilibrium conditions. Under nonequilibrium circumstances we must know the composition of the charged particles in order to calculate the ionisation potential lowering required to truncate the internal partition function. In order to obviate this necessity the ionisation potential lowering has been considered as a parameter in the calculations of the present study. Diagrams of Qint versus T for different values of the ionisation potential lowering are shown in Figs. 4–9 for the species N, N+ , N2+ , N3+ , N4+ , N5+ . Of course, the dependence of the internal partition function on the number of energy levels inserted in the sum changes according to the particular ion under consideration. We can observe that the neutral atom N dramatically depends on the assumed ionisation potential lowering while the multicharged ion N5+ is completely insensitive, in the temperature range considered in the present report, because of the large energies of its electronic levels. 3.4

Thermodynamic properties

The thermodynamic properties of the single species follow in a straightforward manner when the internal partition function and its logarithmic derivatives are known. The internal contribution Eint to the thermodynamic energy and the internal constant pressure specific heat Cp,int are obtained, respectively, from Eint = RT

2




∂ ln Qint ∂T






Cp,int = R 2T

 V

∂ ln Qint ∂T



 V

+T

2

∂ 2 ln Qint ∂T 2

 V

The global quantities are obtained by adding up the translational contributions; they read 3 E = Eint + RT 2 5 Cp = Cp,int + R 2 7

The internal contribution to the thermodynamic energy and the internal constant pressure specific heat depend strongly on the assumed sets of energy levels and on the adopted cut-off criterion. Unfortunately, Drellishak et al. 12, 13 do not report these quantities and we can compare our values only with less accurate calculations. A comparison among the data calculated in the present study and those calculated by other authors 21–25 with respect to the non-dimensional total constant pressure specific heats for O and O+ is shown in Figs 10–11. Our values are those corresponding to an oxygen plasma 17 in chemical equilibrium at p = 1 atm while the values from other authors have been obtained by using either all the levels tabulated in Moore’s tables 7 or a limited number of levels. It is noticeable that the different calculations agree well at low temperature, the agreement becoming less satisfactory as the temperature increases. It should be emphasised that only our data depend on pressure, because the other authors cited in Figs 10–11 did not consider any cut-off criterion in their calculations. The dependence on pressure is better evidenced in Fig. 12, where we have plotted the total constant pressure specific heat of O as a function of temperature for different pressures; once again the data refer to equilibrium oxygen plasmas 17 . In this figure we have also plotted the data calculated by Browne 21 , who used only the levels tabulated in Moore’s tables. Browne’s values are similar to our results for a pressure of 10 atm, i.e. for high values of the ionisation potential lowering. Similar behaviour can be seen in Fig. 13 which shows the total constant pressure specific heat of N and N+ for different pressures according to the cut-off criterion of Margenau and Lewis 10 . In this figure we have also plotted the data at a pressure of 1 atm for the same species but calculated according to the cut-off criterion of Griem 14 and the translational contribution to the specific heat (called ground state in the figure). We can see that the Margenau and Lewis’ criterion gives specific heats greater than those obtained with the Griem’s criterion. The internal energy and the internal constant pressure specific heat of N and its successive ions as a function of temperature for different ionisation potential lowerings are shown in Figs. 14–25. It is worth noticing that the internal specific heat, after reaching a maximum, strongly decreases and practically vanishes at high temperature. This effect, which is a consequence of considering a limited number of levels in the partition function and its derivatives, is similar to the one occurring in the molecules when a finite number of rotovibrational states is considered.

8

4

THE INTERNAL PARTITION FUNCTION OF MOLECULAR SPECIES

The starting point for calculating the internal partition function of molecular species is to solve the Schr¨ odinger equation of a representative molecule to obtain the energy levels corresponding to the independent molecular degrees of freedom. The Schr¨ odinger equation is solved in the Born-Oppenheimer approximation 9 , which separates the motion of the electrons from that of the nuclei. The solution of the Schr¨ odinger equation for the electrons yields the approximate wave functions of the infinite electronic states as well as their degeneracies. The energy corresponding to the nth electronic state is expressed as a function of the internuclear distance r and constitutes the potential energy Vn (r) seen by the internal motion of the nuclei, governed by the nuclear Schr¨ odinger equation. 4.1

Diatomic molecules

In the case of diatomic molecules, the nuclear Schr¨odinger equation generates two angular equations, which can be solved analytically and introduce the magnetic m and the rotational (or azimuthal) J quantum numbers, and a radial (or vibrational) equation with an effective potential UnJ (r) given by UnJ (r) = Vn (r) + J(J + 1)h2 /8π 2 µr 2

(9)

where µ is the reduced mass. The second term on the right-hand side of (9) is the centrifugal potential due to molecular rotation. The vibrational equation is solved for all the values of J up to a maximum Jmax for which the effective potential (9) becomes purely repulsive. Moreover, the number of vibrational levels for each J is limited to vmax (n, J) given by the condition EnJv < UnJ (rb,nJ ) where rb,nJ is the internuclear distance at which the effective potential (9) reaches a relative maximum 3 . Once the complete sets of EnJv are known, the internal partition function can be calculated from Qint =

Jmax (n) vmax (n,J) nmax   1  gn (2J + 1) exp(−EnJv /kT ) σ n v J

(10)

where σ is a symmetry factor that equals one or two for, respectively, heteronuclear and homonuclear diatomic molecules. This method, recently applied by Liu et al. 3 for N2 , is probably the most accurate for obtaining the internal partition function of a diatomic molecule. The expression (10) clearly shows the impossibility of separating independent contributions to EnJv , i.e. electronic, rotational and vibrational energies, especially when ab initio procedures are used for its calculation. This method was also partially used by Jaffe 1 . Let us now examine another interesting method for calculating the internal partition function developed many years ago by Drellishak 11 , by Drellishak et al. 13 and by Stupochenko 26 . In this method, the energy EnJv is split into three contributions: the electronic excitation energy Eel (n), the vibrational energy Evib (n, v) and the rotational energy Erot (n, v, J). Thus EnJv = Eel (n) + Evib (n, v) + Erot (n, v, J) 9

Following Herzberg 27 , the vibrational energy associated with the nth electronic state of a diatomic molecule is expressed in analytical form as Evib (n, v) = ωe (v + 1/2) − ωe xe (v + 1/2)2 + ωe ye (v + 1/2)3 + ωe ze (v + 1/2)4 hc

(11)

where ωe , ωe xe , ωe ye , ωe ze are spectroscopically determined constants 27, 28 for each electronic state, and c is the speed of light. The expression (11) is rewritten with the energy referenced to that of the first vibrational level, which reads 1 1 1 1 Evib (n, 0) = ωe − ωe xe + ωe ye + ωe ze 2 4 8 16 In this case, the expression (11) turns into Evib (n, 0) Evib (n, v) = + ω0 v − ω0 x0 v 2 + ω0 y0 v 3 + ω0 z0 v 4 hc hc

(12)

The constants in (12) are given from combinations of the spectroscopic constants appearing in (11); they read 3 1 ω0 = ωe − ωe xe + ωe ye + ωe ze 4 8 3 3 ω0 x0 = ωe xe − ωe ye − ωe ze 2 2 ω0 y0 = ωe ye + 2ωe ze ω0 z0 = ωe ze Assuming that (11) is valid for all vibrational states up to dissociation, we can determine the maximum permissible value vmax of the vibrational quantum number for each rotationless (J = 0) molecular state from the equation 2 3 4 + ω0 y0 vmax + ω0 z0 vmax = ω0 vmax − ω0 x0 vmax

D0 (n) hc

in which D0 (n) is the dissociation energy of the nth electronic state referenced to the energy of the first vibrational level. The rotational energy for a non-rigid rotator associated with the v th vibrational level of the nth electronic state reads Erot (n, v, J) = Bv J(J + 1) − Dv J 2 (J + 1)2 hc

10

where Bv and Dv are spectroscopically determined constants 11, 13 that depend on the electronic and vibrational quantum numbers. They read Bv = Be − αe (v + 1/2) Dv = De − βe (v + 1/2) The maximum permissible value Jmax of the rotational quantum number for each vibrational quantum number is determined by searching for the zeroes of the first and second derivatives of the effective potential. For this purpose, Vn (r) is taken according to the expression proposed by Morse 29 Vn (r) = De,min [1 − exp(−2βξ)]2 where De,min is the dissociation energy of the given state as measured from the minimum of the potential curve, b is a parameter of the Morse function and ξ is given by ξ = (r − re )/re re being the equilibrium distance for the considered state. Inspection of the potential curves 11, 13, including the centrifugal term, shows the existence of bound states above the dissociation limit of the molecule. In fact, one can see that the different potential curves, corresponding to different values of J, possess a maximum and a minimum: these two points coalesce giving the value Jmax for the considered v. The procedure, which is widely discussed by Drellishak and by Stupochenko, gives us the consistent couples v, Jmax . The internal partition function is calculated from the expression

Qint



v (n)

J (v)

max 1 n Eel (n) max Evib (n, v) max Erot (n, v, J) = gn exp − exp − (2J + 1) exp − (13) σ n kT kT kT v J

In this case, too, notice that the factorisation of the internal partition function in terms of independent contributions cannot be achieved. This method has been applied in the present study + − + to N2 , N+ 2 , O2 , O2 , O2 , NO, NO ; Tables 1–7 list all the input data used in the calculations. In general, these data have been obtained from Jaffe 1 and from Huber and Herzberg 28 . In a few cases, however, it was necessary to calculate them according to standard formulae 27 . For example, we have considered two electronic states for O− 2 . The values of ωe , ωe xe and re relative to the ground state were obtained from Huber and Herzberg. We then calculated Be , De and αe according to the following expressions Be =

h 8π 2 cµre2

(14)

De =

4Be3 ωe2

(15)

11



6 ωe xe Be3 6Be2 αe = − ωe ωe

(16)

For the excited state, Huber and Herzberg give only the value of ωe , ωe xe . The equilibrium distance was estimated after Krauss et al. 30 and the other parameters were calculated from (14), (15), and (16). Another method for calculating the internal partition function, proposed by the Mayers, 31 is based on the following expression Qint





max 1 n θn −1 Eel (n) T = 1 − exp − (1 + γn T ) gn exp − σ n T 1.4388Bn kT

(17)

On the right-hand side of (17), the terms [1 − exp(−θn /T )]−1 and (T /1.4388Bn ) represent the closed forms of, respectively, the vibrational and rotational contributions to the internal partition function associated with the nth electronic state, the term (1 + γn T ) accounting for the effects of anharmonicity and nonrigidity. Spectroscopic data can be used to determine θn , Bn and γn from the following formulae θn ( ◦K) = 1.4388(ωe − ωe xe )n Bn = (Be − 1/αe )n γn =

1 (8Be /ωe + αe /Be + 2ωe xe /ωe )n θn

An expression similar to (17) has been used by Drellishak and coworkers for the calculation of the internal partition function of N2 and N+ 2 . The expression (17) can be further rearranged if one assumes that all the electronic states have the same rotovibrational parameters; in this case, (17) reduces to

Qint




1 T0 = 1 − exp − σ T

 −1




T (1 + γ0 T ) 1.4388B0

 n max n



Eel (n) gn exp − kT

where T0 , g0 and B0 are the rotovibrational parameters corresponding to the ground electronic state. The last formulation is indeed an approximation not only because each electronic state has its own rotovibrational parameters but also because it uses infinite levels in the corresponding vibrational and rotational contributions. The advantage, however, is the separation of the different internal contributions. A better formulation, which in part preserves the separation, is obtained by writing

Qint




1 T0 = 1 − exp − σ T

 −1




T (1 + γ0 T ) 1.4388B0



g0 +

n max n



Eel (n) pn exp − kT



(18)

where the pn values also contain information on the vibrational and rotational structure of the excited states 32 . 12

We are now able to compare the different approaches to the problem of calculating the internal partition function. Unfortunately, Liu and Vinokur 2 and Liu et al. 3 do not report values of the internal partition function but only give diagrams comparing the internal energy of N2 calculated according to the methods described and discussed in there. We can, therefore, compare the internal partition function of N2 calculated by Jaffe 1 , by Drellishak et al. 12, 13 , and by Capitelli et al. 16 with the data obtained in the present study 8. Such a comparison is shown in Fig. 26; the data are relative to the choice of referring the vibrational energies to that of the v = 0 vibrational level of the ground electronic state. Jaffe and the present study used the finite summation method based on (13), but with different sets of rotovibrational levels, while Drellishak and Capitelli used the approach proposed by the Mayers and based on (17) and (18). The agreement among the different calculations is generally satisfactory despite the diversity of the methods used. 4.2

Thermodynamic properties

When the internal partition functions are known, one can determine the thermodynamic properties of the molecular species in the manner described in Section 3.4. The relevant logarithmic derivatives can be obtained by either analytical 5 or numerical 13 differentiation. The different methods for the calculation of the internal energy of N2 up to 50000 K have been compared by Liu and Vinokur 2 and Liu et al. 3 . The ab initio method adopted by them gives results which agree very well with those of Jaffe 1 and Stupochenko 26 in the whole temperature range; on the other hand, the Mayers’ approach begins to deviate from theirs, approximately, above 20000 K. Inspection of Fig. 27 clarifies this point; in this figure, we have compared the internal energy of N2 calculated in the present study 8 with those calculated by Capitelli et al. 16 from (18). The data calculated by Capitelli increase monotonically and reach a quasi plateau, while those from the present study reach a maximum and then decrease. On the other hand, comparison of the non-dimensional internal constant pressure specific heat, shown in Fig. 28, indicates larger differences between the two methods; in particular, the results obtained in the present study show a dramatic decrease of the internal specific heat with increasing temperature, while the results from Capitelli’s old calculation converge to the value two, which represents the excitation of the rotational and vibrational degrees of freedom. The dramatic decrease of the internal specific heat is due to the finite, though large, number of internal energy levels accounted for in the summation of the internal partition function. However, the use of closed forms for the rotational and vibrational contributions to the internal partition function implies the inclusion of an infinite number of energy levels and, therefore, the molecule always has available higher levels for further excitation when the temperature increases. This fact is evidenced by the convergence to the value 2R of the internal specific heat of a diatomic molecule. Coming back to the data obtained in the present study, we are confronted by the apparent paradox that the specific heat of a diatomic molecule converges to that of an atomic species. However, it is worth noticing that the calculations of the present study consider only a finite number of excited electronic states. It should be interesting to study the effect of the addition of several other excited states on the internal specific heat calculated in the framework of the finite summation method. It is interesting to see the behaviour of the different contributions in (18) and the corresponding logarithmic derivatives as reported by Capitelli and Molinari 5 . This has been done for N2 in Fig. 29. The electronic contribution follows the typical trend of the atoms, while the rotovi-

13

brational contribution slightly increases with the temperature. It should be noticed that the rotovibrational contribution cannot be separated in any of the described formulations. Another interesting issue is whether the direct summation method permits the factorisation of the internal partition function; infact, this would be the case if the rotovibrational contribution to the internal partition function were equal for each electronic state. We shall examine this problem in the case of N2 by using the data from the present study 8 which are relative to the inclusion of ten electronic states (see Table 1). The diagram in Fig. 30 indicates clearly that each electronic state makes its own rotovibrational contribution to the internal partition function; in particular, the rotovibrational contribution decreases from low to high-lying excited states. This happens because the dissociation energy decreases when the electronic energy increases (see Table 1). On the other hand, if we now sum the different rotovibrational contributions, modulated with the exponential factor carrying the electronic energy, the contribution of highlying excited states to the internal partition function (Fig. 31), to the internal energy (Fig. 32) and to the internal constant pressure specific heat (Fig. 33) begins to be important only at very high temperatures and, therefore, the rotovibrational contribution of the ground electronic state is the predominant one for low temperatures. These conclusions, however, must be confirmed by a careful examination of the potential energy curves of high-lying electronic states.

14

5

CONCLUSIONS

In this report we have examined and discussed the efforts made in the last 30 years to calculate the high-temperature thermodynamic properties of air species. The analysis has shown the impossibility of calculating the internal partition functions of the atomic species as a function of temperature only; a cut-off criterion, depending either on the pressure or on the number density of the charged particles, needs to be used to truncate the internal partition function and its derivatives, particularly when complete sets of electronic levels, i.e. those experimentally determined and listed by Moore 7 and the missing ones determined via semiempirical laws, are accounted for. No problem, on the contrary, exists concerning the separation of the internal partition function from the translational one. In this case a two-temperature approach can be used to account for thermal non-equilibrium, even though one must be careful to use the appropriate transport properties of the species. We need, in fact, to remember that a large contribution to the internal energy of the atomic species comes from the excited electronic states near the ionisation continuum. These states possess transport collision cross sections much greater than the corresponding cross sections of the ground state. Consequently, use of thermodynamic properties which account for contributions from the excited electronic states to the internal energy should be accompanied by consistent sets of transport cross sections. Unfortunately, only few papers 33–37 have tried in the past to solve this problem, which, in the authors’ opinion, may represent a critical issue for the construction of accurate thermodynamic models suited for hypersonic flows. As regards the diatomic molecules, recent calculations 2, 3 based on the solution of the vibrational Schr¨ odinger equation in the field of an effective potential have shown the accurate way to obtain the internal partition function and the subsequent thermodynamic properties. The obtained results, however, are in excellent agreement with the older method, proposed by Drellishak 11 , by Drellishak et al. 13 and by Stupochenko et al. 26 , based on analytical expressions of the rotovibrational energies. All these studies show how difficult it is to separate the different internal contributions, even in the framework of the Born-Oppenheimer approximation. The construction of thermodynamic models based on different temperatures associated with the internal structure of the molecules (rotational, vibrational and electronic) appears open to question. Only crude models reach the goal of the separation. However, inspection of the different contributions to the internal partition function can help to separate at least the electronic contribution from the rotovibrational one. This last contribution may be separated only for the harmonic-oscillator and rigid-rotator models, the validity of which is strongly questionable at sufficiently high temperatures. The calculation of the internal partition function of diatomic molecules may seem not to be affected by the divergence problem existing for the atomic species: this point is, however, illusory. In fact, while a well-founded cut-off criterion has been applied to the rotovibrational energies of each electronic state, a practical cut-off criterion has been applied for the number of electronic states. The approach is, in fact, to insert only few electronic states in the summation to determine the internal partition function. The justification could be that possible crossings between very excited bound electronic states and repulsive electronic states dissociate the molecule. This point needs further study. As a conclusion we can say that caution should be exercised when the thermodynamic properties of high temperature gases are used in fluid dynamics codes. The pessimism discussed above can be somewhat alleviated if one considers that all the difficulties discussed for both molecules and atoms occur in temperature ranges in which entropic effects favour the dissociation of the 15

molecule or the ionisation of the atom. It is to be hoped that this point, which is true for equilibrium situations, may also hold for nonequilibrium cases, thus eliminating some of the problems discussed above.

16

6

REFERENCES AND NOTES

1

R. L. Jaffe, “The Calculation of High-Temperature Equilibrium and Non-Equilibrium Specific Heat Data for N2 , O2 and NO”, AIAA-87-1633, June 1987

2

Y. Liu, M. Vinokur, “Equilibrium Gas Flow Computations. I. Accurate and Efficient Calculation of Equilibrium Gas Properties”, AIAA-89-1736, June 1989

3

Y. Liu, F. Shakib, M. Vinokur, “A Comparison of Internal Energy Calculation Methods for Diatomic Molecules”, Phys. Fluids A 2, 1884 (1990); AIAA-90-0351, January 1990

4

M. Capitelli, E. Ficocelli Varracchio, “Thermodynamic Properties of Ar-H2 Plasmas”, Rev. int. Htes Temp. et Refract. 14, 195 (1977)

5

M. Capitelli, E. Molinari, “Problems of Determination of High Temperature Thermodynamic Properties of Rare Gases with Application to Mixtures”, J. Plasma Phys. 4, 335 (1970)

6

M. Capitelli, E. Ficocelli V., E. Molinari, “Electronic Excitation and Thermodynamic Properties of High Temperature Gases”, Z. Naturforsch. 26a, 672 (1971)

7

C. Moore, “Atomic Energy Levels as Derived from the Analyses of Optical Spectra”, NSRDSNBS-35-VOL-1, 1971

8

D. Giordano, M. Capitelli, G. Colonna, C. Gorse, “Tables of Internal Partition Functions and Thermodynamic Properties of Air Species from 50 K to 100000 K”, ESA STR-237, 1994

9

P. W. Atkins, Physical Chemistry (Oxford University Press, Oxford, 1987)

10

H. Margenau, M. Lewis, “Structure of Spectral Lines from Plasmas”, Rev. Mod. Phys. 31, 594 (1959)

11

K. S. Drellishak, “Partition Functions and Thermodynamic Properties of High Temperature Gases”, AEDC-TDR-64-22, 1964

12

K. S. Drellishak, D. P. Aeschliman, A. B. Cambel, “Tables of Thermodynamic Properties of Argon, Nitrogen, and Oxygen Plasmas”, AEDC-TDR-64-12, 1964

13

K. S. Drellishak, D. P. Aeschliman, A. B. Cambel, “Partition Functions and Thermodynamic Properties of Nitrogen and Oxygen Plasmas”, Phys. Fluids 8, 1590 (1965)

14

H. R. Griem, “High-Density Corrections in Plasma Spectroscopy”, Phys. Rev. 128, 997 (1962)

15

M. Capitelli, E. Ficocelli, E. Molinari, Equilibrium Compositions and Thermodynamic Properties of Mixed Plasmas: III Argon-Hydrogen Plasmas at 10−2 –103 Atmospheres between 2000 K and 35000 K (Adriatica Editrice, Bari, 1972)

16

M. Capitelli, E. Ficocelli, E. Molinari, Equilibrium Compositions and Thermodynamic Properties of Mixed Plasmas: I He–N2 , Ar–N2 and Xe–N2 Plasmas at One Atmosphere, between 5000 K and 35000 K” (Adriatica Editrice, Bari, 1969)

17

M. Capitelli, E. Ficocelli, E. Molinari, Equilibrium Compositions and Thermodynamic Properties of Mixed Plasmas: II Argon-Oxygen Plasmas at 10−2 –10 Atmospheres between 2000 K and 35000 K (Adriatica Editrice, Bari, 1970)

18

K. B. S. Eriksson and I. Johansson, “Spectra of Neutral Nitrogen Atom”, Arkiv. Fysik 19, 242 (1961)

19

H. Hotop, W. C. Lineberger, “Binding Energies in Atomic Negative Ions. II”, J. Phys. Chem. Ref. Data 14, 731 (1985) 17

20

D. Spence, W. A. Chupka, “Measurement of Resonances in Atomic Oxygen by Electron Transmission Spectroscopy”, Phys. Rev. A 10, 71 (1974)

21

W. G. Browne, “Thermodynamic Properties of Some Atoms and Atomic Ions”, Eng. Phys. Techn. Memo. # 2, General Electric Co., 1962

22

B. McBride, S. Heimel, J. G. Ehlers, S. Gordon, “Thermodynamic Properties to 6000 K for 210 Substances Involving the First 18 Elements”, NASA-SP-3001, 1963

23

D. R. Stull, H. Prophet, “JANAF Thermochemical Tables”, NSRDS-NBS-37, 1971

24

A. Balakrishnan, “Correlations for Specific Heats of Air Species to 50000 K”, AIAA-86-1277, June 1986

25

L. V. Gurvich, I. V. Veyts, C. B. Alcock, Thermodynamic Properties of Individual Substances (Hemisphere Publishing Corporation, New York, 1989)

26

E. V. Stupochenko, I. P. Stakhenov, E. V. Samuilov, A. S. Pleshhanov, I. B. Rozhdestvenskii, “Thermodynamic Properties of Air in the Temperature Interval from 1000 K to 12000 K and Pressure Interval from 10−3 to 103 Atmospheres”, ARS J. supplement 30, 98 (1960)

27

G. Herzberg, Molecular Spectra and Molecular Structure, I. Spectra of Diatomic Molecules (D. Van Nostrand, Inc., New York, 1963)

28

K. P. Huber, G. Herzberg, Constants of Diatomic Molecules (Van Nostrand Reinhold, New York, 1979)

29

P. M. Morse, “Diatomic Molecules According to the Wave Mechanics. II. Vibrational Levels”, Phys. Rev. 34, 57 (1929)

30

M. Krauss, D. Neumann, A. C. Wahl, G. Das, W. Zemke, “Excited Electronic States of O− 2 ”, Phys. Rev. A 7, 69 (1973)

31

J. E. Mayer, M. G. Mayer, Statistical Mechanics (John Wiley, New York, 1977).

32

A. R. Hochstim, “Approximations to High-Temperature Thermodynamics of Air in Closed Form”, in Kinetics, Equilibria and Performance of High Temperature Systems, edited by G. S. Bahn and E. Z. Zukoski (Butterworths Publ. Lim., London, 1960)

33

M. Capitelli, E. Ficocelli, “Collision Integrals of Oxygen Atoms in Different Electronic States”, J. Phys. B 5, 2066 (1972)

34

M. Capitelli, “The Influence of Excited States on the Reactive Thermal Conductivity of an LTE Hydrogen Plasma”, Z. Naturforsch. A 29a, 953 (1974)

35

M. Capitelli, C. Guidotti, U. Lamanna, “Potential Energy Curves and Excitation Transfer Cross Sections of Excited Hydrogen Atoms”, J. Phys. B 7, 1683 (1974)

36

M. Capitelli, “Charge Transfer from Low-Lying Excited States: Effects on Reactive Thermal Conductivity”, J. Plasma Phys. 14, 365 (1975)

37

M. Capitelli, “Transport Properties of Partially Ionized Gases”, J. Phys. (Paris) 38, C3-227 (1977)

18

7

TABLES

19

TABLES

List of Tables 1

Electronic states and spectroscopic data of N2 . . . . . . . . . . . . . . . . . . . .

21

2

Electronic states and spectroscopic data of N+ 2. . . . . . . . . . . . . . . . . . . .

22

3

Electronic states and spectroscopic data of O2 . . . . . . . . . . . . . . . . . . . .

23

4

Electronic states and spectroscopic data of O+ 2. . . . . . . . . . . . . . . . . . . .

24

5

Electronic states and spectroscopic data of O− 2. . . . . . . . . . . . . . . . . . . .

25

6

Electronic states and spectroscopic data of NO. . . . . . . . . . . . . . . . . . . .

26

7

Electronic states and spectroscopic data of NO+ . . . . . . . . . . . . . . . . . . .

27

20

Table 1: Electronic states and spectroscopic data of N2 .

21

Table 2: Electronic states and spectroscopic data of N+ 2.

22

Table 3: Electronic states and spectroscopic data of O2 .

23

Table 4: Electronic states and spectroscopic data of O+ 2.

24

Table 5: Electronic states and spectroscopic data of O− 2.

25

Table 6: Electronic states and spectroscopic data of NO.

26

Table 7: Electronic states and spectroscopic data of NO+ .

27

8

FIGURES

28

FIGURES

List of Figures 1

Internal partition function of H. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Variations of the electronic levels of N with angular momentum quantum number. 32

3

Internal partition function of N and O in chemical equilibrium plasmas at p = 1 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

Internal partition function of N. (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Internal partition function of N+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

Internal partition function of N2+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

Internal partition function of N3+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Internal partition function of N4+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Internal partition function of N5+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

10

Nondimensional constant pressure specific heat of O. . . . . . . . . . . . . . . . .

40

11

Nondimensional constant pressure specific heat of O+ . . . . . . . . . . . . . . . .

41

12

Pressure dependence of the nondimensional constant pressure specific heat of O in an equilibrium plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

4 5 6 7 8 9

31

13

Constant pressure specific heat of N and N+ according to different cut-off criteria. 43

14

Internal energy of N. (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Internal energy of N+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

Internal energy of N2+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

Internal energy of N3+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Internal energy of N4+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Internal energy of N5+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

Internal constant pressure specific heat of N. (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . .

50

15 16 17 18 19 20

29

Internal constant pressure specific heat of N+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . .

51

Internal constant pressure specific heat of N2+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . .

52

Internal constant pressure specific heat of N3+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . .

53

Internal constant pressure specific heat of N4+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . .

54

Internal constant pressure specific heat of N5+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250). . . . . . . . . . . . . . . . . . . . . . . . .

55

26

Internal partition function of N2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

27

Internal energy of N2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

28

Internal constant pressure specific heat of N2 . . . . . . . . . . . . . . . . . . . . .

58

29

Electronic and rotovibrational contributions to the internal constant pressure specific heat of N2 according to the Mayers’ method. . . . . . . . . . . . . . . . . . .

59

Rotovibrational contributions to the internal partition function of N2 for the first ten electronic states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Influence of successive inclusion of the first ten electronic states on the internal partition function of N2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Influence of successive inclusion of the first ten electronic states on the internal energy of N2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Influence of successive inclusion of the first ten electronic states on the internal constant pressure specific heat of N2 . . . . . . . . . . . . . . . . . . . . . . . . . .

63

21 22 23 24 25

30 31 32 33

30

Fig. 1: Internal partition function of H.

31

Fig. 2: Variations of the electronic levels of N with angular momentum quantum number.

32

Fig. 3: Internal partition function of N and O in chemical equilibrium plasmas at p = 1 atm.

33

Fig. 4: Internal partition function of N. (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

34

Fig. 5: Internal partition function of N+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

35

Fig. 6: Internal partition function of N2+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

36

Fig. 7: Internal partition function of N3+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

37

Fig. 8: Internal partition function of N4+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

38

Fig. 9: Internal partition function of N5+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

39

Fig. 10: Nondimensional constant pressure specific heat of O.

40

Fig. 11: Nondimensional constant pressure specific heat of O+ .

41

Fig. 12: Pressure dependence of the nondimensional constant pressure specific heat of O in an equilibrium plasma.

42

Fig. 13: Constant pressure specific heat of N and N+ according to different cut-off criteria.

43

Fig. 14: Internal energy of N. (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

44

Fig. 15: Internal energy of N+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

45

Fig. 16: Internal energy of N2+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

46

Fig. 17: Internal energy of N3+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

47

Fig. 18: Internal energy of N4+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

48

Fig. 19: Internal energy of N5+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

49

Fig. 20: Internal constant pressure specific heat of N. (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

50

Fig. 21: Internal constant pressure specific heat of N+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

51

Fig. 22: Internal constant pressure specific heat of N2+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

52

Fig. 23: Internal constant pressure specific heat of N3+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

53

Fig. 24: Internal constant pressure specific heat of N4+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

54

Fig. 25: Internal constant pressure specific heat of N5+ . (Curve 1: ∆E = 2000 cm−1 ; 2: 1500; 3: 1000; 4: 750; 5: 500; 6: 250).

55

Fig. 26: Internal partition function of N2 .

56

Fig. 27: Internal energy of N2 .

57

Fig. 28: Internal constant pressure specific heat of N2 .

58

Fig. 29: Electronic and rotovibrational contributions to the internal constant pressure specific heat of N2 according to the Mayers’ method.

59

Fig. 30: Rotovibrational contributions to the internal partition function of N2 for the first ten electronic states.

60

Fig. 31: Influence of successive inclusion of the first ten electronic states on the internal partition function of N2 .

61

Fig. 32: Influence of successive inclusion of the first ten electronic states on the internal energy of N2 .

62

Fig. 33: Influence of successive inclusion of the first ten electronic states on the internal constant pressure specific heat of N2 .

63