Abstract
This theoretical paper deals with the thermodynamic equilibrium of multi-temperature gas mixtures. After a brief review of basic notions and methods relative to the chemical equilibrium of gas mixtures in complete thermal equilibrium, the fundamental relations energy and entropy describing the thermodynamics of gas mixtures in disequilibrium with respect to energy and mass exchanges are formally introduced. Possible constraints arising from the subset of state parameters associated with the molecular degrees of freedom are recognized and the importance of the role they play in determining the thermodynamic equilibrium of the system is discussed. In particular, the influence that a partially constrained thermal equilibrium exercises on the equations governing the associated chemical equilibrium is made evident by performing the equilibrium analysis for the cases of entropic and energetic freezing of the molecular degrees of freedom; the analysis shows that the minimization/maximization of energy/entropy may not necessarily lead to the vanishing of the chemical reaction affinities and that, therefore, the uniqueness of the chemical equilibrium equations is lost when the thermal equilibrium is partially constrained. The equivalence of the fundamental relations energy and entropy to determine the equilibrium conditions is shown to be maintained also in multi-temperature circumstances; the problem of the lack of such equivalence for the Helmholtz and Gibbs potentials is briefly mentioned. As an application, the chemical equilibrium of a two-temperature partially ionized gas is considered with the purpose to show how the uncertainty associated with the two-temperature Saha equation can be resolved in the framework of the theory proposed in this work.
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Abbreviations
- F :
-
Helmholtz potential
- f :
-
number of frozen chemical reactions
- G :
-
Gibbs potential
- H :
-
enthalpy
- h :
-
Planck constant
- k :
-
Boltzmann constant
- G :
-
Lagrangian function
- M e :
-
electron mass
- m :
-
total mass
- N A, N A+ :
-
atom, ion number density
- N °a :
-
low temperature atom number density
- N e- :
-
electron number density
- n :
-
number of components
- p :
-
thermodynamic pressure
- q A, q A+ :
-
atom, ion electronic partition function
- R G :
-
universal gas constant, 8.3144 J/K
- r :
-
number of independent chemical reactions
- S :
-
entropy
- T :
-
temperature (thermal equilibrium)
- T e :
-
electron temperature
- T h :
-
heavy species temperature
- U :
-
energy
- V :
-
volume
- θ :
-
arbitrary reference temperature
- λ S , λ Ψ S , λ U :
-
Lagrange multipliers
- φΦΨ:
-
entropic potentials
- A k :
-
chemical affinity
- \( {{\bar{A}}_{k}} \) :
-
no standard name39
- ℓ ε :
-
number of independent degrees of freedom
- \( {{\bar{\ell }}_{ \in }} \) :
-
number of (entropically/energetically) frozen degrees of freedom
- M ε :
-
molar mass
- M ε :
-
molecular mass
- m ε :
-
mass
- m °ε :
-
initial mass
- N e :
-
number density
- p εδ :
-
partial pressure
- q εδ :
-
partition function
- S εδ :
-
entropy
- T εδ :
-
temperature
- U εδ :
-
energy
- ε εδ,i :
-
quantum state energy
- μ ε :
-
chemical potential
- μ εδ :
-
contribution to chemical potential
- V kε :
-
stoichiometric coefficient
- ξ k :
-
progress variable
- δ :
-
degrees of freedom
- ε :
-
components
- k :
-
chemical reactions
- i :
-
quantum states
- (dZ) X,Y :
-
differential of Z with X, Y constant
- \( \mathop{\sum }\limits_{{ \in ,\delta }}^{{nf}} \) :
-
summation extended to non-frozen degrees of freedom
- \( \mathop{\sum }\limits_{{ \in ,\delta }}^{f} \) :
-
summation extended to frozen degrees of freedom
- \( \mathop{\sum }\limits_{k}^{{nf}} \) :
-
summation extended to non-frozen chemical reactions
References and Notes
H. B. Callen (1963) Thermodynamics, John Wiley & Sons, New York.
L. Tisza (1966) Generalized Thermodynamics, The M. I. T. Press, Cambridge, MA.
H. S. Robertson (1993) Statistical Thermophysics, P T R Prentice Hall, Englewood Cliffs (NJ).
L. G. Napolitano (1971) Thermodynamique des Systèmes Composites en Équilibre ou Hors d’Équilibre, Gauthier-Villars Éditeurs, Paris.
C. F. Hansen (1976) Molecular physics of equilibrium gases, NASASP-3096.
K. Denbigh (1955) The Principles of Chemical Equilibrium, Cambridge University Press, Cambridge.
V. I. Golovitchev (private communication).
W. B. White, S. M. Johnson, and G. B. Dantzig (1958) Chemical equilibrium in complex mixtures, J. Chem. Phys. 28, 751.
F. J. Zeleznik and S. Gordon (1960) An analytical investigation of three general methods of calculating chemical-equilibrium compositions, NACA-TN-D-473.
F. J. Zeleznik and S. Gordon (1966) Equilibrium computations for multi-component plasmas, Can. J. Phys. 44, 877.
F. J. Zeleznik and S. Gordon (1968) Calculation of complex chemical equilibria, Ind. Eng. Chem. 60, 27.
Various authors (1970) Kinetics and thermodynamics in high-temperature gases, NASA-SP-239.
M. Capitelli and E. Molinari (1970) Problems of determination of high temperature thermodynamic properties of rare gases with application to mixtures, J. Plasma Phys. 4, 335.
F. Van Zeggeren and S. H. Storey (1970) The Computing of Chemical Equilibria, Cambridge University Press, Cambridge.
R. Holub and P. Vonka (1976) The Chemical Equilibrium of Gaseous Systems, Reidel Publishing Co., Dordrecht.
W. R. Smith and R. W. Missen (1982) Chemical Reaction Equilibrium Analysis: Theory and Algorithms, John Wiley & Sons, New York.
Y. Liu and M. Vinokur, “Equilibrium gaseous flow computations. I. Accurate and efficient calculation of equilibrium gas properties”, AIAA Paper No. 89–1736, 24th Thermophysics Conference, Buffalo, NY, June 12–14, 1989.
T. L. Eddy and K. Y. Cho (1991) A multitemperature model for plasmas in chemical nonequilibrium, in K. Etemadi and J. Mostaghimi (eds.), HTD, Vol. 161: Heat Transfer in Thermal Plasma Processing, ASME, New York, pp. 195–210.
There are instances in the literature where the conditions of chemical equilibrium are found from the imposition of the vanishing of the entropy production; see Sec. 7 at p. 75 of W. G. Vincenti and C. H. Kruger jr. (1965) Introduction to Physical Gasdynamics, John Wiley & Sons, New York. However, this method is not rigorously proper because the vanishing of the entropy production, when the conditions of chemical equilibrium prevail, is a result of the theory and not information available a priori.
The entropic potentials are known in the literature as Massieu functions. See, for example, Sec. 5.4 at p. 101 of Ref. 1 or Sec. 3.2.2 at p. 58 of Ref. 4.
See Sec. 4.4.1 at p. 99 and Sec. 4.4.4 at p. 109 of Ref. 4. See also the section “Conditions d’équilibre chimique” at p. 283 in vol. 1 of L. Sédov (1975) Mécanique des Milieux Continus,Editions Mir (Traduction française), Moscou.
The term “law of mass action” is also used to refer directly to the equations (3).
The term “free energy” is also used to refer to the Gibbs potential.
See Sec. 3.3 at p. 18 of Ref. 17.
A. V. Potapov (1966) Chemical equilibrium of multitemperature systems, High Temp. (USSR) 4, 48.
M. C. M. Van de Sanden (1991) The Expanding Plasma Jet: Experiments and Model, Thesis, Technische Universiteit Eindhoven, Eindhoven.
M. C. M. Van de Sanden, P. P. J. M. Schram, A. G. Peeters, J. A. M. van der Mullen, and G. M. W. Kroesen (1989) Thermodynamic generalization of the Saha equation for a two-temperature plasma, Physical Review A 40, 5273.
A. Morro and M. Romeo (1988) The law of mass action for fluid mixtures with several temperatures and velocities, J. Non-Equilib. Thermodyn. 13, 339.
A. Morro and M. Romeo (1988) Thermodynamic derivation of Saha’s equation for a multi-temperature plasma, J. Plasma Phys. 39, 41.
A. Morro and M. Romeo (1986) On the law of mass action in fluid mixtures with several temperatures, Il Nuovo Cimento 7 D, 539.
K. Chen and T. L. Eddy (1995) Investigation of chemical affinity for reacting flows of non-local thermal equilibrium gases, J. Thermophys. Heat Transfer 9, 41.
A. E. Mertogul and H. Krier (1994) Two-temperature modeling of laser sustained hydrogen plasmas, J. Thermophys. Heat Transfer 8, 781.
D. Giordano and M. Capitelli (1995) Two-temperature Saha equation: a misunderstood problem, J. Thermophys. Heat Transfer 9, No. 4, 803.
D. Kannappan and T. K. Bose (1977) Transport properties of a two-temperature argon plasma, Phys. Fluids 20, 1668.
See references from 6. to 23. in the list given in Ref. 18.
See the introduction to ch. 5 at p. 123 of Ref. 4. Using Napolitano’s notation, when the state parameter m. in the entropic formulation is a progress variable the identification of its associated state equation B r . with the ratio of the reaction affinity A r . to the temperature T relative to the degrees of freedom in mutual equilibrium, i.e. B r . = A r /T, does not appear to be correct.
See Sec. 1.15 at p. 3 of Ref. 4.
A system is defined isolated if it exchanges neither energy nor mass with its environment. Upon a basic postulate, an isolated system cannot exchange any other physical property with its environment. See Secs. 1.5–1.6 at p. 3 of Ref. 4.
In order for the terminology to conform to this more general situation, it seems appropriate to extend the definition of affinity. Given that the chemical potential \( {{\mu }_{ \in }} = \sum\limits_{{\delta = 1}}^{{{{\ell }_{ \in }}}} {{{\mu }_{ \in }}\delta } \) and the quantity \( \sum\limits_{{\delta = 1}}^{{{{\ell }_{ \in }}}} {{{\mu }_{ \in }}\delta /{{T}_{ \in }}\delta } \) are state equations [see (10), (11)] in, respectively, the energetic and entropic schemes, it appears consistent to define the expression (28) as entropic affinity and to rename the expression (29) as energetic affinity. In this way, the equations (21), (27) indicate, respectively, that the chemical equilibrium driven by an entropically/energetically constrained thermal equilibrium is characterized by the vanishing of the energetic/entropic affinities of the non-frozen chemical reactions.
See Sec. 5.4 at p. 130 of Ref. 4.
See Eq. (106) at p. 201 of Ref. 18 and Eq. (21) at p. 50 of Ref. 25.
The choice of the degrees of freedom in mutual thermal equilibrium is not unique but varies from author to author. See the paragraph following Eq. (107) at p. 201 of Ref. 18.
Morro and Romeo assume, implicitly, the mutual thermal equilibrium of the degrees of freedom of the heavy species A and A+; thus, and using their notation, the gas mixture they consider is characterized by the equilibrium temperature of the heavy species θ h and the translational temperature θ e of the electrons.
See Sec. 4 of Ref. 29.
See, for example, the paragraph following Eq. (14) at p. 783 of Ref. 32.
See Eq. (1) at p. 1668 of Ref. 34. However, the equation used by Kannappan and Bose, as it appears in their paper, shows a slight inaccuracy because the ratio of the electron translational temperature to the heavy species temperature, 1/θ in the notation of the reference, is missing in the exponent of the internal partition functions. The same inaccuracy appears also in the corresponding equation reported by Morro and Romeo [See Eq. (28) at p. 47 of Ref. 29] for comparison with their own equation. Kannappan and Bose refer to S. Veis, “The Saha equation and lowering of the ionization energy for a two-temperature plasma” [in Proceedings of the Czechoslovak Conference on Electronics and Vacuum Physics (Prague, 7–12 Oct. 1968), edited by L. Paty, p. 105–110, Karlova Universita, 1968] for the derivation of the two-temperature Saha equation they used.
See Eq. (26) at p. 47 or Eq. (29) at p. 48 of Ref. 29.
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Giordano, D. (1996). Thermodynamic Equilibrium of Multi-Temperature Gas Mixtures. In: Capitelli, M. (eds) Molecular Physics and Hypersonic Flows. NATO ASI Series, vol 482. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0267-1_15
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DOI: https://doi.org/10.1007/978-94-009-0267-1_15
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