Thermodynamic Properties of Plasmas

Abstract

Physical and chemical processes occurring in thermal plasmas, and their interactions with gases, particulate matter, or liquids, are highly complex and require knowledge of composition, thermodynamic, and transport properties of the plasma. In this chapter, the relationships between thermodynamic functions and partition functions for plasmas in local thermodynamic equilibrium (LTE) and local chemical equilibrium (LCE) are discussed. The methods of calculation of these partition functions are given, followed by a review of the different ways to calculate the composition of a plasma, and of the corresponding thermodynamic properties. As has been shown in the chapter 4, Sect. 4.4, even if the plasma core can be assumed to be in complete thermodynamic equilibrium (CTE), this may not be the case in the fringes or in the plume of the thermal plasmas, where two different temperatures can be defined (one for the electrons and the other for the heavy species). Separate discussions for the effects of such deviations from LTE or LCE, on the thermodynamic and transport properties of plasmas, are presented in the chapters 9 and 10 respectively.

Nomenclature and Greek Symbols

a_γ

Frozen velocity

B(T)

Second virial coefficient

B_ij(T)

Second virial coefficient related to the interaction between particles i and j

c_p

Specific heat at constant pressure (kJ/kg.K)

c ^r_P

Reactional specific heat at constant pressure (kJ/kg.K)

c ^f_P

Frozen specifie heat at constant pressure (kJ/kg.K)

d_ij

Distance at which interaction between particles i and j has to be taken into account (m)

dη_i

Rate of production of species i

D_e

Energy difference between the equilibrium position of the nuclei and the free atoms for a diatomic molecule (eV or cm⁻¹)

e

Symbol of electron

e_n

Base of natural logarithm (e_n = 2.7182818)

E_i,s

Energy of the chemical species i in the excited state, s (eV or cm⁻¹)

E ⁰_0,i

Energy of the chemical species i in its ground state related to an absolute reference state (eV or cm⁻¹)

E_e

Electronic energy of a given particle (eV or cm⁻¹)

E_i

Energy of the i state or of the electronic (e), or vibrational (v) or rotational (r) state

B_v(e)

Vibrational energy of a given particle in the electronic state e (eV or cm⁻¹)

E_r(e,v)

Rotational energy of a given particle in the electronic state e and the vibrational state v (eV or cm⁻¹)

E ^D_X

Dissociation energy of the diatomic molecule X₂ (eV)

${\mathrm{E}}_{{\mathrm{X}}^{+}}^{\mathrm{I}}$

Ionization energy of the atom X (eV)

F

Helmholtz free energy (J)

F₀

Reference energy (J)

g_i,s

Statistical weight of the chemical species i in the excited state s

g_v

Vibrational statistical weight

g_r

Rotational statistical weight

g_e

Electronic statistical weight

G

Gibbs free enthalpy (J)

h

Planck’s constant (h = 6.626 × 10⁻³⁴ J.s)

h_g

Specific enthalpy of the gas (kJ/kg)

H_i

Molar enthalpy of the chemical species i (kJ/mole)

H ^D_X

Dissociation enthalpy of the molecule X₂ (J)

H ^I_X

Ionization enthalpy of the molecule X(J)

J

Angular momentum of the spectral level under consideration

i

Index of the chemical species

k

Boltzmann’s constant (1.38 × I0-23 J/K)

K_c(X)

Equilibrium constant to produce the species X

K_p(X⁺)

Partial pressure equilibrium constant for ionization of X

K_p(X)

Partial pressure equilibrium constant for dissociation of X₂

ℓ_ij

Mean free path between collision of particles of type i and j (m)

m_i

Mass of a particle of chemical species i (kg)

M_i

Atomic mass of chemical species i (kg)

n

Principal quantum number

n *

Maximum value of n for the calculation of the partition function

n_i

Number density of chemical species i regardless of their excitation state (m⁻³)

n_i,s

Number density of chemical species, i in the excited state, s (m⁻³)

n ^′_i

Mole number of chemical species i regardless of their excitation state

n_T

Total number density (n_T = p/kT) (m⁻³)

N

Total number of particles

N_av

Avogadro number (Nav = 6.022 141 29 × 10²³)

N_i

Number of particles of chemical species, i whatever their excited state may be

N_i,s

Number of particles of chemical species, i in the excited stat, s

p

Pressure of the gas (Pa)

p_i

Partial pressure of the chemical species, i (Pa)

Qi

Partition function of the chemical species, i

Q ^tr_i

Translational partition function of the chemical species, i

Q ^int_i

Internal partition function of the chemical species, i

Q_tot

Total partition function of the gas $\left({\mathrm{Q}}_{\mathrm{tot}}={\displaystyle {\prod}_{\mathrm{i}}{\mathrm{Q}}_{\mathrm{i}}^{{\mathrm{N}}_{\mathrm{i}}}}/{\displaystyle {\prod}_{\mathrm{i}}{\mathrm{N}}_{\mathrm{i}}}\right)$

s

Index of the excited state

s*

Maximum value of s for the calculation of the partition function

S

Entropy of the gas (kJ/kg.K)

T

Equilibrium temperature (K)

T_h

Heavy species temperature (K)

T_i

Temperature of species i (K)

T_e

Electron temperature (K)

V

Volume of the gas or of the plasma (m³)

V_ij

Interaction potential between particles of type i and j

x_i

Molar fraction of chemical species, i

Z_i

Effective charge of the chemical species, i (Z_i = 0 for atoms or molecules, Z_i = 1 for the first ion, and so on…)

$\upgamma$

Specific heat ratio (c_p / c_v)

$\updelta {\mathrm{E}}_{{\mathrm{X}}^{+}}^{\mathrm{I}}$

Ionization potential lowering for X⁺ ion production (eV)

Δp

Lowering of the pressure (Pa)

ΔH

Enthalpy change of a thermodynamic state at T and p with respect to a reference state (J)

Γ

Isentropic coefficient

ε_o

Vacuum permittivity (ε_o = 8.854 187 817 × 10⁻¹² F/m)

λ_D

Debye length (m)

Λ_o

de Broglie’s wavelength for the chemical species, i (m)

μ_i

Chemical potential for the chemical species, i (J)

μ ^o_i

Part of the chemical potential of chemical species i that depends only on temperature (J)

Υ

Vibrational frequency (s⁻¹)

ω_e

Vibrational energy of the molecule at rest (cm⁻¹)

θ

Ratio of electron to heavy particle temperature