Kinetic Theory of Gases

Abstract

This chapter deals with the fundamentals of the kinetic theory of gases and the basic mechanism behind transport phenomena under plasma conditions. A brief introduction of particles and collisions is presented introducing the concepts of collision cross section and frequencies. This is followed by elementary processes for elastic and inelastic collisions, including excitation and ionization mechanisms. The concept of distribution functions and reaction rates for binary and three-body recombination is also briefly discussed.

Nomenclature and Greek Symbols

A

Surface (m²)

c

Velocity of light (c = 2.998 × 10⁸ m/s)

C_ij

Net rate of increase of particles in the control volume due to collisions

$\overrightarrow{\mathrm{c}}$

Center-of-mass velocity of two particles (m/s)

dA

Elementary surface (m²)

dN(v)

Number of particles with velocities between v and v+dv

e

Electron charge (e = 1.602 × 10⁻¹⁹C)

E

Electric field (V/m)

E_b

Binding energy of the molecular ion

E_i

Energy of the excited level i (EV or cm⁻¹)

E_ion

Ionization energy of an atom (eV)

E_m

Energy of a metastable state (eV)

E_s

Threshold energy (eV)

E*

Energy of an excited state (eV)

f°(v)

Maxwellian distribution function

$\mathrm{f}\left(\overrightarrow{\mathrm{r}},\overrightarrow{\mathrm{v}},\mathrm{t}\right)$

Velocity distribution function at time t at the tip of the position vector $\overrightarrow{\mathrm{r}}$

${\mathrm{f}}_{\mathrm{n}}\left(\overrightarrow{\mathrm{r}},\overrightarrow{\mathrm{v}},\mathrm{t}\right)$

Normalized distribution function

F_i

Incident flux of particles i (m⁻²·s⁻¹)

${\mathrm{F}}_{\mathrm{n}}^{+}$

Flux per unit time and unit surface carried to the (+) side of a surface defined by its normal $\overrightarrow{\mathrm{n}}$ and moving with the gas velocity ${\overrightarrow{\mathrm{v}}}_{\mathrm{y}}$

${\mathrm{F}}_{\mathrm{n}}^{-}$

Flux per unit time and unit surface carried to the (-) side of a surface defined by its normal $\overrightarrow{\mathrm{n}}$ and moving with the gas velocity ${\overrightarrow{\mathrm{v}}}_{\mathrm{y}}$

h

Planck’s constant (h = 6.626 × 10⁻³⁴ J·s, or kg·m²/s)

J

Associated quantum number

$\overrightarrow{\mathrm{J}}$

Total angular momentum $\left(\overrightarrow{\mathrm{J}},=,{\sum}_{\mathrm{i}},{\overrightarrow{\mathrm{J}}}_{ \mathrm{i}}\right)$

k

Boltzmann’s constant (k = 1.38 × 10⁻²³ J/K)

${\overline{\mathrm{k}}}_{12}\left({\overrightarrow{\mathrm{v}}}_{12}\right)$

Mean direct binary collision rate or reaction rate (m³/s)

$\overline{\mathrm{k}}\left({\overrightarrow{\mathrm{v}}}_1,{\overrightarrow{\mathrm{v}}}_2,{\overrightarrow{\mathrm{v}}}_3\right)$

Mean direct three-body collision rate (m⁶/s)

K

Fraction of energy transferred by elastic collision from one particle of mass m to another of mass M averaged over all angles

ℓ

Mean free path (m)

ℓ_i

Mean free path of particles of type i in a mixture of different chemical species (m)

ℓ_ij

Mean free path traversed by a particle of type i between two successive collisions with particles of type j (m)

ℓ

As index, lower energy level

L

Distance between electrodes (m)

m

Mass of a particle (kg)

m_H

Mass of hydrogen atom $\left(\frac{{\mathrm{m}}_{\mathrm{H}}}{{\mathrm{m}}_{\mathrm{e}}},=,1856\right)$

m_i

Mass of the species i (kg)

M

Mass of a particle (kg)

n

Principal quantum number

$\overrightarrow{\mathrm{n}}$

Normal to a surface dA

n_i

Number density of particles of chemical species i (m⁻³)

p

Absolute value of momentum

$\overrightarrow{\mathrm{p}}$

Momentum vector

p_ij(V_ij, θ, φ)

Proportionality factor defining the differential cross section

P(x)

Probability of a free path to exceed a distance x (m)

P(t)

Probability that a particle with relative velocity $\left(\overrightarrow{\mathrm{V}}={\overrightarrow{\mathrm{v}}}_1-{\overrightarrow{\mathrm{v}}}_2\right)$ survives a time t without suffering a collision

P*(t).dt

Probability that a particle, after surviving without collisions for a time t, suffers a collision in the time interval between t and t + dt

Q_i

Total effective cross section for collision processes of particle i with particles j (m⁻¹)

Q

Total effective cross section $\left(\mathrm{Q},=,{\displaystyle \sum_{\mathrm{i}}},{\mathrm{Q}}_{\mathrm{i}}\right)$

u

As index, upper energy level

$\overrightarrow{\mathrm{U}}$

Peculiar velocity of a particle $\left(\overrightarrow{\mathrm{U}}=\overrightarrow{\mathrm{v}}-{\overrightarrow{\mathrm{v}}}_{\mathrm{g}}\right)$ where $\overrightarrow{\mathrm{v}}$ is the particle velocity and ${\overrightarrow{\mathrm{v}}}_{\mathrm{g}}$ the gas velocity (m/s)

U_n

Normal component of the peculiar velocity $\left(\overrightarrow{\mathrm{U}}=\overrightarrow{\mathrm{v}}-{\overrightarrow{\mathrm{v}}}_{\mathrm{g}}\right)$ (m/s)

${\overrightarrow{\mathrm{r}}}_{ \mathrm{i}}$

Position vector of particle i (m)

${\overrightarrow{\mathrm{r}}}_{ \mathrm{j}}$

Position vector of particle j (m)

${\overrightarrow{\mathrm{R}}}_{ \mathrm{i}\mathrm{j}}$

Relative position of particles i and j $\left({\overrightarrow{\mathrm{R}}}_{ \mathrm{i}\mathrm{j}},=,{\overrightarrow{\mathrm{r}}}_{ \mathrm{i}},-,{\overrightarrow{\mathrm{r}}}_{ \mathrm{j}}\right)$

T

Temperature (K)

v

Absolute value of the velocity (m/s)

v_m

Most probable velocity (m/s)

$\overline{\mathrm{v}}$

Mean velocity (m/s)

$\overline{{\mathrm{v}}^2}$

Mean square of the effective velocity (m/s)

${\overrightarrow{\mathrm{v}}}_{\mathrm{i}}$

Velocity vector of a particle of chemical species i (m/s)

${\overrightarrow{\mathrm{v}}}_{\mathrm{g}}$

Mean velocity referred to as the species fluid (or average) velocity relative to some laboratory frame of reference (m/s)

$\mathrm{V}$

Relative velocity of particles of chemical species i and j before collision (m/s)

${\mathrm{V}}^{\prime }$

Relative velocity of particles of chemical species i and j after collision (m/s)

${\overrightarrow{\mathrm{V}}}_{\mathrm{ij}}$

Relative velocity vector of particles i and j before collision (m/s)

${\overrightarrow{\mathrm{V}}}_{\mathrm{ij}}^{\prime }$

Relative velocity vector of particles i and j after collision (m/s)

${\overline{\mathrm{V}}}_{\mathrm{ij}}$

Mean relative velocity of particles i and j (m/s)

w.dt

Probability that a particle suffers a collision between t and t + dt, also called collision rate

X

Designation for the chemical species in its ground state

X*

Designation for the chemical species in an excited state

X^m

Designation for the chemical species in a metastable state

X⁺

Designation for the chemical species in its first ionized state

X⁺⁺

Designation for the chemical species in its second ionized state

X₂

Designation for the molecule made of two X atoms in its ground state

${\mathrm{X}}_2^{+}$

Designation for the ionized molecule

Y

Designation for the chemical species in its ground state

Z'

Number of protons in the nucleus

α

Recombination reaction rate coefficient (m³/s)

ΔE

Energy difference between two excited states (eV or cm⁻¹)

ΔE_tot

Total kinetic energy exchange during a collision (eV, J, cm⁻¹)

ΔJ

Difference in the angular momentum of the atom between its initial and final states

ΔP

Change in angular momentum during collisions (ΔP = h·ΔJ)

θ

Angle in spherical coordinates

λ_i

Maximum incident wavelength for ionization by photons (nm)

Λ

Wavelength (nm)

σ

Wave number (m⁻¹)

σ₀

Total scattering cross section (m²)

σ_ij

Differential scattering cross section (m²)

σ_imax

Maximum value of σ(m⁻¹)

σ₀(V_ij)

Total scattering cross section (m²)

$\upsigma \left({\overrightarrow{\mathrm{V}}}_{\mathrm{ij}},\uptheta, \upvarphi \right)$

Differential scattering cross section (m²)

${\upsigma}_{\mathrm{ij}}\left({\overrightarrow{\mathrm{V}}}_{\mathrm{ij}},\uptheta, \upvarphi \right)$

Differential scattering cross section (m²)

τ

Mean time between collisions also called relaxation time (s)

τ_ij

Mean time between collisions of two particles of types i and j

υ

Frequency of the associated wave (s⁻¹ or Hz)

υ_ij

Collision frequency between two particles of types i and j (s⁻¹)

φ

Azimuthal angle in spherical coordinates

$\upchi \left(\overrightarrow{\mathrm{r}},\overrightarrow{\mathrm{v}},\mathrm{t}\right)$

Particle properties in terms of position $\overrightarrow{\mathrm{r}}$ and velocity $\overrightarrow{\mathrm{v}}$ at time t

Ω

Solid angle (ster.)