Consider a gas mixture of two types of particles ( = 1, 2), each one characterized by

Consider a gas mixture of two types of particles (α = 1, 2), each one characterized by a Maxwellian distribution function

with its own mass, density, and temperature.

(a) Make the following transformation of velocity variables,

where v′_c is a velocity similar to the center of mass velocity, g is the relative velocity between the two species (g = v₁ – v₂), and

Show that the Jacobian of this transformation,

satisfies |J| = 1, so that d³v′c d³g = d³v₁ d³v₂ .

(b) The relative speed between the two species, g = |v₁ – v₂|, when averaged over both their velocity distribution functions, is given by

Transform the variables of integration v₁ and v₂ to v′_c and g, and perform the integrals over v′_c and g to show that

(c) If only one kind of particle is present, so that m₁ = m₂ = m, T₁ = T₂ = T, and n₁ = n₂ = n, show that

where < v > = (8kT/πm)^1/2 is the average speed and μ = m/2 is the reduced mass. If the mutual scattering cross section is a, show that the collision frequency in a homogeneous Maxwellian gas is given by

Transcribed Image Text:

ma r) = na(27) exp(-2kfa fa(v) = 3/2