3.5 Gaseous (Barrier) Diffusion
139
Fig. 3.11 Escape cylinder
for a particle with velocity v
v
y
x
z
t
Δ
The number of atoms contained within the cylinder shown in Fig. 3.11 will be the
volume of the cylinder times the number density of atoms ρ N = N/V, where N is the
number of atoms in the gas and V is the volume of the container (In this section, ρ
designates number density, not mass density). To make number density a meaningful
concept, we have to assume that the density of the gas stays constant as atoms fly in
and out through the sides of the cylinder; we presume that for each atom that leaves
the escape cylinder, one arrives to take its place.
The volume of a cylinder of top area S, slant length v and tilt angle θ is given
by
V cyl = S v( cos θ.
(3.50)
The number of atoms in the escape cylinder will then be
N cyl = ρ N S v ( cos θ.
(3.51)
Now, not all of these N cyl atoms will be moving in the correct direction (θ, φ) to
achieve escape. To account for this, we have to multiply (3.51) by the probability of
an atom having its velocity so directed, which is given by (3.49):
N esc ((t, θ, φ) = N cyl P (dΩ) =
ρ N S v(
4π
cos θ sin θ dθ dφ.
(3.52)
We can account for all possible directions of escape by integrating (3.52) over the
relevant angles: