Environmental Engineering Reference
In-Depth Information
same as in (4.88). Table 4.11 contains the diffusion coefficients of molecular ions
of inert gases in parent gases.
We now consider the mobility of atomic ions in a parent atomic gas taking into
account that the cross section of the resonant charge exchange process exceeds sig-
nificantly the cross section of elastic ion-atom scattering. Then straight trajectories
of colliding ions and atoms give the main contribution to the diffusion cross sec-
tion, so the scattering angle is
# D
π
. Correspondingly, the diffusion cross section
is [52]
Z
(1
σ
D
cos
#
)
d
σ
D
2
σ
res
,
(4.91)
where
res
is the cross section of the resonant charge exchange process. Then,
assuming the resonant charge exchange cross section
σ
res
to be independent of
the collision velocity, we obtain from (4.21) for the ion mobility in a parent gas in
the first Chapman-Enskog approximation [43, 53, 54]
σ
3
e
p
π
16
N
a
K
I
D
res
p
TM
,
σ
where
M
is the mass of the ion or atom, and we assume here the cross section
σ
res
of resonant charge exchange to be independent of the collision velocity. The second
Chapman-Enskog approximation gives a correction [55],
1/40. Accounting for
a weak velocity dependence of the cross section of resonant charge exchange, we
obtain in the second Chapman-Enskog approximation [43, 54]
Δ
D
0.341
e
N
a
p
Tm
a
K
II
D
,
(4.92)
σ
res
(2.1
v
T
)
where the argument of the cross secti
on indicat
es the collision velocity for which
the cross section is taken, and
v
T
D
p
8
T
/(
m
a
) is the average atom velocity. This
formula may be represented in a convenient form [43, 54]
π
1340
p
Tm
a
K
D
,
(4.93)
σ
res
(2.1
v
T
)
10
19
cm
3
;
the mobility is expressed in square centimeters per volt per second, the gas tem-
perature is given in Kelvin, the atom mass
m
a
is expressed in atomic mass units
(1.66
which is related to the normal number density of gas atoms
N
a
D
2.69
10
24
g), and the cross section is given in units of 10
15
cm
2
.
In analyzing the mobility of an atomic ion in a parent gas in a weak electric
field, we ignore elastic ion-atom scattering. We now take it into account assuming
its contribution to be small and is determined by the ion-atom polarization inter-
action [43, 56]. In this case the cross section of resonant charge exchange is given
by (2.46), and the zero-field mobility of an atomic gas in a parent gas is given by [43]
p
α
K
0
e
2
/
T
K
D
,
x
D
res
(
p
7.5
T
/
M
)
,
(4.94)
1
C
x
C
x
2
σ
