Environmental Engineering Reference
In-Depth Information
gas. In this case, approach of ions with charges of different signs results from the
action of the field of a given ion on the ion of the opposite sign, and approach of
ions is braked due to gas resistance through ion collisions with gas atoms. As a
result, the mobilities of ions under the action of the field of another ion give the
rate of ion-ion recombination.
Indeed, if each ion has charge
e
and the distance between ions is
R
,theelectric
field of each ion at the location of another ion is
E
˙
e
/
R
2
,andthisfieldcauses
D˙
ion approach, which is determined by the drift velocity:
eE
(
K
C
C
K
)
w
D
E
(
K
C
C
K
)
D
.
R
2
This is valid for a dense gas if the criterion
R
λ
holds true, and
λ
is the mean
free path of ions in a gas.
To determine the decay rate for positive ions due to recombination, imagine a
sphere of radius
R
around the positive ion, and compute the number of negative
ions entering this sphere per unit time. This is given by the product of the surface
area of the sphere, 4
R
2
,andthenegativeionflux,
j
N
e
(
K
C
C
K
)/
R
2
.Fromthiswehavethebalanceequationforthenumberdensityofpositive
ions is
π
D
N
w
D
dN
C
dt
D
N
C
N
4
π
e
(
K
C
C
K
).
Comparing this with the definition of the recombination coefficient, we derive for
this coefficient the Langevin formula [105]:
α
D
4
π
e
(
K
C
C
K
) .
(4.129)
This formula holds true for a dense gas, and we now combine all the limiting cases
for recombination of positive and negative ions. Below we summarize various cases
of recombination of positive and negative ions in a gas.
Figure 4.21 gives the dependence of the recombination coefficient for positive
and negative ions in a gas on the number density
N
of gas atoms, and we divide
the range of the atom number density into three groups. At low number densi-
ties recombination of positive and negative ions in a gas has the pairwise charac-
ter (2.98), and in accordance with (2.100) the value of the pairwise recombination
coefficient in this range of densities is
2
/
m
e
μ
T
1/2
,since
R
0
α
„
a
0
.Here
1
m
e
is the electron mass,
μ
is the reduced mass of the ions,
T
is the gas (ion) tem-
perature, and
a
0
2
/(
m
e
e
2
) is the Bohr radius. The three body process of ion-
ion recombination (2.90) corresponds to region 2, and the recombination coeffi-
cient for positive and negative ions in a gas is determined on the basis of (2.91),
α
D„
(
Ne
6
/
T
3
)(
e
2
/
)
1/2
,where
μ
is the polarizability of the gas atom,
N
is the
2
number density of gas atoms, and
is the reduced mass of the ion and atom. From
this one can estimate the boundary number density
N
1
of atoms for transition from
region 1 to region 2 in Figure 4.21:
μ
a
0
T
e
2
5/2
1/2
N
1
.
