Environmental Engineering Reference
In-Depth Information
Ta b l e 1 . 6
Parameter 1/(2
x
e
) in (1.73).
Molecule
H
2
OH
CO
N
2
NO
O
2
1/(2
x
e
)
18
22
82
84
68
66
and the minimum number density of excited molecules is given by
N
0
exp
.
„
ω
v
min
2
T
ν
N
(
v
min
)
D
Table 1.6 contains the values of the first factor in (1.73) for some molecules. The
effect considered is remarkable at
v
10 in terms of the distinction between vi-
brational and translational temperatures.
Thus, the special feature of the Treanor effect is that at high vibrational excita-
tions, collisions of molecules with transfer of vibrational excitation energy to trans-
lational energy become effective. This causes a mixing of vibrational and transla-
tional subsystems, and invalidates the Boltzmann distribution for excited states as
a function of vibrational temperature. Note that the model employed is not valid
for very large excitations because of the vibrational relaxation processes.
1.2.11
Normal Distribution
A commonly encountered case in plasma physics as well as in other types of
physics is one in which a variable changes by small increments, each change oc-
curring randomly, and the distribution of the variable after many steps is studied.
Examples of this are the diffusive motion of a particle, or the energy distribution
of electrons in a gas. This energy distribution as it occurs in a plasma results from
elastic collisions of electrons with atoms, with each collision between an electron
and an atom leading to an energy exchange between them that is small because of
the large difference in their masses. Thus, in a general statement of this problem,
we seek the probability that some variable
z
has a given value after
n
1steps,if
the distribution for each step is random and its parameters are given.
Let the function
f
(
z
,
n
) be the probability that the variable has a given value
after
n
steps, with
(
z
k
)
dz
k
the probability that after the
k
th step the change of the
variable lies in the interval between
z
k
and
z
k
'
C
dz
k
. Since the functions
f
(
z
)and
'
(
z
) are probabilities, they are normalized by the condition
1
Z
Z
1
f
(
z
,
n
)
dz
D
'
(
z
)
dz
D
1.
1
1
By definition of the above functions we have
Z
Z
1
1
Y
n
X
n
f
(
z
,
n
)
D
dz
1
dz
n
1
'
(
z
k
),
z
D
z
k
.
(1.74)
k
D
k
D
1
1
1
