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M
2p
3=2
v
x
þ
v
y
þ
v
z
P
ð
v
x
;
v
y
;
v
z
Þ¼
exp
M
(6.12)
RT
2
RT
where P(v
x
, v
y
, v
z
)dv
x
dv
y
dv
z
is the fraction of all molecules having velocities in the range of
(v
x
, v
y
, v
z
)to(v
x
þ
dv
z
) in the Cartesian (rectangular) coordinate axes of x, y,
and z. For an isotropic medium, i.e. the mixture (locally) is uniform, there is directional pref-
erence for any changes. Let us do a change of variables by setting
dv
x
, v
y
þ
dv
y
, v
z
þ
q
v
x
þ
v
y
þ
v
z
v
¼
v
x
¼
v
cos q sin f
(6.13)
v
y
¼
v
sin q sin f
v
z
¼
v
cos f
which corresponding to a transformation from Cartesian space of (v
x
, v
y
, v
z
) to spherical
space of (v,
q
,
f
). As such, 0
f p
,0
q
2
p
. This change of variables applies to
Eqn
(6.12)
, we have
v
2
sin f
ð2pRT
Þ
3=2
e
Mv
2
=ð2RTÞ
d
v
d
f
d
q
P
ð
v
; q; fÞ
d
v
d
f
d
q ¼
(6.14)
=
M
where P(v,
f
,
q
)dvd
f
d
q
is the fraction of all molecules having velocities in the range v to
v
þ
dv, with the directions setting between (
f
,
q
)to(
f þ
d
f
,
q þ
d
q
). Integrating the whole
range of 0
f p
and 0
q
2
p
, we obtain
v
2
4
e
Mv
2
=ð2RTÞ
d
v
P
ð
v
Þ
d
v
¼
(6.15)
Þ
3=2
p
1=2
ð2
RT
=
M
where P(v)dv is the fraction of all molecules having velocities in the range v to v
þ
dv. This is
the Maxwell
Boltzmann distribution, we
obtain other quantities of interest, such as the average velocity and the root-mean-square
velocity. These are listed in
Table 6.1
.
e
Boltzmann distribution. From this Maxwell
e
TABLE 6.1
Properties Derived from the Maxwell
e
Boltzmann Distribution Law
Property
Description
Relation
Value
¼
R
0
r
2
v
Average velocity
v
vP
ð
v
Þ
d
v
RT
pM
2
r
2
Z
v
m
1
2
¼
RT
M
v
m
Median velocity
P
ð
v
Þ
d
v
1:0877
0
3
RT
M
v
2
Second momentum of
distribution
v
2
¼
R
0
r
3
v
2
P
ð
v
Þ
d
v
p
v
2
RT
M
Root-mean-square
velocity
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