Environmental Engineering Reference
In-Depth Information
Here and below we use the notation
d
p
D
dxdydz
.The
quantity
d
p
d
r
is called a differential element of the phase space, and the number of
states in (1.44) is the statistical weight of the continuous spectrum of states because
it is the number of states for an element of the phase space.
Let us consider now some cases of the Boltzmann distribution of particles. First
we analyze the distribution of diatomic molecules in vibrational and rotational
states. The excitation energy for the
dp
x
dp
y
dp
z
and
d
r
D
th vibrational level of the molecule is giv-
v
en by
is the
energy difference between neighboring vibrational levels. On the basis of (1.42),
we have
„
ω
v
if the molecule is modeled by a harmonic oscillator. Here
„
ω
N
0
exp
,
„
ω
v
T
N
v
D
(1.45)
where
N
0
is the number density of molecules in the ground vibrational state. Be-
cause the total number density of molecules is
exp
X
1
X
1
„
ω
v
T
N
0
h
1
exp
„
T
i
N
D
N
v
D
N
0
D
,
v
D
0
v
D
0
the number density of excited molecules is
exp(
„
ω
v
/
T
)
N
v
D
N
.
(1.46)
[1
exp(
„
ω
/
T
)]
The excitation energy of the rotational state with angular momentum
J
is given
by
BJ
(
J
1), where
B
is the rotational constant of the molecule, and the statis-
tical weight of this state is 2
J
C
C
1. On the basis of the normalization condition
P
J
N
v
J
T
(asisusuallydone),thenumberdensityof
molecules in a given vibrational-rotational state is
D
N
v
and assuming
B
T
exp
.
1)
B
BJ
(
J
C
1)
N
v
J
D
N
v
(2
J
C
(1.47)
T
As an example of the particle distribution in an external field, let us consid-
er the distribution of particles in the gravitational field. In this case (1.42) gives
N
(
x
)
U
/
T
), where
U
is the potential energy of the particle in the external
field. For the gravitational field we have
U
exp(
mgz
,where
m
is the mass of the
molecule,
g
is the free-fall acceleration, and
z
is the altitude above the Earth's sur-
face. Formula (1.42) takes the form of the barometric distribution, and then has the
form
D
N
(0) exp
,
mgh
T
T
mg
,
N
(
z
)
D
l
D
(1.48)
where
N
(
z
) is the number density of molecules at altitude
z
.Foratmosphericair
near the Earth's surface at room temperature we have
l
9 km, that is, atmospher-
ic pressure falls noticeably at altitudes of a few kilometers.
