Environmental Engineering Reference
In-Depth Information
Substitution of this expression into (3.6) gives for the collision integral
Z
f
1
f
2
f
1
f
2
j
v
1
d
v
1
d
v
2
d
v
2
.
I
col
(
f
)
D
v
2
j
d
σ
(3.8)
The nine integrations implied in (3.6) are replaced by five integrations in (3.8). This
is a consequence of accounting for the conservation of momentum for the colliding
particles (three integrations) and the conservation of their total energy (one more
integration).
3.1.3
Equilibrium Distribution of Gas Atoms
The kinetic equation (3.4) allows us to analyze the equilibrium state of an atomic
gas. As follows from (3.4), the equilibrium state of a gas
@
f
/
@
t
D
0satisfiesthe
equation
I
col
(
f
)
0. As follows from (3.6) for the collision integral, this equation
requires fulfillment of the following relation for any pair of colliding atoms:
D
f
1
f
2
.
f
1
f
2
D
Let us rewrite this relation in the form
ln
f
(
v
1
)
ln
f
(
v
2
).
ln
f
(
v
1
)
C
ln
f
(
v
2
)
D
C
From this it follows that ln
f
(
v
) is an additive function of the integrals of motion.
Taking into account the conservation of total momentum and total energy of the
atoms, we obtain the general form of the distribution function as
ln
f
(
v
)
D
C
1
C
(
C
2
p
)
C
C
3
ε
,
where
p
and
are the momentum and kinetic energy of an atom. This leads to the
distribution function in the form
ε
(
v
w
)
2
].
f
(
v
)
D
A
exp[
α
This expression corresponds to the Maxwell distribution function, where
A
is the
normalization constant,
w
is the average velocity of the distribution, and
α
D
m
/(2
T
), where
m
is the mass of an atom and
T
is the temperature of the gas.
Thus, the equilibrium is established in a gaseous ensemble of identical particles
as a result of collisions between them, and these collisions lead to the Maxwell
distribution function for particles, which has the form
A
exp
m
(
v
w
)
2
2
T
'
(
v
)
D
,
(3.9)
where
A
is the normalization constant in accordance with normalization condi-
tion (3.1) and
w
is the average velocity of particles.
