Environmental Engineering Reference
In-Depth Information
The cross section
if
may depend on the relative velocity of the particles.
One can give another interpretation for the transition cross section
σ
if
.Letus
place particles
B
on a plane with surface density
n
. Then the probability of the tran-
sition proceeding only between states
i
and
f
is
n
σ
1) after intersection
of this plane by a particle
A
in the direction perpendicular to this plane.
When particles
A
and
B
can have arbitrary velocities, the transition probability
per unit time may be represented in the form
σ
if
(
n
σ
if
ν
D
[
B
]
hj
v
A
v
B
j
σ
iD
[
B
]
h
k
if
i
,
(2.3)
if
if
where the angle brackets signify an average over relative velocities of the particles,
and the rate constant for the process
k
if
Dhj
v
A
v
B
j
σ
i
(2.4)
if
is also a characteristic of the elementary act of collision. The rate constant for the
process is useful when we are interested in the rate of transition averaged over the
velocities of the particles.
It is useful to use the rate constants
k
if
for inelastic collisions in the balance
equation for the number density
N
i
of particles
A
foundinagivenstate
i
if transi-
tions proceeds as a result of collisions with gas particles
B
. This equation has the
form
[
B
]
X
f
[
B
]
N
i
X
f
dN
i
dt
D
k
fi
N
f
k
if
.
(2.5)
The balance equation (2.5) may be extended by including other processes.
In elastic collisions of particles internal states of the colliding particles remain
unchanged. Let us consider the classical case of particle collision that, in particular,
takes place in thermal collisions of atoms in gases. Then the particle motion is
described by Newton's equations
m
1
d
2
,
m
2
d
2
dt
2
D
@
R
1
U
@
R
1
dt
2
D
@
R
2
U
@
R
2
.
Here
R
1
and
R
2
are coordinates of colliding particles,
m
1
and
m
2
are their masses,
and
U
is the interaction potential between the particles and depends on the relative
distance between particles only, that is,
U
D
U
(
R
1
R
2
). Therefore,
@
U
/
@
R
i
is
the force acting on particle
i
due to the other particle, and
@
U
/
@
R
1
D@
U
/
@
R
2
.
Introducing the coordinate of the center of mass
R
c
D
(
m
1
R
1
C
m
2
R
2
)/(
m
1
C
m
2
)
and the relative distance between colliding particles
R
D
R
1
R
2
, we rewrite the
set of Newton's equations in terms of these variables:
m
2
)
d
2
R
c
dt
2
D
d
2
dt
2
D
@
R
U
@
R
(
m
1
C
0,
μ
,
m
2
) is the reduced mass of colliding particles. From this
it follows that the center of mass travels with a constant velocity, and scattering is
μ
D
m
1
m
2
/(
m
1
C
where
