Chemistry Reference
In-Depth Information
∂
f
i
∂
S
i
=
d
3
v
i
t
Coll
d
i
d
3
v
l
d
3
v
k
· |
=
v
l
−
v
k
| ·
σ
klmn
(
|
v
l
−
v
k
|
,
)
·
(
f
k
·
f
l
−
f
m
·
f
n
)
klmn
i
klmn
(3.141)
too.
In most cases the gain and loss terms are expressed by means of the rate
coefficient
k
klmn
±
S
i
=
k
i
klmn
·
n
k
·
n
l
d
3
v
k
·
f
l
klmn
±
=
d
3
v
l
· |
v
l
−
v
k
| ·
σ
i
klmn
(
|
v
l
−
v
k
|
)
·
f
k
·
·
n
k
·
n
l
(3.142)
of the binary collision with
k
i
klmn
=
s
−
1
.
As an example, we assume collisions between two particles with mass
m
1
and
m
2
. The Maxwellian velocity distribution at the same temperature
T
does not depend
on the internal energy states of the particles. The transformation in the center of mass
system using the reduced mass
m
red
and the integration over the relative velocity
v
r
results in following analytical expression of the rate coefficient
k
m
3
·
(
T
)
f
k
(
T
)
=
v
r
·
σ
(
v
r
)
=
(
v
r
)
·
v
r
·
σ
(
v
r
)
·
dv
r
m
red
2π
3
/
2
exp
∞
m
red
·
v
r
=
·
·
v
r
)
·
v
r
·
−
·
4π
σ
(
dv
r
(3.143)
·
k
B
T
2
·
k
B
T
0
with the reduced mass
m
red
=
(
m
1
·
m
2
)/(
m
1
+
m
2
)
and the relative velocity
v
r
=
|
v
2
|
.
In similar way the corresponding rate coefficient
k
v
1
−
(
T
)
is achieved using the
relative translational energy
1
/
2
exp
d
ε
r
∞
1
k
B
T
·
8
ε
r
k
B
T
k
(
T
)
=
·
σ
(
ε
r
)
·
ε
r
·
−
(3.144)
π
·
m
red
·
k
B
T
0
with the relative energy ε
r
=
v
r
.
Another simple case considers two different temperatures
T
1
and
T
2
of the parti-
cles with masses
m
1
and
m
2
, respectively. An effective temperature can be defined in
m
red
/
2
·
the expression of the rate coefficient
k
T
eff
.
m
red
2π
3
/
2
exp
∞
k
T
eff
=
m
red
·
v
r
4π
·
·
σ
(
v
r
)
·
v
r
·
−
·
dv
r
(3.145)
·
·
k
B
T
eff
2
k
B
T
eff
0
T
eff
=
(
m
1
T
2
+
m
2
T
1
)/(
m
1
+
m
2
)
and
m
red
=
(
m
1
·
m
2
)/(
m
1
+
m
2
)
.
