Environmental Engineering Reference
In-Depth Information
widespread cross sections which determine transport coefficients of atomic parti-
cles in gases, we have, accounting for two terms of expansion of these cross sections
over a small parameter, 1/
n
[12-14]
Z
(1
U
(
R
)
ε
σ
D
R
2
cos
#
)
d
σ
D
π
,
D
0.89
I
Z
(1
2
3
π
U
(
R
2
)
ε
(2)
cos
2
R
2
,
σ
D
#
)
d
σ
D
D
0.23 .
(2.15)
These expressions conserve the form of the hard-sphere model and allow us to
determine the sphere radius
R
0
.
Note that if the hard-sphere model holds true for the collision of a test atomic
particle with gas atoms or molecules, one can introduce the mean free path of the
test particle as
λ
D
1/(
N
σ
), where
N
is the number density of gas atoms and
R
0
is the diffusion cross section of the test particle colliding with gas atoms.
The concept of the mean free path was introduced by Clausius [15] and was used
by Maxwell [3-5, 16] for determination of transport parameters of gases.
σ
D
π
2.1.3
Collision Processes Involving Clusters
The hard-sphere model was applied above to the problem of collision of two gas
atoms or molecules with a strongly varied repulsive interaction potential. In reality
this model was designed in the first place for the analysis of many-particle sys-
tems. In particular, this model allows us to describe the properties of dense gases
and condensed systems, allowing us to obtain the thermodynamic parameters of
liquids [17, 18] and solids within the framework of this model. Since one of topics
covered in this topic is processes with clusters in an ionized gas, we continue with
the hard-sphere model for clusters by changing the interaction potential (2.10) of
atoms by adding of a narrow attraction potential at
R
R
0
.Asaresult,weobtain
the liquid drop model that represents a system of interacting particles as a liquid
drop consisting of small incompressible liquid drops. As the hard-sphere model,
the liquid drop model for a system of bound particles has a universal character. In
particular, it was introduced by Bohr [19] in nuclear physics and allows one to an-
alyze various properties of atomic nuclei [20-22], including nuclear fission [23]. In
considering a cluster consisting of many atoms, we reduce the interactions inside
this system to electrostatic and exchange interaction between individual atoms or
ions that chooses an optimal distance between the nearest atomic particles. We use
the analogy of clusters with the macroscopic system of atoms where the compe-
tition between electrostatic and exchange (electron-electron) interactions chooses
the optimal distance between atoms or ions [24, 25]. We then obtain the liquid drop
model for a cluster that has a spherical shape and is cut off a macroscopic system.
Assuming the number density of atoms in the cluster and bulk system to be identi-
cal and taking the number of cluster atoms to be
n
,weobtainfromthisforacluster
D
