Environmental Engineering Reference
In-Depth Information
⎩
1
⎭
N
=
N
exp
−
(
g
−
μ
v
n
)
(19.29)
n
1
n
k T
B
The rate of nucleation is proportional to the number
N
crit
of the critical nuclei [1,11,32].
Frenkel has failed to ind the connection between the size distributions in Equations 19.25 and
19.29. To do this, Lothe and Pound have considered an imaginary process (devised by Gibbs), in
which a cluster embedded in the bulk liquid is transferred to the vapor phase.
In the Gibbs theory of interface the work
W
crit
of formation of critical nucleus is represented by
the following formula [12]:
β
α
W
= −
V P
(
−
P
)
+
σ
A
(19.30)
crit
S
s
S
where
P
β
is the pressure of the reference bulk liquid having the same temperature and chemical poten-
tial as the vapor
A
S
is the area of the surface of tension, which is assumed to be spherical
V
S
is the volume enclosed by this surface
To make the meaning of Equation 19.30 clearer, Gibbs has introduced an imaginary process consist-
ing of two separate stages. Let us consider this process in a nutshell. Initially the system consists of
the bulk liquid reference phase (at pressure
P
β
) and the bulk vapor phase (at pressure
P
α
). Due to the
pressure difference the reference phase is surrounded by an elastic envelope. However, it is assumed
that the envelope is transmittable for the gas molecules. First, some number of molecules from the
vapor is transferred to the bulk liquid. The volume of the reference phase increases by
V
S
due to this
transfer but the surface area is kept constant. During this stage the beneit of work is (
P
β
−
P
α
)
V
S
.
In the next stage an aperture in the envelope opens and then closes so that a volume
V
S
of the liquid
phase is extruded outside and the envelope intrudes inside to decrease the volume by the same mag-
nitude
V
S
. The total work at the second stage is σ
s
A
S
. This work includes different components. For
instance, during the extrusion the drop loses the interaction with the bulk liquid; therefore, σ
s
A
S
con-
tains the potential energy of this interaction. σ
s
A
S
also includes the work of structural relaxation of
the extruded drop and the adsorption of some gas molecules to its surface.
In the Gibbs's thought process, the translational and rotational degrees of freedom of the cluster
is not taken into account, either in the bulk liquid or in the vapor. Lothe and Pound [8] proposed the
way to account the difference in free energy associated with these degrees of freedom is necessary.
They added to
W
crit
the translational and rotational free energies of the cluster in the gas phase and
subtracted the entropy contribution to free energy associated with the vibrational translation and rota-
tion of the embedded cluster with the relative positions of the molecules in the cluster held ixed. The
partition function corresponding to those vibrational modes of luctuation,
q
rep
, is called the replace-
ment partition function, since these modes are replaced by the free translation and free rotation in the
vapor phase. The procedure of translation-rotation correction results in the Lothe-Pound factor [11]:
Q Q
q
Q
Q
Q
Q
tr
rot
tr
rot
Φ
LP
=
=
(19.31)
l
l
rep
tr
rot
where
l
l
q
rep
=
Q Q
(19.32)
tr
rot
Q
t
l
is the partition function of vibrational translations of the embedded cluster
Q
rot
l
is the partition function of vibrational rotations around the center of mass of the cluster
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