Environmental Engineering Reference
In-Depth Information
Equation 19.5 may be written with the supericial density Γ of matter computed with respect to the
surface of tension [33]:
−
1
d
[
lnσ
(
R
)]
2
Γ
Δ
m
⎛
⎜
2
Γ
m
⎞
⎟
S
S
=
1
+
(19.7)
d
[
ln
R
]
R
ρ
R
Δ
S
S
S
where Δρ(g/cm
3
) = ρ
β
− ρ
α
is the difference between the densities of reference bulk phases β and α.
If the function σ(
R
S
) is known, it is possible to solve Equation 19.5 and determine the function δ(
R
S
).
On the other hand, Equation 19.6 gives the relationship between
R
S
and the real radius
R
e
. The radius
R
e
is the radius of the dividing surface which is chosen in such a way that the supericial density
Γ computed with respect to this surface is equal to zero [12].
As follows from the Gibbs theory [12] the work of formation of the critical nucleus is
π σ (
=
4
R
R
)
S
S
S
W
(19.8)
crit
3
The formula used in the CNT is not the expression (19.8) but uses the surface tension σ
∞
for the lat
surface instead of σ(
R
S
) and
R
e
instead of
R
S
which is a rough approximation. Thus, CNT gives for
the nucleation rate [1]:
2
⎛
2
⎞
N
V
2
m
σ
π
1
4
π σ
R
k T
=
⎛
⎞
⎟
1
∞
e
∞
I
exp
−
(19.9)
⎜
⎟
⎜
CNT
ρ
3
⎝
⎠
B
where ρ(g/cm
3
) is the density of incompressible bulk liquid. Besides, the proper account of the
translational and rotational free energies of the critical nucleus is missing in CNT resulting in
underestimation by orders of magnitude in the classical formula for the nucleation rate. In the fol-
lowing sections we will give an accurate derivation of the formula taking into account both the
dependence of the drop surface tension on radius and the contribution to the free energy from the
translational and rotational degrees of freedom of the critical nucleus (correction factor
K
).
19.3 CALCULATION OF ZELDOVICH FACTOR
Our irst task is to derive the expression for the Zeldovich factor which takes into account the
dependence of droplet surface tension on radius. Such a calculation was made recently [7,35] in
the framework of the Nishioka theory; in this section we provide a more transparent derivation.
Following Nishioka and Kusaka [36] we consider a multicomponent system which is a liquid spheri-
cal drop surrounded by a vapor phase. In general the drop is not a critical nucleus. The total number
of molecules for the component
i
in the system is speciied as
N
i
to be governed by the equation
N
N
i i
+ , where
N
i
α
and
N
i
β
are the numbers of molecules in the real system belonging
to the vapor and droplet, respectively. One of the main assumptions of the publication [36] is that the
molecules of interfacial region are regarded as belonging to the drop. In other words, the chemical
potential for surface molecules is equal to the chemical potential μ
i
for the volume molecules of
droplet. In general the droplet and vapor chemical potentials are not equal (μ ≠
β α
α
β
i
=
=
const
i i
).
It was shown in Ref. [36] (as well as in Ref. [37]) that the variation of minimum work necessary
to form a non-critical droplet in a multicomponent system under the non-equilibrium process at
T
, μ
i
α
, and
P
α
being constant is governed by the following equation:
∑
μ
(
)
β
α
β
dW
=
−
μ
dN
(19.10)
i
i
i
i
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