Environmental Engineering Reference
In-Depth Information
where
B
1
=
2
σ
aw
V
w
/RT
and
B
2
=
3
n
i
V
w
/
4
π
. Note that
B
1
(in
μ
m)
≈
0.66/
T
and
B
2
m
3
)
10
13
(in
is the number of ions resulting from solute
dissociation,
m
s
is the mass (g) per particle of solute, and
M
s
is the molecular weight
(g/mol) (Seinfeld and Pandis, 1998).
Note the difference between the equation for a pure water droplet and that for an
aqueous solution droplet.Whereas for a pure water droplet there is a gradual approach
of
P
w
→
μ
≈
3.44
×
ν
m
s
/M
s
, where
ν
P
w
as
r
is increased, in the case of an aqueous solution droplet
P
w
can either
increase or decrease with
r
depending on the magnitude of the second term,
B
2
/r
3
,
which results solely from the solute effects and is a function of its mole number,
m
s
/M
s
. When the two terms on the right-hand side become equal,
P
w
=
P
w
; the
radius at which this is achieved is called the
potential radius
and is given by
3
8
1
/
2
n
i
π
RT
σ
aw
r
∗
=
.
(4.54)
The maximum value of ln
(P
w
/P
w
)
will be reached when the derivative with respect
to the radius goes to zero, which gives a
critical radius
√
3
r
∗
.
3
B
1
r
c
=
B
2
=
(4.55)
A typical plot of
P
w
/P
w
for both pure water droplets and solution droplets (solute
being nonvolatile, e.g., an inorganic salt or an organic compound) is shown in
Figure 4.6.We shall assume that the concentration of
i
is low enough that
σ
aw
remains
unaffected. If the compound is surface active,
σ
aw
will be lower and hence the Kelvin
term will be even smaller. This can be the case especially for many of the hydropho-
bic compounds of environmental interest (Perona, 1992). Although electrolytes also
affect the surface tension of water, their influence is mostly
<
20% within the observed
ranges of concentrations.
For pure water droplets, the curve shows a steep decrease with increasing
r
. For the
solution droplet, there is an initial steeply rising portion till
r
c
is reached. This results
from the solute effects. Beyond this point, the Kelvin term dominates and the slow
decrease in
P
w
/P
w
is apparent. Curves such as these are called
Kohler curves.
They
are useful in estimating the size of an aqueous droplet given the relative humidity
(
P
w
/P
w
) and the concentration of the solute.
Consider a fixed ratio of
P
w
/P
w
. For this fixed value there are two possible radii,
one less than
r
c
and the other greater than
r
c
.For
r < r
c
, the drop is at its equilibrium
state.Ifitaddsanywater,itsequilibriumvaporpressurewillbelargerthantheambient
value and it therefore quickly loses that water through evaporation and reverts to its
original equilibrium. If
r > r
c
, the drop will have a lower than ambient vapor pressure
upon gaining more water molecules. Therefore, it will continue to accumulate water
and grow in size. If the value of
P
w
/P
w
is greater than the maximum value, whatever
be the size of the drop, it always has lower than ambient vapor pressure and hence
it continues to grow in size. A drop that has crossed this threshold is said to be
activated
. With increasing radius of the drop, the height of the maximum decreases
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