Environmental Engineering Reference
In-Depth Information
E
XAMPLE
4.14 F
LUX OF
CO
2
BETWEEN THE
A
TMOSPHERE AND
S
EAWATER
Broeker and Peng (1974) estimated that the mean
K
w
for CO
2
is 11 cm/h in seawater at
20
◦
C. This is representative of the world's oceans. The maximum rate of CO
2
transfer
can be obtained by assuming that
C
i
w
≈
0, that is, CO
2
is rapidly assimilated in the sur-
facewatersoftheoceans.Since
P
i
forCO
2
is0.003 atm,thevalueof
C
i
a
=
0.003
/RT
=
1.25
×
10
−
7
mol/cm
3
.
K
aw
for CO
2
is 1.29. Hence
C
i
a
/K
aw
=
9.7
×
10
−
8
mol/cm
3
.
The flux is therefore
J
Dry
(
G
)
=
11
(
9.7
×
10
−
8
)
=−
1
×
10
−
6
mol/cm
2
h. The flux is
from the atmosphere to seawater.
4.2.5 T
HERMODYNAMICS OF
A
QUEOUS
D
ROPLETS IN THE
A
TMOSPHERE
In Section 3.3.4, it was noted that for pure liquids, the vapor pressure above a curved
air-water interface is larger than that over a flat interface. This was termed the
Kelvin
effect
. The Kelvin equation representing this effect was derived as
ln
P
P
∗
2
σ
aw
r
·
V
w
RT
,
=
(4.50)
where
r
is the radius of the droplet,
P
and
P
∗
are the vapor pressures over the curved
interface and the flat surface, respectively, and
V
w
is the molar volume of water.
Let us now consider the aqueous droplet to contain a nonvolatile species
i
with
molar volume
V
i
. If the number of moles of
i
is
n
i
and that of water is
n
w
, the
total volume of the drop
(
4
/
3
)
r
3
π
=
n
i
V
i
+
n
w
V
w
. The mole fraction of water in the
droplet is given by
x
w
=
n
w
/(n
i
+
n
w
)
. Using these relations, one can write
1
1
+
n
i
V
w
/(
4
/
3
)
π
r
3
−
n
i
V
i
.
x
w
=
(4.51)
If the Raoult's law convention is applied to the case of the flat air-water interface
(refer to Section 3.2.3), the vapor pressure of water above the solution will be given by
P
w
= γ
w
x
w
P
w
,
(4.52)
where
P
w
is the vapor pressure of water over the flat interface. The Kelvin equation
now takes the form (Seinfeld and Pandis, 1998)
ln
P
w
γ
w
x
w
P
w
2
σ
aw
r
V
w
RT
.
=
(4.53)
Substituting for
x
w
, we can rearrange the above equation, and use the dilute solution
definition, that is,
(
4
/
3
)
r
3
n
i
V
i
. Furthermore, for a dilute solution according
to the Raoult's law convention, the activity coefficient
π
γ
w
=
1. Therefore, the final
equation is
ln
P
w
P
w
B
1
r
−
B
2
r
3
,
=
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