McKay, Donald. "Front matter" Multimedia Environmental Models ...

expect a diffusivity of approximately 0.25 ¥ 10–4
m2/s
or 0.25 cm2/s
or 0.1 m2/h,
which is borne out experimentally. The kinetic theory of gases can be used to
calculate B theoretically but, more usefully, the theory gives a suggested structure
for equations that can be used to correlate diffusivity as a function of molecular
properties, temperature, and pressure.
In liquids, molecular motion is more restricted, collisions occur almost every
molecular diameter, and the friction experienced by a molecule as it attempts to
“slide” between adjacent molecules becomes important. This frictional resistance is
related to the liquid viscosity m (Pa s). It can be shown that, for a liquid, the group
(Bm/T)
should be relatively constant and (by the Stokes-Einstein equation) approximately
equal to R/(6pNr),
where N is Avogadro’s number, R is the gas constant,
and r is the molecular radius (typically 10–10
m). B is therefore T R/( m6pNr),
where
the viscosity of water m is typically 10–3
Pa s. Substituting values of R, T, µ, and r
suggests that B will be approximately 2 ¥ 10–9
m2/s
or 2 ¥ 10–5
cm2/s
or 7 ¥ 10–6
m2/h,
which is also borne out experimentally. Again, this equation forms the foundation
of correlation equations.
The important conclusion is that, during its diffusion journey, a molecule does
not move with a constant velocity related to the molecular velocity. On average, it
spends as much time moving backward as forward, thus its net progress in one
direction in a given time interval is not simply velocity/time. In t seconds, the distance
traveled (y) will be 2tB
m. Taking typical gas and liquid diffusivities of 0.25 ¥
10–4
m2/s
and 2 ¥ 10–9
m2/s
respectively, a molecule will travel distances of 7 mm
in a gas and 0.06 mm in a liquid in one second. To double these distances will
require four seconds, not two seconds. It thus may take a considerable time for a
molecule to diffuse a “long” distance, since the time taken is proportional to the
square of the distance. The most significant environmental implication is that, for a
molecule to diffuse through, for example, a 1 m depth of still water requires (in
principle) a time on the order of 3000 days. A layer of still water 1 m deep can thus
effectively act as an impermeable barrier to chemical movement. In practice, of
course, it is unlikely that the water would remain still for such a period of time.
The reader who is interested in a fuller account of molecular diffusion is referred
to the texts by Reid et al. (1987), Sherwood et al. (1975), Thibodeaux (1996), and
Bird et al. (1960). Diffusion processes occur in a large number of geometric configurations
from CO2
diffusion through the stomata of leaves to large-scale diffusion
in ocean currents. There is thus a considerable literature on the mathematics of
diffusion in these situations. The classic text on the subject is by Crank (1975), and
Choy and Reible (2000) have summarized some of the more environmentally useful
equations.
7.3.3 Mass Transfer Coefficients
Diffusivity is a quantity with some characteristics of a velocity but, dimensionally,
it is the product of velocity and the distance to which that velocity applies. In
many environmental situations, B is not known accurately, nor is y or Dy;
therefore,
the flux equation in finite difference form contains two unknowns, B and Dy.
Ignoring
the negative sign,
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