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Encyclopedia of Physical Science and Technology EN009H-407 July 18, 2001 23:34

Mass Transfer and Diffusion 175

of the apparatus. Because the center of mass has moved, where ∇µ 1 is the gradient of the solute’s chemical poten-
there must be some convection. Yet we would expect this tial. One ordinarily expects the concentration to vary with
process to be completely described by diffusion. chemical potential as
In fact, we are right in our expectation. The total flux, 0
µ 1 = µ + k B T ln c 1 γ 1 (25)
1
the sum of the diffusive flux and the convective flux, can
where γ 1 is an activity coefficient. Combining these rela-
be written as
tionships, we find
n 1 = c 1 v 1 (19)
∂ ln γ 1
where c 1 and v 1 are the concentration and velocity of the − j 1 = D 0 1 + ∇c 1 (26)
∂ ln c 1
solute of interest. We can then split off a convective ve- This says that the diffusion coefficient should vary with
0
locity v as:
the activity coefficient.
0 0
n 1 = c 1 v 1 − v + c 1 v (20) The interesting feature of Eq. (26) is that it predicts the
The first term on the right-hand side of this equation is diffusion coefficient will go to zero at a critical point or a
that due to diffusion, so that we can write Eq. (20) as consolute point. This is verified experimentally: the diffu-
sion coefficient does drop from a perfectly normal value
n 1 = j 1 + c 1 v 0 (21)
by more than a million times over perhaps just a few de-
While this much is straightforward, the choice of the ve- grees centigrade (Kim et al., 1997). Curiously, the drop
0
locity v can be complicated, beyond the scope of this occurs more rapidly than predicted by Eq. (26). In many
article (Taylor et al., 1993 and de Groot et al., 1962). For- ways, this is a boon, because the diffusion coefficient is
tunately, this is not normally significant. When we want small only in a very small region of little practical signifi-
a very accurate description, we should consider this addi- cance. However, it is disquieting that we do not understand
tional factor. completely why the drop is faster than it should be.
In addition to convection, we must recognize that Fick’s
law applies exactly to only one solute and one solvent, II. DISPERSION
i.e., to a binary system. In general we should write a more
complete flux equation like (de Groot et al., 1962 and At this point we can benefit from a tangent by discussing
Katchalsky et al., 1967): dispersion, a different effect than diffusion but described
n−1 by the same mathematics. Unfortunately, dispersion is fre-

− j i = D ij ∇c j (22) quently called “diffusion” in some literature. As a result,
j=1 it seems sensible to cover it here, if only to show why the
which is often referred to as a generalized Fick’s law form processes are different.
of multicomponent diffusion equation. For an n compo- A good example of dispersion is a plume of smoke be-
2
nent system, Eq. (22) has (n − 1) diffusion coefficients ing swept away by the wind. This plume will normally as-
of which n(n − 1)/2 are independent. Alternatively, one sume a Gaussian profile, a bell-shaped curve whose width
can use a different form of diffusion equation which for is a function of the dispersion coefficient. If the amount of
ideal gases is (Taylor et al., 1993): smoke emitted per time S is a constant, then the concentra-
n tion of material in the smoke is given by (Seinfeld, 1985)
y i y j

∇y i = (v j − v i ) (23) S z 2
D ij −
j=1 c 1 = e 4Et (27)
4πxE
where y i is the mole fraction of species i and the D ij
where x is the distance down wind, E is the dispersion
here are the binary diffusion coefficients. This equation,
coefficient, z is the direction perpendicular to the wind,
frequently called the Maxwell-Stefan form, is attractive
and t is the time. This has a similar Gaussian dependence
intellectually but can be hard to use. Fortunately, the entire
as that found for diffusion of a pulse, shown in Eq. (18).
subject of multicomponent diffusion is not that important
The dispersion coefficient E shown in Eq. (27) is not
because any solute present at high dilution will follow the equal to the diffusion coefficient defined in the earlier
binary form of Fick’slaw. parts of this entry. The dispersion coefficient does have
The final issue is the validity of Fick’s law itself. On the the same dimensions of length per time as the diffusion
2
basis of irreversible thermodynamics (Taylor et al., 1993;
coefficient. Its function is to describe how fast the smoke
de Groot et al., 1962; and Katchalsky et al., 1967), one
spreads, just as the diffusion coefficient describes how fast
can show that an alternative form of Fick’slawis
the solute spreads. However, the dispersion coefficient E
D 0 c 1 is much more a function of physics and much less a func-
− j 1 = ∇µ 1 (24)
k B T tion of chemistry. For example, we expect the diffusion