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

174 Mass Transfer and Diffusion

TABLE I A Comparison of Diffusion Coefficients and Their Variations
Typical value Variations
2
Phase cm /sec Temperature Pressure Solute size Viscosity Remarks
Gases 10 −1 T 3/2 p −1 (Radius) −2 µ +1 Successful theoretical predications
Liquids 10 −5 T Small (Radius) −1 µ −1 Can be concentration-dependent
Solids 10 −10 Large Small (Lattice spacing ) +2 Not applicable Wide range of values
Polymers 10 −8 Large Small (Molecular weight) (−0.5to −2) Often small Involve different special cases
Note: These heuristics are guides for estimates, but will not always be accurate.

M/A −z 2 of coffee in which we dropped a lump of sugar. We would
c 1 = √ e 4Dt (18)
4πDt describe diffusion as how fast the sugar moved within the
coffee cup, independent of whether the coffee was on the
In fact, this particular problem is not that important for
kitchen table or in an airplane flying at 1000 km/hr. Thus
diffusion itself but as the basis of dispersion, discussed
when we are considering diffusion, we would sensibly
below. As a result, we defer further discussion for now.
subtract any additional motion of the system.
But with diffusion, things are not always quite so sim-
E. Diffusion Coefficients ple. As an example, consider the basic apparatus shown
in Fig. 4 (Cussler, 1997). In this apparatus two identical
So far, we have treated the diffusion coefficients which ap-
bulbs contain different gases. For example, the bulb on the
peared above as parameters which would necessarily need
left might contain nitrogen and the bulb on the right might
to be determined by experiment. As a result of 150 years
contain hydrogen. Because nitrogen’s molecular weight
of effort, the experimental measurements of these coef-
is higher, the initial center of mass would be closer to the
ficients are now extensive. Their general characteristics
nitrogen bulb, as shown in the figure. If we now open the
are shown in Table I (Cussler, 1997). In general, diffu-
valve between the two bulbs and allow diffusion to take
sion coefficients in gases and liquids can be accurately
place, we will wind up with the two bulbs finally con-
estimated, but those in solids and polymers can not. In
taining equal amounts of hydrogen and of nitrogen. That
gases, estimates based on kinetic theory are accurate to
means that the final center of mass will be in the center
around 8%. In liquids, estimates based on the assumption
that each solute is a sphere moving in a solvent continuum
are accurate to around 20%, but can be supplemented by
extensive data and empiricisms (Reid et al., 1997).
Other characteristics are harder to generalize. The typ-
ical values given in Table I are reasonable, for the coeffi-
cients do tend to group around the estimates given. This is
less true for solids than for the other phases. The variation
of diffusion coefficients with temperature is large in solids
and polymers, but small in gases and liquids. Variations of
the coefficients with pressure are small except for gases.
Interestingly, the diffusion coefficient is proportional to
the viscosity in gases, but is inversely proportional to the
viscosity in liquids. Beyond these generalizations, we rec-
ommend using data whenever possible.

F. Problems with this Simple Picture
The simple picture of diffusion given above ignores sev-
eral issues that can be important. These include diffusion-
engendered convection, multicomponent diffusion, and
the limits of Fick’s law. Each of these merits discussion. FIGURE 4 An example of reference velocities. Descriptions of
diffusion imply reference to a velocity relative to the system’s mass
We begin with the diffusion-engendered convection. In
or volume. While the mass often has a nonzero velocity, the vol-
general, the total flux is the sum of the diffusive flux and ume often shows no velocity. Hence, diffusion is best referred to
the convective flux. For example, imagine we had a cup the volume’s average velocity.