Chemistry Reference
In-Depth Information
Eq. (6-7)
cannot be simplified to
Eq. (6-8)
. In this case, solving such a nonlinear
partial differential equation [i.e.,
Eq. (6-7)
] is not trivial if not impossible. Since
solving
Eqs. (6-7) and (6-8)
is not the main interest here, interested readers are
referred to the work of Crank for the solution of the equations subjected to vari-
ous types of boundary and initial conditions
[8]
.
6.3
Diffusion Coefficients
It is clear from Fick's laws that the key to solving mass transfer problems is to
have a prior knowledge of the diffusion coefficient. The diffusion coefficient,
which is usually used in Fick's first law, is the mutual diffusion coefficient which
is sometimes called the interdiffusion coefficient or chemical diffusion coeffi-
cient, depending on the context in which it appears. Nonetheless, in a binary mix-
ture, it quantifies the diffusion rate of species A inside a medium made up of
species A and B. Obviously, the diffusion rate of A depends on its mobility as
well as the mobility of species B. Here, species B can diffuse back into regions
with higher concentrations of A. The process is somewhat similar to a mixing
process but occurring at the molecular level. The mutual diffusion coefficient
essentially signifies the rate of a non-equilibrium mass transfer process (concen-
tration gradients exist). However, diffusion also takes place in equilibrium sys-
tems in which no concentration gradients exist. In the case of pure systems, the
self-diffusion coefficient is the quantity for characterizing the mobility of the
molecules. In binary systems in which one of the components is at extremely low
concentrations, the tracer diffusion coefficient is used to characterize the mobility
of such a component.
At a given temperature, molecules exhibit various modes of motions as a
result of the available thermal (kinetic) energy. In the case of solids, the thermal
energy manifests in the vibration of bonds, but in the case of liquids and gases,
translational motion dominates. These translational motions, coupled with inter-
molecular collisions, naturally cause the molecules to move more or less like a
random walker in the system. Random walks for gaseous molecules have much
longer displacement steps before collisions, while those for liquid molecules are
limited to jiggling inside cages formed by neighboring molecules followed by
sudden jumps (a hopping process). Although in an equilibrium system with a sin-
gle component, the mean displacement of these random motions is zero, the mean
square displacement of the motions is not vanishing. In fact, the mean square dis-
placement of the random walkers is a measure of their mobility and will yield the
self-diffusion coefficient, which will be discussed in the next few sections. The
important point here is to distinguish the term “self-diffusion” from the diffusion
coefficient used in Fick's first law. Unlike the mutual diffusion coefficient which
is defined for nonequilibrium systems in which concentration gradients exist, self-
diffusion is a measure of the mobility of molecules due to their thermal motions
and can be evaluated even in pure systems at equilibrium
[9
11]
.
