Membranes - Introduction to Carbon Capture and Sequestration - page 315

Figure 7.4.3 Labeled particle in a solvent

Now let us look at a different diffusion coeffi cient that is not directly
defi ned by looking at the transport of mass. For this we label a molecule
in our fl uid (see Figure 7.4.3 ). This label is something that does not affect
the properties of the molecule, but allows us to follow this particular mol-
ecule. If we could follow the movement of this molecule, we would
observe that because of collisions with the other molecules it exhibits
some kind of Brownian motion. If we could do this experiment many
times, we could determine the probability of fi nding this particle at a dis-
tance z from the initial point. We can calculate this probability from our
mass balance equation:

ρ

d dj

=−

dt dz

ρ ∗ is the density (in number of particles per unit volume) of our
labeled particle.
We assume that the fl ux is given by Fick's law:

where

D dz ρ

s d

j =−

The diffusion coeffi cient we have introduced here is the self-diffusion
coeffi cient. Of importance is that all particles are identical except that we
tag one of them, and hence defi ne the concentration of the tagged