Membranes - Introduction to Carbon Capture and Sequestration - page 320

that is left empty by molecule i will have a signifi cantly higher probability
of jumping. This is the type of correlation that we can ignore in the limit
of very low loading. In addition, if a particle jumps it leaves a vacancy by
defi nition. Hence, at high loading the next hop has a high probability that
the particle will jump back to its original position. This correlation is much
more important for the self-diffusion coeffi cient than for the collective
diffusion coeffi cient, and explains why the self-diffusion coeffi cient is
always lower.

Diffusion in porous media

Let us now go back to Figure 7.4.1 , our experimental system of hydro-
gen diffusing in the zeolite FAU. The data shown in this fi gure nicely
demonstrate the difference between the various diffusion coeffi cients.
We also see that in the limit of low loading the three diffusion coeffi cients
are identical.
We can now also understand the increase in the diffusion coeffi cient
as a function of loading. Intuitively, one would argue that as we increase
the number of molecules in the pores we restrict the movement of those
molecules. This is indeed what we observe in bulk. What makes the dif-
ference in a porous medium?
Let us fi rst look at the thermodynamic factor. In Box 7.4.2 we have
shown that if the adsorption can be described with a Langmuir isotherm
and the gas phase approximated by an ideal gas, we have:

ρ

0

Γ= RT

ρ−ρ

0

What we see is that this coeffi cient gets very large for high loading. The
thermodynamic factor approaches infi nity at the maximum loading. Indeed,
toward the maximum loading one needs to increase the chemical potential
by an infi nite amount to increase the concentration. Hence, at these condi-
tions a small concentration gradient gives a very large thermodynamic fac-
tor . Or, in more physical terms, if a pore is completely fi lled with

molecules, the addition of yet another molecule at one end of the pore
causes a molecule at the other end to leave (see fi gure in Box 7.4.2 ). As the
transport diffusion coeffi cient is related to the fl ux, it does not matter which
particle leaves the pore, and we observe an infi nitely large diffusion coef-
fi cient. If we remove this thermodynamic contribution from our diffusion

Γ