Membranes - Introduction to Carbon Capture and Sequestration - page 313

Box 7.4.1 Concentration or chemical potential

The idea of Fick diffusion is intuitive: molecules fl ow from a high to a low concen-
tration. Diffusion is, however, not always as simple as this. For example, the fi gure
shows the adsorption of molecules in a porous material. In such a system, we can
have a gas in equilibrium with a dense fl uid inside the pores. If we increase the
pressure, molecules fl ow from a low concentration, the gas phase, to a high con-
centration in the pores.

Molecules diffusing from a low density gas phase to a high density one in the pores.

Let us describe the same experiment in terms of the chemical potential. The
chemical potential has an energetic contribution and an entropy contribution. As
the adsorbed particles have to move in a smaller volume, adsorption causes the
entropy to decrease. The interactions of the particles with the walls inside the
pores, however, can compensate for this entropy loss and the chemical potential
can be lower in the adsorbed phase than the gas phase. According to Onsager,
molecules will fl ow toward a lower chemical potential, which in our example cor-
responds to molecules fl owing from a low to high concentration!
Is the conclusion that using a Fick diffusion coeffi cient is incorrect? No, we
can always use Fick's law as a defi nition of the Fick diffusion coeffi cient. But, we
now understand why we would get for our adsorption case a negative diffusion
coeffi cient.

where D MS is the Maxwell-Stefan diffusion coeffi cient, which is related to
the Fick diffusion coeffi cient by:

Fick

D

MS

D =

Γ

This thermodynamic coeffi cient Γ can be calculated from the experimen-

tal isotherm (see Box 7.4.2 ).