Membranes - Introduction to Carbon Capture and Sequestration - page 312

( z ) ( z + dz )

Figure 7.4.2 Flux through part of the membrane

where ρ is the number of molecules per unit volume. Only in exceptional
cases is the Fick diffusion coeffi cient constant. More typically the diffu-
sion coeffi cient will depend on the temperature and concentration (load-
ing) of guest molecules in the material.
Onsager argued (see Box 7.4.1 ) that the fundamental driving force
for diffusion is not a concentration gradient, but rather a gradient in the
chemical potential, µ , or:

Ld µ d µ

MS

j D
RT dz dz
where L is the so-called Onsager coeffi cient and D MS the Maxwell-Stefan
(or Darken-corrected, or collective) diffusion coeffi cient. The concentra-
tion of the adsorbed molecules in the material and the chemical potential
of these molecules are not independent quantities. This relation is given
by the thermodynamic coeffi cient

=− =− ,

Γ at isothermal conditions:

∂µ
Γ=

∂ρ

T

With this coeffi cient, we can write an expression that relates the fl ux to
the chemical potential gradient:

Fick

d ρ D d ρ d µ

Fick MS

j =− D =− Γ =− D ,

dz Γ dz dz