Membranes - Introduction to Carbon Capture and Sequestration - page 314

Box 7.4.2 Thermodynamic coeffi cient

Let us calculate the thermodynamic coeffi cient for a material of which the

adsorption can be described with a Langmuir isotherm:

Γ

bp
pb

ρ=ρ 0 ,

1

+

where p is the pressure of the gas phase and

0 is the maximum loading in moles
per unit volume. We have in this equation a relation between the pressure of the
gas phase and the number of adsorbed molecules per unit volume. For the ther-
modynamic coeffi cient, however, we need to know the relation between the load-
ing and the chemical potential of the gas phase. If we assume that the bulk gas
pressure is suffi ciently low, we can consider the gas phase to be an ideal gas. For
an ideal gas, the relation between the chemical potential and pressure is given by

ρ

( )

0

µ=µ + RT ln p RT / ,

which gives the Langmuir isotherm in terms of the chemical potential as a function
of loading:

1 ρ

0

µ=µ + RT ln ,

ρ−ρ

bRT

0

and we obtain for the thermodynamic coeffi cient:

ρ

0

Γ= RT

ρ−ρ

0

This gives for the relation between the Maxwell-Stefan and the Fick diffusion
coeffi cient:

ρ

Fick 0 MS

D = T D

ρ−ρ

0

This equation shows that if the loading in the material approaches the maxi-
mum loading ρ → ρ 0 , the Fick diffusion coeffi cient becomes infi nitely large. The

reason is illustrated in the fi gure. If our material is completely saturated and we add
one more molecule, another molecule needs to come out instantaneously, which
corresponds to an infi nitely large diffusion coeffi cient!