Membranes - Introduction to Carbon Capture and Sequestration - page 336

z

Figure 7.6.2 Free energy profi le in the membrane
The free energy as a function of the position of a molecule in the pore for our model
membrane, where the step represents a window between cavities.

contribute to the free energy F at position z . We can see that this free
energy takes two values:

L

cy UkT
y

− /

e z ∈ cavity

cB

L

()

exp − Fz kT / =

B

L

wy UkT
y

− /

e z ∈ window

wB

L

In Figure 7.6.2 , we plot this free energy as a function of the position for
our model membrane. We see that the window forms a barrier to the dif-
fusion of our adsorbed molecule. If this free energy barrier is suffi ciently
high we see that a molecule will be in a cavity most of the time and only
once in a while hop from one cavity to another. This hopping is equiva-
lent to a random walk on a lattice for which we have shown how to
compute a diffusion coeffi cient in the previous section.
The important point is that if we have a situation in which hopping
from one cage to another is a rare event, we can make a very good
estimate of the hopping rate using the transition state theory [7.16]. In
Box 7.6.1 , we show that this theory gives the hopping rate:

L exp − U / k T

1 wy w B

( )

k cavity → cavity = ν

1 2 a

2

2

L exp − U / k T

cz c B