Membranes - Introduction to Carbon Capture and Sequestration - page 334

This formula is used in molecular simulations to compute the (excess)
chemical potential µ, which is closely related to the Henry coeffi cient
[7.16]. To apply this formula to our model membrane we have to know the
energy of a molecule in a pore. We assume that our pore is structureless
and that the energy is simply U c if the molecule is in the cavity and U w in
the window region. The energy would be infi nitely large if we were to
place a molecule in a position occupied by the membrane molecules, and
hence we only get a contribution from the cavity and window regions.
To compute the Henry coeffi cient for the model shown in Figure 7.6.1 ,
we can envision inserting a gas particle at random in either the cavity or
the window. As the contribution to the Henry coeffi cient is proportional
to the volumes of these two regions, we have:

V V V −− V V

1 − UkT / UkT /

c w − c w

H e e 0 ,

= + +

CB wB

kT V V V

B

where V is the volume of a unit cell ( L y × L y × L z ), V c is the volume of the
cavity ( L cy × L cz ), and V w is the volume of the window region ( L wz × L wy ).
From this formula, we can obtain some important insights. Let us
assume that the energies in the cavity and in the window region are both
zero: U c = U w = 0. Then our equation for the Henry coeffi cient reads:

1 VV +

U = 0 cw

H =

kT V

B

At this point, you may wonder why is this interesting. From your ther-
modynamic course, you may recall that equilibrium is the state for which
the free energy takes the minimum value. For a system in which we have
a fi xed volume, number of particles, and temperature, we have to look at
the Helmholtz free energy, which is:

AUTS =−

Question 7.6.1 Henry coeffi cient and temperature

Is the statement that the Henry coeffi cient is dominated by the energy of
the cavity valid at all temperatures? Assume that the energy in the cavity is
-70 kJ/mol and on top of the barrier -30 kJ/mol. What if these energies are
-35 kJ/mol and -30 kJ/mol? Hint: plot the contributions as a function of
temperature.