2702 Tlenkopatchev et al.
Macromolecules, Vol. 38, No. 7, 2005
where V is the volume of the box and p is the pressure.
By the same token, the statistical weight associated
with the removal of a particle from a box containing n
particles would be
Therefore, to optimize the separation of condensable
gases from the less condensable ones requires that DA/
DB , SA/SB, where D is the diffusion coefficient. Work
dealing with the diffusive characteristics of the mem-
branes used in this study will be published later.
(
n - 1)kT
∆F
kT
σnfn-1;re
)
exp
(13)
(
)
Conclusions
pV
The critical interpretation of the solubility of O2, N2,
CO, CO2, CH4, C2H4, C2H6, C3H6, and C3H8 in glassy
membranes prepared by casting from chloroform solu-
tions of poly(N-3,5-bis(trifluoromethl)phenyl-exo-endo-
norbornene-5,6-dicarboximide) shows that gas solution
in the continuous phase of the membrane assumed by
the dual mode model can be determined from the gas
The values of σnfn+1;in obtained by means of eq 11 are
obviously in excess because N is calculated as whether
polymer chains were not present in the cubic box. A
better approach would be to estimate N from Henry’s
constant, which as was shown above, can be fairly well
estimated from eq 6. The only unknown quantity in this
equation is the enthalpic parameter for which a value
of 0.5 could in principle be used. Under a pressure p,
the statistical weight associated with the insertion of
an additional gas molecule in the cubic box already
containing n molecules, can be obtained as
(in the liquid state)-polymer mixing free energy usually
utilized in very concentrated polymer-liquid solutions.
This approach leads to an expression relating Henry’s
contribution to the solubility with both the enthalpic
interaction parameter and the boiling point of the gases.
Reasonable values of the enthalpic interaction gas (in
the liquid state)-polymer parameter are obtained which
exhibit a tendency to increase as the molecule gas size
increases. The Flory-Huggins approach allows to pre-
dict reasonably well the dependence of the Henry’s
constant on the boiling temperature of the gases at 1
bar. Also, the logarithm of the solubility coefficient for
different gases is a linear function of their respective
boiling temperatures. Methods are outlined to improve
the simulation of gas sorption in glassy membranes.
kDpV
∆F
kT
σnfn+1;in
)
exp -
(14)
(
)
n + 1
where it was considered that
N = k pV
(15)
D
The value of N obtained from eq 15 is in most cases the
lower bound for the number of particles inside the box.
Notice that according to eq 1
Acknowledgment. We thank the CONACyT for
limS ) k + bC′ ; limS ) kD
(16)
D
H
pf0
pf∞
generous support of this research with Contract No. NC-
2
04. Financial support provided by the DGICYT (Di-
For the less condensable gases (H2, O2, N2, CO, etc.), S
kD, and therefore, the values obtained for N are in
principle fairly realistic. Under a pressure p, the sta-
tistical weight associated with the removal of a molecule
from the box which already contains n molecules is
given by
recci o´ n General de Investigaci o´ n Cient ´ı fica y Tecnol o´ g-
ica) through Grant MAT2002-04042-C02-02 is also
gratefully acknowledged.
=
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(
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kDpV
∆F
kT
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σnfn-1;re
)
exp
(17)
(
)
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(
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