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Membrane Processes 343

referred to as the chemical potential and the standard chemical poten-
tial of i, respectively. The sum of chemical and electrical potentials
) is called the electrochemical potential.
(E chemi 1 E eleci
Substituting Eqs. 3 through 5 into Eq. 6, and noting that the standard
molar Gibbs energy of any species i is independent of the system con-
ditions, the overall energy change accompanying transport of species i
across a membrane per unit mass of i transported can be computed as:

o
E 5 V P 1 sG i 1 RT ln a d 1 z F
i
i
i
i
5 V P 1 RT ln a 1 z F (7)
i
i
i
where the ’s are taken as the feed value minus the permeate value and
are assumed to be small. As noted previously, a spatial gradient in
,
energy can be interpreted as a force. Therefore, approximating P, a i
and as the changes in those parameters across the thickness of
m
the membrane, the driving force for transport of i across the membrane
from feed to permeate, per mole of i, is:
F 5 Ei 5 V P 1 RT ln a i 1 z F (8)
i
i
i
d
d
d
d
m
m
m
m
Simplifications to Eq. 8 in which one driving force dominates in the trans-
port of a single component of a system give rise to several frequently
encountered expressions such as Fick’s law (concentration gradient),
Ohm’s law (only electrical potential), and D’Arcy’s law (pressure gradient).
Consider, for example, a single component (i 1) where transport is
driven by differences in concentration alone ( P 0). The driving
force per molecule for transport is obtained by dividing Eq. 8 by Avogadro’s
number, N . If we further assume that the molecules rapidly achieve a
A
steady, average diffusion velocity, and that the molecules experience a
force of resistance to movement, F resist , which is proportional to this
average velocity, v diff , the driving force for diffusion must be balanced by
this resistive force:
kT c 1
F resist 5 v diff f 5 F tot,1 5 (9)
d m c 1
where the Boltzmann constant, k, is equal to R/N and f is the propor-
A
tionality factor relating resistive force to velocity. The diffusive flux is
the product of the average diffusion velocity and the concentration of the
diffusing component. Thus, rearranging Eq. 9, the diffusive flux can be
expressed as:
kT c 1
J 5 v diff c 1 5 (10)
1
f d m