Page 362 - Environmental Nanotechnology Applications and Impacts of Nanomaterials

Membrane Processes 347

RT
p 5 ln a w,2 (17a)
V w
c
p < RT s (17b)
where p and a w are the osmotic pressure and activity of water in the
solution of interest and c s is the molar concentration of salt. Eq. 17b
applies if the solution is dilute where we can approximate the activity
of water as being 1 minus the mole fraction of the salt in compartment
2 and note that ln(1 x) < x. Higher molecular weight solutes produce
less osmotic pressure than do small molecular weight compounds. This
can be illustrated by considering the case of a dilute solution of mole-
cules with molecular weight M at a mass concentration C. The osmotic
pressure can be approximated as:

C
p > RT (18)
M
Eqs. 17a, 17b, and 18 are all ways of writing what is known as the van’t
Hoff equation. Note that Eq. 18 predicts that the osmotic pressure will
decrease as the molecular weight of the species increases. For this
reason, the osmotic pressure exerted by larger particles and macro-
molecules can typically be ignored. In contrast, small nanoparticles,
while exerting an osmotic pressure much less than that of an ionic
solute at equal mass concentration, may nonetheless result in a signif-
icant osmotic pressure.
Substituting Eq. 17a into Eq. 6, we obtain the following result for the
difference in available energy of the water on the two sides of the mem-
brane in the absence of an electrical potential:

E 5 V P 1 RT lna 5 V s P 2 pd (19)
w
w
w
w
In other words, a pressure P greater than the osmotic pressure must
be applied across the membrane to create a driving force sufficient to
move water across the membrane. We now write the corresponding
expression for the solute:

(20)
E s 5 V s P 1 RT lna s
p
, so 5 1 . Substituting
Based on Eq. 17, for this system, p < RT c s
RT c s
that relationship into Eq. 20, and assuming that activity is approximated
by concentration, we obtain:
p
E 5 V P 1 RT ln c s (21)
s
s
RT c s