7.3. Beyond equilibrium: Osmotic flow[[Student version, January 17, 2003]] 227
c=0 c=c 0U
(z)p
(z)p(z)zzazz∆p=c 0 kBT∆p>c 0 kBT∆p=0d
c
b
a
Figure 7.6:(Schematic; sketch graphs.) (a)Aliteral model of a semipermeable membrane, consisting of a perforated
wall with channels too small for suspended particles to pass. (b) The force alongˆzexerted by the membrane on
approaching particles is−dU/dz,whereUis the potential energy of one particle. (c)Inequilibrium, the pressure
pis constant inside the channel (between dashed lines), butpfalls in the zone where the particle concentration is
decreasing. (d)Solid curve:If the pressure on both sides is maintained at the same value, osmotic flow through the
channel proceeds at a rate such that the pressure drop across the channel (from viscous drag) cancels the osmotic
pressure jump.Dashed curve:In reverse osmosis, an external force maintains a pressure gradient even greater than
the equilibrium value. The fluid flows oppositely to the case of ordinary osmotic flow, as seen from the reversed slope
of the pressure profile inside the channel.
Gilbert: That’s true, but the effect you mention is random and averages to zero. In contrast, the
membrane exerts only rightward, never leftward, forces on the solute particles. This force doesnot
average to zero. Its effect is to rectify the Brownian motion of the nearby particles.
Sullivan: It still seems like you get something for nothing.
Gilbert: No, the rectification comes at a price: In order to do useful work, the piston must move,
increasing the volume of the side with solute. This costsorder,asrequired by the abstract Idea 6.19
on page 189.
Gilbert has put a finger on where the net momentum flow into the fluid comes from. Particles
constantly impinge on the membrane in Figure 7.6a from the right, never from the left. Each time