7.2. Osmotic pressure[[Student version, January 17, 2003]] 219
z abc
(^0) z 0
zf
solutionc 0
membrane water
Figure 7.1:(Schematic.) (a)Asemipermeable membrane is stretched across a cup-shaped vessel containing sugar
solution with concentrationc 0 .The vessel is then plunged into pure water. Initially the sugar solution rises to a
heightz 0 in the neck of the cup. (b)Solution begins to rise in the vessel by osmotic flow, until (c)itreaches an
equilibrium heightzf.The pressure in the final equilibrium state is the final heightzftimesρm,wg.
byV/Agives
〈V〉=d(−kBTlnZ(p))/dp=dF(p)/dp, (7.6)
wherepis the pressure.
T 2 Section 7.1.2′on page 249 introduces the idea of thermodynamically conjugate pairs.
7.2 Osmotic pressure
7.2.1 Equilibrium osmotic pressure obeys the ideal gas law
Wecan now turn to the problem of osmotic pressure (see Figure 1.3 on page 11). A membrane
divides a rigid container into two chambers, one with pure water, the other containing a solution
ofNsolute particles in volumeV.The solute could be anything from individual molecules (sugar)
to colloidal particles. We suppose the membrane to be permeable to water but not to solute. A
very literal example would be an ultrafine sieve, with pores too small to pass solute particles. The
system will come to an equilibrium with greater hydrostatic pressure on the sugar side, which we
measure (Figure 7.1). We’d like a quantitative prediction for this “osmotic pressure.”
One might think that the situation just described would be vastly more complicated than the
ideal-gas problem just studied. After all, the solute molecules are in the constant, crowded, company
of water molecules; hydrodynamics rears its head, and so on. But examining the arguments of the
preceding subsection, we see that they apply equally well to the osmotic problem. It is true that
the solute molecules interact strongly with the water, and the water molecules with each other. But
in a dilute solution the solute particles don’t interact much witheach other,and so the total energy
of a microstate is unaffected by their locations. More precisely, the integral over the positions of
the solute molecules is dominated by the large domain where no two are close enough to interact
significantly. (This approximation breaks down for concentrated solutions, just as the ideal gas law
fails for dense gases.)