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7.1.2′There is a beautiful symmetry between Equation 7.6 and the corresponding formula from our
earlier discussion:
p=−dF(V)/dV (Equation 7.2); 〈V〉=dF(p)/dp (Equation 7.6).Pairs of quantities such aspandV,which appear symmetrically in these two versions, are called
“thermodynamically conjugate variables.”
Actually, the two formulas just given are notperfectlysymmetrical, since one involvesV and
the other〈V〉.Tounderstand this difference, recall that the first one rested on the entropic-force
formula, Equation 6.17. The derivation of this formula involvedmacroscopicsystems, in effect
saying “the piston is overwhelmingly likely to be in the position.... ” In macroscopic systems there
is no need to distinguish between an observable and its expectation value. In contrast, Equation 7.6
is valid even for microscopic systems. Thus the formulation of Equation 7.6 is the one we’ll need
when we analyze single-molecule stretching experiments in Chapter 9.
7.3.1′
- The discussion of Section 7.3.1 made an implicit assumption, and though it is quite well obeyed
in practice, we should spell it out. We assumed that the filtration constantLpwassmall enough,
and hence the flow was slow enough, that the flow doesn’t significantly disturb the concentrations
on each side, and hence we can continue to use the equilibrium argument of Section 7.2.1 to find
∆p.More generally, the osmotic flow rate will be a power series in ∆c;wehavejust computed its
leading term. - Osmotic effects will occur even if the membrane is not totally impermeable to solute, and indeed
real membranes permitbothsolvent and solute to pass. In this case the roles of pressure and
concentration jump are not quite as simple as in Equation 7.15, though they are still related. When
both of these forces are small we can expect a linear response combining Darcy’s law and Fick’s
law: [
jv
js
]
=−P
[
∆p
∆c]
.
HerePis called the “permeability matrix”. ThusP 11 is the filtration coefficient, whileP 22 is the
solute permeabilityPs(see Equation 4.20). The off-diagonal entryP 12 describes osmotic flow, that
is, solvent flow driven by a concentration jump. Finally,P 21 describes “solvent drag”: Mechanically
pushing solvent through the membrane pulls along some solute.
Thus a semipermeable membrane corresponds to the special case withP 22 =P 21 =0. Ifin
addition the system is in equilibrium, so that both fluxes vanish, then the formula above reduces
toP 11 ∆p=−P 12 ∆c. The result of Section 7.3.1 is then that for a semipermeable membrane
P 12 =−LpkBT.
More generally, L. Onsager showed in 1931 from basic thermodynamic reasoning that solvent
drag isalwaysrelated to solute permeability byP 12 =kBT(c 0 −^1 P 21 −Lp). Onsager’s reasoning
is given for instance in Katchalsky & Curran, 1965. For a concrete model, similar in spirit to the
treatment of this chapter, see Benedek & Villars, 2000b.