Up to this point, we have considered the case of osmotic flow across a semipermeable membrane—that is, one that is permeable to solvent but impermeable to the solute so that interdiffusion is completely prevented. What would happen if the membrane in Fig. 13 could not distinguish at all between a solvent molecule and a solute molecule; that is, both could cross with equal ease? Suppose, for example, that compartment o contains pure water so that the concentration of water in that compartment (CH O) = 55.6 M. And, suppose that compartment i contains a 1-M solution of deuterium oxide (D2O, or heavy water) in water. Under these circumstances, C^o = 54.6 M, so ACh2o = ACDlo = 1 M. Thus, H2O would diffuse from compartment o to compartment i driven by a concentration difference of 1 M, and D2O would diffuse from compartment i to compartment o driven by an equal but oppositely directed concentration difference. Because both species can cross the membrane with equal case, interdiffusion or mixing would not be restricted and would take place with no displacement of volume. In short, when the membrane cannot distinguish between solute and solvent, osmotic flow, Jv = 0 and, obviously, An = 0.
Clearly, the situation in which the membrane is ideally impermeable to the solute and the one in which the membrane cannot distinguish between solute and solvent are extreme examples. In most instances, we are faced with a situation somewhere between these two extremes—that is, a situation in which interdiffusion is to some extent, but not completely, restricted. Under this condition, for a given concentration difference the volume displacement, or osmotic flow, will be somewhere between zero and the maximum that would be observed with an ideal semipermeable membrane. Also, the osmotic pressure (the pressure necessary to abolish osmotic flow) will be between zero and that predicted by van't Hoff's law.
In 1951, the Dutch physical chemist Staverman provided a quantitative expression for the osmotic pressure across nonideal membranes, which was based on the following reasoning: Let us perform the ultrafiltration experiment illustrated in Fig. 14. In the upper cylinder, we place an aqueous solution of a solute at a concentration of Ci. We then apply pressure to the piston, force fluid through the membrane, and collect the filtrate. If the membrane does not restrict the movement of solute relative to that of water, then the concentration of solute in the filtrate (Cf) will be equal to Ci; that is, both components passed through the membrane in the same proportion as they existed in the solution of origin. At the other extreme, if the membrane is impermeable to the solute, the filtrate will be pure water (i.e., Cf = 0). Between these two extremes is the condition where the membrane partially restricts the movement of the solute relative to that of water, and under these conditions Cf will be lower than Ci but greater than zero. We can now define a parameter that tells us something about the relative ease with which water and solute i can traverse the membrane:
FIGURE 14 Determination of the reflection coefficient of a solute i, employing ultrafiltration through a membrane.
where ( is the reflection coefficient of the membrane to i, because it is a measure of the ability of the membrane to "reflect" the solute molecule i; that is, it tells us how perfect the membrane is as a molecular sieve. Clearly, the reflection coefficient must have a value between zero (for the case where the membrane does not distinguish between the solute and water) and unity (for a membrane that is absolutely impermeant to the solute). Staverman then showed that if the same membrane used in the ultrafiltration experiment is mounted in the apparatus shown in Fig. 13, and if two solutions of the same solute at two different concentrations are placed in compartments o and i, the effective osmotic pressure necessary to prevent volume flow is given by:
Because ai, must have a value between zero and unity, the effective osmotic pressure across a real membrane must fall between zero and that predicted by van't Hoffs law for ideal semipermeable membranes; the exact value depends on the concentration difference and the relative permeability of the membrane to water and the solute i given by
In short, Eq. 21 permits quantitation of the effect of interdiffusion between solutes and solvent (water) across membranes on the effective osmotic pressure that is exerted across those membranes. If a membrane is equally permeable to the solvent (water) and the solute (i), then (ji = 0, and the presence of a concentration difference for i across the membrane will not generate an osmotic pressure. Conversely, if the membrane is impermeant to i, then interdiffusion is prohibited, and the effective osmotic pressure across the membrane will be given by van't Hoffs equation.
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