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HOW WE TEACH
Hubert H. Humphrey Center for Experimental Medicine and Cancer Research, Faculty of Medicine, Hebrew University, Jerusalem 91120, Israel
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Abstract |
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Key words: Donnan equilibrium; momentum transfer; semipermeable membranes; solute-pore interaction; van’t Hoff relationship
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Introduction |
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TOPAbstractIntroductionTHE MODELDISCUSSIONAPPENDIXReferences |
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Osmosis depends on the presence of a membrane permeable to solvent, but not solute, molecules—a semipermeable membrane. When such a membrane separates two solutions of different concentrations, solvent molecules will flow from the more dilute to the less dilute side until equilibrium is reached, usually by the ensuing build-up of hydrostatic pressure. The amount of hydrostatic pressure 
required to counterbalance the osmotic flow is given by the empirical expression first proposed for dilute solutions by van’t Hoff in 1885
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where R is the gas constant, T the absolute temperature, and 
C the concentration difference of nonpermeable solutes across the membrane. The initial formulation was designed to agree with the experimental data of Pfeffer (24), but a kinetic model based on an analogy with the behavior of an ideal gas soon followed (28), only to be discarded in favor of more formal treatments based on free-energy considerations.
The free energy of a solution depends on, among other things, the chemical potential and molar concentration of its constituents (the Gibbs-Duhem equation), and one can derive the van’t Hoff relationship directly from the laws of classical thermodynamics (3). Yet, the introduction of the concept of chemical potential difference of the solvent as the driving force in osmosis does not account explicitly for the case where the membrane is only partially permeable to the solute. By turning to irreversible thermodynamics, Kedem and Katchalsky (20) were able to provide a comprehensive analysis of all phenomena involving osmosis, in the form of their well-known equation based on the pioneering work of Staverman (27)
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where Jv is the steady-state volume flux across the membrane, Lp the hydraulic conductivity, P the hydrostatic pressure difference, and 
the reflection (or selectivity) coefficient. For an ideal semipermeable membrane, = 1; for one completely permeable, = 0.
Nevertheless, problems remain. "The fact that the same coefficient Lp is used to describe flow of water through a semipermeable membrane under the action of either a hydrostatic or an osmotic pressure difference has caused great conceptual difficulties among physiologists" (4); or, more broadly, "Few phenomena are so well understood thermodynamically or so ill understood kinetically as the osmotic flow of a solvent through a semipermeable membrane" (23). Still, as Fermi has remarked (7), "... it is sometimes rather unsatisfactory to obtain results without being able to see in detail how things really work, so that in many respects it is very often convenient to complete a thermodynamic result with at least a rough kinetic interpretation."
The search for a microscopic mechanism able to explain qualitatively and quantitatively the phenomenon of osmosis, therefore, is not exactly new, but even after a hundred years, a controversy still exists between those who support the idea of a negative pressure imparted by the solute altering the water at the free surface of the solution and those who reject this notion in favor of a scheme based on a decreased solvent concentration (see APPENDIX).
Far be it from us to attempt to adjudicate between these schools and these scholars. Our objectives are much more modest: to derive an intuitive model of osmotic pressure from elementary kinetic theory. Biologists, for whom osmosis ranks among the fundamentals of cell physiology (5), should find such a mechanistic description particularly appealing.
Other mechanistic models abound, including those based on diffusion (2) and on continuum hydrodynamics (17), but they are considerably more complex.
What we present here, then, is a microscopic model that treats osmosis through the dynamics of momentum transfer and leads quantitatively to the van’t Hoff equation, including those biologically interesting cases in which the membrane is permeable to some of the solutes but not to others. To the best of our knowledge, previous attempts of this nature have been only partially successful (4), although a methodology has been proposed that aims at making osmosis more acceptable to students of the life sciences (15).
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THE MODEL |
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TOPAbstractIntroductionTHE MODELDISCUSSIONAPPENDIXReferences |
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Throughout the solvent side of the membrane and within the solution side away from the pores, the molecules exchange momentum freely. When a solute molecule attempts to enter a pore, however, it strikes the membrane instead and goes from velocity u (in the direction of the membrane) to velocity 0. As it bounces off, it gathers momentum in the opposite direction, away from the membrane, until its original velocity is regained. This event, a solute molecule going from -u to +u over a short time interval t', has no equivalent on the solvent side of the membrane and so gives rise to a pressure difference P(t) over the pore area A during the time t'. The osmotic pressure is just this pressure difference averaged over the time between collisions 
and summed over all solute molecules.
Let ni be the number of solute molecules per unit volume with velocities perpendicular to the membrane that are initially between ui and ui + 
ui. Then, where the subscript i refers to this subset
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Here Pi(t) has been replaced by the force per unit area Fi(t)/A and Fi(t) in turn by the rate of change in the momentum of a particle of mass m and velocity v(t). The mean time between collisions i is just the reciprocal of the collision frequency of these molecules with the pores, niAui, so that
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and
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Here 
is the mean-square velocity and N = 
ni the number of solute molecules per unit volume. But m = 
kT, so that
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where n
N/N0 is the molar concentration, k R/N0 is the Boltzmann constant, and N0 is Avogadro’s Number.
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DISCUSSION |
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TOPAbstractIntroductionTHE MODELDISCUSSIONAPPENDIXReferences |
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= 1), the various solute-solvent, solvent-solvent, and solvent-membrane interactions seem to cancel each other out, leaving only the equivalent of ideal gas kinetics. Is this fortuitous or a necessary consequence of the molecular mechanics of dilute solutions? We don’t know, but it does enable us to present a simplified model that is readily understood by physiology and medical students and yet accords fully with the empirical equation governing osmotic behavior in real systems. As viewed by the model above, the general applicability of the van’t Hoff equation, regardless of the size of the solute molecules or the dimensions of the pores in the membrane—as long as it is semipermeable—arises from the fact that all molecular species acquire the same mean kinetic energy at a given temperature (according to the Boltzmann distribution) and that as the cross section of a pore becomes larger, the rate at which solute molecules impinge on it increases but the pressure at each collision decreases in the same proportion. It should be noted that the equations were developed independently of thermodynamics, and the van’t Hoff relationship was arrived at solely through considerations of momentum transfer.
The model presented here is intended to revive the intuitive approach of van’t Hoff by shifting the focus from the chemical potential of the solvent to the kinetic behavior of the solute molecules. According to this microscopic picture, the osmotic pressure is indeed analogous to a hydrostatic pressure in that the solute molecules suck solvent each time they collide with the membrane pores.
Implicit in the model is the view that every solute-pore interaction is an all-or-none phenomenon. Intermediate degrees of reflection (0 < < 1) can be understood on the basis of inhomogeneous or time-dependent pore size or a permeability that depends on solute orientation.
In the case of electrolytes, if the membrane is impermeable to one of the ions, the other is held back by separation-of-charge forces and so is "reflected" from the membrane to the same extent as the nonpermeable species and contributes its full share to the osmotic pressure.
It has been suggested that, in the classical Donnan system, the excess pressure by the polyelectrolyte is a result of its accumulation at the interface owing to the effect of the electric field (32). Thus Donnan equilibrium is just a special case of the osmotic pressure depending on the kinetics of interaction between membrane and impermeable particles. An analogous situation, in which colloidal particles are subjected to an external magnetic field, has been analyzed by Hobbie (16).
In conclusion, one can think of the solute molecules as micropumps that affect the flow of fluid through the pores of a semipermeable membrane. At osmotic equilibrium, the "pumping power" on both sides is equal. When the membrane becomes permeable to one of the species, as is the case in many biological systems, the pumping ability of these molecules is lost and a net flow of solvent is initiated. The kinetics of this flow are well beyond the scope of our simple equilibrium model and has been considered in depth by others (1, 26), including the case of porous membranes (14, 19).
Our feeling is that a mechanistic view of osmosis like the one presented here will facilitate interpretation of data on the part of the physiologist (25), and the physical chemist, by analyzing the model, will be able to translate deviations from ideal osmotic behavior into physical and molecular terms.
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APPENDIX |
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TOPAbstractIntroductionTHE MODELDISCUSSIONAPPENDIXReferences |
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Acknowledgments |
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Received for publication April 2, 2002. Accepted for publication October 28, 2002.
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References |
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TOPAbstractIntroductionTHE MODELDISCUSSIONAPPENDIXReferences |
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ek K and Sigler K. Osmosis: membranes impermeable and permeable for solutes, mechanism of osmosis across porous membranes. Physiol Res 49: 191–195, 2000.[ISI][Medline]
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