At present, physical chemistry employs the tools of thermodynamics to treat

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1 OSMOSIS: A MACROSCOPIC PHENOMENON, A MICROSCOPIC VIEW S. A. Ben-Sasson, and N. B. Grover Hubert H. Humphrey Center for Experimental Medicine and Cancer Research, Faculty of Medicine, Hebrew University, Jerusalem 91120, Israel At present, physical chemistry employs the tools of thermodynamics to treat osmosis across a semipermeable membrane. We propose a model in terms of momentum transfer, the inherent asymmetry of which leads quantitatively to the van t Hoff relationship; qualitatively, the solute molecules can be looked upon as micropumps that suck solvent through the pores in the membrane. ADV PHYSIOL EDUC 27: 15 19, First published Month 00, 2002; /advan Key words: Donnan equilibrium; momentum transfer; semipermeable membranes; solute-pore interaction; van t Hoff relationship No physical phenomenon has any greater import in biology than does osmosis. Thus did the Editor of The American Journal of Physiology: Regulatory, Integrative and Comparative Physiology introduce the 1979 Forum on Osmosis (31). Important, yes, but difficult and not easily grasped. Fortunately, it is quite simple to construct an elementary model based on momentum transfer that can be followed by undergraduate medical and biology students and yet prove useful to the professional physiologist or physical chemist. 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 RT C 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) / 03 $5.00 COPYRIGHT 2003 THE AMERICAN PHYSIOLOGICAL SOCIETY 15

2 J v L p P RT C where J v is the steady-state volume flux across the membrane, L p 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 L p 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). THE MODEL Let us first consider a simple system, one in which a solution of impermeable solute molecules exists on one side of a semipermeable membrane and pure solvent on the other. As long as the mean velocity of a solute molecule is constant, no net work is done by it or on it. When the molecule approaches the pore, however, it is prevented from entering by its large diameter and strikes the surrounding membrane instead. In a perfect elastic collision, all the kinetic energy of the particle is transferred to the membrane and then back again to the particle. During the rebound phase, the solute molecule performs net work on the molecules of the solution in the direction away from the pore. Thus an asymmetry exists that has no counterpart on the solvent side of the membrane, where solvent molecules are free to enter the pore and impinge on other solvent molecules, and a momentary pressure difference is formed across the pore. The translation of this model into quantitative terms is straightforward, as shown below. 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. 16

3 Let n i be the number of solute molecules per unit volume with velocities perpendicular to the membrane that are initially between u i and u i u i. Then, where the subscript i refers to this subset i 1 t P i t dt 1 t t F i t 1 mdv t dt dt A A dt m u i dv t 2mu i A A i i ui Here P i (t) has been replaced by the force per unit area F i (t)/a and F i (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, n i Au i, so that and i 2mu i A i 2mn i u i 2 i 2m n i u i 2 2mNu 2 Here u 2 is the mean-square velocity and N n i the number of solute molecules per unit volume. But mu 2 1 kt, so that 2 NkT nrt RT C where n N/N 0 is the molar concentration, k R/N 0 is the Boltzmann constant, and N 0 is Avogadro s Number. DISCUSSION A model of osmosis based on the ideal gas analogy was first proposed by van t Hoff (28) but was soon abandoned (29). Clearly, the molecular events that take place around and within the membrane are more complicated (10), but, at least for low concentrations (21) and nonleaky semipermeable membranes ( 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 solutepore interaction is an all-or-none phenomenon. Intermediate degrees of reflection (0 1) can be understood on the basis of inhomogeneous or timedependent 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 17

4 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. APPENDIX An Ongoing Controversy The view that osmosis arises from the negative pressure imparted by the solute altering the water at the free surface of the solution has been reviewed from a historical perspective by Hammel (12). In this comprehensive and eloquent presentation, the author dismisses ideas about reduced solvent concentration, both early (22) and current (11), in favor of the negative pressure model, which is even older (6, 18). Kiil (21), equally eloquent, claims that the kinetic theory is in disrepute (9) and was rejected by van t Hoff himself (29), whereas the concept of a decrease in solvent concentration has been around since 1877 (24) and is supported by venerable experimental evidence (30). He is adamant, although tactful, in dismissing the negative pressure approach: Scholander [and Hammel have] emphasized the significance of the tensile strength of solvents in osmosis, but the failure to distinguish between factors influencing osmotic pressure and osmotic flow may have caused some confusion. Others are less delicate: An attempt to explain osmosis on the basis of tension within the water-solute system... cannot be correct (8) or... to speak of solvent tension in a mixture is nonsense (13). Hammel, in the other camp, is equally blunt (12): The water concentration idea is false and cannot explain osmosis. Both sides marshal support, old and new, and each rejects outright the interpretation of the other. Address for reprint requests and other correspondence: N. B. Grover, Hubert H. Humphrey Center for Experimental Medicine and Cancer Research, Faculty of Medicine, Hebrew University, Jerusalem 91120, Israel ( norman@md.huji.ac.il). Received 2 April 2002; accepted in final form 28 October References 1. Atlan H and Thoma J. Solvent flow in osmosis and hydraulics: network thermodynamics and representation by bond graphs. Am J Physiol Regul Integr Comp Physiol 252: R1182 R1194, Benedek GB and Villars FMH. Physics with Illustrative Examples from Medicine and Biology (2nd ed.). Statistical Physics. Berlin: Springer, 2000, vol. 2, p Chinard FP and Enns T. Osmotic pressure. Science 124: , Dainty J. Water relations of plant cells. Adv Bot Res 1: , Davson H. A Textbook of General Physiology (4th ed.). London: Churchill, 1970, vol. 1, p Dixon HH. A transpiration model. R Dublin Soc Sci Proc 10: , Fermi E. Thermodynamics. New York: Dover, 1956, p. x. 8. Ferrier J. Osmosis and intermolecular force. J Theor Biol 106: , Glasstone S. Textbook of Physical Chemistry (2nd ed.). New York: Van Nostrand, 1946, p Green HS. The Molecular Theory of Fluids. New York: Dover, 1969, p Guyton AC. Textbook of Medical Physiology (8th ed.). Philadelphia, PA: Saunders, 1991, p Hammel HT. Evolving ideas about osmosis and capillary fluid exchange. FASEB J 13: , Hildebrand JH. Forum on osmosis. II. A criticism of solvent tension in osmosis. Am J Physiol Regul Integr Comp Physiol 237: R108 R109, Hill AE. Osmosis in leaky pores: the role of pressure. Proc R Soc Lond B237: , Hobbie RK. Osmotic pressure in the physics course for students of the life sciences. Am J Physics 42: , Hobbie RK. On the interpretation of experiments on osmotic pressure. Proc Nat Acad Sci USA 71: , Hobbie RK. Intermediate Physics for Medicine and Biology (3rd ed.). Berlin: Springer, 1997, p

5 18. Hulett GA. Beziehung zwischen negativem Druck und osmotischem Druck. Z Phys Chem 42: , Janáček K and Sigler K. Osmosis: membranes impermeable and permeable for solutes, mechanism of osmosis across porous membranes. Physiol Res 49: , Kedem O and Katchalsky A. Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. Biochim Biophys Acta 27: , Kiil F. Molecular mechanisms of osmosis. Am J Physiol Regul Integr Comp Physiol 256: R801 R808, Lewis GN. The osmotic pressure of concentrated solutions, and the laws of the perfect solution. J Am Chem Soc 30: , Longuet-Higgins HC and Austin G. The kinetics of osmotic transport through pores of molecular dimensions. Biophys J 6: , Pfeffer W. Osmotische Untersuchungen. Leipzig, Germany: Engelman, 1877, p Riddick DH. Contractile vacuole in the amoeba, Pelomyxa carolinensis. Am J Physiol 215: , Soodak H and Iberall A. Forum on osmosis. IV. More on osmosis and diffusion. Am J Physiol Regul Integr Comp Physiol 237: R114 R122, Staverman AJ. The theory of measurement of osmotic pressure. Rec Trav Chim Pays-Bas 70: , Van t Hoff JH. Die Rolle des osmotischen Druckes in der Analogie zwischen Lösungen und Gasen. Z Phys Chem 1: , Van t Hoff JH. Wie die Theorie der Lösungen entstand. Ber Dtsch Chem Ges 27: 6 19, Vegard L. On the free pressure in osmosis. Proc Cambridge Philos Soc 15: 13 23, Yates FE. Introducing a forum on osmosis. Am J Physiol Regul Integr Comp Physiol 237: R93, Zorof E and Ilani A. Transient pressure changes in Donnan systems displaced from equilibrium. Nature 267: , Downloaded from by on July 5,

DERIVATION OF PRACTICAL KEDEM - KATCHALSKY EQUATIONS FOR MEMBRANE SUBSTANCE TRANSPORT

DERIVATION OF PRACTICAL KEDEM - KATCHALSKY EQUATIONS FOR MEMBRANE SUBSTANCE TRANSPORT M. Jarzyńska Technical High School of Environment Developing, Piotrków Trybunalski, Broniewskiego 16, Poland, e-mail: