Membrane equilibrium. Osmotic pressure

These equalilies are the equilibrium conditions for a multiphase multicomponent system. 

Such a system is stable as a whole, i.e. it is in the state of a stable or mettustablc equilibrium, if the stability conditions hold for each coexisting phase (see Equation 1.1.2 18) (Gibbs, 1928 Rusanov, 1960, 1967) 

The above relationships have been derived without regard for the effects of surface phenomena. The equilibrium conditions of a multiphase (c.g. two-pli2isc) system with allowance made for the interfsice (the subscript n) take the form 

The mechanical equilibrium conditions depend on the degree of the interfeice curvature. E.g. a flat surface implies that 

Consider an isolated system with u components and 7 phases, with all the phases separated from each other with a solid semipermcable membrane. The volume of each phase is therefore fixed. Due to the membrane impermeability to the (v — s) components, variations in the u — s) mole fractions are equal to zero. 

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A semi-permeable membrane, which is unequally permeable to different components and thus may show a potential difference across the membrane. In case (1), a diffusion potential occurs only if there is a difference in mobility between cation and anion. In case (2), we have to deal with the biologically important Donnan equilibrium e.g., a cell membrane may be permeable to small inorganic ions but impermeable to ions derived from high-molecular-weight proteins, so that across the membrane an osmotic pressure occurs in addition to a Donnan potential. The values concerned can be approximately calculated from the equations derived by Donnan35. In case (3), an intermediate situation, there is a combined effect of diffusion and the Donnan potential, so that its calculation becomes uncertain. 

As we discussed in Section 3.2, samples of solution and solvent separated by a semipermeable membrane will be at equilibrium only when the solution is at a greater pressure than the solvent. This is the osmotic pressure. If the solution is under less pressure than the equilibrium osmotic pressure, solvent will flow from the pure phase into the solution. If, on the other hand, the solution is under a pressure greater than the equilibrium osmotic pressure, the pure solvent will flow in the reverse direction, from the solution to the solvent phase. In the last case, the semipermeable membrane functions like a filter that separates solvent from solute molecules. In fact, the process is referred to in the literature by the terms hyperfiltration and ultrafiltration, as well as reverse osmosis (Sourirajan 1970) however, the last term is enjoying common use these days. 

If a solution is placed on one side of a semipermeable membrane and water is placed on the other side, then there is a natural tendency (oj/nwi.t) for waler to diffuse through the membrane lo the solute side until an equilibrium osmotic pressure is reached. 

At the beginning of an osmotic experiment, the difference in heights A/z observed after filling both chambers of the osmometer does not correspond to the osmotic pressure at equilibrium. The equilibrium pressure is only observed after solvent molecules permeate the membrane. If A/z is greater than the equilibrium osmotic pressure, the solvent molecules permeate from the solution chamber into the solvent chamber, and in the reverse direction if A/z is smaller than the equilibrium osmotic pressure. The time taken to reach equilibrium increases with the amount of solvent that must be displaced, i.e., increases with the diameter of the capillaries. Since, experimentally, problems such as dirt in the capillaries, etc., limit the size of capillary that one can go down to, and since the membranes must be tight (semipermeability), the establishment of osmotic equilibrium can take days or weeks. Other problems such as poor solvent drainage in the capillaries, adsorption of solute on the membrane, partial permeation of solute through the membrane, etc., can interfere with the attainment of a true osmotic equilibrium. The absence or presence and allowance for these complications must be individually established. 

Osmotic pressure is a thermodynamic property of the solution. Thus, n is a state variable that depends upon temperature, pressure, and concentration but does not depend upon the membrane as long as the membrane is semipermeable. Osmotic equilibrium requires that the chemical potentials of the solvent on the two sides of the membrane be equal. Note that the solutes are not in equilibrium since they cannot pass through the membrane. Although osmotic pressure can be measured directly, it is usually estimated from other measurements (e.g., Reid, 1966). For an incompressible liquid osmotic pressure can be estimated from vapor pressure measurements. 

When a solution is separated from the pure solvent by a semi-permeable membrane i.e. a membrane that permits the passage of solvent molecules but not of solute molecules, the solvent molecules always tend to pass through the membrane into the solution. This general phenomenon is known as osmosis, and the flow of solvent molecules leads to the development of an osmotic pressure which at equilibrium just prevents further flow. The equilibrium osmotic pressure, n, can be measured using a capillary osmometer such as that shown schematically in Fig. 3.9. 

Fig. 3.9 Schematic representation of an osmotic pressure cell consisting of a polymer solution in equilibrium with pure solvent across a semi-permeable membrane. The osmotic pressure fl can be determined from the difference of the height of the liquids in the capillaries.

In the applications it is however rarely possible to apply the Donnan theory for ideal systems, at least if more than qualitative results are wanted. From the eq. (115, 116, 117 and 118) it is possible to determine the charge on the colloid in three independent ways. This means, that two relations between the three phenomena (distr. of ions, membrane potential, osmotic pressure) should exist. Now usually, when the osmotic pressure is calculated either from analytical data or from the membrane potential it is found to be too low (Hahmarsten effect ). This clearly indicates the need for a more exact treatment of the Donnan equilibrium, in the first place the introduction of activity coefficients. 

We consider this system in an osmotic pressure experiment based on a membrane which is permeable to all components except the polymeric ion P that is, solvent molecules, M" , and X can pass through the membrane freely to establish the osmotic equilibrium, and only the polymer is restrained. It does not matter whether pure solvent or a salt solution is introduced across the membrane from the polymer solution or whether the latter initially contains salt or not. At equilibrium both sides of the osmometer contain solvent, M , and X in such proportions as to satisfy the constaints imposed by electroneutrality and equilibrium conditions. 

The deduction adopted is due to M. Planck (Thermodynamik, 3 Aufl., Kap. 5), and depends fundamentally on the separation of the gas mixture, resulting from continuous evaporation of the solution, into its constituents by means of semipermeable membranes. Another method, depending on such a separation applied directly to the solution, i.e., an osmotic process, is due to van t Hoff, who arrived at the laws of equilibrium in dilute solution from the standpoint of osmotic pressure. The applications of the law of mass-action belong to treatises on chemical statics (cf. Mel lor, Chemical Statics and Dynamics) we shall here consider only one or two cases which serve to illustrate some fundamental aspects of the theory. 

We can prevent the flow of solvent by placing a piston on top of the chamber on the left and applying a pressure p to maintain equilibrium so that the liquid levels on the two sides of the membrane stay equal. The difference in pressure p - p° is known as the osmotic pressure II so that... 

A 0.020 vi C6H 206(aq) solution (glucose) is separated from a 0.050 M CO(NH2)2(aq) solution (urea) by a semipermeable membrane at 25°C. For both compounds / = 1. (a) Which solution has the higher osmotic pressure (b) Which solurion becomes more dilute with the passage of H20 molecules through the membrane (c) To which solution should an external pressure be applied to maintain an equilibrium flow of H20 molecules across the membrane (d) What external pressure (in atm) should be applied in part (c) ... 

The situation changes when there is a concentration imbalance. Figure 12-15 shows red blood cells immersed in solutions of different concentrations. When the fluid outside the cell has a higher solute concentration, the result is slower movement of water through the membrane into the cell. The net result is that water leaves the cell, causing it to shrink. When the fluid outside the cell has a lower concentration, movement of water into the cell increases. The extra water in the cell causes an increase in internal pressure. Eventually, the internal pressure of the cell matches the osmotic pressure, and water transport reaches dynamic equilibrium. Unfortunately, osmotic pressures are so large that cells can burst under the increased pressure before they reach equilibrium. 

This condition expresses the fact that the two solutions are under different pressures, px and p2, as a result of their, in general, different osmotic pressures. An analogous equation cannot be written for the non-diffusible ion as it cannot pass through the membrane and the equilibrium concentrations cannot be established. 

Osmosis is the passage of a pure solvent into a solution separated from it by a semipermeable membrane, which is permeable to the solvent but not to the polymeric solute. The osmotic pressure n is the pressure that must be applied to the solution in order to stop the flow. Equilibrium is reached when the chemical potential of the solvent is identical on either side of the membrane. The principle of a membrane osmometer is sketched in Figure 2. 

If you were to place a solution and a pure solvent in the same container but separate them by a semipermeable membrane (which allows the passage of some molecules, but not all particles) you would observe that the level of the solvent side would decrease while the solution side would increase. This indicates that the solvent molecules are passing through the semipermeable membrane, a process called osmosis. Eventually the system would reach equilibrium, and the difference in levels would remain constant. The difference in the two levels is related to the osmotic pressure. In fact, one could exert a pressure on the solution side exceeding the osmotic pressure, and solvent molecules could be forced back through the semipermeable membrane into the solvent side. This process is called reverse osmosis and is the basis of the desalination of seawater for drinking purposes. These processes are shown in Figure 13.1. 

Mauritz et al., motivated by these experimental results, developed a statistical mechanical, water content and cation-dependent model for the counterion dissociation equilibrium as pictured in Figure 12. This model was then utilized in a molecular based theory of thermodynamic water activity, aw, for the hydrated clusters, which were treated as microsolutions. determines osmotic pressure, which, in turn, controls membrane swelling subject to the counteractive forces posed by the deformed polymer chains. The reader is directed to the original paper for the concepts and theoretical ingredients. 

At least two different techniques are available to compress an emulsion at a given osmotic pressure H. One technique consists of introducing the emulsion into a semipermeable dialysis bag and to immerse it into a large reservoir filled with a stressing polymer solution. This latter sets the osmotic pressure H. The permeability of the dialysis membrane is such that only solvent molecules from the continuous phase and surfactant are exchanged across the membrane until the osmotic pressure in the emulsion becomes equal to that of the reservoir. The dialysis bag is then removed and the droplet volume fraction at equilibrium is measured. 

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