Osmotic pressure description

Both the concentration polarization and osmotic pressure descriptions can be applied to polymer solutions that form well-defined gels at high concentrations. In a gel the thermodynamic osmotic pressure results from the solvent-mediated interactions between the randomly moving gel monomers, and this pressure tends to swell the gel. Both descriptions have been calculated in some detail for gelling macromolecular solutions and shown to produce similar behaviors (Probstein et al. 1979, Trettin and Doshi 1981). Actually a relatively simple argument shows that the two approaches are equivalent if the diffusion... 

Chapters 7 to 9 apply the thermodynamic relationships to mixtures, to phase equilibria, and to chemical equilibrium. In Chapter 7, both nonelectrolyte and electrolyte solutions are described, including the properties of ideal mixtures. The Debye-Hiickel theory is developed and applied to the electrolyte solutions. Thermal properties and osmotic pressure are also described. In Chapter 8, the principles of phase equilibria of pure substances and of mixtures are presented. The phase rule, Clapeyron equation, and phase diagrams are used extensively in the description of representative systems. Chapter 9 uses thermodynamics to describe chemical equilibrium. The equilibrium constant and its relationship to pressure, temperature, and activity is developed, as are the basic equations that apply to electrochemical cells. Examples are given that demonstrate the use of thermodynamics in predicting equilibrium conditions and cell voltages. 

Two main approaches for osmotic pressure of polymeric solutions theoretical description can be distinguished. First is Flory-Huggins method [1, 2], which afterwards has been determined as method of self-consistent field. In the initial variant the main attention has been paid into pair-wise interaction in the system gaped monomeric links - molecules of solvent . Flory-Huggins parameter % was a measure of above-said pair-wise interaction and this limited application of presented method by field of concentrated solutions. In subsequent variants such method was extended on individual macromolecules into diluted solutions with taken into account the tie-up of chain links by Gaussian statistics [1]. 

For description of the osmotic pressure 7t of polymeric solutions the virial decomposition is used in the Flory-Huggins method... 

The explanation of the pressure-independent region during the ultrafiltration of macromolecules requires the arbitrary introduction of the concept of a gel-layer in the film model. A more complete description of the dependence of the membrane permeation rate on the applied pressure may be given by considering the effect of the osmotic pressure of the macromolecules as described by Wijmans et alS18 Equation 8.2 may then be written as ... 

Analysis of polyelectrolytes in the semi-dilute regime is even more complicated as a result of inter-molecular interactions. It has been established, via dynamic light-scattering and time-dependent electric birefringence measurements, that the behavior of polyelectrolytes is qualitatively different in dilute and semi-dilute regimes. The qualitative behavior of osmotic pressure has been described by a power-law relationship, but no theory approaching quantitative description is available. 

In comparison with the qualitative description of diffusion in a binary system as embodied by Eqs. (11), (12) or (14), the thermodynamic factors are now represented by the quantities a, b, c, and d and the dynamic factors by the phenomenological coefficients which are complex functions of the binary frictional coefficients. Experimental measurements of Dy in a ternary system, made on the basis of the knowledge of the concentration gradients of each component and by use of Eqs. (21) and (22), have been reviewed 35). Another method, which has been used recently36), requires the evaluation of py from thermodynamic measurements such as osmotic pressure and evaluation of all fy from diffusion measurements and substitution of these terms into Eqs. (23)—(26). 

Virial Isotherm Equation. No isotherm equation based on idealized physical models provides a generally valid description of experimental isotherms in gas-zeolite systems (19). Instead (6, 20, 21, 22) one may make the assumption that in any gas-sorbent mixture the sorbed fluid exerts a surface pressure (adsorption thermodynamics), a mean hydrostatic stress intensity, Ps (volume filling of micropores), or that there is an osmotic pressure, w (solution thermodynamics) and that these pressures are related to the appropriate concentrations, C, by equations of polynomial (virial) form, illustrated by Equation 3 for osmotic pressure ... 

For truly high rejection reverse osmosis membranes, the solution-diffusion description of this process is the most popular and probably the most realistic. In this case, the high osmotic pressure difference between the... 

Sucrose in a concentration of 1,000 mg/L has an osmotic pressure of 7.24 kNa (kiloNewtons absolute). Thus, the reverse pressure to be applied must be, theoretically, in excess of 7.24 kNa for a sucrose concentration of 1,000 mg/L. For NaCl, its osmotic pressure in a concentration of 35,000 mg/L is 2744.07 kNa. Hence, to reverse the flow in a NaCl concentration of 35,000 mg/L, a reverse pressure in excess of 2744.07 kNa should be applied. The operation just described (i.e., applying sufficient pressure to the tip of the tube to reverse the flow of water) is the fundamental description of the basic reverse osmosis process. 

Pure liquids and solutions have probably received a major portion of the experimental effort devoted to the nonspectroscopic methods of detection. The liquid phase is susceptible to simple techniques and is the naturally occurring state for many substances. The principal methods of study are vapor pressure measurements, cryoscopy, solubility, and partition studies. To a lesser degree parachor, refractive index, thermal and acoustic conductivity, osmotic pressure, and magnetic susceptibility measurements have been applied to H bonded materials. Unfortunately, the difficulty of giving an adequate description of the liquid state sometimes produces problems of interpretation. 

In spite of the partial success in theoretical description, we believe that more realistic models are needed for the theory to have a predicting power. For example, measurements usually take place in the presence of a large excess of simple electrolyte. The electrolyte present is often a buffer, a rather complicated mixture (difficult to model perse) with several ionic species present in the system. Note that many effects in protein solutions are salt specific. Yet, most of the theories subsume all the effect of the electrolyte present into a single parameter, the Debye screening parameter n. In the case of the Donnan equilibrium we measure the subtle difference between the osmotic pressures across a membrane permeable to small ions and water but not to proteins. We believe that an accurate theoretical description of protein solutions can only be built based on the models which take into account hydration effects. 

Studies of various kinds of equilibria provide a wealth of information about polymer systems. Classical thermodynamics, which is concerned with the macroscopic properties of a system and the relations that hold between them at equilibrium, form a sufficient basis for description of these equilibria in polymer systems. We shall consider in a major part of this chapter methods of study of polymer solutions that deal with equilibria and can be fully described by thermodynamic relations. These include vapor pressure, osmotic pressure, and phase separation in polymer-solvent systems. 

C. Arguments Against Flory s Formula and Other Descriptions for the Osmotic Pressures... 

Cake filtration could be used for some of these materials, but a cake of l-/rm particles would have a high resistance to flow, and the filtration rate would be very low. Ultrafiltration (UF) covers a wider size range, from 1-pm particles down to molecules about 10 /rm in size (Af= 300). The term hyperfiltration is sometimes used for separation of small molecules or ions, but reverse osmosis is a more descriptive term, because the osmotic pressure has a major effect on the flux. Furthermore, the separation in reverse osmosis occurs by a solution-diffusion mechanism in the dense polymer rather than by a screening action at the membrane surface (see Chap. 26). 

A full description of various osmometers and the necessary experimental technique can be found in the book edited by Allen. More-recent types of osmometer have been described, but the outstanding problem in osmometry is still the preparation of suitable semi-permeable membranes. Methods of preparing membranes claimed to be suitable for materials of low molecular weight have been described, and there are several reports of comparisons of the behavior of different types of membranes in osmometryThe problem of correcting observed osmotic pressures for any solute diffusion which may occur has been considered theoretically, - and a suitable technique established. ... 

A second explanation for a pressure-independent permeate rate could be a strong osmotic pressure dependence on concentration with the osmotic pressure approaching the applied pressure. This description is viable only where the osmotic pressure has meaning. In a system in which solidlike particulates coalesce, the osmotic pressure model would not be a good one. 

The Osmotic Pressure Model, as shown in (3.6), is an equivalent description for macromolecules according to Wijmans et al. (1985). AfT is the osmotic pressure difference across the membrane. 

The dialysis equipment, shown schematically in Figure 1, is similar to that used by Tanner and Berry, without explicit description. Dilute solutions were brought to osmotic equilibrium, using osmotic pressure membranes, over a several hour period by continuous feed of fresh mixed solvent with the desired concentration to the solvent side of the dialysis cell. About 20 ml of solution was retained on the solution side, agitated slowly with a magnetic stirring bar. 

There are three main modes of interaction between a polymer solution and a solid surface. The first interaction mode is depletion [2,3]. If the monomers are repelled by the surface (or in other words if the attractive interaction between the solvent molecules and the surface is larger than the interaction between the monomers and the surface), the polymer concentration in solution decreases as the surface is approached and a region depleted in polymer exists in the vicinity of the surface. The size of this region is the size of the polymer chain if the solution is dilute and the size of the correlation length of the solution if the solution is semidilute (if the polymer chains overlap). When two surfaces are brought in close contact, the density in the gap between the surfaces is smaller than the bulk concentration and the osmotic pressure in the gap is smaller than the bulk osmotic pressure. This osmotic pressure difference induces an attraction between the surfaces. The depletion interaction is not specific to polymers and exists with any particle with a size in the colloidal range [4]. It has sometimes been used to induce adhesion between particles of mesoscopic size such as red blood cells. The only limitation to this qualitative description of the depletion force is that at equilibrium the polymer chains (or any other particles) must leave the gap as the surfaces get closer. There is no attractive depletion force if they remain trapped in the gap. We will not consider further the depletion interaction. 

There have been a number of attempts to revise and elaborate Brown s description of the film formation process [11,30]. For an overview of these approaches the reader is refmed to Mazur s review [25]. Perhaps the most important insight is that in a strict thermodynamic sense, there is no distinction betweai the osmostic pressure, II, and the mean capillaiy pressure, pc, which develops when the meniscus intercepts the surface of the particles. As the system becomes more concentrated, there will be some contribution of capillary forces to 11 before the particles come into contact and before the particle-water meniscus develops. Ultimately, n must converge to pc once the particles are in contact This continuity of n and Pc is the main point of the theoretical analysis by Crowley et al. [31], and they showed that when the osmotic pressure rises to little as 1% of the equilibrium vapour pressure of water, the force should be sufficirait to densify the film. 

Van t Hoffs equation is very similar to the general gas law. In fact, both equations can be interpreted in the same way. Here we need to keep in mind that the forces of attraction between the A particles keep the liquid together (compare Sect. 11.1, keyword cohesion pressure ). The contribution of the external pressure p is comparatively small. The B particles that drift far away from each other and scarcely influence each other cause a pressure like that of a dilute gas. However, in this case the pressure is not compensated by the container walls but by the cohesion of the A particles. When the osmotic pressure posm is higher than the external pressure—a condition that is often attained— the liquid A behaves as if it were under negative pressure. If we calculate the potential pa of the liquid for a pressure reduced by Posm and keep in mind that for dilute solutions V a Wa and b/ a. b. we again end up with Eq. (12.3). This demonstrates that both descriptions are equivalent ... 

When 0 is larger than typically a few percent, the above description no longer applies. In the case of hard spheres, the osmotic pressure is well represented by the Carnaham and Starling formula up to large volume fractions [21]. The friction coefficient is more difficult to evaluate because the hydrodynamic interactions are long-range. Microemulsion droplets behave as hard spheres in many circumstances. However, in W/O microemulsions, droplets frequently exhibit supplementary attractive interactions. It has been proposed that the osmotic pressure... 

Detailed studies on thermo-osmosis using highly selective cellulose acetate membrane in the presence and absence of osmotic pressure difference have also been carried out [25]. Using general description of thermo-osmosis based on irreversible thermodynamics, it was shown that coupling between the flow of heat and the flow of water is quite loose possibly on account of thermal leak between the compartments. Whatever the detailed stmctural interpretation, it was argued that in annealed, less-permeable membranes, the water-matrix interaction is increased relative to the water-water interaction and with only this type of interaction strong thermo-osmosis is expected. 

Accordingly, we shall pass next to a description of the influence of temperature upon osmotic pressure. 

Even at concentrations which are still very low, interactions between these particles take place and result in a rapid rise of osmotic pressure and viscosity with concentration. It will be appropriate, therefore, to pass to the description of conditions at higher concentrations. 

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