Osmotic pressure defined

In the preceding derivation, the repulsion between overlapping double layers has been described by an increase in the osmotic pressure between the two planes. A closely related but more general concept of the disjoining pressure was introduced by Deijaguin [30]. This is defined as the difference between the thermodynamic equilibrium state pressure applied to surfaces separated by a film and the pressure in the bulk phase with which the film is equilibrated (see section VI-5). 

The pressure difference between the high and low pressure sides of the membrane is denoted as AP the osmotic pressure difference across the membrane is defined as Att the net driving force for water transport across the membrane is AP — (tAtt, where O is the Staverman reflection coefficient and a = 1 means 100% solute rejection. The standardized terminology recommended for use to describe pressure-driven membrane processes, including that for reverse osmosis, has been reviewed (24). 

Feed characteri2ation, particularly for nondesalination appHcatioas, should be the first and foremost objective in the design of a reverse osmosis plant. This involves the determination of the type and concentration of the main solutes and foulants in the stream, temperature, pH, osmotic pressure, etc. Once the feed has been characteri2ed, a reaHstic process objective can be defined. In most cases, some level of pretreatment is needed to reduce the number and concentration of foulants present in the feed stream. Pretreatment necessitates the design of processes other than the RO module, thus the overaH process design should use the minimum pretreatment necessary to meet the process objective. Once the pretreatment steps have been determined and the final feed stream defined, the RO module can be selected. 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

We may assume, as a close approximation, that the osmotic pressure is defined by the equation ... 

The sodium chloride equivalent of a chemical is defined as the amount of sodium chloride (in grams or grains) that has the same osmotic pressure as that of 1 g of the chemical. The sodium chloride equivalents are symbolized by the letter E. The quantities of two substances that are isotonic equivalents are proportional to the molecular weight of each multiplied by the i value of the other. Thus, if the molecular weight and i value of a given chemical are known, one can calculate the sodium chloride equivalent, E, of that chemical as follows ... 

The difference in pressures, P — Pq, required to maintain osmotic equilibrium is defined as the osmotic pressure and is denoted by IT. Equation (15.44) thus becomes... 

To account for their data (Fig. 2.7), Mondain-Monval et al. hypothesized that these two forces simply add and that the repulsion between micelles and droplets increases the effective diameter of the droplets (or micelles) [22]. This force is derived by integrating the osmotic pressure Posm over the accessible zone for micelles of diameter 2r (r = 2.35 nm) from 6 = n to 9 = 7t -Oi, with 9i defined in Fig. 2.6. The distance at which the small micelles are excluded from the gap between the droplets is evidently influenced by the electrostatic micelle-droplet repulsion. To account for this repulsion, droplets (or micelles) may be considered as particles of effective radius (a + S) [or micelles of radius (r + 5)]. From... 

The osmolahty of a contrast agent solution is proportional to the number of independent particles in the solution and is strongly influenced by both the concentration of the contrast agent (or any other constituents) and the temperature of the solution. The osmotic pressure of a contrast agent preparation is given in milliosmol kg water (mosm kg ), in Megapascal (MPa) or in atmospheres (at). Conversion between the different units follows the equation 1 osm kg = 1000 mosm kg = 2.58 MPa = 25.5 at. In a multi-component system, the osmolality is defined as... 

The properties of solutions of macromolecular substances depend on the solvent, the temperature, and the molecular weight of the chain molecules. Hence, the (average) molecular weight of polymers can be determined by measuring the solution properties such as the viscosity of dilute solutions. However, prior to this, some details have to be known about the solubility of the polymer to be analyzed. When the solubility of a polymer has to be determined, it is important to realize that macromolecules often show behavioral extremes they may be either infinitely soluble in a solvent, completely insoluble, or only swellable to a well-defined extent. Saturated solutions in contact with a nonswollen solid phase, as is normally observed with low-molecular-weight compounds, do not occur in the case of polymeric materials. The suitability of a solvent for a specific polymer, therefore, cannot be quantified in terms of a classic saturated solution. It is much better expressed in terms of the amount of a precipitant that must be added to the polymer solution to initiate precipitation (cloud point). A more exact measure for the quality of a solvent is the second virial coefficient of the osmotic pressure determined for the corresponding solution, or the viscosity numbers in different solvents. 

The influence of neutral salts as well as of acids and bases on the swelling of gelatine which we have seen can be attributed to an apparent change in the solvation of the gel fibrils and may be interpreted in the light of Donnan s theory of the effect of a non-diffusible ion on the osmotic pressure differences between the two phases, is likewise to be noted in the alteration of the viscosity and alcohol precipitation values of protein solutions. From the considerations already advanced there should exist two well-defined maxima in the viscosity and alcohol precipitation curves when these properties are plotted as functions of the Ph, the maxima coinciding with the points of maximum dissociation of the salts... 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, using software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Rtf. To facilitate these calculations, the following data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sh, is defined as Sh = 0.04 Re0 75 Sc0-33, where Sc is the Schmidt number (2) osmotic pressure follows van t Hoff s equation, ie, 7r = iCRgTy where i is the number of ions (3)... 

Here, the quantities jn ° and ji are, respectively, the chemical potentials of pure solvent and of the solvent at a certain biopolymer concentration V is the molar volume of the solvent and n is the biopolymer number density, defined as n C/M, where C is the biopolymer concentration (% wt/wt) and M is the number-averaged molar weight of the biopolymer. The second virial coefficient has (weight-scale) units of cm mol g. Hence, the more positive the second virial coefficient, the larger is the osmotic pressure in the bulk of the biopolymer solution. This has consequences for the fluctuations in the biopolymer concentration in solution, which affects the solubility of the biopolymer in the solvent, and also the stability of colloidal systems, as will be discussed later on in this chapter. 

It is important to note that the concept of osmotic pressure is more general than suggested by the above experiment. In particular, one does not have to invoke the presence of a membrane (or even a concentration difference) to define osmotic pressure. The osmotic pressure, being a property of a solution, always exists and serves to counteract the tendency of the chemical potentials to equalize. It is not important how the differences in the chemical potential come about. The differences may arise due to other factors such as an electric field or gravity. For example, we see in Chapter 11 (Section 11.7a) how osmotic pressure plays a major role in giving rise to repulsion between electrical double layers here, the variation of the concentration in the electrical double layers arises from the electrostatic interaction between a charged surface and the ions in the solution. In Chapter 13 (Section 13.6b.3), we provide another example of the role of differences in osmotic pressures of a polymer solution in giving rise to an effective attractive force between colloidal particles suspended in the solution. 

Elementary and advanced treatments of such cellular functions are available in specialized monographs and textbooks (Bergethon and Simons 1990 Levitan and Kaczmarek 1991 Nossal and Lecar 1991). One of our objectives in this chapter is to develop the concepts necessary for understanding the Donnan equilibrium and osmotic pressure effects. We define osmotic pressures of charged and uncharged solutes, develop the classical and statistical thermodynamic principles needed to quantify them, discuss some quantitative details of the Donnan equilibrium, and outline some applications. 

The thermodynamic preliminaries and concepts needed for defining osmotic pressure are discussed in Sections 3.2a-c. The nonideality of colloidal solutions can be appreciable since the solvent and solute particles are so different in size. Classical thermo-... 

The easiest way to extend these considerations to the osmotic pressure of nonideal solutions is to return to Equation (22), which relates ir to a power series in mole fraction. This equation applies to ideal solutions, however, since ideality is assumed in replacing activity by mole fraction in the first place. To retain the form and yet extend its applicability to nonideal solutions, we formally include in each of the concentration terms a correction factor defined to permit the series to be applied to nonideal solutions as well ... 

This same conclusion may also be reached by the following argument. The product nJA, in Equation (43) equals the weight of component i in the solution the total weight of solute in the solution equals L,n,M,. The experimental osmotic pressure depends on and therefore measures the total number of moles of solute The ratio of the total weight to the total number of moles of solute defines the number average molecular weight. 

The experimental result obtained was explained by the formation of ion pairs between the charges of a network and counter ions. The theoretical analysis of this problem has shown that the degree of ion pairs formation very strongly (exponentially) depends on e (cf. Sect. 2.2). Thus, if the precipitant has a small dielectric constant e (e.g. dioxane) the degree of dissociation of ion pairs is sufficiently small and this fact leads to the decrease of the osmotic pressure of counter ions which defines the swelling of the gel and the point of the transition in the collapsed state. As a result, in this case the degree of swelling of the gel near the transition point is less pronounced than for other solvents and only a relatively small amount of the precipitant is required to reach this point. In... 

When solvent and solution are separated by a semipermeable membrane that permits solvent molecules to pass, an osmotic pressure is developed in the solution. This pressure, tt, is defined as the mechanical pressure that must be applied to the solution to prevent solvent molecules from diffusing into it. For water solutions the relationship between tt and the molal concentration m is given by the equation... 

The osmotic pressure of a solution is defined by Glasstone (1) as the excess pressure which must be applied to a solution to prevent the passage into it of solvent when they are separated by a perfectly semipermeable membrane. Actually no membrane is... 

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