Osmotic pressure, concentration

The similarity between the plots of c/r vs. c shown in Figs. 47 and 48 and those for tc/c vs. c shown in Figs. 38 and 39 is apparent. Deviations from ideality (i.e., the changes in iz/c and in c/r with c) have the same origin for both types of measurements. As with the osmotic pressure-concentration ratio, the change of c/r with c may be reduced by choosing a poor solvent. A further advantage of a poor solvent enters because of the smaller size assumed by the polymer molecule in a poor solvent environment, which reduces the dissymmetry correction. 

They showed further that the limiting slope (RTA2) of the plot of the osmotic pressure-concentration ratio tz/c against the polymer concentration in a binary solvent mixture should be proportional to the value of the quantity on the left side of Eq. (17),f with V2 representing the volume fraction of solvent in the nonsolvent-solvent mixture which is in osmotic equilibrium with the solution. The composition of the liquid medium outside the polymer molecules in a dilute solution must likewise be given by V2. The composition of the solvent mixture within the domains of the polymer molecules may differ slightly from that outside owing to selective absorption of solvent in preference to the nonsolvent. This internal composition is not directly of concern here. If the solution is made sufficiently dilute, the external nonsolvent-solvent composition v2 = l—Vi) will be practically equal to the over-all solvent composition for the solution as a whole. Hence... 

Fig. 145.—Osmotic pressure-concentration ratios ( in g./cm and c in g./lOO ml.) for poly-(4-vinylpyridine) in alcohol, O, coordinates left and below poly-(N-butyl-4-vinylpyridinium bromide) in alcohol, coordinates right and above and the same polymer in alcoholic 0.61 N lithium bromide, 3 coordinates left and below. °> ...

In reverse osmosis, where the solutes retained are relatively low in molecular weight and have a significant osmotic pressure, concentration polarization can result in osmotic pressures considerably higher than those represented by the bulk stream concentration. Higher pressures are required to overcome the osmotic pressure (Figure 6). 

Polymer solutions, even at very low concentrations, show considerable deviations from this law, largely due to their abnormally high entropy of dilution. It has been calculated on this basis7 that the osmotic pressure-concentration equation should be of the continued series form ... 

It should be pointed out at this juncture that strict thermodynamics treatment of the film-covered surfaces is not possible [18]. The reason is difficulty in delineation of the system. The interface, typically of the order of a 1 -2 nm thick monolayer, contains a certain amount of bound water, which is in dynamic equilibrium with the bulk water in the subphase. In a strict thermodynamic treatment, such an interface must be accounted as an open system in equilibrium with the subphase components, principally water. On the other hand, a useful conceptual framework is to regard the interface as a 2-dimensional (2D) object such as a 2D gas or 2D solution [ 19,20]. Thus, the surface pressure 77 is treated as either a 2D gas pressure or a 2D osmotic pressure. With such a perspective, an analog of either p- V isotherm of a gas or the osmotic pressure-concentration isotherm, 77-c, of a solution is adopted. It is commonly referred to as the surface pressure-area isotherm, 77-A, where A is defined as an average area per molecule on the interface, under the provision that all molecules reside in the interface without desorption into the subphase or vaporization into the air. A more direct analog of 77- c of a bulk solution is 77 - r where r is the mass per unit area, hence is the reciprocal of A, the area per unit mass. The nature of the collapsed state depends on the solubility of the surfactant. For truly insoluble films, the film collapses by forming multilayers in the upper phase. A broad illustrative sketch of a 77-r plot is given in Fig. 1. 

The increase of n(A] with decreasing A, at constant temperature, is the two-dimensional analogue of am osmotic pressure-concentration Isotherm. Such surface pressure isotherms are the prime source of information about the orientational and/or conformational properties of the molecules in the monolayer they reflect their dimensional properties as well as interactions between them. In this respect, x(A) isotherms have about the same function as adsorption Isotherms. This matter will be discussed in more detaill in secs. 3.4 and 5. 

The effect of osmotic pressure in macromolecular ultraflltra-tlon has not been analyzed in detail although many similarities between this process and reverse osmosis may be drawn. An excellent review of reverse osmosis research has been given by Gill et al. (1971). It is generally found, however, that the simple linear osmotic pressure-concentration relationship used in reverse osmosis studies cannot be applied to ultrafiltration where the concentration dependency of macromolecular solutions is more complex. It is also reasonable to assume that variable viscosity effects may be more pronounced In macromolecular ultra-filtration as opposed to reverse osmosis. Similarly, because of the relatively low diffuslvlty of macromolecules conqiared to typical reverse osmosis solutes (by a factor of 100), concentration polarization effects are more severe in ultrafiltration. 

The possibility of such proof became available from laws governing the relation between vapor pressure and mole fraction, or between osmotic pressure, concentration, temperature, and molecular weight, which were discovered by Raoult (1882-1885) and van t Hoff (1887-1888). With these methods, very high molecular weights (between 10,0(X) and 40,000) were subsequently obtained for rubber, starch, and cellulose nitrate. Other authors found similarly high values for the same materials e.g., Gladstone and Hibbert " found 6000-12,000 for rubber, and Brown and Morris obtained cryoscopically about 30,000 for a product of starch obtained by degradation hydrolysis. 

The counterions form a diffuse cloud that shrouds each particle in order to maintain electrical neutrality of the system. When two particles are forced together their counterion clouds begin to overlap and increase the concentration of counterions in the gap between the particles. If both particles have the same charge, this gives rise to a repulsive potential due to the osmotic pressure of the counterions which is known as the electrical double layer (EDL) repulsion. If the particles are of opposite charge an EDL attraction will result. It is important to realize that EDL interactions are not simply determined by the Columbic interaction between the two charged spheres, but are due to the osmotic pressure (concentration) effects of the counterions in the gap between the particles. 

Detailed osmotic studies by Mandel et al. [140] on NaPSS yield an exponent of 9/8 at low concentration and 9/4 at high concentration of the osmotic pressure-concentration power law (n c ) which again was interpreted as a dilute-semidilute concentration transition. A recent literature study [66] confirmed the experimental scaling exponents but clearly demonstrated that the cross-over concentration does not depend on the molar mass of the polyions. 

Example 6.3.10 Expressions (6.3.154c) and (6.3.154d) developed for if for a solution-diffusion RO membrane are not tmly predictive of R, since the right-hand side has a Air term which contains C,p. Develop an expression for R that contains only membrane transport parameters, feed conditions and AP assume that the osmotic pressure-concentration relation is given by tr(C.j) = bCy (relation (3.4.61b)). 

The number average molecular weight of a polysaccharide may be calculated from its osmotic pressure in solution, according to the Van t Hoff formula, but any association of the molecules in solution will give spurious high values. This may be overcome by using a limiting value of osmotic pressure/concentration extrapolated to zero concentration. Alternatively, a modified Van t Hoff expression ... 

It must be kept in mind that both pictures are modelistic and invoke extrather-modynamic concepts. Except mathematically, there is no such thing as a two-dimensional gas, and the solution whose osmotic pressure is calculated is not uniform in composition, and its average concentration depends on the depth assumed for the surface layer. 

This subject has a long history and important early papers include those by Deijaguin and Landau [29] (see Ref. 30) and Langmuir [31]. As noted by Langmuir in 1938, the total force acting on the planes can be regarded as the sum of a contribution from osmotic pressure, since the ion concentrations differ from those in the bulk, and a force due to the electric field. The total force must be constant across the gap and since the field, d /jdx is zero at the midpoint, the total force is given the net osmotic pressure at this point. If the solution is dilute, then... 

Theta conditions in dilute polymer solutions are similar to tire state of van der Waals gases near tire Boyle temperature. At this temperature, excluded-volume effects and van der Waals attraction compensate each other, so tliat tire second virial coefficient of tire expansion of tire pressure as a function of tire concentration vanishes. On dealing witli solutions, tire quantity of interest becomes tire osmotic pressure IT ratlier tlian tire pressure. Its virial expansion may be written as... 

Figure C2.1.5. Reduced osmotic pressure FT / (RTc as a function of the weight concentration c of polystyrene (M = 130 000 g mor ) in cyclohexane at different temperatures. At 7"= 35 °C and ambient pressure, tire solution is at tire 0 conditions. (Figure from 1741, reprinted by pennission of EDP Sciences.)...

Figure C2.1.8. Reduced osmotic pressure V l(RTc as a function of the polymer weight concentration for solutions of poly(a-metliylstyrene) in toluene at 25 °C. The molecular weight of poly(a-metliylstyrene) varies...

Since capillary tubing is involved in osmotic experiments, there are several points pertaining to this feature that should be noted. First, tubes that are carefully matched in diameter should be used so that no correction for surface tension effects need be considered. Next it should be appreciated that an equilibrium osmotic pressure can develop in a capillary tube with a minimum flow of solvent, and therefore the measured value of II applies to the solution as prepared. The pressure, of course, is independent of the cross-sectional area of the liquid column, but if too much solvent transfer were involved, then the effects of dilution would also have to be considered. Now let us examine the practical units that are used to express the concentration of solutions in these experiments. 

Figure 8.9 Osmotic pressure data plotted as n/RTc2 versus concentration for nitrocellulose in three different solvents. [Data from A. Dobry,/. Chem. Phys. 32 50 (1935).]...

The ratio n/c2 is called the reduced osmotic pressure-and can be plotted with or without the RT-and the zero-intercept value (subscript 0) is the limiting value of the reduced osmotic pressure. Quite an assortment of different pressure units are used in the literature in reporting n values, and the units of R in Eq. (8.88) must be reconciled with these pressure (as well as concentration) units. 

These results show more clearly than Fq. (8.126)-of which they are special cases-the effect of charge and indifferent electrolyte concentration on the osmotic pressure of the solution. In terms of the determination of molecular weight of a polyelectrolyte by osmometry. ... 

By describing the concentration dependence of an observable property as a power series, Eq. (9.9) plays a comparable role for viscosity as Eq. (8.83) does for osmotic pressure. 

Although Eq. (10.50) is still plagued by remnants of the Taylor series expansion about the equilibrium point in the form of the factor (dn/dc2)o, we are now in a position to evaluate the latter quantity explicitly. Equation (8.87) gives an expression for the equilibrium osmotic pressure as a function of concentration n = RT(c2/M + Bc2 + ) Therefore... 

Thus we have finally established how light scattering can be used to measure the molecular weight of a solute. The concentration dependence of r enters Eq. (10.54) through an expression for osmotic pressure, and this surprising connection deserves some additional comments ... 

In Chap. 8 we saw how the equilibrium osmotic pressure of a solution is related to AG for the mixing process whereby the solution is formed. Any difference in the concentration of the solution involves a change in AG j, ... 

When the superfluid component flows through a capillary connecting two reservoirs, the concentration of the superfluid component in the source reservoir decreases, and that in the receiving reservoir increases. When both reservoirs are thermally isolated, the temperature of the source reservoir increases and that of the receiving reservoir decreases. This behavior is consistent with the postulated relationship between superfluid component concentration and temperature. The converse effect, which maybe thought of as the osmotic pressure of the superfluid component, also exists. If a reservoir of helium II held at constant temperature is coimected by a fine capillary to another reservoir held at a higher temperature, the helium II flows from the cooler reservoir to the warmer one. A popular demonstration of this effect is the fountain experiment (55). 

Ultrafiltration. Membranes are used that are capable of selectively passing large molecules (>500 daltons). Pressures of 0.1—1.4 MPa (<200 psi) are exerted over the solution to overcome the osmotic pressure, while providing an adequate dow through the membrane for use. Ultrafiltration (qv) has been particulady successhil for the separation of whey from cheese. It separates protein from lactose and mineral salts, protein being the concentrate. Ultrafiltration is also used to obtain a protein-rich concentrate of skimmed milk from which cheese is made. The whey protein obtained by ultrafiltration is 50—80% protein which can be spray dried. 

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