Reduced osmotic pressure concentration

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...

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. 

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. 

For this reason, the relationship between the reduced Osmotic pressure p/C and the concentration is generally expressed in the form of virial equation as given below ... 

Fig Reduced Osmotic pressure p/C as a function of the concentration of the solution C (1) polymer solution, (2) ideal solution. 

Figure 2 Plot of the reduced osmotic pressure versus concentration for PS-SO3Li-40, ( O ) in cyclohexane at T = 30°C, ( ) in THF at T = 25 C.

Figure 8 Concentration dependence of the reduced osmotic pressure of PS-S03Li 80 under conditions. The full line represents the model calculations with = 10 9 (mol/1) and N = 11, the dashed line indicates = 6,000.

We have already seen that the second virial coefficient may be determined experimentally from a plot of the reduced osmotic pressure versus concentration. Since all other quantities in Equation (99) are measurable, the charge of a macroion may be determined from the second virial coefficient of a solution with a known amount of salt. As an illustration of the use of Equation (99), we consider the data of Figure 3.6 in Example 3.5. 

Figure 2.8 Dependence of reduced osmotic pressure on concentration (1) a linear high polymer in a good solvent (2) the same polymer in a poor solvent (3) a globular protein in aqueous solution...

Fig. 1.5 Variation of the reduced osmotic pressure tt/c, (tt in cm of solvent, c in g dl ) with the concentration c for three poly(pentachlorophenyl methacrylate) fractions in benzene solutions at 40° C ( -solvent). (From ref. [44])...

Figure 3. Reduced osmotic pressure fve. reduced total amphlphllar concentration y. The curves are labeled for various values of the parameter A. The values of the other parameters are o — 4.5, n = 40, and n = 50.

Fig. 6. Reduced osmotic pressure vs. reduced concentration, plotted on a log-log scale, measured through osmometry by Noda el al. (1981). Symbols denote various molecular weights of poly(a-methylstyrene) in toluene at 25 C. The solid line has a slope of 1.32.

Figure 5.7(b) demonstrates that the functional form of Eq. (5.43) reduces osmotic pressure data at various M and 6 (or c) to a universal curve. The limiting scaling laws of II (/> or II are only valid sufficiently far from the overlap concentration. Near (f> (and more generally near any crossover point), a more complicated functional form than a simple power law is needed. For osmotic pressure in a good solvent (and many other examples) the full functional form of Eq. (5.43) is well described by a Simple sum of the two limiting behaviours ... 

The critical micelle concentrations (erne s) were determined with a slightly modified methodology of Floriano et al. (1999). Figure 7 illustrates the concept. The reduced osmotic pressure is obtained from the logarithm of the grand partition function, a quantity which can in turn be determined from the simulations to within an additive constant (Floriano et al., 1999). For the system H2T4 (12), L = 10, we obtain the two curves shown in Fig. 7 at the two temperatures indicated there. There is a clear break, indicating the... 

Figure 4.6 Reduced osmotic pressure (II/c) versus concentration (c) plots for the same polymer sample in different solvents of increasing solvent power.

Figure 1. Dependence of the reduced osmotic pressure on the polymer concentration, Cp. of sodium pectate in 0.1 M ionic strength at 27 C (O ) pH 6.5 ( ) pH 3.5.

It should be noted that the values of the Mu, reported in the Figure 6, are calculated from the reduced osmotic pressure extrapolated to zero polymer concentration. whereas all the other results reported in this work have been obtained at finite polymer concentration and therefore may include terms arising from the concentration dependence of the investigated property. 

Fig. 14.1. The reduced osmotic pressure tlMjRTp is plotted against the mass concentration p, for various temperatures (according to Strazielle 1 polystyrene M = 130000 solvent cyclohexane temperatures 40 °C, 35 °C, 30 °C). The temperature T corresponding to 35 °C is close to Tf (the Flory temperature).

Figure 16.13 shows, first of all, the existence of an interval in which the reduced osmotic pressure is a universal function of the rate of overlap. This result is a confirmation of those which were obtained in Chapter 15, Section 4. On the other hand, when the concentration becomes large (i.e. CXi > 10, in Fig. 16.13), the universal behaviour disappears. 

In particular, for the same rate of overlap, the deviation of the osmotic pressure of a solute with respect to the universal function is much larger if it is made of short chains than if it is made of long chains. Now we note that, for a fixed rate of overlap, the volume fraction cp of solute increases when the molecular mass of the chains decreases. This fact appears in Fig. 16.14. Thus, when the concentration is large enough [see (8.2.14)], the volume fraction cp is the essential parameter which determines the osmotic pressure and this result looks very reasonable. Actually the authors quoted above21 showed that under such conditions, the measured pressures can easily be interpreted with the help of the Flory-Huggins equation (14.5.4). Accordingly, the existence of a concentration domain where the reduced osmotic pressure is a universal function of the rate of overlap appears all the more remarkable. 

Colloid Osmotic Pressure When a colloidal system is separated from its equilibrium liquid by a semipermeable membrane, not permeable to the colloidal species, the colloid osmotic pressure is the pressure difference required to prevent transfer of the dissolved noncolloidal species. Also referred to as the Donnan pressure. The reduced osmotic pressure is the colloid osmotic pressure divided by the concentration of the colloidal species. See also Osmotic Pressure. 

Figure 25.9 Reduced osmotic pressure as a function of reduced concentration.

The dependence of k on pin the mean field theory (p ) is different from that in the scaling theory by a factor of Such difference is related to a correlation effect given by the number of contacts between monomers. The reduced osmotic pressure (K/Kjdeai) plotted as a function of the reduced concentration pip ), in a double logarithm scale, displays a curve that shows the change in slope to 5/4 for pip > 1 (Fig. 25.9). 

Figure 9-1. The concentration dependence of the reduced osmotic pressure 11/c of a poly(methyl methacrylate) in chloroform, dioxane, and m-xylene at 20°C (according to G. V. Schulz and H. Doll).

Fig. 5.3. Reduced osmotic pressure of poly(

For the reduced osmotic pressure irfRTci, the state equation of an ideal dilute solution (Equation 28) shows no concentration dependence... 

At the same time, the slope of the curves in the linear region increases with the solvent quality and the onset of the non-linear behavior is shifted to lower concentrations. The slope of the reduced viscosity is equivalent to the product K x [ of the Huggins constant and the intrinsic viscosity squared according to the Huggins Eq. (4.9). The slope is also formal equivalent to a second virial coefficient like A2 in the equation of the reduced osmotic pressure H/c ... 

For polymer chains in a -solvent the scaling exponent y takes its mean-field value y — 1. The polymer concentration derivative of the reduced osmotic pressure follows from (4.29) as... 

Many macromolecules in aqueous solution are polyelectrolytes. The remarkable changes in the conformation of linear polyelectrolytes as a function of concentration, ionic strength, and pH are discussed. The various theories of chain expansion are reviewed. The thermodynamic properties of polyelectrolyte solutions reveal dramatic behavior. The large increase in the reduced osmotic pressure, jr/c, as the solution is diluted is explained in terms of the entropy of the counterions. The strong dependence of the conformation of the chains with solution conditions also leads to large changes in the viscosity. The viscosity is also explained in terms of the coil size and the interactions of the chains. 

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