Theory osmotic coefficient

The papers in the second section deal primarily with the liquid phase itself rather than with its equilibrium vapor. They cover effects of electrolytes on mixed solvents with respect to solubilities, solvation and liquid structure, distribution coefficients, chemical potentials, activity coefficients, work functions, heat capacities, heats of solution, volumes of transfer, free energies of transfer, electrical potentials, conductances, ionization constants, electrostatic theory, osmotic coefficients, acidity functions, viscosities, and related properties and behavior. 

Keywords Rod-like polyelectrolytes Poisson-Boltzmann theory Osmotic coefficient Electric birefringence SAXS... 

Results for 1-1 Electrolytes. Here we present some of the most recent data obtained for electrolyte solutions within the framework of the MSA. We have shown(5,12) that using the MSA as an empirical theory, osmotic coefficients and activity coefficients of 1-1 electrolytes can be calculated within the experimental accuracy if a density dependent cation radius is used in connection with Pauling crystallographic radii for the anions. Figures 2 and 3 show the kind of... 

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between ([ly and ((ij is a measure of the accuracy of the approximation. The osmotic coefficients calculated via the energy equation in the MS approximation are comparable in accuracy to the HNC approximation for low valence electrolytes. Figure A2.3.15 shows deviations from the Debye-Htickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

Equation (7.45) is a limiting law expression for 7 , the activity coefficient of the solute. Debye-Htickel theory can also be used to obtain limiting-law expressions for the activity a of the solvent. This is usually done by expressing a in terms of the practical osmotic coefficient

electrolyte solute, it is defined in a general way as... 

Debye-Hiickel theory 333-50 in electrochemical cells 481-2, 488 and osmotic coefficient 345-8 parameters 342... 

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

In this paper the authors propose to make a general derivation for the work function of a particle having the characteristics of an ion-dipole. Further, it is planned to apply this result to obtain equations for the activity coefficient and osmotic coefficient for an ion-dipole particle. The theory will be checked by applying it to published data on the activity and osmotic coefficients for ions, dipole, and ion-dipoles. 

Integral equations theories are another approach to incorporate higher order correlations, and consequently also lead to lowered osmotic coefficients. There are numerous variants of these theories around which differ in their used closure relations and accuracy of the treatment of correlations [36]. They work normally very well at high electrostatic coupling and high densities, and are able to account for overcharging, which was first predicted by Lozada-Cassou et al. [36] and also describe excluded volume effects very well, see Refs. [37] for recent comparisons to MD simulations. 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

Recently, Nyquist et al. [39] tried to develop a theory for rods of finite size. These authors used a two-state model for the counterions and employed a random phase approximation in order to calculate the osmotic coefficient (j) of rod-like polyelectrolytes [39]. An important goal of this work was to reproduce in the zero density limit the correct osmotic coefficient of 1 instead of the Manning limiting value which is due to the unphysical infinite rod assumption employed. The model presented in ref. [39], however, seems to overestimate considerably the osmotic coefficient when compared to experimental data (see below Sect. 4.2). 

As already indicated in Sect. 2, the osmotic coefficient 0 provides a sensitive test for the various models describing the electrostatic interaction of the counterions with the rod-like macroion. It is therefore interesting to first compare the PB theory to simulations of the RPM cell model [26, 29] in order to gain a qualitative understanding of the possible failures of the PB theory. In a second step we compare the first experimental values 0 obtained on polyelectrolyte PPP-1 [58] quantitatively to PB theory and simulations [59]. 

Figure 6 gives the comparison of the osmotic coefficient predicted by the PB-theory to simulated data [26, 60]. The simulation system is not strictly a cell system, rather we considered an infinite array of parallel aligned rods which sit on a hexagonal lattice. The rod diameter a was of the same size as the counterions a, the line charge density X had the value 1=0.9593 e0 o, and the density and the Bjerrum length was varied. For details of the simulations we refer to Ref. [26, 60]. 

Fig. 6 Osmotic coefficient 0 versus reduced density n/a3 for monovalent counterions. Heavy dots mark the measurements, while the solid lines are fits which merely serve to guide the eye. The dotted lines are the prediction of PB theory. From top to bottom the Bjerrum length lB/o varies as 1,2,3. The errors in the measurement are roughly as big as the dot size [29]...

Up to now, only two sets of data of the osmotic coefficient of rod-like polyelectrolytes in salt-free solution are available 1) Measurements by Auer and Alexandrowicz [68] on aqueous DNA-solutions, and 2) Measurements of polyelectrolyte PPP-1 in aqueous solution [58]. A critical comparison of these data with the PB-cell model and the theories delineated in Sect. 2.2 has been given recently [59]. Here it suffices to discuss the main results of this analysis displayed in Fig. 8. It should be noted that the measurements by the electric birefringence discussed in Sect. 4.1 are the most important prerequisite of this analysis. These data have shown that PPP-1 form a molecularly disperse solution in water and the analysis can therefore assume single rods dispersed in solution [49]. 

First of all, the comparison of the PB-theory and experiment shown in Fig. 8 proceeds virtually without adjustable parameters. The osmotic coefficient (j) is solely determined by the charge parameter polyelectrolyte concentration. The latter parameter determines the cell radius R0 (see the discussion in Sect. 2.1) Figure 8 summarizes the results. It shows the osmotic coefficient of an aqueous PPP-1 solution as a function of counterion concentration as predicted by Poisson-Boltzmann theory, the DHHC correlation-corrected treatment from Sect. 2.2, Molecular Dynamics simulations [29, 59] and experiment [58]. 

Fig. 8 Osmotic coefficient as a function of counterion concentration cc for the poly(p-phenylene) systems described in the text. The solid line is the PB prediction of the cylindrical cell-model, the dashed curve is the prediction from the correlation corrected PB theory from Ref. [58]. The full dots are experiments with iodine counterions and the empty dots are results of MD simulations described in ref. [29,59]. The Manning limiting value of l/2 is also indicated...

Both the correlation-corrected DHHC theory as well as the simulations that capture in principle all kinds of ion correlations (see Sect. 2.2) show a decrease in the osmotic coefficient when compared to the prediction of the PB-theory. Since these two totally different approaches agree so well, it becomes clear that they indeed give a good description of the influence of the correlations. However, they do not lower the osmotic coefficient sufficiently to reach full agreement with the experimental data. Moreover, the deviation from the Poisson-Boltzmann curve increases for higher densities, which is true for the DHHC and the simulations as well as for the experiment. This appears plausible if one recalls that correlations become more important at higher densities. 

The fact that Poisson-Boltzmann theory overestimates the osmotic coefficient is well-known in literature. Careful studies of typical flexible polyelectrolytes in solution ([2, 23] and further references given there) indicated that agreement of the Poisson-Boltzmann cell model and experimental data could only be achieved if the charge parameter was renormalized to a higher value. To justify this procedure it was assumed that the flexible polyelectrolytes adopt a locally helical or wiggly main chain in solution. Hence,... 

The osmotic coefficient obtained experimentally from polyelectrolyte PPP-1 having monovalent counterions compares favorably with the prediction of the PB cell model [58]. The residual differences can be explained only partially by the shortcomings of the PB-theory but must back also to specific interactions between the macroions and the counterions [59]. SAXS and ASAXS applied to PPP-2 demonstrate that the radial distribution n(r) of the cell model provides a sufficiently good description of experimental data. 

Within PB theory [2] and on the level of a cell model the cylindrical geometry can be treated exactly in the salt-free case [3, 4]. The Poisson-Boltzmann (PB) solution for the cell model is reviewed in the chapter in this volume on the osmotic coefficient. The PB approach can provide for instance new insights into the phenomenon of Manning condensation [5-7]. For example, the distance up to which counterions can be called condensed can be conveniently found via the inflection point in the log plot of the integrated radial distribution function P(r) of counterions [8, 9], defined as... 

Activity coefficients of H4[SiWi204o] obtained in 0.0004 to 0.04 M concentrations in queous solutions have also been reported11S. The results obtained agree with the Debye-Huckel theory for 1—4 electrolytes. Osmotic coefficients of 12-tungsto-silicic acid have also been reported116. ... 

There are many measurement techniques for activity coefficients. These include measuring the colligative property (osmotic coefficients) relationship, the junction potentials, the freezing point depression, or deviations from ideal solution theory of only one electrolyte. The osmotic coefficient method presented here can be used to determine activity coefficients of a 1 1 electrolyte in water. A vapor pressure osmometer (i.e., dew point osmometer) measures vapor pressure depression. 

For an ideal solution, Jq = I and is unity. Then Eq. (9) is consistent with Eq. (10 11), since the total molality of all solute species is vm for a completely dissociated solute of molality m. For ionic solutions, the Debye-Hiickel theory predicts a value of yo different from unity and therefore a deviation of g from unity. A treatment of this aspect of the Debye-Hiickel theory is beyond the scope of this book, and we shall merely state the result. The osmotic coefficient g at 0°C for dilute solutions of a single strong electrolyte in water is given by... 

For hydrochloric acid, a strong electrolyte, calculate an experimental value of with Eq. (9) for each of the concentrations studied. In addition, use Eq. (13) to obtain a value of the osmotic coefficient based on the Debye-Hiickel theory for each concentration. Compare these experimental and theoretical values. 

Three systems were selected for examination, namely the solubilities of oxygen, carbon dioxide, and methane in water -1- sodium chloride. An accurate semiempirical equation [64] was used to express the composition dependence of the osmotic coefficient in water-r sodium chloride. The results of the calculations are presented in Fig. 1 and Table 1. One can see that Eq. (26) provides an accurate correlation for the gas solubility in solutions of strong electrolytes. In addition, the fluctuation theory allows one to use the experimental solubility data to examine the hydration in water (l)-gas (2)-cosolvent (3) mixtures. 

Figure 2. Osmotic coefficient and equivalent conductivity A of LiCUAt solution in DMC as the functions of electrolyte concentration c. - experimental values [29, 33], solid line -theoretical prediction from the AMSA theory [13,14].

Before presenting numerical results, it is worth summarizing the main characteristics of the experimental results for the osmotic pressure of polyelectrolyte solutions [9, 17, 18, 57, 107], The measured osmotic coefficients most often exhibit strong negative deviations from ideality. The measured values are a) lower than it was predicted by the cylindrical cell model theory, b) rather (but not completely) insensitive to the nature of the counterions, and c) also insensitive to the polyelectrolyte concentration in a dilute regime and/or for... 

Deserno, M., Holm, C., Blaul, J., Ballauff, M., and Rehahn, M. The osmotic coefficient of rod-like poly electrolytes Computer simulation, analytical theory, and experiment. European Physical Journal E, 2001, 5, No. 1, p. 97-103. 

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