Counterions activity coefficient

Significantly, counterion activity coefficients generally are not strongly affected by polyelectrolyte concentration. In contrast, simple salts exhibit a monotonic increase in activity upon dilution. This effect may be attributed to the fact that the charge density in the vicinity of the polyelectrolyte coil is insensitive to dilution, while dilution of a simple salt solution results in uniform separation of all of the charges and hence weaker ionic interactions. 

Figure 14 shows the experimental counterion activity coefficients for DADMAC, PDADMAC of different molar masses Mn (with Mw/Mn 1.5), as well as the theoretical concentration dependence for a charge distance of b=0.5 nm, calculated in terms of different theoretical expressions from Table 9. 

Fig. 14. Concentration dependence of the counterion activity coefficients, fa, of DADMAC, PDADMAC with different molar masses, and comparison with theoretical predictions (T=20 °C DADMAC PDADMAC with Mn 12,000 g mol"1 72,000 g mol"1 170,000 g mol-1 .Manning ------Gueron) (Data taken from [38])...

The agreement between theory and experimental results has been found to depend on the chemical structure of the repeat unit. Figure 15 represents the concentration dependence of counterion activity coefficients for two polyelectrolytes, PDADMAC and MEAC/AA (50 50) copolymer, differing in chemical structure of the cationic monomer unit but having the same charge distance (0.5 nm). 

Table 10. Experimental and calculated counterion activity coefficients (fa) for DADMAC/AAM copolymers with different charge distances (b)...

Fig. 15. Influence of the chemical structure of the ionic monomer unit on the concentration dependence of the counterion activity coefficients fa charge distance b= 0.5 nm (T=20 °C copolymer of acrylamide with 50 mol % 2-(methacyloyloxy)ethyltrimethylammonium...

The influence of molar mass, charge density as well as chain branching was also determined in the presence of low molecular mass salt. As seen in Fig. 16, the differences between theory and experiment are more important to low molar masses. In Fig. 16 the concentration dependence of the activity of the low molecular salt has been taken into account when calculating fac=fexp/fo [H4, 126], where fac and fexp are calculated and experimentally determined counterion activity coefficients, respectively f0 is the activity coefficient of the added low molecular salt in aqueous solution without polyelectrolyte. 

Species listed above (sodium salts) have been characterized in water and/or aqueous IMaCl solution in terms of Na counterion activity coefficient, heat of dilution, and heat of Cu ions binding. These experiments are part of a systematic investigation on the relationship between "charge-density" along poly-electrolyte chains and (metal] ion-binding, comparing experimental evidence with existing theories (J, 3). ... 

Counterion Activity Coefficients of Ionic Dextran Derivatives in Water at 25 C... 

In these equations and are the osmotic coefficient and counterion activity coefficient, respectively, in the absence of added salt (measured at the same equivalent polyelectrolyte concentration... 

Fig. 4. Activity coefficients of univalent small ions in the presence of added simple salt. The solid lines are the limiting laws for (top to bottom) the co-ion activity coefficient 72, the counterion activity coefficient yi, and the square of the mean activity coefficient y y2. The curves are drawn for = 1.85. The dashed line is the additivity rule for yi and 7172 the additivity rule for 72 coincides with the top baseline 72 = 1. The data points are measurements of 7ci (Reference [44]) for a polyvinylalcohol sulfate sample of degree of substitution corresponding to = 1.85 and NaCl concentration 1 x lO" M.

By combining these ions with other counterions, single ion transfer activity coefficients are calculated. By these techniques transfer free energies or activity coefficients have been determined for many ions and nonelectrolytes in a wide variety of solvents.Parker has discussed the extrathermodynamic assumptions that lead to single ion quantities. 

No specific ionic selectivity is really admitted in pectins with monovalent counterions due to the relativity low charge parameter a very interesting behaviour is observed when divalent counterions are considered. Specially, it was demonstrated that when DM<50% the activity coefficient of magnesium is much larger than that of calcium. The transport parameters (f) were found following the order [45] ... 

Of the preponderance of small ions, the colligative properties of polyelectrolytes in ionising solvents measure counterion activities rather than Molecular weight. In the presence of added salt, however, correct Molecular weights of polyelectrolytes can be measured by membrane osmometry, since the small ions can move across the membrane. The second virial coefficient differs from that previously defined, since it is determined by both ionic and non-ionic polymer-solvent interactions. 

The activity coefficient for the counterions was taken to be a constant value of 0.862. This value was derived from the experimental value of 0.745 for NaCJl, which was reported by Moore (20). The sodium ion activity coefficient was obtained as the square root of the sodium chloride activity coefficient. 

V is the voltage applied to the system. Unit activity coefficients and zero standard counterion potentials in both phases are assumed for the sake of simplicity. As was pointed out in 4.3, the solution of (4.4.15)-(4.4.22) with voltage V given corresponds to potentiostatic experimental conditions and yields the concentration, electric potential fields, and the ionic fluxes as functions of V. We could alternatively fix the total current density... 

If a membrane separates two solutions with mixtures of counterions — in which each counter-ion is present only on one side of the membrane — and the same co-ion, we meet with a so-called multi-ionic system. These are also treated by F. Helfferich (53, 55) (ref. 55, page 327). An explicit solution of the flux equations in this case is obtained if the flow of co-ions is neglected and if all the counter-ions possess the same valency. Gradients of activity coefficients in the membrane and convection are also neglected. Diffusion coefficients and concentration of active groups are considered to be constant. It is assumed that there is equilibrium between the salt solution and the membrane surface on either side of the membrane. 

The activity coefficients of the polymers are much lower than those of the monomer. Theories predict, within their limits, a concentration, molar mass, and chemical structure independence of the counterion activity. As seen from Fig. 14, the experimental curvatures differ from the theoretical predictions. The concentration dependence and the absolute values of fa change with the molar mass. Further, the activity coefficient has been found to be reciprocally related to the molar mass [38]. To obtain reliable results a minimization of the salt out-... 

In the previous discussion, activity coefficients have been totally neglected, for simplicity, but they should be included in a proper treatment. However, for ionic amphiphiles, where activity corrections are expected to be most important, additional complications arise. If the counterions are explicitly incorporated in the equilibria the number of possible chemical species is greatly enhanced making a detailed analysis even more complex. Furthermore a description of a process in terms of an equilibrium constant is only really suitable when the forces involved are of a short range... 

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