Counterions free fraction

On the contrary, that an excess of monovalent ions can compete with condensed divalent ones is out of the range of validity of Manning s model. However, the model of Iwasa (l. ) which introduces an entropic term contributing to the free energy of polyelectrolyte systems can explain that an excess NaCl displaces condensed divalent counterions. In the experiment described in Fig. 6, NaCl was added to a solution of chondroitin sulfate containing Sr " " counterions, a fraction of which are condensed. The variation of Dg shows that Sr2+ are "decondensed" by an excess NaCl, in good agreement with Iwasa s model. 

If the free fraction of counterions is determined, the number of counterions bound can be deduced. In fact the selectivity must be attributed to a certain number of these atmospheric counterions. 

The results are recorded in Figure 1 for monovalent and divalent counterions. The experimental results are compared to the free fractions of counterions calculated by the treatment of Oosawa (yo) [5] and to the activity of the counterions given by Manning (y ) [6] these parameters are given for infinite dilution by ... 

The free fraction of counterions has been obtained [3,7] by different techniques the set of results given in Table I is for monovalent counterions and in Tables I and II for divalent counterions. As shown, at equilibrium, the active fraction of counterions is quasi independent of the methods used especially, the osmotic coefficient is very close to the activity coefficient. The value is almost independent of the nature of the counterions nevertheless, it seems that there is a sequence related to the ionic selecti-vity yu < 7Na < Tk < ycs (Table I). 

Free fractions of divalent counterions by conductimetry and potentiometry... 

Using specific electrodes for Na" " and Ca " ions respectively, for exact neutralization by a mixture (NaOH-fCa (011)2) with different compositions, the free fractions of both counterions have been determined at the same time for each equivalent fraction of the hydroxide. Results are given in Table IV [13]. The same experiment should be realized with two monovalent counterions. In fact, the displacement of Na" " on Naform of the polyelectrolyte has been tested by addition of a salt XCl for a ratio Na/X = l, the fraction has been calculated [11] the results on CMC DS 2.9 for different counterions are given in Table V. 

These results are in agreement with the selectivity Li>Na>K>Cs the inde-pendance of the free fraction of the counterions from its nature must be assumed. 

The determination needs also the knowledge of the variation of the free fraction of counterions with the ionic strength in addition, we assume that the activity of the neutral salt is unaffected by the polyion. 

Free fractions of monovalent and divalent counterions for different stoechiometric compositions [13]... 

For conceptual purposes it is very important to note the following about and cj) these quantities do not represent a free fraction of counterions nor do the quantities 1 — and l—cj) represent a bound fraction of counterions. In the first place, yi 4>. Secondly, even though at (J = zi " for example, no counterions are condensed, the values of y and (j) are 0.61 and 0.50, respectively in other words, the Debye-Hiickel effect is quite large in this case. The only meaningful statement about binding is that for the fraction l — of the counterions is bound (condensed). 

The free fraction of counterions is determined with specific electrodes for Na ", Ca, Sr ", Mg. The measurements are sensitive to H ions, so the data are obtained for complete neutralization (7calibration curve is drawn which relates the electrochemical potential to the concentration its independance of the anion is assumed. The activity coefficient y assimilated to the free fraction of counterions is given by ... 

Free fraction of monovalent counterions -The results obtained with Na as counterions and different CMC are plotted on Figure 2a the experimental points (not represented)... 

Free fraction of divalent counterions - With CMC, the results obtained give in Figure 2b a single curve one can also see that for 2> 1, 7Na/7ca = 2. Figure 4 compares the carboxylic and sulphonic families the particularly low value characterizing PES could well be attributed to the formation of intramolecular bridges with Ca " . The experimental data are collected in Table I. 

Free fraction of monovalent and divalent counterions in mixture - CMC is completely neutralized by a series of mixed solutions of sodium and calcium hydroxides, the equivalent ionic fraction varying from zero to one, (X a + = 1) in the resulting... 

The behaviour is directly related to the nature of the ionic site (—SO3H or — COOH). In the particular case of divalent counterions, one can assume that the free fraction of... 

Experimental values of the free fraction of counterions in a CMC solution, obtained by potentiometry and conductimetry are given in Table II one can conclude that both techniques give similar results for divalent ions. 

Free fraction of divalent counterions obtained by potentiometry and conductimetry [10b]... 

The free fraction of counterions cj) is essentially dependent on the linear charge density the role of the rigidity of the backbone is clearly shown. The phenomenon is purely electrostatic with an exception for Ca-PES. 

Fig. 17. Free fraction of counterions obtained with different definitions as function of the charge parameter A [13]--------------------calculated values O experimental data.

The activity coefficient of free fraction of counterions y is given by... 

The points of Figure 26 were calculated from Equation (25) using a= 12 A and assimilating the potentiometric results to the free fraction y. Oosawa s treatment best fits the experimental data for 2>1. The ion-selectivity [20] (Figure 25) or the ultrasonic absorption [21] increases with the number of counterions whilst the effective ionisation approaches the y2 1 limit. 

Free diffusion of Cs" was measured [22] and calculated the values are given just under and compared with the free fraction of counterions obtained by conductimetry (as proposed by Manning [16])... 

A set of experimental results was discussed first in terms of charge density the essential laws of polyelectrolytes are shown. The systematic theoretical investigation permits to discuss the validity of the different theories. From the resolution of the Poisson-Boltzmann equation, we have shown that only DIDq) or 0C2 is a good approximation for the free fraction of counterions measured by potentiometry. 

The MAM described here is a generalization of the model previously published (10). Hence, only a summary of the derivation will be given here. Details can be found elsewhere (17). The basic equations are the surfactant and counterion material balances and the minimization of the Gibbs free energy of the system with respect to the micelle concentration c , and mole fraction x (11). Equation 4 from Ref. (11) has been changed to... 

Figure I. Linear dependence of the apparent bimolecular rate constant of living polypropylene sulfide propagation on the fraction of free ions a with Na + [222] as counterion at -30°C in THF ( ) in THP (V) with fBNa + [222] in...

In order to elucidate this point, viscosity measurements of living and deactivated PDMS solutions were performed In toluene, with Li+ + [211] as counterion. As no significant change was observed, It can be deduced that the fraction of aggregates Is negligible (< 1% for [C] 10 mole. " ). Moreover, conductance measurements made on model sllanolates In THF Indicate that the fraction of free Ions Is very low. In our system (benzene) we conclude therefore that the contribution to the reactivity from free ions can be neglected. Thus the main Ionic species are cryptated ion pairs, and... 

It is apparent that CMC values can be expressed in a variety of different concentration units. The measured value of cCMC and hence of AG c for a particular system depends on the units chosen, so some uniformity must be established. The issue is ultimately a question of defining the standard state to which the superscript on AG C refers. When mole fractions are used for concentrations, AG c directly measures the free energy difference per mole between surfactant molecules in micelles and in water. To see how this comes about, it is instructive to examine Reaction (A) —this focuses attention on the surfactant and ignores bound counterions — from the point of view of a phase equilibrium. The thermodynamic criterion for a phase equilibrium is that the chemical potential of the surfactant (subscript 5) be the same in the micelle (superscript mic) and in water (superscript W) n = n. In general, pt, = + RTIn ah in which... 

The solution thus consists of different particles denoted as contact ion pairs, solvent-separated ion pairs and free ions. The fraction of the individual particles depends on the type of salt, type of solvent, polymerization system, temperature, and salt concentration. The catalytic effect of these particles may be very different as is evident in anionic polymerization of vinyl monomers. For instance, free polystyryl anion is 800times more reactive than its ion pair with the sodium counterion 60 . From this fact it follows that, although the portion of free ions is small in the reaction system, they may play an important role. On the other hand, anionic polymerization and copolymerization of heterocycles proceeds mostly via ion pairs. This is due to a strong localization of the negative charge on the chain-end heteroatom which strongly stabilizes the ion pair itself62. Ionic dissociation constants and ion contributions to the reaction kinetics are usually low. This means that for heterocycles the difference between the catalytic effect of ion pairs and free ions is much weaker than for the polymerization of unsaturated compounds. This is well documented by the copolymerization of anhydrides with epoxides where the substi-... 

For most samples investigated, the temperature interval during which precipitation occurs, is quite small, i.e. 2-5 °C, as shown in Fig. 10. The upper dotted curve in Fig. 10 represents the temperature dependence which would be expected for the polyion solution if no precipitation occurs and if the fraction of free counterions would not increase with temperature, i.e. the increase in conductivity is solely caused by the decrease of the viscosity of water. Act may be utilized to determine the effective charge density f by the equation... 

The condensed fraction, of density nc, is localized on a two-dimensional (2D) planar surface [51]. The fraction of condensed counterions in 2D is x = Z0enc/ao, and the reduced surface charge density is enR = u0 — Z0enc. The free energy of the condensed counterions per unit area is [51]... 

Minimizing the total free energy + Tid, with respect to the fraction x of condensed counterions, leads to a nonlinear equation [51]... 

To explain the folding characteristics of Tetrahymena ribozyme, a simple generalization of counterion condensation model was proposed [108, 109], In this two state model, equilibrium is considered between condensed and free counterions [108]. The chemical potential for each phase is approximately calculated as follows. If the volume fraction occupied by the counterions is denoted by < ), then the chemical potential of the free counterions is [108]... 

---