Counterions fraction bound

The specific case to be considered is that of the Cl" counterion surface in contact with a dilute salt solution—e.g., at 10 2N NaCl where 0 = 28° (Table II). Data of Stigter (33) show that the fraction of Cl" counterions outside the shear surface of dodecyl ammonium chloride micelles is about 0.4 in the range 0.02-0.05M NaCl plus surfactant at the critical micelle concentration (CMC). This value appears to rise toward 0.5-0.6 as NaCl concentration is lowered. Conductance data of Robins and Thomas (30) indicate that 73% of Cl" counterions are bound to 2-dodecyl aminoethanol hydrochloride micelles at the CMC in the absence of added salt. 

If there is an ion selectivity, the experimental specific conductivity Xe ip can be interpreted using the equivalent fractions of counterion 1 and X 2 = 1 — 1 of counterion 2 free in solution and Zj, Z2 for equivalent fractions bound on the polyelectrolyte... 

For strongly charged polyelectrolytes such as DNA, a fraction of counterions is bound to the chain due to the strong electrostatic attraction. This phenomenon is known as counterion condensation or Manning condensation. It reduces the repulsive electrostatic interactions between polyelectrolyte chains and leads to an effective charge smaller than expected based on stoichiometry. 

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). 

A polyelectrolyte solution contains the salt of a polyion, a polymer comprised of repeating ionized units. In dilute solutions, a substantial fraction of sodium ions are bound to polyacrylate at concentrations where sodium acetate exhibits only dissoci-atedions. Thus counterion binding plays a central role in polyelectrolyte solutions [1], Close approach of counterions to polyions results in mutual perturbation of the hydration layers and the description of the electrical potential around polyions is different to both the Debye-Huckel treatment for soluble ions and the Gouy-Chapman model for a surface charge distribution, with Manning condensation of ions around the polyelectrolyte. 

Consider the formation of a mixed micelle in aqueous solution from a binary surfactant solution consisting of a nonionic and an anionic surfactant. The process is depicted as the aggregation of ng molecules of nonionic surfactant B, of n molecules of anionic surfactant A", and in addition there will be counterions, C" ", of the anionic surfactant in the amount of an where a is the fraction of the counterions associated or bound with the surfactant anions in the micelle. The process as depicted is... 

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 Mass Action Model The mass action model represents a very different approach to the interpretation of the thermodynamic properties of a surfactant solution than does the pseudo-phase model presented in the previous section. A chemical equilibrium is assumed to exist between the monomer and the micelle. For this reaction an equilibrium constant can be written to relate the activity (concentrations) of monomer and micelle present. The most comprehensive treatment of this process is due to Burchfield and Woolley.22 We will now describe the procedure followed, although we will not attempt to fill in all the steps of the derivation. The aggregation of an anionic surfactant MA is approximated by a simple equilibrium in which the monomeric anion and cation combine to form one aggregate species (micelle) having an aggregation number n, with a fraction of bound counterions, f3. The reaction isdd... 

The micellar surface has a high charge density and the stability of the aggregate is heavily dependent on the binding of counterions to the surface. From the solution of the Poisson-Boltzmann equation one finds that a large fraction (0.4—0.7) of the counterions is in the nearest vicinity of the micellar surface300. These ions could be associated with the Stern layer, but it seems simpler not to make a distinction between the ions of the Stern layer and those more diffusely bound. They are all part of the counterions and their distribution is primarily determined by electrostatic effects. 

Significant differences have also recently been reported in the extent which Na+ counterions bind to isotactic and syndiotactic PMA (20). The results are summarized in Fig. 5 and 6. The data in Fig. 5 indicate that the fraction of Na+ ions bound to conventional (and syndiotactic)... 

Fraction of counterions bound to micelle, P = 1 — a Critical micelle concentration... 

Since an increase in the surfactant concentration results in a higher fraction of substrate bound to the aggregate, an increase of the rate effects is expected, as seen in the above example, as [surfactant] increases and when [surfactant] >cac. Consequently, an enzymelike rate versus concentration profile is anticipated with a tendency of the curve to plateau when all the substrate is transferred into the aggregate. However, this is correct only in case (3) and in case (1) when the counterion of the added ionic surfactant is the reactive species (Figure 7, right). In case (1) when the reactive ion is not added as the counterion of the ionic surfactant (and is, consequently, kept constant in concentration) and in case (2), reaction profiles go through a maximum as the concentration of surfactant is increased (Figure 7, left). 

Dependence of Adsorption Parameter K on Salt Concentration The physical meaning of Equation 5.48 can be revealed by chemical-reaction considerations. For simplicity, let us consider Langmuir-type adsorption i.e., we treat the interface as a two-dimensional lattice. We will use the notation Bg for the fraction of the free sites in the lattice, 0, for the fraction of sites containing adsorbed surfactant ion S , and 02 for the fraction of sites containing the complex of an adsorbed surfactant ion + a bound counterion. Obviously, we can write 0q -1- 0i -1- 02 = F The adsorptions of surfactant ions and counterions can be expressed in the form ... 

If we assume that the contribution of the hydrophilic head group AG(—W) and the fraction of counterions bound to the micelle, a, do not change with increase in the length of the hydrophobic group, then for any homologous series of surfactants, the relations between the CMC and the number of carbon atoms in the hydrophobic group can be put into the form... 

Several methods revealed that PECs between strong polyelectrolytes have a 1 1 end point stoichiometry and also a 1 1 stoichiometry of ionic binding under full release of the low molecular counterions at nonstoichiometric mixing ratios. However, it remains an open question whether the major component in such systems is bound in excess in the PEC structures, giving them a net excess charge. To solve this problem, viscometry, analytical or preparative ultracentrifugation, and fractionation techniques in combination with analyzing methods can be employed. 

The fraction of counterions bound to CKC, 0, was calculated from the equation... 

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