Donnan phase

The expression for the potential variation with x within the Donnan phase is approximated by [17] ... 

In an analogous, though not rigorous, approach 115), the internal solution was considered to consist of an equipotential Donnan phase (denoted by subscript or superscript D) next to the pore walls and a bulk phase (subscript or superscript B) electrically identical with the external solution (i.e. with Av / = 0) in the centre of the pore. Then, bearing in mind that S8 = wB = w — wD,... 

The thickness of the Donnan phase layer rD was equated to the Debye length so that wD/w = rD/r = l/r /pCs. However, experimental values of wD/w calculated from measurements of SP and SN in cellulose-KCl by means of a combination of Eq. (42), the corresponding equation for the counterion and the proper version of Eq. (37), namely SpS = w2D, did not agree very well with the aforesaid theoretical rD values 116). Recently, this difficulty has been resolved by showing 110) that Eqs. (41) and (42) become identical under the proper conditions (namely at high Cs), upon setting... 

Donnan phase to the aqueous phase extending away from the barrier must satisfy the Nernst potential (ENj) for that ion. 

The Donnan potential can also be regarded as a special case of a diffusion potential. We can assume that the mobile ions are initially in the same region as the immobile ones. In time, some of the mobile ions will tend to diffuse away. This tendency, based on thermal motion, causes a slight charge separation, which sets up an electrical potential difference between the Donnan phase and the bulk of the adjacent solution. For the case of a single species of mobile cations with the anions fixed in the membrane (both assumed to be monovalent), the diffusion potential across that part of the aqueous phase next to the membrane can be described by Equation 3.11 n — El = (u- — u+)/(u + w+)](i 77F)ln (c11/ 1) that we derived for diffusion toward regions of lower chemical potential in a solution. Fixed anions have zero mobility (u = 0) hence (u — u+)/(u — u+) here is —uJu+> or —1. Equation 3.11 then becomes En — El = — (RT/F) In (cll/cl)> which is the same as the Nernst potential (Eq. 3.6) for monovalent cations [—In = In (cVc11)]. 

Thus the Donnan potential can also be regarded as a diffusion potential occurring as the mobile ions tend to diffuse away from the immobile charges of opposite sign, which remain fixed in the Donnan phase (Fig. 3-8). 

This equation indicates that the ApK term defined at a specified a value reflects the fi%e H -ion concentration ratio between the Donnan phase and the bulk solution phase. 

The following relationship is derived for the distributions of ions M " " and Na+ between the polyion domain (Donnan phase) and the bulk solution phase with the Gibbs-Donnan model. 

By combining Eqs. (32) and (33), and by relating the quantity of the complexed species in the Donnan phase to their mass-action-based expression, the following equation is obtained ... 

In the asymmetrical ion-exchange equilibrium system, where Z > 2, the volume term in the left-hand side of Eq. (50) remains and its value can be determined by use of the values of and [Na]. In the case of polyion gels, electroneutrality persists and there is no term to consider, whereas in the case of the linear polyelectrolyte, electroneutrality is not reached in the Donnan phase and the ( )p term is needed to describe the site vacancy of the polyions [16]. The (j)p fja values used in this calculation were taken from the values determined for these salt-free polyions and reported by Katchalsky and his co-workers (Fig. 12) [31]. It is reasonable to justify the assumption of ( >p a uniqueness in the presence of added salt with the additivity rule. ... 

The single-ion activity coefficients of and Na" " ions in the Donnan phase in Eq. (50) have been estimated by use of the published mean activity coefficients of KCl, NaCl, CaCl2, and LaClj, respectively [64] as described earlier. The ionic strength in the Donnan phase, Iq, needed for these computations of the (y,)q values, is calculated by assuming homogeneous distribution of ions in this phase and... 

As the first approximation, the first Vjj/np value is calculated by assuming the activity coefficient quotient in Eq. (50) is unity. Then, by use of the first approximated Vu/Op value, the first value is calculated with Eq. (51) to permit computation of the activity coefficients of the respective species in the Donnan phase. The second approximated Vjj/np value is then obtained with Eq. (50). By this iterative procedure, a self-consistent Vjj/np value is finally determined. 

In the preceding discussion of the relationship of the Donnan potential and the Donnan phase volume terms to the structural parameter of a linear polyion, a/b (A ), it has been shown that the elearostatic effect on the binding equilibria of ionic polysaccharides can be predicted quantitatively by use of the universal curves shown in Figs. 13 and 14. Also, by separating the overall binding equilibria into two processes, i.e., (1) the concentration of counterion in the polyion domain, territorial binding. 

Also the contribution to 0 by site-bound Ag ions, (0Ag)site Ag expressed by use of an intrinsic stability constant, (Pi)d, of the monodentate complex, [Ag(P03)]°, and the free phosphate unit concentration in the Donnan phase, [Pj. ... 

Since (0Ag)si,e is defined as the difference between the amounts of the total and the free silver ions in the Donnan phase, (0Ag)si,e can be expressed with the following equation as... 

The polyion domain volume can be computed by use of the acid-dissociation equilibria of weak-acid polyelectrolyte and the multivalent metal ion binding equilibria of strong-acid polyelectrolyte, both in the presence of an excess of Na salt. The volume computed is primarily related to the solvent uptake of tighdy cross-linked polyion gel. In contrast to the polyion gel systems, the boundary between the polyion domain and bulk solution is not directly accessible in the case of water-soluble linear polyelectrolyte systems. Electroneutrality is not achieved in the linear polyion systems. A fraction of the counterions trapped by the electrostatic potential formed in the vicinity of the polymer skeleton escapes at the interface due to thermal motion. The fraction of the counterion release to the bulk solution is equatable to the practical osmotic coefficient, and has been used to account for such loss in the evaluation of the Donnan phase volume in the case of linear polyion systems. 

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