Gibb-Donnan model

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. 

The (Pi)d values as estimated for the complexes of various metal ions with CM-Sephadex C-25 and C-50 gels at a = 1 are listed in Tables 1 and 2. It is apparent that the (Pj)q values of respective metal complexes determined for both gel systems are, within experimental error, in agreement with each other. This documents the validity of the Gibbs-Donnan model for interpretation of the electrostatic effect encountered in the cross-linked polyion systems. 

Only trends in ion-exchange selectivity were predicted by this approach when it was used to examine the interpretive quality of the Gibbs Donnan model for analysis of the ion-exchange phenomenon in flexible, cross-linked ion-exchange resins... 

It has been well documented by the research performed in this laboratory that use of the Gibbs-Donnan model for the interpretation of ion-exchange equilibria provides a most useful avenue for the accurate anticipation of counter ion distribution in charged polymeric systems. The activity coefficient ratio of competing ions in the polymeric phase which is the essential parameter to be assessed (see Equation 1) for successful use of this model has, as we have pointed out, so far been unaccessible by a straight-forward computation. This deficiency of our... 

Use of Eq. (18) permits identification of the log(y +/Yp,+ ) term as the one contributing most importantly to differences in affinity of pairs of univalent metal ions, M and N, for a cation-exchange resin. For example, for the equilibrium distribution of Li and Na ions between dilute solutions of Li and Na" " chloride (myQ + m jci = 0.010 m) and the much more highly concentrated Dowex-50 (8% cross-linked by weight with divinylbenzene) phase (niy, + iUnj, = 4.5 m) the p(Vy+ - Vjjj,+ )/2.3 RT and the 2 log(yyQ/y j(-i) term yield a small sum (< 0.04) while the value of log (Yu /Ynj,+ ) approaches 0.35 once correction for interaction between the two metal ions is made. Correlation between experiment and computation of the Gibbs-Donnan-based terms is strongly supportive of the model. 

In the preceding section, the remarkable salt concentration effect on the acid dissociation equilibria of weak polyelectrolytes has been interpreted in a unified manner. In this treatment, the p/( ,pp values determined experimentally are believed to reflect directly the electrostatic and/or hydrophobic nature of polyelectrolyte solutions at a particular condition. It has been proposed that the nonideality term (Ap/Q corresponds to the activity ratio of H+ between the poly electrolyte phase and the bulk solution phase, and that the ion distribution equilibria between the two phases follow Donnan s law. In this section, the Gibbs-Donnan approach is extended to the equilibrium analysis of metal complexation of both weak acidic and weak basic polyelectrolytes, i.e., the ratio of the free metal ion activity or concentration in the vicinity of polyion molecules to that of bulk solution phase is expressed by the ApAT term. In Section III.A, a generalized analytical treatment of the equilibria based on the phase separation model is presented, which gives information on the intrinsic complexation equilibria at a molecular level. In Secs. B and C, which follow, two representative examples of the equilibrium analyses with weak acidic (PAA) and weak basic (PVIm) functionalities have been presented separately, in order to validate the present approach. The effect of polymer conformation on the apparent complexation equilibria has been described in Sec. III.D by exemplifying PMA. 

Numerous attempts to formulate ion-exchange equilibria have been reported (1-12) and a rather complete discussion of the spectrum of theoretical approaches and models that have been employed for this purpose has been presented by Helfferich (13). One of the most successful of these is the Gibbs-Donnan (14,15) model. In this model, K M, the selectivity coefficient, i.e.,... 

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