Donnan’s theory

The influence of neutral salts as well as of acids and bases on the swelling of gelatine which we have seen can be attributed to an apparent change in the solvation of the gel fibrils and may be interpreted in the light of Donnan s theory of the effect of a non-diffusible ion on the osmotic pressure differences between the two phases, is likewise to be noted in the alteration of the viscosity and alcohol precipitation values of protein solutions. From the considerations already advanced there should exist two well-defined maxima in the viscosity and alcohol precipitation curves when these properties are plotted as functions of the Ph, the maxima coinciding with the points of maximum dissociation of the salts... 

Chemical equilibrium in heterogeneous systems (from the thermodynamic standpoint) when capillary or electrical effects are of importance—Adsorption—Donnan s theory of membrane equilibria—Micelle theory of colloidal electrolytes... 

In the case of the sodium-calcium ferrocyamde solutions a somewhat unexpected result was obtained Whereas equation (b), which refers to the concentrations of the two salts, holds within the limit of experimental error, it was found that equation (a) does not accurately represent the relationship between the ionic concentrations of the calcium and sodium salts on the two sides of the membrane The activities of the ions in this case appear to be more closely related to the molar than to the ionic concentrations The difficulty here encountered is not to be regarded as a failure of Donnan s theory of distributional equilibrium, but a failure m the means possessed at the present time for determining with accuracy the true activities of ions... 

Donnan s theory is also applicable to complex ions having a high molecular weight, where there is little tendency to penetrate membranes. 

Based on Donnan s [18, 19] and Onsager s [41, 42] fundamental works, the theories for Donnan dialysis systems were developed [20-26, 32-36]. The BAHLM system could be considered as two DD systems, operating in consecutive order, continuously in one module (see Fig. 6.2) the first is composed of feed/LM and the second is composed of LM/strip compartments, separated by ion-exchange membranes. Therefore, the Kedem-Katchalsky equations [43, 44] can be applied to our case ... 

Suppose a vessel is separated into two compartments by a semipermeable membrane which permits water and crystalloids, but not colloidal particles, to pass through. If water is placed in both the compartments and then some NaCl is added to one compartment, the NaCl will diffuse through the membrane and after a time become equally distributed in the water of both the compartments. However, if an ion which cannot pass through is placed on one side of the membrane, the distribution of a freely diffusible electrolyte like NaCl, becomes unequal in the solutions on the opposit sides of the membrane. This observation made in 1911 by Donnan is known after his name as Donnan s equalibrium theory. A theroretical derivation of this generalisation based on considerations of kinetics is given below. 

Prediction from Counterion Condensation Alone. Katchalsl s theory does not consider counterion condensation. The question then is how important is the counterion condensation in determining the salt partitioning Can the counterion condensation alone account for all the non-ideality of the Donnan equilibrium To examine this problem, we will calculate the salt absorption by only considering the counterion condensation from both linear and nonlinear counterion condensation theories and compare the predictions with experimental results. 

New Expression for Salt Partitioning. As mentioned before, Katchalsky s theory does not consider the counterion condensation. We will show later that die counterion condensation theories alone, both linear and nonlinear theories, cannot account for all the non-ideality of the Donnan equilibrium. Therefore, we seek an expression that can combine both contributions. We follow a similar approach of Katchalsky by writing die Donnan equOibrium as ... 

Figure 2. Conq)arison of experimental salt absorption with theoretical predictions based on ideal Donnan equUibrium (Cal-I), nonlinear counterion condensation theory (Cal-N), linear counterion condensation theory (Manning s theory) (Cal-L)...

Figure 4. Comparison of expaimental salt abswption with Aeoretical predictions based on ideal Donnan equilibrium (Cal-I), Katchalsky s theory (Cal-K), and our...

Neale endeavoured to explain the swelling of cellulose in sodium hydroxide solutions (which shows a marked maximum at a concentration of about 2 mole/liter) on the basis of the Donnan theory, thereby considering cellulose as a weak acid capable of splitting off hydrogen ions. Though the results were consistent with the experimental data, the behaviour of this system seems to require another explanation, since Hess and coworkers have shown that the results of experiments on the distribution of added neutral salts lead to the conclusion that the theory does not hold. Moreover, the assumption that cellulose acts as a weak acid seems not to be sufficiently justified. It is true that carboxylic endgroups frequently occur in cellulose, but these cannot be held responsible for the acid character postulated by Neale, since they have a much larger dissociation constant than the one required by Neale s theory. 

Gibbs s adsorption equation was tested by W. C. McC. Lewis, F. W. Donnan and J. T. Barker, and (in a more satisfactory way) by J. W. McBain and C. W. Humphries and McBain and R. C. Swain,who confirmed it for an air-solution interface. Zawidzki found that the concentration of saponin in the foam of a solution was 1 26 to 1 33 times that in the bulk of the solution there can be no doubt that the concentration is actually greater in the surface layer of a solution than in the interior. Ramsden observed the formation of a solid film on the surface of some solutions and suspensions, and Metcalf the formation of a skin of peptone on a water surface. These phenomena are connected with Gibbs s theory. 

Donnan, from Laplace s theory of the internal pressure of liquids, arrived at a theory of negative surface tension which was supposed to explain why substances disperse to form colloidal rather than true solutions, a minimum size of particle being stable. Donnan and Barker investigated adsorption (see p. 742), Donnan and Le Rossignol the kinetics of the reaction between ferri-cyanide and iodide (see p. 660), and Donnan and Miss K. A, Burke the kinetics of the reaction between silver nitrate and alkyl iodides. 

In Sect. 2 we reviewed the original Tanaka s treatment of ions in gels. More precise theory should properly account for the chemical dissociation equilibrium in the interior of gels and the Donnan equilibrium at the gel-solvent boundary where an electric double layer is formed [31,97,98]. 

The procedure used for testing the ideal Donnan theory is applicable to any model that decouples ionic effects from network elasticity and polymer/solvent interactions. Thus we require that nnet depend only on EWF and not C. While this assumption may seem natural, several models which include ionic effects do not make this assumption. For example, the state of ionization of a polymer chain in the gel and the ionic environment may affect the chain s persistence length, which in turn alters the network elasticity [26]. Similarly, a multivalent counterion can alter network elasticity by creating transient crosslinks. 

Qualitative Considerations of the Membrane Behaviour on the Basis of M.S.T. Theory and Donnan Equilibrium... 

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