Donnan equilibrium theory

This theory will be demonstrated on a membrane with fixed univalent negative charges, with a concentration in the membrane, cx. The pores of the membrane are filled with the same solvent as the solutions with which the membrane is in contact that contain the same uni-univalent electrolyte with concentrations cx and c2. Conditions at the membrane-solution interface are analogous to those described by the Donnan equilibrium theory, where the fixed ion X acts as a non-diffusible ion. The Donnan potentials A0D 4 = 0p — 0(1) and A0D 2 = 0(2) — 0q are established at both surfaces of the membranes (x = p and jc = q). A liquid junction potential, A0l = 0q — 0P, due to ion diffusion is formed within the membrane. Thus... 

Earlier, Gavach et al. studied the superselectivity of Nafion 125 sulfonate membranes in contact with aqueous NaCl solutions using the methods of zero-current membrane potential, electrolyte desorption kinetics into pure water, co-ion and counterion selfdiffusion fluxes, co-ion fluxes under a constant current, and membrane electrical conductance. Superselectivity refers to a condition where anion transport is very small relative to cation transport. The exclusion of the anions in these systems is much greater than that as predicted by simple Donnan equilibrium theory that involves the equality of chemical potentials of cations and anions across the membrane—electrolyte interface as well as the principle of electroneutrality. The results showed the importance of membrane swelling there is a loss of superselectivity, in that there is a decrease in the counterion/co-ion mobility, with greater swelling. 

This result is expressed mathematically in the Donnan equilibrium theory by the following relationships. First, the potassium and chloride ion concentrations outside the gel are equal in the starting condition by electroneutrality ... 

The Donnan equilibrium theory implies that dilution of a clay/water system containing monovalent and divalent cations displaces the equilibrium in such a manner that the absorption of divalent ions increases, whereas the absorption of monovalent ions decreases. The ionic charge is not the only determining factor in the absorption effect. Factors such as temperature, pH, and specific ions also play important roles. Hydration energy, which appears to be one of the most important factors for the absorption and fixation of cations, displaces the ionic equilibria in a manner opposing the Donnan equilibrium theory. According to Sawhney (1972), "cations with low hydration energy such as Ca, Mg and Sr, produce expanded interlayers and are not fixed". 

According to the Donnan equilibrium theory, the concentration of sorbed electrolyte in the membrane increases with increasing external solution concentration which tends to result in lower current efficiency at higher external solution concentrations (45). 

Concentration of Electrolyte Myer and Sievers"" applied the Donnan equilibrium to charged membranes and developed a quantitative theory of membrane selectivity. They expressed this selectivity in terms of a selectivity constant, which they defined as the concentration of fixed ions attached to the polymer network. They determined the selectivity constant of a number of membranes by the measurement of diffusion potentials. Nasini etal and Kumins"" extended the measurements to paint and varnish films. 

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]. 

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

Previous to giving a quantitative elaboration of the Nemst-Planck equations for the different membrane processes, at first a qualitative treatment of membrane phenomena will be given here on the basis of M.S.T. theory and Donnan equilibrium. 

Sep. 5,1870, Colombo, Ceylon (British Empire), now Sri Lanka - Dec. 16,1956, Canterbury, Kent, UK). Donnan was a British chemist who greatly contributed to the development of colloid chemistry, physical chemistry, and electrochemistry [i—iii]. In different periods of his life, he was working with van t - Hoff, -> Ostwald, F. W., and Ramsay. In electrochemistry, he studied (1911) the electrical potential set-up at a semipermeable membrane between two electrolytes [iv], an effect of great importance in living cells [v], Donnan is mostly remembered for his theory of membrane equilibrium, known as - Donnan equilibrium. This equilibrium results in the formation of - Donnan potential across a membrane. 

Method (a), the use of the position of the coulombic attraction theory minimum with the Od = 0 value for g, leads to the same mathematical formula for s as that expressing the Donnan equilibrium. However, we cannot say that this constitutes a derivation of the Donnan equilibrium from the coulombic attraction theory because it does not correspond to a physical limit. If Od = 0 really were the case, there would be no reason for the macroions to remain at the minimum position of the interaction potential. Nevertheless, the identity of the two expressions is an interesting result. Because Equation 4.20 is derived in the case in which there is no double layer overlap and Equation 4.1 (the Donnan equilibrium) is likewise derived without reference to the overlap of the double layers, it is precisely in this limit that the calculation should reproduce the Donnan equilibrium. The fact that it does gives us some confidence that our approximations are not too drastic and should lead to physically significant results when applied to overlapping double layers. 

For low surface potentials ( s < 1), the salt-fractionation factors calculated (a) via the coulombic attraction theory and the electric integral solved by Equation 4.20 (i.e., via the Donnan equilibrium) and (b) via the coulombic attraction theory and the electric... 

Our experimental conclusion — that s is constant and equal to 2.6 0.4 in the range 3 mM < cex <120 mM — accords well with the prediction from the new generalized Donnan equilibrium made in Chapter 4. We recall that the coulombic attraction theory, with the constant surface potential boundary condition s = 70 mV, predicts that. v is constant and equal to 2.8. A factor of x40 in c provides a severe test of the prediction and it passes, although the quantitative agreement between the theoretical and experimental values of s in this case should be treated with caution because of the severity of the approximations used in deriving the theoretical result. The pure Donnan prediction that s = 4.0 for s = 70 mV is definitely invalidated by... 

As far as I am aware, independent experimental evidence for the values of the surface potential and salt fractionation factor have not been obtained for any system other than the n-butylammonium vermiculite gels. For this isolated system, the predicted values of 5 from the Donnan equilibrium and the new equilibrium based on the coulombic attraction theory, namely 4.0 and 2.8, respectively, are definitely distinguished by the experimental results. It would be highly desirable to obtain further tests of our prediction for 5 in systems of interacting plate macroions, both in clay science and lamellar surfactant phases. 

Donnan EG. Theory of membrane equilibrium and membrane potential in the presence of non-dialysing electrol3des A contribution to physical-chemical physiology. Z Elektrochem. Angewandte Phys. Chem. 1911 17 572-581. 

Since both sides have high electrolyte strength, the activity coefficient term can be assumed identical for a given species in the two solutions, in accordance with the correlation based on modified Debye-Huckel theory proposed by Davies [28]. Therefore, the concentration term can replace the activity coefficient term and the condition for Donnan equilibrium becomes... 

Theoretical discussion of the osmotic pressure of polyelectrolytes has been made by two methods, one using the Donnan equilibrium and the other the McMillan and Mayer theory. Both methods are equivalent but in order to obatin explicitly the osmotic pressure we should know in the former case the activities of component systems and in the latter case the potential of average force between the solute molecules. 

Because of its simplicity, the concept of Donnan equilibrium is an attractive approach for the modelling of fibre-ion interactions. However, despite its general success in many cases, the Donnan theory alone is not sufficient to describe the experimentally observed distributions. In those cases, it is necessary to use ion-specific complexation equilibria with the acidic groups as a part of the model to describe the observed phenomena satisfactorily. Such behaviour has been shown to exist, for example, with chemically modified highly charged fibres. An apparent systematic deviation in the ionic distribution from the predictions of the Donnan theory has also been noted with common oxygen-delignified kraft pulps, partly, but... 

This section considers the Donnan equilibrium which is established by the equilibrium distribution of a simple electrolyte between an aqueous protein-electrolyte mixture and an aqueous solution of the same simple electrolyte, when the two phases are separated by a semipermeable membrane. A difference in osmotic pressure is estabhshed across the membrane permeable to all other species but proteins. This difference is measurable and provides important information about the protein-protein interaction in solution [37, 109-112, 116]. The principal goal of the theory is to explain how factors such as protein concentration, pH, protein aggregation, salt concentration and its composition, influence the osmotic pressure. At the moment this goal seems to be too ambitious these systems are often complicated mixtures of highly concentrated electrolytes and protein molecules, and the principal forces are not easy to identify [117]. 

In spite of the partial success in theoretical description, we believe that more realistic models are needed for the theory to have a predicting power. For example, measurements usually take place in the presence of a large excess of simple electrolyte. The electrolyte present is often a buffer, a rather complicated mixture (difficult to model perse) with several ionic species present in the system. Note that many effects in protein solutions are salt specific. Yet, most of the theories subsume all the effect of the electrolyte present into a single parameter, the Debye screening parameter n. In the case of the Donnan equilibrium we measure the subtle difference between the osmotic pressures across a membrane permeable to small ions and water but not to proteins. We believe that an accurate theoretical description of protein solutions can only be built based on the models which take into account hydration effects. 

The difference is explained by Gibbs theory of equilibria and was studied experimentally by Donnan the overall process is known as the Gibbs-Donnan equilibrium. 

Neale J. Text. Inst., 1929, 373 1930, 225 1931, 349) attributes the attraction of cellulose towards sodium ions to the formation of non-diffus-ible anions of the type (CeHioOj)- or (CgH 9O5.H2O)-. This, it has been suggested, causes a distribution of sodium and hydroxyl ions on each side of the fibre boundary in accordance with the Donnan membrane equilibrium theory. The Donnan theory provides thermodynamic proof that if a solution of an electrolyte containing two diffusible ions is separated by a membrane from another solution containing a salt with a non-diffusible ion, then the equilibrium distribution of the former will not be equal on the two sides of the membrane . This state is illustrated diagrammatically in Fig. 3.12 where, on the side marked A of the membrane (which in this case is the... 

Because of this the Donnan membrane theory states that the activity quotient inside the gel and outside the gel are equal at equilibrium. In mathematical terms then, the following equation is true ... 

Figure 7. Computed current efficiency as a function of EW according to the absolute reaction rate theory. The closed circles and open squares are results of the Cluster-Network (CN) model and classical Donnan equilibrium (DE), respectively. In comparison the experimental trend (----------) is also shown. The adjustable param-...

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... 

D.G. Donnan, The theory of membrane equilibria, Chem. Rev., 1924, 1, 73-90 R.M. Wallace, Concentration and separation of ions by Donnan membrane equilibrium, Ind. Eng. Chem. Process Design Dev. 1967, 6, 423. 

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