Donnan membrane equilibrium equations

One of the earliest, and reasonably successful, approaches to quantitatively predicting selectivity behaviour was through the thermodynamic treatment of ion exchange systems as a Gibbs-Donnan membrane equilibrium. Such a description is given by equation 5.29 which for the sake of simplicity is shown in terms of single ion activity coefficients ... 

Polyelectrolyte molecules in highly dilute aqueous solutions exert strong electrical repulsions on each other. These repulsive forces are long range (proportional to l/r ) by comparison with normal dispersion forces (proportional to 1/r ), and as a consequence the intermolecular interactions persist down to the lowest measured concentrations. In osmotic-pressure measurements on polyelectrolytes, the Donnan membrane equilibrium must be satisfied and experimental results indicate that the second virial coefficient in the osmotic-pressure equation (p. 915) becomes very large. 

The difference between Equations (55) and (60) may be qualitatively understood by comparing the results with the Donnan equilibrium discussed in Chapter 3. The amphipathic ions may be regarded as restrained at the interface by a hypothetical membrane, which is of course permeable to simple ions. Both the Donnan equilibrium (Equation (3.85)) and the electroneutrality condition (Equation (3.87)) may be combined to give the distribution of simple ions between the bulk and surface regions. As we saw in Chapter 3 (e.g., see Table 3.2), the restrained species behaves more and more as if it was uncharged as the concentration of the simple electrolyte is increased. In Chapter 11 we examine the distribution of ions near a charged surface from a statistical rather than a phenomenological point of view. 

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 ... 

In reality, a state of equilibrium is reached where the ionic attraction of the counterions by the polyanion is just balanced by diffusion into the external solution, which is driven by the chemical potential gradient, which in turn arises from the difference in counterion concentrations between the two domains. The overall effect closely resembles Donnan membrane equilibria (4) (Figure 3). As a consequence of the difference in counterion concentrations, the osmotic pressure inside the polymer domain exceeds that of the external solution, and the expansion of the polyelectrolyte can be equated to the difference in osmotic pressures of the intramolecular and intermolecular solutions. 

This equation, called the Donnan equation, is named after the British chemist Frederick George Donnan who published his important paper about the theory of membrane equilibrium in 1911. 

Equation (31) is true only when standard chemical potentials, i.e., chemical solvation energies, of cations and anions are identical in both phases. Indeed, this occurs when two solutions in the same solvent are separated by a membrane. Hence, the Donnan equilibrium expressed in the form of Eq. (32) can be considered as a particular case of the Nernst distribution equilibrium. The distribution coefficients or distribution constants of the ions, 5 (M+) and B X ), are related to the extraction constant the... 

A semi-permeable membrane, which is unequally permeable to different components and thus may show a potential difference across the membrane. In case (1), a diffusion potential occurs only if there is a difference in mobility between cation and anion. In case (2), we have to deal with the biologically important Donnan equilibrium e.g., a cell membrane may be permeable to small inorganic ions but impermeable to ions derived from high-molecular-weight proteins, so that across the membrane an osmotic pressure occurs in addition to a Donnan potential. The values concerned can be approximately calculated from the equations derived by Donnan35. In case (3), an intermediate situation, there is a combined effect of diffusion and the Donnan potential, so that its calculation becomes uncertain. 

The movement of solute across a semipermeable membrane depends upon the chemical concentration gradient and the electrical gradient. Movement occurs down the concentration gradient until a significant opposing electrical potential has developed. This prevents further movement of ions and the Gibbs-Donnan equilibrium is reached. This is electrochemical equilibrium and the potential difference across the cell is the equilibrium potential. It can be calculated using the Nemst equation. 

At equilibrium there is a zero free-energy change, AG=0, that takes place between compartments separated by a membrane, with the free-energy change being dependent on the difference in concentration of various ions and the electrical potential difference that exists across the membrane. The relationships among sodium, potassium, and chloride ions, pH, and electrolytic potential have become known as Donnan equilibria. The concentrations and electrolytic potentials are related by the following equation ... 

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. 

Equation (6.148) is the well-known Donnan equilibrium of salt across a membrane in the presence of a polyelectrolyte, to which the membrane is permeable. It demonstrates the characteristic properties of the chemical potentials of neutral salts. 

The conditions of Donnan equilibrium derived in Equation 34.6 can be understood in a slightly different way. For the coupled transport described previously, the heterogeneous exchange of cation A in the aqueous phase and cation B in the membrane phase is represented by the following equation ... 

When the mobile ion in the solid membrane is an anion, the basic equations giving the membrane potential differ with respect to the sign of the contributing terms. The equilibrium giving rise to the Donnan potential is now... 

The equilibrium determining the Donnan potential is given by equation (9.8.4). Assuming that there is no diffusion potential in the membrane, the Donnan potentials on either side of the membrane are given by the same equations obtained for the ion-exchanging system, namely, equations (9.8.36) and (9.8.38). In addition, the expression for the membrane potential is the simple result given by equation (9.8.40). 

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... 

This rather trivial case is useful as an introduction to the Donnan equilibrium, since equation (7.79) will still be obeyed even if the system contains a nondiffusible ion in addition to sodium and chloride ions. For example, suppose that initially we have the situation represented in Figure 7.11 . On the left-hand side, there are sodium ions and nondiFusible anions, P . On the right-hand side there are sodium and chloride ions. Since there are no chloride ions on the left-hand side, spontaneous diffusion of chloride ions from right to left will occur. Since there must always be electrical neutrality on each side of the membrane, an equal number of sodium ions must also diffuse from right to left. Figure 7.11c shows the situation at equilibrium ... 

Although the large scale industrial utilisation of ion-exchange membranes began only 20 years ago, their principle has been known for about 100 years [1]. Beginning with the work of Ostwald in 1890, who discovered the existence of a "membrane potential" at the boundary between a semipermeable membrane and the solution as a consequence of the difference in concentration. In 1911 Donnan [2] developed a mathematical equation describing the concentration equilibrium. The first use of electrodialysis in mass separation dates back to 1903, when Morse and Pierce [3] introduced electrodes into two solutions separated by a dialysis membrane and found that electrolytes could be removed more rapidly from a feed solution with the application of an electrical potential. 

Donnan Equilibrium and Electroneutrality Effects for charged membranes are based on the fact that charged functional groups attract counter-ions. This leads to a deficit of co-ions in the membrane and the development of Donnan potential. The membrane rejection increases with increased membrane charge and ion valence. This principle has been incorporated into the extended Nemst-Planck equation, as described in the NF section. This effect is responsible for the shift in pH, which is often observed in RO. Chloride passes through the membrane, while calcium is retained, which means that water has to shift its dissociation equilibrium to provide protons to balance the permeating anions (Mallevialle et al. (1996)). 

Integration of this equation over the whole membrane thickness at the equilibrium conditions leads to the expression for electrical potential difference developed across the membrane, known as Donnan potential,... 

In Section 3.6, we have derived that a concentration difference across a (virtual) semipermeable membrane gives rise to an osmotic pressure. Similarly, the ion concentration difference, (c, + Cj + zc + c) - (c + Cn), characterizing the Donnan equilibrium, leads to an osmotic pressure n, the colloid-osmotic pressure, which follows from Equation 3.37... 

The Donnan equilibrium allows the evaluation of the distribution of M and X" over both sides of the semipermeable membrane. If electrodes responding to either M+ or X" were inserted at either side of the membrane there would be no potential difference between them. This is a consequence of the system being in equilibrium which implies that no work can be performed. Nevertheless, because of the different ion concentrations, the potentials at the respective electrodes i/o,i and /o n are not equal. Consequently, there must be a compensating potential difference across the membrane, Axi/. If the electrodes respond reversibly to the ion concentrations so that Nernst s law (Equation 9.14) applies, it follows for the membrane potential... 

This equation gives the ionic or Donnan equilibrium of charged solutes in the presence of a charged membrane (or charged macromolecules) possessing a fixed charge density R. ... 

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