Donnan equilibrium equation

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. 

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. 

The significance of the Donnan equilibrium is probably best seen as follows. Combining Equations (86) and (88) yields... 

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. 

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. 

The values for g obtained from Equation 4.22 do not seem to be very different from those obtained from Equation 4.20, as shown in Table 4.1 and Figure 4.3a. It is easy to see that g must be equal to /2 as Os tends to zero by expanding the exponentials in the linear approximation (Debye limit). Naturally, Equation 4.15 gives us i = 1 in this limit, as an uncharged layer does not expel co-ions and salt is equally distributed between regions I and II. However, as shown in Table 4.2 and Figure 4.3b, the predicted salt-fractionation effect obtained by substituting Equation 4.22 into Equation 4.15 is markedly different from the Donnan equilibrium. 

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... 

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 ... 

If m moles of AI were transferred from cell I to cell II until Donnan equilibrium was achieved, it would mean that 3m moles of H" " would be transferred from cell 11 to cell I, as the number of equivalents will be preserved in both the cells to maintain zero current conditions. This transfer would satisfy the conditions of Equation 34.5. Substituting for the valence term in this equation, m can be calculated. It would lead to the foUowing values at equilibrium ... 

Eriksson applied Donnan equilibrium calculations to heterovalent exchange, reasoning that clays in a salt solution could be thought of as an ion species restricted from free diffusion. His equation was... 

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 ... 

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)). 

For a polyelectrolyte gel, however, the ionic strength or salt concentration in the gel is not known. In diis case, usually we first use linear counterion condensation tiieory to find the counterion concentration, then we use Donnan equilibrium to calculate the sdt concentration. From the counterion concentration and salt concentration we can find the ionic strength. Then the ionic strengA is used to solve the nonlinear Poisson-Boltzmann equation to find the real counterion condensation and effective degree of... 

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