Nernst-Donnan equation

Building on initial work [47], the main focus of SECM in the study of ET at ITIES has been to identify and understand the potential-dependence of ET rates. In these studies, the potential drop across an ITIES has been controlled by varying the concentration of potential-determining ions in the two phases. The potential drop across an ITIES follows the Nernst-Donnan equation [74,75],... 

The Galvani potential difference across the oil/water interface A< q (f e potential of the water (W) phase referred to that of the oil (O) phase) when both the O and W phases contain a common ion M which is transferable across the interface is given by the Nernst-Donnan equation [31, 39-42],... 

In ion-exchange resins, diffusion is further complicated by electrical coupling effec ts. In a system with M counterions, diffusion rates are described by the Nernst-Planck equations (Helfferich, gen. refs.). Assuming complete Donnan exclusion, these equations canbe written... 

UF and RO models may all apply to some extent to NF. Charge, however, appears to play a more important role than for other pressure driven membrane processes. The Extended-Nemst Planck Equation (equation (3.28)) is a means of describing NF behaviour. The extended Nernst Planck equation, proposed by Deen et al. (1980), includes the Donnan expression, which describes the partitioning of solutes between solution and membrane. The model can be used to calculate an effective pore size (which does not necessarily mean that pores exist), and to determine thickness and effective charge of the membrane. This information can then be used to predict the separation of mixtures (Bowen and Mukhtar (1996)). No assumptions regarding membrane morphology ate required (Peeters (1997)). The terms represent transport due to diffusion, electric field gradient and convection respectively. Jsi is the flux of an ion i, Di,i> is the ion diffusivity in the membane, R the gas constant, F the Faraday constant, y the electrical potential and Ki,c the convective hindrance factor in the membrane. 

Thus, the diffusible ions are distributed as follows K C = Ki C i, according to the Donnan equation. The potential that develops at the surface of the cell as a result of the uneven distribution of the cations can be calculated with the aid of the Nernst equation ... 

MacGillivray AD (1968) Nernst-Planck equations and the electroneutrality and Donnan equilibrium assumptions. J Chem Phys 48 2903-2906... 

No attempt has been made so far to relate the membrane phctse concentration of an ion to its external concentration. A relation of the type (3.3.120e, f) may be derived by using the relation (3.3.118b) for Donnan equilibrium as well as the electroneutrality relations in the membrane and in the external solution. The ionic flux of any ion through an ion exchange membrane may then be obtained from the Nernst-Plank equation (3.1.106). If is to be eliminated from such an equation, then the following procedure may be adopted. Write down the flux for /J and /j. If there is some constraint like Jl 0 (as in (3.4.121)), the problem is easily solved. In some cases, /J = (for example, in battery separators for alkaline Ni/Zn batteries, KOH is the electrolyte and/k+ = /oh-)- This will also allow elimination of V(i. Then one can express / in terms of the concentrations of two solutions on the two sides of the membrane by integrating across the battery separator membrane at steady state. 

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... 

The Donnan potential can also be regarded as a special case of a diffusion potential. We can assume that the mobile ions are initially in the same region as the immobile ones. In time, some of the mobile ions will tend to diffuse away. This tendency, based on thermal motion, causes a slight charge separation, which sets up an electrical potential difference between the Donnan phase and the bulk of the adjacent solution. For the case of a single species of mobile cations with the anions fixed in the membrane (both assumed to be monovalent), the diffusion potential across that part of the aqueous phase next to the membrane can be described by Equation 3.11 n — El = (u- — u+)/(u + w+)](i 77F)ln (c11/ 1) that we derived for diffusion toward regions of lower chemical potential in a solution. Fixed anions have zero mobility (u = 0) hence (u — u+)/(u — u+) here is —uJu+> or —1. Equation 3.11 then becomes En — El = — (RT/F) In (cll/cl)> which is the same as the Nernst potential (Eq. 3.6) for monovalent cations [—In = In (cVc11)]. 

Donnan dialysis The BAHLM systems with ion-exchange membranes, based on Donnan dialysis [18,19], will be considered below. Donnan dialysis is a continuously operating ion-exchange process. There are many theoretical models describing transport mechanisms and kinetics of DL) [18-26]. All transport kinetics models are based on Fick s or Nernst-Planck s equations for ion fluxes. In both cases, the authors introduce many assumptions and simplifications. 

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... 

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