Membrane Equilibrium Donnan Potentials

The liquid junction potential was defined with no membrane separating the two media. 

A membrane separating an eleetrolyte in two eompartments is often selectively permeable (e.g., rather open to water), but less permeable to certain ions or larger charge carriers. 

The selectivity may be due to the mechanical dimensions of pores, or eharge dependent forces. 

With different concentrations on each side, such membranes generate an osmotic pressure difference. With different ionic concentration also an electrical potential difference is generated. This is called the Donnan potential difference, 

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When the membrane of polystyrene beads is in thermodynamic equilibrium, Donnan potentials exist at the pore walls of the porous membrane, where the protein is immobilized. At high charge densities, the resulting fields can extend to the pore centers, and consequently the membrane itself will have a potential with respect to the bulk solution. 

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). 

At each phase boundary there exists a thermodynamic equilibrium between the membrane surface and the respective adjacent solution. The resulting thermodynamic equilibrium potential can then be treated like a Donnan-potential if interfering ions are excluded from the membrane phase59 6,). This means that the ion distributions and the potential difference across each interface can be expressed in thermodynamic terms. 

The equilibrium conditions for homogeneous systems with membranes were first formulated in this form by Frederick G. Donnan in 1911. Hence, such equilibria are often called Donnan equilibria, and the membrane potentials associated with them are called Donnan potentials. Sometimes these terms are used as well for the equilibria arising at junctions between dissimilar solutions (Section 5.3). 

Cells of the type in Scheme 10 represent the simplest case of an ion-selective liquid cell its EMF is often called a membrane, or monoionic, potential [3]. The first term is too narrow due to the fact that the membrane potential corresponds to the behavior of a number of cells, including those of Schemes 8 to 11, and to the cells with solid membranes and with Donnan equilibrium. 

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. 

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... 

The equilibrium (also known as the Donnan effect) established across a semipermeable membrane or the equivalent of such a membrane (such as a solid ion-exchanger) across which one or more charged substances, often a protein, cannot diffuse. Diffusible anions and cations are distributed on the two sides of the membrane, such that the sum of concentrations (in dilute solutions) of diffusible and nondiffusible anions on either side of the membrane equals the sum of concentrations of diffusible and nondiffusible cations. Thus, the diffusible ions will be asymmetrically distributed across the membrane and a Donnan potential develops. 

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. 

The double-layer model of the membrane consists of many particles (assume a diameter of 0.1 gm) that are impenetrable for solution and carry a surface charge. Electrical double layers exist around each particle, and because the dimensions of the membrane pores are of the same order as the double layers around the particles, double layers exist throughout the membrane pores. The potential measured by the underlying ISFET, with respect to the bulk potential, is on one hand determined by the mean pore potential, which is the net result of the contribution of all surface potentials of the charged particles, and on the other hand by the pH at the membrane-1 SFET interface. The measured ISFET response in equilibrium is therefore the same as that of an ISFET without a membrane, because the distribution of the protons between membrane and solution results in a pH difference, which compensates the mean membrane potential (this is the same mechanism as in the Donnan model). The relation between the surface charge on the particles a (C/cm2) and the surface potential jx of each particle is given by... 

First, assume that the surface charge on the membrane particles does not interact with the mobile protons (no proton release or uptake). An ion step will result in an increase in the double-layer capacitances of the particles and consequently in a decrease of the surface potentials fr, because the charge densities remain constant. The ISFET will measure a transient change in the mean pore potential. As a result of the potential changes, an ion redistribution will take place and the equilibrium situation is re-established. The theoretical maximum ion step response is the change in the mean pore potential. This is comparable with the Donnan model where the theoretical maximum is determined by the change in the Donnan potential at the membrane solution interface. 

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. 

In general, the chemical potentials of the cation are not equal in a given solution and in the membrane. At equilibrium the electrochemical potentials of the cation must be equal in the two phases. As a result, a potential difference called the Donnan potential is established at each interface. Moreover, the concentration of the cation on the left-hand side of the membrane is not always the same as that on the right and the cation diffuses from the location of high concentration to the one where it is lower. The non-equilibrium diffusion process gives rises to a diffusion potential. 

The activity coefficient of the cation within the membrane is assumed to be constant independent of location, so that it is not explicitly written in the equilibrium expression. is the equilibrium constant for the ion exchange process. This potential difference is called the Donnan potential. It arises essentially because the activities of the ion M are not the same in the solution and in the membrane. 

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 transport kinetics is driven mainly by Donnan equilibrium coupling in steps 1,2,6, and 7 and by liquid membrane facilitation (LMF) potential and Donnan equilibrium coupling in step 4. 

When the Donnan equilibrium of the counter-ion is attained at membrane-solution interfaces, the Donnan potential at the interface 1, E, is... 

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)). 

In order to explain this transmembrane potential, a number of authors have proposed membrane potential theories. Historically, the first membrane potential theory for biological systems was the use of the concept of the Donnan membrane equilibrium. 

NF is closely related to RO, and is sometimes called loose RO (Table 1.3). The basic principles of the RO process discussed above are appficable to NF except that the rejection of solutes depends both on molecular size and Donnan exclusion effects, which are due to the acid groups, e.g., carboxyfrc or sulphonic acid groups, on the polymer backbone. The equilibrium between the charged membrane and the bulk solution is characterised by an electric potential called the Donnan potential. Ions smaller than the pore size are rejected because of Donnan exclusion [6, 7, 17]. 

Donnan dialysis In Donnan dialysis, a cation-exchange membrane separates the donor and receptor solutions. Cationic metal species are transported across the membrane driven by the negative electrostatic potential (the Donnan potential) across the membrane, until equilibrium is achieved. Matching of the ionic strengths of donor and acceptor solutions is necessary. Since cationic species exchange readily compared to neutral and anionic species, it is claimed that the measurement more closely relates to the free metal ion. 

The ISE membrane thus behaves as a permselective membrane and the signal measured is given by the Donnan potential, which depends on the selectivity of the interfacial interaction, given by the equilibrium constants of the interactions of the membrane with the analyte(s) and interferents. Moreover, a diffusion potential develops across the membrane, as charge is passed on the passage of ions between the solution and the membrane. 

The rejected solute is enriched in the donor solution. To take into account the electric charge Zi of the solutes and of the separation membrane, the Donnan effect has to be considered. The chemical potential p,-is extended by the electric potential gradient, Acj). When a membrane separates two solutions of a dissociating salt BA (BA B +A ), one of which contains a rejected ion X, an electric potential difference Acj) = a is built up between the solutions (Figure 3). The thermodynamic equilibrium constant can be calculated in the absence of osmotic pressure differences according to eqn [12] with the... 

In the true membrane arrangement, when the conducting ion-exchange polymer is situated between two solutions (1 and 11) of different concentrations, a Donnan potential also arises at the other membrane-solution interface, which can be expressed similarly however, no equilibrium exists in this case. 

Membrane phenomena cover an extremely broad field. Membranes are organized structures especially designed to perform several specific functions. They act as a barrier in living organisms to separate two regions, and they must be able to control the transport of matter. Moreover, alteration in transmembrane potentials can have a profound effect on key physiological processes such as muscle contraction and neuronal activity. In 1875, Gibbs stated the thermodynamic relations that form the basis of membrane equilibria. The theory of ionic membrane equilibrium was developed later by Donnan (1911). From theoretical considerations, Donnan obtained an expression for the electric potential difference, commonly known as the membrane potential between two phases. 

The Gibbs-Donnan potential occurs when a nonpermeant ion (for biological systems, usually a polyion such as a protein) is unequally distributed between the two electrolyte solutions separated by a permselective membrane, which allows certain electrolyte ions to move freely between the two solutions. The second law of thermodynamics and the principle of electroneutrality restrict this movement. The first restriction requires that each permeant ion species moves only down its electrochemical potential gradient, the latter requires the sum of all positive charges (cations) to be equal to that of all negative charges (anions) in each solution. When the system reaches its equilibrium, there is no flux of any permeant species i, = 0. 

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,... 

It is important to consider that the protonation/deprotonation equilibrium involves the concentration (activity) of protons in the polymer. However in experiments the activity of i.e., the pH of the bathing solution, is controlled. Therefore the membrane state of the polymer, i.e., the Donnan potential, which affects the peutitioning... 

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