Donnan potential derivation

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. 

Here, b is the distance between the nearest unit charges along the cylinder (b = 0.34nm for the ssDNA and b = 0.17nm for the dsDNA), (+) and (—) are related to cations and anions, respectively, and a = rss for the ssDNA and a rds for the dsDNA. The expressions (5) and (6) have been obtained using the equations for the electrostatic potential derived in [64, 65], where a linearization of the Poisson-Boltzmann equation near the Donnan potential in the hexagonal DNA cell was implemented. 

Figure 2. Sodium and chloride uptake across an idealised freshwater-adapted gill epithelium (chloride cell), which has the typical characteristics of ion-transporting epithelia in eukaryotes. In the example, the abundance of fixed negative charges (muco-proteins) in the unstirred layer may generate a Donnan potential (mucus positive with respect to the water) which is a major part of the net transepithelial potential (serosal positive with respect to water). Mucus also contains carbonic anhydrase (CA) which facilitates dissipation of the [H+] and [HCO(] to CO2, thus maintaining the concentration gradients for these counter ions which partly contribute to Na+ import (secondary transport), whilst the main driving force is derived from the electrogenic sodium pump (see the text for details). Large arrow indicates water flow...

Even for this simplest of all situations we had to make a fairly drastic assumption of high concentration of NaCl, in order to get from (6.11) to (6.13). The situation is considerably more complicated when different multivalent ions are present in the solution, although the basic argument is the same. In biological fluids, such as whole blood, the value of the Donnan potential across the dialysis membrane can be tens of millivolts. (In the above derivation of the Donnan potential, concentrations instead of activities have been used for purely historical reasons.)... 

Donnan Potential. Colon exclusion and counterion condensation on the charged vesicle surface creates the well known Donnan potential. The Donnan potential can be derived either by a kinetic or by a thermodynamic approach (32). Using the kinetic approach, the mass transport equation is written by ... 

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)]. 

Consider the case where the thickness of the surface layer d is much larger than the Debye length 1/k. In this case, the electric field (d J//dx) and its derivative (f j/ldx ) deep inside the surface layer become zero so that the potential deep inside the surface layer becomes the Donnan potential i/ don. given by Eq. (4.18). It must be noted here that this is the case only for d 1/k. If the condition d I/k does not hold, there is no region where the potential reaches the Donnan potential. When d 1/k, by replacing y —d) by Tdon in Eq. (4.35), we obtain... 

Equations (13.67) and (13.68) correspond to the assumption that the potential far inside the membrane is always equal to the Donnan potential. Expressions for yooNi and yDON2 can be derived by setting the right-hand sides of Eqs. (13.59) and (13.61) equal to zero, namely,... 

The electrical potential generated between membrane and solution phases, the Donnan potential, is derived from Eqs. (2.17) and (2.18),... 

This is seen to be significantly higher than the rates attained for non-electrolytes. The comparison is, of course, not valid the derivation assumes that only the concentration difference in (CrOJ) provides the driving force. Instead, the work of Wallace [11], Smith [12] and Melsheimer et al. [13], indicates that the driving force is the Donnan potential, which drives the system toward the equilibrium ... 

The results of Volta-potential measurements on the emersed PMPy-coated electrodes are summarized in Fig. 3.22, which presents the results at various electrode potentials derived from the preceding measurements. The increasingly positive slope indicates that the polymer assumes the state of an anion exchanger as it is oxidized. In Fig. 3.23 the expected Donnan potentials were calculated with Eqn. 18 the polaron/bipolaron density, obtained by integrating the oxidation current in Fig. 3.21, was assumed to constitute the fixed-charge density c. At the electrolyte concentration = 0.1 M, the calculated and experimental values (see Fig. 3.22) were matched, and at lower concentrations, theoretical and experimental values were plotted. The agreement can be considered quite good. Based on this analysis. Fig. 

The expression for the second virial coefficient of polyelectrolytes, A c = Kp/ic, is derived from the Donnan potential [64]. It originates from a non-uniform spatial distribution of the counterions in order to fulfill the local electroneutrality condition. As a consequence the concentration of the counterions is enhanced in the vicinity of the polyion. 

A theoretical derivation of the potential in a system with a combination of a Donnan layer and a diffuse layer has been given by Ohshima and Kondo [14-17]. The basic step is to write Poisson s equation in terms of all relevant charge carriers. For the Donnan layer, it reads ... 

To circumvent the above problems with mass action schemes, it is necessary to use a more general thermodynamic formalism based on parameters known as interaction coefficients, also called Donnan coefficients in some contexts (Record et al, 1998). This approach is completely general it requires no assumptions about the types of interactions the ions may make with the RNA or the kinds of environments the ions may occupy. Although interaction parameters are a fundamental concept in thermodynamics and have been widely applied to biophysical problems, the literature on this topic can be difficult to access for anyone not already familiar with the formalism, and the application of interaction coefficients to the mixed monovalent-divalent cation solutions commonly used for RNA studies has received only limited attention (Grilley et al, 2006 Misra and Draper, 1999). For these reasons, the following theory section sets out the main concepts of the preferential interaction formalism in some detail, and outlines derivations of formulas relevant to monovalent ion-RNA interactions. Section 3 presents example analyses of experimental data, and extends the preferential interaction formalism to solutions of mixed salts (i.e., KC1 and MgCl2). The section includes discussions of potential sources of error and practical considerations in data analysis for experiments with both mono- and divalent ions. 

In the definitions of T, two variables in addition to the ion chemical potential must also be specified as constant. In an equilibrium dialysis experiment, these are temperature and the chemical potential of water. This partial derivative is known as the Donnan coefficient. (Note that the hydrostatic pressure is higher in the RNA-containing solution.) In making connections between T and the Gibbs free energy, it is more convenient if temperature... 

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. 

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... 

The solute transport is driven by solute concentration gradient, by Liquid membrane facilitation potential (LMF), K, and by Donnan equilibrium coupling, Kj. is denoted as an internal LM carrier driving force coefficient, derived from extraction distribution constants for solute between... 

From electric potential measurements Loeb could derive the pu of the liquid. The results were in conformity with the Donnan equilibrium. From the experimental data the osmotic pressure difference could be calculated. Plotted against the pH of the outer liquid, the maximum value of A coincided with the pH corresponding to maximum swelling, as shown by the following selected figures. 

This statement regarding the uniform chemical potential of a species applies to both a substance and an ion, as the following argument explains. The derivation in this section begins with Eq. 9.2.37, an expression for the total differential ofU. Because it is a total differential, the expression requires the amount , of each species i in each phase to be an independent variable. Suppose one of the phases is the aqueous solution of KCl used as an example at the end of the preceding section. In principle (but not in practice), the amounts of the species H2O, K+, and Cl can be varied independently, so that it is vahd to include these three species in the sums over i in Eq. 9.2.37. The derivation then leads to the conclusion that K+ has the same chemical potential in phases that are in transfer equilibrium with respect to K+, and likewise for Cl . This kind of situation arises when we consider a Donnan membrane equilibrium (Sec. 12.7.3) in which transfer equilibrium of ions exists between solutions of electrolytes separated by a semipermeable membrane. 

Eirst, let us show how to derive the required porous electrode RC theory ( transmission line-theory ) from our results in Section 15.4.3. At high ionic strength conditions, the diffuse (or, Donnan) layer potential will be snfficiently low such that we are in the low-potential limit where we have no net salt (electro-)adsorption in the EDLs (see Equation 15.4). This is the fundamental reason... 

Use of equation (8) requires knowledge of the concentration of mobile ions inside the polymer network which originated in the solution (c s)- This may be found from Donnan ion exclusion theory, which calculates the distribution of ions which develops to maintain the equality of electrochemical potentials of all ions In the system, both inside and outside the gel. Detailed derivation of this may be found elsewhere [23, 24]. The key result is that the distribution of the ions between the gel and the solution is governed by the following equation, where the primes indicate concentrations in the gel phase ... 

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