The Donnan equilibrium

The term Donnan equilibrium refers to the distribution of ions between two solutions in contact through a semipermeable membrane, in one of which there is a polyelectrolyte, such as NavP (with P a polyanion), and where the membrane is not permeable to the large charged macromolecule. This arrangement is one that actually occurs in living systems, where we have seen that osmosis is an important feature of cell operation. The thermodynamic consequences of the distribution and transfer of charged species across cell membranes is explored further in Chapter 5. 

Consider a situation in which a high concentration of a salt such as NaCl is added to the solution on both sides of the membrane so that the number of cations that P provides is insignificant in comparison with the number sup-phed by the additional salt. Apart from small imbalances of charge close to the membrane (which have important consequences, as we shall see in Chapter 5), electrical neutrality must be preserved in the bulk on both sides of the membrane if an anion migrates, a cation must accompany it. For simplicity, we take the volumes of the solutions on each side of the membrane to be equal. 

On one side of the membrane—call it the left-hand side—there are P , Na, and Cl ions. In the right-hand side there are Na and Cl ions. The condition 

If we ignore activity coefhcients and interpret [Na+]/c and [Ch]/c as [Na+] and [Ch], respectively, the two expressions are equal when [Na+]L[Ch]L = [Na+]j [Ch]j. As the Na+ ions are supplied by the polyelectrolyte as well as the added salt, the conditions for bulk electrical neutrality lead to the charge-balance equations [Na+]L = [Ch]L -I- v[P ] and [Na+]j = [Ch]j. We can now combine these three conditions to obtain expressions for the differences in ion concentrations across the membrane. For example, we write 

It follows from the definition [Cl ] = ([Cl ]l-I- [Cl ]r) and the charge-balance equations that 

A matter of considerable importance in biological systems is the ionic equilibrium which exists between two solutions separated by a membrane, such as a cell wall. Complications arise with ions that are too large to diffuse through the membrane, and the diffusible ions then reach a special type of equilibrium known as the Donnan equilibrium. Its theory was first worked out in 1911 by the British physical chemist Frederick George Donnan (1870- 1956). 

I where the terms are the Gibbs energy dilRerences across the membrane (e.g., AG is 

In other words, at equilibrium we have equal concentrations of the electrolyte on each side of the membrane. 

Under certain circumstances, the establishment of the Donnan equilibrium can lead to other effects, such as changes in pH. Suppose, for example, that an electrolyte NaP (where P is a large anion) is on one side of a membrane, with pure water on the other. The Na ions will tend to cross the membrane and, to restore the electrostatic balance, H ions will cross in the other direction, leaving an excess of OH ions. Dissociation of water molecules will occur as required. There will thus be a lowering of pH on the NaP side of the membrane, and a raising on the other side. 

1 The following are some conventional standard enthalpies of ions in aqueous solution, at 25 C  

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This last result describes the Donnan equilibrium condition as it applies to the system under consideration. Like other ionic equilibrium expressions, it requires the equality of ion products in equilibrium solutions. 

Concentration of Electrolyte Myer and Sievers"" applied the Donnan equilibrium to charged membranes and developed a quantitative theory of membrane selectivity. They expressed this selectivity in terms of a selectivity constant, which they defined as the concentration of fixed ions attached to the polymer network. They determined the selectivity constant of a number of membranes by the measurement of diffusion potentials. Nasini etal and Kumins"" extended the measurements to paint and varnish films. 

When the Donnan equilibrium is operative the entry of ions into the membrane is restricted. Consequently as the concentration of ions in the solution increases the resistance of the membrane remains constant until the concentration of ions in the solution reaches that of the fixed ions attached to the polymer network. At this point their effect will be swamped and the movement of ions will be controlled by the concentration gradient. 

Films of a pentaerythritol alkyd, a tung oil phenolic and an epoxypolyamide pigmented with iron oxide in the range 5-7% p.v.c. were exposed to solutions of potassium chloride in the range 0.0001-2.0 m. It was found that in all cases the resistance of the films steadily decreased as the concentration of the electrolyte increased. Since the resistances of the films were at no time independent of the concentration of the electrolyte, it was concluded that the Donnan equilibrium was not operative and that the resistance of the films were controlled by the penetration of electrolyte moving under a concentration gradient. 

Membrane Efficiency The permselectivity of an ion-exchange membrane is the ratio of the transport of electric charge through the membrane by specific ions to the total transport of electrons. Membranes are not strictly semipermeable, for coions are not completely excluded, particularly at higher feed concentrations. For example, the Donnan equilibrium for a univalent salt in dilute solution is ... 

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... 

However, as soon as at the eluate-side H ions are replaced with an equivalent amount of Na or K ions, which elute, the then asymmetric cell acquires a potential that reflects the Donnan equilibrium potential on the basis of the ion mobilities concerned. Hence the potential change as a function of time represents the ionic chromatogram and the peaks concerned yield the alkali metal ion contents via calibration. 

Overbeek, J. Th. G., The Donnan equilibrium, in Progress in Biophysics and Biophysical Chemistry, Vol. 6, Pergamon Press, London 1956. 

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... 

For homopolyelectrolyte, we first studied the ellipsometric measurement of the adsorption of sodium poly(acrylate) onto a platinum plate as a function of added sodium bromide concentration (5). We measured the effect of electrolyte on the thickness of the adsorbed layer and the adsorbances of the polyelectrolyte. It was assumed that the Donnan equilibrium existed between the adsorbed layer and the bulk phase. The thickness was larger and the adsorbance of the polyelectrolyte was lower for the lower salt concentration. However, the data on the molecular weight dependence of both the adsorbance and the thickness of the adsorbed polyelectrolyte have been lacking compared with the studies of adsorption of nonionic polymers onto metal surfaces (6-9). 

Since the components in the adsorbed polyelectrolyte layer are considered to be the same as the bulk phase with a three component system which consists of polyelectrolyte, simple salt, and water, we calculate the adsorbances of polyelectrolyte and salt by assuming the Donnan equilibrium between the bulk phase and the adsorbed polyelectrolyte layer, as described previously (5). 

A related phenomenon occurs when the membrane in the above-mentioned experiment is permeable to the solvent and small ions but not to a macroion such as a polyelectrolyte or charged colloidal particles that may be present in a solution. The polyelectrolyte, prevented from moving to the other side, perturbs the concentration distributions of the small ions and gives rise to an ionic equilibrium (with attendant potential differences) that is different from what we would expect in the absence of the polyelectrolyte. The resulting equilibrium is known as the Donnan equilibrium (or, the Gibbs-Donnan equilibrium) and plays an important role in... 

Elementary and advanced treatments of such cellular functions are available in specialized monographs and textbooks (Bergethon and Simons 1990 Levitan and Kaczmarek 1991 Nossal and Lecar 1991). One of our objectives in this chapter is to develop the concepts necessary for understanding the Donnan equilibrium and osmotic pressure effects. We define osmotic pressures of charged and uncharged solutes, develop the classical and statistical thermodynamic principles needed to quantify them, discuss some quantitative details of the Donnan equilibrium, and outline some applications. 

We conclude the chapter with a discussion of the Donnan equilibrium and the thermodynamic behavior of charged colloids, particularly with respect to osmotic pressure and molecular weight determination (Section 3.5), and some applications of osmotic phenomena (Section 3.6). 

This expression describes what is known as the Donnan equilibrium. It does not say that the activity of M+ and/or X is the same on both sides of the membrane, but that the ion activity product is constant on both sides of the membrane. In the sense that an ion product is involved, the Donnan equilibrium clearly resembles all other ionic equilibria. 

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

The combined effects of electroneutrality and the Donnan equilibrium permits us to evaluate the distribution of simple ions across a semipermeable membrane. If electrodes reversible to either the M+ or the X ions were introduced to both sides of the membrane, there would be no potential difference between them the system is at equilibrium and the ion activity is the same in both compartments. However, if calomel reference electrodes are also introduced into each compartment in addition to the reversible electrodes, then a potential difference will be observed between the two reference electrodes. This potential, called the membrane potential, reflects the fact that the membrane must be polarized because of the macroions on one side. It might be noted that polarized membranes abound in living systems, but the polarization there is thought to be primarily due to differences in ionic mobilities for different solutes rather than the sort of mechanism that we have been discussing. We return to a more detailed discussion of the electrochemistry of colloidal systems in Chapter 11. 

What is Donnan equilibrium Give at least three examples of applications for which the Donnan equilibrium is important. 

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. 

A terminological remark is due. An equilibrium between two media with different fixed charge density (e.g., an ion-exchanger in contact with an electrolyte solution) is occasionally termed the Donnan equilibrium. The corresponding potential drop between the bulks of the respective media is then termed the Donnan potential. By the same token, we speak of the local Donnan equilibrium and the local Donnan potential, referring, respectively, to the local equilibrium and the interface potential jump at the surface of discontinuity of the fixed charge density, considered in the framework of the LEN approximation. 

In Sect. 2 we reviewed the original Tanaka s treatment of ions in gels. More precise theory should properly account for the chemical dissociation equilibrium in the interior of gels and the Donnan equilibrium at the gel-solvent boundary where an electric double layer is formed [31,97,98]. 

The contribution fl>3 corresponds to the difference between the osmotic pressure in the ionized gel and in external solution (mixing of ions with the slovent). From the Donnan equilibrium we obtain... 

When an ion exchanger is placed in an electrolyte solution, the concentration of electrolyte is higher outside the resin than inside it. The equilibrium between ions in solution and ions inside the resin is called the Donnan equilibrium. 

The Donnan equilibrium is the basis of ion-exclusion chromatography. Because dilute electrolytes are excluded from the resin, they pass through a column faster than nonelectrolytes, such as sugar, which freely penetrates the resin. When a solution of NaCl and sugar is applied to an ion-exchange column, NaCl emerges from the column before the sugar. 

Cell membranes are impermeable to most ions. Only a small number of ions can enter cells readily and these usually do so with the assistance of protein channels or pores. The principal anion of plasma (Box 5A) is Cl-, which passes through membranes readily by virtue of the presence of channel-forming proteins. Chloride ions are often distributed across membranes passively according to Eq. 8-5, which describes the Donnan equilibrium.167,479,480... 

In these experiments respiring mitochondria are observed to take up the K+ or Rb+ to give a high ratio of K+ inside to that outside and consequently a negative Em. There are problems inherent in the method. The introduction of a high concentration of ion perturbs the membrane potential, and there are uncertainties concerning the contribution of the Donnan equilibrium (Eq. 8-5) to the observed ion distribution.184... 

An electrochemical system, important particularly in biological systems, is one in which the species are ions and the system is separated into two parts by a rigid membrane that is permeable to some but not all of the species. We are interested in the conditions attained at equilibrium, the Donnan equilibrium. Two cases, one in which the membrane is not permeable to the solvent (nonosmotic equilibrium) and the other in which the membrane is permeable to the solvent (osmotic equilibrium), are considered. The system is at constant temperature and, for the purposes of discussion, we take sodium chloride, some salt NaR, and water as the components. The membrane is assumed to be permeable to the sodium and chloride ions, but not to the R-ions. We designate the quantities pertinent to the solution on one side of the membrane by primes and those pertinent to the solution on the other side by double primes. 

The concentration of a co ion in an ion-exchange membrane can be calculated from the Donnan equilibrium. For a monovalent salt and a dilute salt solution and assuming the activity coefficients of the salt in the membrane and the solution to be 1, the co ion concentration in the membrane is given to a first approximation by... 

Note that this treatment inherently takes into account the effect of the Donnan equilibrium. The osmotic coefficient obtained therefore is that of the polyelectrolyte with no further Donnan correction term being necessary. 

At low ionic strength the local ion concentration is essentially constant and increases at larger added salt concentration. The cs-dependence follows a trend that is characteristic for the Donnan equilibrium. As derived by Hariharan et al. [41] there should be a simple relation... 

It is insightful to view counterion association through the Donnan equilibrium experiment. Let cq be the concentration of polyion, and let c3 be the salt concentration in the sample chamber [65-70, 74], In the presence of... 

Adsorbed films between two immiscible liquids. The question of the meaning of the term pn in the surface layer has been raised by Crax-ford, Gatty, and Teorell,2 without, however, coming to any very clear decision. Danielli s estimate was a very rough one, based on the application of the Donnan equilibrium between the surface layer and the interior, and suffers from the difficulties always attending an attempt to consider concentrations in the surface layer in a similar way to concentrations in a bulk phase the surface layer is not homogeneous. pH is closely related to, and is determined by, the electrochemical potential (see Chap. VIII, pp. 304 ff.), and this depends on the electrostatic potential, which varies rapidly at different levels near to the surface it appears possible that the only satisfactory definition of pa in the surface may be one which varies rapidly at different depths. The question appears one which would repay... 

Under the condition of electroneutrality, Eq. (10.1) describes the Donnan equilibrium across a membrane, which separates solutions containing nonpermeating ions. With the Donnan equilibrium, differences of pressure and electric potential will appear. If the nonpermeating components are electrically neutral, only the pressure difference occurs. 

Figure 4.1b is a typical illustration of the Donnan equilibrium [1], A membrane impermeable to macroions (P" ) but permeable to small ions (M+, X ) and solvent molecules (S) divides a solution into two regions. The situation is a common one in colloid science, and the fact that the equilibrium salt concentration in region II (the simple electrolyte solution), [X ]n, is greater than that in region I (the region occupied by the macroions), [X ]I( has been used in countless dialysis experiments. It is also well known [2] that equilibrium involves the establishment of not only a pressure difference but also an electrical potential difference across the membrane and that, in the simple case where the mobile ions behave as ideal solutes, the equilibrium condition is expressed as... 

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. 

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