Membrane equilibria

In this section, we present a discussion of the phenomena observed when a rigid, heat-conducting membrane separating two electrolyte solutions is permeable to some of the components but not to others. We restrict our attention to systems in which the electrolytes on the two sides of the membrane are dissolved in the same solvent. 

A typical example of the systems under consideration is the cell 

Since the electrodes are identical, the measured emf of the cell of Eq. (13-150) is given by 

Equation (13-156) holds for all ions to which the membrane is permeable. Therefore we may write 

The existence of a membrane potential implies that the membrane must be impermeable to some of the components. 

The integrand is not dependent upon x and the integration is easily performed. 

The emf of a cell with transference of the type discussed but with different electrodes is readily obtained. Consider the cell 

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. 

When the membrane is not permeable to the solvent, the pressures on the two parts of the system are independent and need not be the same. The conditions of equilibrium for this case are 

These equations imply that a difference of electrical potential exists across the membrane. However, we can add the two equations to give 

For dilute aqueous solutions, we assume that the standard activity for each species and the electrochemical standard potential for each species are the same in each phase at the same temperature and pressure 

These equations are based on the integration of the partial molar Gibbs energy equation dG = —SdT + VkdP for component k at constant T and volume Vk of pure component. Substituting Eq. (10.2) into Eq. (10.1), we have the equilibrium condition 

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. 

In biological systems with dilute aqueous solutions, the last term in Eq. (10.3) disappears, since zw = 0 and the activity of the species determines the osmotic pressure (II). For water, we have 

We may introduce the following approximations. First, for ideal solutions, the activity coefficients are unity (yk = 1), and concentrations are equal to mole fractions ak = xk. Second, using the definitions 

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Donnan membrane equilibrium This concerns the distribution of ions on each side of a membrane separating two portions of a solution of... 

Membrane permeability for the Cl ions is not in contrast to the conclusion that a simple membrane equilibrium such as that described in Section 5.4.1 is established at the membrane. In fact, the membrane potential calculated for the example above with Eq. (5.26) from the Cl ion concentration ratio is exactly -90mV (i.e., the d ions in the two solutions are in equilibrium, and there is no unidirectional flux of these ions). 

Membrane Equilibrium density in sucrose g/cm3 %w/w Marker enzyme/substance... 

The behavior of an ion type is described quantitatively by the Nernst equation (3). A /g is the membrane potential (in volts, V) at which there is no net transport of the ion concerned across the membrane (equilibrium potential). The factor RT/Fn has a value of 0.026 V for monovalent ions at 25 °C. Thus, for K, the table (2) gives an equilibrium potential of ca. -0.09 V—i. e., a value more or less the same as that of the resting potential. By contrast, for Na ions, A /g is much higher than the resting potential, at +0.07 V. Na" ions therefore immediately flow into the cell when Na channels open (see p. 350). The disequilibrium between Na" and IC ions is... 

This exclusion is based on size of the ion and leads to the formation of the Don-nan potential, first mentioned in Chapter 2. Its origin can be explained using the simplest case, involving a uni-univalent electrolyte (NaCl) and a large polyelectrolyte anion V, which is present only in the left compartment (marked with a in Fig. 6.4) and carries z negative charges. We again recall two conditions that must be satisfied at this membrane equilibrium ... 

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. 

When a membrane system has two phases, m number of permeating components, and zk ionic valences, the thermodynamic state of the composite system is determined uniquely by T, PA, PB, mole fraction xk in the two phases, and the electric potential difference i(/B - ii/A across the membrane. These all add up to 1 + 2 + 2m + 1 = 4 + 2m variables. These variables are restricted by m equilibrium relations (Eq. (10.1)), so that the degrees of freedom are 4 + m. This is a special form of the Gibbs phase rule for electrochemical or chemical membrane equilibrium. 

Example 10.1 Membrane equilibrium An aqueous solution (phase A) of 100 mmol/L of NaCl is in equilibrium across a protein-tight membrane with an aqueous solution (phase B) of NaCl and protein. The protein concentration is 5 mmol/L with a negative ionic valency of 10. Determine the difference in electric potential and hydrostatic pressure across the membrane when both solutions are assumed to be ideal and the temperature is 25°C. Figure 10.1 shows the membrane system with the phases A and B. 

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. 

As far as the spread molecules are concerned, the system is closed in the thermo-d5mamic sense, (even if colloquially these substances are referred to as surfactants ). Upon compression or expansion they cannot leave or enter the monolayer because they are insoluble in the liquid (although we continue calling it solvent or water ). This layer is, via a barrier, separated from a surface that does not contain surfactants. However, water molecules and molecules dissolved in it, including electrolytes, can pass underneath the barrier, so for these components the system is open. The ensuing stationary state is a typical example of a membrane equilibrium, that is an equilibrium between two phases when (at least) one of the components is present in one of the phases only (sec. 1.2.12). 

One of the earliest, and reasonably successful, approaches to quantitatively predicting selectivity behaviour was through the thermodynamic treatment of ion exchange systems as a Gibbs-Donnan membrane equilibrium. Such a description is given by equation 5.29 which for the sake of simplicity is shown in terms of single ion activity coefficients ... 

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