Membranes external balance equations

Since Ki is expressed as a ratio, any consistent measure of composition in the membrane and external phases may be used in Equation 7.2. When K> 1, the membrane acts as a concentrator that attracts component i from the external phase and makes it available at the membrane surface for transmembrane movement. Intermolecular forces of solvation and mixing that are responsible for the partitioning process may be entropic as well as enthalpic in origin. The balance of these forces acting between the membrane and external phase can cause either a higher or lower concentration of a given solute inside the membrane relative to the external phase. If the tendency to enter the membrane is negligible, the partition coefficient approaches zero, that is, Kj —> 0. 

Equation 3.6, the Nernst equation, is an equilibrium statement showing how the internal and the external activities of ionic species / are related to the electrical potential difference across a membrane (Fig. 3-2). At equilibrium, a 10-fold difference in the activity of a monovalent ion across some membrane is energetically equivalent to and can balance a 59-mV difference in electrical potential (at 25°C). Hence, a relatively small electrical potential difference can energetically balance a large difference in activity or concentration across a membrane. For instance, if the external activity were 1% of the internal activity (aj/ax- = 0.01), the Nernst potential would be -118 mV for K+ and +118 mV for Cl- (Fig. 3-2). For some calculations, y° /y) is set equal to 1 (a less stringent assumption than setting both y° and yj equal to 1). Under this condition, a°fct- in Equation 3.6 becomes the ratio of the concentrations, c°/cj (a = yff Eq. 2.5). Such a substitution may be justified when the ionic strengths on the two sides of a membrane are approximately the same, but it can lead to errors when the outside solution is much more dilute than the internal one, as occurs for Chara or Nitella in pond water. 

In reality, a state of equilibrium is reached where the ionic attraction of the counterions by the polyanion is just balanced by diffusion into the external solution, which is driven by the chemical potential gradient, which in turn arises from the difference in counterion concentrations between the two domains. The overall effect closely resembles Donnan membrane equilibria (4) (Figure 3). As a consequence of the difference in counterion concentrations, the osmotic pressure inside the polymer domain exceeds that of the external solution, and the expansion of the polyelectrolyte can be equated to the difference in osmotic pressures of the intramolecular and intermolecular solutions. 

In a review of their work, Caplan and Naparstek pointed out that a simpler system might have been an enzyme-free membrane separating the alkahne BAEE solution from a small chamber containing the papain [56]. The chamber could be treated as homogeneous, and quasi-steady-state ordinary differential equations could account for transport of substrate, acid, and base across the membrane as well as for the enzyme-catalyzed reaction. By fixing the external concentrations of BAEE and H+ (and hence OH since the dissociation product is constant), it was shown that conditions exist under which diffusional and reaction fluxes that balance each other are unstable, and the system is directed to a hmit cycle. This simplified membrane-chamber system was further investigated theoretically by Ohmori et al [59], who identified regions of parameter space that are predictive of pH oscillations for compartmentalized papain and other proteolytic... 

Membrane structure and external conditions determine water sorption and swelling. The resulting water distribution determines transport properties and operation. Water sorption and swelling are central in rationalizing physical properties and electrochemical performance of the PEM. The key variable that determines the thermodynamic state of the membrane is the water content k. The equilibrium water content depends on the balance of capillary, osmotic, and electrostatic forces. Relevant external conditions include the temperature, relative humidity, and pressure in adjacent reservoirs of liquid water or vapor. The theoretical challenge is to establish the equation of state of the PEM that relates these conditions to A.. A consistent treatment of water sorption phenomena, presented in the section A Model of Water Sorption, revokes many of the contentious issues in understanding PEM structure and function. 

For the MIEC membranes without external circuit, the overall charge balance is applied or Zz,-/,- = 0 and the local velocity of inert marker is negligible, 0 = 0. Accordingly, the transport flux of charged defects in the MIEC membrane at steady state can be derived (one-dimensional model) from Equations [Al] to [A3] as ... 

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