Flux equations facilitated diffusion

Facilitated diffusion has certain general characteristics. As already mentioned, the net flux is toward a lower chemical potential. (According to the usual definition, active transport is in the energetically uphill direction active transport may use the same carriers as those used for facilitated diffusion.) Facilitated diffusion causes fluxes to be larger than those expected for ordinary diffusion. Furthermore, the transporters can exhibit selectivity (Fig. 3-17) that is, they can be specific for certain molecules solute and not bind closely related ones, similar to the properties of enzymes. In addition, carriers in facilitated diffusion become saturated when the external concentration of the solute transported is raised sufficiently, a behavior consistent with Equation 3.28. Finally, because carriers can exhibit competition, the flux density of a solute entering a cell by facilitated diffusion can be reduced when structurally similar molecules are added to the external solution. Such molecules compete for the same sites on the carriers and thereby reduce the binding and the subsequent transfer of the original solute into the cell. 

For convenience, we have been discussing facilitated diffusion into a cell, but the same principles apply for exit and for fluxes at the organelle level. Let us assume that a transporter for K+ exists in the membrane of a certain cell and that it is used as a shuttle for facilitated diffusion. Not only does the carrier lead to an enhanced net flux density toward the side with the lower chemical potential, but also both the unidirectional fluxes and i ut can be increased over the values predicted for ordinary diffusion. This increase in the unidirectional fluxes by a carrier is often called exchange diffusion. In such a case, the molecules are interacting with a membrane component, namely, the carrier hence the Ussing-Teorell equation [Eq. 3.25 = c /(ctjeljFEM/RT)] is not obeyed because it does not consider... 

Both active and passive fluxes across the cellular membranes can occur simultaneously, but these movements depend on concentrations in different ways (Fig. 3-17). For passive diffusion, the unidirectional component 7jn is proportional to c°, as is indicated by Equation 1.8 for neutral solutes [Jj = Pj(cJ — cj)] and by Equation 3.16 for ions. This proportionality strictly applies only over the range of external concentrations for which the permeability coefficient is essentially independent of concentration, and the membrane potential must not change in the case of charged solutes. Nevertheless, ordinary passive influxes do tend to be proportional to the external concentration, whereas an active influx or the special passive influx known as facilitated diffusion—either of which can be described by a Michaelis-Menten type of formalism—shows saturation effects at higher concentrations. Moreover, facilitated diffusion and active transport exhibit selectivity and competition, whereas ordinary diffusion does not (Fig. 3-17). 

Figure 1 Computer-generated curves of the flux vs. increasing concentrations of substrate transported by either a simple diffusion process fit to the equation J = 5(S) or by a facilitated diffusion process fit to the equation J = 100(S)/(0.4 + S).

The mathematical analysis of this model is facilitated by the pseudo-steady-state (PSS) assumption, i.e., the interface remains stationary while the mass flux equations are written. This is generally satisfied for gas-solid reactions. The following equations for makers of A diffusing per unit time for a single pellet, can be written ... 

As applied to drug transfer from outside the cell (Co) to inside the cell (C,), the flux F) of drug into the cell (via facilitated diffusion or active transport) can be described by the following equation ... 

An interface separates a medium of high permeability from an enzyme-layer of lower permeability to pure diffusion. For example, a gas-phase limiting an enzyme solution or membrane. The boundary-concentration on the gas-phase side will be constant for both pure and facilitated diffusion the concentration gradient at the liquid side of the interface indicates the substrate-diffusion flux which is easy to calculate with the help of the already exposed equations. 

Using this simplified model, CP simulations can be performed easily as a function of solution and such operating variables as pressure, temperature, and flow rate, using software packages such as Mathcad. Solution of the CP equation (eq. 8) along with the solution—diffusion transport equations (eqs. 5 and 6) allow the prediction of CP, rejection, and permeate flux as a function of the Reynolds number, Rtf. To facilitate these calculations, the following data and correlations can be used (/) for mass-transfer correlation, the Sherwood number, Sh, is defined as Sh = 0.04 Re0 75 Sc0-33, where Sc is the Schmidt number (2) osmotic pressure follows van t Hoff s equation, ie, 7r = iCRgTy where i is the number of ions (3)... 

The simultaneous movement of ionic and electronic charge carriers under the driving force of a gradient in the electrochemical potential of oxygen facilitates transport of oxygen in the oxide bulk. The flux density of oxide anions is given (Figure 8.12) [77-79,109] by the ambipolar diffusion equation (see Section 5.7.6) [110,111]... 

The solution of Pick s second law is facilitated by the use of Laplace transforms, which convert the partial differential equation into an easily integrable total differential equation. By utilizing Laplace transforms, the concentration of diffusing species as a function of time and distance from the diffusion sink when a constant normalized current, or flux, is switched on at f = 0 was shown to be... 

The characteristic of a facilitated or carrier-mediated transport is the occurrence of a reversible chemical reaction or complexation process in combination with a diffusion process. This implies that either the diffusion or the reaction is rate limiting The total flux of a permeant A will thus be the sum of both the Fickian diffusion and the carrier-mediated diffusion as illustrated in Equation 4.19 [46] ... 

In this equation, the first term on the right-hand side is the flux contributed by the solution-diffusion mechanism, while the second term is due to the facilitated transport mechanism. The nonreacting gases, like H2, N2, and CO, do not have chemical association with carriers and therefore can only be transported by diffusion, which is limited by their low solubility in the highly polar sites in the membranes.16... 

The first component ofEq. (3), fCsAcs. represents the solute diffusive flux, driven by Donnan equihbrium couphng with facilitation by lEM and LMF potentials. Comparison ofEq. (3) with the equations in the model for the BOHLM systems (see Chapter 5 and [46]) shows that the diffusive mass-transfer coefficient corresponds to the diffusive overaU mass-transfer coefficients, fCp/E on the feed side and Ke/r on the strip side of the BOHLM system with hydrophihc or ion-exchange membranes ... 

Smith and Quinn (35) and Hoofd and Kreuzer (46) Independently developed analytical solutions for the facilitation factor which holds over a range In properties and operating conditions. Smith and Quinn obtained their solution by assuming a large excess of carrier. This allowed them to linearize the resulting differential equations. Hoofd and Kreuzer separated their solution into two parts a reaction-limited portion which is valid near the interface and a diffusion-limited portion within the membrane. Both groups obtained the same result for the facilitation factor. Hoofd and Kreuzer ( T) then extended their approach to cylinders and spheres. Recently, Noble et al. (48) developed an analytical solution for F based on flux boundary conditions. This solution allows for external mass transfer resistance and reduces to the Smith and Quinn equation In the limit as the Sherwood number (Sh) becomes very large. 

A non-linear least-squares regression analysis was used to fit the observed ethene fluxes to Equation 5. From this analysis, K" = 0.7 0.1 atm and c = (1.4 0.1)xl0. Based on data reported by Trueblood and Lucas (6), this result is of the same order as in bulk silver nitrate aqueous solutions. A water content, f, of about 14 weight % was determined for the silver ion-exchanged fibers. Based on ICP analysis, the silver ion concentration, Ct, within the fibers was estimated to be about 5 M, which corresponds to an ion exchange of sodium with silver of about 80 %. The estimated effective diffusion coefficient for ethene in the facilitated mode, calculated from c in Equation 6, was on the order of 2x10 cm s. Table HI shows a summary of the constants obtained for the module with an effective membrane area of 25 cm. ... 

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