Permeability membrane, equations

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

Let us proceed with the second interesting example concerning application of the ROZ equations. We would Hke first to mention that simple fluids confined to slits with permeable membranes have been studied by both computer simulations and theory, see, e.g.. Refs. 49-52. The simplest way is to visualize a permeable membrane as a barrier of finite height and width. To our best knowledge, no studies of a system containing multiple barriers of a more sophisticated geometry than the sHt-Hke have been undertaken so... 

Our main focus in this chapter has been on the applications of the replica Ornstein-Zernike equations designed by Given and Stell [17-19] for quenched-annealed systems. This theory has been shown to yield interesting results for adsorption of a hard sphere fluid mimicking colloidal suspension, for a system of multiple permeable membranes and for a hard sphere fluid in a matrix of chain molecules. Much room remains to explore even simple quenched-annealed models either in the framework of theoretical approaches or by computer simulation. 

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. 

The apparent permeability of 11 test compounds was measured in the presence and absence of human serum albumin in the donor compartment, and by solving the differential equations describing the kinetics of membrane permeability, membrane retention and protein binding, the authors were able to obtain the Kd. With the protein in solution rather than immobilized and without the need for mass balance or equilibrium conditions, this approach provides an attractive alternative to existing methods with the potential to be applied to an array of other soluble proteins. 

The concept of permeability. Pm, described first in Section 4.3.2.2 also applies to membranes. Equation (4.77) relates the permeability to the diffusion coefficient and solubility. Some representative values of permeabilities for common gases in common polymer films are given in Table 4.17. The units of permeability in Table 4.17 are obtained when diffusivity is in units of m /s, and gas solubility is in units of m gas m /(m soUd-N). Note that carbon dioxide permeabilities are generally 3-4 times... 

Equation (9.1) is the preferred method of describing membrane performance because it separates the two contributions to the membrane flux the membrane contribution, P /C and the driving force contribution, (pio — p,r). Normalizing membrane performance to a membrane permeability allows results obtained under different operating conditions to be compared with the effect of the operating condition removed. To calculate the membrane permeabilities using Equation (9.1), it is necessary to know the partial vapor pressure of the components on both sides of the membrane. The partial pressures on the permeate side of the membrane, p,e and pje, are easily obtained from the total permeate pressure and the permeate composition. However, the partial vapor pressures of components i and j in the feed liquid are less accessible. In the past, such data for common, simple mixtures would have to be found in published tables or calculated from an appropriate equation of state. Now, commercial computer process simulation programs calculate partial pressures automatically for even complex mixtures with reasonable reliability. This makes determination of the feed liquid partial pressures a trivial exercise. 

Gibbs considered the statistical mechanics of a system containing one type of molecule in contact with a large reservoir of the same type of molecules through a permeable membrane. If the system has a specified volume and temperature and is in equilibrium with the resevoir, the chemical potential of the species in the system is determined by the chemical potential of the species in the reservoir. The natural variables of this system are T, V, and //. We saw in equation 2.6-12 that the thermodynamic potential with these natural variables is U[T, //] using Callen s nomenclature. The integration of the fundamental equation for yields... 

Potentiometric sensors are based on a membrane that separates the sample solution of a reference solution contained within the electrode. The membranes are permeable to particular types of ions (ISEs) or gases (gas-permeable membrane sensors). These electrodes generate a potential that is proportional to the concentration of a single analyte. This proportionality is expressed by an equation... 

The irreversibility accompanying a steady flow process may be computed for any zone by noting the fluxes of entropy into and out of the zone. Leaving aside the problems posed by semi-permeable membranes, which introduce ambiguities into the meanings of heat and work, (2), equation (3) provides such a balance ... 

The permeability coefficient depends on the characteristics of the membrane and solute, and can vary considerably for various solutes. For example,/) = 10-21 m/s for sucrose and 10 4 m/s for water in the human red blood cell membrane. Equation (1) may be generalized by including the effect of pressure gradient APm = P(0) - P(L), and we have... 

Isothermal chemistry in fuel cells. Barclay (2002) wrote a paper which is seminal to this book, and may be downloaded from the author s listed web site. The text and calculations of this paper are reiterated, and paraphrased, extensively in this introduction. Its equations are used in Appendix A. The paper, via an equilibrium diagram, draws attention to isothermal oxidation. The single equilibrium diagram brings out the fact that a fuel cell and an electrolyser which are the thermodynamic inverse of each other need, relative to existing devices, additional components (concentration cells and semi-permeable membranes), so as to operate at reversible equilibrium, and avoid irreversible diffusion as a gas transport mechanism. The equilibrium fuel cell then turns out to be much more efficient than a normal fuel cell. It has a greatly increased Nernst potential difference. In addition the basis of calculation of efficiency obviously cannot be the calorific value of the... 

Figure 4.4. Driving forces in ion diffusion across selectively permeable membranes, a Entropy promotes diffusion down a con-centrationgradient,regardlessof charge, b Electroneutrality will oppose entropy, c The Nemst equation describes the membrane potential that ensues when entropy and electroneutrality are in equilibrium.

When two electrolyte solutions at different concentrations are separated by an ion--permeable membrane, a potential difference is generally established between the two solutions. This potential difference, known as membrane potential, plays an important role in electrochemical phenomena observed in various biomembrane systems. In the stationary state, the membrane potential arises from both the diffusion potential [1,2] and the membrane boundary potential [3-6]. To calculate the membrane potential, one must simultaneously solve the Nernst-Planck equation and the Poisson equation. Analytic formulas for the membrane potential can be derived only if the electric held within the membrane is assumed to be constant [1,2]. In this chapter, we remove this constant held assumption and numerically solve the above-mentioned nonlinear equations to calculate the membrane potential [7]. 

By inserting Henry s law (Equation 4.6) into Pick s law (Equation 4.1), integrating across the membrane and remembering the definition of the permeability coefficient (Equation 4.5), Equation 4.2 was developed as the standard equation for transport through a dense polymeric membrane. 

Osmotic pressure The osmotic pressure, IT, represents the amount of pressure that can be created between a concentrated solution and pure water separated by a permeable membrane. The equation given below represents the osmotic pressure for a single electrolyte solution [18]. 

Gas-sensing electrodes are examples of multiple membrane sensors these contain a gas-permeable membrane separating the test solution from an internal thin electrolyte film in which an ion-selective electrode is immersed. For example, for the ammonia sensor, the pH of the recipient layer is determined by the Henderson-Hasselbach equation [Eq. (18)], derived from the chemical equilibrium between solvated ammonia and ammonium ions ... 

Potentiometric transducers measure the potential under conditions of constant current. This device can be used to determine the analytical quantity of interest, generally the concentration of a certain analyte. The potential that develops in the electrochemical cell is the result of the free-energy change that would occur if the chemical phenomena were to proceed until the equilibrium condition is satisfied. For electrochemical cells containing an anode and a cathode, the potential difference between the cathode electrode potential and the anode electrode potential is the potential of the electrochemical cell. If the reaction is conducted under standard-state conditions, then this equation allows the calculation of the standard cell potential. When the reaction conditions are not standard state, however, one must use the Nernst equation to determine the cell potential. Physical phenomena that do not involve explicit redox reactions, but whose initial conditions have a non-zero free energy, also will generate a potential. An example of this would be ion-concentration gradients across a semi-permeable membrane this can also be a potentiometric phenomenon and is the basis of measurements that use ion-selective electrodes (ISEs). 

Whenever a solution is separated from a solvent by a membrane that is permeable only to solvent molecules (referred to as a semi-permeable membrane), there is a passage of solvent across the membrane into the solution. This is the phenomenon of osmosis. If the solution is totally confined by a semipermeable membrane and immersed in the solvent, then a pressure differential develops across the membrane, which is referred to as the osmotic pressure. Solvent passes through the membrane because of the inequality of the chemical potentials on either side of the membrane. Since the chemical potential of a solvent molecule in solution is less than that in pure solvent, solvent will spontaneously enter the solution until this inequality is removed. The equation which relates the osmotic pressure of the solution. If, to the solution concentration... 

Boiling point elevation AT = mK[, (the constants have been tabulated) Freezing point depression AT = —mK (the constants have been tabulated) A solution in contact with its pure solvent across a semi-permeable membrane experiences an increase in pressure as pure solvent flows through the membrane into the solution. This osmotic pressure can be measured quite accurately, and through the equation ttV = nRT permits determination of the molecular weight of the solute. 

The equilibrium constant Kp (see equations above) decreases with increasing temperature. Thus, low temperatures favor product formation in the WGS reaction. The equilibrium can also be shifted towards the right by increasing the steam concentration or removing a product from the reaction mixture (i.e., hydrogen through a hydrogen permeable membrane). 

PAMPA-pKa fiux optimized design (pOD)-permeabiiity Iso-pH mapping unstirred PAMPA was used to measure the effective permeability, Pe, as a function of pH from 3 to 10, of five weak monoprotic acids (ibuprofen, naproxen, ketoprofen, salicylic acid, benzoic acid), an ampholyte (piroxicam), five monoprotic weak bases (imipramine, verapamil, propranolol, phenazopyridine, metoprolol), and a diprotic weak base (quinine). The intrinsic permeability, Po, the UWL permeability, Pu, and the apparent pKa (pKa.fiux) were determined from the pH dependence of log Pg. The underlying permeability-pH equations were derived for multiprotic weak acids, weak bases, and ampholytes. The average thickness of the UWL on each side of the membrane was estimated to be nearly 2000 p, somewhat larger than that found in Caco-2 permeability assays (unstirred). As the UWL thickness in the human intestine is believed to be about forty times smaller, it is critical to correct the in vitro permeability data for the effect of the UWL. Without such correction, the in vitro permeability coefficient of lipophilic molecules would be indicative only of the property of water. In single-pH PAMPA (e.g., pH 7.4), the uncertainty of the UWL contribution can be minimized if a specially selected pH (possibly different from 7.4) were used in the assay. From the analysis of the shapes of the log Pe-pH plots, a method to improve the selection of the assay pH, called pOD-PAMPA, was described and tested. From an optimally selected assay pH, it is possible to estimate Pg, as well as the entire membrane permeability-pH profile. 

Other sugars were also employed, and in later papers an account is given of similar very accurate measurements both of osmotic pressure and lowering of vapour pressure, due to calcium ferrocyanide m water, this salt being a very soluble one, and one which at the same time is practically stopped by the copper-ferrocyanide semi-permeable membrane (Earl of Berkeley, E G J Hartley, and C V Burton, PM Trans, 209 A, 177, 1909 Dilute solutions ofthe same solute were also investigated by the Earl of Berkeley, E G J Haitley, and J Stephenson, ibid, p 319 ) For details the reader is again referred to the ongmal papers The object of the work referred to was to test Porter s equation This equation will be taken later... 

The solution and solvent are placed m a vessel and separated by a semi-permeable membrane (Fig 32) The space above is also separated into two parts by a partition semi-permeable to the vapour of the solvent, but not to an inert gas A pressure difference p - Po - Yp is maintained between the two sides by aid of an inert gas Then, unless the vapour pressures itp and ir0/,0 are equal, a flow of vapour will occur with such consequent evaporation and condensation on the two fluids respectively as to upset the initial osmotic equilibrium in a direction which will maintain the difference of vapour pressures and thus cause perpetual flow, the possibility of which we are entitled to deny This conclusion may be taken as a check upon the equations which we have derived... 

The permeability coefficient, k, is a characteristic of the membrane. Equation (17.4) is valid for a constant temperature. Furthermore, it assumes that the substance in question is small enough to pass through the pores of the membrane and that it is non-ionic. 

Suppose one could create an interface between two electrolyte phases across which only a single ion could penetrate. A selectively permeable membrane might be used as a separator to accomplish this end. Equation 2.3.34 would still apply but it could be simplified by recognizing that the transference number for the permeating ion is unity, while that for every other ion is zero. If both electrolytes are in a common solvent, one obtains by integration... 

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