Pressure, chemical potential

Figure 2.6 Chemical potential, pressure, and solvent activity profiles through an osmotic membrane following the solution-diffusion model. The pressure in the membrane is uniform and equal to the high-pressure value, so the chemical potential gradient within the membrane is expressed as a concentration gradient...

Pervaporation is a separation process in which a multicomponent liquid is passed across a membrane that preferentially permeates one or more of the components. A partial vacuum is maintained on the permeate side of the membrane, so that the permeating components are removed as a vapor mixture. Transport through the membrane is induced by maintaining the vapor pressure of the gas on the permeate side of the membrane at a lower vapor pressure than the feed liquid. The gradients in chemical potential, pressure, and activity across the membrane are illustrated in Figure 2.12. 

Figure 2.12 Chemical potential, pressure, and activity profiles through a pervaporation membrane following the solution-diffusion model...

When looking for an economically feasible enzymatic system, retention and reuse of the biocatalyst should be taken into account as potential alternatives [98, 99]. Enzymatic membrane reactors (EMR) result from the coupling of a membrane separation process with an enzymatic reactor. They can be considered as reactors where separation of the enzyme from the reactants and products is performed by means of a semipermeable membrane that acts as a selective barrier [98]. A difference in chemical potential, pressure, or electric field is usually responsible from the movement of solutes across the membrane, by diffusion, convection, or electrophoretic migration. The selective membrane should ensure the complete retention of the enzyme in order to maintain the full activity inside the system. Furthermore, the technique may include the integration of a purification step in the process, as products can be easily separated from the reaction mixture by means of the selective membrane. 

The system is provided with a membrane of suitable molecular cut-off, which acts as a selective barrier for the retention of the enzyme. Permeable substrates and products are taken out from the reaction mixture by the action of a gradient (chemical potential, pressure) through the membrane. Based on the combination of membranes and enzyme reactors two main configurations are considered, as shown in Fig. 6.6.2. In the first configuration, the enzyme may be immobilized by covalent binding between an activated group of the membrane and a functional group of the protein... 

This chapter focuses on two types of biocatalyst membrane reactors, namely, enzymatic membrane reactors and cell-immobilized membrane reactors. The fundamental characteristic of an enzymatic membrane reactor is the separation of enzymes (biopolymers) from products and/or substrates by a semipermeable membrane (Jochems et al, 2011 Lopez et al, 2002). Permeable substrates and products can be selectively separated from the reaction mixture by the action of a driving force across the membrane (chemical potential, pressure, electric field) that causes the movement (diffusion, convection, electrophoretic migration) of solutes. In an enzymatic membrane reactor, the biocatalyst (enzyme) is retained within the system by... 

For the statistical mechanical problem to be well posed, a choice of ensemble is essential. To this point, we have assumed the canonical or NVT ensemble, that is one in which the number of particles, N, the volume, V, and the temperature T are held constant, while the conjugate variables chemical potential, pressure, and energy are allowed to fluctuate. The magnitude of these fluctuations can be related to thermodynamic... 

In this section are briefly reviewed some technical problems of the simulation of dense many-chain systems, such as the sampling of intensive variables such as chemical potential, pressure etc., but also entropy, which are not straightforward to obtain as averages of simple quantities. Some of the standard recipes developed for computer simulation of condensed phases in general have difficulties here, due to the fact that the primary unit, the polymer chain, is already a large object and not a point particle. But knowledge of quantities such as the chemical potentials are necessary, e.g., for a study of phase equilibria in polymer solutions. ... 

In an irreversible process the temperature and pressure of the system (and other properties such as the chemical potentials to be defined later) are not necessarily definable at some intemiediate time between the equilibrium initial state and the equilibrium final state they may vary greatly from one point to another. One can usually define T and p for each small volume element. (These volume elements must not be too small e.g. for gases, it is impossible to define T, p, S, etc for volume elements smaller than the cube of the mean free... 

Here p is the chemical potential just as the pressure is a mechanical potential and the temperature Jis a thennal potential. A difference in chemical potential Ap is a driving force that results in the transfer of molecules tlnough a penneable wall, just as a pressure difference Ap results in a change in position of a movable wall and a temperaPire difference AT produces a transfer of energy in the fonn of heat across a diathennic wall. Similarly equilibrium between two systems separated by a penneable wall must require equality of tire chemical potential on the two sides. For a multicomponent system, the obvious extension of equation (A2.1.22) can be written... 

The chemical potential now includes any such effects, and one refers to the gmvochemicalpotential, the electrochemical potential, etc. For example, if the system consists of a gas extending over a substantial difference in height, it is the gravochemical potential (which includes a tenn m.gh) that is the same at all levels, not the pressure. The electrochemical potential will be considered later. 

If there are more than two subsystems in equilibrium in the large isolated system, the transfers of S, V and n. between any pair can be chosen arbitrarily so it follows that at equilibrium all the subsystems must have the same temperature, pressure and chemical potentials. The subsystems can be chosen as very small volume elements, so it is evident that the criterion of internal equilibrium within a system (asserted earlier, but without proof) is unifonnity of temperature, pressure and chemical potentials tlu-oughout. It has now been... 

In experimental work it is usually most convenient to regard temperature and pressure as die independent variables, and for this reason the tenn partial molar quantity (denoted by a bar above the quantity) is always restricted to the derivative with respect to Uj holding T, p, and all the other n.j constant. (Thus iX = [right-hand side of equation (A2.1.44) it is apparent that the chemical potential... 

On the other hand, in the theoretical calculations of statistical mechanics, it is frequently more convenient to use volume as an independent variable, so it is important to preserve the general importance of the chemical potential as something more than a quantity GTwhose usefulness is restricted to conditions of constant temperature and pressure. 

Note that a constant of integration p has come mto the equation this is the chemical potential of the hypothetical ideal gas at a reference pressure p, usually taken to be one ahnosphere. In principle this involves a process of taking the real gas down to zero pressure and bringing it back to the reference pressure as an ideal gas. Thus, since dp = V n) dp, one may write... 

Given this experimental result, it is plausible to assume (and is easily shown by statistical mechanics) that the chemical potential of a substance with partial pressure p. in an ideal-gas mixture is equal to that in the one-component ideal gas at pressure p = p. 

For precise measurements, diere is a slight correction for the effect of the slightly different pressure on the chemical potentials of the solid or of the components of the solution. More important, corrections must be made for the non-ideality of the pure gas and of the gaseous mixture. With these corrections, equation (A2.1.60) can be verified within experimental error. 

Note that this has resulted in the separation of pressure and composition contributions to chemical potentials in the ideal-gas mixture. Moreover, the themiodynamic fiinctions for ideal-gas mixing at constant pressure can now be obtained ... 

The coefficient of dE is the inverse absolute temperature as identified above. We now define the pressure and chemical potential of the system as... 

The coexisting densities below are detennined by the equalities of the chemical potentials and pressures of the coexisting phases, which implies that tire horizontal line joining the coexisting vapour and liquid phases obeys the condition... 

The van der Waals p., p. isothenns, calculated using equation (A2.5.3), are shown in figure A2.5.8. It is innnediately obvious that these are much more nearly antisynnnettic around the critical point than are the conespondingp, F isothenns in figure A2.5.6 (of course, this is mainly due to the finite range of p from 0 to 3). The synnnetry is not exact, however, as a carefiil examination of the figure will show. This choice of variables also satisfies the equal-area condition for coexistent phases here the horizontal tie-line makes the chemical potentials equal and the equal-area constniction makes the pressures equal. 

Ultimately, the surface energy is used to produce a cohesive body during sintering. As such, surface energy, which is also referred to as surface tension, y, is obviously very important in ceramic powder processing. Surface tension causes liquids to fonn spherical drops, and allows solids to preferentially adsorb atoms to lower tire free energy of tire system. Also, surface tension creates pressure differences and chemical potential differences across curved surfaces tlrat cause matter to move. 

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