Phase Transport Equilibria

The parent p-f-butylcalix[n]arenes (n = 4, 6, 8) are almost completely insoluble in water. However, their resemblance to crown ethers and spherands makes them interesting from the point of view of applications as phase transfer catalysts (Section 3.8.2). Table 3.20 shows the selectivity of calixarene 3.118 and its hexameric and octameric homologues for the extraction of various metal hydroxides into an organic receiving phase such as chloroform. Fortunately, in aqueous base the calixarenes are sufficiently soluble to act as phase transfer catalysts as a consequence of deprotonation of one of their phenolic hydroxyl groups. This solubility contrasts to [18]crown-6, which is much more effective in neutral solution. 

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Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

Latent heat associated with phase change in two-phase transport has a large impact on the temperature distribution and hence must be included in a nonisothermal model in the two-phase regime. The temperature nonuniformity will in turn affect the saturation pressure, condensation/evaporation rate, and hence the liquid water distribution. Under the local interfacial equilibrium between the two phases, which is an excellent approximation in a PEFG, the mass rate of phase change, ihfg, is readily calculated from the liquid continuity equation, namely... 

Partitioning coefficients describe the phase equilibrium of one solute between the two phases. At equilibrium, there is no net transfer. When there is a net transfer, it is proportional to the difference from equilibrium. Equilibrium is therefore always important, because it is the result of transport it is where the chemical concentrations are going. Example 8.1 will show us that even a common term, such as relative humidity, can be related to Henry s law. 

In the following, we will often be concerned with ternary systems. Heterogeneous binary systems have two phases in equilibrium and are nonvariant (at given P and T). When two ternary phases are in contact, the system still has one (thermodynamic) degree of freedom. A ternary phase has three independent transport coefficients iee., Ln,L22, and /. ). 

Cryptands of type 7-9 and derivatives thereof carry alkali cations [6.4], even under conditions where natural or synthetic macrocycles are inefficient. The selec-tivities observed depend on the structure of the ligand, the nature of the cation and the type of cotransported counteranion. Designed structural changes allow the transformation of a cation receptor into a cation carrier [6.1, 6.4]. The results obtained with cryptands indicated that there was an optimal complex stability and phase-transfer equilibrium for highest transport rates. Combined with data for various other carriers and cations, they give a bell-shaped dependence of transport rates on extraction equilibrium (Fig. 11), with low rates for too small or too large... 

Solid-Phase Chemical Equilibrium. For the growth of multicomponent films, the solid film composition must be predicted from the gas-phase composition. In general, this prediction requires detailed information about transport rates and surface incorporation rates of individual species, but the necessary kinetics data are rarely available. On the other hand, the equilibrium analysis only requires thermodynamic data (e.g., phase equilibrium data), which often are available from liquid-phase-epitaxy studies, as discussed by Anderson in Chapter 3. 

The final colligative property, osmotic pressure,24-29 is different from the others and is illustrated in Figure 2.2. In the case of vapor-pressure lowering and boiling-point elevation, a natural boundary separates the liquid and gas phases that are in equilibrium. A similar boundary exists between the solid and liquid phases in equilibrium with each other in melting-point-depression measurements. However, to establish a similar equilibrium between a solution and the pure solvent requires their separation by a semi-permeable membrane, as illustrated in the figure. Such membranes, typically cellulosic, permit transport of solvent but not solute. Furthermore, the flow of solvent is from the solvent compartment into the solution compartment. The simplest explanation of this is the increased entropy or disorder that accompanies the mixing of the transported solvent molecules with the polymer on the solution side of the membrane. Flow of liquid up the capillary on the left causes the solution to be at a hydrostatic pressure... 

Solubilization of lipid digestion products in intestinal mixed micelles enhances their dissolution and dramatically increases the GI lumen-enterocyte concentration gradient that drives absorption by means of passive diffusion. Micelles, however, are not absorbed intact [8, 9], and lipids are thought to be absorbed from a monomolecular intermicellar phase in equilibrium with the intestinal micellar phase [10], The dissociation of monomolecular lipid from the micellar phase appears to be stimulated by the presence of an acidic microclimate associated with the enterocyte surface [11,12], In addition to passive diffusion, growing evidence suggests that active uptake processes mediated by transport systems located in the enterocyte membrane are also involved in the absorption of (in particular) fatty acids into the enterocyte [4],... 

Martin has considered the chemical and physical factors affecting partition coefficients.20 Restricting his discussion to ideal solutions, he considered a solute, A, distributed between two phases in equilibrium with each other. The partition coefficient, a, of the solute A is related to the free energy required to transport one mole of A from one phase to... 

This chapter describes basic physico-chemical relations between the gas phase transport of atoms and molecules and their thermochemical properties, which are related to the adsorption-desorption equilibrium. These methods can either be used to predict the behavior of the adsorbates in the chromatographic processes, in order to design experiments, or to characterize the absorbate from its experimentally observed behavior in a process. While Part I of this chapter is devoted to basic principles of the process, the derivation of thermochemical data is discussed in Part n. Symbols used in the following sections of Part I are described in Section 5. For results, which were obtained applying the described evaluation methods in gas-adsorption chromatography, see Chapters 4 and 7 of this book. 

In the K-L model, reaction occurs within the bed s phases, and material is continuously transferred between the phases. Two limiting situations thus arise. In one, the interphase transport is relatively fast and transport equilibrium is maintained, causing the system performance to be controlled by the rate of reaction. In the other, the reaction rate is relatively fast and the performance is controlled by interphase transport. It will be shown that the ammonia oxidation example used above is essentially a reaction-limited system. 

Intrapellet transport restrictions can limit the rate of removal of products, lead to concentration gradients within pellets, and prevent equilibrium between the intrapellet liquid and the interpellet gas phase. Transport restrictions increase the intrapellet fugacity of hydrocarbon products and provide a greater chemical potential driving force for secondary reactions. The rate of secondary reactions cannot be enhanced by a liquid phase that merely increases the solubility and the local concentration of a reacting molecule. Olefin fugacities are identical in any phases present in thermodynamic equilibrium thus, a liquid phase can only increase the rate of a secondary reaction if it imposes a transport restriction on the removal of reacting species involved in such a reaction (4,5,44). Intrapellet transport rates and residence times depend on molecular size, just as convective transport and bed residence time depend on space velocity. As a result, bed residence time and molecular size affect chain termination probability and paraffin content in a similar manner. 

In another study by Nishiyama et al. [53], the Vapour-phase Transport method was applied on alumina supports. No permeation of 1,3,5-triisopropylbenzene (kinetic diameter 0.85 nm) could be observed through the 10 pm thick membrane. Mordenite has parallel channels with an elliptical pore dimension of 0.65 x 0.7 nm. Pervaporation of benzene-p-xylene (molar ratio 0.86) at 22°C resulted in a separation factor of 164 (total flux 1.19 10" mol.m s ). The theoretical value based on the gas-liquid equilibrium amounts to 11.3. Apparently, the mordenite-based membrane shows high selectivity for aromatic hydrocarbons. 

A catalyst is a substance that increases the rate at which a chemical reaction approaches equilibrium without, itself, becoming permanently involved in the reaction. The key word in this definition is permanently since there is ample evidence showing that the catalyst and the reactants interact before a reaction can take place. The product of this interaction is a reactive intermediate from which the products are formed. This substratexatalyst interaction can take place homogeneously with both the reactants and the catalyst in the same phase, usually the liquid, or it can occur at the interface between two phases. These heterogeneously catalyzed reactions generally utilize a solid catalyst with the interaction taking place at either the gas/solid or liquid/solid interface. Additional phase transport problems can arise when a gaseous reactant is also present in the liquid/solid system. 

Many important chemical reactions take place in the aqueous component of the atmospheric aerosol or in fog droplets. An example is the solution-phase oxidation of SO2 to SO. Such reactions may drive the dilTusioiial transport of reactants from the gas to the particles followed by absorption and chemical reaction. If the chemical reactions are slow compared with the gas- and aero.sol-phase transport rates, the dissolved reactive species will be nearly in equilibrium between the gas and particles. 

Wet deposition encompasses the removal of gases and particles from the atmosphere by precipitation events, through incorporation into rain, snow, cloud, and fog water, followed by precipitation (Hales, 1986). As in the case of dry deposition, wet deposition is a complex phenomenon which in this particular case involves transport to the surface of a droplet, absorption, and possible aqueous-phase chemical conversion. Wet removal of gases is frequently approximated by assuming that the species is in equilibrium between the gas and aqueous phases. The equilibrium partitioning is represented in terms of a washout ratio, Wg = [C]drop/[C]air, where [C]drop and [C]ajr are the concentrations of the chemical in the aqueous and gas phases (Mackay, 1991). 

Because of negligible liquid-side resistance the flux is governed by the gas phase transport. Note that mxA,i = yAwhich implies that the interfacial gas-phase composition is in equilibrium with (he bulk liquid composition. Even if the mass transfer coefficients in the iwo phases are computable. Eqs. (2,4-IOa) aed (2.4-1 Ob) show that phase equilibrium considerations can cause one or the other phase to control." For a very soluble gas (m is small), the gas phase may control while for a sparingly soluble gas such as 02 in water the liquid phase transport generally will govern. 

Figure 12.5 depicts schematically the gas- and aqueous-phase concentrations of A in and around a droplet. The aqueous-phase concentrations have been scaled by HART, to remove the difference in the units of the two concentrations. This scaling implies that the two concentration profiles should meet at the interface if the system satisfies at that point Henry s law. In the ideal case, described by (12.45), the concentration profile after the scaling should be constant for any r. However, in the general case the gas-phase mass transfer resistance results in a drop of the concentration from cA(oo) to cA(Rp) at the air-droplet interface. The interface resistance to mass transfer may also cause deviations from Henry s law equilibrium indicated in Figure 12.5 by a discontinuity. Finally, aqueous-phase transport limitations may result in a profile of the concentration of A in the aqueous phase from [A(/ ,)J at the droplet surface to [A(0)] at the center. All these mass transfer limitations, even if the system can reach a pseudo-steady state, result in reductions of the concentration of A inside the droplet, and slow down the aqueous-phase chemical reactions. 

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